modeling of atmospheric flow II

8/16/2019 modeling of atmospheric flow II

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Energy MeteorologyInstitute of Physics / ForWindCarl von Ossietzky Universität Oldenburg

Detlev Heinemann

National Cheng Kung University, Tainan, Taiwan – 10 September 2015

- Overview- Model Classes (linear, RANS, LES, ..)

- Scales of Atmospheric Motion

MODELING OF ATMOSPHERIC FLOW (II)

DAAD/NCKU Summer School – Lecture 4



MODELING OF ATMOSPHERIC FLOW

MODELING OF TURBULENT FLOW

2National Cheng Kung University, Tainan, Taiwan – 10 September 2015

! Di"culty in modeling turbulent flows: wide range of length andtime scales --> most approaches are not feasible

! Models of turbulent flow 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 thesimulation has to increase, and the computational cost will also

! Modeling (i.e., not resolving) 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




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

NUMERICAL MODELS OF TURBULENT FLOW





Energy spectrum of turbulent kinetic energy (TKE)

ENERGY OF THE TURBULENT FLOW


! The energy spectrum indicates how much of thetotal TKE is associated with each eddy scale.

! The total TKE is given by the area under the

curve.

! Permanent generation of TKE from shear orbuoyancy at large scales.

! TKE cascades through medium-size eddies to bedissipated by molecular viscosity at the small-eddy scale (TKE is not conserved!).

Stull (2006)




LINEARIZED MODELS

! Definition of a potential function, $(x, z,t), as a continuousfunction that satisfies conservation of mass and momentum,assuming incompressible, inviscid and irrotational flow.

! Vector identity states for any scalar $, " ! "$ = 0

! By definition, for irrotational flow, " ! v = 0

! Therefore v = "$

where $ = $(x, y, z,t) is the velocity potential function! The components of velocity are

u = %$/%x, v = %$/%y, w = %$/%z

! Potential functions $ can be defined for various simple flows.





LINEARIZED MODELS

! Famous example: WAsP (Wind Atlas Analysis and ApplicationProgram) from Risø based on the concept of linearised flowmodels (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

! Poor description of flow separation and recirculation

! Limitations in more complex terrain regions due to the linearityof the equation set





DIRECT NUMERICAL SIMULATION DNS


! Resolves the entire range of turbulent length scales

! E& ect of models is marginalized

! Extremely computationally expensive: computational costs

~ Re3

.! Intractable for flows with complex geometries or flow

configurations




DIRECT NUMERICAL SIMULATION DNS

! Direct numerical simulation of the Navier-Stokes equations fora full range of turbulent motions for all scales („brute force“)

! Only approximations which are numerically necessary tominimise discretisation errors

! Clear definition of all conditions (initial, boundary and forcing)! 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 informationfor verifying and revising turbulence models





LARGE EDDY SIMULATION LES


! Removing the smallest scales of the flow through a filteringoperation

! E& ect of small scale motion is described using subgrid

scale models! Largest and most important scales of turbulence are

resolved

! Greatly reducing the computational e& orts incurred by thesmallest scales

! Requiring greater computational resources than RANSmethods, but far less than DNS




LARGE EDDY SIMULATION LES


Separation of scales:

Large scales: contain most of the energy and fluxes, significantlya& ected by the flow configuration, are explicitly calculated

Smaller scales: more universal in nature & with little energy are

parameterized (SFS model)LES solution supposed to be insensitive to SFS model

Turbulent flow

Energy-containing eddies

Subfilter scale eddies

(not so important)

(important eddies)




Equations:

SFS

Apply filter G

LARGE EDDY SIMULATION


SFS: Subfilter scale



EXAMPLE

Convective Updraft (Moeng, NCAR)







! 100 x 100 x 100 points

! grid sizes < tens of meters

! time step < seconds

! higher-order schemes, not too di& usive

! spin-up time ~ 30 min

! simulation time ~ hours

! massive parallel computers



Typical configuration




! Realistic surface

complex terrain, land use, waves

! Inflow boundary condition

! SFS e# ect near irregular surfaces

! Proper scaling; representations of ensemble mean

! Computational challenge resolve turbulent motion @ ~ 1000 x 1000 x 100 grid points

Massive parallel computing

LES: CHALLENGES I





LES: CHALLENGES II

Using for

! Understand turbulence behavior & di& usion properties

! Develop/calibrate ABL models, i.e. Reynolds averaged models

! Case studies of wind flow in technical environments

Future Goals

! Understand ABL in complex environment and improve itsparameterization (turbulent fluxes, clouds, ...)

! Application of LES for ‘real-world’ wind flow modeling, e.g. inlarge wind farms


ABL: Atmospheric Boundary Layer




REYNOLDS-AVERAGED NAVIER-STOKES RANS


f

non-turbulent

Applyensemble average

Time-averaged equations of motion

Reynolds stressterm





! 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 di& erent levels of closure

! For instationary flows:

Turbulence models used to close the equations are validonly as long as the time over which these changes in the

mean occur is large compared to the time scales of theturbulent motion containing most of the energy






! unknown Reynolds stress terms -> problem of closure

! from four unknowns with four equations we have tenunknowns with still four equations

! Navier-Stokes equations are no longer solvable directly--> RANS

! Turbulence models must be introduced to solve the flowproblem

! inherently di"cult to develop reliable Reynolds stress

models! RANS based CFD codes remain the most practical tools

! Hybrid model incorporating LES: Detached Eddy Simulation(DES)


O G O OS C O




CLOSURE PROBLEM IN THE RANS E UATION I


! Averaging introduces non-linear term from the convectiveacceleration (Reynolds stress):

! Closing the RANS equation requires modeling of Rij

! Simple concept of eddy viscosity: Relating the turbulentstresses to the mean flow to close the system of equations

with't: turbulent eddy viscosity K= 0,5ui‘2: turbulent kinetic energy (ij: Kronecker delta.

Rij = ui‘u j‘

-ui‘u j‘ ='

t ( + ) - ( K + '

t ) (

ij

!ui !u j 2 !uk

!x j !xi 3 !xk





CLOSURE PROBLEM IN THE RANS E UATION II


! eddy viscosity is modeled by analogy with molecularviscosity:

with mixing length lm.

't = lm

2!u

!z





k-$-MODEL FOR TURBULENCE CLOSURE


! most widely used and validated

! low computational costs

! high numerical stability

! good performance, when Reynolds stresses are lessimportant (rarely the case in wind engineering)

! use is superior to other models in simple flow regimes (i.e.,low hills)





! The atmosphere features a wide range of circulation types,with a wide variety of di& erent behaviours. Ex.: Turbulence <—> Planetary waves

! Typically, these circulations are classified according to theirsize (spatial or length scale) and/or their oscillation periodor duration (time scale)

SCALES OF ATMOSPHERIC MOTION












Scale Category Time Scale Spatial Scale Examples

microscale

seconds to

minutes meters to 1 km

turbulence, small

cumulus clouds

mesoscaleminutes to hours

to 1 daykilometers to

hundreds of km

thunderstorms, seabreezes, mountain

circulations

synoptic scale days to weeks thousands of kmfronts, cyclones,

anticyclones

planetary scale weeks to months globalplanetary waves,

el niño







Large Eddy Simulation (LES) Model

MesoscaleTurbulence Cumulus Cumulunimbus convective Extratropical Planetary clouds clouds systems cyclones waves

Mesoscale Model

Numerical Weather Prediction

(NWP) Model

Global Climate Model

S u b g r i d

Trend: Model boundaries shift towards smaller scales

‘ZOO‘ OF ATMOSPHERIC MODELS






Di& erent models need di& erent levels of parametrization!

ATMOSPHERIC

MODELS


Model dx/dy dz timeClimate models 200 km 500 m 100 years

Global weather prediction 20 km 200 m 10 days

Limited area weather prediction 5 km 100 m 2 days

Cloud resolving models 500 m 500 m 1 day

Large eddy models 50 m 50 m 5 hours




EXAMPLE: GLOBAL SCALE

Climate patterns

(e.g., El Niño / La Niña)

Planetary-scale waves










Rossby waves at the boundarybetween polar and mid-latitudeair masses (polar front)

http://aoss.engin.umich.edu/faculty/nrenno/earth.html

EXAMPLE: GLOBAL SCALE




EXAMPLE: SYNOPTIC SCALE

Most of ‘everyday weather’High and low pressuresystems, warm and cold fronts







Sea breeze circulations



EXAMPLE: MESO-SCALE




EXAMPLE: MICRO-SCALE

Mountain circulations (lee vortices in this case)







Individual storms and theircomponent parts

Thunderstorms/collections ofthunderstorms







Small cumulus clouds /turbulent eddies

Boundary layer turbulence






Regional ModelGrid distance 5-10 kmHydrostatic

Mesoscale ModelEx.: WRF, non-hydrostatic

NWP ModelEx.: ECMWF,grid distance: 16 km

ATMOSPHERIC MODELS





HYDROSTATIC VS NON-HYDROSTATIC MODEL

Hydrostatic approach

If H/L " 1 ) vertical velocity is relatively small

Hydrostatic equation valid

Flat terrain; over sea

First generation NWP models were hydrostatic

ATMOSPHERIC MODELS





Non-hydrostatic approach

If H/L ~ 1 ) W of similar order of U

Full equation required

Strong updraft; mountainous regions


ATMOSPHERIC MODELS

HYDROSTATIC VS NON-HYDROSTATIC MODEL




PARAMETERIZATION

Why parameterization?

! Small scale processes are not resolved by large scale

models (because they are sub-grid…)

! E& ect of sub-grid processes on large scale can only berepresented statistically

! Procedure of expressing the net e& ect of a sub-grid

process is called parameterization

ATMOSPHERIC MODELS





PARAMETERIZATION

What is a parameterization and why is it needed?

! Standard Reynolds decomposition and averaging leads to

co-variances that need ‘closure’ or ‘parameterization’.Not each small eddy can be modelled!

! Radiation absorbed, scattered and emitted by molecules,aerosols and cloud droplets play an important role in theatmosphere and need parameterization.

Not each scattering process can be modelled!

! Cloud microphysical processes need ‘parameterization’.Not each droplet can be modelled!

ATMOSPHERIC MODELS





PARAMETERIZATION

ATMOSPHERIC MODELS





PARAMETERIZATION

ATMOSPHERIC MODELS





Thank You!

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modeling of atmospheric flow II