1. 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. CENTRAL AEROHYDRODYNAMIC INSTITUTE named after Prof. N.E. Zhukovsky (TsAGI) 2

3. 3 Explicit approximation – 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

4. CENTRAL AEROHYDRODYNAMIC INSTITUTE named after Prof. N.E. Zhukovsky (TsAGI) Implicit scheme. Smoother 4 Roe linearization«explicit» part Linear system: Next step value:

5. CENTRAL AEROHYDRODYNAMIC INSTITUTE named after Prof. N.E. Zhukovsky (TsAGI) Linearization 5

6. CENTRAL AEROHYDRODYNAMIC INSTITUTE named after Prof. N.E. Zhukovsky (TsAGI) Localization 6 Example: 1-D Euler (for simplification) Gauss-Zeidel

7. CENTRAL AEROHYDRODYNAMIC INSTITUTE named after Prof. N.E. Zhukovsky (TsAGI) Zonal approach 7 P0P0 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

8. CENTRAL AEROHYDRODYNAMIC INSTITUTE named after Prof. N.E. Zhukovsky (TsAGI) Explicit-implicit combination 8 ExplicitImplicit

9. CENTRAL AEROHYDRODYNAMIC INSTITUTE named after Prof. N.E. Zhukovsky (TsAGI) Calculation speed-up: multigrid 9 Fine grid High solution accuracy Low convergence speed Coarse grid Low solution accuracy High convergence speed

10. CENTRAL AEROHYDRODYNAMIC INSTITUTE named after Prof. N.E. Zhukovsky (TsAGI) RANS in rotating frame 10 - 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

11. CENTRAL AEROHYDRODYNAMIC INSTITUTE named after Prof. N.E. Zhukovsky (TsAGI) RANS on rotating mesh 11 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

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

13. CENTRAL AEROHYDRODYNAMIC INSTITUTE named after Prof. N.E. Zhukovsky (TsAGI) Features of the implicit scheme on rotating meshes 13 The matrix of the Roe matrix eigenvalues: Rotating rate ​​ is accounted in the stabilizing matrixes

14. CENTRAL AEROHYDRODYNAMIC INSTITUTE named after Prof. N.E. Zhukovsky (TsAGI) 14

15. CENTRAL AEROHYDRODYNAMIC INSTITUTE named after Prof. N.E. Zhukovsky (TsAGI) Implicit smoother test case 15 Boundary layer on plate M = 0.8 Re = 22.8×10 6 NACA 0012 M = 0.8 α = 0° Re = 9×10 6 CPU time Acceleration : 27 times Acceleration : 20 times COMGLEI (Combination of Global and Local tau type with Explicit and Implicit schemes)

16. CENTRAL AEROHYDRODYNAMIC INSTITUTE named after Prof. N.E. Zhukovsky (TsAGI) Multigrid test case 16 Onera M6 wing M = 0.8395 α = 3.06° Re = 11.72×10 6 Residual Friction drag coefficient Lift coefficient Fivefold solution convergence acceleration

17. CENTRAL AEROHYDRODYNAMIC INSTITUTE named after Prof. N.E. Zhukovsky (TsAGI) Rotating mesh test case Computation PSP Precision on most considered regimes – 3 - 4 %

18. CENTRAL AEROHYDRODYNAMIC INSTITUTE named after Prof. N.E. Zhukovsky (TsAGI) 18

19. CENTRAL AEROHYDRODYNAMIC INSTITUTE named after Prof. N.E. Zhukovsky (TsAGI) The thrust reverser impact on aircraft aerodynamics 19 velocity Lift “Jump” in Lift magnitude due to the flow structure reconfiguration

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

21. CENTRAL AEROHYDRODYNAMIC INSTITUTE named after Prof. N.E. Zhukovsky (TsAGI) WT modeling 21 Brand new estimation of corrections for CL_max caused by the WT walls are obtained

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

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

24. CENTRAL AEROHYDRODYNAMIC INSTITUTE named after Prof. N.E. Zhukovsky (TsAGI) Conclusion 24 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. CENTRAL AEROHYDRODYNAMIC INSTITUTE named after Prof. N.E. Zhukovsky (TsAGI) 25