Development of a mesoscopic simulation method for atmospheric convection

Lian-Ping Wang1, Wojciech W. Grabowski2, Zhaoli Guo3, and Ryo Onishi4�

1Department of Mechanical Engineering, University of Delaware, USA 2Japan Agency for Marine-Earth Science and Technology, Japan

3National Center for Atmospheric Division, USA 4Huazhong University of Science and Technology, China

lwang@udel.edu

10:50 – 11:10, December 1 (Thursday) The 4th International Workshop on Nonhydrostatic Models

Nov. 30 (Wed) - Dec. 2 (Fri), 2016 The Prince Hakone Lake Ashinoko, Hakone, Japan

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Outline

v  Background, motivation, and objectives

v  The DUGKS method

v  Some results

v  Summary

3

Background & Motivation ² Mesoscopic methods based on the Boltzmann equation to

solve complex turbulent flows have been rapidly developed over the last 30 years

² But very limited applications to geophysical flows Overall objectives ² Simulate high-Rayleigh number convection flows using

DUGKS

² How to treat unresolved local gradients? Controlled local numerical diffusion (TVD, monotone, ….) Explicit SGS model

² Compare DUGKS and Navier-Stokes based solvers in terms of accuracy, numerical stability, and efficiency

4

Thermal convection flows Macroscopic world∂ρ∂t+∂∂x j

ρu j( ) = 0

∂ui∂t

+ u j∂ui∂x j

= −1ρ∂!p∂xi

+ βg θ −θ0( )δi3 +ν∂2ui∂x j∂x j

∂θ∂t

+∂∂x j

u jθ( ) = ∂∂xi

κ∂θ∂xi

"

#$$

%

&''

θ = T −T

u0 = gβΔTH

Ra= gβΔTH3

νκ

Pr = νκ= 0.71

Mesoscopic world

∂t f +ξ ⋅ ∇f = !Ω ≡ −f − f eq

τ v+Φ, Φi ≈

β!gθ ⋅ !ei −

!u( )RT

fi( eq )

∂th +ξ ⋅ ∇h = Ψ ≡ −h − heq

τ c, θ = hi

i∑

ρ = hii∑ , ρ !u = fi

i∑ ξi +

δt2ρβ!gθ , p = ρRθ

f eq = ρ wi 1+!ei ⋅!u

cs2+ !ei ⋅!u( )2

2cs4

−!u ⋅ !u2cs

2

+

,

---

.

/

000, heq =θ wi 1+

!ei ⋅!u

cs2+ !ei ⋅!u( )2

2cs4

−!u ⋅ !u2cs

2

+

,

---

.

/

000

5

²  Based directly on the Boltzmann equation with BGK collision model

²  Gas Kinetic Scheme combined with certain good features of LBM (simplicity and low numerical dissipation)

²  The kinetic advection is treated as fluxes through cell interfaces

²  Fluxes are computed using distributions at the half time step, with streaming and collision coupled together (low numerical dissipation)

²  A much more general approach than LBM − Key advantages: all flow types, all Kn and Ma − Non-uniform grid

K Xu and J-C Huang, J. Comp. Phys. 229, 7747 (201) ZL Guo, K Xu, RJ Wang, PRE 88, 033305 (2013); PRE 91, 033313 (2015) P Wang, S Tao, ZL Guo, Computers & Fluids, 120 70-81 (2015).

DUGKS: Discrete Unified Gas Kinetic Scheme

DUGKS: finite-volume discretization for hydrodynamic velocity

∂t f +ξ ⋅ ∇f = !Ω ≡ −f − f eq

τ v+Φ

ξ → ei , Φi ≈β!g T −T( ) ⋅ !ei −

!u( )RT

fi( eq )

Updating rule for cell-center distribution functions

fin+1 x( )− fin x( )+ dt

Vjfin+0.5 xs( )ei ⋅ dS

dS∑ =

dt2!Ωin+1 + !Ωi

n+,

-.

Constructing fin+0.5 xs( ) : Integrate from tn to tn + 0.5dt along the characteristic path

fin+0.5 xs( )− fin xs − 0.5dt ei( ) = dt4

!Ωin+0.5 xs( )+ !Ωi

n xs − 0.5dt ei( )+,

-.

x = cell center, xs = cell boundary

Mid-point Trapezoidal

For nearly incompressible flow, D3Q19

K Xu and J-C Huang, J. Comp. Phys. 229, 7747 (201) ZL Guo, K Xu, RJ Wang, PRE 88, 033305 (2013); PRE 91, 033313 (2015)

Similar transformations as in LBM are used to make all explicit

7

DUGKS: Boundary conditions – all done to distributions at cell interfaces

No-slip wall: bounce-back

Free-slip wall: mirror reflection

Fixed temperature

hαn+0.5 xs( ) = −hαn+0.5 xs( )+ 2Wα

Twall

Zero heat flux

hαn+0.5 xs( ) = hαn+0.5 xs( )

in out

in out

8

DNS of turbulent flows using DUGKS Homogeneous isotropic turbulence in a periodic box Wang P, Wang L-P, Guo ZL, 2016, Comparison of the LBE and DUGKS methods for DNS of decaying turbulent flows. Phys. Rev. E., 94, 043304 Turbulent channel flow [non-uniform mesh with large grid aspect ratios] Bo YT, Wang P, Guo ZL, Wang L-P, 2016, Parallel implementation and validation of DUGKS for three-dimensional Taylor-Green vortex flow and turbulent channel flow, Computers & Fluids, submitted. DNS of thermal convection in an enclosure Wang L-P, Wen X, Geneva N, Wang P, Guo ZL, 2016, Simulations of high-Rayleigh-number convection flows using mesoscopic methods, Discrete Simulation of Fluid Dynamics 2016. Ra ~ 1011 in 3D MPI, non-uniform mesh, external forcing, boundary conditions

The physical problem: Rising and evolution of a warm dry bubble Background ρ0 =1.0, θ0 = 300K , ΔT = 2 K

θ =θ0 +ΔT cos π r

2000

!

"#

$

%&

'

()

*

+,

2

r <1000 m

0, otherwise

!

"#

$#

Centered at 1040 m height

Domain: 3200 m H, 5000m V

H =1000 m, Pr = 0.71, ν=1.57×10−5

Ra = PrGΔTH3

ν 2=1.89×1017

Periodic BC in horozontal Vertical boundaries: zero heat flux & stress free

Evolution at a given grid resolution of dx = 10 m No numerical limiter applied

0 min 5 min 10 min 15 min

How about increasing the grid resolution? No numerical limiter applied

dx=1.25 m dx=2.5 m dx=5.0 m

12

-  The higher the resolution, the more rapid the smallest scale is formed

Maximum local temperature gradient and maximum local vorticity à Rapid evolution of the smallest spatial scale

The van-Leer slope limiter (the monotonized central limiter)

σ jn =minmod 2

U j+1n −U j

n

Δx,U j+1n −U j−1

n

2Δx,2U jn −U j−1

n

Δx

#

$%%

&

'((

where

minmod a1,a2 ,a3( ) =sign a1( )min a1 , a2 , a3( ) , if sgn a1( ) = sgn a2( ) = sgn a3( )0, otherwise

)*+

,+

Variation

σ jn = sgn s1( )+ sgn s2( )-

./0s1 s2s1 + s2

, where s1 =U j+1n −U j

n

Δx, s2 =

U jn −U j−1

n

Δx

Slope in a cell for field reconstruction

(Form 1)

(Form 2)

B. van Leer, J. Comput. Phys. 23, 276 (1977).

14

Time =12.0 min, dx = 10 m, comparison of T field

No limiter van Leer limiter Piecewise parabolic method Carpenter et al (1990)

Carpenter et al., 1990, Monthly Weather Rev. 118: 586 – 612.

15

Time =16.0 min, dx = 10 m, comparison of T field

No limiter van Leer limiter Piecewise parabolic method Carpenter et al (1990)

Codes to be compared NCAR Grabowski: 2nd-order finite difference code with MPDATA (the multidimensional positive definite advection transport), set to constant background density JAMSTec Onishi: atmospheric background density WAF2nd: 2nd-order Weighted-Averaged Flux method with SUPER-BEE flux limiter WS5th: Wicker-Skamarock 5th-order upwind scheme

The center of the thermal

Zc =θ −θ0( ) z dV∫

θ −θ0( )dV∫

Vertical profiles at 5 min

w

T −T0

Vertical profiles at 10 min

w

T −T0

Vertical profiles at 15 min

w

T −T0

Summary

Demonstrated the feasibility of using DUGKS for high Ra convection flows − DUGKS has low numerical diffusivity and provides accurate solution for convection flow − DUGKS code is stable even when the flow is not resolved − When the flow is unresolved, local oscillations occur − Numerical limiter help suppress oscillations

At later times, the solution could depend sensitively on the treatment of the advection term, details of the numerical limiter, grid resolution etc., or even how grids are set up relative to the center of the bubble − How do we develop benchmark solutions at later times?

Potential benefits: fast scalable computation, low numerical dissipation, ….

Next steps − Directional splitting may be tested − Better slope / flux limiter scheme: MPDATA, WENO − Atmospheric background density / temperature