Swirling flows - Thayer School of Engineering at Dartmouth

Swirling flows
“When I meet God, I am going to
ask him two questions: why
relativity and why turbulence? I
really believe he will have an
answer for the first”
Werner Heisenberg
OutLine
I.
II.
III.
Swirling flows
Turbulence modeling of swirling
flows
Application of swirling flows
Rotation
An essential ingredient in many industrial processes:
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mixing,
separation
stabilisation
However, in some cases is an inevitable by product causing damage
and financial loss:
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Temperature-differences between ocean and atmosphere
leading to thunderstorms and tornadoes
vortices generated by wings of large airplanes leading to delay
during landing-procedures.
Swirling flows:
Many Engineering applications involve
swirling or rotating flow:
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In combustion chambers of Jet
engines
Turbomachinery
Mixing tanks
2D Swirling or Rotating flows
Also known as the axisymmetric flow with swirl or rotation:
Assumption: no circumferential gradients in the flow.
Solving this problem includes the prediction of the circumferential or swirl velocity.
The momentum conservation equation for swirl velocity is given by :
Where x: is the axial coordinate
r: the radial coordinate
u: the axial velocity
v: the radial velocity
w: the swirl velocity
Physics of the swirling flows:
In swirling flow, conservation of angular momentum results in the
creation of a free vortex flow in which circumferential velocity w
increases as the radius decreases and then decays to zero at r=0
due to the action of viscosity.
For an ideal free vortex: The circumferential forces are in
equilibrium with radial pressure gradient
For non-ideal vortices the radial pressure gradient also changes
affecting the radial and axial flows.
Turbulence modeling in swirling flow:
Turbulent flows with significant amount of swirl: swirling
jets or cyclone flows.
The strength of the swirl is gauged by the swirl number S,
defined as the ratio of the axial flux of angular
momentum to the axial flux of the axial momentum.
Where is R is the hydraulic radius*
Turbulence modeling in swirling flow:
Realizable k-epsilon model
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Realizable: model meets certain
mathematical constraints on the Reynold’s
stresses that correspond to the physics of
the turbulent flows.
Considered as an improvement of the kepsilon standard model.
Characterized by both the new formulation of
the turbulent viscosity and a new equation
for the dissipation rate epsilon that is derived
from the transport of the mean square
vorticity fluctuation.
Provides improved predictions for the
spreading rate of both planar and round jets.
Exhibits superior performance for flows
involving rotation, boundary layers under
strong adverse pressure gradients,
separation, and recirculation.
RNG k-epsilon model
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Developed using Renormalisation Group (RNG)
methods to renormalize the Navier-Stokes
equations to account for small scales of motion.
This is different from the original k-epsilon model
where the eddy viscosity is determined from a
single turbulence length scale.
Mathematically similar to the k-epsilon model, but
it has a different epsilon equation that accounts
for different scales contribution to the production
term.
Reynolds Stress Models:
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Also known as the Reynolds stress
Transport (RST)
Usually used for high level turbulence
models
The method of closure used is called
Second Order Closure
The Eddy viscosity approach is not used
and the Reynold stresses are directly
calculated using differential transport
equation.
The calculated Reynold’s stresses are
then used to obtain closure for the
Reynolds’ averaged momentum equation.
Application of the swirling flow:
Simulating the internal flow of a pressure swirl fuel
injector
Fuel injectors
High velocity liquid fuel⇒ atomization and oxidation
with air⇒ evaporation ⇒ combustion
Swirl injectors:
Hollow cone spray ⇒ more fuel droplets
exposed to the hot air in the combustion
chamber⇒ shorter evaporation time
The Physics of the atomization inside a PSI:
1. Film formation:
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Liquid fuel is introduced through the tangential
ports into the swirl chamber.
The swirling motion pushes the liquid to the walls
of the injector which constitutes the origin of the
thin film
2. Free Sheet :
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At the exit of the nozzle, the free sheet is formed
in the shape of a cone.
3. Atomization:
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The liquid free sheet is an unstable structure. As it
interacts with air, it starts to break down into
ligaments, these ligament disintegrate into small
droplets [1].
Mathematical formulation
Favre averaging:
Favre averaging is a time averaging method that takes into account a
changing density:
Applying the Favre averaging on Y, the gas mass fraction :
Mathematical formulation
Conservation of mass:
Conservation of momentum:
Conservation of energy
Closure: Homogeneous relaxation model
● Used to study thermal non equilibrium
two phase flow
● Assumes adiabatic conditions
● Provides an equation for the return or the
relaxation of the quality to the equilibrium
value
Computational methods
The geometry
Creating the mesh
Boundary conditions:
Fuel Inlets:
-zero pressure gradient
Outlets:
-atmospheric pressure
-zero velocity
Walls:
-zero velocity
-zero pressure gradient
Post-processing:
A swirling velocity field
Simulation results:
Density field
Volume fraction
Pressure field
Temperature field
Spray angle predictions:
Problems and challenges:
Schmidt Number consideration:
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Swirling flows have a higher critical Re to transition from laminar flow to turbulent flow:
relatively stable
The Schmidt number is the ratio of the momentum diffusion rate over the mass
diffusion rate.
If there is no mass diffusion between the liquid phase and the gas phase (no mixing)
then the Schmidt number goes to infinity
A Schmidt number of 1: means that both types of diffusion are occurring at the same
rate.