Spectral Difference Solution of Unsteady Compressible Micropolar - - PowerPoint PPT Presentation

▶
spectral difference solution of unsteady compressible
SMART_READER_LITE
LIVE PREVIEW

Spectral Difference Solution of Unsteady Compressible Micropolar - - PowerPoint PPT Presentation

Dec 18, 2023 40 likes •507 views

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

slide-1
SLIDE 1

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

1Assistant Professor, George Washington University

Applied and Computational Mathematics Division seminar series at NIST in Gaithersburg, MD on Oct. 18th, 2011

1/33

slide-2
SLIDE 2

Spectral Difference Solution of Unsteady Compressible Micropolar Equations on Moving and Deforming Grids Outline

1

Motivation

2

Why spectral difference method? Element-wise polynomial reconstruction High-order accuracy even with curved boundary

3

Mathematical Formulation

4

Transform Navier-Stokes and Micropolar equations

5

Elements of the SD method

6

Verification

7

Flow past an oscillating cylinder

8

Flow around a heaving and pitching airfoil past an oscillating cylinder

9

Concluding remark

2/33

slide-3
SLIDE 3

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

slide-4
SLIDE 4

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

slide-5
SLIDE 5

Spectral Difference Solution of Unsteady Compressible Micropolar Equations on Moving and Deforming Grids Motivation

Plunge-Pitch airfoil for lift generation

5/33

slide-6
SLIDE 6

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

slide-7
SLIDE 7

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

slide-8
SLIDE 8

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

J = ∂(x, y, t) ∂(ξ, η, τ) =   xξ xη xτ yξ yη yτ 1   (1) Key Universal reconstruction

7/33

slide-9
SLIDE 9

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

2nd order SD 4th order SD

8/33

slide-10
SLIDE 10

Spectral Difference Solution of Unsteady Compressible Micropolar Equations on Moving and Deforming Grids Mathematical Formulation

Compressible Navier-Stokes equations

∂Q ∂t + ∇Finv(Q) − ∇Fv(Q, ∇Q) = 0 (2) Q =        ρ ρu ρv E        , fi =        ρu ρu2 + p ρuv u(E + p)        , gi =        ρv ρuv ρv2 + p v(E + p)        (3) fv µ =        2ux + λ(ux + vy) vx + uy ufv[2] + vfv[3] + Cp

Pr Tx

       , gv µ =        vx + uy 2vy + λ(ux + vy) ugv[2] + vgv[3] + Cp

Pr Ty

      

9/33

slide-11
SLIDE 11

Spectral Difference Solution of Unsteady Compressible Micropolar Equations on Moving and Deforming Grids Mathematical Formulation

Linear constitutive relation

tij = (−p + λamm)δij + (µ + κ)aij + µaji (4) For Navier-Stokes equations, aij = vj,i; Micropolar formulation has two deformation tensors aij = vj,i + ejikωk; bij = ωi,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: E =

p γ−1 + 1 2ρ(u2 + v2) for Navier-Stokes formulation;

E =

p γ−1 + 1 2ρ(u2 + v2) + 1 2ρjω2 for Micropolar formulation.

10/33

slide-12
SLIDE 12

Spectral Difference Solution of Unsteady Compressible Micropolar Equations on Moving and Deforming Grids Mathematical Formulation

Micropolar formulation

∂Q ∂t + ∇Finv(Q) − ∇Fv(Q, ∇Q) = S (6) Q =            ρ ρu ρv ρjω E            , fi =            ρu ρu2 + p ρuv ρjωu u(E + p)            , gi =            ρv ρuv ρv2 + p ρjωv v(E + p)            (7) S =              κ

  • ∂vy

∂x − ∂vx ∂y − 2ω

            (8)

11/33

slide-13
SLIDE 13

Spectral Difference Solution of Unsteady Compressible Micropolar Equations on Moving and Deforming Grids Mathematical Formulation

Viscous fluxes of Micropolar equations

fv =            (2µ + κ)ux + λ(ux + vy) µ(vx + uy) + κ(vx − ω) Γωx ufv[2] + vfv[3] + ωfv[4] + µCp

Pr Tx

           (9) gv(Q, ∇Q) =        µ(vx + uy) + κ(uy + ω) (2µ + κ)vy + λ(ux + vy) Γωy ugv[2] + vgv[3] + ωgv[4] + µCp

Pr Ty

       (10) Ref: Chen, Lee, Liang (2011), JNFM; Chen, Liang, Lee (2011), JNN.

12/33

slide-14
SLIDE 14

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 ˜ G ˜ Q   = |J |   ξx ξy ξτ ηx ηy ητ 1     F G Q  

13/33

slide-15
SLIDE 15

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

∂ ˜ Q ∂τ + ∂ ˜ F ∂ξ + ∂ ˜ G ∂η = 0 (13) ∂|J | ∂τ + ∂(|J |ξt) ∂ξ + ∂(|J |ηt) ∂η = 0 (14) Final set of equations ∂Q ∂τ = 1 |J |

  • Q

∂(|J |ξt) ∂ξ + ∂(|J |ηt) ∂η

  • −
  • ∂ ˜

F ∂ξ + ∂ ˜ G ∂η

  • .

(15) A five-stage fourth-order Runge-Kutta method for time advancement.

14/33

slide-16
SLIDE 16

Spectral Difference Solution of Unsteady Compressible Micropolar Equations on Moving and Deforming Grids Elements of the SD method

Locating flux and solution points

∂ ˜ Q ∂τ + ∂ ˜ F ∂ξ + ∂ ˜ G ∂η = 0

Figure: Solution and flux points for a fourth-order SD scheme

15/33

slide-17
SLIDE 17

Spectral Difference Solution of Unsteady Compressible Micropolar Equations on Moving and Deforming Grids Elements of the SD method

Locating flux and solution points

∂ ˜ Q ∂τ + ∂ ˜ F ∂ξ + ∂ ˜ 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

slide-18
SLIDE 18

Spectral Difference Solution of Unsteady Compressible Micropolar Equations on Moving and Deforming Grids Elements of the SD method

Locating flux and solution points

∂ ˜ Q ∂τ + ∂ ˜ F ∂ξ + ∂ ˜ 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

slide-19
SLIDE 19

Spectral Difference Solution of Unsteady Compressible Micropolar Equations on Moving and Deforming Grids Elements of the SD method

Locating flux and solution points

∂ ˜ Q ∂τ + ∂ ˜ F ∂ξ + ∂ ˜ 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

slide-20
SLIDE 20

Spectral Difference Solution of Unsteady Compressible Micropolar Equations on Moving and Deforming Grids Elements of the SD method

Locating flux and solution points

∂ ˜ Q ∂τ + ∂ ˜ F ∂ξ + ∂ ˜ 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

slide-21
SLIDE 21

Spectral Difference Solution of Unsteady Compressible Micropolar Equations on Moving and Deforming Grids Elements of the SD method

Compute interface fluxes

Eigenvalues of ∂Fi/∂Q are Vn − c, Vn, and Vn + c for N-S equations.

16/33

slide-22
SLIDE 22

Spectral Difference Solution of Unsteady Compressible Micropolar Equations on Moving and Deforming Grids Elements of the SD method

Compute interface fluxes

Eigenvalues of ∂Fi/∂Q are Vn − c, Vn, and Vn + c for N-S equations. Eigenvalues of ∂Fi/∂Q are Vn − c, Vn, Vn and Vn + c for Micropolar equations.

16/33

slide-23
SLIDE 23

Spectral Difference Solution of Unsteady Compressible Micropolar Equations on Moving and Deforming Grids Elements of the SD method

Compute interface fluxes

Eigenvalues of ∂Fi/∂Q are Vn − c, Vn, and Vn + c for N-S equations. Eigenvalues of ∂Fi/∂Q are Vn − c, Vn, Vn and Vn + c for Micropolar equations. Rusanov flux ˆ Finv = 1

2

  • (FL

i + FR i ) · nf − |Vn + c| ·

  • QR − QL

16/33

slide-24
SLIDE 24

Spectral Difference Solution of Unsteady Compressible Micropolar Equations on Moving and Deforming Grids Elements of the SD method

Compute interface fluxes

Eigenvalues of ∂Fi/∂Q are Vn − c, Vn, and Vn + c for N-S equations. Eigenvalues of ∂Fi/∂Q are Vn − c, Vn, Vn and Vn + c for Micropolar equations. Rusanov flux ˆ Finv = 1

2

  • (FL

i + FR i ) · nf − |Vn + c| ·

  • QR − QL

Viscous interface flux – using averaging approach.

16/33

slide-25
SLIDE 25

Spectral Difference Solution of Unsteady Compressible Micropolar Equations on Moving and Deforming Grids Elements of the SD method

Locating solution and flux points

N solution points (One can actually position arbitrarily) Chebyshev-Gauss points N+1 flux points Legendre-Gauss quadrature points plus two end points of 0 and 1. key difference from Kopriva Pn(ξ) = 2n − 1 n (2ξ − 1)Pn−1(ξ) − n − 1 n Pn−2(ξ) (16) where n = 1, . . . , N − 1, P−1(ξ) = 0 and P0(ξ) = 1 Ref: H. T. Huynh, AIAA paper, 2007-4079, Van den Abeele, Lacor, Wang, JSC, 2008,

  • A. Jameson, JSC, 2010.

17/33

slide-26
SLIDE 26

Spectral Difference Solution of Unsteady Compressible Micropolar Equations on Moving and Deforming Grids Verification

Verification study for order of accuracy on unstructured grids

  • No. of cells

DOFs L2-error Order L1-error Order 3rd order SD 2 18 8.247E-4

  • 7.376E-4
  • 8

72 1.501E-4 2.46 1.244E-4 2.57 32 288 1.865E-5 2.99 1.675E-5 2.89 4th order SD 2 32 2.531E-4

  • 1.93E-4
  • 8

128 2.19E-5 3.53 1.927E-5 3.55 32 512 1.825E-6 3.585 1.641E-6 3.32

Table: L2 and L1 errors and orders of accuracy for planar Couette flow

18/33

slide-27
SLIDE 27

Spectral Difference Solution of Unsteady Compressible Micropolar Equations on Moving and Deforming Grids Flow past an oscillating cylinder

Computational conditions

Re = ρU∞D

µ+κ = 185

Freestream Mach number = 0.2 j = 1e-6 Γ = 1e − 8 Oscillation amplitude Ay/D = 0.2, Reduced frequency fD/U∞ = 0.2145.

19/33

slide-28
SLIDE 28

Spectral Difference Solution of Unsteady Compressible Micropolar Equations on Moving and Deforming Grids Flow past an oscillating cylinder

Solution of Navier-Stokes equations

20/33

slide-29
SLIDE 29

Spectral Difference Solution of Unsteady Compressible Micropolar Equations on Moving and Deforming Grids Flow past an oscillating cylinder

Solution of Micropolar equations (µ/κ = 0.544)

21/33

slide-30
SLIDE 30

Spectral Difference Solution of Unsteady Compressible Micropolar Equations on Moving and Deforming Grids Flow past an oscillating cylinder

Micropolar effect on lift coefficient

µ/κ = 0.544 case v.s. Navier-Stokes

Only one shedding frequency is obtained from Micropolar solution!

22/33

slide-31
SLIDE 31

Spectral Difference Solution of Unsteady Compressible Micropolar Equations on Moving and Deforming Grids Flow past an oscillating cylinder

Solution of Micropolar equations (µ/κ = 2.6)

23/33

slide-32
SLIDE 32

Spectral Difference Solution of Unsteady Compressible Micropolar Equations on Moving and Deforming Grids Flow around a heaving and pitching airfoil past an oscillating cylinder

Computational condition for Case I

Re = ρU∞D

µ+κ = 500,

Mach = 0.2, µ = 8e − 4, κ = 1e − 3, j = 1e-6, Γ = 1e − 8, Oscillation amplitude Ay/D = 0.25, Reduced frequency fD/U∞ = 0.1.

24/33

slide-33
SLIDE 33

Spectral Difference Solution of Unsteady Compressible Micropolar Equations on Moving and Deforming Grids Flow around a heaving and pitching airfoil past an oscillating cylinder

Vorticity contour

25/33

slide-34
SLIDE 34

Spectral Difference Solution of Unsteady Compressible Micropolar Equations on Moving and Deforming Grids Flow around a heaving and pitching airfoil past an oscillating cylinder

Computational condition for Case II

Re = ρU∞D

µ+κ = 500,

Mach = 0.2, µ = 8e − 4, κ = 1e − 3, j = 1e-6, Γ = 1e − 8, Oscillation amplitude Ay/D = 0.25, Reduced frequency fD/U∞ = 0.5.

26/33

slide-35
SLIDE 35

Spectral Difference Solution of Unsteady Compressible Micropolar Equations on Moving and Deforming Grids Flow around a heaving and pitching airfoil past an oscillating cylinder

Vorticity contour from Navier-Stokes solution

27/33

slide-36
SLIDE 36

Spectral Difference Solution of Unsteady Compressible Micropolar Equations on Moving and Deforming Grids Flow around a heaving and pitching airfoil past an oscillating cylinder

Vorticity contour from Micropolar solution (µ/κ = 0.54)

28/33

slide-37
SLIDE 37

Spectral Difference Solution of Unsteady Compressible Micropolar Equations on Moving and Deforming Grids Flow around a heaving and pitching airfoil past an oscillating cylinder

Gyration contour

29/33

slide-38
SLIDE 38

Spectral Difference Solution of Unsteady Compressible Micropolar Equations on Moving and Deforming Grids Concluding remark

Concluding remarks

We introduced a new formulation for compressible flow different from Navier-Stokes equations.

30/33

slide-39
SLIDE 39

Spectral Difference Solution of Unsteady Compressible Micropolar Equations on Moving and Deforming Grids Concluding remark

Concluding remarks

We introduced a new formulation for compressible flow different from Navier-Stokes equations. SD method is successfully formulated and implemented for unsteady Micropolar flow.

30/33

slide-40
SLIDE 40

Spectral Difference Solution of Unsteady Compressible Micropolar Equations on Moving and Deforming Grids Concluding remark

Concluding remarks

We introduced a new formulation for compressible flow different from Navier-Stokes equations. SD method is successfully formulated and implemented for unsteady Micropolar flow. Optimal order of accuracy is obtained.

30/33

slide-41
SLIDE 41

Spectral Difference Solution of Unsteady Compressible Micropolar Equations on Moving and Deforming Grids Concluding remark

Concluding remarks

We introduced a new formulation for compressible flow different from Navier-Stokes equations. SD method is successfully formulated and implemented for unsteady Micropolar flow. Optimal order of accuracy is obtained. Extension is successfully made to moving and deformable grids.

30/33

slide-42
SLIDE 42

Spectral Difference Solution of Unsteady Compressible Micropolar Equations on Moving and Deforming Grids Concluding remark

Acknowledgement

George Washington University Faculty Startup Fund

31/33

slide-43
SLIDE 43

Spectral Difference Solution of Unsteady Compressible Micropolar Equations on Moving and Deforming Grids Concluding remark

Acknowledgement

George Washington University Faculty Startup Fund GW Graduate Fellowship to James Chen

31/33

slide-44
SLIDE 44

Spectral Difference Solution of Unsteady Compressible Micropolar Equations on Moving and Deforming Grids Concluding remark

Acknowledgement

George Washington University Faculty Startup Fund GW Graduate Fellowship to James Chen Dr James Chen was co-advised by Prof. James D. Lee and Chunlei Liang

31/33

slide-45
SLIDE 45

Spectral Difference Solution of Unsteady Compressible Micropolar Equations on Moving and Deforming Grids Concluding remark

Mesh for a Plunge-Pitch Airfoil

32/33

slide-46
SLIDE 46

Spectral Difference Solution of Unsteady Compressible Micropolar Equations on Moving and Deforming Grids Concluding remark

Publications of the SD method

Liu, Vinokur, Wang, J. Comput. Physics, 2006; Wave Equations. Wang, Liu, May, Jameson, J. Scientific Computing, 2007; Euler Equations. Liang, Jameson, Wang, J. Comput. Physics, 2009; N-S equations. Chen, Liang, Lee, Computers & Fluids, 2011. Micropolar Equations. Multidomain staggered spectral method

Kopriva, J. Comput. Physics, 1998.

33/33