Atmospheric flow modeling III

WIND ENERGY METEOROLOGY

UNIT 7
ATMOSPHERIC FLOW MODELING III:
LARGE-EDDY SIMULATION

Detlev Heinemann

ENERGY METEOROLOGY GROUP
INSTITUTE OF PHYSICS
OLDENBURG UNIVERSITY
FORWIND – CENTER FOR WIND ENERGY RESEARCH

Mittwoch, 15. Juni 2011

ATMOSPHERIC FLOW MODELING III: LES

SIMULATION OF TURBULENT ATMOSPHERIC
FLOW
We know:
‣ Turbulence consists of three-dimensional, chaotic, or random
motion that spans a range of scales that increases rapidly with
Reynolds number
‣ Complete numerical integration of the exact equations
governing the turbulent velocity field (Navier–Stokes equations)
is known as direct numerical simulation (DNS).
‣ Because of limited computing power, DNS is restricted to low-
Reynolds-number turbulence, which exists in laboratory flows,
e.g., in wind tunnels.

2


FLOW
TURBULENCE MODELING HIERARCHY

Direct Numerical Simulation (DNS)
• Solution of the Navier-Stokes equations without use of
an explicit turbulence – limited to low Reynolds numbers
• Powerful research tool
Large Eddy Simulation (LES)
• Direct resolution of the large, energy-containing scales
of the turbulent flow, model only the small eddies
• High computational cost in boundary layers
Reynolds-average Navier-Stokes (RANS)
• Model the entire spectrum of turbulent motions
• Uneven performance in flows outside of the calibration
range of the models



FLOW

• Solution of the Navier-Stokes equations without use of increase
in cost
increase in
• Uneven performance in flows outside of the calibration empiricism

range of the models



FLOW

• Solution of the Navier-Stokes equations without use of increase
in cost
“hybrid” methods
combine RANS
and LES (e.g.,
Detached-Eddy
Simulation)
increase in
• Uneven performance in flows outside of the calibration empiricism

range of the models



FLOW

l η = l/Re3/4

Direct numerical simulation (DNS)

Large eddy simulation (LES)

Reynolds averaged Navier-Stokes equations (RANS)

4


FLOW

spatial and temporal resolution of scales in “inertial subrange”

5


ENERGY SPECTRUM OF TURBULENCE

6


ENERGY SPECTRUM OF TURBULENCE

7


FLOW
Example:
‣ PBL: largest turbulent eddies are on the order of kilometers and
the smallest on the order of millimeters
--> spectrum of turbulent motion spans more than six orders
of magnitude
‣ To numerically integrate the Navier–Stokes equations for this
turbulent flow would require at least 1018 numerical gridpoints
(today 1010 is possible...)

8


FLOW
Consequence:
‣ Only a portion of the scale range can be explicitly resolved,
--> larger eddies or most important scales of the flow
‣ Remaining scales must be roughly represented or parameterized
in terms of the resolved portion
‣ philosophy behind large-eddy simulation (LES)
‣ PBL turbulence:
Large eddies contain most of the turbulent kinetic energy (TKE)
--> energy-containing eddies
they are responsible for most of the turbulent transport

9


FLOW
‣ Explicite calculation of these large eddies and approximate
representation of the effects of smaller ones
‣ Accuracy of LES increases as the grid resolution becomes finer
‣ LES is a compromise between DNS, in which all turbulent
fluctuations are resolved, and the traditional Reynolds-averaging
approach in which all fluctuations are parameterized and only
ensemble-averaged statistics are calculated
‣ With increasing computer power a much broader application of
LES to more complicated geophysical turbulence problems is
anticipated

10


THE LES TECHNIQUE

Basis for an LES of the PBL:
Navier–Stokes equations for an incompressible fluid

where ui satisfy the continuity equation:

ui: flow velocities in the three spatial Xi: ith-component of body forces
directions p: pressure fluctuation
ρ: air density t time
ν: kinematic viscosity of the fluid xi spatial coordinates

11


THE LES TECHNIQUE

‣ PBL applications:
‣ major body forces are gravity and Coriolis forces
‣ Xi can be approximated as
‣ where the gravitational acceleration gi is nonzero only in the x3
(or z) direction, θ is the virtual potential temperature, T0 is the
temperature of some reference state, and f is the Coriolis
parameter.
‣ An additional transport equation is required for θ if buoyancy is considered.

‣ The numerical integration of these equations is DNS
‣ for LES they need to be spatially filtered

12


THE LES TECHNIQUE

Deriving the volume-filtered Navier-Stokes equations:
first decomposing all dependent variables, e.g., ui, into a volume
~
average, ui, and a subgrid-scale (SGS) (or subfilter) component, ui‘‘:

Here the volume-averaged or resolved-scale variable is defined as

where G is a three-dimensional (low-pass) filter function, e.g.,
Gaussian, top-hat or sharp wave cutoff filter.

13


THE LES TECHNIQUE

Filtering process
Example: One-dimensional random signal

Solid curve:
Total signal (fluctuating in x).
Dashed curve:
Smoother field after applying the filter
operator G (so-called filtered field or
resolved-scale motion).

Difference between total and resolved
signals representing the SGS fluctuations.
(Partitioning between resolved and SGS components
depends on the filter; i.e., cutoff scale and sharpness)

14


THE LES TECHNIQUE

Filtering process

‣ Filtering is a local spatial
averaging over the filter
width Δ
‣ Increasing Δ
‣ removes more scales
from the velocity field
and
‣ increases the
contribution of τij
--> filter width should be part of the
expressions for the models of τij

15


THE LES TECHNIQUE

Filters

‣ Sharp Fourier cutoff filter in wave space

‣ Gaussian

‣ Tophat filter in physical space

16


THE LES TECHNIQUE

Effect of Filters

Unfiltered and filtered velocity spectra

17


THE LES TECHNIQUE

Applying the filtering procedure, term-by-term, to the Navier-
Stokes equation leads to equations that govern large (resolved-
scale) eddies:

~
- first term on the right-hand side: advection of ui by the resolved-scale
~
motion uj
- second term: SGS contribution
- remaining terms: identical to their counterparts in NS, except that they
depend on filtered (resolved-scale) fields

18


THE LES TECHNIQUE

An alternative version can be derived using the identity

and expressing the SGS stress (or flux) tensor as

19


THE LES TECHNIQUE

‣ Both equations equally describes the evolution of the LE field.
They differ in their forms of the resolved advection and SGS
terms.
‣ First eq.: SGS term consists of two kinds of influences: cross-
products of resolved-SGS components (i.e., ) and a
nonlinear product of SGS–SGS components (i.e., )
‣ Second eq.: SGS term includes all of these influences plus a
resolved scale contribution:
‣ --> in principle different SGS models should be used.
‣ For geophysical turbulence, the molecular viscosity term is
negligibly small compared with the advection terms and can be
neglected.

20


THE LES TECHNIQUE

‣ So far in deriving the LE equations, no approximations have been
made.
‣ Because of the spatial filtering procedure, the LE equations
contain SGS terms that are unknown and must be modeled in
terms of the resolved fields.
‣ Because the magnitudes of SGS terms depend on the filter, its
modeling in principle should depend on the filter size and shape.

--> To solve the equations, the SGS terms need to be parameterized.

21


SUBGRID-SCALE PARAMETERIZATION

‣ Parameterization introduces uncertainty in LES, particularly in
regions where small eddies dominate, i.e., near the surface or
behind an obstacle.
‣ In regions where energy-containing eddies are well resolved,
LES flow fields are rather insensitive to SGS models
(In the interior of PBL, the SGS motions serve mainly as net energy sinks that
drain energy from the resolved motions)

‣ Most widely used SGS closure scheme: Smagorinsky–Lilly (S–L)
model
(Most PBL–LESs adopt a similar scheme)

22


SUBGRID-SCALE PARAMETERIZATION:
SMAGORINSKY–LILLY (S–L) MODEL
‣ Relating SGS stresses to resolved-scale strain tensors by

with the strain tensor
‣ SGS heat fluxes are similarly related to local gradients in the
resolved temperature field by

‣ The SGS eddy viscosity KM and diffusivity KH are expressed as

23



‣ the Smagorinsky constant cS remains to be determined
‣ Δs is a filtered length scale often taken to be proportional to the
grid size
‣ the magnitude of the strain tensor, S, is (2SijSij)1/2
‣ Pr (~1/3) is the SGS Prandtl number
‣ Important: the SGS fluxes are nonlinear functions of the resolved
strain rate
(different from the viscous (molecular) stress–strain relationship)

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Extension to include local buoyancy effects:

‣ KM is modified to depend on local Richardson number Ri (the
ratio of buoyancy to shear production terms of TKE budget):

where Ric is the critical Richardson number often set between
0.2–0.4, and n = 1/2 is often used
‣ When Ri reaches Ric, turbulence within that grid cell vanishes
and the eddy viscosity is shut off.

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Extension: Explicit calculation of the SGS–TKE e

‣ Relating KM and KH to e via

where
- cK is a diffusion coefficient to be determined
- ℓ is another SGS length scale, which is often taken as the
minimum of two length scales

(assuming a direct effect of local stability on the local SGS length scale)

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The SGS TKE e evolves from the following equation:

‣ Terms on the right-hand side:
- advection of e by the resolved-scale motion
- turbulent and pressure transports
- local shear production (nonlinear scrambling)
- local buoyancy production
- molecular dissipation
‣ are approximated by (s. slide 15):
transport terms: molecular dissipation rate
with cε: dissipation coefficient

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SGS model parameters:
cS, cK, and cε are usually chosen to be consistent with Kolmogorov
inertial-subrange theory, i.e., assuming that the SGS motions are
isotropic with a k-5/3 spectral slope.
Commonly used values are:
cS ~ 0.18,
cK ~ 0.10
cε ~ 0.19 + 0.74 ℓ/Δs.
With these model parameters, LESs are in a way forced – in an
ensemble-mean sense – to drain energy at a rate sufficient to
produce a k-5/3 spectral slope near the filter cutoff scale.
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Problem:
Above SGS models are based on ensemble average concepts but
are used inside LES on an instantaneous basis, i.e., to represent
SGS effects at every gridpoint and time step. However, small-scale
turbulent motion is anisotropic and intermittent, and locally the
energy transfer can either be forwardscatter (from large to small
scales) or backscatter (from small to large scales), which causes
deviations from the equilibrium k-5/3 law.
Eddy viscosity SGS models also assume that SGS stresses and
strains are perfectly aligned, and hence the local dissipation rate ε
= - τij Sij is always positive, thus preventing backscatter of energy

29



These deficiencies of eddy viscosity models have motivated
continued development of new SGS models, including
‣ stochastic models where a random field is imposed at the SGS
level, thus permitting a backscatter of energy,
‣ dynamic models where the Smagorinsky coefficient is
dynamically predicted using a resolved field filtered at two
different scales
‣ velocity estimation models that attempt to model the SGS
velocity fluctuations ui‘‘ instead of SGS stresses τij.

30


NUMERICAL SETUP

‣ Choice of LES grid and domain sizes depend on the physical
flow of interest and the computer capability
‣ Grid-scale motion in LES is nearly isotropic
-> Requiring a grid mesh close to isotropic
‣ From the chosen gridpoints, say 100x100x100, an LES domain is
chosen to resolve several largest (dominant) turbulent eddies
and at the same time resolve eddies as small as possible into the
inertial-subrange scales.
Example: For a convective PBL with 1 km depth, a 5 km x 5 km x 2 km domain
of LES with 100x100x100 gridpoints would cover 3 to 5 large dominant
eddies in each horizontal direction and at the same time resolve small eddies
down to about 100 mx100 mx40 m in size, assuming model resolution is
twice the grid size. For the stable PBL where dominant eddies are smaller, a
smaller domain (and consequently a finer grid) is preferred.

31


BOUNDARY CONDITIONS

Surface boundary conditions
‣ LES cannot resolve the viscous layer close to the surface; its
lowest grid level lies in the surface layer
‣ M–O similarity theory is used as a surface boundary condition to
relate surface fluxes to resolved-scale fields at each grid point
just above the surface
‣ The primary empirical input parameter to these formulas is the
surface roughness
‣ Different from the smooth-wall condition in engineering flows
‣ M–O theory describes ensemble-mean flux–gradient
relationships in the surface layer and may not apply well at the
local LES grid scale. (Especially when LES horizontal grid size is comparable
to or smaller than the height of the first grid level.)
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BOUNDARY CONDITIONS

Upper boundary conditions
‣ The upper boundary of a typical LES domain is usually set to be
well above the PBL top, in order to avoid influences on
simulated PBL flows from artificial upper boundary conditions.
At the top of the domain, turbulence is negligible and a no-
stress condition is applicable. Because turbulent motions in the
PBL may excite gravity waves in the stably stratified inversion
layer, a means of handling gravity waves is often applied.
Typically, a radiation condition, which allows for an upward
escape of gravity waves, or a wave- absorbing sponge layer is
used at the top of the simulation domain.

33


BOUNDARY CONDITIONS

Lateral boundary conditions
‣ Most PBL LESs use periodic boundary conditions.
(inflow at each gridpoint on a sidewall is equal to the outflow on opposite
sidewall)

‣ appropriate for PBLs with homogeneous terrain
‣ no explicit statement of the sidewall boundary (turbulence)
conditions --> computational convenience
‣ no simulation of realistic meteorological flows with
inhomogeneous surface!

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