Modeling of turbulent flow

WIND ENERGY METEOROLOGY

UNIT 5
MODELING OF TURBULENT FLOW

Detlev Heinemann

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

Dienstag, 17. Mai 2011



‣ Difficulty in modeling turbulent flows: wide range of length and
time scales
--> most approaches are not feasible
‣ Turbulence models can be classified based on the range of these
length and time scales that are modeled and/or resolved
‣ If more turbulent scales are resolved, the resolution of the
simulation has to increase, and the computational cost will also
‣ Modeling all or most of the turbulent scales:
--> very low computational cost
--> decreased accuracy

‣ Additional problem: non-linear terms in the governing equations
--> Numerical solution with appropriate boundary and initial
conditions
2


NUMERICAL MODELS OF TURBULENT FLOW
Numerical methods of studying (turbulent) motion:

‣ Linearized flow models
‣ Reynolds-average modeling (RANS)
‣ Modeling ensemble statistics
‣ Direct numerical simulation (DNS)
‣ Resolving all eddies
‣ Large eddy simulation (LES)
‣ Intermediate approach

3


LINEAR MODELS
‣ Famous example: WAsP (Wind Atlas Analysis and Application
Program) from RISØ based on the concept of linearised flow
models (Jackson and Hunt, 1975)

‣ Developed initially for neutrally stable flow over hilly terrain
‣ Contains simple models for turbulence and surface roughness
‣ Best suited to more simple geometries
‣ Quick and accurate for mean wind flows
‣ Poorly predict flow separation and recirculation

‣ Limitations in more complex terrain regions due to the linearity
of the equation set

4


DIRECT NUMERICAL SIMULATION (DNS)
‣ Direct numerical simulation of the Navier-Stokes equations for
a full range of turbulent motions for all scales („brute force“)
‣ Only approximations which are necessary numerically to
minimise discretisation errors
‣ Clear definition of all conditions (initial, boundary and forcing)
and the production of data for every single variable
‣ Only simple geometries and low Reynolds numbers will be
modelled
‣ Very large computational requirements
‣ No practical engineering tool (--> fundamental research)
‣ Basic computations using DNS provide very valuable
information for verifying and revising turbulence models
5


LARGE EDDY SIMULATION
Separation of scales:
Large scales: contain most of the energy and fluxes, significantly
affected by the flow configuration, are explicitly calculated
Smaller scales: more universal in nature & with little energy are
pameterized (SFS model)
LES solution supposed to be insensitive to SFS model

Energy-containing eddies
(important eddies)
turbulent flow
Subfilter scale eddies
(not so important)

6


Equations:

Apply filter G

SFS

7



Convective Updraft (Moeng, NCAR)

8



‣ 100 x 100 x 100 points
‣ grid sizes < tens of meters
‣ time step < seconds
‣ higher-order schemes, not too diffusive
‣ spin-up time ~ 30 min, no use
‣ simulation time ~ hours
‣ massive parallel computers

9


LES: Challenges (I)

‣ Realistic surface
complex terrain, land use, waves
‣ Inflow boundary condition
‣ SFS effect near irregular surfaces
‣ Proper scaling; representations of ensemble mean

‣ Computational challenge
resolve turbulent motion @ ~ 1000 x 1000 x 100 grid points
Massive parallel computing

10


LES: Challenges (II)
Using for
‣ Understand turbulence behavior & diffusion property
‣ Develop/calibrate PBL models, i.e. Reynolds average models
‣ Case studies of wind flow in technical environments

Future Goals
‣ Understand PBL in complex environment and improve its
parameterization (turbulent fluxes, clouds, ...)
‣ Application of LES for „real-world“ wind flow modeling e.g. in
large wind farms

11


REYNOLDS-AVERAGED NAVIER-STOKES (RANS)
Time-averaged equations of motion

f

Apply
ensemble average

non-turbulent

12



‣ oldest approach to turbulence modeling
‣ Solving an ensemble version of the governing equations,
introducing new apparent stresses: „Reynolds stresses“
‣ This adds a second order tensor of unknowns
--> various models with different levels of closure
‣ For instationary flows:
Turbulence models used to close the equations are valid only
as long as the time over which these changes in the mean
occur is large compared to the time scales of the turbulent
motion containing most of the energy

13



‣ unknown Reynolds stress terms -> problem of closure
‣ from four unknowns with four equations we have ten unknowns
with still four equations

‣ Navier-Stokes equations are no longer solvable directly --> RANS
‣ Turbulence models must be introduced to solve the flow
problem

‣ inherently difficult to develop reliable Reynolds stress models
‣ RANS based CFD codes remain the most practical tools
‣ Hybrid model incorporating LES: Detached Eddy Simulation
(DES)
14


CLOSURE PROBLEM IN THE RANS EQUATION

‣ Averaging introduces non-linear term from the convective
acceleration (Reynold‘s stress):
Rij = vi‘vj‘
‣ Closing the RANS equation requires modeling of Rij
‣ Simple concept of eddy viscosity: Relating the turbulent
stresses to the mean flow to close the system of equations

∂vi ∂vj 2 ∂vk
-vi‘vj‘ = νt ( + ) - ( K + νt ) δij
∂xj ∂xi 3 ∂xk
with
νt: turbulent eddy viscosity
K= 0,5vi‘2: turbulent kinetic energy
δij: Kronecker delta.
15


CLOSURE PROBLEM IN THE RANS EQUATION (II)

‣ eddy viscosity is modeled by analogy with molecular viscosity:
∂u
νt = lm2
∂z

with mixing length lm.

16


k-ε-MODEL FOR TURBULENCE CLOSURE

‣ most widely used an validated
‣ low computational costs
‣ high numerical stability
‣ good performance, when Reynolds stresses are less important
(rarely the case in wind engineering)
‣ use is superior to other models in simple flow regimes (i.e., low
hills)

17


DIRECT NUMERICAL SIMULATION (DNS)

‣ Resolves the entire range of turbulent length scales
‣ Effect of models is marginalized
‣ Extremely computationally expensive: computational costs
~ Re .
3

‣ Intractable for flows with complex geometries or flow
configurations

18


LARGE EDDY SIMULATION (LES)

‣ Removing the smallest scales of the flow through a filtering
operation
‣ Effect of small scale motion is described using subgrid scale
models
‣ --> Largest and most important scales of turbulence are resolved
‣ greatly reducing the computational efforts incurred by the
smallest scales
‣ Requiring greater computational resources than RANS methods,
but far less than DNS

19

Modeling of turbulent flow

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