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
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
The University of Delaware is advancing
leading-edge research that solves mysteries,
tackles problems, and invents new
technologies - all aimed at enhancing our
quality of life. UD has distinguished faculty,
capable staff, and outstanding students,
supported by state-of-the-art facilities. Join us
as we explore the latest frontiers in science,
engineering, business, the arts, and
humanities. Your adventure starts here!Feature: 5 of 8
August 23, 2010
University of DelawareResearch magazine published
The second issue of "University ofDelaware Research" highlights healthsciences research at UD, including thegrowing collaborations of the DelawareHealth Sciences Alliance, a partnership ofUD, Christiana Care Health System,Nemours/Alfred I. duPont Hospital forChildren, and Thomas Jefferson University.
Read More >>
Debessay wins Ernst and Young award for inclusiveness
URS to decommission former Chrysler plant for science,
technology campus
QPS, a Delaware-based business, expands and goes global
Standard Solar brings solar energy deployment to UD
Satellite receiving stations provide real-time environmental data
$1 million Teaching American History grant awarded
Scholars urged to carry McNair experience into grad school
UD holds first research, service celebratory symposium
It is our office's mission to
encourage and support the
research, scholarship, and
creative activity of UD's faculty,
staff, and students. This Web site
is designed to provide practical
information to UD researchers, as
well as offer the public an exciting
gateway to UD's latest discoveries, inventions, and
K–12 classroom resources. Visit often!
2
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