CFD in Industrial Applications and a Mesh Improvement Shock-Filter for Multiple Discontinuities Capturing. Lakhdar Remaki

Transcription

1 CFD in Industrial Applications and a Mesh Improvement Shock-Filter for Multiple Discontinuities Capturing Lakhdar Remaki

2 Outline What we doing in CFD? CFD in Industry Shock-filter model for mesh adaptation Motivation The model Brief insight in the mathematical analysis Numerical tests Conclusions

3 What we doing in CFD? Navier-Stokes Equations t ρ + ( ρv ) = 0 Continuity equation t ( ρv ) + ( ρ v v ) + p = ρf + ( λ v ) + ( 2 µ D ) Momentum equation t ( ρe ) + [ ( ρe + p )v ] + p = ρf v + (κ θ ) + (λ ( v )v ) + (2 µdv ) Energy equation

4 Bloodhound SSC Project Constructing a vehicle to take the World Land Speed Record to 1000 mph

5 Bloodhound SSC Project Thrust SSC: 763mph Bloodhound SSC: 1000mph

6 Governing Equations d α α Qd V = F ( Q ) n d S G ( Q ) n ds α α dt V S S ρ ρu 1 Q = ρu 2 ρ u 3 E 2 u τ βα = µ k δ βα 3 xk ρu α ρu1uα + pδ α 1 Fα = ρu 2 uα + pδ α 2 ρu 3uα + pδ α 3 ( E + p ) u α u β uα + µ + x xβ α Stress tensor α = 1,2,3 0 τ 1α Gα = τ 2α τ 3α u βτ 3α qα qα = k T xβ Average heat flux

7 Equations Discretization Finite Volume Method (FLITE system) Cell vertex finite volume solution (Dual mesh) Time discretization: explicit multi-stage Runge Kutta Convergence acceleration to steady state by local time stepping and an agglomerated multigrid process.

8 Inviscid flux Approximation α F ( Q ) n ds α S ~ F IJ J Λ I ρqij 1 ρu1qij + pnij ~ FIJ = ρu2 qij + pnij2 ρu q + pn31 IJ 3 IJ ( E + p )qij Solve a Riemann problem ~ Q + FIJ (Q) = 0 t s QI if s < 0 Q (0, s ) = QJ if s > 0 qij = uα nαij

9 Numerical Results Transonic viscous flow over an ONERA M6 wing M= , AoA=3.06, Re=11.720e6, tetrahedral, 10 prismatic layers Transonic flow over ONRA M6 wing. Used mesh and Cp profile

10 Numerical Results Comparison to Experimental Results ONERA M6: Cp chordwise profile at 44% of wing semispan ONERA M6: Cp chordwise profile at 65% of wing semispan

11 Numerical Results Comparison to Experimental Results ONERA M6: Cp chordwise profile at 80% of wing semispan ONERA M6: Cp chordwise profile at 90% of wing semispan

12 Numerical Results Supersonic inviscid flow over an F15 fighter M=2.0, AoA=3.0, tetrahedral

13 Numerical Results Residual curve

14 Numerical Results M=3.0, AoA=3.0 Cp Profile Residual curve

15 Spay Drag Model for Bloodhound SSC Supersonic Vehicle: Motivation Pressure Shocks around the vehicle

16 Governing Equations Fluid phase (Navier stokes-equations) tφ f ρ f + (φ f ρ f U f ) = 0 on Ω [ 0.T ] t (φ f ρ f U f ) + (φ f ρ f U f U f ) = φ f Pf +.τ β (U p U f ) + φ f ρ f g Solid phase tφ p + (φ pu p ) = 0 on Ω [ 0.T ] t (φ pu p ) + (φ pu p U p ) = φp +φ f = 1 φp ρf Pf + β (U p U f ) + φ p (1 ) g ρp ρp φpρ f U p U f φ p2 µ f φ f D p2 Dp β = 3 φ p ρ f U p U f 2.65 φp 4 Cd D p R ( Re p ) if Re p < 1000 Cd = e p Drag coefficient 0.4 else Re p = if φ f < 0.8 if φ f 0.8 D pφ U p U f f νf Particle Reynolds number g Gravity

17 Validation Experimental Apparatus: (a) General Flow System, (b) Geometry of the curved square duct Y. Kliafas, M. Holt, LDV measurements of a turbulent airsolid twophase fow in a 90 bend. Experiments in Fluids 5, 7385.

18 Validation Hybrid used mesh and Volume Fraction profile for the curved duct

19 Validation Residual convergence

20 Validation Station θ=0 Station θ=15 Station θ=45 Station θ=30 90 Bend case: Mean Stream fluid and particles velocity comparison to experimental results for different stations

21 Application to Bloodhound SSC Bloodhound SSC Supersonic Car: Hybrid mesh The delimited area using normal velocity gradient criterion

22 Application to Bloodhound SSC Volume fraction profile

23 Application to Bloodhound SSC Residual convergence (a) Drag convergence before and after injecting sand particles: (a) Volume Fraction= 1.5e-3, Drag increases by 5%. (b) Volume Fraction = 5e-3, Drag increases by 10%. (b)

24 Bloodhound SSC Project Aerodynamic Optimization Initial Geometry Final Geometry

25 Tricycle recreational vehicle (CTA-BRPUDES)

26 CAD Mesh

27 CAD Mesh

28 CFD Model CAD/CATIA Meshing/CCM + CAD Fix Flow simulation/ccm+ Suitable mesh parameters Optimizatio n Validation/Experimental 28

29 Velocity profile 29

30 Objective: Minimizing Drag by Fender Shape modification Drag drop 26.15% Drag drop13.18%

31 Shock-filter model for mesh adaptation The CFD solutions are mesh dependant Poor mesh quality leads to a non-accurate CFD solution Generating good mesh quality is a difficult task Solution: Automatic mesh adaptation

32 Example: supersonic flow around a NACA0012

33 Process of an Anisotropic SolutionMesh Adaptation OptiGrid Initial grid Solver OptiGrid Solver OptiGrid Solver Final grid Solver

34 Mesh adaptation overview, 1 Adaptation principle: Objective: Estimation Minimize the computational error u u h u uh X < C1 u rh u X over the domain X Interpolation error For linear interpolation we have: u rh u X < C 2 h 2 D 2u h X Procedure: Build a new mesh from the previous one by Equidistributing the interpolation error bound h 2 D 2u h ( x ) of a selected scalar variable

35 Mesh adaptation overview, 3 To obtain an anisotropic mesh we try to equi-distribute the error along the edges The v second derivative along an edge 1 D u ( x) = 2 < Hv, v > v2 2 v h Derive The M = H = P 1 D P a metric error along and edge v 1 T ε (v ) = v M ( s )v ds 0 H Hessian

36 Mesh adaptation overview, 4 Optimization operations include: nodes movement, refinement by cutting edges with large error and coarsening by concatenating edges with small errors

37 Multiple shocks problem Problem: the aim of mesh adaptation is to start with an arbitrary and coarse initial mesh, taking less time and effort to generate. However, when the initial grid is too coarse discontinuities are badly defined, especially in the case of multiple shocks, whatever the solver used. The proposed solution: Enhance the discontinuities of the selected scalar variable before adaptation. How? Using a shock filter.

38 Shock-filter model, 1 Rudin filter : Nonlinear hyperbolic equation ut + F (u xx ) u x = 0 ( x, t ) (Ω IR) IR + u (0, t ) = u0 ( x) Initial condition sf ( s ) > 0 s Ω Steady state

39 Shock-filter model, 2 The proposed shock-filter ut + a1 ( x, y ) F1 ( u, u x ) x f1 (u ) + a2 ( x, y ) F2 ( u, u y ) y f 2 (u ) = 0 ( S ) (t, x, y ) IR Ω u (0, x, y ) = u0, srf1 ( s, r ) > 0 srf2 ( s, r ) > 0 ( s, r ) Ω F1, F2, f1, f 2 C a1, a2 L (Ω)

40 Mathematical Analysis Existence and uniqueness of a solution of equation (S) in a suitable space Classical frame of Distribution space Obstacle: The product of two distribution is not defined in general (Schwartz theorem) Generalized functions framework (Colombeau algebra): More suitable

41 Insight into the generalized functions algebra, 1 Intuitive idea Microscopic aspect ε u U ε Macroscopic aspect { } U = Calss u ε, 0 < ε < 1 Generalized function

42 Insight into generalized functions algebra, 2 Construction of the algebra The reservoir of representatives : c Es (Ω) = R C (] 0,1] Ω, IR) / D, n IN, c > 0, x Ω, ( DR)(ε, x) n ε The ideal I s (Ω) = { R C (] 0,1] Ω, IR) / D, q IN, cq > 0 / x Ω, ( DR)(ε, x) cqε q } Definition of the Algebra Gs (Ω ) = E s (Ω) I s (Ω )

43 Insight into generalized functions algebra, 3 Some definitions and properties g1, g 2 Gs (Ω), R1, R 2 Es (Ω) / g1 = R1 et g 2 = R 2 The association operation g1, g 2 are said to be associated ( g1 g 2 ) if and only if for two representatives R1, R2 [ ( R (ε, x) R (ε, x))ϕ ( x)] dx 0 quand ε 0 Ω 1 2 D(Ω) The space of ϕ D(Ω) C function with a compact support

44 Insight into generalized functions algebra, 4 Some definitions and properties Product of two generalized functions g1 g 2 = R1 R 2 Derivatives of a generalized function g = R (ε,.)

45 Insight into generalized functions algebra, 5 Some definitions and properties Regularized derivatives g = ( R(ε,.)) ρ h (ε ) ρ D(Ω) or step functions / ρdx = 1, ρ h (ε ) = 1 ρ ( x / h(ε )) n h (ε ) h : ] 0,1] ] 0,1] / h(ε ) 0 qd ε 0

46 Insight into generalized functions algebra, 6 Some definitions and properties Injective mapping L (Ω) Gs (Ω) f L (Ω) f ρε ( x) Es (Ω) How distributions are related to generalized functions T D' (Ω), g Gs (Ω) / g T J.F. Colombeau, Multiplication of Distribution, Lecture Notes in Mathematics, Vol. 1532, Springer-Verlag, 1992.

47 An existence and uniqueness result of the proposed PDE within the generalized functions algebra, 1 A rough Idea of the proof Consider the 1D case ut + a ( x) F (u xx, u x ) x f (u ) = 0 ( S ) (t, x) IR Ω u (0, x) = u0, Interpretation of (S) in u0ε = u0 ρε a ε = a ρε Gs (Ω) U 0 = u0ε, A = a ε U t + A( x) F ( xxu, xu ) x f (U ) 0 ( SG ) (t, x) IR Ω U (0, x) = U 0,

48 An existence and uniqueness result of the proposed PDE within the generalized functions algebra, 2 Proving the existence of a generalized function by proving the existence of a representative u ε t + a ε ( x) F ( xx u ε, x u ε ) x f (u ε ) = 0 ( S E ) ε u (0, x) = u0ε, (t, x) IR Ω Show that exists and ε Show the uniqueness u U1 and U 2 and then u ε Es are two solutions of ODE equation U = uε ( S G ), then U1 U 2

49 An existence and uniqueness result of the proposed PDE within the generalized functions algebra, 3 Theorem [1,2]: Under the assumptions: u L ( Ω ) L (Ω ) 0 1 a L (Ω) f, G C (Ω) L (Ω) Equation (Sg) admits a unique solution in Gs ( IR + Ω) Proposition [3]: The shock-filter equation (S) admits a steady-state u ε L (Ω)

50 Numerical Analysis of the PDE, 1 Results Finite difference scheme u i +1ε, n 2u i ε, n + u i 1ε, n ε, n +1 n h2 ui =u i rmax 0, a i F ε, n ε,n u u i 1 i h u i +1ε, n 2u i ε, n + u i 1ε, n 2 h rmin 0, a i F ε,n ε,n u u i i + 1 h ui ε,0 = u ε,0 (ih), ε,n ε,n ε,n f ' ( u ) u u i 1 i i (, ε,n ε,n ε,n f ' ( u ) u u i i i +1 ( ) )

51 Numerical Analysis of the PDE, 1 Results Proposition : Under the assumptions a BV (Ω) L (Ω) 1 2 The previous scheme is stable for the norm L, for the total variation in space and for the total variation in time in Tonnelli-Cesari sense (i ) u iε,n u 0 L ( ℜ ), i Z, n IN ε,n ε,n (ii ) u i +1 u i TV (u 0 ), n IN i Z (iii ) u ε,n +1 u ε,n TV (u ), n IN i i 0 i Z CFL condition r aff ' <

52 References [1] L. Remaki and M. Cheriet, Enhanced and Restored Signal as a Generalized Solution for Shock Filter Models: Part-I: Existence and Uniqueness result of the Cauchy Problem, Journal of Mathematical Analysis And Applications. Vol. 279, No 1, pp , [2] L. Remaki and M. Cheriet, Enhanced and Restored Signal as a Generalized Solution for Shock Filter Models: Part-II: Numerical Study, Journal of Mathematical Analysis And Applications. Vol. 279, No 2, pp , [3] L. Remaki and W.G. Habashi 3D-mesh adaptation on multiple weak shocks and boundary layers, SIAM J. Sci. Comupt. Vol. 28, No. 4, pp

53 Behaviour of the Shock-filter : 1D case

54 Application to Mesh Adaptation, 1 The scalar solution of adaptation The steady-state of the shock filter equation is a piecewise constant function. Initial solution Steady state Solution of adaptation = Average solution ((original + steady state)/2)

55 Application to Mesh Adaptation, 2 Shock-filter effect on a transonic profile. Transonic Flow around NACA0012: Mach profile The shock filter effect

56 Numerical tests: Inviscid supersonic flow on a ramp: AoA=0, Mach =2 Initial grid Mach profil

57 Numerical tests: Inviscid supersonic flow on a ramp: AoA=0, Mach =2 Grid after 2 cycles of adaptation without pre-processing Grid after 2 cycles of adaptation with preprocessing Mach profile Mach profile

58 Numerical tests: Inviscid supersonic flow on a ramp: AoA=0, Mach =2 Corresponding residuals curves

59 Impact of the Shock filter pre-processing Inviscid test case: NACA 0012, AoA=1, Mach = 0.85 Original grid Cp profile

60 Adaptation without shock-filter preprocessing. The different cycles and corresponding grids The different adapted grids

61 Adaptation without shock-filter preprocessing. The different cycles and corresponding Cp profiles, 1 Different Cp profiles

62 Adaptation without shock-filter preprocessing. The different cycles and corresponding Cp profiles, 2 The chordwise pressure distributions

63 Adaptation with shock-filter pre-processing. The different cycles and corresponding grids and Cp profiles, 1 Different grids and corresponding Cp profiles

64 Adaptation with shock-filter pre-processing. The different cycles and corresponding grids and Cp profiles, 2 The chordwise pressure distributions

65 Inviscid test case: NACA 0012, AoA=1.25, Mach = 0.8 Initial grid Cp profile

66 Adaptation without shock-filter preprocessing. The different cycles and corresponding grids Figure 10-a: Different adapted grids Different adapted grids

67 Adaptation without shock-filter preprocessing. The different cycles and corresponding Cp profiles, 1 Different Cp profiles

68 Adaptation without shock-filter preprocessing. The different cycles and corresponding Cp profiles, 2 The chordwise pressure distributions

69 Adaptation with shock-filter pre-processing. The different cycles and corresponding grids and Cp profiles