multigrid accelerated numerical methods based on implicit scheme for moving control volumes for wt...

CENTRAL AEROHYDRODYNAMIC INSTITUTE named after Prof. N.E. Zhukovsky (TsAGI)

Multigrid accelerated numerical methods based on implicit scheme for moving control volumes for WT flows simulating

E.Kazhan, I.Kursakov, A.Lysenkov


2

Method description


3

Explicit approximation

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– convective & diffusion fluxes

– turbulent model source terms

•convection: Godunov-Kolgan-Rodionov(Russian TVD)•diffusion: central-difference approximation•sources: local-implicit scheme•stable only with CFL ≤ 1


Implicit scheme. Smoother

4

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Roe linearization«explicit» part

Linear system:

Next step value:


Linearization

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Localization

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Example: 1-D Euler (for simplification)

Gauss-Zeidel


Zonal approach

7

P0

Zone separation:•main (inviscid) area - explicit scheme•thin layer near wall - implicit scheme

Zonal approach:•Ignoring small-scale processes in boundary layer, assuming them as quasi-steady•Correct global-scale processes description•Global-scale processes predominate the behavior of boundary layer

Explicit scheme area

Implicit scheme area

Implicit scheme:•Large computation time per step•CFL can be larger than 1•Effective only with huge CFL•Results: incorrect description of global-scale processes


Explicit-implicit combination

8

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Explicit Implicit


Calculation speed-up: multigrid

9

kjikji

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t,,

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t QFUU

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1,, KJI

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Fine gridHigh solution accuracy

Low convergence speed

Coarse gridLow solution accuracy

High convergence speed


RANS in rotating frame

10

WxF

tU

i i

)()(

0))(2())(2(

))(2(0

SqS

ruvruwrvw

W zyx

yzx

xzy

,

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wn

qn

knnxnxnxn

xnz

xny

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TQ

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TwIvIuIpEQIpsQwIpsQvIpsQu

Q

)()/( F

a

z

y

x

a - axis of rotation

- rotating rate

This system can be solved, but there are difficulties in the calculation of the far field at long distance to the axis of rotation

Additional terms appear in the sources

No change of flows


RANS on rotating mesh

11

ri i

r WxF

tU

)()(

0)()(

)(0

SqS

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W yx

zx

zy

r

,

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UVVV zyxr

)( FF

,0xV ,zVy yVz

For rotation around the axis X:

Additional terms are entered into the calculation of flows associated with the flow due to the grid rotation. In the source term is the correction to the Coriolis force.

The flow through the rotating mesh faces

An amendment to the Coriolis force

Special thanks to Dr. V.Titarev


Solver modifications for solutions on rotating mesh

12

• The solution of the Riemann problem of the discontinuity decay on moving mesh

• Modification of the boundary conditions: slip condition - given by the rotation rate impermeability condition –

condition is stated for the "Riemann" condition – mesh rotation rate is taken into

account in determining the flow direction• Time step correction for the explicit scheme• Roe matrixes are modified for implicit scheme

sideflow VVV


Features of the implicit scheme on rotating meshes

13

rotn

rotn

rotn

rotn

rotn

rotn

rotn

VcVVcV

VVVV

VVVV

VV

000000000000000000000000000000000000000000

The matrix of the Roe matrix eigenvalues:

Rotating rate is accounted in the stabilizing matrixes


14

Testing


Implicit smoother test case

15

Boundary layer on plate M = 0.8

Re = 22.8×106

NACA 0012 M = 0.8α = 0°

Re = 9×106

CPU time CPU time

Acceleration: 27 times Acceleration: 20 times

COMGLEI (Combination of Global and Local tau type with Explicit and Implicit schemes)


Multigrid test case

16

Onera M6 wing M = 0.8395 α = 3.06°

Re = 11.72×106

Residual

Friction drag coefficient

Lift coefficient

Fivefold solution convergence acceleration


Rotating mesh test case

0 10 20 30 40 50 60 70 80

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

PSP

Расчёт

Эксперимент

V, м/с

Computation PSP Precision on most considered regimes – 3 - 4 %


18

Applications


The thrust reverser impact on aircraft aerodynamics

19

velocity

Lift

“Jump” in Lift magnitude due to the flow structure reconfiguration


The thrust reverser impact on aircraft aerodynamics

20

Higher pressure zones

• Landing devices impacts on the reversed jets propagation

• Calculations considered landing devices allow determining the high loads zones


WT modeling

21

Brand new estimation of corrections for CL_max caused by the WT walls are obtained


Propeller characteristics calculation approach application

22

WT Т-104

Propeller test rig VP-107

• Obtaining the integral characteristics of propeller: thrust, torque

• Propeller and airframe interference• Experimental data corrections :

Calculation of the shaft cone and propeller blades interference

Calculation of the influence of the experimental setup elements on the propeller characteristics

Reynolds number influence


Propellers calculation features

23

Flow separation Mesh refinement at the blade end is required

The maximum propeller thrust mode is alike the flow separation regime. Separation from the propeller blades should be well predicted.


Conclusion

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1. Combined method based on the Godunov-Kolgan-Rodionov is proposed.2. Acceleration:

«Boundary layer on plate» ‑ 27 times; «Profile NACA0012» – up to 20 times;

3. Use of the multigrid approach demonstrates that the convergence of the solution is fivefold accelerated.4. The solvers developed in this work allow to solving the wide class of stationary problems of computational aerodynamics.


25

Thank you for your attention

multigrid accelerated numerical methods based on implicit scheme for moving control volumes for wt...

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