Spectral Difference Solution of Unsteady Compressible Micropolar - PowerPoint PPT Presentation

Spectral Difference Solution of Unsteady Compressible Micropolar Equations on Moving and Deforming Grids Spectral Difference Solution of Unsteady Compressible Micropolar Equations on Moving and Deforming Grids Chunlei Liang 1 1 Assistant Professor, George Washington University Applied and Computational Mathematics Division seminar series at NIST in Gaithersburg, MD on Oct. 18th, 2011 1/33

Spectral Difference Solution of Unsteady Compressible Micropolar Equations on Moving and Deforming Grids Outline Motivation 1 Why spectral difference method? 2 Element-wise polynomial reconstruction High-order accuracy even with curved boundary Mathematical Formulation 3 Transform Navier-Stokes and Micropolar equations 4 Elements of the SD method 5 Verification 6 Flow past an oscillating cylinder 7 Flow around a heaving and pitching airfoil past an oscillating 8 cylinder Concluding remark 9 2/33

Spectral Difference Solution of Unsteady Compressible Micropolar Equations on Moving and Deforming Grids Motivation Sea turtle swimming Four flippers Front flippers for thrust generation. Back flipper for steering. 3/33

Spectral Difference Solution of Unsteady Compressible Micropolar Equations on Moving and Deforming Grids Motivation Oscillating wing wind- and hydro- power generator Hydrodynamically controlled wing Aerohydro Research and Technology Associates. 4/33

Spectral Difference Solution of Unsteady Compressible Micropolar Equations on Moving and Deforming Grids Motivation Plunge-Pitch airfoil for lift generation 5/33

Spectral Difference Solution of Unsteady Compressible Micropolar Equations on Moving and Deforming Grids Why spectral difference method? Element-wise polynomial reconstruction p-refinement No re-meshing 6/33

Spectral Difference Solution of Unsteady Compressible Micropolar Equations on Moving and Deforming Grids Why spectral difference method? Element-wise polynomial reconstruction p-refinement No re-meshing Poor boundary representation 6/33

Spectral Difference Solution of Unsteady Compressible Micropolar Equations on Moving and Deforming Grids Why spectral difference method? High-order accuracy even with curved boundary element mapping with high-order curved boundary ∂ ( x, y, t ) J = ∂ ( ξ, η, τ )  x ξ x η x τ   (1) = y ξ y η y τ  0 0 1 Key Universal reconstruction 7/33

Spectral Difference Solution of Unsteady Compressible Micropolar Equations on Moving and Deforming Grids Why spectral difference method? High-order accuracy even with curved boundary High-order scheme is attractive for vortex dominated flow 4th order SD 2nd order SD 8/33

Spectral Difference Solution of Unsteady Compressible Micropolar Equations on Moving and Deforming Grids Mathematical Formulation Compressible Navier-Stokes equations ∂ Q ∂t + ∇ F inv ( Q ) − ∇ F v ( Q , ∇ Q ) = 0 (2)       ρ ρu ρv       ρu 2 + p       ρu ρuv       Q = , f i = , g i = (3) ρv 2 + p ρv ρuv             E u ( E + p ) v ( E + p )        0   0          f v 2 u x + λ ( u x + v y ) , g v v x + u y     µ = µ = v x + u y 2 v y + λ ( u x + v y )     uf v [2] + vf v [3] + C p ug v [2] + vg v [3] + C p     P r T x P r T y     9/33

Spectral Difference Solution of Unsteady Compressible Micropolar Equations on Moving and Deforming Grids Mathematical Formulation Linear constitutive relation t ij = ( − p + λa mm ) δ ij + ( µ + κ ) a ij + µa ji (4) For Navier-Stokes equations, a ij = v j,i ; Micropolar formulation has two deformation tensors a ij = v j,i + e jik ω k ; b ij = ω i,j . The same linear relation for heat flux in both N-S and Micropolar formulations, i.e. Fourier’s Law: σ = ν Pr · gradT. (5) Pressure-Energy Relation: 2 ρ ( u 2 + v 2 ) for Navier-Stokes formulation; γ − 1 + 1 p E = 2 ρ ( u 2 + v 2 ) + 1 2 ρjω 2 for Micropolar formulation. γ − 1 + 1 p E = 10/33

Spectral Difference Solution of Unsteady Compressible Micropolar Equations on Moving and Deforming Grids Mathematical Formulation Micropolar formulation ∂ Q ∂t + ∇ F inv ( Q ) − ∇ F v ( Q , ∇ Q ) = S (6)       ρ ρu ρv    ρu 2 + p          ρu ρuv                   ρv 2 + p Q = ρv , f i = ρuv , g i = (7) ρjω ρjωu ρjωv                         E u ( E + p ) v ( E + p )         0     0         0 S = (8) � � ∂v y ∂x − ∂v x κ ∂y − 2 ω           0   11/33

Spectral Difference Solution of Unsteady Compressible Micropolar Equations on Moving and Deforming Grids Mathematical Formulation Viscous fluxes of Micropolar equations  0     (2 µ + κ ) u x + λ ( u x + v y )        µ ( v x + u y ) + κ ( v x − ω ) f v = (9) Γ ω x       uf v [2] + vf v [3] + ωf v [4] + µC p    P r T x   µ ( v x + u y ) + κ ( u y + ω )      (2 µ + κ ) v y + λ ( u x + v y )   g v ( Q , ∇ Q ) = (10) Γ ω y    ug v [2] + vg v [3] + ωg v [4] + µC p  P r T y   Ref: Chen, Lee, Liang (2011), JNFM; Chen, Liang, Lee (2011), JNN. 12/33

Spectral Difference Solution of Unsteady Compressible Micropolar Equations on Moving and Deforming Grids Transform Navier-Stokes and Micropolar equations Transform conservative equations ∂F/∂x = ∂F/∂ξ · ∂ξ/∂ x + ∂F/∂η · ∂η/∂ x + ∂F/∂τ · ∂τ/∂ x (11) ∂G/∂y = ∂G/∂ξ · ∂ξ/∂ y + ∂G/∂η · ∂η/∂ y + ∂G/∂τ · ∂τ/∂ y (12) ˜ Q = |J | · Q ˜       F ξ x ξ y ξ τ F ˜  = |J | G η x η y η τ G      ˜ 0 0 1 Q Q 13/33

Spectral Difference Solution of Unsteady Compressible Micropolar Equations on Moving and Deforming Grids Transform Navier-Stokes and Micropolar equations Transformed conservative equations + Geometric conservation law ∂ ˜ ∂τ + ∂ ˜ ∂ξ + ∂ ˜ Q F G ∂η = 0 (13) ∂ |J | + ∂ ( |J | ξ t ) + ∂ ( |J | η t ) = 0 (14) ∂τ ∂ξ ∂η Final set of equations � � �� ∂ ˜ ∂ξ + ∂ ˜ ∂Q 1 � ∂ ( |J | ξ t ) + ∂ ( |J | η t ) � F G = Q − (15) . ∂τ |J | ∂ξ ∂η ∂η A five-stage fourth-order Runge-Kutta method for time advancement. 14/33

Spectral Difference Solution of Unsteady Compressible Micropolar Equations on Moving and Deforming Grids Elements of the SD method Locating flux and solution points ∂ ˜ ∂τ + ∂ ˜ ∂ξ + ∂ ˜ Q F G ∂η = 0 Figure: Solution and flux points for a fourth-order SD scheme 15/33

Spectral Difference Solution of Unsteady Compressible Micropolar Equations on Moving and Deforming Grids Elements of the SD method Locating flux and solution points ∂ ˜ ∂τ + ∂ ˜ ∂ξ + ∂ ˜ Q F G ∂η = 0 solution points store ˜ Q , ξ flux points store ˜ F and η flux points store ˜ G . Figure: Solution and flux points for a fourth-order SD scheme 15/33

Spectral Difference Solution of Unsteady Compressible Micropolar Equations on Moving and Deforming Grids Elements of the SD method Locating flux and solution points ∂ ˜ ∂τ + ∂ ˜ ∂ξ + ∂ ˜ Q F G ∂η = 0 solution points store ˜ Q , ξ flux points store ˜ F and η flux points store ˜ G . 4 solution points in 1D Figure: Solution and flux points for a fourth-order SD scheme 15/33

Spectral Difference Solution of Unsteady Compressible Micropolar Equations on Moving and Deforming Grids Elements of the SD method Locating flux and solution points ∂ ˜ ∂τ + ∂ ˜ ∂ξ + ∂ ˜ Q F G ∂η = 0 solution points store ˜ Q , ξ flux points store ˜ F and η flux points store ˜ G . 4 solution points in 1D 5 flux points in 1D Figure: Solution and flux points for a fourth-order SD scheme 15/33

Spectral Difference Solution of Unsteady Compressible Micropolar Equations on Moving and Deforming Grids Elements of the SD method Locating flux and solution points ∂ ˜ ∂τ + ∂ ˜ ∂ξ + ∂ ˜ Q F G ∂η = 0 solution points store ˜ Q , ξ flux points store ˜ F and η flux points store ˜ G . 4 solution points in 1D 5 flux points in 1D The reconstructed field using polynomials is continuous within the cell but discontinuous across the cell interfaces. Figure: Solution and flux points for a fourth-order SD scheme 15/33

Spectral Difference Solution of Unsteady Compressible Micropolar Equations on Moving and Deforming Grids Elements of the SD method Compute interface fluxes Eigenvalues of ∂ F i /∂Q are V n − c , V n , and V n + c for N-S equations. 16/33

Spectral Difference Solution of Unsteady Compressible Micropolar Equations on Moving and Deforming Grids Elements of the SD method Compute interface fluxes Eigenvalues of ∂ F i /∂Q are V n − c , V n , and V n + c for N-S equations. Eigenvalues of ∂ F i /∂Q are V n − c , V n , V n and V n + c for Micropolar equations. 16/33