High-Reynolds Number Flow Simulations on Embedded …

High-Reynolds Number Flow Simulations on Embedded-Boundary Cartesian Grids

in collaboration with Michael Aftosmis,

NASA Ames

Marsha BergerNew York University

Monday, July 27, 15

Talk Outline

• Background

• Cart3D - automated cut cell method for inviscid flows in complicated geometries

• Extension to viscous turbulent (SA model) flow

• Sub-grid wall functions / wall models

• Diffusion model

• Coupling of model and Cartesian grid

• Wall function and results

• BVP Model and results

• Conclusions

Monday, July 27, 15

• Cartesian cut cell method generates grids around arbitrarily complex configurations described by surfaces triangulations with cut cells at wall(mesh described by face areas, normals, centroids)

• 2nd order finite volume method for compressible Euler eq. with cut cell algs. for gradient & limiting

• steady flows: multigrid with RK time-stepping and automatically generated coarse grids.unsteady flows: BDF2 and leverage the above

• highly scalable on-the-fly partitioning using SFC for any number of processors

• adjoint-based error estimation and refinement (Nemec & Aftosmis)

Cart3D for Inviscid Flow

M!= 0.5 M!= 1.1 M!= 1.3

" = -25°

" = -20°

" = -12°

" = -8°

" = -4°

" = -2°

" = 0°

" = 2°

" = 4°

" = 6°

Monday, July 27, 15

Baseline Viscous Finite Volume Solver

M! = 0.5

ReL = 5000

NACA 0012Leading edge

Trailing edge

(a) (b)

0 0.2 0.4 0.6 0.8 1U/U

!

0

2

4

6

8

10

12

"

Rex = 5000 (~13 cells)Rex = 10000 (~17 cells)Rex = 50000 (~40 cells)

0 0.2 0.4 0.6 0.8v/U

! sqrt(Rex)

0

2

4

6

8

10

"

Rex = 5000 (~13 cells)Rex = 10000 (~ 17 cells)Rex = 50000 (~ 40 cells)

0 0.2 0.4 0.6 0.8 1U/U

!

0

2

4

6

8

10

12

"

Rex = 5000 (~13 cells)Rex = 10000 (~17 cells)Rex = 50000 (~40 cells)

0 0.2 0.4 0.6 0.8v/U

! sqrt(Rex)

0

2

4

6

8

10

"

Rex = 5000 (~13 cells)Rex = 10000 (~ 17 cells)Rex = 50000 (~ 40 cells)

u/U�

v/U��

Rex

u/U�

v/U��

Rex

��

��

u

v

15° Non-aligned plate

no-slip

slip

M!!= 0.5

ReL = 5000

" = 15°

Tangential velocity Tangential velocity

Normal velocity Normal velocity

No Limiter van Leer Limiter

Blasius profile

Monday, July 27, 15

Baseline Turbulent Flow Solver

�xx

=2

3(µ+ µ

t

)(2ux

� vy

)

�xy

= (µ+ µt

)(uy

+ vx

) = �yx

�yy

=2

3(µ+ µ

t

)(2vy

� ux

)

Extend to RANS equations:

Spalart-Allmaras turbulence model: additional equation for νSA (μt ~ ρνSA) dνSA /dt + d(uj νSA) /dxj = Diffusion + Prod. - Dest.

SA model:• need 2nd order advection on non-streamlined mesh (often implemented with 1st

order method for positivity)• improved model for negative transients• multigrid using 3-5 levels (but robustness can deteriorate)

Monday, July 27, 15

u+ =u

uτy+ =

yuτ

ν

Baseline Turbulent Flow Solver

10-1 100 101 102 103 104

y+0

5

10

15

20

25

30

u+

14lev mesh @ Rex = 5.e6Spalding profile

Aligned turbulent flat plate, M=0.5, ReL = 5 x 105

Comparison of flat plate boundary layer computation with approximate solution using Spalding profile

y+ = u+ + e�kB

✓eku

+

� 1� ku+ � (ku+)2

2� (ku+)3

6

◆

Monday, July 27, 15

Can embedded boundary meshes compute high Re flows?

• Boundary layer zoning not possible without body-fitted meshes

• Computational cost unaffordable with uniform refinement

• Use subgrid-scale wall modeling - wall function or wall model - for 1d resolution, and conservatively couple to background Cartesian grid

Subgrid convection is modeled in non-conservative form, as shown above [Eqs.

(3)–(6)] since, in an isothermal situation where there is no wall heating, the con-

servative form of the convection term can lead to a non-zero source/sink term in the

subgrid temperature equation, due to mass flux imbalance through the cell faces.

Convection parallel to the wall is calculated using neighboring subgrid values.

The wall-normal V-velocity (required to evaluate the wall-normal convection) is

calculated from subgrid continuity and is then scaled to ensure that the subgrid

velocity at the outer boundary (shown as position n in Figure 1b) matches the main-

grid velocity at that location. Although continuity is not rigorously satisfied in the

Figure 1. (Top) Subgrid arrangement within the near-wall main-grid control volume. (Bottom) Subgrid

boundary conditions and pressure gradient.

308

T. J. CRAFT ET AL.VI. Turbulent wall-modelling

Linear interpolations are the basic ingredient of low Reynolds immersed boundary methods based on adiscrete forcing approach. Moreover they are e�ective only if the forcing points are located in the linearregion of the boundary layer. As a consequence, for medium/high Reynolds number flows a wall modellingis proposed. A two-layer approach, based on a decomposition of the near wall region,29 has been developed.The basic idea is to have a external layer governed by the usual RANS equations and a near wall zone

Figure 4. Two-layer wall model.

modelled by three-dimensional thin boundary layer equations

⇧

⇧x2

�(µ + µt)

⇧ui

⇧x2

⇥= Fi, i = 1, 3 (10)

with x2 the direction normal to the wall and (x1, x3) the tangential to the wall directions. In general theright hand side Fi is equal to sum of the unsteady, convective and pressure gradient terms

Fi =⇧

⇧t(⇥ui) +

⇧

⇧xj(⇥ujui) +

⇧p

⇧xi(11)

Here a simplified version of eqn. (10) is adopted, in which the unsteady and convective terms are neglected

⇧

⇧x2

�(µ + µt)

⇧ui

⇧x2

⇥=

⇧p

⇧xi, i = 1, 3 (12)

For a flat plate with zero pressure gradient the universal law of the wall is recovered and the profiles of theboundary layer quantities collapse in the region between the wall and the outer edge of the logarithmic layerif properly scaled with the friction velocity u�

u+ =u

u�, x+

2 =⇥u�

µx2, µ+ =

µt

µ, p+ =

µ

⇥2u3�

⇧p

⇧x2,

k+ =k

u2�

, ⇤+ =µ

⇥u2�

⇤, g+ =

⇤⇥u2

�

µg

9 of 24

American Institute of Aeronautics and Astronautics

Monday, July 27, 15

Diffusion Model

Leading term NS equations: d/dy ((μ + μt ) du/dy) = 0

Integrate: (μ + μt ) du/dy = constant = ρ uτ2 (v + vt) du/dy = uτ2

(1 + vt /v) du/dy = uτ2 / vIn wall coordinates: (1 + vt /v) du+/dy+ = 1 (u+ = u / uτ; y+ = y uτ / v)

Two regimes: near wall: vt /v << 1, ⇒ du+/dy+ = constant in viscous sublayer, or u+∝ y+

log layer: vt /v >> 1, (vt /v) du+/dy+ = 1 assume a mixing length model for vt /v = k y+ solve to get law of the wall: u+ = (1/k) log (y+) + B

Spalding wall function: functional form combining both regimesNew SA wall function: solve equivalent SA form exactly (due to Allmaras)

Monday, July 27, 15

100 101 102 103 104

y+0

5

10

15

20

25

30

u+

SA wall model Spalding profile

u+ = y

u+ = 2.44 ln(y+) + 5.0

Transitionregion

Diffusion Model: Spalding vs. SA Model

u+ =u

u⌧

y+ =yu⌧

�

u+(y+) =

¯B + c1 log((y+ + a1)2 + b21) � c2 log((y+ + a2)2 + b22)

� c3 arctan(b1

y++a1) � c4 arctan(

b2y++a2

)

SA:

Sp:

Monday, July 27, 15

10-1 100 101 102 103 104

y+

0

5

10

15

20

25

30u+

SA profile13 level, quadratic, 0o

10 level, wall model, 0o

11 level, wall model, 15o

Integrate-to-wall

Aligned wall model

Non-aligned(with model)

Wall Functions

• 3 fewer levels of refinement needed for aligned case.• 2 fewer levels in non-aligned case

Turbulent flat plate, M=.5, ReL = 5 x105.

Monday, July 27, 15

Coupling of Wall Model and Cartesian Grid

Choose pt. F a fixed distance from wall (using Capizzano terminology) (wall functions very sensitive to loc. of 1st point)

Cartesian grid provides:

• pt. F data to wall model

Newton iteration to find friction velocity uτ

i.e. (u→u+, y→y+) lie on SA curve u+ − SA(y+) = 0

Wall model provides:

• du/dy at wall

• velocity gradient at cell centroid

• tangential velocity at cell centroid

• viscous gradients at cut faces

C d

F

D

h

Monday, July 27, 15

Wall Function Results

Comparison with Langley Turbulence Modeling Resource solutions for NACA 0012

M∞ = 0.15 ReL = 6x106

Mach contours

Monday, July 27, 15

Wall Function Results

Comparison with Langley Turbulence Modeling Resource solutions for NACA 0012

M∞ = 0.15 ReL = 6x106 Mach contours

Monday, July 27, 15

Wall Function Results

Comparison with Langley Turbulence Modeling Resource solutions for NACA 0012

M∞ = 0.15 ReL = 6x106

Monday, July 27, 15

Wall Function Results

Comparison with Langley Turbulence Modeling Resource solutions for NACA 0012

M∞ = 0.15 ReL = 6x106

Monday, July 27, 15

Wall Function Results

Comparison with Langley Turbulence Modeling Resource solutions for NACA 0012

M∞ = 0.15 ReL = 6x106

Monday, July 27, 15

• Functional form doesn’t allow flow separation (can’t divide by 0)• Diffusion model has small y+ range• Mixing length model has small y+ range

y/40 80

1

x-velocityEddy viscosity

y/40 80

1

x-velocityEddy viscosity

Problems with SA Wall Function

• Use discrete model to add more terms, extend range

y+ ~ 102y+ ~ 102y+ ~ 102y+ ~ 102y+ ~ 102y+ ~ 102y+ ~ 102y+ ~ 102y+ ~ 102y+ ~ 104

Monday, July 27, 15

• Functional form doesn’t allow flow separation (can’t divide by 0)• Diffusion model has small y+ range• Mixing length model has small y+ range

Problems with SA Wall Function

• Use discrete model to add more terms, extend range

Monday, July 27, 15

Simplest BVP: equivalent to diffusion model solve for u, assuming mixing length model for μt = ρ νt

d/dy ((μ + μt ) du/dy) = 0 u(0) = 0 u(F) = uF

BVP Models

{

• Each cut cell discretizes and solves its own linelet. • Using Shampine’s BVP_M solver with augmented stucture, modified for thread safety, tolerances, linking with C, initialize with wall fn. solution

F

C

Use data at pt. F to solve 2 point bvp with different levels of fidelity:generic bvp: q�� = f(q, q�, t, y)

q(0), q(F ) specified

Monday, July 27, 15

Next level: SA version solve for u and vSA

d/dy ((μ + μt ) du/dy) = 0 d/dy ((v + vSA ) dvSA/dy) = Diffusion + Prod. - Dest.

u(0) = 0, u(F) = uF

vSA (0) = 0 vSA (F) = vSAF

BVP Models

{Finest level: Full NS equations cut cell linelets not smooth enough to compute streamwise derivatives

Captures more curvature of vSA. Would still like larger y+

F

C

Monday, July 27, 15

Intermediate level of fidelity? Some add to diffusion eq. No help, sometimes worse.

BVP Models

• At wall, u = v = 0, so p balances shear stress

• At edge of boundary layer, shear stress ~ 0, and p balances convective terms.

∇p

extracted at x=1.2 from cfl3D results

from TMR soln for bump in channel

Monday, July 27, 15

BVP Models

• At wall, u = v = 0, so p balances shear stress

• At edge of boundary layer, shear stress ~ 0, and p balances convective terms.

Looking again at bvp: d/dy ((μ + μt ) du/dy) = px + u ux + v uy

• need way to shut off pressure

• use convective balance taken from background Cartesian grid

• use shut-off function for rhs

Monday, July 27, 15

BVP Models

• At wall, u = v = 0, so p balances shear stress

• At edge of boundary layer, shear stress ~ 0, and p balances convective terms.

Looking again at bvp: d/dy ((μ + μt ) du/dy) = px + u ux + v uy

d/dy ((μ + μt ) du/dy) = px + S(uux + vuy) S = shutoff fn. from 0 to 1

Monday, July 27, 15

BVP Results

use of pressure gradient in bvp corrects centering of skin friction profile

Monday, July 27, 15

BVP Results

x = 0.2

x = 0.75

x = 1.2

• vSA comparison using bvp, wall fn., CFL3D

• CFL3D solution from TMR bump in channel

Monday, July 27, 15

Conclusions

Have demonstrated:• Fully conservative coupling of wall model to mesh

• Wall fn./model reduces resolution req. by 3-4 levels of refinement in 2D

• Wall model improvements over SA wall fn.

Next steps:• Improve bvp (wake fn., smoother interp.)

• improve code (mesh interfaces, stronger smoother)

• 3D much more complicated

• Time dependent

Monday, July 27, 15