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The Spalart Allmaras turbulence model

The main equation

The Spallart Allmaras turbulence model is a one equation model designed especially foraerospace 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:


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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:


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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 toconverge 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.


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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 users 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.


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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 0,281289 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.


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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


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Figure 2: Cp distribution for the section situated at 50% of

the span


the span


the span


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 dataAUSM data

Roe data

x/c

Cp

0 0.2 0.4 0.6 0.8 1

-0.5

0

0.5

1

1.5


Roe data

x/c

Cp

0 0.2 0.4 0.6 0.8 1

-1

-0.5

0

0.5

1

1.5


Roe data

x/c

Cp

0 0.2 0.4 0.6 0.8 1

-1

-0.5

0

0.5

1


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


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Figure 7: Skin friction coefficient for the section located at

50% of the span


68% of the span


80% of the span


89% of the span

Figure 11: Skin friction coefficient for the section located at 96% of the span

x/c

Skin

friction

coeffic

ient

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

Skin

friction

coeffic

ient

0 0.2 0.4 0.6 0.8 1

0.002

0.004

0.006

0.008

0.01 Roe

AUSM

x/c

Skin

friction

coefficient

0 0.2 0.4 0.6 0.8 1

0.002

0.004

0.006

0.008

0.01 Roe

AUSM

x/c

Skin

friction

coefficient

0 0.2 0.4 0.6 0.8 1

0.002

0.004

0.006

0.008

0.01

Roe

AUSM

x/c

Skin

friction

coefficient

0 0.2 0.4 0.6 0.8 1

0.002

0.004

0.006

0.008

Roe

AUSM


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Vortex analysis

The above figure present the visualizations of the computed wake structure using iso-surfaces ofvorticity, 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


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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 -30Figure 17: Tip vortex shading for -40

Figure 18: Tip vortex shading for -50




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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






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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 sincethe 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.


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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

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