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Edge-Based Reconstruction Schemes in CFD and CAA

Edge-Based Reconstruction Schemes in CFD and CAA. Ilya ABALAKIN, Tatiana KOZUBSKAYA CAA Laboratory of Keldysh Institute of Applied Mathematics of RAS, Moscow, Russia. Pavel BAKHVALOV Moscow Institute of Physics and Technology, Dolgoprudny, Russia. Outline. Basic 1D high accuracy scheme

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Edge-Based Reconstruction Schemes in CFD and CAA

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  1. Edge-Based Reconstruction Schemes in CFD and CAA Ilya ABALAKIN, Tatiana KOZUBSKAYA CAA Laboratory of Keldysh Institute of Applied Mathematics of RAS, Moscow, Russia Pavel BAKHVALOV Moscow Institute of Physics and Technology, Dolgoprudny, Russia

  2. Outline Basic 1D high accuracy scheme Scheme for linear transport equation on uniform meshes Scheme for linear transport equations on non-uniform meshes Scheme for nonlinear hyperbolic equation EBR schemes for unstructured meshes Conservative vertex-centred formulation Edge-Based Reconstruction on translationally symmetric meshes Edge-Based Reconstruction on arbitrary triangular/tetrahedral meshes Meshes providing the highest theoretically reachable order of accuracy Numerical illustrations WENO-EBR schemes Basic 1D WENO scheme WENO-EBR Scheme for unstructured meshes Numerical illustrations Numerical results Introduction Concluding remarks

  3. Introduction Nowadays high accuracy of CFD and, especially, CAA is of strong demand from industry Unstructured meshes become more and more attractive Two main streams: FE, mostly DG and FV polynomial-based schemes compete Both are still rather expensive, to be widely used for solving applied problems EBR (Edge-Based Reconstruction) schemes provide a good compromise as a lowER cost and highER accuracy numerical method EBR schemes efficiency is particularly evident in the nonlinear case when shock-capturing techniques are to be implemented

  4. Basic 1D high accuracy scheme

  5. Basic 1D high accuracy scheme Uniform mesh Linear transport equation Semi-discrete approximation Uniform mesh Computational cell Vertex-centered approach should be defined

  6. 3 third order schemes for linear transport equation Amplification factor for third-order schemes 2, 1 and 3

  7. Fifth order scheme for linear transport equation Amplification factor for fifth-order scheme

  8. Fifth order scheme for linear transport equation Positive/negative advection velocity Linear transport equation Semi-discrete approximation

  9. Basic 1D high accuracy scheme Non-uniform mesh Linear transport equation Semi-discrete approximation Non-uniform mesh Computational cell Such representation coincides with HO approximation on uniform mesh (i) and is exact for linear functions (ii)

  10. hybrid Remark Basic 1D high accuracy scheme Nonlinear equation Semi-discrete approximation Numerical flux is defined by the Reimann solver. Consider two options:

  11. EBR schemes for unstructured meshes

  12. - cell around vertex i - its volume - neighbors of vertex i Conservative vertex-centred formulation Euler equations Vertex-centred formulation implies the existence of cells corresponding to the mesh nodes (vertices). - its boundary

  13. ● ● ● ● ● Conservative vertex-centred formulation Under the rectangular rule • numerical flux defined with the help of some Riemann solver (Roe or Huang or hybrid solver in our case) is somewhat defined in single point of intersection of cell face ik with the edge ik • sum of oriented areas of subfaces of face ik The problem of scheme construction is reduced to the problem of definition (approximation) of in red points

  14. For instance, they can be built by skewing “Cartesian” mesh Reconstruction on translationally symmetric meshes Def.: A mesh is said to be translationally symmetric (TS) if it is translationally invariant with respect to the vectors of all edges of the mesh How do they look like?

  15. Translationally symmetric (TS) meshes The edges of TS meshes form 3 groups of equal-spaced parallel lines (in 2D) Not only triangles in 2D, parallelograms are still possible In 3D, tetrahedral TS-meshes can be produced by any linear transformation of cubes decomposed uniformly into six tetrahedrons. A set of tetrahedrons included to the cube is characterized by seven different edges, so that the TS-mesh remains the same with respect to translation on each of them.

  16. 1D HO Reconstruction on TS-meshes - 6-points stencil of 1d HO interpolation Numerical flux can be defined with the help of left (L) and right (R) reconstructed values: Such reconstruction coincides with HO reconstruction on uniform mesh (1) and is exact for linear functions (2)

  17. it is not a real stencil since it does not consist of mesh vertices Edge-Based Reconstruction on arbitrary triangular/tetrahedral meshes Direction of 1D reconstruction

  18. Edge-Based Reconstruction on arbitrary triangular/tetrahedral meshes Unstructured analogues of divided differences Direction vector

  19. Stencil of quasi-1D reconstruction Edge-Based Reconstruction on arbitrary triangular/tetrahedral meshes Quasi-1D reconstruction Def.: A reconstruction is said to be quasi-1D if the following two conditions are satisfied: 1) the reconstruction coincides with the 1D HO reconstruction on translational symmetric meshes; 2) the reconstruction coefficients are continuous with respect to the mesh deformation

  20. Theoretical estimation Highest possible theoretical order (of 5th-6th) is reachable on the following meshes: 1) translationally symmetric meshes (with the same way of cell construction for each node); 2) “Cartesian”* and other meshes if their cells including the vertices are translationally superposable Examples of translationally superposable barycentric (left) and orthocentric (right) cells Experimental estimation In practice EBR schemes provide highER accuracy of order form 2nd to 5-6th depending on mesh quality Edge-Based Reconstruction on arbitrary triangular/tetrahedral meshes Order of accuracy

  21. Numerical results Evolution of 2D initial disturbance of Gaussian shape Linearised Euler Equations Initial problem Parameters: Gaussian-pulse half-width 6 and 12 points per half-width 3 and 6 points per half-width Gaussian-pulse amplitude Time Numerical results at

  22. Numerical results Experimental estimation of order of accuracy Numerical results obtained on two meshes (coarse and fine) are compared with the exact solution Procedure of refinement

  23. Numerical results Evolution of 2D initial disturbance of Gaussian shape 6 / 12 points per half-width 3 / 6 points per half-width

  24. Numerical results Evolution of 2D initial disturbance of Gaussian shape 6 / 12 points per half-width 3 / 6 points per half-width

  25. Numerical results Evolution of 2D initial disturbance of Gaussian shape 6 / 12 points per half-width 3 / 6 points per half-width

  26. Numerical results 3D simulation of monopole point source in subsonic flow case (to assess the numerical tools used by partners) Mass source

  27. Numerical results 3D simulation of monopole point source in subsonic flow Mesh in use Far-field noise directivity Pressure pulsation (left) and pressure time-derivative (right) along the centre-line

  28. Intermediate concluding remarks EBR scheme can be considered as a FD quasi-1D method In other words, they are closer to FD than to FV content If we adopt the quasi-1D philosophy of scheme construction, a lot of useful further developments can be done • Among them are: • quasi-1D higher-accuracy scheme for cell-centered formulation • quasi-1D shock capturing techniques (TVD, WENO) • quasi-1D higher-accuracy scheme in polar/spherical coordinates • for better treatment of curvilinear body shapes • quasi-1D higher-accuracy scheme for unstructured meshes of arbitrary elements • quasi-1D higher-accuracy scheme for turbulence closure models

  29. WENO-EBR schemes

  30. Semi-discrete approximation Fifth order scheme 3 third order schemes Basic 1D WENO scheme Remind the HO schemes for linear transport equation

  31. EBR5: smoothing monitors WENO-EBR: WENO-EBR schemes on TS meshes 3 third-order 1D reconstructions from each (L/R) side

  32. WENO-EBR schemes on arbitrary meshes 3 quasi-1D reconstructions from each (L/R) side smoothing monitors

  33. WENO-EBR schemes on arbitrary meshes Characteristic-wise quasi-1D approach Calculate Roe average Jacobian along direction Characteristic variables Reconstruct characteristic variables and go back to

  34. 100 M tetras MPI, 8 OpenMP threads NOISEtte in-house code for solving aerodynamics and aeroacoustics problems on unstructured meshes ●Euler based family of models EE, NSE, NLDE, LEE Non-inertial reference frame is available ●Turbulence modeling RANS, LES, DES, DDES, IDDES ●Unstructured tetrahedral meshes ●Higher accuracy numerical scheme EBR multi-parameter vertex-centered scheme (up to 6th order) Finite-element approach for diffusive terms WENO-EBR schemes for shock capturing ●Implicit and explicit time integration Explicit Runge-Kutta up to 4-th order Implicit 2-nd order Newton with preconditioned BICG-stab solver ●Far field acoustics FW/H method ●Boundary conditions at open boundaries Characteristic-based flux splitting, Tam non-reflecting, periodicity ●Hybrid two-level MPI+OpenMP parallelization Heterogeneous parallel model including GPU-OpenCL under development

  35. Numerical results Transonic flow around airfoil (2D) RANS-SA model WENO-EBR5 scheme Local Mach number of flow around NACA 23012 at M=0.85

  36. Numerical results Transonic flow around airfoil (2D) RANS-SA model WENO-EBR5 scheme Local Mach number of flow around biconvex airfoil at M=0.9: numerical results (top), Schlieren photography (bottom)

  37. Numerical results Transonic flow around airfoil (3D) RANS-SA model WENO-EBR5 scheme Surface М=1 in grey Pressure isolines in central cross-section Pressure field on blade surface time t* = 0.2TperM(t* ) =0.85

  38. Numerical results Subsonic turbulent flow around helicopter blade IDDES hybrid model EBR5 scheme Mesh ~ 700000 nodes ~ 4M tetrahedrons 2-criterion for vorticity visualization: Zero-isosurface of 2nd Eigen value of tensor S2+W2 (S=(ui,j+uj,i)/2, W=(ui,j-uj,i)/2)

  39. Visualization of Numerical results Subsonic turbulent flow around helicopter blade IDDES hybrid model EBR5 scheme

  40. Numerical results Subsonic turbulent flow around rotating helicopter blade Euler equations WENO-EBR5 scheme rad/s m/s

  41. Numerical results Subsonic turbulent flow around cylinder Re=56000 (D=0.012 [m], v=70 [m/s]) IDDES model EBR6 scheme Implicit time integration of 2nd order : Newton linearization, BiCGStab solver Far field: FFWH 192 MPI X 8 OpenMP (1536 cores) on Lomonosov supercomputer Computational set-up

  42. Numerical results Subsonic turbulent flow around cylinder instantaneous flow fields

  43. Numerical results Subsonic turbulent flow around cylinder

  44. Numerical results case 1: gap-turbulence interaction DDES model EBR5 scheme Вид сверху Turbulent viscosity in central cross-section Вид сбоку Computational set-up Computational set-up m/s Isosurfaces of turbulent viscosity

  45. Numerical results Gap-turbulence interaction case Вид сверху Вид сбоку

  46. Numerical results Two-struts case DDES model EBR5 scheme Computational set-up Multi-block tetrahedral mesh (8 times coarsened) Вид сбоку

  47. Numerical results case 4: two-struts Вид сверху Вид сбоку

  48. Averaged velocity fields (0°) Horizontal velocity PIV Vertical velocity PIV

  49. Local Mach number (0° and 10°) PIV PIV

  50. Turbulent stresses (0°) PIV PIV

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