introduction to classical large eddy simulation (les) of turbulent flows andrés e. tejada-martínez...

Introduction to classical large eddy simulation (LES) of turbulent flows

Andrés E. Tejada-Martínez

Center for Coastal Physical Oceanography

Old Dominion University

Outline

• Part I: Theory

• Part II: Computations

- introduction to spatial filters

- equations governing the large eddies (filtered (LES) equations)

- LES of isotropic turbulence and unstratified/stratified channel flows

- animations (flow over an airfoil, flow over a cavity)

- subgrid-scale (SGS) models/approximations in the filtered equations

- importance of numerical discretization (i.e. the numerical solver)

- LES of Langmuir turbulence

Steps in large-eddy simulation (LES)

• The Navier-Stokes equations are filtered with a low-pass filter

• Filtering presents a closure problem as an unknown residual (subgrid-scale) stress which may be modeled or approximated

• The modeled filtered Navier-Stokes equations governing the largest scales (or the large eddies) are numerically solved.

- filtering attenuates smallest (residual) scales, while preserving largest

- stress represents effect of attenuated smallest scales on largest scales

Large-eddy simulation

large eddies

resolved in LES

• Residual scales not resolved in LES, and must be modeled

Large-eddy simulation

• Residual scales are modeled under assumption of isotropy

Filtering in real space

• A filtered function is defined as:

• Examples of homogeneous filter kernels, , are

• Filtering attenuates scales less than and splits the function as

dyyfyxGxf )(),()(

fff

yx x+hx-h

yx x+hx-h

1/2h 1/hbox hat

large component

residual (small) component

|)(| yxG

h2

)(O

Filtering in real space

• Filtering attenuates or removes (depending on the shape of the filter kernel) scales on the order of the filter width:

• Note that in general except for the sharp cutoff filter 0u

Scales < )(O

Filter kernels in real space

• The width of G(r) may be defined with respect to the box filter as 2/1

2 )(12

drrGr

• See Turbulent Flows by S.B. Pope for functional forms of kernels

Filters in Fourier space: transfer function

• The Fourier transform (F.T.) of a filtered function is the transfer function of the filter multiplied by the F.T. of the un-filtered function:

].[.]).[.2(].[. uTFGTFuTF

• For the sharp cutoff filter but in general this is not true 0u

ln(E(k))

ln(k))

filtered spectrumusing box filter

un-filtered spectrumIn the low wavenumber range

Sketch of filtered energy spectra

Ck

filtered spectrumusing cutoff filter

• Filtering with the box filter leads to an attenuation of scales around

• Filtering with the sharp cutoff filter preserves scales at less than and completely erases scales at higher wavenumbers

Ck

Ck

Both filters leavelow wavenumber content untouched

un-filtered spectrumIn the high wave-number range

LES and other approaches • For a turbulent flow, the Navier-Stokes (N-S) equations contain a large

number of scales.

• While solving these equations numerically, the grid must contain a great number of points in order to represent (resolve) all of the scales present.

• In LES we filter the equations, thereby suppressing the smaller scales. With fewer scales, the filtered equations need less grid points.

• In direct numerical simulation (DNS) no filtering is performed, as the simulation attempts to represent all scales down to the dissipative ones

• In Reynolds-averaged N-S simulation (RANSS), we solve the ensemble-averaged N-S. Averaging suppresses all of the scales except for the largest, thus RANSS requires much fewer grid points than LES and DNS.

• LES resolves many more scales than RANSS, but not as many as DNS.

4/9Re/ LNo. of grid points

Unfiltered equations

32

2

)(Re

1)(ib

j

i

ij

jii Rix

u

x

p

x

uu

t

u

0

i

i

x

u

Unfiltered equations

0 ),(0* txi

),(),(),( 3 txtxtx ibi

32

2

)(Re

1)(ib

j

i

ij

jii Rix

u

x

p

x

uu

t

u

0

i

i

x

u

2

2)(

ii

i

xx

u

t

21 ,3 ),(),(xxib txtx

1PrRe

Filtered momentum equation

2

2

Re

1)(

j

i

ij

jii

x

u

x

p

x

uu

t

u

• Filter the momentum eqn. with an arbitrary homogenous filter of width .Homogeneity of filter allows commutation with differentiation:


2

2

Re

1)(

j

i

ij

jii

x

u

x

p

x

uu

t

u


jijijiji uuuuuuuu


2

2

Re

1)(

j

i

ij

jii

x

u

x

p

x

uu

t

u


jijijiji uuuuuuuu • leads to

j

ij

j

i

ij

jii

xx

u

x

p

x

uu

t

u

2

2

Re

1)(

jijiij uuuu

• is an unknown stress accounting for the effect of the filtered-out small scales on the large scales governed by the filtered equation

ij

Residual (subgrid-scale (SGS)) stress

isoij

dijij • Decompose the SGS stress as

• Note that in general: kkijijdij AAA )3/1(


isoij

dijij

kkij

kkkkijjijidij uuuuuuuu

)(3

1)(

kk

kkkkijisoij uuuu

)(3

1

• Decompose the SGS stress as



isoij

dijij

kkij


)(3

1)(

kk

kkkkijisoij uuuu

)(3

1


deviatoric (trace-free) component isotropic component



isoij

dijij

kkij


)(3

1)(

kk

kkkkijisoij uuuu

)(3

1


• This decomposition leads to

deviatoric (trace-free) component isotropic component

j

dij

j

i

ij

jii

xx

u

x

P

x

uu

t

u

2

2

Re

1)(

• The modified filtered pressure contains the isotropic part of the SGS stress


Filtered equations

0

i

i

x

u

i

i

iii xxx

ut

2

2

)(Re

12

2

bj

dij

j

i

ij

ij

i Rixx

u

x

P

x

uu

t

u

Filtered equations

)(3

1)()( kkkkijjiji

djiji

dij uuuuuuuuuuuu

0

i

i

x

u

i

i

iii xxx

ut

2

2

)(Re

12

2

bj

dij

j

i

ij

ij

i Rixx

u

x

P

x

uu

t

u

SGS stress:

SGS stress

Filtered equations

)(3

1)()( kkkkijjiji

djiji

dij uuuuuuuuuuuu

0

i

i

x

u

i

i

iii xxx

ut

2

2

)(Re

12

2

bj

dij

j

i

ij

ij

i Rixx

u

x

P

x

uu

t

u

SGS density flux: iii uu

SGS stress:

(obtained in same way as the SGS stress)

SGS density flux

SGS stress

Comments on the filtered equations

• The SGS stress and SGS density flux present closure problems and must be modeled or approximated in terms of filtered variables only

• The filtered equations are numerically solved for the filtered variablesdescribing the large scales ),,( Pui

• In theory, the filter used to obtain the filtered equations is arbitrary

• In practice, the filter is inherently assumed by the discretization (i.e. the numerical method used to solve the filtered equations and the SGS models)

• The discretization can only represent (resolve) down to scales on the order of 1,2, or 3 times the grid cell size, h, thereby “filtering-out” smaller scales.

)31( hhO

ln(E(k))

ln(k))

3

-5

dissipative scales

spectrum based on u

)(u

Sketch of energy spectrum in LES )(u

most energetic scales usually resolved in RANSS and general ocean circulation simulations

ln(E(k))

ln(k))

3

-5

scales ininertial range

dissipative scales

spectrumbased on

spectrum based on u

u



ln(E(k))

ln(k))

3

-5

resolved (large) scales sub-grid (residual) scales

scales ininertial range

dissipative scales

spectrumbased on

spectrum based on u

u

)(u )(u



)/(/ hO

)31( hhO

• We choose the grid size, h, to fall within the inertial range to facilitate SGS modeling

ln(E(k))

ln(k))

Spectrum of obtainedwith discretization B

Role of discretization in LES

u

u

• Discretization A behaves more like a sharp cutoff filter, while B behavesmore like a box filter

• Ideally we would aim for a discretization like A

Spectrum of obtainedwith discretization A

u

)/(/ hO

Spectrum of

Both A and B do a good jobrepresenting low wavenumber spectrum of u

Smagorinsky SGS model

• Recall that the SGS stress and density buoyancy flux must bemodeled or approximated



Smagorinsky (1967) model: ijTd

jijidij Suuuu 2)(

Both are trace-free




jijidij Suuuu 2)(

||)( 2 SCST

i

j

j

iij x

u

x

uS

2

1ijij SSS 2||

Smagorinsky coefficient

Both are trace-free




jijidij Suuuu 2)(

||)( 2 SCST

i

j

j

iij x

u

x

uS

2

1

Analogously: )/( iTiii xuu

||)( 2 SCT

ijij SSS 2||

Smagorinsky coefficient

Both are trace-free

The eddy (turbulent) viscosity

• The turbulent viscosity has units . Because we are working with the smallest resolved scales, we can set

TL /2

• In LES the SGS range starts at the inertial range, thus we may invoke Kolmogorov’s 2nd hypothesis:

L

Statistics of scales of size, say, within the inertial range have a universal Lform uniquely determined by the rate of energy transfer,

• And we may have 3/43/13/12 )/( CT T

• In a global sense, the rate of energy transfer within the inertial range is roughly equal to the SGS dissipation. Here we assume it locally:

|||| 22/32

2

SCSSSC

TTijdij

Difficulties with the Smagorinsky model

Smagorinsky model:

||)( 2 SCST

• For isotropic turbulence, Lilly (1967) showed that 16.0SC

• The constant coefficient allows for a non-vanishing turbulent viscosityat boundaries and in the presence of relaminarization

• The Smagorinsky coefficient should be a function of space and time

• In 1991, Germano and collaborators derived a dynamic expressionfor the Smagorinsky coefficient

Major difficulty:

ijTd

jijidij Suuuu 2)(

Dynamic Smagorinsky model • Recall filtering the N-S equations with an homogeneous filter of width

j

dij

j

i

ij

ij

i

xx

u

x

P

x

uu

t

u

2

2

Re

1ijS

djiji

dij SSCuuuu ||)(2)( 2


j

dij

j

i

ij

ij

i

xx

u

x

P

x

uu

t

u

2

2

Re

1

• Consider a new filter made up from successive applications of the 1st filter (above) and a new “test” filter. This “double” filter has width

• Application of this “double” filter is denoted by a “bar-hat” in the form of f

ijSd

jijidij SSCuuuu ||)(2)( 2


j

dij

j

i

ij

ij

i

xx

u

x

P

x

uu

t

u

2

2

Re

1



j

dij

j

i

ij

ij

i

x

T

x

u

x

P

x

uu

t

u

2

2 ˆ

Re

1ˆˆˆ

ˆ

ijSd


• With this new filter, the filtered momentum equation becomes:


j

dij

j

i

ij

ij

i

xx

u

x

P

x

uu

t

u

2

2

Re

1



j

dij

j

i

ij

ij

i

x

T

x

u

x

P

x

uu

t

u

2

2 ˆ

Re

1ˆˆˆ

ˆ

ijSd



ijSd

jijid

ij SSCuuuuT ˆ|ˆ|)ˆ(2)ˆˆ( 2


j

dij

j

i

ij

ij

i

xx

u

x

P

x

uu

t

u

2

2

Re

1



j

dij

j

i

ij

ij

i

x

T

x

u

x

P

x

uu

t

u

2

2 ˆ

Re

1ˆˆˆ

ˆ

ijSd



ijSd

jijid

ij SSCuuuuT ˆ|ˆ|)ˆ(2)ˆˆ( 2

• Scale invariance: Both and are in the inertial range, thus SS CC

Dynamic Smagorinsky model

• Consider the following tensor proposed by Germano : dij

dij

dij TL



dij

dij TL

djiji

djiji

djiji

dij uuuuuuuuuuuuL )ˆˆ()()ˆˆ( (resolved)



dij

dij TL

djiji

djiji

djiji


ijSijSdij SSCSSCL ||)(2ˆ|ˆ|)ˆ(2 22 (modeled)



dij

dij TL

djiji

djiji

djiji



klkl

ijijS MM

MLC

2)( 2

• Minimization of the difference between these two with respect to Cs leads to:

jijiij uuuuL ˆˆ

ijijij SSSSM ˆ|ˆ|||

2ˆ

- Averaging in statistically homogenous direction(s)



dij

dij TL

djiji

djiji

djiji



klkl

ijijS MM

MLC

2)( 2

• Minimization of the difference between these two with respect to Cs leads to:

jijiij uuuuL ˆˆ

ijijij SSSSM ˆ|ˆ|||

2ˆ

- Averaging in statistically homogenous direction(s)

• Explicit application of test filter (denoted by a “hat”) is required, unlike 1st filter

ln(E(k))

ln(k))

Spectrum based on


u

)/(/ hOk

Unresolved, subgrid scales Resolved scales

Sketch of spectra

ln(E(k))

ln(k))

Spectrum based on


uu

)/(/ hOk

Spectrum based on

Unresolved, subgrid scales Resolved scales

Sketch of spectra

ln(E(k))

ln(k))

Spectrum based on


u

• By applying the test filter, the Germano formulation samples the field betweenthe subgrid scales and the subtest scales in order to obtain the model coeff.

u

)/(/ hOk

Spectrum based on

)2/(ˆ/ hOk

Unresolved, subgrid scales

Subtest scales

Resolved scales

Sketch of spectra

Dynamic mixed model

djiji

dij uuuu )( • Recall and uuu ii

• Inserting the former into the latter leads to

djiji

dij uuuu )( + subgrid-scale terms

• The subgrid-scale terms can be approximated via the Smagorinsky model

subgrid component

• A dynamic coefficient in the Smagorinky model can be derived here as well

• This mixed approach leads to a modeled SGS stress better correlated withthe true SGS stress. Both approaches lead to good correlation with true SGS energy dissipation

true SGS stress

LES methodology used in computations

ijSdij SSC

T

||)(2 2

0

i

i

x

u

i

i

iii xxx

ut

2

2

)(Re

112

2

bij

dij

j

i

ij

ij

i RiFxx

u

x

P

x

uu

t

u

SGS density flux model: ii xSC

T

/||)( 2

SGS stress model:

SGS density flux

SGS stress

• Model coefficients in SGS models are computed dynamically as described

• Horizontal derivatives (in x and y) are treated spectrally

• Vertical (z-) derivatives are treated with 6th or 5th order implicit stencils

• To prevent spurious high wavenumber content not resolvable by the grid, advection terms are:

• The high order accuracy of this discretization allows for it to behave like the sharp cutoff filter

Numerical scheme used in computations

- restriction to periodic boundaries in x and y

- allows Dirichlet and Neumann boundaries in z

1. de-aliased in x and y2. filtered in z with a high order implicit filter

• Isotropic turbulent fluctuations decay in time due to viscous dissipation in the absence of energy source

• Comte-Bellot & Corrsin (1971) studied this flow by passing air at

U=10 m/s through a bar grid with cells of size M = 2 in.

• Data in the form of energy spectra is available at tU/M=42,98

• Each side of computational box is 55cm. Grid is 33x33x33 periodic.

• Size of smallest resolved scales about 35 x (size of dissipative scales (0.07cm))

• Energy spectrum of initial condition matches that recorded in experiment at tU/M=42. Numerical solution will be compared to data at tU/M=98.

LES of decaying isotropic turbulence

Decaying isotropic turbulence behind a bar grid

isotropic far field

bar grid

Effect of sub-grid scale (SGS, residual) model

3-D energy spectra

• Channel geometry:

• Reynolds No. based on friction velocity,

• Periodicity in x and y. No-slip velocity and fixed density at walls

• Grid is 33x33x65 (# of points in x, y, and z), stretched in z.

• DNS of Kim, Moin & Moser (192x160x129) is compared with our results

y

z

x

xLyL

hLx 4hLy )3/4(

2800Re180Re

hUhu bb

:u

h2

Geometry and grid for LES of channel

2/wallF

bodyF

2/wallF

z

xwallwall Fu Re

Body force

• Flow is driven by a body force. There are 2 ways to determine this force:

1. Force control to achieve desired Re

hz

hz

2/wallF

bodyF

2/wallF

z

xwallwall Fu Re

Body force

• Flow is driven by a body force. There are 2 ways to determine this force:

1. Force control to achieve desired Re

bRe2. Mass control to achieve desired

- Body force is dynamically adjusted towards desired bulk velocity

hz

hz

uuu

yxt ,,

• Recall the classical Reynolds decomposition and let

Basic relationships

uuu

yxt ,,

uuu


• In LES:

Basic relationships

uuu

yxt ,,

uuu


• In LES:

Basic relationships

turbulence intensities (not subgrid components)

uuu

2uurms

wu

yxt ,,

uuu

2 rms


• In LES:

• We want to study:

resolved Reynolds shear stress

Basic relationships

turbulence intensities (not subgrid components)

Wall forces in unstratified channel flow

• Theoretical mean force = 0.435

• DNS mean force = 0.426

Mean velocity in unstratified channel

Variances in unstratified channel flow

Shear stresses in unstratified channel

1-D spectra in unstratified channel

• Spectra obtained from autocorrelation in x in the LES with force control

• Our high order discretization behaves like a sharp cutoff filter

1-D spectra in unstratified channel

• Cases 9, 10, and 11 use the same # of grid points as our LES.

• Low order discretization used by Najjar and Tafti behaves like a box filter

as spectra deviates from DNS data at around kx=4.

Results from Najjar and Tafti, Physics of Fluids, 1996

Ref. 34 is DNS data usedearlier from Kim, Moin & Moser, J. Fluid Mech., 1987

Stratification effects

dibbi

b

RiFuPuut

u

312 )(

Re

10u

dut

2


dibbi

b

RiFuPuut

u

312 )(

Re

10u

dut

2

)/( 20 bb UghRi

strength of stratification

)/( 20 ughRi

Ri based on bulk velocity Ri based on friction velocity


dibbi

b

RiFuPuut

u

312 )(

Re

10u

dut

2

)/( 20 bb UghRi

strength of stratification

)/( 20 ughRi

240,120,60,16,0Ri

Ri based on bulk velocity Ri based on friction velocity

• Ri based on friction velocity is more convenient to track

• We will look at with fixed density at top and bottom

Wall Forces

• Theoretical mean force = 0.435

• DNS mean force = 0.426

High/low speed streaks near wall

x

y

Fully turbulent

Laminarized

Mean velocity

Instantaneous streamwise velocity contours

x

z

Ri = 0

Ri = 60

Root mean squares of fluctuations

Shear stress and turbulence structure

Nusselt numbers

Nusselt No. = measure of wall

mass transport due to turbulence

Density statistics

Instantaneous density contours

x

z

Ri = 0

Ri = 60

Mixing Efficiency

Observations

• Stratification suppresses turbulence intensities.

• Stratification leads to a density interface separating density into two layers.

• Within the two layers density is well mixed

• Stratification leads to a quasi-periodic flow structure in the core co-existing with turbulent structures near the boundaries

Langmuir cells in wind-driven channelz

xhz

0zsurface

wallno-slip wall

hLx 4hLy )3/4(


xhz

0zsurface

wall

• Surface stress is applied such that 180/Re hu

• Craik-Leibovich (C-L) vortex forcing is added to the filtered momentum equation to account for Langmuir cells

no-slip wall

hLx 4hLy )3/4(


xhz

0zsurface

wall



• Simulations are distinguished by the turbulent Langmuir number

180/ ST uuLa Su - Stokes drift vel. in vortex forcing

no-slip wall

hLx 4hLy )3/4(


xhz

0zsurface

wall



• Simulations are distinguished by the turbulent Langmuir number

180/ ST uuLa Su - Stokes drift vel. in vortex forcing

- Simulations with TLa 1TLa5.0TLa

and use (33x33x65)-grid with z-stretching

- Simulation with uses (33x33x97)-grid with z-stretching

no-slip wall

hLx 4hLy )3/4(

Mean velocity

• Langmuir cells tend to homogenize the upper water column

Variances

• Langmuir cells increase horizontal fluctuations in the upper water column

Time-averaged x-component of vorticity

z/h

y/h

TLa

5.0TLa

• Counter rotating cells are seen in the case with C-L vortex forcing, 5.0TLa

Engineering applications of LES

Simulation by Kenneth Jansen, atRensselaer PolytechnicInstitute

• Velocity contours on 4 planes parallel to wing and 1 plane normal to wing

• High/low speed streaks appear on parallel planes closer to wing

• LES of flow around an airfoil

introduction to classical large eddy simulation (les) of turbulent flows andrés e. tejada-martínez...

Documents

filter kernel scales

filtered function

attenuation of scales

fewer scales

scales present

smaller scales

large number of scales

smallest residual scales