NASCART-GT - Georgia Tech

Transcription

Georgia Institute of Technology
School of Aerospace Engineering
Computational Capabilities
Stephen M. Ruffin, http://www.ae.gatech.edu/~sruffin
NASCART-GT: A Viscous SolutionNASCARTSolutionAdaptive Cartesian Grid Flow Solver
Stephen M. Ruffin
[email protected]
Associate Professor, School of AE
Atlanta, GA
Overview of Cartesian Grid
Approach
Introduction
Overview of Cartesian Grid Approach
•
•
•
•
Unstructured method with cell faces
aligned with coordinate directions
Control volume approach used to
solve governing equations
Internal cells created by subdividing
parent cells
Boundary cells “cut” by surface
Overlayed Cartesian Cell
Cut Cell
Introduction
Overview of Cartesian Grid Approach
• Benefits
– Automated grid generation and geometry definition separated
from grid resolution selection
– Less truncation error for cells since have orthogonal control
volumes
– Well-suited for high-order schemes due to orthogonal, regularlyspaced cells
– Fewer terms in equations (example: Navier-Stokes momentum
equation – 14 terms vs. 94 terms for generalized structured grid,
Meakin 1997)
• Drawbacks
– Solid surfaces required more computational work due to the
generalized finite volume formulations
– Like other unstructured grid approaches, more bookkeeping
needed to preserve grid topology than for structured grids
– Novel boundary condition treatments needed to handle viscous
derivatives near surface of cut cells.
Viscous Flux Stencils
Introduction
Prior Viscous Cartesian Related Efforts
• Navier-Stokes on Pure Cartesian
Grids
– Coirier and Powell (1993, 1996)
demonstrated that traditional viscous
flux formulations failed due to the nonsmoothness of the grids
– The tradeoff between accuracy and
positivity in the viscous flux stencil
indicates that a new treatment for
surface cells is needed
– Non-physical fluctuation and instability
due to non-body-fitted boundary cells
• Existence of cell centers inside of
the wall boundary
• Sharp fluctuation of the distance
from cell centers to the wall
Viscous Wall BC
• Example of initial attempts
.
.
-1.5
-1.5
Direct implementation
Smoothing treatment
-1
-1
-0.5
Cp
Cp
-0.5
0
.
.
0
0.5
AGARD Experiment
Smoothing treatment
0.5
1
1
1.5
0
0.2
0.4
0.6
0.8
1
x/c
Laminar: Cp over NACA0012 airfoil,
Rec = 2.0E5, M∞ = 0.3, α = 3.59°
0
0.2
0.4
0.6
0.8
1
x/c
Turbulent: Cp over NACA0012 airfoil,
Rec = 1.86E6, M∞ = 0.3, α = 3.59°
Boundary Condition Development
Embedded Boundary Method
•
•
Basic Idea: Interpolate to find state vector
at reference cells (e.g. locations C3 and
C5 in schematic) along surface normals to
set state vector at “ghost cells” (e.g. cells
3 and 5) and perform finite volume
integration all cells except “ghost cells”
– Utilize flat wall and curve wall
extrapolations similar to Extrapolation
Method to set state vector at ghost
cells
– In finite volume integration over
surface cells (e.g. cell 1), are treated
as if uncut
Benefits:
– Eliminates viscous flux stencil
positivity problem due to perfectly
uniform cell centroids near surface
(Extrapolation Method was slightly
non-uniform)
– Eliminates time integration limitation
associated with the cut cells
– All cells (surface and flow cells) utilize
C3
4
2
1
3
C5
5
Ghost Cells
Body Surface
NACA0012: Laminar
• Fine grid solutions
– Casalini : structured grid
– NASCART : Embedded Boundary Interpolation Method
-1.5
0.6
Casalini 1999
NASCART
-1
0.4
-0.5
Cf
Cp
0.2
0
0
0.5
-0.2
Casalini 1999
NASCART
1
1.5
-0.4
0
0.2
0.4
0.6
0.8
x/c
< Pressure coefficient >
1
0
0.2
0.4
0.6
0.8
x/c
< Skin friction coefficient >
1
Inviscid & Laminar Wall BC
• Embedded boundary ghost cell
approach using reference point
1
2
– Primitive variables at reference point (B, D)
interpolated from its 3 closest neighbor cells
using Linear Least Square Interpolation
5
6
B
3
4
7
8
δr
D
– Pressure
p g = p ref
=0
:
δg
V N,ref
δr
12
w
∂T
=0
∂η
)
κ : curvature
p g = p ref
∂p
−
∂η
(δ
g
δr
+ Tw
+δg )
13
14
15
w
Tref = Tw = Tg
)δ
r
δg
(
≈ κ ρV
δg
V N, g = −
A
C
2
T ref
Isothermal wall : Tg = (Tw − Tref
– Normal velocity
11
w
– Temperature
Adiabatic wall :
10
9
δr
∂p
∂η
Curved wall : ∂p
∂η
Flat wall
δr = cell diagonal length
δg = distance from wall
– Tangential velocity
Inviscid :
VT ,g = VT ,ref
Laminar :
VT ,g = −
δg
VT ,ref
δr
16
Turbulent Wall BC
• Wall function with standard k-ε model
– Introduced by Launder and Spalding (1974)
– Very grid efficient method for solving the RANS equations
– Widely used in structured and tetrahedral unstructured grid
solvers
– Smooth variation of grid cell distance from the wall required
– Conventional methods are not compatible with non-bodyfitted immersed Cartesian cells
• Unphysical fluctuation and separation / unreal flow acceleration
in transient solution
• Complicated coordinate transformation required to integrate
viscous flux → Increased truncation error & attenuate the
advantage of Cartesian grid topology
New Turbulent Wall BC
• The law of the wall
– Spalding’s formulation :
+ 2

+
u
κ
κu +
+
+
−κB
κu
+
y = u + e e − 1 − κu −
−
2
6

τw
δρ u
u
uτ ≡
y+ = w τ
u+ = t
ρw
uτ
µw
( ) ( ) 
3

κ : von Karman constant,
0.41
B : related to roughness
parameter
– Valid for log-law layer, buffer layer and viscous sublayer
– Excellent agreement with various experimental data, even for
y+>300
– Wall temperature from Crocco-Busemann equation :
Tw = Tref
2
r VT ,ref
+
2 cp
– Use of iterative method, i.e. Newton’s method to find wall shear
VT ,ref
stress from
• Application of wall shear stress
– Numerical approximation :
τ
xy ,1+
1
2
≈
1
[(µ l + µ t )1 + (µ l + µ t )2 ] u 2 − u1
2
y 2 − y1
– Computed wall shear stress ≠ Actual
wall shear stress
– Computed wall shear stress should
be corrected to the actual value
obtained from the law of the wall
Viscous Wall BC
• Results of initial attempts of shear stress
transformation in embedded Cartesian solver
– Slip wall condition
• Observed non-physical flow acceleration near wall in
transient solution
• Large CPU time required to converge
– No-slip wall condition
• Observed good solution in body-fitted Cartesian grid
system, i.e. flat plate
• Found non-physical fluctuation and instability in nonbody-fitted boundary cells
• Fluctuations could not be removed in many attempts
• Requirements
– Remove non-physical fluctuation
– Yield realistic transient solution
– Eliminate complicated coordinate transformation
– Give stable solution with relatively coarse grid at y+>300
• Development of new approach
– When tangential velocity of ghost cell corrected to satisfy
actual wall shear stress, this would eliminate the need for the
coordinate transformation
– Cells near wall would remain in numerically linear region
Sample results from new model for NACA0012, Turbulent
•
-1.5
Flow conditions
–
M∞ = 0.3, α = 3.59°,
Rec = 1.86 ×106
–
•
B = 5.0 (smooth wall)
Computational grid
–
5 chords apart from airfoil
–
22×20 root cells
–
Refinement level : 8, 9, 10 for coarse,
medium and fine grids
Pressure distribution
–
Very good correlation observed for
fine & medium grids
–
Coarse grid solution nicely correlated
except slight under-prediction at
suction peak
-0.5
Cp
•
-1
0
0.5
Experiment
NASCART, Fine (y+ = 81.1)
NASCART, Medium (y+ = 154.2)
NASCART, Coarse (y+ = 287.2)
1
1.5
0
0.2
0.4
0.6
x/c
0.8
1
Hemispheroid Turbulence Validation
•
Validation of k-epsilon turbulence
model in NASCART-GT
conducted for flat plates and
axisymmetric bodies
Flow conditions for
hemispheroid tests
0.01
– M∞ = 0.063 in
experiment
0.008
= 0.3 in
computation
0.006
– ReL = 2.0×106
– B = 5.5 (from
0.004
experiment)
Experiment, theta=0 (deg)
Experiment, theta=180 (deg)
NASCART
Cf
•
Ramaprian’s (1981)
hemispheroid
Computed vs
Experimental
Skin friction
• Good skin friction
correlation observed
0.002
– Discrepancy at first 0
station caused by
transitional flow (not-0.002 0
modeled) at that
location
0.2
0.4
0.6
x/c
0.8
1
NASCART-GT
NASCART-GT: Flow Modeling Overview
www.ae.gatech.edu/~sruffin/nascart
•
Governing Equations:
– Navier-Stokes
• Laminar
• Turbulent, k-Epsilon model
– Euler + Integral Boundary Layer
• Laminar, transition, & turbulent
• Uncoupled viscous/inviscid
• Coupled viscous/inviscid
– Euler
Thermal Models
– Ideal gas, Equilibrium & NonEquilibrium Reacting flow
Time Integration:
– 2nd order, explicit
– LU-SGS, implicit
Solution-adaption parameters:
– Divergence, vorticity, entropy
magnitude, turbulent k gradient,
turbulent epsilon gradient
Inviscid Fluxes:
– 2nd, 3rd or 5th order
– Roe FVS
– AUSM/PW+ (for hypersonic)
•
Boundary Conditions
– Isothermal, Adiabatic, radiative
equilibrium thermal BC
– Moving & flexible body BC’s
– Actuator Disk model
– Riemann invariants at
Inflow/Outflow for External Flows
– Slip and no-slip surface conditions
– Supersonic exhaust, Characteristic
based for subsonic internal flows*
*Future capability
Sample Results
NASCART-GT
Sample Perfect Gas Results
Current Capability
Results – Cessna 172
Mach=0.2, AoA=2.5 [deg], Alt=SL
Pressure Contours
9.900e+4 [N/m2]
1.025e+5 [N/m2]
Current Capability
Results – Transonic Inviscid NACA 0012
Current Capability
Results – Transonic Inviscid NACA 0012
Extrapolation Method
Circular Cylinder Drag Coefficient
Circular Cylinder Analysis
100
Measurements by C.
Wieselsberger
Numerical results by
Hamielec
Asymptotic formula for Re
near zero
10
Cd
NASCART-GT
Solution-adapted grid & entropy contours
[NASCART-GT], Re=1.0e4
Computed: Sr=0.21 [NASCART-GT]
Experimental: Sr=0.21 [Schlicting]
1
0.1
1.0E-1
1.0E+0
1.0E+1
1.0E+2
1.0E+3
1.0E+4
1.0E+5
Re
•
•
NASCART-GT viscous solutions developed using solution adaption
technique for vortex shedding
Viscous boundary treatment of cut cells yields good agreement with
experimental drag and Strouhal Number (Sr)
1.0E+6
Extrapolation Method
Circular Cylinder Analysis
Predicted entropy contours
[NASCART], Re=1.0e4
Predicted entropy contours
[NASCART-GT], Re=140
Experimental Karman vortex street
[Van Dyke], Re=140
Moving Geometry Capability
Moving Geometry Capability
•
Three routines installed for analysis of moving and deforming geometries
– Geometry motion module for each grid surface node and rigid body movement
– Computational grid adaptation module
– Flow boundary condition module
•
Developed and tested interface to LS-DYNA and performed aeroelastic
simulation using modified Newtonian option of NASCART-GT
Computed solution around NACA0012
Airfoil with active trailing-edge flap
Computed Mach number & adapted grid for
wing/pylon/store configuration at M=0.95
Rotorcraft Applications
GT Rotor Model
• 3 test cases
– Euler, no flap
– RANS, no flap
– RANS, flap
• Flow conditions
– M∞ = 0.3, α = 0°,
ReL = 9.196 ×105
– CT = 0.009045
h/R = 0.3
• Computational grid
– 5 body length apart from airfoil
– 26×22×20 root cells, 8 level
refinement
– y+max = 169.9 (no flap)
= 209.9 (flap)
Rotor blade : NACA 0015
< GT rotor configuration >
• Actuator disk model
– Trimming on collective pitch angle
only (no cyclic pitch in
experimental model)
GT Rotor Model
• Final grid configuration
& Entropy contours
– Cells refined around
fuselage and disk
– Entropy is a measure of
vorticity
– Vortices are well captured
and cells adapted around
vortex core
– Stronger tip vortex and faster
descending of vortex core in
advancing side than
retreating side due to BET
GT Rotor Model
• Upper centerline Cp
Experiment
Outline
Euler, no flap
RANS, no flap
RANS, flap
2
– Euler & RANS results are
close except reduced
pressure peak due to
viscous dissipation and
wakes
– Peak pressure and its
location estimated better by
adding flap effect
– Inconsistency with
experiment near nose and
local peak & drop at x/R=0.3
– Limitation of actuator disk
model : not analyzing
unsteady wake effect
Cp
1
0
-1
0
1
2
x/R
3
GT Rotor Model
• Iso-surface of entropy
– Vortex rollup at fore
disk propagates as a
vortex sheet &
dissipates rapidly
– Tip vortex rollup at
lateral sides of disk
merges to strong
line vortex &
propagates further
– In reality, individual blade generates strong vortex filaments by tip
vortex rollup to form helical shape line vortex (illustrated by O’Brien
and Smith from overset method, 2005) → Unsteady effect
– Actuator disk model based on time-averaged loading cannot produce
these unsteady effect → Model limitation
Current Capability
Results – Robin Rotorcraft
Test Case 1
• Fuselage alone
Realistic Configuration
• Vinf = 81.7 knots
Described by Super-Ellipse Equations • AOA = 0.0o
•
Re/L = 2.88x106/m
Extensively Tested
Model Features:
Test Case 2
• Fuselage with rotor
• Vinf = 81.7 knots
• AOA = 2.68o
• Advance ratio = 0.15
NASCART-GT Grid and Computed Pressure
ROBIN Fuselage/Rotor Systems
•
Successful 3-D tests of vortex refinement with k-epsilon turbulence
model in NASCART-GT conducted
•
Flow conditions for ROBIN test
– M∞ = 0.3, α = 0°,
1.2° nose left,
Re2L = 1.312 ×106
– Rotor: µ = 0.051,
CT = 0.00636
•
Effective
mesh
refinement
demonstrate
d in rotor
wake for 3-D
configuration
s Entropy Contours,
Cartesian Grid and
Actuator Disk for
ROBIN tests
ROBIN Model
• Cp on upper centerline
6
4
2
Cp
– Offset of pressure
tap location
makes two
distinct value at
same x-location
– Both solutions
well correlated
with experiment
– Two solutions
very similar
except pylon
region where flow
separation occurs
0
Experiment, averaged unsteady
Experiment, steady
Euler
RANS
-2
-4
0
0.4
0.8
1.2
x/L
1.6
2
ROBIN Model
• Sectional Cp
0.15
0.1
– Well correlated with
experiment
– Discrepancy on upper
surface at x/L=1.17 &
on lower surface at
x/L=1.354 seems to be
due to strut & rotor
shaft in experiment
z/L
0
x/L=1.170
x/L=0.353
-0.05
-0.1
0.15
-0.15
x/L=1.540
0.1
0.05
x/L=1.354
z/L
– Difference between two
solutions increased in
x-dir due to enhanced
viscous effect
downstream
0.05
0
Exp. RHS
Exp. LHS
Euler
RANS
-0.05
-0.1
-0.15
-12
-8
-4
0
Cp
4
-12
8
-8
-4
0
Cp
4
8
NASCART-GT Summary
Integral Boundary Layer Option
•
•
Rapid determination of viscous effects for laminar, transitional or turbulent flow
Integration conducted along surface streamlines from 3-D solution to include
pressure gradient effects (for attached flow)
Robin Fuselage
turbulent
•
Rexe
–
–
 22,400  0.46
 Re xe
ReθTr = 1.1741 +
Re

xe 
–
•
laminar
Laminar: Twaites Method
Transition: Michel’s Criterion,
Turbulent: Head’s Method
Separation predicted by shape factor criteria.
Cf
Parallelization
Intro: Space Filling Curve
• Idea:
- Fills every nodes of multidimensional space while preserving locality
- Reduce the multidimensional space into 1D
- Domain decomposition: “cutting” this 1D line
Hilbert:
Morton:
•Why use a Space Filling Curve (SFC) for parallelization?
•Communication Overhead: SFC maintains cells in contiguous blocks thus
lowering inter-CPU communication time.
•Load Balancing: SFC allows the number of cells on each processor to be
equal providing efficient use of each computer node.
•Simplicity: Does not take an iterative method to decompose domains
Intro: Space Filling Curve
• History of SFC
- First described by Italian mathematician, Giuseppe Peano, 1890
- A year later, Hilbert curve was described by David Hilbert
- Then several more versions are described by Jordan, Morton, Moore
- Recently adopted by wide range of disciplines: image compression,
vision sensing, ground mapping for GPS, motion pictures, CAD, CFD
• Hilbert curve outperforms the others in preserving locality
- Always connects the closest two nodes
• Why use a Space Filling Curve (SFC) for parallelization?
- Communication Overhead: SFC maintains cells in contiguous blocks
thus lowering inter-CPU communication time.
- Load Balancing: SFC allows the number of cells on each processor
to be equal providing efficient use of each computer node.
- Simplicity: Does not take an iterative method to decompose domains
ONERA M6 Test Case
14 CPUs
1.2 Million Cells
0.9 Million Surface Panel
Mach Number = 0.84
Angle of attack = 3.06
Velocity Divergence Solution Adaption
Inviscid mode
Pressure Contours
Sectional view of
domain decomposition
Smooth communication
across CPU boundaries
Hub
Mid-section
Tip
Parallelization Results
Decomposed domain distribution to 8 CPUs while performing solution adaption
Initial Grids over 3D Sphere
Grids after 39th Solution Adaption over 3D Sphere
M = 1.2, A.O.A. = 3 degree
Hypersonic and Reacting
Flows
Thermo--Chemical Nonequilibrium
Thermo
Development
• Governing Equations
- Euler or Euler + Integral boundary layer method
- Navier Stokes with two equation (k-epsilon) turbulence model
- Species conservation equations
- Vibrational energy equation (Two Temperature Model), L-T vibrational relaxation model
• Thermodynamic Model
- Calorically perfect gas
- Chemically reacting thermally perfect gas: (NASA Glenn Thermodynamic Curve Fit)
• Chemistry Model
- Air: Dunn/Kang, Gupta, Park85, Park87, Park91
- Titan N2-CH4-Ar: Nelson, Gokcen
• Flux Calculation
- Inviscid flux:
AUSMPW+ scheme with variable ratio of specific heat for reacting gas
MUSCL data reconstruction
-Species flux + Chemistry Source Term: Point Implicit
• Solution Adaption
- Velocity Divergence, Vorticity, Gradient of K-epsilon, Species Gradient
Equations for Thermochemical N.E.
Total Energy:
Governing Equations:
Chemical Source Term:
Point Implicit Formulations:
Vibrational Source Term:
AUSMPW+ Scheme
Idea : Fluxes splitting based on intuitively defined cell interface Mach number
PW+ : Pressure Weighted, Critical Speed of Sound
2D Cylinder Verification Results for Thermochemical N.E. (Air5
(Air5--Park93)
Freestream Conditions : M ∞ = 12.7, ρ ∞ = 1.60 × 10−3 kg / m 3 , T∞ = 196K , P∞ = 90 pa
Present
Max: 5.96E+3
Min: 1.96E+2
Present
Max: 1.54E-2
Min: 1.60E-3
Present
Max: 1.88E+4
Min: 9.04E+1
Present
Max: 4.77E+3
Min: 1.93E+2
DPLR
Max: 6.11E+3
Min: 1.96E+2
DPLR
Max: 1.55E-2
Min: 1.60-3
DPLR
Max: 1.90E+4
Min: 9.04E+1
DPLR
Max: 4.84E+3
Min: 1.96E+2
Temperature
Density
Pressure
Vibrational T
(Air5--Park93)
Present
Present
Present
Present
Present
DPLR
DPLR
DPLR
DPLR
DPLR
N2
O2
NO
O
N
Present
Max: 7.67E-1
Min: 7.26E-1
Present
Max: 2.33E-1
Min: 1.81E-2
Present
Max: 8.01E-1
Min: 0.00
Present
Max: 1.95E-2
Min: 0.00
Present
Max: 1.49E-1
Min: 0.00
DPLR
Max: 7.67E-1
Min: 7.24E-1
DPLR
Max: 2.33E-1
Min: 1.93E-2
DPLR
Max: 8.71E-1
Min: 0.00
DPLR
Max: 1.94E-2
Min: 0.00
DPLR
Max: 1.43E-1
Min: 0.00
(Air5--Park93)
Huygen Probe Verification Results for Thermochemical N.E. (N2(N2-CH4
CH4--Ar 13)
Freestream Conditions : M ∞ = 18.86, ρ ∞ = 2.96 × 10−4 kg / m 3 , P∞ = 15.621 pa
Present Cartesian Grids
60 x 40 Structured Grids
CH4--Ar)
Present
DPLR
Temperature
Present
DPLR
Vibrational Temperature
CH4--Ar)
DPLR
Present
Density
Present
Pressure
DPLR
CH4--Ar)
DPLR
Present
N2
DPLR
Present
CN
Shock Structure Comparison for
Trailing Toroidal Ballute
•
Calorically Perfect gas comparison with
LAURA Navier-Stokes Solver
– Good Agreement in Flow Structure: shock
locations and shock interaction
– Agreement in computed CD value within 5%
– NASCART run in Euler mode so differences
exist in Ballute trailing edge, separated flow
region relative to viscous Laura result
shown.
Temperature Distribution in plane of symmetry at
M = 26.3 calculated from LAURA Navier-Stokes solver, Cd = 1.39
(by P. Gnoffo, NASA LRC)
Temperature Distribution in plane of symmetry at
M = 26.3 calculated from NASCART-GT Euler solver, Cd = 1.318
Validation to Wind Tunnel Test
• Coupled Solution obtained from
NASCART-GT & LS-DYNA
• Coupled Solution matches well
with the wind tunnel data in axial
displacement and shock shape
Shock structure over deformed clamped ballute at freestream of
Mach = 5.73, Density = 1.46x10e-3 kg/m^3, Temp = 254k
by Reuben R. Rohrschneider
Titan Clamped Ballute (Results)
Temperature
• Total of 13 species N2-CH4-Ar chemistry with
two-temperature model
• 187,043 active flow cells
• Solution adaption based on velocity divergence
& species gradient
• CD = 1.393
• No shock interactions are observed
• Max. Temp outside of T.B.L. = 7.87E+3 K
• Max. Pressure at Surface = 1.09E+2 pa
Pressure
Vibrational Temperature
Modified Newtonian
Titan Clamped Ballute (Results)
N2 and N Mole
Fraction Contours,
X-plane: N2
Y-plane and Surface: N
CN, H2 and H Mole
Fraction Contours,
X-plane: CN,
Y-plane: H2
Surface: H
CH4, CH and C Mole
Fraction Contours,
X-plane: CH4,
Y-plane: CH,
surface: C
• Max. N2 Dissociation = 8.25 % (by volume)
• Max. CN(Cyano Radical) = 0.875% (by volume)
• All the hydrocarbon species disappear right after the shock wave
• Only C, N2, N, H exist near the surface
NASCART-GT: Next Steps
• Continue reacting flow CFD development and
validation for hypersonic flows.
• Develop validations of coupled viscous
approach for skin friction and heat transfer
including separated flows
– Demonstrate efficiency & accuracy of approach.
• Enhance efficiency of parallelized code to
utilize multi-processor computational clusters.
• Develop adjoint methodology for coupling with
vehicle optimization tools.
• Develop 6 DOF integration for store separation
and moving body problems.
• Develop flexible geometry integration with
structural analysis solvers.

NASCART-GT - Georgia Tech

Transcription

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