an immersed boundary method enabling large-eddy simulations of complex terrain in the wrf model
DESCRIPTION
Performance Measures x.x, x.x, and x.x. An Immersed Boundary Method Enabling Large-Eddy Simulations of Complex Terrain in the WRF Model. Presented at: University of Utah April 4, 2012. Katherine A. Lundquist 1,2 , Fotini K. Chow 2 1 Lawrence Livermore National Laboratory - PowerPoint PPT PresentationTRANSCRIPT
Lawrence Livermore National Laboratory, P.O. Box 808, Livermore, CA 94551
This work was performed under the auspices of the U.S. Department of Energy byLawrence Livermore National Laboratory under Contract DE-AC52-07NA27344.
An Immersed Boundary Method Enabling Large-Eddy
Simulations of Complex Terrain in the WRF Model
Performance Measures x.x, x.x, and x.x
Presented at:
University of Utah
April 4, 2012
Katherine A. Lundquist1,2, Fotini K. Chow2
1 Lawrence Livermore National Laboratory2 University of California, Berkeley
Numerical Weather Prediction in Complex Terrain
Vertical coordinate systems used in numerical weather prediction models.
Non-orthogonalorthogonal
sigma, or terrain-followingeta, or “step mountain”
Numerical Weather Prediction in Complex Terrain
Nested grids used in mesoscale models allow integration of physical processes across a range of scales
50 m grid1 km grid 300 m grid
Numerical Weather Prediction in Complex Terrain
Higher grid resolution leads to steeper slopes in complex terrain
Problems with Terrain-Following Coordinates in Complex Terrain
Grid skewness leads to significant errors in horizontal derivatives
• Advection• Diffusion• Pressure gradient
),(),(yxzzyxzzz
httop
httop
Mapping function for coordinate transformation (Gal-Chen and Sommerville ,1975)
New terms are introduced into the governing equations (metric terms)
zzppp z
When discretized, the metric terms have additional truncation errors
)(1 xOxF
xFF
i
ii
Problems with Terrain-Following Coordinates in Complex Terrain
In addition to truncation errors, numerical inconsistencies arise when:
zht
ht
Δzzx
zxx zzz
In the transformed coordinate η, the horizontal derivative is:
A first order finite difference approximation yields:
kixF
, xkiFkiF
),(),1(
zkiiFkiF
xz
),()1,1(
Problems with Terrain-Following Coordinates in Complex Terrain
Numerically inconsistent derivatives are more likely at large aspect ratios
Horizontal grid spacing of 1km with a vertical grid of 50 m leads to an aspect ratio of 20
Moral of the story: Don’t stretch grids too much
Example: 1 km horizontal spacing, 50 m vertical -- max allowed slope is 3 degrees
How can we quantify numerical errors due to terrain-following coordinates? Model fails at steep slopes, but when does solution quality
deteriorate?• What slope?• What grid aspect ratio?
Numerical Weather Prediction in Complex Terrain
Vertical coordinate systems used in numerical weather prediction models.
Non-orthogonalorthogonal
orthogonal
sigma, or terrain-followingeta, or “step mountain”
immersed boundary
Use immersed boundary method • Can eliminate the terrain-following coordinate
transformation• Quantify numerical errors through direct
comparison of the two solutions
Scalar Advection Test Case (Schär et al. 2002)
Domain Set-Up & Initialization Peak Height = 3 km Elevated Velocity Shear Layer at
5 km (terrain is very steep, but isolated)
Inviscid flow- no mixing Stable Stratification
Domain size 300 km x 25 km Grid spacing 1 km x 0.5 km 5th order horizontal and 3rd order vertical
advection scheme is used
Scalar Advection Test CaseGrid Configuration
WRF
IBM-WRF
Grid distortion
Scalar Advection Test CaseVelocity Comparisons
WRF
IBM-WRF
WRF
IBM-WRF
U (m/s) at t = 10000 s W (m/s) at t = 10000 s
Hei
ght
(km
)
x (km)
Hei
ght
(km
)
x (km)
Scalar Advection Test CaseScalar Concentration and Error
Effects of terrain slope in WRF
Change max mountain height
Error = max(|WRF-exact|)Er
ror
Terrain slope (degrees)
Atmosphere At Rest
3D hill, quiescent atmosphere, no forcing 10 degree slope (not very steep!) Stable atmosphere No flow should develop at all
U (m/s)
θ(K)
Atmosphere at RestSpurious Flow Develops
WRF coord. diffusion
WRF horiz. diffusion
IBM-WRF
U (m/s) U (m/s)
θ(K)θ(K)
Max velocity 1.7 m/s
Max velocity 0.28 m/s
Max velocity 3.8 e-5 m/s
Flow Over 3D Hill
Compare WRF and IBM-WRF• max slope of 10, 20, 30 deg, grid aspect ratio 1
Geostrophic pressure gradient forcing No-slip boundary condition Zero flux condition on temperature Run to steady state
Flow Over 3D HillVelocity Difference – WRF vs IBM-WRF
Absolute velocity difference – slice through peak of hill
x (km) x (km)x (km)
10 slope 20 slope 30 slope
Max u diff 1.0 m/s
Max u diff 1.8 m/s
Max u diff 3.1 m/s
ht (
km)
Increased turbulent eddy viscosity
Overwhelms numerical errors Absolute differences between WRF and IBM-WRF decrease
Low viscosity
Hei
ght
(km
)
x (km)
High viscosity
x (km)
Immersed Boundary Method
The effects of the body on the flow are represented by the addition of a forcing term in the momentum equation.
0
2
U
FUpUUtU
The immersed boundary method is a technique for representing boundaries on a non-conforming grid
Immersed Boundary MethodFormulation of the Forcing Term
Source: Peskin (1977)RHStUVFn
IBM was first used by Peskin (1972) and (1977) to simulate blood flow through the mitral valve of the heart.
Direct (or Discrete) Forcing proposed by Mohd-Yusof (1997) and used to model laminar flow over a ribbed channel
where VΩ is the desired Dirichlet boundary condition
FUpUUtU
2
tUV
tUU nnn
1
Immersed Boundary MethodBoundary reconstruction
Boundary is coincident with computational nodes
Boundary effects must be interpolated to
computational nodes
Stair step or nearest neighbor grid
Immersed Boundary Method grid
Immersed Boundary MethodBoundary Reconstruction
Using a ghost cell method, the forcing term is applied within the solid domain.
UΩ2
U2U1
Ughost cell
cellghostdcellghostccellghostbacellghost
d
c
b
a
xzwzwxwwUUUUU
wwww
xzzxxzzxxzzxxzzx
____
2
1
2
1
222
111
222
111
1111
UΩ1
Scalar Fluxes at the Immersed Boundary
Bt FFVt
2
Use the immersed boundary method to impose boundary conditions on temperature, moisture, passive scalars, etc.
Or a flux boundary condition can be imposed with IBM.
fn ˆ
A Dirichlet boundary condition
Immersed Boundary MethodBoundary Reconstruction
Neumann boundary conditions are set by modifying the interpolation matrix to include the boundary condition
δΦ/δnΩ1
Φ2Φ1
Φghost cell
cellghostdcellghostccellghostbacellghost
n
n
d
c
b
a
xzwzwxww
wwww
xzzxxzzxzxzx
____
2
1
2
1
222
111
2222
1111
11
sincoscossin0sincoscossin0
fn ˆ
δΦ/δnΩ2
Idealized Valley Simulations
Thermal Slope flow induced by diurnal heating
Uncoupled simulations with specified surface heating
Coupled simulations using atmospheric parameterizations
WARM
Idealized Valley Simulations
Set-up and Initialization ΔX = ΔY = 200 m, ΔZ ~100 m (U,V,W) = (0,0,0) Stable Potential Temp. 40% Relative Humidity Sandy Loam, Savannah Soil Moisture, 20% saturation
rate Soil Temperature, equal to
atmospheric temperature
Uncoupled Ideal Valley
Integrate from 6:00 to 18:00 UTC
Specified heat flux Zero moisture flux at
surface No atmospheric physics No surface properties Constant t
Uncoupled Ideal Valley Evolution of Potential Temp.
Potential Temp at Valley Center
Coupled Ideal ValleyComparison of Velocity Profiles
Comparison of instantaneous velocity profiles for IBM-WRF (red) and WRF (black)
Coupled Ideal Valley
Fully Coupled Model• RRTM Longwave Radiation• MM5 Shortwave Radiation• MM5 Surface Layer Model• NOAH Land Surface Model
Each atmospheric physics module has been modified to account for the immersed boundary
Coupled Ideal Valley Initialization
Date: March 21, 2007 Location: 36°N, 0°E Sandy Loam, Savannah Soil Moisture, 20% saturation rate Soil Temperature, equal to atmospheric temperature
Coupling IBM to the Atmospheric Physics Models
Surface physics models interact with the lowest coordinate level when terrain-following grids are used
The atmospheric physics models used here are column models
For radiation models the vertical integration limits are modified to exclude any portion of the atmosphere below the terrain
For surface physics a modified reference height is calculated and used with similarity theory
Coupled Ideal Valley Radiation Models
Domain averaged incoming radiation (longwave and
shortwave) differ by less than 0.43% during the simulation.
Spatial variation in radiation at 12:00. Error from IBM coupling is negligible in
comparison to changes with terrain height.
Coupled Ideal Valley Surface Physics
Domain averaged heat flux differs by less than 5.4%, and moisture fluxes by less than 0.74%
Coupled Ideal ValleyLand-Surface Properties
IBM provides boundary condition to both WRF and NOAH simultaneously.
Complex Terrain Owens Valley, CA
Valley terrain can be extremely steep with slopes
of up to 60 degrees.
IBM allows explicit resolution of this terrain.
Verification with Field Campaign Dataset
Joint Urban 2003 Oklahoma City Field Campaign
Source: Allwine and Flaherty (2006)
Joint Urban 2003 Oklahoma City Terrain
Problems with Boundary Reconstruction
Matrix is singular
Flux is prescribed in the incorrect direction
Cannot find eight appropriate neighbors
cellghostdcellghostccellghostbacellghost
d
c
b
a
xzwzwxwwUUUUU
wwww
xzzxxzzxxzzxxzzx
____
2
1
2
1
222
111
222
111
1111
UΩ2
U2U1
Ughost cell
UΩ1
Immersed Boundary MethodBoundary Reconstruction
p
n
nn RR
RRw
max
max
nnn
nn
o Fww
F 1
Inverse distance weighing is an interpolation method developed for scattered data (Franke 1982)
Immersed Boundary MethodBoundary Reconstruction
Inverse Distance Weighting is used for the interpolation which determines the forcing applied at the ghost node to
enforce the Dirichlet boundary condition
p
n
nn RR
RRw
max
max
nnn
nn
image Uww
U 1
imageghost UUU 2
Immersed Boundary MethodBoundary Reconstruction
Inverse Distance Weighting preserves local maximum and minimum values. For Neumann boundary conditions, the probe length must be extended, so that the ‘image’ point is
surrounded by neighbors.
p
n
nn RR
RRw
max
max
nnn
nn
image ww
1
nd probeimageghost ˆ
Verification with Flow Over 3D Hill
Geostrophic pressure gradient forcing No-slip boundary condition Zero flux condition on temperature Run to steady state
VerificationGeostrophic Flow over a 3D Hill
Differences are larger for inverse distance weighing than for trilinear interpolation, however both methods produce accurate results. Inverse distance weighting has the added advantages of being easier to implement and using a flexible interpolation stencil.
VerificationGeostrophic Flow over a 3D Hill
Changing the vertical grid in WRF produces much larger differences than those seen between WRF and IBM-WRF
Joint Urban 2003 Oklahoma City Terrain
Joint Urban 2003 Oklahoma CityNested Domain
Mesoscale models are usually run in a nested mode. Here the mesoscale domain is nested down to an urban domain,
Joint Urban 2003 Oklahoma CityOne-way Nested Domain
One-way nesting is used to run the Oklahoma City domain within a channel flow simulation
Parent Domain Nested Domain
Joint Urban 2003 Oklahoma CitySet-Up and Parent Domain Flow
• IOP 3• Outer Domain: ΔX, ΔY = 6 m and ΔZ is stretched form 1 to 3 m• Inner Domain: ΔX, ΔY = 2 m and uses the same ΔZ• Δt = 1/20 s on the outer domain, and 1/60 s on the inner domain• Domains are run in concurrent mode• No atmospheric physics• Static Smagorinsky closure
Joint Urban 2003 Oklahoma City Instantaneous Velocity Field
IBM-WRF (LES)
FEM3MP (RANS)
Joint Urban 2003 Oklahoma City Verification with Observations
IBM-WRF
FEM3MP
Joint Urban 2003 Oklahoma City Verification with Observations
FACx = Predictions within a factor of XFB = Fractional biasMG = Geometric mean biasNMSE = Normalized mean square errorSAA = Scaled average angle
Joint Urban 2003 Oklahoma City Verification with Observations
IBM-WRF
FEM3MP
Joint Urban 2003 Oklahoma City Verification with Observations
FACx = Predictions within a factor of XFB = Fractional biasMG = Geometric mean biasNMSE = Normalized mean square error
Summary
An immersed boundary method was developed in WRF that eliminates numerical errors caused by terrain-following coordinates
Errors arising from terrain-following coordinates were quantified
Two different interpolation cores were developed for use in the IBM, each with different strengths
The IBM has been verified for use in both 2D and 3D terrain through canonical cases
JU2003 was used for verification for real urban terrain Atmospheric physics parameterizations have been
coupled to the IBM to provide surface fluxes of heat and moisture
Immersed Boundary MethodNearest Neighbor Algorithm
A) Numerical instabilities in previous IBMs are avoided by choosing the nearest neighbors of an image point (instead of the ghost point)
B & C) By choosing boundary points (and not ghost points) as nearest neighbors the solution for the ghost point is independent. Iterative procedures used in previous IBMs are not necessary.
D) With isobaric coordinates this point often moves between the fluid and solid domain. Flexibility is added to the algorithm, so that a fluid point in close proximity to the boundary can be a ghost point.