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, thought to depend only on the vorticity magnitude and was expressed in this way: was
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 stream allows a fully turbulent behaviour in any region where shear is present. in the free
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 AUSM for 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.
y-plus 100 80 Y+ section 50 Y+ section 68 Y+ section 80 Y+ section 89 Y+ section 96 60 40 Y+=30 20 Y+=5 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.
Cp Cp Cp Cp Cp -0.5 Experimental data AUSM data Roe data -0.5 Experimental data AUSM data Roe data 0 0 0.5 0.5 1 1 1.5 Figure 2: Cp distribution for the section situated at 50% of the span 1.5 Figure 3: Cp distribution for the section situated at 68% of the span -1 Experimental data AUSM data Roe data -1 Experimental data AUSM data Roe data -0.5-0.5 0 0 0.5 0.5 1 1 1.5 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 -1 Experimental data AUSM data Roe data -0.5 0 0.5 1 Figure 6: Cp distribution for the section situated at 96% of the span
Skin friction coefficient Skin friction coefficient Skin friction coefficient Skin friction coefficient Skin friction coefficient 0.012 0.01 Roe AUSM 0.01 Roe AUSM 0.008 0.008 0.006 0.006 0.004 0.004 0.002 0.002 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 0.01 0.01 Roe AUSM 0.008 Roe AUSM 0.008 0.006 0.006 0.004 0.004 0.002 0.002 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 0.008 Roe AUSM 0.006 0.004 0.002 Figure 11: Skin friction coefficient for the section located at 96% of the span
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. Z X Y Figure 12: Vorticity magnitude visualisation for a rotational speed of 2350 rpm
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.