Doru Caraeni CD-adapco, USA CFD Futures Conference, August 6-8, - PowerPoint PPT Presentation

Doru Caraeni CD-adapco, USA CFD Futures Conference, August 6-8, 2012

Why I did Residual-based schemes research ? - (1996) Leading the CFD/CAE group (Centrifugal Compressors) at COMOTI Bucharest - Challenge: to perform LES of turbulence inside high-PR CC - Write a new CFD code (together with Aerospace Department at “ Polytechnica ” Institute Bucharest) for industrial LES - Found a few papers about early Residual-Distribution schemes - Learned more about these scheme at a VKI Advanced CFD course - Went to Lund Institute to learn LES of turbulence and develop a (hopefully) best-in-class LES algorithm, for industry (Dec.1997)

A short early history of MU-RDS Multidimen idimensional ional Upwind d Residual dual Distribut bution ion scheme me : - Fluctuation-Splitting (RDS) proposed in 1986 by professor Phil Roe - Developed by professors and students at Michigan University (Roe), VKI (Deconinck), Bordeaux University (Abgrall), Polytechnica di Bari, Lund (Caraeni), Univ. of Leeds (Hubbard) etc. - Compact matrix distribution schemes for steady Euler and Navier-Stokes equations (E.van der Weide, H. Paillere), 1996. - Second order RD scheme for LES of turbulence using a residual- property preserving , dual time-step approach (Caraeni, 1999).

A short early history of MU-RDS (cont.) Multidimen idimensional ional Upwind d Residual dual Distribut bution ion scheme: me: - Second order space-time RD scheme for unsteady simulations (2000, VKI) (using space-time integration/residual-distribution to achieve accuracy) - Third order RD scheme for steady inviscid flow simulations (2000, LTH) (node gradient-recovery for quadratic solution representation) - Third order RD scheme for the unsteady turbulent flow simulations (2001, LTH). (node gradient-recovery and residual-property satisfying) - Third order results with above gradient-recovery idea reported by Rad and Nishikawa (2002, MU). - High-order (>3) RD scheme for scalar transport equations (2002, BU & MU). (sub-mesh reconstruction for high-order solution representation) “ Third-order non-oscillatory fluctuation schemes for steady scalar conservation laws ” M. Hubbard, 2008.

Ricchiuto, CEMRACS, 2012

A 3 rd Order Residual-Distribution scheme for Navier-Stokes simulations (Residual-property satisfying formulation) (A Third Order Residual-Distribution Method for Steady/Unsteady Simulations: Formulation and Benchmarking, including LES, Caraeni, VKI, 2005)

High-order RD scheme for Navier-Stokes equations      ( ) 0 u , ,     t j j c v  U F F U         , , , , j j j j t  ( ) ( ) u u u p , , , , i t i j j i ij j         ( ) ( ) ( ) ( ) e u h u T    , , , , , t j j i ij j j j U dv ,          t    c v ( , , , , ) U u u u e [( ) ] F F U dv 1 2 3 j j , j , t  Jameson dual-time   C V U F F , , , t j j j j algorithm    u 0 j        u u p  1 1 j j 1 j         C  V  F u u p F 2 2 j j j j 2 j       u u p   3 j 3 j 3 j           u h u u u T   j 1 1 j 2 2 j 3 3 j , j

High-order RD scheme (cont.) for Navier-Stokes equations    c c Convection flux cell-residual F . dv T , j T    v v Diffusion flux cell-residual F . dv T , j T    uns Unsteady term cell-residual . U dv , T t T Update scheme for steady/unsteady simulations (Caraeni):    1 ,    n k 1 , 1  1 ,       n k n k T c v uns ( ) U U B T T T i i i V   , T i T i   T B I Upwind matrix residual T B i i  Distribution coefficient (bounded) i T (conservativity)

High-order RD scheme (cont.) for Navier-Stokes equations Distribution schemes (for preconditioned system): 1 1      L ow D iffusion A ( LDA ) 1 K F n R R   , , i j U j i i i i 3 3  ,   T LDA 1 ( ) B K K 1 i i j      1 K R R j i i i i 3 1 L ax- W endroff ( LW )      1 K R R i i i i 3 1 1  ,     T LW 1 [ ( ) ] B I K K i j i 4 2 j Computes the convective 4    cell residual with second c K U T i i order accuracy (linear data) 1

High-order RD scheme (cont.) How to construct a 3rd-order RDS (Ph.D. 2000, LTH): 1. Use (upwind or upwind-biased) uniformly bounded residual-distribution coefficients (linearity/accuracy preserving RD scheme), and apply to total cell- residual 2. Compute the total cell residual (convective + diffusive + unsteady terms) with the required accuracy: - we used condition-1 + linear solution, second order accurate integration for 2 nd order RDS - we need to use condition-1 + use quadratic reconstruction, 3 rd order accurate integration for 3 rd order RDS The idea is to use the same accuracy-preserving RD scheme, as for second order schemes, but compute the total cell residual with 3 rd order accuracy.

High-order RD scheme (cont.) Convection residual discretization, 3rd order. Use parameter variable Z and   ( 1 , , , , ) Z u u u H assume a quadratic variation 1 2 3 over the tetrahedral cell. Z ,j computed with 2 nd order accuracy Cell-residual in integral form: (multi-step algorithm)  c     c c . . F dv F dS , T j i i  i i 0 1  ( ) 0 1  Z Z T T Z Z i i ( mid )    1 0 , , j j ( ) r r Z   Z Z 2 8 0 j       Z Z p 2  2  2 1 j 1 j     Z Z Z 1      c   1 2 3 ( ) F Z Z p p Z Z 2 2 j j  0 4   2   Z Z p   3 j 3 j   2 u Z Z     i 4 H c T j p 2

High-order RD scheme (cont.) Convection residual discretization, 3rd order. 4 n        c c , j i c . { ( ) }  F dS F d   face ( ) I i Z Z d T j n k j  Z Z  1 i , T face j i J k i face i       ( ) n Z Z d j , i 0 j      ( ) Z Z d face i   k j       ( ) n Z Z p d , j i face 1 1 j j i   face i   6          c  i i   ( ) ( ) ( ) ( ) F n d n Z Z p d i i { } 0 1 Z Z H H d , , 0 1 j i j i 2 2 j j j   k j face face i i    , 1 i i face  0 1    i ( ) n Z Z p d   , j i 3 j 3 j   face i       ( ) ( ) j k  [ ] H H d A I ( ) n Z Z d   j , i , face HH j k 4 j i   face face i i [ ] I Pre-computed matrix , HH j k

High-order RD scheme (cont.) Diffusion residual discretization, 3rd order.      v v v . F dv F ds , T j  T T Assuming a quadratic variation of the Z variables over the cell, the diffusive flux vector integral can be computed over the cell-face. Use the values of the Z variable and its gradients, defined in the nodes of the high-order FEM tetrahedral-cell. 4      v [( ) ( ) ] u u    v v    . k . , , F ds F n i i u ( ) face k  ,  T i   1 k T

High-order RD scheme (cont.) Unsteady residual discretization , 3rd Order.     1 , 1 n k n n 3 4 U U U  2 nd order discretization in time, U  , t 2 . and 3 rd order in space: t 9         ( ) v ( ) . . U dv Q U dv , , T t t   0 T T    9 ( )  . ; I Q dv      ( ) v  ( ) . U Q dv , T t T   0     T 0 . 05 ; 0 ,.., 3 I V  T     0 . 20 ; 4 ,.., 9 I V  T

High-order RD scheme (cont.) Monotone shock capturing   2  h  . 2 2 V   p       1. Shock detection or or ( ) ( ) # nodes   2   N p V av j 1 2. Blending between the high-order scheme and a first order positive RD scheme (the N-scheme)          T LDA N = 0 for a smooth flow ( 1 ). . i i i = 1 (discontinuity detected)    : ( , ,..) where f P

Summary of this 3 rd order RD algorithm - Uses a Multi-D Upwind Residual-Distribution scheme - Formulated for fully unstructured grids (tetrahedrons), - Compact scheme, highly efficient parallel algorithm. - Implicit time integration (dual time-stepping algorithm). - 3 rd - order accuracy in space (using FEM integration) - 2 nd - order time discretization (BDF2 scheme) - Acceleration techniques: preconditioning , point- implicit relaxation, geometric multi-grid, etc.