Solution Algorithms for High-Speed Viscous Flows Antony Jameson Department of Aeronautics and Astronautics Stanford University, Stanford CA

Transcription

1 Solution Algorithms for High-Speed Viscous Flows Antony Jameson Department of Aeronautics and Astronautics Stanford University, Stanford CA 13 th Conference on Finite Element for Flow Problems University of Wales at Swansea, United Kingdom April 4-6, 2005

2 Flo3xx Computational Aerodynamics on Arbitrary Meshes 2

3 Support for Arbitrary Meshes Examples of mesh types which are being used in computational aerodynamics Structured Unstructured Cell-Centered Unstructured Cell-Vertex Nested Cartesian With Cut Cells In Flo3xx a unified mesh-blind formulation supports all of these in one code Designed to meet the following objectives: Platform for automatic mesh adaptation Migration path to emerging mesh generation technologies A robust algorithm that is tolerant to bad meshes 3

4 Support for Arbitrary Meshes Conservation laws are enforced on discrete control volumes Fluxes of conserved variables are exchanged through interfaces between these cells Independent of the mesh topology, each interface separates exactly two control volumes (on the right, face N separates cells A and B) All algorithms are expressed in terms of a generic interface-based data structure 4

5 Treatment of Structured Meshes Interface Flux First Neighbors Second Neighbors Associate first and second neighbors with each face Allows implementation of standard schemes with five-point stencils (Jameson-Schmidt-Turkel JST, SLIP) in the same code Eliminates the need for gradient reconstruction Numerical experiments verify 25% overhead due to indirect addressing in comparison with standard structured-code implementation (FLO107) 5

6 Flo3xx in Action Mach Number CL FLO-XX, Mach 3.25 FLO-XX, Mach 2.50 Manual, Mach 3.25 Manual, Mach 2.50 Lift: Angle-of-Attack sweep Angle of Attack w.r.t Fuselage (deg) Geometry Courtesy of Lockheed Skunk Works Lockheed SR71 at M= 3.2, - Euler calculation with 1.5 Million grid points From IGES definition to completed result in one week, including CAD fixes, mesh generation We need to be able to compute extreme test cases This concerns both complexity of geometry and flow conditions 6

7 Validation of Unified Solution Algorithm in Flo3xx: Inviscid Transonic Flow Onera M6 wing at M = 0.84, a = 3.06 o 7

8 Convergence Using Automatic Multigrid Engineering Accuracy 8

9 Analysis of Formation Flight of UAVs using Flo3xx Mach Number Contours Mach number and Upwash 9

10 Initial Validation for Viscous Flow: Zero-Pressure-Gradient Boundary Layer Hexahedral Mesh (84k Nodes) Cell-Centered Algorithm 10

11 RANS Results Using FLO107-MB For Drag Prediction Workshop Statistical Evaluation DPW1 All Participants Flo107-MB (DPW2) Accurate drag prediction for complex geometries in transonic flow is still very hard Flo3xx is currently in viscous validation phase. FLO107-MB has been thoroughly validated. Results of right figure were obtained with CUSP scheme and k- turbulence model w 11

12 Flo3xx Payoffs Highly flexible platform for all applied aerodynamics problems and other problems governed by conservation laws Fast turnaround through convergence acceleration techniques Framework can be used to support advanced research, such as the BGK method or the Time-Spectral Method, which will be addressed in this talk This means, take advanced research out of a laboratory setting and apply it to problems of practical engineering interest, which is ultimately the only way to make an impact on the state-of-the art 12

13 Non Linear Symmetric Gauss-Seidel Multigrid Scheme

14 Nonlinear Symmetric Gauss-Seidel (SGS) Scheme Forward and reverse sweeps: w For 1D case : t + x f w Sweep (1) : Increasing j ( 1 w ) 0 j = w j Ê ( ) - A -1 Á f Ë ( 01 f ) ( 0 = f w ) ( j,w 1 ) j-1 j- 1 2 ( ) ( 00 ) - f j A = f w Sweep (2) : Decreasing j ( 2 w ) 1 j = w j Ê ( ) - A -1 Á f Ë ( 12 ) - f j ( ) = 0 ( 10) j- 1 2 ( 11) j Flux evaluations in each double sweep Cost per iteration similar to 4 - stage Runge - Kutta scheme ˆ ˆ f (10) (00) (1) (1) (0) (0) (0) w w w w w x x x x x f j x x x x x j f (11) (12) (1) (1) (1) (2) (2) w w w w w f 14

15 Solution of Burgers Equation on 131,072 Cells in Two Steps With 15 Levels of Multigrid Converged 0 15

16 Solution of 2D Euler Equations: Convergence for NACA0012 The convergence history shows the successive computation on meshes of different sizes The convergence rate is independent of the mesh size Convergence rate ~.75 per cycle 16

17 Solution of 2D Euler Equations NACA0012 Airfoil Solution after 3 multigrid cycles Solution after 5 multigrid cycles Solid lines: fully converged result 17

18 Symmetric Gauss-Seidel Scheme with Low Mach Preconditioner A. Jameson and D. Caughey ( ) 18

19 Euler Calculation at Mach 0.8 Converge to 10-9 within 60 iterations C P Distribution Convergence History 19

20 Euler Calculation at Mach Converge to within 100 iterations C P Distribution Convergence History 20

21 Face-based Gauss Seidel (FBGS) Scheme (Following a suggestion by John Vassberg) On an arbitrary grid, loop over faces instead of looping over cells Update the cells adjacent to a face as you go along Updated state will be used on next visit to a cell 21

22 The Finite-Volume BGK Scheme Using Statistical Mechanics to Enhance Computational Aerodynamics 22

23 A Major Conceptual Difference Between Continuum Mechanics and Statistical Mechanics In continuum mechanics the unknown solution variables are assumed to have precise values: U = U(x, y,z,t) In statistical mechanics the solution variables exist only as moments of a statistical distribution in physical and phase space, or as expectation values : U = Ú y f (x, y,z,u,v,w,x,t) du dvdwdx 23

24 The Key Idea of behind Gas-Kinetic Finite-Volume Schemes Consider a finite-volume discretization of the Navier-Stokes equations w n +1 = w n - 1 V Dt Ú 0 Ú r F da r Compute the fluxes for the Navier-Stokes equations at interface N from the distribution functions in cells A and B A distribution function needs to be constructed at each time step for each cell r r F ( x, t) = Ú uy f ( x, t, u, v, w, x ) dudvdwdx Ê where y = 1,u,v,w, 1 ˆ Á 2 (u2 + v 2 + w 2 + x 2 ) Ë T 24

25 Finding the Distribution Function The equilibrium distribution function for ideal gases is known from Boltzmann statistics: f eq = g( r x, r u,,x) = A( r x )e -l( r { } x ) ( U r -u r ) ( U r -u r )+x 2 The nonequilibrium distribution function is unknown, but its evolution is given by the Boltzmann equation: f t + u f x + v f y + w f z = Q( f, f ) Global numerical solution infeasible, because of high dimensionality Collision Integral 25

26 A Crucial Simplification (Bhatnagar, Gross & Krook - BGK) [Phys. Rev. Vol. 94, 1954] Replace the Collision Integral Q with a linear relaxation term: Q = - f - g t fi Collision Time This equation can be solved analytically: f t + u f x + v f y + w f z = - f - g t f ( r x, r u,t,x) = t Ú 0 g( x r - u r (t - t ), u r -(t- t ), t t,x)e d t + e - t t f 0 ( x r - u r t, u r,x) 26

27 27 Key Step: The Construction of the Distribution Function Ú = t t t t u ut x f e dt e t u t t u x g t u x f 0 0 ) ( ),, ( ),, ), ( ( ),,, ( x x x t t r r r r r r r r Ó Ì Ï = 0, ) ( 0, ) ( 0 s ug g s g g s ug g s g g f r x r t r x r l x l t l x l t t ( ), ) ( ) ( t g s g s H s g s H g g t x x = + - The initial nonequilibrium distribution can be approximated from the Chapman-Enskog expansion of the equilibrium distribution in cells A and B. Similarly, the relaxation state at the cell interface is obtained as H(x) = 0, x > 0 1, x 0 Ï Ì Ó s [Prendergast, Xu,1993; Xu,2001]

28 Constructing the Fluxes The Flux normal to the face is computed as F = Ú u s Ê 1 ˆ Á Á u s Á v s f (x Á f,t,u s,v s,w s,x)du s dv s dw s dx Á w s Á 1 2 (u 2 s + v 2 s + w 2 s + x 2 ) Ë Rotate Fluxes back to (x,y,z) coordinates Update the solution according to r W n +1 = W r n - 1 V Dt Ú 0 Ú r F da r dt r F = (F,G,H) T 28

29 F E Transport Coefficients By Chapman-Enskog expansion the Navier Stokes equations can be recovered from the BGK equation, with the viscosity coefficient F E + m =t p By setting the collision time appropriately, Navier-Stokes fluxes can be computed directly from the distribution function Prandtl number is fixed at 1, correction needed Method 1: Fix at level of discretized equations, Expensive! ( 1/ Pr-1) q where q = Ú ( U - u) ( U - u) fdx Method 2: Scale Temperature gradient during reconstruction, this requires one floating point operation 1 2 r r r r g x g = r x r - 3 2Pr T x T + 1 2RT 2 Pr T x (u -U)2 + 1 RT (u -U)U x 29

30 Payoff It is not necessary to compute the rate of strain tensor in order to calculate viscous fluxes This eliminates the need to perform two levels of numerical differentiation, which is difficult on arbitrary meshes Improved accuracy and reduced sensitivity to the quality of the mesh Automatic upwinding via the kinetic model, with no need for explicit artificial diffusion, thus reduced computational complexity 30

31 Validation of the BGK Scheme: 3D Inviscid Transonic Flow Finer Mesh (316k Nodes) Coarser Mesh (94k Nodes) Onera M6 wing at M = 0.84, a = 3.06 o 31

32 Convergence The BGK scheme can be used with standard convergence acceleration techniques: Multigrid Local Time Stepping 32

33 Validation of the BGK Scheme using Flo3xx: 3D Inviscid Transonic Flow Density from to 1.1 Falcon Business Jet at M=0.8, a = 2 deg 33

34 Validation of the BGK Scheme using Flo3xx: 3D Inviscid Transonic Flow Density Mach Number a SR71 at M=3.2, = 3.2 deg 34

35 Viscous Validation of the BGK Scheme: Zero-Pressure-Gradient Boundary Layer Velocity Profile (Incompressible), M=0.2, Re=6000 Temperature Profile (Compressible), M=0.6 Similarity solution for entire plate shown 3 Sections along plate Computed on a Structured Mesh 35

36 Boundary Layer on a Stretched Triangular Mesh The plot shows the similarity solution for the entire plate Mesh-independent discretization for the Navier-Stokes equations on a simple next-neighbor stencil 36

37 Viscous Validation of the BGK Scheme: Resolved Shock Structure (M=10) Velocity Heat Flux Prandtl number fix works perfectly. Can set Prandtl number as input parameter 37

38 Viscous Validation: NACA0012 Extreme viscous Testcase: Re = 73, M = 0.8 Results match the conventional Methods Pressure peaks at trailing edge due to singularity (sharp edge) 38

39 Viscous Validation of the BGK Scheme NACA0012 at Re = 5000, M = 0.5, a = 0 o 39

40 Viscous Validation: Turbulent Boundary Layer Solve for eddy viscosity using turbulence model Adjust collision time according to effective viscosity Apply Prandtl number correction according to effective Prandtl number 40

41 High-Order Methods for High-Speed Flow Georg May Antony Jameson

42 What are the Specs for a High- Order Scheme as we see it? High-Order Accuracy in Mesh Refinement (>3) Applicable to hyperbolic PDEs with discontinuities in the solution Ideally retains nominal accuracy everywhere Robust ( Industrial Strength ) Efficient Structured and Unstructured Meshes 42

43 Methods Numerical schemes Spectral Difference Discontinuous Galerkin Treatment of discontinuities Limiting of reconstruction. Simple, robust, but loss of accuracy at discontinuities No Limiting. Try to retrieve high-order information by re-expansion in Gibbs complementary basis (Gegenbauer Polynomials, Freud polynomials). Recovers nominal accuracy everywhere, but difficult in higher dimensions Identifying discontinuities Standard, robust gradient sensor sufficient for limiting Edge detection procedures for reconstruction in Gibbs complementary basis 43

44 The Spectral Difference Method (Liu, Vinokur, and Wang 2004) Use Conservative Form of Equations u t + F(u) = 0 Pseudo-spectral representation of both flux function F(u) and solution variables u in each cell i u i ( r x ) = F i ( r x ) = M u  j=1 M F  j=1 L ij ( r x )u ij M ij ( r x )F ij fi F i ( r x ) = M ij ( r x )F ij Use analytical flux at interior flux-collocation points (if there are any), Riemann flux on cell boundaries M F  j=1 44

45 Example for Stencils Linear Elements F ij u ij Quadratic Elements F ij u ij 45

46 The Spectral Difference Method Conservation follows from the use of the right quadrature formulas Enforce local discrete conservation by appropriate choice of collocation points Fluxes unambiguous in 1D within each cell. For higher dimension special treatment of flux points on corners (2D), edges(3d) is required Can use collocated points instead of staggered points. Needs more points for collocation of u, but saves reconstruction, and thus both memory and CPU time 46

47 2D Algorithm - Fluxes Use analytical fluxes at interior flux collocation points: Element i F = F(u i,k ) Use Riemann fluxes at flux collocation points located on edges edge j n j F = F Riemann (n j,u j,k,l,u j,k,r ) x j,k u j,k,l u j,k,r Each edge has associated left and right states, L and R, at each flux integration point (from polynomial representation) 47

48 2D Algorithm - Modified Riemann Fluxes at Element Corners One can show that flux points at a corner of an element can be split with both incident edges F = F Riemann (n j,n j +1,u j +1,n f,l,u j +1,n f,r) F = F Riemann (n j,n j +1,u j,1,l,u j,1,r ) n j+1 x j +1,n f x j,1 n j edge j+1 edge j Each edge has associated left and right states, L and R The Riemann fluxes at the two points will be function of different left- and right states (depending on which edge they are associated with) But both fluxes are functions of both edge normals 48

49 2D Algorithm Use edge-based data structure Loop over edges, scatter flux contributions from edges to solution collocation nodes The weighting coefficients used in the scattering are differentiation matrix elements Loop over cells, add fluxes from interior nodes and update solution Fluxes from edges Fluxes from interior points F = F Riemann (n j,n j +1,u j,1,l,u j,1,r ) x j,1 F = F Riemann (n j,u j,2,l,u j,2,r ) F = F(u i,k ) F = F Riemann (n j-1,n j,u j,3,l,u j,3,r ) 49

50 Validation of Spectral Difference Method: Linear Advection Equation Maximum Error in Mesh Refinement Solution at T=0.3 u t + u x + 2u y = 0 u(x, y,0) = sin(2p(x + y)) 50

51 Burgers Equation - Discontinuous Case T=0.45 u t + uu x + uu y = 0 u(x, y,0) = sin(p(x + y)) Use Limiters to prevent oscillations But: Using Limiters destroys global accuracy 51

52 Treatment of Discontinuities: Represent Solution in Gibbs-Complementary Space Do not prevent oscillations, just stabilize by high-order diffusion or filtering. Try to retrieve information from spectral coefficients of oscillatory solution use Gibbs-complementary basis to expand spectral coefficients We seek the best basis to retrieve the information from whatever is available to us Example: Gegenbauer or Freud Polynomials for Fourier, Chebyshev Representations Re-project Solution onto new space, and you have a nonoscillatory, globally accurate, solution 52

53 Example: Gegenbauer Polynomials (Gottlieb et.al.) Rigorous theory available for resolution of Gibbs' phenomenon through expansion in Gegenbauer polynomials: Assume p exact spectral coefficients are given Recover exponential convergence for spectral representation of piecewise smooth functions as the number of spectral modes p is increased ( prefinement ) max a x b f - f p < e ap (a < 0, p Æ ) Gegenbauer polynomials form Gibbs-complementary basis, with respect to many standard basis sets, such as trigonometric basis functions and Chebyshev polynomials 53

54 Textbook Gegenbauer Projection (Gottlieb et.al.) Expand spectral representation of f in each smooth subdomain in Gegenbauer polynomials m f m,l l p (x) = Âg n C l n (x(x)) where g l n = 1 f l p (x(x))c(x)dx h n and f p (x) = p  k= 0 n= 0 c k f k (x(h)) where c k is obtained by collocation or Galerkin projection Ú For the Fourier Series the Gegenbauer coefficients can be evaluated by analytical integration Alternative: Freud Polynomials. Removes some of the numerical difficulties associated with Gegenbauer polynomials. Otherwise same approach 54

55 Gegenbauer/Freud Procedure and Numerical Schemes Only approximate spectral coefficients are available from numerical solution Theory incomplete for this case Furthermore: Choice of artificial diffusion in riemann fluxes could be critical Boundary conditions 55

56 Burgers Equation with Spectral Difference and Gegenbauer Re-Projection U(x,t=1) Pointwise Error in [0;1] for varying mesh size u t + uu x = 0, x(x,0) = sin(px) 5 th order Spectral Difference Method using Gauss-Lobatto points with Gegenbauer Procedure. Gegenbauer coefficients computed by quadrature or analytical integration after expansion in Fourier Series 56

57 Burgers Equation (cont d) Global Error Solution with and without Re-projection Recover nominal accuracy (5 th order) in maximum norm over a wide range of mesh refinement Slope on a log scale is not constant because of varying Gegenbauer parameters Error reduction beyond 10-8 not possible due to numerical difficulties when increasing order of Gegenbauer polynomial beyond N=25 57

58 Gibbs-Complementary Postprocessing II: Freud Polynomials Fourier Spectral Method: p-refinement Spectral Difference 5 th Order: h-refinement Expansion in Freud polynomials numerically more benign than using Gegenbauer polynomials 58

59 Euler Equations - Sod Shock tube Case with SD and Gegenbauer Re-Projection 5 th order Spectral Difference Method with Gegenbauer Procedure 59

60 Sod Case with and without Re-Projection Stabilize numerical solution by high order diffusion This allows oscillations, but retains accuracy in spectral coefficients Use re-projection to extract information 60

61 Numerical Experiments: Euler Equations - Sod Shocktube Case with BGK Flux computation 5 th order Spectral Difference Method with Gegenbauer Procedure using gas-kinetic flux at cell boundaries 61

62 Fast Time Integration Methods for Unsteady Problems Arathi Gopinath Matt McMullen Antony Jameson

63 Potential Applications Flutter Analysis, Flow past Helicopter blades, Rotor-Stator Combinations in Turbomachinery, Zero-Mass Synthetic Jets for Flow Control 63

64 Dual Time Stepping BDF The kth-order accurate backward difference formula (BDF) is of the form D t = 1 Dt k 1 Â (D - ) q q q=1 where D - w n +1 = w n +1 - w n The non-linear BDF is solved by inner iterations which advance in pseudo-time t* The second-order BDF solves dw dt + È 3w - 4wn + w n-1 + R(w) * Í = 0 Î 2Dt Implementation via RK dual time stepping scheme with variable local Dt * (RK-BDF) Nonlinear SGS dual time stepping scheme (SGS-BDF) with Multigrid 64

65 Test Case: NACA64A010 pitching airfoil (CT6 Case) Mach Number Pitching amplitude Reduced Freq. Reynolds Number /- 1.01deg million Cycling to limit cycle Pressure Contours at Various Time Instances (AGARD 702) Results of SGS-BDF Scheme (36 time steps per pitching cycle, 3 iterations per time step ) 65

66 Payoff of Dual-time Stepping BDF Schemes Accurate simulations with an order of magnitude reduction in time steps. For the pitching airfoil: from ~ 1000 to 36 time steps per pitching cycle with three sub-iterations in each step. 66

67 Frequency Domain and Global Space- Time Multigrid Spectral Methods Application : Time-periodic flows Using a Fourier representation in time, the time period T is divided into N steps. Then, ˆ w k = 1 N N-1 w n e -ikndt  n= 0 The discretization operator is given by D t w n = 2p T N 2-1  k= -N 2 ikw ˆ k e ikndt 67

68 Method 1 (McMullen et.al.) : Transform the equations into frequency domain and solve them in pseudo-time t* d ˆ w k dt * + 2p T ik w ˆ k + R ˆ k = 0 Method 2 (Gopinath et.al.) : Solve the equations in the time-domain. The space-time spectral discretization operator is N 2-1 D t w n = Â d m w n +m, d m = 2p T m=- N (-1)m +1 cot( pm N ),m 0 This is a central difference operator connecting all time levels, yielding an integrated space-time formulation which requires simultaneous solution of the equations at all time levels. 68

69 Comparison with Experimental Data - C L vs. a (CT6 Case) RANS Time-Spectral Solution with 4, 8 and 12 intervals per pitching cycle Computed Results Experimental Data 69

70 3D Test Case NLR LANN Pitching Wing - RANS Mach number Mean Alpha. Pitching Amplitude Pitching axis Red. Frequency Reynolds Number deg 0.25deg 62% RChord million Pressure Contours on the Wing C L vs. a plot with 4 and 8 time intervals 70

71 Application : Vertical-Axis Wind Turbine(VAWT) Objective : To maximize power output of the VAWT by turbine blade redesign. and various parametric studies. 71

72 VAWT : NACA0015 Airfoil Single-Blade Inviscid 3D model. Free-stream Mach Number Blade Tip-Speed / V_inf Turbine Radius / Blade Chord Coefficient of Power generated by the VAWT as a function of Rotation Angle - Time Spectral Method with 4,8 and 16 time intervals 72

73 Payoff of Time Spectral Schemes Engineering accuracy with very small number of time intervals and same rate of convergence as the BDF. Spectral accuracy for sufficiently smooth solutions. Periodic solutions directly without the need to evolve through 5-10 cycles, yielding an order of magnitude reduction in computing cost beyond the reduction already achieved with the BDF, for a total of two orders of magnitude. 73

74 Filtering the Navier-Stokes Equations with an Invertible Filter 74

75 Consider the incompressible Navier--Stokes equations where r u i t + ru j u i + r p = m 2 u i (1) x j x i x i x j u i x i = 0 In large eddy simulation (LES) the solution is filtered to remove the small scales. Typically one sets u i (x) = Ú G(x - x )u( x )dx (2) where the kernel G is concentrated in a band defined by the filter width. Then the filtered equations contain the extra virtual stress t ij = u i u j - ui u j (3) because the filtered value of a product is not equal to the product of the filtered values. This stress has to be modeled. 75

76 A filter which completely cuts off the small scales or the high frequency components is not invertible. The use, on the other hand, of an invertible filter would allow equation (1) to be directly expressed in terms of the filtered quantities. Thus one can identify desirable properties of a filter as 1. Attenuation of small scales 2. Commutativity with the differential operator 3. Invertibility Suppose the filter has the form which can be inverted as ui = Pu i (4) Qui = u i (5) where Q = P -1. Moreover Q should be coercieve, so that for some positive constant c. Qu > c u (6) 76

77 Note that if Q commutes with x i then so does Q -1, since for any quantity f which is sufficiently differentiable Also since Q commutes with (Q -1 f ) = Q -1 Q (Q -1 f ) x i x i x i, = Q -1 x i (QQ -1 f ) = Q -1 x i ( f ) u x i = 0 (7) As an example P can be the inverse Helmholtz operator, so that one can write Ê Qui = 1-a 2 2 ˆ Á ui = u i (8) Ë x k x k where a is a length scale proportional to the largest scales to be retained. One may also introduce a filtered pressure p, satisfying the equation Ê Qp = 1-a 2 2 ˆ Á p = p (9) Ë x k x k 77

78 Now one can substitute equation (8) and(9) for u i and p in equation (1) to get r Ê 2 ˆ Ê Á t 1-a2 ui + r 1-a 2 2 ˆ Á u j Ë x k x k Ë x k x k = m 2 Ê 1-a 2 2 ˆ Á x j x j Ë x k x k Ê 1-a 2 2 ˆ Á ui + Ê 1-a 2 2 ˆ Á p x j Ë x l x l x i Ë x k x k Because the order of the differentiations can be interchanged and the Helmholtz operator satisfies condition(6), it can be removed. The product term can be written as r Ï Ê 1-a 2 2 ˆ Ê Á ui 1-a 2 2 ˆ Ì Á u j x j Ó Ë x k x k Ë x l x l = r x j Ï Ì uiu j -a 2 ui Ó = r Ï 2 Ì uiu j -a 2 x j Ó x k x k = rq x j 2 u j x k x k -a 2 u j ui 2 ui x k x k + a 4 2 ui x k x k 2 u j x l x l ( ui u j ) + 2a 2 ui u j 2 + a 4 ui x k x k x k x k Ï Ê uiu j + a 2 Q -1 2 ui u j 2 + a 2 ui 2 u ˆ j Ì Á Ó Ë x k x k x k x k x l x l 2 u j x l x l According to condition (6), if Qf = 0 for any sufficiently differentiable quantity f, then f = 0. 78

79 Thus the filtered equation finally reduces to with the virtual stress r u i t + r x j ( ui u j ) + p = m 2 ui - r t ij (10) x i x k x k x j Ê t ij = a 2 Q -1 2 ui u j + a 2 2 ui 2 u ˆ j Á (11) Ë x k x k x k x k x l x l The virtual stress may be calculated by solving Ê 1-a 2 2 ˆ Ê Á t ij = a 2 2 ui u j + a 2 2 ui Á Ë x k x k Ë x k x k x k x k 2 u ˆ j (12) x l x l Taking the divergence of equation (10), it also follows that p satisfies the Poisson equation 2 p + r x i x i x i x j ui u j ( ) + r 2 x i x j t ij = 0 (13) 79

80 In a discrete solution scales smaller than the mesh width would not be resolved, amounting to an implicit cut off. There is the possibility of introducing an explicit cut off off in t ij. Also one could use equation (8) to restore an estimate of the unfiltered velocity. In order to avoid solving the Helmholtz equation (12), the inverse Helmholtz operator could be expanded formally as ( 1-a 2 D) -1 =1+ a 2 D + a 4 D where D denotes the Laplacian the approximate virtual stress tensor assumes the form 2 x k x k.now retaining terms up to the fourth power of a, t ij = 2a 2 ui x k u j x k È Ê + a 4 2D ui u j ˆ Í Á + DuiDu j (14) Î Ë x k x k One may regard the forms (11) or (14) as prototypes for subgrid scale (SGS) models. 80

81 The inverse Helmholtz operator cuts off the smaller scales quite gradually. One could design filters with a sharper cut off by shaping their frequency response. Denote the Fourier transform of f as where (in one space dimension) ˆ f (k) = f (x) = ˆ f = Ff 1 2p 1 2p Then the general form of an invertible filter is F Pf Ÿ f (x)e -ikx dx Ú - f ˆ (k)e ikx dk Ú - = S(k) ˆ f (k) Ÿ 1 F Qf = S(k) ˆ f (k) where S(k) should decrease rapidly beyond a cut off wave number inversely proportional to a length scale a. 81

82 In the case of a general filter with inverse Q, the virtual stress follows from the relation Then Qu i u j = u i u j = Qu i Qu j t ij = u i u j - u i u j = Q -1 (Qu i Qu j - Q(u i u j )) This formula provides the form for a family of subgrid-scale models. 82