The Spalart Allmaras turbulence model
The main equation The Spallart Allmaras turbulence model is a one equation model designed especially for
aerospace applications; it solves a modelled transport equation for kinematic eddy viscosity
without calculating the length scale related to the shear layer thickness. The variable
transported in the Spalart Allmaras model is which is assimilated, in the regions which are
not affected by strong viscous effects such as the near wall region, to the turbulent kinematic
viscosity. This equation has four versions, the simplest one is only applicable to free shear flows
and the most complicated, which is written below, can treat turbulent flow past a body with
laminar regions.
This transport equation bring together the turbulent viscosity production term and the
destruction term . The physics behind the destruction of turbulence occurs in the near wall
region, where viscous damping and wall blocking effects are dominants. The other terms or
factors are constants calibrated for each physical effect which needs to be modelled. This
equation allows to determinate for the computation of the turbulent viscosity , which is
for interest for us, from:
The production terms In the transport equation for kinematic eddy viscosity, the production term is modelled in this
way:
Where
is a scalar measure of the deformation tensor. During the development of the formula, was
thought to depend only on the vorticity magnitude and was expressed in this way:
Where is the mean rate-of-rotation tensor and is defined by
In Fluent this formulation is used when the option Vorticity based is selected in the turbulence
model definition box.
Nowadays it is known that it is necessary to take into account the effect of the mean strain on the
turbulence production and has been modify by J. Dacles-Mariani, G. G. Zilliac, J. S. Chow, and
P. Bradshaw and incorporated to FLUENT:
Where
With the mean strain rate, , defined as
Those formulations are used in FLUENT when the Strain Vorticity based option is selected in the
turbulence model definition box. Its effect is to reduce the eddy viscosity in region where
measure of vorticity exceeds that of strain rate.
Destruction terms As it has already been pointed out, the destruction terms are only active in the region where
shear is present and hence where viscosity effects are strong. The destruction terms are
modelled in this way:
The constants of the models are , and . There values have
been calibrated and can be summarized as follow:
0.1355 0.622 2/3 7.1
0.3 2 0.4187
Boundary conditions The boundary conditions have been set and tested to match theory and experiments with a good
convergence of the code. The wall condition is , it have been tested and it results that the
turbulence viscosity term begins at the transition trips and spreads genteelly. The ideal value for
the turbulence viscosity in the free stream is zero but some codes might have some difficulties to
converge because of round-off errors and it is often used:
This is in the case of a calculation initialize with the trip term but setting up in the free
stream allows a fully turbulent behaviour in any region where shear is present.
Convergence
The solution has achieved 26000 iterations for both cases, the Roe and the AUSM numerical method.
The baseline model has been set in first order with a low courant number of about 0.25 until it has
converged and then it has been implemented to a second order scheme and to the third order
scheme. The convergence strategy for rotating applications advised in the FLUENT user’s manual has
been used. The rotational speed has then been slowly increased until the case speed was reached.
The implementation to a higher order scheme has been done at 13000 and 18000 iterations
respectively. Unfortunately the residual plot is not available because it has not been saved but the
residuals have converged to values of 0.001 and 0.01 before stabilising. The above graph show a
comparison of the symmetric planes in terms of static pressure and it is possible to see that the
solution is very similar, however it is not identical and the solution might not be converged enough.
Results
Pressure distribution and skin friction coefficient
A comparison of the pressure coefficient on the five sections of the blades located respectively at 50,
68, 80, 88 and 96 percent of the span and for a blade tip Mach number of M=0.827 has been carried
out. This comparison investigates the effects of two numerical methods, the Roe and the AUSM
method, both in third order MUSCL.
The calculations yield results in accordance with the experimental data, even though the suction
peak is not accurately captured and is slightly under-estimated for both solutions. As moving toward
the tip the local velocity of the flow increase and the section peak increases until the flow over the
blade becomes transonic. A shock forms at 89% of the span and is located at 23% of the chord. The
AUSM numerical method tends to predict with more accuracy the shock position. However the
prediction for the last section seems not to predict the shock observed in the experiment, the
solution calculated is more diffusive.
From the skin friction plots it is possible to analyse the shock boundary layer interaction and see if
any separation is induced. No experimental data is provided to compare so only the Roe and AUSM
data are plotted. No separation seems to be predicted for both method but it is possible to see that
the skin friction coefficient is very low, about 0.002, for all section from 60% of the chord to the
trailing edge. This illustrates the fact that the flow is at the edge of separation. At the shock location
the skin friction decreases meaning that the velocity gradient is lower in the boundary layer and that
there is an increase of the boundary layer thickness after the sock. The numerical method which
seems to predict separation the more accurately is the AUSM method.
Table 1: Lift coefficient comparison for the five sections with the Roe and AUSM numerical method
Section location on the span (%)
Cl for AUSM
Errors (%) Cl for Roe Errors (%) Cl experimental
50 0,216 8,94 0,19 19,90 0,2372
68 0,2789 0,50 0,2767 1,28 0,2803
80 0,3074 -9,32 0,3307 -17,60 0,2812
89 0,3157 -4,95 0,339 -12,70 0,3008
96 0,2937 8,05 0,2881 9,80 0,3194
The above table compare the lift coefficient from the experimental data with the calculation done
with the Roe and AUSM numerical method. The lift coefficient for those two method have been
calculated with a trapezoidal rule inducing great errors (errors for the trapezoidal rules can be of
about 60%) and have to be analysed with care. However the lift coefficients predicted by the AUSM
method are much more accurate with errors of about 9% for the worst case whereas the Roe
calculation yields errors of 20%. Those errors can come from the mist prediction of the pressure
distribution which would be reflected in the lift coefficient calculation.
Figure 1: Y+ distribution for the five sections of the blade
The differences between the CFD calculation and the experimental data could be due to the grid
resolution. Indeed a analysis of the Y+ distribution on the five section of the blade show that a large
portion of the blade has Y+ comprised in-between 30 and 5. It is known that a Y+ located in this
region is not the optimum configuration of the grid for accurate results because the turbulent models
are then not resolving the entire boundary layer, hence mist calculate the velocity distribution within
the boundary layer. This affects the pressure distribution, and the skin friction distribution which are
dependent on the velocity gradients in the boundary layer. This would explain why the recovery in
pressure at the leading edge is not very well predicted; indeed the displacement thickness of the
boundary layer must be under-estimated affecting the actual camber of the airfoil. For better results
the grid must be modify so the centroid of the near wall cells does not lay in the region of 5<Y+<30
which would help for a better resolution of the boundary layer.
x/c
y-plus
0 0.2 0.4 0.6 0.8 1
20
40
60
80
100
Y+ section 50
Y+ section 68
Y+ section 80
Y+ section 89
Y+ section 96
Y+=30
Y+=5
Figure 2: Cp distribution for the section situated at 50% of the span
Figure 3: Cp distribution for the section situated at 68% of the span
Figure 4: Cp distribution for the section situated at 80% of the span
Figure 5: Cp distribution for the section situated at 89% of the span
Figure 6: Cp distribution for the section situated at 96% of the span
x/c
Cp
0 0.2 0.4 0.6 0.8 1
-0.5
0
0.5
1
1.5
Experimental data
AUSM data
Roe data
x/c
Cp
0 0.2 0.4 0.6 0.8 1
-0.5
0
0.5
1
1.5
Experimental data
AUSM data
Roe data
x/c
Cp
0 0.2 0.4 0.6 0.8 1
-1
-0.5
0
0.5
1
1.5
Experimental data
AUSM data
Roe data
x/c
Cp
0 0.2 0.4 0.6 0.8 1
-1
-0.5
0
0.5
1
Experimental data
AUSM data
Roe data
x/c
Cp
0 0.2 0.4 0.6 0.8 1
-1
-0.5
0
0.5
1
Experimental data
AUSM data
Roe data
Figure 7: Skin friction coefficient for the section located at 50% of the span
Figure 8: Skin friction coefficient for the section located at 68% of the span
Figure 9: Skin friction coefficient for the section located at 80% of the span
Figure 10: Skin friction coefficient for the section located at 89% of the span
Figure 11: Skin friction coefficient for the section located at 96% of the span
x/c
Skinfrictioncoefficient
0 0.2 0.4 0.6 0.8 1
0.002
0.004
0.006
0.008
0.01
0.012
Roe
AUSM
x/c
Skinfrictioncoefficient
0 0.2 0.4 0.6 0.8 1
0.002
0.004
0.006
0.008
0.01 Roe
AUSM
x/c
Skinfrictioncoefficient
0 0.2 0.4 0.6 0.8 1
0.002
0.004
0.006
0.008
0.01 Roe
AUSM
x/c
Skinfrictioncoefficient
0 0.2 0.4 0.6 0.8 1
0.002
0.004
0.006
0.008
0.01
Roe
AUSM
x/c
Skinfrictioncoefficient
0 0.2 0.4 0.6 0.8 1
0.002
0.004
0.006
0.008
Roe
AUSM
Vortex analysis
The above figure present the visualizations of the computed wake structure using iso-surfaces of
vorticity, for the case with a tip Mach number of 0.827. It is possible to see the wake generated from
the blades and the vortex generated at the tip. The solution has considerable noise because it is not
converged enough yet, even though the tip vortex is resolved until 120°. However, the wake of the
blade is not yet well resolved and the calculation must be carried on for more iteration to be able to
capture it with accuracy.
Figure 12: Vorticity magnitude visualisation for a rotational speed of 2350 rpm
X
Y
Z
The above figures show the vortex shading for different ages going from 0° to 80° with a step of 10°
for the Roe and the AUSM numerical method calculation. At an age of 0° it is possible to see the
vortex been created at the tip of the blade. As the vortex becomes older it grows, becoming less
strong and migrates down toward the root of the blade. Both numerical method yield similar results
and a deeper analysis is done in Figure 31 which plots the age of the vortex versus its Z and Y
direction non dimensionalised by the radius of the blade R=1.142m.
Vortex shading for the Roe calculation:
Figure 13: Tip vortex shading for 0°
Figure 14: Tip vortex shading for -10°
Figure 15: Tip vortex shading for -20°
Figure 16: Tip vortex shading for -30°
Figure 17: Tip vortex shading for -40°
Figure 18: Tip vortex shading for -50°
Figure 19: Tip vortex shading for -60°
Figure 20: Tip vortex shading for -70°
Figure 21: Tip vortex shading for -80°
Vortex shading for the AUSM calculation:
Figure 22: Tip vortex shading for 0°
Figure 23: Tip vortex shading for -10°
Figure 24: Tip vortex shading for -20°
Figure 25: Tip vortex shading for -30°
Figure 26: Tip vortex shading for -40°
Figure 27: Tip vortex shading for -50°
Figure 28: Tip vortex shading for -60°
Figure 29: Tip vortex shading for -70°
Figure 30: Tip vortex shading for -80°
Figure 31: Wake geometry measurements for a rotor speed of 2350 rpm and comparison with classical data
A comparison with the data from F. X. Caradonna and C. Tung, 1981 is carried out for both solution
calculated with the Roe and the AUSM method. Vortex ages from 0° to 120° have been plotted since
the solution was not fully converged to predict the vortex location at a further age. The results are
not very satisfactory and diverge for high vortex ages. This might be due to a convergence problem
and the solution might be run for several more iteration in order to predict the vortex migration with
more accuracy. However, the AUSM solution is much closer to the experimental data and seems to
predict the vortex migration with more accuracy.
Conclusion
This report has focused on the simulation of hovering rotor tip vortices and rotor wake convection
using the Spalart and Allmaras one equation turbulence model for two numerical methods, the Roe
and the AUSM method. The vortex sheet is a relatively weak feature of the flow that descends in a
tightening helical pattern below the rotor. The root and tip vortices follow contracting helical
trajectories below the rotor disc. This behaviour has been observed for both calculation carried out
but the AUSM model tend to be closer to the experimental solution. The tip vortices and the wake
influence strongly the pressure distribution of the blades in a hovering rotor generating vibration and
noise. The result of this analysis tend to show that the results for the pressure distribution are in
accordance with the experimental data but that the resolution of the mesh is of importance and that
further calculations must be carried out with a better grid resolution for a better accuracy of the
results. For a better capture of the vortex trajectory and wake calculation the actual solution is not
converged enough and it is a critical parameter to analyse the vortex migration for advanced ages. In
any case, the numerical method which seems to predict with the more accuracy this type of
problems is the AUSM numerical method but a particular care must be taken toward the Y+
distribution on the wing to avoid near wall cells comprised in the region of 5<Y+<30 for the
turbulence models to apply accurately.