A SHOCK SENSOR BASED SECOND ORDER BLENDED (BX) UPWIND RESIDUAL DISTRIBUTION SCHEME FOR STEADY AND UNSTEADY COMPRESSIBLE FLOW Jiˇ r ´ ı Dobeˇ s Department of Technical Mathematics, Faculty of Mechanical Engineering, Czech Technical University, Prague, Czech Republic Herman Deconinck Von Karman Institute for Fluid Dynamics, Belgium Hyperbolic Problems, July 17–21, 2006 Lyon, France 1
Outline Introduction and motivation RDS intro (steady & unsteady) Linear schemes Non-linear schemes Bx scheme for steady problems Numerical examples Bx scheme for unsteady problems Numerical examples Conclusions Hyperbolic Problems, July 17–21, 2006 Lyon, France 2
The problem System of Euler equations in d spatial variables u t + ∇ · � f = 0 x, t ) : R d + 1 → R q u conserved set of variables, u ( � u h approximation of the solution � f vector of flux functions, � f ( u ) : R q → R q × d Ideal gas assumption Initial and boundary conditions Vector of Jacobian matrices λ = ∂ � f � ∂ u Hyperbolic Problems, July 17–21, 2006 Lyon, France 3
The problem System of Euler equations in d spatial variables u t + ∇ · � f = 0 x, t ) : R d + 1 → R q u conserved set of variables, u ( � u h approximation of the solution � f vector of flux functions, � f ( u ) : R q → R q × d Ideal gas assumption Initial and boundary conditions Vector of Jacobian matrices λ = ∂ � f � ∂ u Scalar advection equation in d spatial dimensions u t + � � λ ∈ R d λ · ∇ u = 0, Hyperbolic Problems, July 17–21, 2006 Lyon, France 3-a
RD schemes Introduction An alternative to the FV and FEM schemes Unstructured meshes, continuous representation of variables Very strong stability properties (in L ∞ ) Second order accurate on arbitrary meshes, compact stencil Do not rely on 1D physics (no Riemann solver) Hyperbolic Problems, July 17–21, 2006 Lyon, France 4
RD schemes Introduction An alternative to the FV and FEM schemes Unstructured meshes, continuous representation of variables Very strong stability properties (in L ∞ ) Second order accurate on arbitrary meshes, compact stencil Do not rely on 1D physics (no Riemann solver) RD schemes (steady version) Introduced almost 25 years ago by P. Roe E. van der Weide: PhD thesis (1998) Abgrall: JCP (2001), Abgrall & Mezine: JCP (2004) Hyperbolic Problems, July 17–21, 2006 Lyon, France 4-a
RD schemes Introduction An alternative to the FV and FEM schemes Unstructured meshes, continuous representation of variables Very strong stability properties (in L ∞ ) Second order accurate on arbitrary meshes, compact stencil Do not rely on 1D physics (no Riemann solver) RD schemes (steady version) Introduced almost 25 years ago by P. Roe E. van der Weide: PhD thesis (1998) Abgrall: JCP (2001), Abgrall & Mezine: JCP (2004) RD schemes (unsteady version) FEM formulation (mass matrix): Ferrante: VKI report (1997) Space-time approach 2001 (VKI, U. Bordeaux) Abgrall & Mezine: JCP (2003), Ricchiuto & Cs ´ ık & Deconinck: JCP (2005) Hyperbolic Problems, July 17–21, 2006 Lyon, France 4-b
RD schemes Introduction An alternative to the FV and FEM schemes Unstructured meshes, continuous representation of variables Very strong stability properties (in L ∞ ) Second order accurate on arbitrary meshes, compact stencil Do not rely on 1D physics (no Riemann solver) RD schemes (steady version) Introduced almost 25 years ago by P. Roe E. van der Weide: PhD thesis (1998) Abgrall: JCP (2001), Abgrall & Mezine: JCP (2004) RD schemes (unsteady version) FEM formulation (mass matrix): Ferrante: VKI report (1997) Space-time approach 2001 (VKI, U. Bordeaux) Abgrall & Mezine: JCP (2003), Ricchiuto & Cs ´ ık & Deconinck: JCP (2005) VKI CFD Lecture Series 2003, 2005 Hyperbolic Problems, July 17–21, 2006 Lyon, France 4-c
RD schemes Introduction u t + ∇ · � f = 0 Scheme for steady problem (u t → 0 ) Compute element residual � ϕ E ≈ ∇ · � f d x E φ Hyperbolic Problems, July 17–21, 2006 Lyon, France 5
RD schemes Introduction u t + ∇ · � f = 0 Scheme for steady problem (u t → 0 ) Compute element residual � ϕ E ≈ ∇ · � f d x E Distribute it to nodes with distribution φ φ coefficient (matrix) ϕ E i = β i ϕ E Hyperbolic Problems, July 17–21, 2006 Lyon, France 5-a
RD schemes Introduction u t + ∇ · � f = 0 Scheme for steady problem (u t → 0 ) Compute element residual � ϕ E ≈ ∇ · � f d x E Distribute it to nodes with distribution φ φ φ φ coefficient (matrix) φ ϕ E i = β i ϕ E φ φ φ Update nodal solution � u n + 1 = u n ϕ E i − α i i i E ∈ i The whole task is to define β i (or ϕ E i ). Hyperbolic Problems, July 17–21, 2006 Lyon, France 5-b
RD schemes Linear schemes Linear schemes: LDA scheme High accuracy (2nd order accurate) Enough dissipative even for some transonic flows This is not the case for unlimited FV method with linear reconstruction! N scheme Positive Quite accurate for 1st order schemes Very robust Hyperbolic Problems, July 17–21, 2006 Lyon, France 6
RD schemes Linear schemes Linear schemes: LDA scheme High accuracy (2nd order accurate) Enough dissipative even for some transonic flows This is not the case for unlimited FV method with linear reconstruction! N scheme Positive Quite accurate for 1st order schemes Very robust Amazing iterative convergence Very robust Unfortunately, cannot be positive and 2nd order accurate Hyperbolic Problems, July 17–21, 2006 Lyon, France 6-a
RD schemes Non-linear schemes N-modified scheme (PSI): Abgrall & Mezine: JCP (2004) 2nd order accurate for steady problems Positive for scalar problems Unsteady version avaliable B scheme: Cs ´ ık & Deconinck & Poedts: AIAA J. (2001) 2nd order accurate for steady problems Does not show oscillatory behavior Performs very similarly Hyperbolic Problems, July 17–21, 2006 Lyon, France 7
RD schemes Non-linear schemes N-modified scheme (PSI): Abgrall & Mezine: JCP (2004) 2nd order accurate for steady problems Positive for scalar problems Unsteady version avaliable B scheme: Cs ´ ık & Deconinck & Poedts: AIAA J. (2001) 2nd order accurate for steady problems Does not show oscillatory behavior Performs very similarly Persisting problems: Accuracy in smooth parts of the solution Poor iterative convergence High nonlinearity of the scheme (bad for implicit method) Hyperbolic Problems, July 17–21, 2006 Lyon, France 7-a
New Bx scheme (steady) Basic idea Use the LDA scheme everywhere except the shock waves Linear stability is sufficient in smooth parts of the flow Use the N scheme only in shock waves Hyperbolic Problems, July 17–21, 2006 Lyon, France 8
New Bx scheme (steady) Basic idea Use the LDA scheme everywhere except the shock waves Linear stability is sufficient in smooth parts of the flow Use the N scheme only in shock waves Blending coefficient (LDA 0 ≤ θ ≤ 1 N) Bx = θ · N + ( 1 − θ ) · LDA wish list: Order θ = O ( h ) in smooth parts of the solution Error of the N scheme will be multiplied by O ( h ) 2nd order of accuracy Very smooth Good convergence towards the steady state solution Implicit method (good Jacobian approximation) Hyperbolic Problems, July 17–21, 2006 Lyon, France 8-a
New Bx scheme (steady) Shock sensor Shock sensor � � � + � � + ∇ p · � � v 1 � sc = � ≈ v · ∇ p d x , δ pv δ pv µ ( E ) T Compression sc > 0 , expansion sc = 0 Smooth solution sc = O ( 1 ) , shock sc ≫ O ( 1 ) Hyperbolic Problems, July 17–21, 2006 Lyon, France 9
New Bx scheme (steady) Shock sensor Shock sensor � � � + � � + ∇ p · � � v 1 � sc = � ≈ v · ∇ p d x , δ pv δ pv µ ( E ) T Compression sc > 0 , expansion sc = 0 Smooth solution sc = O ( 1 ) , shock sc ≫ O ( 1 ) Blending coefficient θ = min ( 1, sc 2 · h ) Order θ = O ( h ) in smooth flow Hyperbolic Problems, July 17–21, 2006 Lyon, France 9-a
New Bx scheme (steady) Chanel Ma iso = 0.675 10 % circular bump chanel Ma iso = 0.675 Mesh h = 1/30 Mach number isolines 5 1 1.5 2 1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 0 0 0.5 0.5 1 1 1.5 1.5 2 2 2.5 2.5 3 3 Hyperbolic Problems, July 17–21, 2006 Lyon, France 10
New Bx scheme (steady) Chanel Ma iso = 0.675 0.95 1.45 0.9 Bx Bx N N 1.4 LDA LDA B B 0.85 N modif N modif FV Barth FV Barth 1.35 Mach Mach 0.8 1.3 0.75 1.25 0.7 1.2 0.65 1.7 1.75 1.8 1.5 1.6 1.7 1.8 x x Hyperbolic Problems, July 17–21, 2006 Lyon, France 11
New Bx scheme (steady) Chanel Ma iso = 0.675 0 -2 FV Barth -4 N modif Log(res) B Bx -6 -8 LDA N -10 -12 20 40 60 80 Iterations Hyperbolic Problems, July 17–21, 2006 Lyon, France 12
New Bx scheme (steady) Ma ∞ = 0.38 flow past a cylinder Sub-critical flow, Ma ∞ = 0.38 Mach number isolines ∆ Ma = 0.02 LDA Bx Hyperbolic Problems, July 17–21, 2006 Lyon, France 13
New Bx scheme (steady) Ma ∞ = 0.38 flow past a cylinder B N-modif Remark: FV2 + Barth – similar to the B scheme result Hyperbolic Problems, July 17–21, 2006 Lyon, France 14
New Bx scheme (steady) Mach 20 bow shock Mach 20 bow shock One can apply convergence fix (do not decrease blending coeff.) θ n i = max ( θ i , θ n − 1 ) i N-modified scheme Bx scheme Hyperbolic Problems, July 17–21, 2006 Lyon, France 15
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