evaluation of numerical methods for turbomachines based on

Transcription

UNIVERSIDAD PONTIFICIA COMILLAS
ESCUELA TÉCNICA SUPERIOR DE INGENIERÍA (ICAI)
INGENIERO SUPERIOR INDUSTRIAL
PROYECTO FIN DE CARRERA
EVALUATION OF NUMERICAL
METHODS FOR TURBOMACHINES
BASED ON EXPERIMENTAL DATA
FROM A FRANCIS PUMP-TURBINE IN
PUMP MODE.
AUTOR:
ANA FABA TORTOSA
MADRID, junio de 2008
EVALUACIÓN
DE
MÉTODOS
NUMÉRICOS
PARA
TURBOMÁQUINAS BASADO EN DATOS EXPERIMENTALES DE
UNA BOMBA-TURBINA FRANCIS EN SENTIDO BOMBA
Autor: Faba Tortosa, Ana.
Director: Braun, Olivier.
Entidad Colaboradora: École Polytecnique Fédérale de Lausanne.
RESUMEN DEL PROYECTO
El objetivo de este proyecto es la validación, a partir de su comparación con datos
reales experimentales, de la capacidad de los métodos CFD para modelar el
comportamiento del flujo dinámico para fenómenos transitorios en turbomáquinas
hidráulicas.
El caso estudiado concierne a una bomba-turbina Francis en sentido bomba. Un
modelo a escala reducida de dicha máquina con una velocidad específica ν = 0.19 es
el utilizado en las medidas de laboratorio.
La turbomáquina está compuesta de 9 álabes en la rueda y 20 canales en el difusor.
Dos puntos de operación distintos, uno de carga parcial y otro de elevada tasa de
descarga, son descritos y analizados por métodos de simulación, en términos de
fluctuaciones de presión a lo largo de los canales del estator. Estas fluctuaciones
generadas por la interacción entre el rotor y el estator (RSI) y los fenómenos que
afectan al comportamiento del flujo a causa de ello forman parte del estudio
realizado en este proyecto.
La simulación numérica de los flujos transitorios está representada gracias al paquete
de software de ingeniería ANSYS CFX.11. para cuatro dominios de computación
diferentes: 3 posibles dominios parciales y también la máquina completa.
Aunque inicialmente se realizan análisis generales para los cuatro casos, tras la
evaluación de la calidad de los resultados, se detalla la investigación para el último
caso exclusivamente. Solamente un dominio de computación que incluya la máquina
entera se considera que minimice los errores de CFD y que consiga así una
simulación fiable.
Esos resultados numéricos precisos y detallados son comparados con las medidas
obtenidas en los test del laboratorio, con la intención de validar el método y
determinar los principales fallos de los métodos CFD.
Las medidas de presión en el marco estacionario, cuyos puntos suponen básicamente
todo el núcleo del análisis, están tomadas mediante diminutos sensores de presión
piezoresistivos, situados en diferentes puntos de dos de los veinte canales del
distribuidor: el primero y el último.
En general, se descubre muy buena concordancia entre la simulación y los resultados
del laboratorio para el punto de operación con elevado caudal de descarga pero
existen algunas diferencias para el de carga parcial.
Las posibles causas y la descripción del comportamiento del flujo están desarrolladas
a lo largo de este proyecto. Por ejemplo, torbellinos y pequeños remolinos pueden
nacer debido a la falta de adaptación entre la geometría de la máquina y los
triángulos de velocidad del agua en cargas parciales.
Este tipo de fenómenos aleatorios e instantáneos crean un régimen no permanente y
un grado de inestabilidad en el comportamiento del flujo que no llega a predecirse ni
a simularse completamente bien con los actuales métodos de CFD.
Este proyecto puede ofrecer una primera referencia de análisis para más detallados
estudios en el futuro.
De acuerdo con esto, las diferencias más relevantes entre los resultados de CFD y los
datos experimentales pueden ayudar a advertir dónde y cómo los métodos numéricos
pueden ser mejorados y corregidos. Por otra parte, como los la simulación en CFD
ofrece resultados de la presión y todas las variables y propiedades del flujo en
cualquier lugar y posición de toda la máquina, para cualquier canal y cualquier punto
en la bomba-turbina, de la observación de los resultados que parezcan insospechados
o más sorprendentes pueden ayudar a sugerir dónde colocar un nuevo sensor de
presión para el próximo test de laboratorio.
Para concluir, la comparación de ambos resultados puede ayudar a mejorar el
desarrollo de las máquinas hidráulicas, tanto encontrando los puntos débiles de las
máquinas actuales, sus riesgos y problemas de operación así como corrigiendo los
métodos de CFD con la intención de mejorarlos y aplicarlos al diseño y rediseño de
dichas máquinas
EVALUATION
OF
NUMERICAL
METHODS
FOR
TURBOMACHINES BASED ON EXPERIMENTAL DATA FROM A
FRANCIS PUMP-TURBINE IN PUMP MODE
The aim of this project is to validate, by comparison with real experimental data, the
ability of CFD methods to model dynamic flow behaviour for unsteady phenomena
in hydraulic turbomachines.
The study case concerns a Francis pump-turbine in pump mode. A reduced-scale
model ν = 0.19 is used for the laboratory measurements. It is composed of 9 runner
blades and 20 diffuser channels. Two different operating points, one of partial load
and another of high discharge, are described and analysed by simulation methods, in
terms of pressure fluctuactions along the stator channels. These fluctuacions
generated by Rotor Stator Interaction (RSI) and the phenomena that affect the flow
behaviour caused by them are part of the study of this paper.
The numerical simulation of the unsteady flow is performed with ANSYS
CFX.11 for four computing domains: 3 possible partial domains and also the entire
machine.
Although previous general analysis are done for the four cases, after the evaluation
of the quality of their results, detailed investigations are finally made for the last one
exclusively. Only a computing domain of the entire machine is considered to be the
best to minimize the CFD errors and to get a reliable simulation.
These computer results are compared with laboratory measurements in order to to
validate the method and to determine main CFD simulation failures. The pressure
measurements in the stationary frame, whose points are the mainly analysis issue, are
performed with piezoresitive miniature pressure sensors located in several locations
at two of the 20 distributor channels: the first and the last one.
Very good agreement between simulation and laboratory results is found in general
for the high discharge operating point but some discrepancies are discovered for the
partial load one.
Posible causes and the description of the flow behaviour are included in this paper.
For example, eddies and small whirls can be born due to the lack of fit between the
machine geometry and water velocity triangles at partial load. This type of
phenomena creates instability and unsteadiness that it is not completely well
simulated by CFD methods.
This project can provide a first reference analysis for more detailed studies in the
future.
According to this, most relevant differences between CFD results and experimental
data can help to advice where and how the numerical method should be improved
and checked. On the other hand, as CFD results are obtained for the whole machine,
all around the channels and at all locations in the pump-turbine, unexpected or
surprising results discovered by the observation of CFD analysis for any point would
suggest to include a sensor at that location during the next laboratory experimental
test.
To sum up, the comparison of both results can help to improve the hydraulic
machines development, by finding current machines weaknesses, risks and operating
problems as well as by checking the CFD methods errors, in order to improve them
for machine design and redesign.
Table of Content
TABLE OF CONTENTS
1
Introduction. ..................................................................................2
1.1 THE HYDRODYNA PROJECT.....................................................................3
1.2 Brief overview of hydro power sector situation. ......................................3
1.3
Introduction to Hydraulic Turbomachines. ................................... 7
1.3.1
Variables definition. Classification of Turbomachines ...... 7
1.3.2
Nomenclature............................................................................ 12
1.3.3
1.3.4
Power conversion and Balance............................................... 14
Euler equations ......................................................................... 18
1.3.5
1.3.6
Rotor-stator Interaction in Francis Pump-Turbines. .......... 22
Frequency Analysis. ................................................................. 26
2
Numerical Simulation and CFD Methods ............................ 28
2.1
Introduction to Computational Methods...................................... 28
2.2
CFD (Computational Fluid Dynamics) ......................................... 30
2.2.1
The Choice of the Physical Approach. .................................. 31
2.2.2
The mathematical model ......................................................... 32
2.2.3
2.2.4
2.3
2.4
2.5
3
4
The Discretization Method..................................................... 33
The implementation of numerical algorithms and Solution.
36
CFD for Turbomachinery. ............................................................... 37
CFD historic evolution.................................................................... 38
CFD Error Estimation ...................................................................... 39
Introduction to the study case ................................................. 43
3.1
3.2
Description of the Hydromachine.................................................. 44
Partial simulations and Studied points ......................................... 46
3.3
Expected rotor-stator behaviour in Hydrodyna .......................... 47
Numerical Simulation. Procedure and Results.................... 50
4.1
Introduction. The Procedure followed ........................................ 50
4.2
Previous study: Turbine mode results .......................................... 51
4.2.1
Parametric studies of the Model............................................. 51
4.2.1.1 Mesh Sensitivity ...................................................................... 51
4.2.1.2 Time Step Sensitivity .............................................................. 52
4.2.1.3 Convergence Sensitivity.......................................................... 52
4.2.2
Conclusions for the turbine mode.......................................... 53
4.3
Pump mode ....................................................................................... 54
Table of Content
4.3.1
Complete machine simulation................................................ 54
4.3.1.1 Generation of Velocity triangles ............................................ 57
4.3.1.2 Results . Rotor-Stator Interaction (RSI) ................................. 62
4.3.1.2.1 STUDY FOR ϕ = @0.043 .............................................. 65
4.3.1.2.2 STUDY FOR ϕ = @0.028 ............................................... 70
4.3.1.2.2.2. Conclusions for simulation with ϕ = @0.028 ............ 77
4.3.2
5
Partial simulations.................................................................... 78
Experimental Data Results ....................................................... 85
5.1
5.2
Measurement techniques................................................................. 85
Experimental Results for ϕ = @0.043 ........................................... 87
5.2.1
Frequency Analysis .................................................................. 90
5.2.2
Conclusions for ϕ = @0.043 ..................................................... 92
5.3
Experimental Results for ϕ = @0.028 ............................................ 93
5.3.1
5.3.2
6
Comparaison of results. Conclusions .................................. 101
6.1
6.2
7
8
9
frequency Analysis................................................................... 95
Conclusions for ϕ = @0.028 ..................................................... 99
Detailed Comparaison of Results................................................. 101
Final Conclusions ........................................................................... 107
Attached documents ................................................................ 109
7.1
Sensors locations............................................................................. 109
7.2
7.3
7.4
7.5
Hydrodyna’s Nomenclature ........................................................ 111
Hydrodyna’s hill chart................................................................... 112
Example of a List of set-up expressions ...................................... 113
Code for creating Velocity triangles session............................. 119
7.6
7.7
Matlab file: frequency analysis and multiple comparisons..... 129
Comparison CFD and Experimental Results Plots.................... 135
REFERENCES ........................................................................... 140
ACKNOWLEDGEMENTS ..................................................... 144
1
Introduction. The Hydrodyna
Project
Introduction
1 INTRODUCTION.
Flow in a turbomachine is three dimensional, turbulent and unsteady.
Furthermore, fluid dynamic interactions appear between the flow fields in
both rotationary and stationary parts, resulting in a more unsteady and
secondary flow fields. In particular, the interaction between impeller blades
and guide vane, known as Rotor-Stator Interaction (RSI), can generate
pressure fluctuations of high amplitude in pump-turbines, especially in the
points situated next to the rotor-stator gap, becoming one of the most
important sources of vibration or blade cracking in this kind of
turbomachines. As a consequence of these fluctuations not only runner
blades can be serious damaged but also other machine components. The
importance of the consequences of the noise, the erosion and the vibrations,
depends on the turbomachine use. For example, it can cause hydraulic and
electric instability in machines installated in power production plants, or
disturb the confortability in ship passengers in propulsion engines. Anyway,
erosion provokes a mass loss in runner blades so that it must be repaired and
periodically inspected, with the following costs.
In order to lessen this damage and reduce these problems it is important to
find the main causes and try to combat them.
Extensive numerical and experimental studies are required to understand
and predict the flow phenomena. In general, not enough experimental data
referred to this problem exist nowadays and computational methods for
unsteady flow have to be still improved, calibrated and validated.
There is still much research to do and more experimental data to analyse
enabling to validate the unsteady numerical calculations and helping to find
solutions to the problems linked to this phenomena.
2
Introduction
1.1
3
THE HYDRODYNA PROJECT
Hydrodyna Project objectives[EURE99] are to improve the availability and
the reliability of hydraulic turbomachinery, to make posible the optimization
of the nowadays technical and scientific challenge in electricity market: The
need of extending the operating range and the reliability of modernized
hydro units according to variable demand.
Development
of
advanced
fluid
and
mechanical
instrumentation,
computation of fluid mechanics for a deeper knowledge are necessary in
order to achieve these goals. They are to be obtained by developing a base of
knowledge from experimental and computational results got from measures
and tests in a Francis pump-turbine.
The Hydrodyna Project intends to study dynamic behaviour of hydraulic
machines to bring under control damage and risks derived from it and to
upgrading efficiency and design technologies especially for pump-turbines.
The aim of this paper is to help to achieve some of this objectives from the
study of numerical and experimental results.
1.2
BRIEF OVERVIEW OF HYDRO POWER
SECTOR SITUATION.
Changes in the electricity market over the last decade create a demand for
widing and becoming more flexible the operating range in hydroelectric
plants.
Nowadays power demand varies considerably and continuously, so a better
energetic ressources gestion will allow to adapt electric production to the
demand. Changes in demand oblige to start and stop electricity production
plants in operating conditions far from the one they were designed to work
in.
Hydraulic power becomes the most convenient generation form to respond
to this challenges at the moment. The great combination of its characteristics
Introduction
and properties (renewable green energy, moderated price, variable
production rate and especially storage capability ) makes hydrounits full of
advantages.
When possible, modern plants tend to use reversible pump-turbines that can
be run in one direction as turbines and in the other direction as pumps. The
system is joined to reversible electric motor/generators. During the storage
part of the cycle the motor drives the pump, while the generator produces
the electricity during the hydraulic discharge from an upper reservoir.
The refurbishment and the upgrading of the hydraulic power plants is the
major concern in the energetic domain for the next years. Some real factors
influenciating this fact are :
•
The capability of hydraulic storage installations to answer to quickly
electricity demand variations ( the only ones together with gas
turbines). Besides, they represent the only possibility in these
moments to store large energy quantities with high global efficiency
(around 80%).
•
The difficulty and the drawbacks of solar and wind energy
accumulation.
•
The obligation of greenhouse gas emission reduction –Kyoto
commitment [ERKM04]
•
The polemic and problematic issues of the nuclear energy source.
Though the development reached for the hydroelectric resources in Europe
during last century, it remains economical interest in the renovation of the
existing power plants, as investors know it is quite profitable. Hydroelectric
production coming from pump-storage installations functioning with power
grid electricity dominates renewable electric sector in Europe, as shown in
the figure (fig. 1a). [ENER07]
4
Introduction
5
figure (1.a) Distribution of electric renewable generation sources in Europe in 2005
In Switzerland, the hydropower production represents 56% (from which 32%
of storage installations) of the total electricity production, compared to 40%
from nuclear and the rest from other energy sources such as solar, wind, fuel
and gas [ASEU04]. (fig. 1.b)
Electricity Production Plants in Switzerland in 2003
5
4.3
24
Hydraulic Plants
1
Storage Hydraulic Plants
2
Nuclear Plants
3
4
40
Thermal Plants
32
5
Others
figure (1.b) Distribution of electric generation sources in Switzerland in 2004 by
installation type.
Introduction
6
There are someones who consider hydro-power the most reliable way to
obtain the green energy increase, but the main advantage of this electric
ressource is that it is the only one capable to adapt to the peaks in the
demand or to profit the economic periods to store.
However, most European hydro power plants were designed many years
ago
and
are
not
well
adapted
to
the
current
operating
requirements.Innovations and new design technologies for future plants and
for the modernization of old power stations can increase considerably their
efficiency as well as their power. The liberalisation of Energy Markets makes
it profitable and reasonable to leverage the hydraulic potential of Southern
Central Europe to cover peak consumption instead of continuous load.
A special challenge in the refurbishment of existing power plants is to fit a
machine with a wider operating range in the space available in the civil
engineering installation, which has to be unchanged for cost and infeasibility
reasons.
These are the basis which motivated the Hydrodyna Project.
A first part of this Project has already been done and it was focused in the
turbine mode of the Francis machine. During the initial experimental study
case pressure measures were also taken for some operating points in the
pump mode.
These measures will be analysed in this paper and they will be compared
with numerical results from CFD methods in order to calibrate and validate
them, in an attempt to derive conclusions which could help even to predict,
prevent, control and reduce unsteady flows from causing damage and
lowing efficiency.
Introduction
1.3
7
INTRODUCTION TO HYDRAULIC TURBOMACHINES.
In this paragraph a general overview about hydraulic turbomachines will be
tackled. It is not the aim of this paper to explain in detail fluids mechanical
theory or turbomachines lessons; however, some basic variable definitions,
nomenclature and conclusions dealing with the subject must be introduced.
Turbomachines are rotating machines that transfer energy between a rotating
part (called rotor) and fluid. Depending on the energy transfer direction we
distinguish turbines (transferring energy from a fluid to the rotor and
normally the rotor to a transmission shaft) or a compressor (it exchanges
energy from the rotor to the fluid).
For hydraulic turbomachines the fluid is liquid water, and compressors are
known as pumps.
Energy transfer is obtained from a simultaneous, “reaction machines”,
exchange of pressure potential energy and kinetic energy. In some machines
(the most famous ones are Pelton turbines) only a conversion of kinetic
energy takes place. They are called “action machines”.
1.3.1
VARIABLES DEFINITION. CLASSIFICATION OF TURBOMACHINES
Definitions for main physical magnitudes will be treated in the following
paragraphs.
- Hydraulic power Ph , also known as flow power, is the power between high
and low pressure machine sections.
It is calculated as follows:
b
c
Ph = ρQB gH I @gH @I = ρQ B E
[1.3.1.a]
Introduction
8
where ρ is density, Q is the discharge (see formula 1.3.1.d) , g is gravity
acceleration and H is installation head (see formula 1.3.1.e), with subscript I
for high pressure region and
@
I
for low pressure (figure 2).
Specific
hydraulic energy (E) definition is included in this formula too.
figure 2. Section showing high and low pressure in a turbine installation.
As has just been discussed before, depending on hydraulic energy transfer
direction, we find turbines or pumps.
-
Turbines are work-generating, so fluid hydraulic power is decreased.
Power is transferred from the water to the rotor (by the runner blades)
and then to the transmission shaft to the generator (typically electric
generators).
-
Pumps are work-absorbing, so the rotor transfers power (taken from a
external motor) to the fluid, increasing the hydraulic power of the
flow. Pumps are receptors.
According to this, hydraulic power is defined as positive for turbines and
negative for pumps.
Ph > 0
Turbines
Introduction
Ph < 0
9
Pumps
- Mechanic Power P is defined for turbomachines as :
@Q
@
@Q @
T
P= w
where
[1.3.1.b]
@
@Q
@
@
wQ is the rotation speed and
T the total moment in the
turbomachine.
- Transferred Power P t is defined as the real power converted inside the
rotor, the runner blades. It does not include friction couples or leakage. The
formula to calculate it from T t (the moment resulting from the flow action
over the runner blades) is:
@
@Q @Q
Pt = T t @
w
[1.3.1.c]
@
where @
wQ is the rotation speed of the runner.
- Flow rate, or discharge, is measured in [m3/s] . It is calculated as:
@
@
Q =Z Z @
cQ @
nQ dA
[1.3.1.d]
A
- Installation Head multiplied by acceleration due to gravity is called Mean
Specific Hydraulic Energy. It is defined as :
h
i
@
@
Q2 @
@
@
p
c
cQ @
nQ
ffff
fffffffffffffffffffff
k fffffffffffffffffffffffffffffffffffff
gH =Z Zj + gZ +
dA
ρ
2
Q
A
[1.3.1.e]
Introduction
10
- Specific velocity is an essential non-dimensional parameter used in
turbomachines.
It owes its importance to its definition variable, since it
combines flow rate, rotation speed, and hydraulic energy. Moreover it gives
an idea of the type and geometry of the runner blades. These properties
make specific velocity a reference parameter, serving to classificate machines
and to choose a particular machine for an installation.
There are several specific velocity definitions, depending on the units in
which the variables are expressed and the relation among them.
The most important one is
ν =ω
w
w
w
w
w
w
w
w
w
w
w
w
w
w
w
w
w
w
w
w
dw
e
Q
fffffff
s
π
fffffffffffffffffffffff
`
a34ffff
[1.1.3.f]
2E
but also are commonly used:
- For turbines:
w
w
w
w
w
w
w
ns =
p P
N
H
5ffff
4
[1.1.3.g]
where N is in [rev/min] and P is in [KW] or [CV].
- For pumps and sometimes also for turbines
w
w
w
w
w
w
w
w
nq =
qQ
N
H
3ffff
4
[1.1.3.h]
Introduction
11
A classification for the most typical examples of hydraulic turbomachines is
made in table 1. Extra characteristics information is given in the same table.
In figure 3 a chart with these examples distributed by its specific velocity and
its installation height is provided.
Name
TURBINES
Type
Flow
Specific
geometry
Velocity
Pelton
Action
Tangential
Low
Francis
Reaction
Radial and
Medium
radial-axial
Deriaz
Reaction
Diagonal
Medium
Kaplan,
Reaction
Axial
High
Reaction
Radial and
Low
Bulb
PUMPS
Radial
radial-axial
Diagonal
Reaction
Diagonal
Medium
Axial
Reaction
Axial
High
Table 1. Hydraulic turbomachines classification
Introduction
Figure 3. Turbomachines classification by specific velocity and Installation Head
1.3.2
NOMENCLATURE
Now that turbomachines have been introduced, it is the time to meet their
main components and their positions.
As the study-case in this paper deals with a Francis pump-turbine, a picture
containing its parts is given in picture 4.
12
Introduction
13
Figure 4.Francis turbine section. [IMAG07]
For the rest of types of machines, nomenclature is the same, with the
exception of pumps where, due to the role they play, some parts receive
differents names. Table 2 shows the correspondances.
TURBINE
PUMP
Runner
Impeller
Diffuser/Draft tube
Inlet Pipe/Bend/Casing
Distributor/ Guide Vane
Vaned /Diffuser
Spiral Casing
Volute
Table 2. Differents names for the machine components in pumps and turbines.
Introduction
1.3.3
14
POWER CONVERSION AND BALANCE.
All power definitions above suggest a global energetic balance. Hydraulic
power results from the total dissipated power by the fluid in differents parts
of the machine and the power exchanged in the runner.
Ph = Prsc + Prv + Pro + Pt + Prq + Prb + Prd
[1.3.3.a]
where :
Ph is the hydraulic power
Prsc is power losses in the spiral casing
Prv is power losses in the state vanes
Pro is power losses in guide vanes
Pt is power transferred by the runner
Prb is power losses in the runner blades by friction and others
Prq is power losses by leakage
Prd is power losses in diffuser
Power is defined from energy. Tranferred energy in the runner is associated
to the homologous power and the flow rate going through the runner Qt:
Pt = ρQt Et
[1.3.3.b]
In turbines, the total energy in the runner is
E = Et + Erb
[1.3.3.c]
E = Et @Erb
[1.3.3.d]
While in pumps is
Introduction
15
In the following two pages there are four pictures (pictures 5a. 5b. 5c. 5d.)
representing the power conversion for both, turbines and pumps. While 5a.
and 5c. Show only the mechanical exchange 5b and 5d provide a complete
and detailed power balance.
The legend for the variables in these pictures is:
-
Pm Runner mechanical power
-
P Machine power
-
PLm External mechanical Power Losses
-
Pt Extracted power (turbine) or Supplied Power (pump)
-
Prm Internal mechanical power losses
-
X
Pre Flow power dissipation in turbine
-
X
Prq Leakage flow power in turbines
-
X
Pq Leakage flow power in pumps
-
X
Pr Flow power dissipation in pumps
It is essential to highlight that since power direction is different between
pumps and turbines, efficiency gets different definitions too.
In turbines
η=
P
ffffff
Ph
[1.3.3.e]
η=
Pffffff
h
P
[1.3.3.f]
While in pumps
Introduction
16
Figure 5a. Turbine Mechanical Balance
Figure 5b. Turbine Power Balance
Introduction
17
Figure 5c. Pump mechanical balance
Figure 5d. Pump Power Balance
Introduction
1.3.4
18
EULER EQUATIONS
Euler equations describe in fluid dynamics how pressure, density and
momentum are related in a moving fluid. They are a coupled system of
differential equations, since they have to be solved simultaneously because
all the dependent variables appear in all the equations. Although they appear
to be very complex, they are actually simplifications of the Navier-Stokes
equations. The Euler equations are only valable for inviscid flow, since they
consider zero viscosity. The solution gives therefore only an approximation
of the reality. [NASA06]
They represent the conservation of mass, flow and momentum.
` a
∂fffffff
ρ
+ 5 Apu = 0
∂t
[1.3.4.a]
b` a
c
∂fffffffffff
ρu
+ 5 ρu N u + 5 p = 0
∂t
d
[1.3.4.b]
e
b
c
∂ffffffff
E
+5 u E + p = 0
∂t
[1.3.4.c]
where E is the total energy per unit volume
b
E = ρe +
c
ρfffffffffffffffffffffffffffffffffffffff
u 2 +v 2 + w 2
2
[1.3.4.d]
and u, v,w are the velocity components, e is the total internal energy per unit
mass flow, p is pressure, ρ is density and u is the fluid velocity. [WIKI07]
There are only three equations and four unknowns, so it is required to add
one more equation to solve the system. Normally, an “equation of state”
linking pressure and density is used.
Introduction
19
In practice more simplifications are considered. In hydraulic turbomachines,
from long mathematical developments, simplification assumptions and
concrete power balance for this kind of machines, the called “Euler’s Global
Equation” is derived for the runner.
b
jjjjkjjj jjjjjjkjjj
Et = Kcu c 1e u 1e @Kcu
1E
d
c
ffff
1E
jjjjkjjfffjj jjjjjjjkjjfffj
e
c 1 e u 1 e @Erb
[1.3.4.e]
where K are flow distribution coefficients depending on the machine design,
kjj
jkjjj
c and u are respectively the fluid velocity and rotation linear velocity and
fff
the subscripts 1 and 1 mean high and low pressure section for the external
(e) streamline. It could be perfectly applied to the internal streamline (i) too.
Machine and flow characteristics, such as energy transferred or discharge
rate, are inferred from this equation and its geometrical meaning (velocity
components: sizes and projections) and representated by “velocity triangles”.
These triangles provide a graphic implementation of this equation, from
which Figure 6 gives an example.
Introduction
20
Figure 6. Example of Velocity Tirangles [WIKI07]
In partial loads velocity triangles are modified in the following way. The
β
angle between rotational and relative velocity (called
) is theorically
considered to be imposed by the geometry of the runner and to be
independent of the flow, so that for partial flow as the cu component is
decreased all the rest of parameters from the velocity triangle vary as shown
in figure 7.
From
this
hypothesis,
“Euler
Characteristic”
correspondance between energy transfer
ψ
(fig.
and the flow
ϕ
8)
shows
(for a fixed
the
β
, so
for a concrete machine) as a straight line.
But this approach is just ideal, due to the fact that in reality
β
is not constant,
because the runner geometry can not dominate completely the flow
behaviour. In pumps, the flow against pressure gradient is complicated and
recirculation and detachments appear
degenerating the ideal “Euler
Characteristic”.
Later studies in this paper will take this idea up again and check it results.
(see chapter 4, paragraph 4.4.1)
Introduction
21
Cu1
Cm1
W1
C1
α1
β1
U1
Figure 7. Velocity triangles at the outlet for differents flow rates
ψb
2

Et
1 Q1 
= 2 1 −

2
U1b
 tg β1b A1U1 
Q1
ϕ1 =
πωR13
ψ 1b = 2

ψ 1b = 2  1 −

1
π R12
tg β1b A1
tg β1b

ϕ1 

φ
Figure 8. Euler’s Characteristic
The also better known Bernoulli equation is derived by integrating Euler's
equations along a streamline. This one, under the assumption of constant
density, the equation of state and neglecting the losses, is usually easily
applied to solve fluid circuits in pipes and for simple geometric problems.
When a relative observer is considered for the rotor study in turbomachines
the same equation is called “Rothalpy”. For further information on these
topics see [WHIT03].
Introduction
22
Finally, an important conclusion to draw related to this paragraph is the fact
that to upgrade the power of a machine at a given radius, the increase of its
speed is needed. At the same time, as a result, pressure fluctuations will be
increased too , and consequently all the problems in terms of the structure
damage and instability.
1.3.5
ROTOR-STATOR INTERACTION IN FRANCIS PUMP-TURBINES.
The heart of this paper will be focused on the RSI analysis. Here an
introduction to the phenomena behaviour sets the basis for the later studies
in detail.
The phenomenology of rotor-stator interaction is a consequence of the
combination of inviscid and viscous flow and it is believed to be influenced
by the design of the machine (for example, too large wicket gate thickness,
high rotation speed, etc.)
With respect to viscous effects,
flow incidence angle in the distributor
channels, blockage and flow detachments related to partial load and wakes,
play major roles in the RSI behaviour. Effects of inviscid flow , also called
potential effects, are associated to relative blades motion: the flow in
distributor channels is periodically perturbed by the rotating impeller blades,
and this effect is propagated in both upstream and downstream directions of
the flow, generating pressure fluctuations and disturbances troughtout the
entire machine flow fields.
Once transient effects have finished, an expression for stationary and
periodic pressure fields can be stablished for the rotor ( pr) and stator ( ps)
flows. It must be highlighted that they are only right expressions if no
unsteady effects appear for the concrete operating point.
Thanks to the help of Fourier Series [GARC98]
complex pressure fields in the following way:
we can
express these
Introduction
23
b
c
1
b
p s θs , t =X Bn cos nzo θs + φn
c
[1.3.5.a]
n=1
b
c
1
b
p r θr , t = X Bm cos nzb θr + φm
c
[1.3.5.b]
m=1
where m and n are the armonics orders, Bx and
φx the amplitude and the
phase shift for the appropiate harmonic, θx the angle coordinates for the
rotative (r) or stationary (s) system and zb and zo the number of blades and of
guide vanes respectively.
In the limit regions between stator and rotor the pressure field is a
combination of the two ones above
[ZOBE07], [NICO06], [TANA90],
[OHUR90] whose expression is got from the product of both of them for each
value of m and n from 1 to 1 .
With the help of the trigonometrics functions properties for the product
` a
` a
cos a cos b =
a
`
aC
1fffB `
cos a + b + cos a @b and considering the stator reference
2
system, those products result in the following expression :
b
c
p mn θs , t =
b
c A
b
c
`
a
`
a
Affffffffffffff
ffffffffffffff
mn
cos mz b ωt @ mz b @nz o θs + φn @φm + mn cos mz b ωt @ mz b + nz o θs + φn @φm
2
2
[1.3.5.c]
The connection between the two reference systems is given by the rotation
velocity, since θr = θs @ω t and this equation is a function of two variables:
time and space.
Introduction
24
Flow field distorsion caused by Flow field distorsion caused
the runner pressure field
Combination of
the guide vane effects
both
figure 9. Modulation process between impeller blade and guide vane flow fields
Numerous studies made by Ohura et al. [OHUR90], Tanaka et al. [TANA90]
and Chen [CHEN61] describe how this interaction induces pressure
fluctuactions that are propagated in the entire machine.
From their
conclusions two different types of pressure waves are distinguished:
-
Diametrical mode rotating in the region between the guide vane and
the impeller blades.
-
Standing waves in the spiral case.
The first type is more relevant to our concrete study case.
In fact, the modulated pressure form
p mn is the combination of two
diametrical pressure modes, with high and low order numbers, k1 and k2:
k 1 = mzb @nzo
[1.3.5.d]
k 2 = mzb +nzo
[1.3.5.e]
Positive values for mode number indicate that the wave, the diametrical
mode, is rotating in the same direction as the rotor, while negative values
indicate it rotates in the opposite direction.
The lowest absolute values of these numbers represent the highest energy for
the diametrical modes. It means that the pressure fluctuations are stronger,
have more energy and propagate further along the entire machine for the
Introduction
25
lowest k absolute values. As a result, it is usually more important the impact
of k1 than k2 because of the definition as a subtraction.
Finally, to illustrate it figure 10a gives an example representing the
diametrical modes shape in 3D.
Figure 10a. 3D pictures of diametrical modes.
[OHAS94]
Depending on the value and the sign of k, in figure 10b, areas marked with –
mean they are dominated by a
low pressure wave and + denote high
pressure wave domain. The direction of the arrow is related to the rotation
speed direction: the same as the diametrical mode for positive k and the
opposite on the contrary.
k= -1
k= 2
Figure 10b. Diametrical modes shape patterns according to k values
Introduction
1.3.6
26
FREQUENCY ANALYSIS.
As a conclusion of the paragraph above, periodical waves are expected in the
analysis of pressure fluctuactions inside the hydraulic machine. As has been
explained, several frequency will take part of these pressure signals so that to
understand and better analyse them a descomposition will be necessary.
Based on the Fourier Series Theory [GARC98], [STOR02] the Fourier
Transformation enables the decomposition of periodic wave or signals into
the sum of several sinusoidal functions whose frequency are a multiple of the
wave main frequency.
The general mathematical expression to express this transformation is :
` a
1
p t = a0 + X
n=1
d
b
c
b
ce
a n B sin 2πfB t + b n B cos 2πfB t
[1.3.6.a]
where p(t) is the signal in the time domain, an and bn are unknown
coefficients of the series. From them, the amplitude and the phase of the
signal for each frequency can be calculated. The parameter f, that has units of
Hertz[Hz], corresponds to the fundamental frequency of the wave.
From the values obtained for the unknown coefficients for this
decomposition the signal is “translated” to the “frequency domain” which
enables to infer which are the most important frequencies defining the wave
and its propagation throught the entire machine.
A Matlab code has been developped for the frequency analysis of the
pressure signals obtained in the simulation and in the experimental data too.
It will be used for the comparisons and conclusions, as well. It is attached in
chapter
7.
2
Numerical Simulation Methods.
CFD Methods.
2 NUMERICAL SIMULATION AND CFD METHODS
2.1
INTRODUCTION TO COMPUTATIONAL METHODS
With the development of computer technology, numerical simulation, also
called computational methods, has turned out to be more and more widely
used in fields of our society.
Simulation techniques not only play very
important roles in engineering studies , taking part on what it is called
Computer Aided Engineering (CAE), but also many other sciences take
advantage of them for their studies. Simulation is applied in biologic and
physic sciences, for example for the study of atmospheric, oceanic or internal
earth flows or also for chemical reactions as combustion. Even social sciences
can use these methods for their studies. For instance, in finances they can be
used for banking investigations or for the Stock Market predictions.
Numerical simulation is the kind of simulation that uses numerical methods
to quantitatively represent the evolution or the state of a concrete system.
Numerical simulation can deal with many processes at the same time. Even
nonlinear processes can be solved by these methods.
The evolution of the system must obey some rules that govern the real
processes in the simulated situation. From the outcomes of such simulation
we will be able to draw proper conclusions and to get a deeper knowledge of
the system. It is important to emphasize that knowledge of background
processes and understanding of the simulated subject can be obtained from
the results.
In this paper these methods will be applied for the concrete study case of
hydraulic turbomachines. Computational Fluid Dynamics (CFD), the specific
computational simulations for this scientific area, has become an essential
part of the engineering design and analysis environment of many industries
who require the ability to predict the performance of new designs or
Numerical Simulation Methods. CFD
29
processes before they are ever manufactured or implemented or who want to
redesign and improve old versions. Fluid dynamics are used in different
technologies including aerospatial, locomotive, power generation, air
conditioning, heating and ventilation devices, chemistry and biomedical, oil
and gas conductions, etc. Fluid dynamics are applied in all measureable
technology scales, for a wide range of cases, from ventilation installations in
huge structures to the smallest micro-pumps and the most accurate
nanotechnology.
Physical models are translated into mathematical models and performed by
complicated computer algorithms are used to solve these engineering
problems.
All
these
techniques
have
experimented
a
marvellous
developement recently, stimulated by the increasing computer power, so that
now they can provide their advantages to analysis, investigation and design
tasks, removing and replacing traditional expensive experiments.
These simulation methods bring some advantages in relation to the old
experimental practices. For example, money invested in creating the test
reduced models for the experiences is saved, as well as the time
and
personal employed in them; it also avoids accidents ocurred during them
and eases the sometimes complicated acquisition of measuress which,
moreover, are not always reliable. Another important role for numerical
simulation is that it enables extrapolation of the observations from a region
in which measuress have already been made to unknown regions. If running
a numerical simulation it accurately predicts the observed values of
parameters in the regions where the values were known, it gives confidence
to the simulation outside this region.
Furthermore, it is important to point out that simulation makes possible to
find new phenomena and processes that might be not found in experimental
or theoretical study.
However, limitations of numerical simulation still exist. The development of
numerical simulations started after the computer technology, so it is a fairly
young method. As with theoretical studies, numerical simulation results
need to be confirmed by experimental data, which will be one of the most
important aims of this paper.
Though numerical simulations can deal with much more complex problems
than analytical study, sometimes they can not include all the physical
processes that exist because their equations are not known or the numerical
simulation is not feasible. In these cases, models appear and with them
approximations. In fact, in the majority of cases there are many
simplifications made in numerical simulations, which leads to errors in the
results. The methodology follows a sequence of steps where approaches and
asumptions are required and whose consequences in accuracy will be
mentioned below.
Focusing on the CFD methods we will now present these steps and some of
the approximations they introduce. [FERZ99] [TUTO05]
2.2
CFD (COMPUTATIONAL FLUID DYNAMICS)
The methodology that follows CFD concerns these four main steps:
1. The choice of the physical approach for the real problem.
2. The choice of the mathematical model for the simulation.
3. The discretization method.
4. The implementation of numerical algorithms to solve the equation
systems.
The following pages will introduce the theorical aspects of these steps in
order to help to understand the procedure followed to the studied case in
30
31
this paper and because the possibility of manipulating and adjusting them
could produce interesting improvements in the CFD methods.
2.2.1
THE CHOICE OF THE PHYSICAL APPROACH.
The first step of the procedure is the determine the governing equations for the fluid
dynamics problem. When these equations are know accurately, in theory exact
solutions can be achieved with any accuracy desired. Normally the
governing equations of physical real problems are groups of Ordinary
Differential Equations (ODE) or Partial Differential Equations(PDE).
In our case, for fluid mechanics, these equations are the Navier-Stokes
conservation laws for the mass, the momentum and the energy. They are a
complex system of partial differential, in time and in space , equations.
Numerous conditions and fluid properties influence the flow behaviour. The
conservation laws for the mass, energy and the momentum apply to all flows
but they are non-linear and coupled, which make them difficult to solve.
In the domain of hydraulic turbomachines, flow is assumed to be isothermal,
incompressible and Newtonian. Such incompressible CFD simulations can
not reproduce the propagation of the hydroacoustics waves generated by
RSI, although they can provide boundary conditions for acoustic modeling of
these phenomena [NICO06].
These simplifications from the general conservation laws provides the wellknown incompressible Navier-Stokes equations [2.2.1.a]. Considered a real
description of the Newtonian flows, they constitute the fundamental basis of
our hydraulic problem.
[2.2.1.a]
32
However, the real problem might not follow some of the initial physics
hypothesis. For instance, we cannot assure that the flow rate is constant all
the time or that there are no leakage, etc. These assumptions made in the
physical approach are the first introduction of approximations in simulation
methods.
2.2.2
THE MATHEMATICAL MODEL
A mathematical model is an abstract representation of aspects of an
existing/designed system using mathematical language to describe its
behaviour. A mathematical model usually describes a system by a set of
variables and a set of equations that establish relationships between the
variables. The objectives and constraints of the system can be represented as
functions of the output variables or state variables. In conclusion, the model
presents knowledge of that system in usable form.
Mathematical model and physical approaches are strongly linked. Physical
approaches are chosen and simplified in order to make possible a
mathematical modelisation. Sometimes it translates to a easier definition of
the geometry, as for example 2-D problems, or considering constant
or
neglecting a variable.(density for incompressible flows or inviscid flows,
respectively).
These kind of simplifications of the physical truth transforms the NavierStokes into other easier. It is the case of the Euler equations for inviscid flow
or potential equation for irrotational velocity flows.
All these simplifications can result appropiate in certain and special flow
conditions, whithout meaning a loss of accuracy. If not, these approximations
introduce more inaccuracy in the method.
One of the most important mathematical model in CFD and especially for
our study case is the “Turbulence model”. Approaches to predict turbulent
fluid flows exist but they have still to be improved and calibrated. The most
important ones (and the ones we use for our calculations) are ReynoldsAveraged Navier-Stokes equations (RANS), which consist in the averaging of
motion equations over time or over a coordinate in which the mean flow
does not vary. This approach leads to a set of partial differential equations
which is not closed and, as a consequence, requires a mathematical model to
approximate it, known as “turbulence models”. They introduce an artificial
viscosity to model the turbulent effects of the flow. The most famous models
are the κ @ε
and the κ @ω . Main consequences of this approximation,
deduced from comparaison to experimental data, are: excessive production
of shear stress, suppression of separation along curved walls, excessive level
of turbulence in regions of strong normal stress and wrong response to swirl.
We could also mention other modeling approaches as Large Eddy
Simulation. There are also Direct Numerical Simulation (DNS) which don’t
include any approximation to Navier-Stokes equations, so they are very
accurate but extremely expensive and still out of reach for turbomachinery
flows for the next few decades.
Besides, for meshfree techniques (which will be mentioned in the next
section), the Vortex method turbulent model is used.
2.2.3
THE DISCRETIZATION METHOD.
After the mathematical mode is set up, it is necessary to transform the
continuous nature of the elements (the space, the time...) to a finite quantity
of elements (grid or mesh generation) upon which the model equations will
be placed. Discretization method approximate these equations by a system of
33
algebraic equations of variables. This discrete representation is necessary to
set a problem computationally solvable. Regrettably, both of these two steps,
force to build in some tolerances, so more errors are acummulated.
In applications dealing with PDE on a finite volumetric domain, the most
usual approach is to genere a grid or a mesh that discretize the domain
volume, composed by a certain (usually very large) number of small
elements, which are typically called cells. Variables are computed at nodes
located on cell center and there the governing equations will be calculated.
They can be structured (regular) grids, block-structured grids or
unstructured grids, and each one is more appropiated to a kind of problem
and suggest a different discretization method.
Depending on what is represented in a cell, two main types are
distinguished: the Lagrangian-grid and the Eulerian-grid. Lagrangian grid is
associated to a material description of the model frame, to the substance
particles (figure 11) [IMAG07], while Eulerian grid is fixed to the time and the
space.
figure11 . The lagrangian-grid can be deformated with the particles.
Both are applicable to CFD problems. However, as in fluids deformations are
frequent, sometimes the Lagrangian grid becomes more complicated and is
more associated to solid elements in CSM (Computational Solid Mechanics).
Eulerian-grid are more tipically used for CFD. It was the one used to
34
generate the mesh of the Francis pump-turbine that will be studied in this
paper. (figure 12)
figure12. A partial view of the mesh used for the Hydrodyna Pump-Turbine
There are other special methods to proceed without a mesh-based method.
These alternatives, called meshfree or meshless methods, are based on the
idea of setting randomly distributed nodes with don’t follow any kind of
grid and but which have been well defined to make sure that the equations
of the model for all the domain, including the boundary and limit conditions,
are satisfied. The best example for this kind of methods in fluid dynamics is
SPH (Smoothed Particle Hydrodynamics).
As has already been said, for the computational procedure time will also
have to be discretized. In unsteady problems, the time step choice is another
parameter of inaccuracy for the results.
Once the grid and time steps are defined, it is the time for the discretization
methods. Three main discretization approaches are used nowadays for CFD.
In this case, discretization refers to the differential equations, which are
converted into several algebraic evaluations. They are:
35
•
The Finite Difference Method (FD), which replaces the partial
derivatives by approximations of the nodal values of the function for
each grid cell.
•
The Finite Volume Method (FV), which first subdivide the domain in
many control volumes (CV) and then applied the interpolated
conservation laws to each one. This method is the most stable because
every each of its cells must verify the conservative laws.
•
The Finite Element Method (FE), similar to FV but multipling the
equations by a “weight function”.
For further references see [FERZ99].
2.2.4
THE IMPLEMENTATION OF NUMERICAL ALGORITHMS AND
SOLUTION.
Once the discretization is made, it is the time to translate the equations into a
computational code that allows to solve the mesh conditions for the model.
The algorithms can be written in several differents programming languages,
it depends on the engineer or the programmer preferences, who will try to
make them the simplest but the most accurate too.
This kind of numerical algorithms are considerably complex and the
enormous quantity of equations and variables to be taken into account
obliges to utilize modern sophisticated and powerful processors and clusters
to implement and solve the millions of calculations they consist on.
However, to solve the equation systems iterative methods are used, as they
are too complicated and not convenient to use direct methods. introducing
once more an error, related to the convergence residual value. Iterative
methods need a stop criteria, i.e. a admissible value of a variable to stop the
iterations. This value can be fixed to any desired level of accuracy.
36
Finally, once the algorithms are implemented and solved, one can use the
solution to obtain conclusions for the real initial problem. Outcomes are
usually analysed with proper post-processing programs to get the maximum
benefit of them.
2.3
CFD FOR TURBOMACHINERY.
In the domain of turbomachinery, CFD is today’s essential tool. The major
challenge for CFD in solving problems of this kind of machines is related to
the relative motion between rotor and stator.
Once the physical and mathematical models are set up the problem arise
when defining the relation between the rotationary and the stationary
meshes, as well as the rotor-stator interface. While the rotor is calculated by a
reference system in movement, the stator uses a stationary frame. The
procedure followed, called “Multiple Frame of Reference”, connects them. It
consists in linking at each cell of the rotor interface mesh the conditions for
the cells in the corresponding stator interface mesh at that precise moment.
Moreover, difficulty is increased for partial simulations approaches.
Although this procedure can save time to the simulation process, it usually
provides less accurate solutions, due to the fact than only the full domain
takes into account all the aspects that influence the flow behaviour.
Boundary conditions, turbulence sources, periodical surfaces and interfaces
are seriously perturbed by this kind of approaches. Flow throughout the
entire machine is connected, so that interactions make that what happens in
no matter what region has a considerable impact on the rest.
The skip of part of the complete machine volume provokes disarranges on
the results, because it is not true to solve the flow in one part and then
reproduce translated periodic conditions for the rest. The number of
37
channels of stator and rotor is normally unequal to avoid resonance
phenomena, and to connect the meshes in partial simulations, the rotor-stator
region has to be critically modified.
To represent a portion of the machine that includes both, rotating and fixed
parts, it is necessary to fit a relation between the number of impeller blades
and diffuser vanes, usually with no common division. The decision of this
relation, known as pitch ratio, plays a decisive role in the triumph of the
outcomes in partial simulations. The more the pitch ratio approaches to the
real ratio between the total number of impeller blades and the number of
guide vanes, the better. On the contrary, if a large different number is chosen,
periodicity effects will be severely perturbed and badly representated by the
model. Mistakes and fake results will possible appear in the outcomes.
The approach in these cases used in CFD programs for approximating to
have periodic conditions in partial simulations used to be called “sliding
mesh” .
2.4
CFD
HISTORIC EVOLUTION
As has already been said, CFD method are quite young. However, they have
experimented a fast evolution and development.
At the beginning, these techniques could only resolve linearized potential
equations in two dimensions, later Euler equations were solved and huge
progresses have been done so that nowadays these methods manage to solve
the Navier-Stockes equations. Firstly, only 2-D codes were developed and
recently 3-D codes are available in numerous software packages.
Figure 13 shows the fast development in computer technology since 1993.
38
Figure 13. Computers performance evolution from 1993-2007 [TOPL07]
From the observation of the first picture one can see that for example, in only
three years, between 2001 and 2004, supercomputers increased 10 times its
power. Just a last remark to emphasize the revolution in computers world:
todays laptops have approximately the same computing power of a 1995’s
super-computer.
2.5
CFD ERROR ESTIMATION
It has already been explained above that even in the most accurate numerical
simulation a few approximations are introduced. The differences between
reality and numerical simulation results can be due to them:
-
The differential equations governing the problem may contain
approximations or idealizations.
-
Discretization approaches are approximations, by definition.
-
Iterative methods to solve the equation systems introduce the
“truncation error”. (maximal residual value of convergence)
39
-
Round-off error, related to the machines precision. It means that
computer precission has a finite number of decimals.
To close this chapter, we can summarize the CFD procedure as normally it is
followed in practice by software packages:
-
The preprocessing, where the physical model equations, the geometry
and and the boundary conditions are described and fixed. In case of a
transient problems the initial conditions must be here defined. The
discretization of the volume and generation of the mesh is part of the
preprocessing.
-
The computational simulation. In many cases the simulation is repeated
and many iterations are done till a good result is obtained. Different
causes, coming from the preprocessing phase or not, can provoke this
need to repeat the calculations; However, in most of the cases these
repetitions are made to control the quality of the results, as in
parametric sentivity studies.
-
The post-processing of the results. It is during this part when the real goal
of computational methods is attained. Many developped tools are
employed to draw out the maximum information and conclusions
from the results.
40
Pre-processing
Processing:
Computational simulation
Post-processing
41
•
•
•
•
Choice of the models
Mesh generation
Physical properties and constraints
Initial and boundary conditions
Iterations to solve equations system
Analysis of the results
Figure 14. CFD sofware procedure
3
Study Case: Introduction and
description.
Study Case: Introduction and description
3
43
INTRODUCTION TO THE STUDY CASE
In the first chapter the Hydrodyna Project was introduced. To get deeper
knowledge in fluid turbulent flows in pump-turbines several studies and
experiments have been made.
As it was said in the first chapter, turbine mode study has already been
investigated, so in this paper we will focus fundamentally in the pump
mode.
The design of pumps is mainly based on steady flow assumptions
throughout all the machine components (runner, diffuser...). This kind of
approach suits reasonably for design operating conditions, but to understand
and to take into account wider ranges in pump designs it is necessary to
increase the knowledge of unsteadyness and RSI.
Two main tasks should be covered according to the project objectives: the RSI
phenomenon and the fluid/structure coupling.
Undesirable events like flow detachments, ungovernable turbulence and
cavitation phenomena, structure vibrations, wakes transport and dissipation
will be studied in the context of experiments in hydraulic turbomachines.
Experimental data is currently very required for a better optimisation of the
pump design. The best option to improve pump design is to develop
powerful
numerical
simulation
methods
of
internal
flow.
Small
improvements to rotating machinery design can translate into large
operating savings and other important advantages. However, only
validating these design methods from their comparison to the observations
obtained in laboratory tests, a real production increase, longevity
improvements and waste decrease will be possible to come true.
As an objective of this paper, all the analysis and the results obtained from
experiments in pump-mode will be explained for a better perception of the
phenomena ocurred in the machine.
3.1
44
DESCRIPTION OF THE HYDROMACHINE
The turbomachine used for the experiments is a reduced scale model of a
Francis pump-turbine (figure 13) composed by 20 stay vanes, 20 guide vanes
and 9 runner blades. The machine casing is the spiral type and its specific
speed is ν = 0.19 (or what is the same nq= 29.98) so it belongs to Francis low
speed range. See picture 3 in chapter 1 to locate it.
Its Efficiency and Energy/Discharge coefficient in pump-mode are given in
figures 16 and 17 only for the 20º vane opening angle position this study will
work with.
Plots containing more other positions are attached in chapter 7. See
“Hydrodyna’s Hill charts”.
figure 15. Cross section of Hydrodyna reduced scale-model.
Runner Type
Francis
Number of
Number of
Number of
Nominal
Specific
blades
guide vanes
stay vanes
flow rate
velocity
channels
channels
20
20
0.228 [m3/s]
ν=0.19
9
nq=29.98
Table 3. Machine definition characteristics
45
η [-]
HYDRODYNA
0.9
0.89
0.88
0.87
0.86
0.85
0.84
0.83
0.82
0.81
0.8
0.79
0.78
0.77
0.76
0.75
0.74
0.73
0.72
0.71
0.7
-0.05
20 deg
-0.045
-0.04
-0.035
-0.03
-0.025
-0.02
-0.015
ϕ [-]
figure 16. Hydrodyna’s efficiency chart
HYDRODYNA
1.2
1.15
1.1
1.05
ψ [-]
1
0.95
0.9
0.85
0.8
20 deg
0.75
0.7
-0.05
-0.045
-0.04
-0.035
-0.03
-0.025
-0.02
-0.015
ϕ [-]
Figure 17 Hydrodyna
ψ.
ϕ coefficients chart
To have detailed views of the scale-reduced Francis model see “Attached
documents” in chapter 7.
3.2
46
PARTIAL SIMULATIONS AND STUDIED POINTS
The choice of points was decided from experimental data measures. During
previous studies made in generating mode, some measures were also taken
for the pump-mode. Among the existing measures, there was a special
interest in the comparison of two of them, due to the different behaviours
observed during the rig experiments.
These points were corresponding to ϕ = @0.028 and ϕ = @0.043 . All the
studies in this paper for both results, computational methods results and
experimental data measures, will be only about them.
The characteristics of these two Hydrodyna’s pump mode studied points are
summarized in table 4.
ψ EXP
ψ CFX
ηEXP
ηCFX
ϕ = @0.028
1.07
1.028
0.855
0.879
ϕ = @0.043
0.83
0.729
0.863
0.795
Table 4. Hydrodyna’s characteristic values for the studied points
The explanation for the differences between these values between
experimental measures and computational simulations has its basis on the
power balance. As it has introduced in the first chapter (see figures 5), there
are many different types of losses throught all the energy transfer between
the machine components. Some of these losses are not considered for the
model in CFD, so the real values calculated from the laboratory measures can
not be exactly equal to the ones calculated from the simulations results.
With respect to the computer domain of study, three partial computational
domains studies were made.
47
It has already been commented that partial simulations allow to solve a
smaller problem by assuming periodicity of the flow, this means identical
flow in all the channels.
Another issue of such partial simulation is that the portions of the rotorstator interface do not cover the same angle αB on the impeller side and αo on
the diffuser side. For the study case of this paper with a real relation of 9
blades/20 guide vanes, the following combinations are considered:
Pitch ratio
ZB
αB
Zo
αo
1
40º
2
36º
1.111
2
80º
4
72º
1.111
3
120º
7
126
0.952
αfffffff
B
αo
Table 5. Parameter values of the three partial simulation studied cases.
The legend for this table is:
ZB: number of blades.
α B :Angle between blades.
Zo :Number of guide vanes channels.
α o : Angle between guide vanes channels.
Conclusions based on the quality of their results compared to the full
simulation will be commented too.
3.3
EXPECTED ROTOR-STATOR BEHAVIOUR IN HYDRODYNA
Recently, RSI has been investigated in depth. The conclusions of all those
studies and investigations have been introduced above. Particularly, the
48
diametricals mode behaviour explained in the first chapter in the rotor-stator
zone and the pressure propagation up and downstream is expected to be
applicable to our case.
Hydrodyna, with 9 impeller blades and 20 guide vanes, will present the
following order values for the diametrical modes, calculated in table 6:
n
m
k1
k2
fs/ fb= mzb
1
2
-2
38
18
1
3
7
47
27
2
4
-4
76
36
2
5
5
85
45
Table 6. Expected rotor-stator diametrical modes for Hydrodyna.
The fifth column includes the frequency expected to dominate the pressure
fluctuactions created by this phenomenon. It is calculated from the number
of blades in the machine and the harmonic order corresponding to the rotor
(m) for the minimum k number.
Since k1 is negative and m value is 2, the wave is expected to rotate in the
opposite direction to the rotor and to have the fundamental frequency for a
value twice the blade passage frequency (BPF). In this case 18 times the
rotating frequency.
4
Numerical Simulation. Procedure
and Results
Experimental Data Results
50
4 NUMERICAL SIMULATION. PROCEDURE AND RESULTS
4.1
INTRODUCTION. THE PROCEDURE
FOLLOWED
As it was discussed in previous chapters, modern software packages and
powerful processors are vital tools to resolve CFD problems of the type we
are dealing with in this paper.
The program used for the calculations was the version ANSYS CFX-11.0 of
ANSYS CFX, containing three subproducts (Pre, Solver-Manager and Post).
The three of them were applied to the problem. [LMH07]
The program enables the analyst to choose among multiple possibilities for
the conditions and the models that will be applied and considered in the
resolution of the problem (angular average velocity profile, constant average
pressure at the outlet, Log law for the solid surface, k-epsilon turbulent
models, stage, frozen rotor or transient rotor-stator simulations...).
The set-up plays a major role in the success of the problem. An input file
created for one of the first simulations of this particular problem is attached
in chapter 7. See “Example of a List of set-up expressions”.
Once the set-up is completely and correctly defined, it is sent to the solver to
be solved and to generate a solution. For this study, numerous simulations
were calculated. Partial domains, different points of work, correction of
errors in the initialization... all of them thought to get the most of the
information possible. These calculations take several days to be finished
even for the high-speed supercomputers, so it is convenient to make sure the
set-up is carefully made in order not to waste time, money and energy.
Before considering the outcomes as ultimate it is always recommended to
check them and judge them, to assure that the set-up was correct and that no
mistakes will be committed in studying these results. Then continue with the
next and final analysis step.
At the end of the procedure, if everything was properly done, the objectives
of the method can be reached. The post simulation tools are used for
analysing in depth the results.
4.2
PREVIOUS STUDY: TURBINE MODE RESULTS
4.2.1
PARAMETRIC STUDIES OF THE MODEL
As the pump mode will make use of many conclusions and tools already
used during the turbine study, parametric study conclusions from the model
in this generating direction are enclosed here.
These parametric simulation results should not disagree with the pump
mode ones as they are general for the global model and they should not vary
within the machine mode.
To take subjective approximations for decisions about the constraints
imposed to the problem conditions without analysing before the influences
of some variables, can lead to overkill the process with calculs and larger
time computing or , on the contrary, have not strong or not reliable
results.[KUEN99]
The most significant parameters to study their sensitivity on the problem are
the mesh, the time step and the convergence.
4.2.1.1
MESH SENSITIVITY
Mesh sensivity was investigated on a partial computational domain because
of computational memory limitations. Structured meshes were alwasys used.
They included two stay vanes and guide vanes channel, one impeller blade
to blade channel and the corresponding part of the cone. The three sensivity
51
52
propositions had the same shape but different grid refinement. The
possibilities were:
Mesh
Stator/channel
Rotor/channel
Coarse
30,000
70,000
Medium
60,000
170,000
Fine
130,000
400,000
Table 7. Mesh sizes for size sensitivity analysis
The conclusion was that, weighting up the accuracy of results and the
computational effort, a medium mesh was enough and satisfactory.
4.2.1.2
TIME STEP SENSITIVITY
For the time step study, a medium mesh was applied to three possible time
steps and at three different points of the mesh. Time steps corresponding to
0.5º, 1º and 2º of the impeller rotation and at two points from the guide vane
and one at the blade leading edge.
Results suggested to use the time step corresponding to 0.5º.
4.2.1.3
CONVERGENCE SENSITIVITY
Convergence sensitivity refers to the maximum residual values for the
numerical methods. Results for Cp obtained using three different residual
values ( 10-2 , 10-3 and 10-4 ) were compared. The mesh used was medium and
the simulation time step was the corresponding to 0.5º according to what had
already been set. The evaluation of the influence of this parameter for the
three cases concluded to establish 10-3 as maximum residual value.
4.2.2
53
CONCLUSIONS FOR THE TURBINE MODE
The turbine mode investigation for the maximum discharge operating
condition present the following major conclusions:
•
Pressure fluctuations for the distributor channel obtained from the
numerical simulation are in very good agreement with experimental
data. However, fluctuactions amplitudes in the rotor frame are around
25% higher for the simulations results than the experimental value.
•
A computing domain modelling the entire machine geometry enables
to minimize the errors. It overcomes the influence of boundary
conditions, pitch-ratio of rotor-stator interface and non-uniformities in
spiral-case flow.
•
The numerical result analysis shows the variation of the pressure
fluctuations at blade passage frequency (BPF) and its harmonics along
a distributor channel of the Francis pump-turbine.
•
The maximum pressure amplitude of BPF takes place in the rotorstator region. However it decreases quickly, backward to the stay
vanes.
•
The diametrical mode resulting from the modulation of RSI flow field
corresponding to the highest energy, has a k number value of -2,
which
means
that
the
pressure
amplitude
dominating
the
fluctuactions generated in the rotor-stator zone spreads to the spiral
casing with a dominating first harmonic of 2 BPF. Third and fourth
frequency modes are still visible in the guide vane but dissapear in the
stay vane channels.
4.3
PUMP MODE
From this point, the crux of this chapter will be handled : The pump mode
simulation and its results.
The same structured mesh that had been generated for the machine in
turbine mode was applied for the inverse mode. Only the vanes angle
position of the diffuser had to be adapted to the position of the operating
point that was going to be studied. It was simple to do, just consisting in a
rotation of the mesh. Two adaptations were made: one for an opening guide
vane angle of 18º and a second one for an opening guide vane angle of 20º.
At the end, only this last one was studied in depth, because there were no
experimental pressure measures for 18º.
4.3.1
COMPLETE MACHINE SIMULATION
As it has been mentioned ANSYS CFX-11.0 was used for the simulations.
The physical model was fixed as incompressible flow, unsteady RANS
(Reynolds Averaged Navier-Stokes) equations and “sliding mesh” for RotorStator interfaces. Between the available turbulence models, in this case, the
SST was chosen, which combinates κ @ε and κ @ω characteristics. Boundary
conditions imposed in the simulation software set up are given in table 8.
54
55
Rotating Frame
Outlet
Inlet
Stationary Frame
Figure 17. The complete machine computational domain.
Boundary conditions
Location
Characteristic
Inlet
Draft tube inlet
Constant mass flow rate
Uniform velocity
Outlet
Spiral casing
Constant average static pressure
Walls
Solid surfaces
Log Law
Table 8. Boundary conditions
For the space discretization, the Finite Volume (FV) method is applied. It
uses a blended discretization scheme for hybrid meshes. These meshes (figure
18), which were taken from the turbine study, are mostly structured, except
for a small region of the spiral casing. The characteristics for each component
of the machine are summarized in table 9. In total, around 4,500,000 nodes
were simulated.
56
Nodes per
Channel
Minimum
Angle
Mean
y+
Spiral casing
500,000
23°
120
Stay vanes
100,000
35°
100
Components
Mesh
Software
ICEMCFD
Guide vanes
Impeller
185,000
39°
60
Cone
135,000
41°
270
Mesh
topology
Structured
5
Table 9. Mesh characteristics
Figure 18. Complete machine runner meshes
To obtain the complete grid from only the basic mesh containing one blade
for the runner and for the stator two vaned diffuser channels, it was
neccesary to make nine copies of the impeller blades and displace them and
analogously for the diffuser channels. Besides, the rest of the regions
composing the complete machine (the cone and the spiral casing) were
added and defined.
The time discretization used a second order backward euler, which is
implicit and quite stable, allowing CFL >1 . The second order approach takes
57
into account tn-1, tn, tn+1 and the time step was set to 1º (although for
parametrics studies 0.5º was recommended).
The convergence criteria for coefficient loop presented a RMS residual value
of 510-5 and a maximum value of 5 10-3.
12 runner revolutions were simulated to arrive to statistically steady state
solutions.
This complete domain simulation was only done for an opening vane angle
of 20º but at the couple of different discharge points:
ϕ = @0.028 and
ϕ = @0.043 .
As it has been already explained, this points were selected because of
existing experimental data measures at them. The different behaviours
observed during the experiments created a special interest in their
comparison and their simulation, in an attempt to find the causes.
4.3.1.1
GENERATION OF VELOCITY
TRIANGLES
Velocity triangles are a specific graphic representations for turbomachines. It
has been mentioned in the beggining of this paper, as a important tool
commonly used, since it contains much information in a simple graph. It
should be added that it can also be used fot detecting users error, just
checking it at the beggining to verify the operating points and main variables
had been well defined in the set-up.
They are derived by the Euler’s Global Equation and its geometrical
meaning. Just having a quickly look to a velocity triangle (its angles, the
length of its sides, the comparison between two triangles of the same
machine or different machines) is possible to approximate and get an idea of
the current operating point or the energy transferred. Therefore, velocity
triangles are really useful and illustrative analysis tools and worth to be
taken into account for any turbomachine investigation.
A procedure to get a representation of velocity triangles for these two points
(and capable to be used for represent any other case) was designed and
implemented using the simulation software. The algorithm, not so evident at
the beginning, enables to see the triangle in 3-D situated between the rotorstator and sorrounded but all the rest of the components. It was something
new, created for the first time for this study. Pictures 19 and 20 show the
outcomes for the particular operating points:
Figure 19. Velocity triangle for ϕ = @0 .028
58
59
Figure 20. Velocity triangle for ϕ = @0 .043
α
β
cfffff
u
u
ϕ = @0 .028
9.8º
ϕ = @0 .043
18.82º
13.67º
15.65º
0.5782
0.4513
Table 10. Summarized velocity triangles characteristics
Angles between the velocities have been included and summarized in table
10 . Alpha
α
is the angle between rotation velocity and flow absolute
velocity, while beta
β is the angle between relative velocity of the flow seen
by a rotative observater situated in the rotor and the rotation velocity.
Velocity triangle for ϕ = @0.028 shows its correspondent α value of around
9.8º and its
β value of 13.67º .
It is a point of partial load.
Velocity triangle for ϕ = @0.043 has important differences. For this point,
while
β is very similar with a value of 15.65º, α
has bigger value, around
double from before: 18.82º. It is very important to highlight that it is quite
well aligned in relation with the guide vanes opening angle of 20º; it is a high
discharge point.
60
On the other hand, first disagreements with the ideal “Euler characteristic”
theory, introduced in the first chapter, appear here. One can see that
β is not
constant, so that it means the runner blade does not always transfer all the
energy it was designed to, because of the slip derived from flow going
against pressure gradient. It behaves different in partial loads and non-linear
losses appear. The “Euler’s characteristic” is perturbed as shown in figure 21.
ψ
E = Et − ∑Er
∂p
>0
∂l
ϕBEP
φ
Figure 21. Euler’s Characteristic Losses Perturbation
Back to our two concrete cases, their differences and resemblaces, having in
mind that rotation speed is the same for both, and the relation to the energy
of the flow and the discharge rate in each operating conditions provide the
following physic interpretations.
In what respects to energy, as it was explained in the first chapter for the
Euler Global Equation, the energy transformed in the runner can be
calculated as:
b
61
jjjjkjjj jjjjjjkjjj
d
c
Et = Kcu c 1e u 1e @Kcu
1E
ffff
1E
jjjjkjjfffjj jjjjjjjkjjfffj
e
c 1 e u 1 e @Erb
[4.3.1.1.a]
Neglecting again losses term and knowing that inlet is designed to present
axial flow (so that energy transferred is optimized), only the first term in the
right part of the equation is considered. Hence, energy is directly
proportional to the circumferential component of the flow absolute velocity.
According to the distribution of the flow velocity components mentioned
above depending on α values, the point ϕ = @0.028 presents smaller
its circumferential component
α , so
for the absolute velocity of the flow ( cu ) is
bigger than for ϕ = @0.043 . In conclusion, ϕ = @0.028 energy transformation
must be greater.
A bigger
α
(for ϕ = @0.043 ) with similar
β
and the same rotation velocity
means that radial components of both, the absolute and the relative velocity
of the flow, are increased; However, only the circumferential component of
the relative velocity grows, while that component decreases for the absolute
velocity as α increases.
As absolute radial velocity component is associated to mass flow rate, if it is
bigger at the point corresponding to ϕ = @0.043 the corresponding discharge
must be bigger than at ϕ = @0.028 . Indeed, this is truth: ϕ = @0.043 presents a
flow rate of 0.231 [m3/s] and ϕ = @0.028 only 0.152[m3/s].
The code written to generate the session in CFX-11 is attached in chapter 7.
See “ Code for creating velocity triangles session”.
4.3.1.2
RESULTS . ROTOR-STATOR INTERACTION (RSI)
“Monitors Points” were set in varied locations of the mesh surface. They
function as sensors, allowing to study the pressure, the vorticity, the velocity,
etc. in the points where they are located. The analysis of these variables using
the tools of CFX and exporting the data files to other external programs
provided a large amount of pictures, results and plots. However only some
of them, considered the most conclusive and/or interesting ones, will be
shown in this paper.
There were 10 monitors situated on a blade wall, reproducing the same
number of the sensors in both faces of a blade, and 8 monitors in each of two
stator channels (so 16 monitors in the distributor). For each of these channels,
4 monitors are in the external part of the stator (next to the shroud) and other
4 at identical positons but in the interior part (next to the hub).
At the beginning all these points were simulated and analysed, but realising
that pressure fluctuations at the blade did not present any interesting
remark, the analysis were focused on the stator points, especially on the ones
at the rotor-stator interaction zone. To show the “ideal” behaviour plots
coming from five monitors points belonging to a streamline at the blade
(figure 22) are exposed in figure 23 and figure 24, for the ϕ = @0.028 and the
ϕ = @0.043 operating points, respectively.
Figure 22. Points from a streamline at the blade
62
63
Cp
0.1
Cp [-]
0.05
0
0
40
80
120
160
200
240
280
320
360
-0.05
-0.1
Bbb10
-0.15
Bbb11
Bbb7
Bbb8
Bbb9
figure 23. Cp fluctuations during a runner revolution for points along a streamline
from the blade at ϕ = @0.028 operating point
Cp
0.025
0.02
0.015
Cp [-]
0.01
0.005
0
-0.005 0
40
80
120
160
200
240
280
320
360
-0.01
-0.015
-0.02
-0.025
Bbb10
Bbb11
Bbb7
Bbb8
Bbb9
figure 24. Cp fluctuations during a runner revolution for points along a streamline
from the blade at ϕ = @0.043 operating point
On the following, only the points on the guide and stay vane will be studied
carefully, so these are the only ones appearing and named in figure 25.
To understand the place where the rest of monitoring points were located,
see the pictures in chapter 7: “Attached documents”. Not all of the pressure
sensors installed in the real machine for the experiments had been
monitorized for the simulation. The table with the correspondency between
them is also attached in that final chapter.
64
Figure 25. Monitoring points location in the distributor
As the aim of this paper is the comparison of experimental and numerical
results and as the region that presents the greater unsteadyness and major
difficulties for the measurements is the rotor-stator gap, the analysis will pay
specific attention to the monitor points located closer to the gap, especially to
A and B, in both external (e) or internal (i) walls.
Furthermore, the study will be focused on the pressure, the velocity fields on
this area, as to investigate what are the causes of vibration and flow
detachments and how they can be prevented.
The non-dimensional parameter Cp defined as
pffffffffffffffffffffffffffffff
@p mean
Cp =d e
1fffff
ρ Au 2
2
[4.3.1.2.a]
were p is the pressure in the current point in the precise moment, pmean is the
average pressure for the point, ρ is the density of the water and u is the
linear velocity for the rotation speed.
Cp will be used all the time in our analysis. It is a coefficient that gives a
normalized idea of the pressure fields behaviour in a point of in relation to
the flow kinetic energy.
Results are completely different for the two operating points of study. The
first example, with a ϕ = @0.028 is much more instable than ϕ = @0.043 ,
whose Cp fluctuactions are cleaner and almost perfectly periodic.
Compared to the rest of the stator points, points next to the rotor-stator
interaction region experience more important pressure fluctuations. Their
amplitude peak to peak of the Cp periodic signal is bigger than for the points
far from this region. However, the absolute values it fluctuates between are
smaller. This is due to the fact that a pump-diffuser converts kinetic energy
into pressure energy, so, in pump mode, it must be increased along the stator
channel.
Certains of these conclusions and more detailed ones will be exposed on the
next paragraph.
4.3.1.2.1
STUDY FOR ϕ = @0.043
For this operating point, corresponding as shown in velocity triangles to an
α
angle of 18.82º, the pressure fluctuactions during a runner revolution are
representated by the coefficient Cp and shown in figure 26.
65
66
Cp
Cp [-]
0.04
-0.01 0
-0.06
40
80
120
160
Point A
200
240
280
320
Point B
figure 26. Cp in the stator side for points A and B near the rotor-stator gap for
ϕ = @0.043
From the graph, a first glance gives already the phase-shift between the
points A and B. They are very similar waves, fluctuating between the same
values with similar periods and amplitudes, only displaced by a phase
component.
As the X axis represents rotation degrees, one can see that peaks are
separated by around 20º, agreeing to the real 18º between point A and point
B.
For each of the points individually periodicity is found every 40º, exactly the
phase shift between two blades (there are 9 runner blades for 360º). There is
the first RSI consequence. Fluctuations at this couple of points in the stator
side are completely determinated by the blade passage frequency (BPF). It
will be studied and proved by the following frequency analysis of the
signals.
More points were studied and compared between them, between the
external and the internal walls and also for the whole streamlines in the
channels, but no sign of unexpected phenomena was found to be shown as
360
67
interesting. See some pictures showing the “perfect” flow behaviour in
figures 27 and 28.
Cp [-]
More details and comparison will be exposed later.
Cp
0.025
0.02
0.015
0.01
0.005
0
-0.005 0
-0.01
-0.015
-0.02
-0.025
-0.03
-0.035
-0.04
-0.045
40
80
120
160
D
200
B
F
240
280
320
360
H
Figure 27. Cp curves for a stator streamline during one runner revolution from the
simulation results for ϕ = @0.043
Cp
0.09
A_(e)
Cp [-]
0.06
A_(i)
0.03
0
0
40
80
120
160
200
240
280
320
360
-0.03
-0.06
Figure 28. Comparison of pressure fluctuactions for one runner revolution between
the internal and external points located in section A for ϕ = @0.043 .
4.3.1.2.1.1
68
FREQUENCY ANALYSIS.
As it has been introduced, the expression for the Fourier Transform is :
` a
1
p t = a0 + X
n=1
d
b
c
b
ce
a n B sin 2πfB t + b n B cos 2πfB t
[4.3.1.2.1.1..a]
The parameter f, in Hertz [Hz], is the fundamental frequency of the wave,
and in this study, is calculated corresponding to one runner blade passage
for the current machine rotation speed: 1/9 w
The developped matlab code (see attached documents) for these frequency
analysis allows multiple comparisons at the same time. From all the
possibilities only some concrete results will be put into show here.
First of all, figure 29 compares frequency spectra for all points in the section
next to the rotor stator interface. So, it includes both external and internal
points A and B, located in the two stator channels.
Figure 29. Pressure Spectra for section next to the RS gap for ϕ = @0.043
69
Conclusions are clearly visible: all the points have very similar frequency
spectra and no relevant differences are found between external and internal
points at this location. There are small amplitude variations between the
channels, with bigger amplitude at the first BPF for channel 20,
corresponding to A section. (to remind nomenclature, see Attached
Documents chapter 7).
Also it is possible to appreciate that while for the first BPF is higher the
amplitude of the external points for the second BPF (18) is the contrary, so
that internal points have higher amplitude.
The following figure shows the propagation along the channel.
Lower
frequencies tend to “dissapear” and also their amplitudes are lower as one
goes far from the rotor-stator interface. For point B which is in this limit
region, the amplitude is still relevant till the fourth harmonic.
Figure 30. Pressure Spectra for points from the stator channel for ϕ = @0.043
70
4.3.1.2.1.2. CONCLUSIONS FOR SIMULATIONS
IN
ϕ = @0.043
For this high load operating point , ϕ = @0.043 , the simulation results satisfy
really well the expected behaviour.
According to the frequency analysis, the fluctuations wave is not propagated
very far upstream the stator diffuser. Only the limiting region next to the the
rotor-stator gap presents relevant Cp amplitudes fo the BPF harmonics, till
the third and forth one.
Just a remark can be done according to flow rate distribution between the
channels. It has been observed that the Cp amplitude is slightly stronger for
the channel 20 than in channel one. It can be due to the influence of the
tongue. At channel 20 is the last one, the pressure waves are blocked while in
the first one where they can be easier propagated to the volute.
The expected behaviour of the pump-turbine is verified from the CFD
results. No sign of detachment flow are shown by the simulation for this
operating point.
Explanations to this fact could be the closeness of this point to the BEP (Best
Efficiency Point. In the hill chart corresponding to 20º, for a ϕ = @0.043 the
efficiency takes a value around 0.863, while the best efficiency point arrives
to 0.88). In these cases, near to the BEP,
for which the design was
optimalized, velocity triangles are quite well adapted to the machine
geometry: it was shown that
α
angle was of 18.82º , almost the same as the
guide vane opening angle of 20º. In these cases, unsteady phenomena does
not develop and RSI behaves as expected.
4.3.1.2.2
STUDY FOR ϕ = @0.028
Now it is the time to analyse the ϕ = @0.028 operating point. Corresponding
to
α
angle of 9.8º, much smaller than the previous one, its pressure
71
fluctuactions during a runner revolution and for the same points as before
(Points A and B) are shown in figure 31.
Cp
0.14
Point A
Point B
Cp [-]
0.09
0.04
-0.01
0
40
80
120
160
200
240
280
320
360
-0.06
figure 31. Cp in the stator side for the points A and B near the Rotor-Stator gap for
ϕ = @0.028
Waves are not so similar now, neither the phase-shift is so evident between
the two points. At least, a kind of in-stable periodic tendency is inferable for
each one, still taking a value of approximately 40º, according to the BPF. To
make easier the analysis for these waves a frequency study will be neccesary.
72
Cp
0.08
0.06
Cp [-]
0.04
0.02
0
0
20
40
60
80
100 120 140 160 180 200 220 240 260 280 300 320 340 360
-0.02
-0.04
D
-0.06
B
F
H
simulation results for ϕ = @0.028
In this case, as the results are more disturbing, more analysis and
comparisons are really necessary in order to find the reasons.
Cp
0.08
Cp [-]
0.04
0
0
40
80
120
160
200
240
280
320
-0.04
-0.08
-0.12
A_(e)
A_(i)
Figure 33.. Comparison of pressure fluctuactions for one runner revolution between
the internal and external points located in section A for ϕ = @0.028
360
Looking for a detachment, pressure fluctuactions from the same points but
belonging to the external (e) and internal(e) region of the pump are
compared.
Results in figure 33 show clear differences compared with the perfect
superposition of cp fluctuactions observed for ϕ = @0.043 .
From this graph, one can see that pressure fields in the hub and the shroud
for this partial load point present a very different behaviour in various
moments during one revolution. The several peaks in the graph show these
strong differences in pressure tendency and its reasons worth a deeper
analysis.
4.3.1.2.2.1. FREQUENCY ANALYSIS
As has been discussed and explained in previous paragraphs, it is possible to
translate the Cp signal into “frequency domain” to extract the most relevant
frequencies appearing on it.
For this particular operating point , it will be really useful to help to find
what can be the main causes of the mess in the pressure signals and to
understand why it happens.
The results are shown and commented in the following pictures:
73
Figure 34. Pressure Spectra for section next to the RS gap for ϕ = @0.028
There is already here an interesting remark: one can see that in CFD
simulation for ϕ = @0.028 at section B are quite different than in section A,
and which is more important, they do not present the same amplitude,
especially between external and internal faces in A. While for BPF the
external A has higher amplitude for 2 BPF is higher the internal. The most
important detail to be remarked is the fact that the internal A has higher
amplitude for the 2nd
frequency than for the 1st. It corresponds to the
diammetrical modes.
To continue the study for these RSI effects in A, a comparison including the
following point C in its same channel is shown in figure 35.
From this picture we can observe that this behaviour occurs only for the
internal point of section A, while the external present Cp amplitudes
decreasing for lower amplitudes. The same observations can be done for
points at C section, both internal and external.
74
Figure 35. Pressure Spectra comparison for points external and internal next to the
RS gap of channel number 20 for ϕ = @0.028
Checking the other channel, (picture 36), one can find out the similar
behaviour: 2nd BPF is more relevant than 1st . However, in this channel for
both external and internal points in section B. On the contrary, point D
behaves as in the first studied ϕ case.
75
In order to finish this analyse , to check if it only this effect is found in the
points next to the rotor stator gap and to have a global overview of the whole
stator streamlines, plots in figures 37 and 38 are attached.
Figure 37. Pressure Spectra for points from the stator channel 1 for ϕ = @0.028
76
Comparing both channels, conclusions about their differences must be
drawn.
There are no special comments to do about the points in the stay vane ( E, F,
G, H) in either channel 20 or channel 1.
On the other hand, point A, B, C and D have differents Cp amplitudes. While
channel 20 shows decreasing tendency for lower amplitudes, in channel 1
point B experiments higher amplitude for the 2nd BPF than for the 1st . This
effect should be taken into account (first be verified by experimental data), in
order no to excite this frequency.
Also, we could remark higher values in Cp amplitude for point D than for
point B, which is the nearest tothe gap region.
4.3.1.2.2.2. CONCLUSIONS FOR SIMULATION WITH ϕ = @0.028
The results for this partial load case point out that something different and
not designed is happening. Some explanations can be based on the analysis
of velocity triangles, frequency domain and the theoric knowledge, as well as
the comparison with the other studied points.
As it has already been said during the frequency analyse, particular and
different phenomena on the domain frequency of the signal were found.
Differences between channels and between external and internal points
amplitudes, have been shown in several of the compared cases, not only in
frequency domain but also in time domain.
As a first conclusion according to all the CFD observations, it is possible to
infer that this disturbance must be due to the fact that in partial loads, far
from the designed point (BEP) of the operation, a detachment of the flow
takes place in the rotor-stator gap. It is caused by the blockage created by offdesign incidence angle between the flow and the vanes (it can be detected
77
78
from the velocity triangle at this point, which shows a
α
angle of 9.8º, quite
far from the 20º of the guide vane opening).
These blockage effects are increased in pump-turbines where the gap is
normally bigger and the vanes are thicker. [NICO02], [NICO06]. Wakes,
eddies and other unsteady phenomena are developped then, causing
vibration, noise and blade cracking.
Other unsteady effects influence the irregular distribution of the flow rate
among the channels. Further studies could be done on this matter.
4.3.2
PARTIAL SIMULATIONS
Three partial computational domain approximations were set and calculated
for Hydrodyna Pump. Their analyse can be interesting as they can be used to
prove the capacity (or not) of partial approaches performance, in order to
evaluate the quality and the power of the numerical simulation methods
used.
However, it is considered more necessary to evaluate first the full pumpturbine domain simulations and later try to ratify them for the partial
approaches.
This paper is focused on the first part, so that partial solutions will be only
put on show and only for ϕ = @0.028 .
Three possible approaches were
studied and introduced in table 5 in paragraph 3.2.
Keeping in mind that Hydrodyna has 9 runner blades and 20 guide vanes,
the first and more basic approach consisted in two guide vane chanels for one
blade impeller (see figure 39).
Its pitch ratio was admissible and the pre-processing and the processing
steps much quicklier.
79
Figure 39. View of the first restricted domain simulated (1/2)
Initial pressure fluctuactions were already studied for this simulation results
at several points.
An example, showing the results of Cp calculs (this parameter will be
explained in detail later; see the following paragraph about the complete
machine simulation) for four points (See figure 25) in the stator channel on
the right is shown in the following plot :
Cp
0.04
0.03
0.02
Cp [-]
0.01
0
1
21
41
61
81
101
12 1
1 41
1 61
18 1
2 01
2 21
2 41
2 61
28 1
3 01
3 21
3 41
36 1
3 81
401
42 1
44 1
46 1
4 81
501
52 1
54 1
5 61
5 81
601
62 1
6 41
6 61
68 1
7 01
7 21
-0.01
-0.02
-0.03
-0.04
-0.05
D
B
F
H
Figure 40. Cp curves for a streamline during a runner revolution from the results of
the partial approach 1/2
We can easily appreciate lack of any kind of periodicity and no correlation
between the curves. This approach should not be taken into account.
A second partial approach was made, this time from two blades and four
guide vanes (see figure 41). The pitch ratio is the same as before but the rest of
global factors influencing partial simulations will change from the first case.
Results obtained in this case are shown in the following picture 42. To be
compared with the first approach results, the pressure fluctuactions at the
same points (see figure 25) are chosen; now corresponding in the figure to the
stator upper streamline.
Figure 41. View of the second restricted domain simulated (2/4)
80
81
Cp
0.02
0.015
0.01
Cp [-]
0.005
0
1
21
41
61
81
101
121
141
161
181
201
221
241
261
281
301
321
341
361
381
401
421
441
461
481
501
521
541
561
581
601
621
641
661
681
701
721
-0.005
-0.01
-0.015
-0.02
-0.025
-0.03
D
B
F
H
Figure 42. Cp curves for stator streamline during a runner revolution from the
results of the partial approach 2/4
Curves are not completely periodic yet , as it had happened for the first
approach. On the other hand, some tendencies are already more clearly
visible. It is possible to distinguish kind of periods related to the BPF for
points B and D, both the nearest to the Rotor-Stator region, which are
normally the most influenced by pressure fluctuactions generated by RSI.
The last partial simulation was made. The computational domain in this case
was composed by three runner blades and seven guide vanes. Pitch ratio was
better than in the previous partial cases so that the simulation becomes more
reliable and its results expected to be more accurate and in better agreement
with reality.
The following pictures (43 and 44) show the corresponding
computer domain and a example of the analyse of the results.
82
Figure 43. View of the third restricted domain simulated, (3/7)
Cp
0.02
0.015
0.01
Cp [-]
0.005
0
1
21
41
61
81
101
121
141
161
181
201
221
241
261
281
301
321
341
361
381
401
421
441
461
481
501
521
541
561
581
601
621
641
661
681
701
-0.005
-0.01
-0.015
-0.02
-0.025
D
B
F
H
Figure 44. Cp curves for a stator streamline during a runner revolution from the
results of the partial approach 3/7
The same streamline (see figure 25) as before is used for the pressure plot . In
this case, some changes arise. The waves seem more periodical and a
modulated wave is present. Especially, points B and D, corresponding to the
721
ones nearest to the rotor-stator gap on the selected streamline, present a more
visible periodic behaviour and BPF can be inferred in them.
As a conclusion, we must say that the second and especially the third
approximation could allow to have a first impression of the phenomena
located in the rotor-stator gap and to distinguish some aspects of the
pressure fluctuactions and peaks associated to the RSI . Besides, some
parameters as efficiency and ψ were quite well approximated by this partial
approaches.
However, more investigations for future applications and improvements at
this matter are still needed.
83
5
85
5 EXPERIMENTAL DATA RESULTS
Tests for Hydrodyna Project were done on a reduced scale model of the
pump-turbine. This practice is very common for turbomachines, since large
sizes of real machinery make difficult their setting on laboratories or their
reproduction in 1:1 scales.
Hydrodynamic similarity theory proves that results obtained from models
are applicable to original devices. These theories include similarity properties
not only in a geometric domain but also in other operating variables such as
velocity, installation conditions or change of fluid. [TANA90].
For the experiments in this paper, the similarity is applied in geometrical
terms. The reduced model keeps geometrical rules and as the same time, the
laboratory tests were carried under controlled conditions, stablished to
reproduce a concrete operating point. Hence, results obtained for the model
will be accurately the same as the expected for the prototype.
5.1
MEASUREMENT TECHNIQUES
Pressure measurements were made in the EPFL laboratory (LMH) test rig.
LDV(LaserDopplerVelocimeter)
[WIKI07]
and
PIV
(Particle
Image
Velocimeter) [WIKI07] measurements were done for various operating
conditions of the Francis model. Not the same techniques could be used for
the stator and rotor measurementss. The signal transmission between them
was done by wireless photodiodes. The synchronization of the data sampling
is performed through a master-slave scheme in the rotating parts. Triggers
were led by a tachometer signal.
86
In the distributor channels, 48 piezoresistive miniature sensors were located
to catch the unsteady pressure in their walls. This kind of sensors are applied
for hydro and aerodynamic pressure measurements. In this case of hydraulic
pressure measurements, the active zone of the sensor must be waterproof.
The sensitive chips are made of silicon and mounted in a Wheatstone bridge.
In order to receive the deformation signal, the pressure indicator pieces are
located in the limits regions of the membrane where traction and
compression effects are expected. The Wheatstone bridge will experiment a
disequilibrium in the moment of the pressure application, and it will create a
variation of voltage which will be measured and registered.
The stator measures used for this paper were taken at a sample frequency of
51.2 kHz.
For the rotating parts, an instrumented shaft was developed by the EPFL to
acquire the unsteady pressure fluctuactions at the runner blades walls. It
consisted in a signal conditioning electronic composed of 32 preamplifiers
and filters and installed on board. Main technical characteristics of the
applied system are given in table 11.
To see all their positions in the model, have a look on chapter 7 pictures:
“Sensor locations”.
Amplification Factor Range
From 1 to 1000
Acquisition Boards Location
Pump-turbine Shaft
Maximum Sampling Frequency
20 KHz
Memory Storage capacity
64000 samples/channel
Digitized Data Transfer Rate
1.5 Mbits/s
Table11.. Characteristics of measurement techniques for the rotor
Simulations for the pump-mode were calculated after experiments, so that
we already knew which points to investigate as being the ones measured.
87
Although more points than ϕ = @0.043 and ϕ = @0.028 were tested, only these
two suggested a deeper analyse after the measures. Laboratory technicians
noticed during them changes in the machine behaviour that make them
interesting.
In the following, results for those two operating conditions will be shown in
a similar way as it was done in the last part of the chapter before.
5.2
EXPERIMENTAL RESULTS FOR
ϕ = @0.043
As it has already been explained, the interest was finally focused on the
points near the rotor-stator gap, so the experimental data results concerning
only those points will be shown.
To make easier comparisons, the same graphs and parameters as for the
simulation are analysed here.
It has already been mentioned that the sample frequency for this experience
had a value of 51200 Hz. Having in mind that pump speed rotation was of
900 rpm, to have a impeller revolution 3414 pressure measures are needed.
Starting by the Cp plot for the points corresponding to the simulated A and
B, the pressure fluctuactions for a runner revolution present the following
shape (figure 45):
88
Cp
Cp [-]
0.04
-0.01 1
191
381
571
761
951 1141 1331 1521 1711 1901 2091 2281 2471 2661 2851 3041 3231
Point A
-0.06
Point B
figure 45. Cp pressure fluctuactions for points A and B during a runner revolution
from experimental measures for ϕ = @0.043
Although the noise gives the signal a less regular form, one can see the
strong similarities between this plot and the same one corresponding for the
simulation results. (see figure 26). A filter is usually applied to this curves to
ease the analysis.
A priori, the agreement between numerical solution and experimental looks
to be admissible and quite satisfactory for this operating point. In these
results one can appreciate once again the RSI expected consequences:
- Individually, each point experiments a periodic fluctuaction, mostly
defined by the blade passage frequency. It has a value corresponding
to 40º, which using the relation between the sample frequency and the
rotational speed of the machine, corresponds approximately 380
samples. On purpose, the sample-step between two marks used in the
graph is 190, so that one can perceive easier the periodicity separated
by 2 intervals.
-As for the simulation results, phase shift of around 18º (due to equally
geometric distribution of the 20 vane guides) is also conserved
between
pressure measurement at point A and point B.
89
Approximately the phase shift correspond to a one interval between
two marks
Further analysis will be done for this operating point to compare with the
“perfect” behaviour results from CFD calculations to verify if in real
experiments it behaves as expected.
Cp [-]
Cp
0.04
0.035
0.03
0.025
0.02
0.015
0.01
0.005
0
-0.005
-0.01
-0.015
-0.02
-0.025
-0.03
-0.035
-0.04
-0.045
-0.05
1
191 381 571 761 951 1141 1331 1521 1711 1901 2091 2281 2471 2661 2851 3041 3231
D
B
F
experimental results for ϕ = @0.043
Cp
0.09
Cp [-]
0.06
0.03
0
1
191
381
571
761
951
1141
1331
1521
1711
1901
2091
2281
2471
2661
2851
3041
3231
-0.03
-0.06
A_(e)
A_(i)
Figure 47. Comparison of pressure fluctuactions measured for one runner revolution
between the internal and external points located in section A for ϕ = @0.043
(Remark: Point H is not included as it used to be for the CFD calculations,
because there was no pressure sensor at this point. Anyway, their
fluctuations are not really relevant. )
It would be convenient to filter the signals for a global comparaison between
them and the CFD ones. Anyway, the agreement with what had been
observed for ϕ = @0.043 from the CFD simulations is quite good, leaving the
most interesting part of tha analysis to the other operating point ϕ = @0.028 .
Figure 46 shows the propagation of the pressure wave is quite normal along
the streamline. As one goes further from the Rotor stator gap the pressure
fluctuactions dissapear.
Figure 47 shows no differences between pressure fluctuaction amplitude in
the external point A and the internal. So, according to measures, there is no
sign of recirculation or whirls during the revolution for this point.
5.2.1
FREQUENCY ANALYSIS
Basically the same procedure that was applied for numerical data will be
applied for the experimental measurements in order to analyse the frequency
domain of the signals. As it has already been shown in time domain,
laboratory measurements have much noise; hence in many cases a phase
average over the real pressure measurements has been done to obtain a
cleaner signal.
First of all, in figure 48 there is a comparison of the frequency spectra for all
points in the section next to the rotor stator interface. It includes both
external and internal points A and B, located in the two stator channels.
90
91
Figure 48. Pressure Spectra of pressure measurement for section next to the RS gap
for ϕ = @0.043
For ϕ = @0.043 no differences between any of these points are found. They all
verify the decreasing tendency as expected for the Cp amplitude with
frequencies from the first BPF. Their values are precisely similar between
external and internal and between one channel and the other.
For a global idea about the whole channel behaviour, figure 49 is shown.
Everything happens as expected: there is a global tendency to strongly
decrease in Cp amplitude for the higher frequencies, which means
attenuation.
Near the rotor-stator region this amplitude is relevant, dissapearing from the
3rd BPF, but the propagation is not so strong to be important at the stay
vanes.
Figure 49. Pressure Spectra for points from the stator channel for ϕ = @0.043
5.2.2
CONCLUSIONS FOR ϕ = @0.043
As the experimental analysis provides almost the same results as the CFD
ones, the conclusions for this operating point are the same.
Furthermore, this perfect agreement between simulation methods and
experimental measures puts into show that CFD is able to model the real
phenomena ocurring in the turbomachines, even for the rotor-stator
interaction effects, at least when no unsteady effects are present in the flow.
Now, CFD availability has to be still verified for all the operating ranges. The
partial load point will be studied from the analysis of the following
measurement results at ϕ = -0.028, which according to CFD results should
present signs of undesirable phenomena.
92
93
EXPERIMENTAL RESULTS FOR ϕ = @0.028
5.3
In figure 50 Cp fluctuactions in A and B are shown as usual. This time, the
plot seems to have more noise than for the case before, and it is harder to
determine the periodicity or the phase gap between the signals.
Some similar problems were found for the simulation results but in the
experimental data analysis the shape are more perturbed and disarranged.
Cp
Point A
Point B
Cp [-]
0.04
-0.01
1
191
381
571
761
951
1141 1331 1521 1711 1901 2091 2281 2471 2661 2851 3041 3231
-0.06
Figure 50. Cp fluctuactions for points A and B during a runner revolution from
experimental measures for ϕ = @0.028
Pressure fluctuactions for the whole stator streamline are also provided to be
compared with the CFD one. More chaotic behaviour than for the high
discharge point is found. More particularly for point B, next to the gap,
instantaneous and not repeated fluctuation are registered.
94
Cp [-]
Cp
0.045
0.04
0.035
0.03
0.025
0.02
0.015
0.01
0.005
0
-0.005
-0.01
-0.015
-0.02
-0.025
-0.03
-0.035
-0.04
-0.045
1
191
381
571
761
951
1141
1331
1521
D
1711
1901
B
2091
2281
2471 2661
2851
3041 3231
F
experimental results for ϕ = @0.028
Following the same procedure as for the simulations, pressure comparison
between external and internal points of the walls were carried out. The plot
obtained was the following (Figure 52)
Cp
0.04
0.03
Cp [-]
0.02
0.01
0
-0.01
1
191 381 571 761 951 1141 1331 1521 1711 1901 2091 2281 2471 2661 2851 3041 3231
-0.02
-0.03
-0.04
A_(e)
A_(i)
Figure 52. Comparison of experimental fluctuations for one impeller revolution
between the internal and external points located in section A for ϕ = @0 .028
95
This plot presents many fluctuactions for both points, internal and external.
In fact, it does not agree perfectly with the one obtained from the simulation
solution, where the period was longer but the differences in peaks amplitude
value between external and internal points were larger. To distinguished
properly these differences the frequency analysis is followed.
5.3.1
FREQUENCY ANALYSIS
Frequency analysis can help to find and to understand the reasons of what
happened during the experience at this operating point.
The first graph shows the frequency spectra for the measurements in points
near to the rotor-stator gap.
The most relevant aspect from this graph is the differences between external
and internal Cp amplitude as well as the bigger Cp amplitud for the 2nd
frequency than for the 1st . Differences in pressure amplitude between
external and internal points can be a sign of detachment and recirculation, as
it has already been explained for the CFD results.
In what respects to the importance of the second frequency, which is
observed at the four points,
it can be explained for the RSI and the
diammetrical modes that where theorically introduced in chapters before.
It had already been predicted by the CFD calculations, which suposses a
great achievement of this method. It is remarkable to know when operating
the machine at this partial load point that the 2 BPF has the highest
amplitude, in order to avoid this frequencies.
Now, differences in amplitude for external and internal points for both
channels are shown in figure 53, giving strength to the possibility of a
detachment in the flow.
96
Figure 53. Pressure Spectra of pressure measurement for section next to the RS gap
for ϕ = @0.028
To closer analysis, one channel is selected and the two points nearest to the
rotor-stator gap are studied. (Figure 54)
The highest amplitude for the second BPF dissapears at point D, and it has
no sign of differences between external or internal position either.
Just to check if it something different takes place for the other channel, a
similar graph is shown in figure 55 for channel 20.
Cp amplitudes of points A and B, internal and external, are the same as for
the corresponding points in channel number 1. Only a slight difference in Cp
amplitude value for external and internal A compared to B could be
mentioned as a sign of differences in the flow rate distribution among the
channels.
Cp amplitudes in frequency domain all along the stator channel 20 and
channel 1 are considered in the two following figures and their analysis draw
the following conclusion:
97
Figure 56. Pressure Spectra for points from the other stator channel 20 for ϕ = @0.028
Both plots show the same characteristic: At this operating point, the highest
Cp amplitude is the one of the 2nd BPF frequency for the points at the rotorstator interaction zone. On the other hand, it attenuates for the points
belonging to the stay vanes, where the 1st BPF has the higher amplitude.
98
5.3.2
CONCLUSIONS FOR ϕ = @0.028
Conclusions for experimental data are the same as for CFD results. Although
the agreement between them must be detailed in the next chapter in general
aspects they provide an almost identical final conclusion: For the point of ϕ=
-0.028, frequency and time domain analysis show unsteady phenomena in
the flow. Pressure differences found between channels and between near
points from the same channel or even between symetrical points at different
faces of the channel put into show the possible appearance of wakes, flow
detachment or swirls and eddies, which are increased by the the pumpturbines design, as it has already been explained for the conclusion in the
CFD results.
The presence of highest Cp amplitudes for the 2 BPF at the points near the RS
interaction region for the laboratory measurements ratifies the diammetrical
modes theory, which had already predicted this RSI effect over the flow in
the difusser.
99
6
Comparaison of results and
Conclusions
Comparison of Results. Conclusions
101
6 COMPARAISON OF RESULTS. CONCLUSIONS
Finally, it is the time for the evaluation of the methods and the
understanding of the results.
It has also to be said that once the simulation has been done and having all
the measurements, there are a huge amount of possible points and moments
to be studied in order to find differences between channels, between Cp
amplitudes, between particular moments...
Some of the most interesting graphics analysed during the preparation of this
paper,
are attached in chapter 7 for further detailed examples and
information.
Though the aim of this paper was basically the evaluation of the methods, it
has provided an initial analysis to this study case, in order to make easier
future further studies and measures. In fact, conclusions in chapters before
and all the theory and explanations to the dynamic flow phenomena
included along all the paper gives evidence of it.
6.1
DETAILED COMPARAISON OF RESULTS
As the individual characteristics of each point have already been described
and analysed in the two chapters before, now only the differences between
CFD and Experimental results will be highlighted.
It has already been said that the number of possible comparisons has become
too large, the selected points to be compared were the following:
-
102
Four points: the two ones in section A and the two ones in B, from
different channels but all at the limit region and in both internal and
external faces.
-
A whole streamline line in the stator channel, more precisely, the
external one in channel number 1. (points B, D, F and H in the external
face)
103
Firstly, the high discharge case ϕ = -0.043 compared results are shown:
figure 58. Comparison for ϕ=- 0.043 for points in the RS region
The agreement is quite accurate in both time and frequency domains for all
the points. The amplitudes are perfectly estimated, except for the 2nd
harmonic which according to measures is lower than CFD prediction.
104
figure 59. Comparison for channel 1 at ϕ = -0.043
The good agreement between experimental and CFD results is confirmed by
figure 59. It is also remarkable the fact that CFD predicts the the highest Cp
amplitude for the 2nd BPF at point F, which is ratified by the experimental
measures.
105
Now, the partial load case ϕ = -0.028 results are evaluated:
Figure 60. Comparison for ϕ=- 0.028 for points in the RS region
The most remarkable event here is the disagreement about the mean
pressure level between channels. While CFD predicts a similar level for both
channels, measurements result show a lower mean pressure level for channel
20 (points A) than for channel 1 (points B).
In what respects to frequencies, although there is agreement about the
highest amplitude for the 2nd BPF, in general CFD had predicted higher
values than what measures results show, especially for the 1st harmonic.
106
figure 61. Comparison for channel 1 at ϕ = -0.028
In time domain, the same points present higher pressure recovery than in
CFD simulation results. It is especially remarkable between points F and H.
In frequency domain, there is a large difference between CFD amplitudes
value prediction at the 1st BPF and measurements results. However, CFD
has quite well prediction for the propagation from B to D and for the fact that
at point D the 1st BPF dominates over the rest.
6.2
107
FINAL CONCLUSIONS
•
The capacity of numerical methods for representing operating points
where no unsteady behaviour takes place is validated from
experimental data. It is proved to be well developped according to the
strong agreement between numerical and experimental data for the
entire computational domain simulations at high discharge point.
•
Operating points presenting unsteady behaviour, as usually occurs for
partial loads in pumps, are quite well simulated by CFD methods.
However, some disagreements are found with experimental data in
what respects to amplitude and to attenuation of the flow pressure
fluctuations.
•
The presence of diametrical modes has been validated from these
analysis. The RSI is well simulated by the CFD methods, detecting the
majority of the cases where the 2 BPF had the highest Cp amplitude.
•
Partial load affects critically pump-turbines pressure fluctuactions.
Flow detachment is developped near the rotor-stator interaction zone,
creating a source of disturbance and instability at this region that
propagates perturbing the whole machine flow.
•
Investigations must improve and evaluate partial simulations as it has
been introduced in this paper. Admissible results are achieved for the
partial simulation 3/7, suggesting the possibility of full development
for these approaches.
7
Attached documents
Attached documents
7 ATTACHED DOCUMENTS
7.1
SENSORS LOCATIONS
109
Attached documents
110
Attached documents
7.2
111
HYDRODYNA’S NOMENCLATURE
Stator
Upper (i)
Lower (l)
EXP(#ch)
Sensor
CFX
XL Col EXP(#chl) Sensor
CFX
XL Col
Ch 20
47
78
UpDist22
AV
21
122
LowDist22
V
Section o_2
A
B
Ch 1
39
59
UpDist7
AN
24
107
LowDist7
Y
Ch 20
44
53
UpDist1
AS
20
101
LowDist1
U
Section o_3
C
D
Ch 1
36
54
UpDist2
AK
23
102
LowDist2
X
E
F
Ch 20
43
29
UpSt1
AR
19
41
LowSt1
T
Section v_4
Ch 1
35
30
UpSt2
AJ
22
42
LowSt2
W
Ch 20
42
35
UpSt7
AQ
18
47
LowSt7
S
Section v_5
G
H
Ch 1
34
36
UpSt8
AI
-
48
LowSt8
-
Rotor
Blade #1
EXP(# ch)
12
13
14
15
16
17
Sensor
P12
P13
P14
P15
P16
P17
CFX
Bba1
Bba2
Bba3
Bba4
Bba5
Bba6
XL Col
M
N
O
P
Q
R
Blade #2
EXP(# ch)
1
2
3
4
5
6
7
8
9
10
11
Sensor
P1
P2
P3
P4
P5
P6
P7
P8
P9
P10
P11
CFX
Bbb1
Bbb2
Bbb3
Bbb4
Bbb5
Bbb6
Bbb7
Bbb8
Bbb9
Bbb10
Bbb11
XL Col
B
C
D
E
F
G
H
I
J
K
L
Blade #3
EXP(# ch)
19
20
18
Sensor
P19
P20
P18
CFX
Bbc10
Bbc11
Bbc7
XL Col
T
U
S
Attached documents
7.3
112
HYDRODYNA’S HILL CHART
η [-]
HYDRODYNA
0.9
0.89
0.88
0.87
0.86
0.85
0.84
0.83
0.82
0.81
0.8
0.79
0.78
0.77
0.76
0.75
0.74
0.73
0.72
0.71
0.7
-0.05
12 deg
14 deg
16 deg
18 deg
20 deg
22 deg
24 deg
26 deg
28 deg
-0.045
-0.04
-0.035
-0.03
-0.025
-0.02
-0.015
ϕ [-]
HYDRODYNA
1.2
1.15
1.1
1.05
12 deg
14 deg
ψ [-]
1
16 deg
0.95
18 deg
0.9
20 deg
22 deg
0.85
24 deg
26 deg
0.8
28 deg
0.75
0.7
-0.05
-0.045
-0.04
-0.035
ϕ [-]
-0.03
-0.025
-0.02
-0.015
Attached documents
7.4
EXAMPLE OF A LIST OF SET-UP EXPRESSIONS
LIBRARY:
CEL:
EXPRESSIONS:
R1 = 0.261875 [m]
Aref = pi*R1^2
omega = 652.9 [rev min^-1]
omegaDL = omega/1.0 [rad]
DTime = 1.0/abs(omegaDL)
DTimeNr = 360
DTimeAngle = 360 [deg]/ DTimeNr
DTimeFr = 0.5[rad]/abs(omega)
DTimeTrn = DTimeAngle/abs(omega)
nb = 9
rho = ave(Density)@Asc I
zb = 9
P1 = -1.0 * massFlowInt(ptotstn)@Int b o Side 2 / rho * zb / nb
Q1 = -1.0 * massFlow()@Int b o Side 2 / rho * zb / nb
E1 = P1/(Q1*rho)
nv = 20
zv = 20
P2 = massFlowInt(ptot)@Int b o Side 1 / rho * zv / nv
Q2 = massFlow()@Int b o Side 1 / rho * zv / nv
E2 = P2/(Q2*rho)
P3m = massFlowInt(ptotstn)@Ad 3m/ rho * zb / nb
Q3m = massFlow()@Ad 3m/rho * zb / nb
E3m = P3m/ (Q3m*rho)
P5 = -massFlowInt(ptot)@ Asc I/ rho * zv / nv
Q5 = -massFlow()@Asc I/rho * zv / nv
E5 = P5/ (Q5*rho)
ERef = 0.5*omegaDL^2* R1^2
TB = (torque_z()@Bb + torque_z()@Sb i +torque_z()@Sb e )*zb/nb
Pm = -1.0 *TB* omegaDL
113
Attached documents
Eta1 = (P2 - P3m)/ Pm
Eta5 = (P5 - P3m)/ Pm
Phi = 0.02853948
QPhi = pi*omegaDL*R1^3*Phi
Phi b 1 = - massFlow()@REGION:RS ROTOR / rho * zb / QPhi
Phi b 2 = - massFlow()@REGION:RS ROTOR 2 / rho * zb / QPhi
Phi v 1 = - massFlow()@REGION:AV56 / rho * zv / QPhi
Phi v 10 = - massFlow()@REGION:AV56 10 / rho * zv / QPhi
Psi1 = (E1-E3m) / ERef
Psi2 = (E2- E3m) / ERef
Psi5 = (E5- E3m) / ERef
Qb = QPhi/ (2.0 * zb)
R1carre = 0,0685785 [m^2]
URef = abs(omegaDL)*R1
END
MATERIAL: Water
Material Description = Water (liquid)
Material Group = Water Data, Constant Property Liquids
Option = Pure Substance
114
Attached documents
Thermodynamic State = Liquid
PROPERTIES:
Option = General Material
Thermal Expansivity = 2.57E-04 [K^-1]
ABSORPTION COEFFICIENT:
Absorption Coefficient = 1.0 [m^-1]
Option = Value
END
DYNAMIC VISCOSITY:
Dynamic Viscosity = 8.899E-4 [kg m^-1 s^-1]
Option = Value
END
EQUATION OF STATE:
Density = 997.0 [kg m^-3]
Molar Mass = 18.02 [kg kmol^-1]
Option = Value
END
REFERENCE STATE:
Option = Specified Point
Reference Pressure = 1 [atm]
Reference Specific Enthalpy = 0.0 [J/kg]
Reference Specific Entropy = 0.0 [J/kg/K]
Reference Temperature = 25 [C]
END
REFRACTIVE INDEX:
Option = Value
Refractive Index = 1.0 [m m^-1]
END
SCATTERING COEFFICIENT:
Option = Value
Scattering Coefficient = 0.0 [m^-1]
END
SPECIFIC HEAT CAPACITY:
Option = Value
115
Attached documents
Specific Heat Capacity = 4181.7 [J kg^-1 K^-1]
Specific Heat Type = Constant Pressure
END
THERMAL CONDUCTIVITY:
Option = Value
Thermal Conductivity = 0.6069 [W m^-1 K^-1]
END
END
END
END
[.................................................]
SIMULATION TYPE:
Option = Transient
EXTERNAL SOLVER COUPLING:
Option = None
END
INITIAL TIME:
Option = Automatic with Value
Time = 0 [s]
END
TIME DURATION:
Number of Timesteps per Run = 180
Option = Number of Timesteps per Run
END
TIME STEPS:
Option = Timesteps
Timesteps = DTimeTrn
END
END
INITIALISATION:
Frame Type = Stationary
Option = Automatic
116
Attached documents
INITIAL CONDITIONS:
Velocity Type = Cylindrical
CYLINDRICAL VELOCITY COMPONENTS:
Velocity Axial Component = 0 [m s^-1]
Velocity Theta Component = -5 [m s^-1]
Velocity r Component = -3 [m s^-1]
AXIS DEFINITION:
Option = Coordinate Axis
Rotation Axis = Coord 0.3
END
END
EPSILON:
Eddy Length Scale = 0.001 [m]
END
K:
Fractional Intensity = 0.05
END
STATIC PRESSURE:
Relative Pressure = 0 [Pa]
END
[.....................................................]
OUTPUT FREQUENCY:
Option = Timestep Interval
Timestep Interval = 40
END
END
MONITOR POINT: Bba2
Cartesian Coordinates = 219.1267024 [mm], -111.3269521 [mm], \
0.006628096 [mm]
117
Attached documents
Option = Cartesian Coordinates
Output Variables List = Pressure,Total Pressure,Velocity,Velocity in \
Stn Frame,Total Pressure in Rel Frame,Velocity u,Velocity \
v,Velocity w,Turbulence Kinetic Energy
END
[..............................................]
CONVERGENCE CRITERIA:
Residual Target = 0.0001
Residual Type = MAX
END
TRANSIENT SCHEME:
Option = Second Order Backward Euler
TIMESTEP INITIALISATION:
Option = Automatic
END
END
END
118
Attached documents
7.5
CODE
119
FOR CREATING VELOCITY TRIANGLES SESSION
turbo more_vars
! $RSInterfaceRFR = "Int b o Side 1";
! $CScaleMax = 100.0;
! $ArrowScale = 3;
! $LOff = 0.017 * $ArrowScale;
! $RPnt = 0.265;
! $ThetaPnt = 10 * 3.1415927 / 180.0;
! $XPnt = $RPnt * cos($ThetaPnt);
! $YPnt = $RPnt * sin($ThetaPnt);
! $XOff = -1.0 * $LOff * sin($ThetaPnt);
! $YOff = $LOff * cos($ThetaPnt);
! $XBeta = $XPnt + $XOff;
! $YBeta = $YPnt + $YOff;
LIBRARY:
CEL:
EXPRESSIONS:
CPlot = sqrt(($CScaleMax [m s^-1])^2 - Cu 1^2 - Cr 1^2) * step(0.04(X^2+Y^2)/\
1[m^2] )
Cr 1 = Q1 / area()@$RSInterfaceRFR
Cu 1 = massFlowAve(Velocity in Stn Frame Circumferential)@$RSInterfaceRFR
U 1 = ave(Rotation Velocity)@$RSInterfaceRFR
UPlot = sqrt(($CScaleMax [m s^-1])^2 - U 1^2 ) * step(0.04-(X^2+Y^2)/1[m^2] )
URef = abs(omegaDL)*R1
Attached documents
120
WPlot = sqrt(($CScaleMax [m s^-1])^2 - (Cu 1 - U 1)^2 - Cr 1^2) * step(0.04(X^2+\
Y^2)/1[m^2] )
BetaAve1 = atan2(Cr 1, U 1 - Cu 1) * 1 [rad]
AlphaAve1 = atan2(Cr 1, Cu 1) * 1 [rad]
END
END
END
USER VECTOR VARIABLE:C Ave 1
Boundary Values = Conservative
Calculate Global Range = On
Recipe = Expression
Variable to Copy = Pressure
X Expression = Cu 1 * Theta Direction X + Cr 1 * Radial Direction X
Y Expression = Cu 1 * Theta Direction Y + Cr 1 * Radial Direction Y
Z Expression = CPlot
END
USER VECTOR VARIABLE:U Ave 1
Recipe = Expression
X Expression = U 1* Theta Direction X
Y Expression = U 1 * Theta Direction Y
Z Expression = UPlot
END
USER VECTOR VARIABLE:W Ave 1
Recipe = Expression
X Expression = (Cu 1 - U 1)* Theta Direction X + Cr 1 * Radial Direction X
Attached documents
Y Expression = (Cu 1 - U 1) * Theta Direction Y + Cr 1 * Radial Direction Y
Z Expression = WPlot
END
POINT:Point 1
Apply Instancing Transform = On
Colour = 1, 1, 0
Colour Map = Rainbow
Colour Mode = Constant
Colour Scale = Linear
Colour Variable = Pressure
Colour Variable Boundary Values = Hybrid
Culling Mode = No Culling
Domain List = All Domains
Draw Faces = On
Draw Lines = Off
Instancing Transform = Default Transform
Lighting = On
Line Colour = 0, 0, 0
Line Width = 2
Max = 0 [Pa]
Min = 0 [Pa]
Node Number = 1
Normalized = Off
Option = XYZ
Point = $XPnt [m], $YPnt [m], 0 [m]
Point Symbol = Crosshair
Range = Global
Specular Lighting = On
Surface Drawing = Smooth Shading
Symbol Size = 1.0
Transparency = 0.0
Variable = Pressure
Variable Boundary Values = Hybrid
121
Attached documents
Visibility = Off
OBJECT VIEW TRANSFORM:
Apply Reflection = Off
Apply Rotation = Off
Apply Scale = Off
Apply Translation = Off
Principal Axis = Z
Reflection Plane Option = XY Plane
Rotation Angle = 0 [degree]
Rotation Axis From = 0 [m], 0 [m], 0 [m]
Rotation Axis To = 0 [m], 0 [m], 0 [m]
Rotation Axis Type = Principal Axis
Scale Vector = 1 , 1 , 1
Translation Vector = 0 [m], 0 [m], 0 [m]
X = 0 [m]
Y = 0 [m]
Z = 0 [m]
END
END
VECTOR:Vector c
Add Sample Vertex Normals = On
Colour = 0.75, 0.75, 0.75
Colour Mode = Use Plot Variable
Colour Variable = C Ave 1
Coord Frame = Global
Direction = X
Domain List = Vbd
Draw Faces = On
Draw Lines = Off
122
Attached documents
Lighting = On
Line Width = 3
Location List = Point 1
Locator Sampling Method = Vertex
Max = 30 [m s^-1]
Maximum Number of Items = 100
Min = 0 [m s^-1]
Normalized = Off
Number of Samples = 100
Projection Type = None
Random Seed = 1
Range = User Specified
Reduction Factor = 1.0
Reduction or Max Number = Reduction
Sample Spacing = 0.1
Sampling Accuracy = High
Sampling Aspect Ratio = 1
Sampling Grid Angle = 0 [degree]
Surface Sampling = Off
Symbol = Line Arrow
Symbol Size = $ArrowScale
Transparency = 0.0
Variable = C Ave 1
Visibility = On
Apply Scale = Off
Principal Axis = Z
123
Attached documents
X = 0 [m]
Y = 0 [m]
Z = 0 [m]
END
END
VECTOR:Vector u
Colour = 0.75, 0.75, 0.75
Colour Mode = Variable
Colour Variable = Rotation Velocity
Direction = X
Domain List = All Domains
Draw Faces = On
Draw Lines = On
Lighting = On
Line Width = 3
Max = 30 [m s^-1]
124
Attached documents
Min = 0 [m s^-1]
Normalized = Off
Random Seed = 1
Range = Global
Symbol = Line Arrow
Transparency = 0.0
Variable = U Ave 1
Visibility = On
Apply Scale = Off
Principal Axis = Z
125
Attached documents
X = 0 [m]
Y = 0 [m]
Z = 0 [m]
END
END
VECTOR:Vector w
Colour = 0.75, 0.75, 0.75
Colour Mode = Use Plot Variable
Colour Variable = W Ave 1
Direction = X
Domain List = Vbd
Draw Faces = On
Draw Lines = Off
Lighting = On
Line Width = 3
Max = 30 [m s^-1]
Min = 0 [m s^-1]
Normalized = Off
Random Seed = 1
Range = Global
126
Attached documents
Symbol = Line Arrow
Transparency = 0.0
Variable = W Ave 1
Visibility = On
Apply Scale = Off
Apply Translation = On
Principal Axis = Z
Translation Vector = $XOff [m], $YOff [m], 0 [m]
X = 0 [m]
Y = 0 [m]
Z = 0 [m]
END
END
TEXT: Text Alpha
Colour = 0, 0, 0
127
Attached documents
Font = Sans Serif
Position Mode = Three Coords
Text Colour Mode = Default
Text Height = 0.018
Text Position = $XPnt [m], $YPnt [m], 0 [m]
Text Rotation = 0 [degree]
Visibility = On
X Justification = Center
Y Justification = None
TEXT: Text Beta
Colour = 0, 0, 0
Font = Sans Serif
Position Mode = Three Coords
Text Colour Mode = Default
Text Height = 0.018
Text Position = $XBeta [m], $YBeta [m], 0 [m]
Text Rotation = 0 [degree]
Visibility = On
X Justification = Center
Y Justification = None
128
Attached documents
7.6
MATLAB FILE:
129
FREQUENCY ANALYSIS AND MULTIPLE
COMPARISONS
%% Initialisation and data aquisitation
clear ;
% Definition of operating points and corresponding data sources (all
XL at
% the moment
OPS = ['028'; '043'];
Phis = [0.028, 0.043];
Types = {'CFD','Raw Data','Phase Average'};
PSources = struct('File',
{'MonP_vvo_3640_Phi028.xls','MonP_vvo_1440_Phi043.xls'}, ...
'Desc', 'CFD',...
'Type', 'XLColumn',...
'Start', {2+18, 2+18},... % -6 Times steps phase
shift (40 - 6 = 34)
'Length', {3600, 1080},...
'NsPerRev', {360, 360},...
'XL2Pa', 1.000,...
'WLLow', 1,...
'CircShift', 0);
PSources = [PSources; struct('File',
{'538_08_22S_0_phi028.xls','538_08_16S_0_phi043.xls'}, ...
'Desc', 'Exp',...
'Start', {10, 10},...
'Length', {3*3414, 3*3414},...
'NsPerRev', {3414, 3414},...
'XL2Pa', 101300.0,...
'WLLow', 10,...
'CircShift', 0)];
PSources = [PSources; struct('File',
{'538_08_22_PhaseAve_Stator.xls','538_08_16_PhaseAve_Stator.xls'},
...
'Desc', 'Exp',...
'Start', {12, 12},...
'Length', {720, 720},...
'NsPerRev', {720, 720},...
'XL2Pa', 101300.0,...
'WLLow', 1,...
'CircShift', 200)];
% Definition of Sensor Locations in structures with all necessary
info
Secs = ['o2'; 'o3'; 'v4'; 'v5'];
SecCell = {'o2'; 'o3'; 'v4'; 'v5'};
MnC = {'Dist', 'Dist', 'St', 'St'};
Attached documents
130
Sfcs = ['i'; 'e'; 'b'];
MnS = {'Up';'Low'};
Chls = [20 1];
SLog = strcmp(cellstr(Secs),'o2');
% Declaration and dummy definition of Structure for a
Monitor/pressure tap
% point conctaining info about:
% names in different namig schemes, channel numbers and so on
% Structured info about its location etc.
% the first index represents the section in the machine
% the second: if it is exterior (Lower) or Interior (Upper)
% the third: which diffuser channel we're in (20 or 1 at the time
being)
MPts = repmat(struct('Sec', 'o2', 'Sfc', 'i', 'Chl', 20, ...
'Cnl', 0, 'XlE', 'A', 'Pnt', 0, ...
'MnP', 'LowDist' , 'XlC', 'A', 'Mnn', 1 ),...
[4 2 2]);
% Selon la structure des indexes choisis les données sont placées
%section [ ... o_2 ... ¦
...o_3...
¦ ... v_4 ... ¦ ... v_5
...]
%HubShr [
i
¦ e
¦
i
¦ e
¦ i
¦
e
¦
i
¦
e ]
%channel [20 01 20 01
20 01 20 01 20 01
20 01 20 01
20 01]
Cans
= [47 39 21 24
44 36 20 23 43 35
19 22 42 34
18 34]
Pnts
= [78 59 122 107 53 54 101 102 29 30
41 42 35 36
47 48]
MnCX = [22 7
22 7
1
2
1
2 1
2
1
2
7
8
7
8]
ColCFD = ['K' 'M' 'C' 'E' 'J' 'L' 'B' 'D' 'N' 'O' 'F' 'G' 'P' 'Q'
'H' 'I']
RepName = ['A' 'B' 'A' 'B' 'C' 'D' 'C' 'D' 'E' 'F' 'E' 'F' 'G' 'H'
'G' 'H']
for i = 1:4
for j = 1:2
for k = 1:2
MPts(i,j,k).Sec = Secs(i,:);
MPts(i,j,k).Sfc = Sfcs(j);
MPts(i,j,k).Chl = Chls(k);
MPts(i,j,k).Cnl = Cans(4*(i-1)+2*(j-1)+k);
MPts(i,j,k).XlE = Can2Col(Cans(4*(i-1)+2*(j-1)+k));
MPts(i,j,k).Pnt = Pnts(4*(i-1)+2*(j-1)+k);
MPts(i,j,k).XlC = ColCFD(4*(i-1)+2*(j-1)+k);
MPts(i,j,k).RpN = RepName(4*(i-1)+2*(j-1)+k);
if MnCX(4*(i-1)+2*(j-1)+k)==0
MPts(i,j,k).MnP = 'None';
else
MPts(i,j,k).MnP =
strcat(MnS(j),MnC(i),sprintf('%d',MnCX(4*(i-1)+2*(j-1)+k)));
end
Attached documents
131
end
end
end
for i = 1:16
MPts(i)
end
%%
rho= 997;
u= 25.062;
%%
% Choices of Pressure Data to display in Raw Data Plot
OPs = [1 1 1 1]; % 1:028, 2:043
Tps = [1 2 1 2]; % '1:CFD, 2:Raw Exp, 3:Phase Average'
nplots = numel(OPs);
SubMean = 1:nplots; %Which mean to substract, default 1 2 3 4... Use
0 for none, do not use higher
SubMean = [1 2 1 2]; %Example substract own mean from two first, and
same two from 3rd and 4th data set CFD-EXP ...CFD-EXP
%SubMean = [1 1 1 1]
%PAve = [360 360 360 360]; %Do additional phase ave on data, 0 do
nothing, for averageing of CFD Data
% PAveM = [1 1 1 1]; % Reappend Phase Averaged data n times for
graphics representation
PAve = [0 0 0 0]; %Do additional phase ave on data, 0 do nothing,
for averageing of CFD Data
PAveM = [1 1 1 1]; % Reappend Phase Averaged data n times for
graphics representation
iPs=[];
iPs (1,1)=
iPs (1,2)=
iPs (1,3)=
iPs (1,4)=
PInd('o2','e',
PInd('o2','e',
PInd('o2','i',
PInd('o2','i',
20,
20,
20,
20,
Secs,
Secs,
Secs,
Secs,
Sfcs,
Sfcs,
Sfcs,
Sfcs,
Chls,
Chls,
Chls,
Chls,
size(MPts));
size(MPts));
size(MPts));
size(MPts));
figtype = '-dpng';
Title = sprintf('Comparison of Results CFD vs EXP in Rotor-Stator
gap for Phi=%5.3f',...
Phis(OPs(1)));
PlotName = 'Cp_o2_i_e_Ch20_EXP_PhAve_028';
RawPlotName = [PlotName '_Time'];
%%
for i = 1:nplots
MPts(iPs(i))
LStart = PSources(Tps(i), OPs(i)).Start;
Length(i,1) = PSources(Tps(i), OPs(i)).Length;
Attached documents
NsPerRev(i,1) = PSources(Tps(i), OPs(i)).NsPerRev;
switch Tps(i)
case 1
XLCol = MPts(iPs(i)).XlC;
case 2
XLCol = MPts(iPs(i)).XlE;
case 3
XLCol = MPts(iPs(i)).XlE;
end
fscale(i,1) = {linspace(0,NsPerRev(i,1)/2,Length(i,1)/2+1)};
XLR = sprintf('%s%d:%s%d',XLCol, LStart, XLCol, LStart +
Length(i,1)-1);
Tmp = PSources(Tps(i), OPs(i)).XL2Pa * xlsread(PSources(Tps(i),
OPs(i)).File, XLR );
if PSources(Tps(i), OPs(i)).WLLow > 1
wSize = PSources(Tps(i), OPs(i)).WLLow;
Tmp=filtfilt(ones(1,wSize)/wSize,1,Tmp)
end
if PSources(Tps(i), OPs(i)).CircShift ~= 0
Tmp=circshift(Tmp,PSources(Tps(i), OPs(i)).CircShift);
end
if PAve(i) ~= 0
q = PAve(i);
nph = floor(numel(Tmp)/q);
for j = 1:(nph-1)
Tmp(1:q) = Tmp(1:q) + Tmp(j*q+1:j*q+q);
end
Tmp(1:q)=Tmp(1:q)/nph;
Tmp(q+1:end,:)=[];
Length(i,1) = q;
for j = 1:PAveM(i)-1
Tmp(j*q+1:j*q+q) = Tmp(1:q)
end
Length(i,1) = PAveM(i)*q;
end
if i==1
PMean1 = mean(Tmp);
end
PMean(i) = mean(Tmp);
Times(i,1) = {(0:Length(i,1)-1)/NsPerRev(i,1)};
if SubMean(i) ~= 0
cP(i,1) = {(Tmp-PMean(SubMean(i)))/(0.5*rho*u^2)};
else
cP(i,1) = {Tmp/(0.5*rho*u^2)};
end
end
%% Raw Data Plot
figraw = figure('Name','Cp fluctuation
(time)','NumberTitle','off',...
'Position',[50 50 800 600], ...
'PaperPosition',[0 0 8 6], ...
'PaperUnits','inch', ...
'PaperPositionMode', 'manual');
tmp = Times{1,1};
for i = 1:nplots
plotdata(2*i-1)= {Times{i,1};};
plotdata(2*i)= {cP{i,1}};
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end
h_cp = plot(plotdata{1,:}, 'LineWidth', 1.0);
for i = 1:nplots
%
LegString = sprintf('P_{%s-%s-%d} - %s', MPts(iPs(i)).Sec,
MPts(iPs(i)).Sfc, MPts(iPs(i)).Chl, char(Types(Tps(i))));
LegString = sprintf('{%s-%s} - %s', MPts(iPs(i)).RpN,
MPts(iPs(i)).Sfc, char(Types(Tps(i))));
set(h_cp(i),'DisplayName',LegString)
end
legend(gca,'show');
xlabel('t f_n','fontsize',12,'fontweight','b');
ylabel('c_p','fontsize',12,'fontweight','b');
axis([0 1 -inf inf]);
%axis([0 1 -inf 0.5]);
title(Title);
print (figtype, RawPlotName);
saveas(figraw, RawPlotName, 'fig');
%% Choices of Pressure Data amongst the former to display for
frequency analysis
nfft = [1 2 3 4]; % Referrring to the Data formerly loaded, choice
of as opposed to the 1..nplots loop
FFTPlotName = [PlotName '_FFT'];
% Fix Sample Rate per revolution to downsample higher sampled
signals
p = 360
% Fix number of periods to phase average longer time series
nrev = 1 % only use 1 at the moment
q=nrev*p
fscale = linspace(0,p/2,q/2+1);
fmax = 90 + 1;
for i = nfft
if NsPerRev(i,1) ~= p
FftTmp = resample(cP{i,1},p,NsPerRev(i,1));
else
FftTmp = cP{i,1};
end
rslength = numel(FftTmp);
if rslength > q
nph = floor(rslength/q);
for j = 1:(nph-1)
FftTmp(1:q) = FftTmp(1:q) + FftTmp(j*q+1:j*q+q);
end
FftTmp(1:q)=FftTmp(1:q)/nph;
FftTmp(q+1:end,:)=[];
end
fft_tmp = fft(FftTmp) / q;
fft_tmp(1) = 0.0;
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Attached documents
cP_Red(:,i) = FftTmp;
cP_Cmplx(:,i) = fft_tmp(1:fmax);
cP_Amps(:,i) = abs(fft_tmp(1:fmax));
cP_Phase(:,i) = angle(fft_tmp(1:fmax));
end
%% FFT Amplitudes Stem Plots
ftick = 9;
xticks = 0:ftick:floor(fscale(fmax));
MCList = {'blue'; 'white'; 'red'; 'green'};
MList = {'o';'s';'d';'p'};
MEdgeList = {'b';'k';'r';'g'};
figfft = figure('Name','Pressure Spectra','NumberTitle','off',...
'Position',[100 100 800 600], ...
'PaperPosition',[0 0 8 6], ...
'PaperUnits','inch', ...
'PaperPositionMode', 'manual');
h_ch01 = stem(fscale(1:fmax)',cP_Amps, 'MarkerSize', 5);
for i = nfft
%
LegString = sprintf('P_{%s-%s-%d} - %s', MPts(iPs(i)).Sec,
MPts(iPs(i)).Sfc, MPts(iPs(i)).Chl, char(Types(Tps(i))));
LegString = sprintf('{%s-%s} - %s', MPts(iPs(i)).RpN,
MPts(iPs(i)).Sfc, char(Types(Tps(i))));
set(h_ch01(:,i),'MarkerFaceColor',char(MCList(i)),...
'MarkerEdgeColor',char(MEdgeList(i)),...
'Marker',char(MList(i)),...
'DisplayName',LegString )
end
axis([0 fscale(fmax) 0 0.01]);
xlabel('f_n','fontsize',12,'fontweight','b');
ylabel('c_p Amplitude','fontsize',12,'fontweight','b');
set(gca, 'XTick',xticks, 'FontSize',11);
legend(gca,'show');
title(Title);
print (figtype, FFTPlotName);
saveas(figraw, FFTPlotName, 'fig');
%% FFT Transfer Function for BPF's
ntp = size(nfft,2)-1;
ftick = 9;
xticks = 0:ftick:floor(fscale(fmax));
nt = size(xticks, 2)-1;
cP_Transfer = zeros(nt,ntp);
for i = nfft(2:end)
cP_Transfer(:,i-1) =
cP_Cmplx(xticks(2:end)+1,i)./cP_Cmplx(xticks(2:end)+1,1);
end
cP_TAmp = abs(cP_Transfer);
cP_TPhase = angle(cP_Transfer);
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Attached documents
7.7
COMPARISON CFD AND EXPERIMENTAL RESULTS PLOTS
135
Attached documents
136
Attached documents
137
Attached documents
138
8
References
References
140
8 REFERENCES
[EURE99]
www. eureka.be/inaction/ACcshowProjectOultine.do
[ERKM04]
Erkman, S. , « Vers une ecologie Industrielle « , Mayer Charles
Leopold Eds, 2004.
[ASEU04]
Office federale de l’energie , « Statistique Suisse de l’electricité
2003 »Bulletin ASE/UCS n.8 , Berne, 2004.
[WHIT03]
White, F. M., “Mecánica de Fluidos “, McGraw Hill
Professional, 2003.
[GARC98]
Garcia, A., A. López de la Rica y A. De la Villa. “Cálculo I:
Teoría y problemas de Análisis Matemático en una variable”
CLAGSA, Madrid, 1998 and “Calculo II: Teoría y problemas
de funciones en varias variables” CLAGSA, Madrid, 2002.
[ZOBE07]
Zobeiri, A., J.L. Kueny, M. Farhat, F. Avellan, «Unsteady
Pressure due to Rotor-Stator Interactions in Generating Mode
of a Pump-Turbine: Numerical and Experimental
Investigations”, 2007.
[NICO06]
Nicolet, C., N. Ruchonnet and F. Avellan. “Hydroacoustic
Modeling of Rotor Stator Interaction in Francis PumpTurbine”, 2006.
[TANA90]
Tanaka, H. “Vibration behaviour and dynamic stress of
runners of very high head reversible pump-turbines.”
Proceedings, 15th IAHR Symposium, Belgrade,
Yugoslavia,1990.
References
[OHUR90]
141
Ohura, Y., M. Fujii, O. Sugimoto, H. Tanaka and I. Yamagata,
“Vibration of the powerhouse structure of pumped storage
power plant.”, Proceedings, 15th IAHR Symposium, Belgrade,
section U2., 1990.
[CHEN61]
Chen, Y. N., “Water-Pressure Oscillations in the Volute
Casings of Storage Pumps.”, Sulzer Technical Review,
Research Number,pp. 21-34, 1961
[OHAS94]
Ohashi, H. , “Case Study of Pump failure due to Rotor-Stator
Interaction”, International Journal ofRotating Machinery,
1994, Vol. I, No. 1, pp. 53-60.
[BREN94]
Brennen, C.E., “Hydrodynamics of Pumps.”, NREC, 1994.
[FERZ99]
Ferziger, P., M. Peric, “CFD Computational Methods for
Fluids Dynamics”. Springer. 1999.
[STOR02]
Storey, B.D., “ Computing Fourier Series and Power Spectrum
with Matlab” , 2002
[KUEN99]
Kueny, J.L. and A. Guedes, “Identification of Rotor-Stator
Unsteady Parameters”, September 1999.
[JURI05]
Juric, M., “Rotor-Stator Interaction”, 2005.
[WIKI07]
www.Wikipedia.com, 2007
[NASA06]
www .grc.nasa.gov/WWW/K-12/airplane/nseqs.html, 2006.
[LMH07]
www. lmh. ch, 2007.
[TUTO05]
Tutorials CFX 11.0, 2005.
References
[ENER07]
142
Picture taken from :
http://www. energies-renouvelables.org
[IMAG07]
Picture taken from:
Images .google
[WIKI07]
Picture taken from:
wikimedia.org
[IMAG07]
Pictures taken from:
Images.google
[TOPL07]
Picture taken from:
www.top500.org/lists/2007/06/performance_development
[NICO02]
Nicolet, C., F. Avellan. P. Allenbachd, A. Sapin and J.-J.
Simond, “Proceeding of the XXI st IAHR Symposium and
Systems” Vol. II. 2002, p.p. 799, 800,814-818, 823-828, 834, 848856, 881.
9
ACKNOWLEDGEMENTS
Acknowledgements
144
9 ACKNOWLEDGEMENTS
Finally, I will like to thank all the people who have helped me to do this
project.
Especially I thank my coordinator Olivier Braun, who was working with me
sometimes till very late and who have dedicated many hours to solve my
questions and my problems.
Thank you also to the rest of the Numerical Methods Team in LMH, Alireza
Zobieri and Cecile Muench, whom I have worked with during the first
months and who have provided me many information and all the
explanations and help when I needed it. I especially thank Alireza Zobieri,
who had already studied the turbine mode, whose results I used in many
times as a reference.
I will like also to thank previous investigators (Christophe Nicolet, Mohamed
Farhat, Jean-Louis Kueny..) because I have learnt much from their results and
their publications.
I would like to thank M. Avellan and my university coordinators José Ignacio
Linares too, for offerring and allowing me this opportunity, respectively.
I do not forget all the LMH staff , whom I was talking to and seeing everyday
during the project and who get to make me an easier and more pleasant
work.
Last but not least, I thank all my family and friends, because without them,
without their support and encouragement I would have not succeed to finish
this project.

evaluation of numerical methods for turbomachines based on

Transcription

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