Analysis of High-Order Residual-Based Dissipation for Unsteady Compressible Flows

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A comprehensive study of the numerical properties of high-order residual-based dissipation terms for unsteady compressible flows leads to the design of well-behaved, low dissipative schemes of third-, fifth- and seventh-order accuracy. The dissipation and dispersion properties of the schemes are then evaluated theoreticaly, through Fourier space analysis, and numerically, through selected test cases including the inviscid Taylor-Green Vortex flow.

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Analysis of High-Order Residual-Based Dissipation for Unsteady Compressible Flows

  1. 1. DynFluidFluid Dynamics Laboratory Analysis of high-order residual-based dissipation for unsteady compressible flows Karim GRIMICH Joint Work with: Paola CINNELLA Alain LERAT ICCFD-2804 1
  2. 2. Introduction• Two usual ways for constructing high-accurate schemes on a structured grid:  Extend the grid stencil  use compact approximations (directional compact, DG, RBC …)• Owing to their narrow stencil, compact approximations: greatly reduce the error coefficient with respect to a non-compact approximation of the same accuracy order  improve treatment of irregular meshes• Here we use centered compact approximation:  With no filter,  No artificial viscosity or limiter• A Residual-Based Compact (RBC) scheme [Lerat, Corre, JCP 2001] can beexpressed only in terms of approximations of the exact residual (sum of all theterms in the governing eqns), even for the numerical dissipation.• Use of residual based approach enables genuinely multi dimensional dissipation. DynFluid Fluid Dynamics Laboratory 2
  3. 3. Aims and motivations• A queer feature is that the residual-based dissipation behaves as the discreteresidual, i.e. it is consistent with a null function  When residual tends to zero high-order dissipation terms are left.• Need for identifying the actual dissipation operator through suitable expansions  finding the necessary and sufficient conditions ensuring that this operator is really dissipative for all flow conditions.  studying the spectral properties of RBC schemes for a multidimensional problem to get information on the evolution of Fourier modes supported on a given grid. DynFluid Fluid Dynamics Laboratory 3
  4. 4. Summary I. High-order RBC schemes II. The χ-criterion for RBC dissipation III. Spectral properties of RBC schemes IV. Numerical experimentsDynFluidFluid Dynamics Laboratory 4
  5. 5. Summary I. High-order RBC schemes II. The χ-criterion for RBC dissipation III. Spectral properties of RBC schemes IV. Numerical experimentsDynFluidFluid Dynamics Laboratory 5
  6. 6. High-order RBC schemes Initial value problem on a uniform mesh ~ ~ ~ ~ Mean and difference operators:DynFluidFluid Dynamics Laboratory 6
  7. 7. High-order RBC schemes Exact residual:Residual-Based Compact schemes: Third-order of accuracy using 3x3 points: RBC3 Fifth-order of accuracy using 5x5 points: RBC5 Seventh-order accuracy using 5x5 points: RBC7DynFluidFluid Dynamics Laboratory 7
  8. 8. High-order RBC schemesSpatial approximation of the main residual: Pade approximation of first order derivatives where Applying to the all left hand side gives Which truncation error isDynFluidFluid Dynamics Laboratory 8
  9. 9. High-order RBC schemesResidual based dissipation operator: Mid-point residuals using the same technique as for the main residual Where and are  designed once for all [Lerat, Corre, JCP 2001]  with no tuning parameter  depending only on A=df/dw, B=dg/dw, andDynFluidFluid Dynamics Laboratory 9
  10. 10. High-order RBC schemesA peculiar feature of unsteady RBC schemes:  A mass matrix appears due to difference operators applied to wt. Operators applied to Operators applied to spatial temporal derivatives derivativesDynFluidFluid Dynamics Laboratory 10
  11. 11. High-order RBC schemesThe residual-based operator is consistent with the first order and dissipativeoperator [Lerat, Corre, JCP 2001]: For an exact solution the residual is null is consistent with some higher order term  To which high order term is it consistent with?  Is this high-order term dissipative or not?  How this high-order dissipation behaves? DynFluid Fluid Dynamics Laboratory 11
  12. 12. Summary I. High-order RBC schemes II. The χ-criterion for RBC dissipation III. Spectral properties of RBC schemes IV. Numerical experimentsDynFluidFluid Dynamics Laboratory 12
  13. 13. The χ-criterion for dissipationRole of the x-discretization in r1 and the y-discretization in r2: We define With Doing Taylor expansion of r1x around the point (j+1/2,k) Similar result forDynFluidFluid Dynamics Laboratory 13
  14. 14. The χ-criterion for dissipationFully discretized RBC dissipation: The full dissipation of a RBCq scheme can be expressed as: with:DynFluidFluid Dynamics Laboratory 14
  15. 15. The χ-criterion for dissipationThe χ-criterion for dissipation: In the linear case: WithSince all derivatives in being even, its Fourier symbol is real.Definition: The operator dq is said to be dissipative ifDynFluidFluid Dynamics Laboratory 15
  16. 16. The χ-criterion for dissipationThe χ-criterion for dissipation [Lerat, Grimich, Cinnella, submitted to JCP 2012]: Theorem: The operator dq is dissipative for any order q=2p-1, any pair (A,B) and any functions Φ1 and Φ2 such that if and only if Criterion easily satisfied through a proper choice of Pade coefficients  RBC3 and RBC7 presented in [Corre, Falissard, Lerat, C&F 2007] do not satisfy the χ-criterion (weak instability for unsteady flows)  RBC5 presented in [Corre, Falissard, Lerat, C&F 2007] satisfies the χ-criterionAnalogous results proved in 3DDynFluidFluid Dynamics Laboratory 16
  17. 17. Summary I. High-order RBC schemes II. The χ-criterion for RBC dissipation III. Spectral properties of RBC schemes IV. Numerical experimentsDynFluidFluid Dynamics Laboratory 17
  18. 18. Spectral properties of RBCRBC schemes spatial discretization expresses asand can be rewritten asWhere is the operator applied to wt and the operator applied to w.In a more compact way:withalong with initial condition these ordinary differential equations define a Cauchyproblem that is studied in the following.DynFluidFluid Dynamics Laboratory 18
  19. 19. Spectral properties of RBCWe study the linear problemIts Fourier transform is (1)Where is the velocity vector and is a 2D wavevector.We introducewhich is the reduced wave number aligned aligned with advection directionwhere and are the CFL numbersand is the reduced wave number vectorThus (1) rewrites:DynFluidFluid Dynamics Laboratory 19
  20. 20. Spectral properties of RBCTaking the Fourier transform of the spatial discretization of RBC schemes (2)Introducing the modified reduced wave number along the advection direction(2) re-writes We define the error with respect to the exact wave numberDynFluidFluid Dynamics Laboratory 20
  21. 21. Spectral properties of RBC  Cauchy stability of RBC schemes  Dispersion properties of RBC schemes  Dissipation properties of RBC schemesIf RBC schemes were infinitely accurate we would haveDynFluidFluid Dynamics Laboratory 21
  22. 22. Spectral properties of RBC Cauchy stability of RBC schemes unstable unstable stable stable RBC7 with χ≠0 RBC7 with χ=0 DynFluid Fluid Dynamics Laboratory 22
  23. 23. Spectral properties of RBCAdvection along a mesh direction  Dispersion and dissipation properties of RBC schemes (example of RBC7) 1D cut 1D cut Dispersion Dissipation DynFluid Fluid Dynamics Laboratory 23
  24. 24. Spectral properties of RBCAdvection along a mesh direction  1D cut Dispersion Dissipation DynFluid Fluid Dynamics Laboratory 24
  25. 25. Spectral properties of RBCAdvection along the mesh diagonal  Dispersion and dissipation properties of RBC schemes (example of RBC7) Dispersion Dissipation DynFluid Fluid Dynamics Laboratory 25
  26. 26. Spectral properties of RBCResolvability : cutoff reduced wavelength for an error lower than 10-3 : minimum number of points per wavelength for an error lower than 10-3 DynFluid Fluid Dynamics Laboratory 26
  27. 27. Summary I. High-order RBC schemes II. The χ-criterion for RBC dissipation III. Spectral properties of RBC schemes IV. Numerical experimentsDynFluidFluid Dynamics Laboratory 27
  28. 28. Numerical experimentsThe discretization has to be completed by selecting a timeapproximation methodHere, we choose the 2nd-order accurate, A-stable Gear schemeStability region includes the complex stable half-plane • Coupled with a dissipative scheme, this results in an unconditionally stable discretizationAssociated nonlinear system solved via a dual-time stepping techniqueDynFluidFluid Dynamics Laboratory 28
  29. 29. Advection of a sine wave Validation of the χ-criterion for dissipation: χ≠0 χ≠025x25 mesh λ χ=0 χ=0 DynFluid Fluid Dynamics Laboratory 29
  30. 30. Advection of a sine waveResolvability of RBC schemes: 6 points per wavelength 8 points per wavelength 16 points per wavelengthDynFluidFluid Dynamics Laboratory 30
  31. 31. Inviscid Taylor-Green vortexRepresentative model to within the inertial range Periodic boundary conditions + t=0 t=3 t=5 RBC5 scheme 1283 mesh Q-criterion colored by kinetic energy M=0.3 (Shu et al, J.Sci.Comput. 2005) DynFluid Fluid Dynamics Laboratory 31
  32. 32. Inviscid Taylor-Green vortexTime evolution of the total kinetic energy 643 mesh 1283 mesh DynFluid Fluid Dynamics Laboratory 32
  33. 33. Inviscid Taylor-Green vortex RBC5 schemePhysics well captured: vortex stretching 1283 mesh vorticity contours t=3 t=5 DynFluid Fluid Dynamics Laboratory 33
  34. 34. Conclusion• RBC schemes unsteady high-order dissipation has been identified  the χ-criterion for dissipation has been demonstrated ensuring that this operator is really dissipative for all flow conditions.  the spectral properties of RBC schemes for a multidimensional problem gave information on the evolution of Fourier modes supported on a given grid: stability, dissipation, dispersion.• Numerical experiments confirmed the validity of the χ-criterion.  the low dissipative and dispersive behavior of high-order RBC schemes  the high resolvability of high-order RBC schemes• Perspectives  LES, DES simulations  design a suitable high-order time integration technique DynFluid Fluid Dynamics Laboratory 34
  35. 35. Thank you for your attention Any question?DynFluidFluid Dynamics Laboratory 35
  36. 36. ReferencesMain references on previous works on RBC schemes: DynFluid Fluid Dynamics Laboratory 36
  37. 37. Converging cylindrical shockConverging cylindrical shock [Chisnell, JFM 1957]  analytical estimation of the pressure behind a moving cylindrical shock No filter No Artificial Viscosity No Tuning parameter Shock strength becomes infinite when it reaches the axis DynFluid Fluid Dynamics Laboratory 37

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