Dynamic subgrid-scale modeling in large-eddy simulation of turbulent flows with a

stabilized finite element method

Andrés E. Tejada-Martínez

Thesis advisor: Kenneth E. Jansen

Department of Mechanical, Aerospace, & Nuclear Engineering

Rensselaer Polytechnic Institute

Outline

• Part I:

• Part II:

- dynamic subgrid-scale modeling in large-eddy simulation

- new models accounting for implicit filter characteristic

- spatial filters for dynamic modeling (test filters)

- physical and numerical energy dissipation

- new dynamic model accounting numerical dissipation associated to the discretization (the stabilized method)

of the discretization

Large-eddy simulation (LES)

large eddies resolved in

LES

• Subgrid-scales not resolved in LES, and must be modeled

Large-eddy simulation (LES)

• In direct numerical simulation (DNS) all scales are resolved

Filtering operation

• Consider f(x) with a wide variety of scales. A filtered function is defined as:

• Scales of and less are damped and f(x) is decomposed into resolved and residual components:

• Two homogenous filters we use are the box and hat filters.

dyyfyxGxf )(),,()(

)(O)( f )( f fff

filter width

yx x+hx-h yxx+h x-h

1/2h 1/hbox hat

The filtered Navier-Stokes equations

• Homogenous arbitrary kernel is used.

• Continuity:

• Momentum:

drij

dij

jij

jii

xx

P

x

uu

t

u )()(1

0

i

i

xu

ijd

ij S 2)(

i

j

j

iij x

u

xu

S21

),,( yxG

ijTijr

kkr

ijdr

ij Sv23

1 )()()(

jijir

ij uuuu )(

)(

3

1 rkkpP

deviatoric subgrid-scalestress at the -level

SCv ST22 ijij SSS 2

• Smagorinsky model for the SGS stress:

subgrid-scale stress at the -level

Dynamic model for• Thus, we have:

• Application of a homogenous, secondary (test) filter, , to the once-filtered eqns. creates a second stress tensor defined as

• This stress is associated to , obtained from successive applications of the primary filter and test filter.

• Assuming scale-invariance, the deviatoric portion of can be modeled by Smagorinsky as

jijir

ij uuuuT ˆˆ)(

ijSdr

ij SSCT ˆˆˆ2 22)(

)ˆ,,(ˆ

yxG

subgrid-scale stressat the -level

)ˆ,,(ˆ

yxG

22SC

)(rijT

ijSdr

ij SSC 22)( 2

Model continued• The Germano identity between and is defined as

• Least squares min. between -resolved and -modeled w.r. t. to leads to a dynamic expression for for use in

)( )()( drij

drij

dij TL

djiji

djiji

djiji

dij uuuuuuuuuuuuL )ˆˆ()()ˆˆ(

ijijs

dij SSSSCL 22 ˆˆˆ2

dijL -resolved

dijL -modeled

dijL

SC 22SC

ijSdr

ij SSC 22)( 2

dijL

Dynamic model with standard test filter

)(2

)( 2 stdklkl

ijijstdS f

MM

MLC

ijijij SSSSM ˆˆ2ˆ

- Averaging in statistically homogenous direction(s)

jijiij uuuuL ˆˆ

filter width ratiosquared

standard test filter )ˆ,,(ˆ

yxG

Dynamic model with wide test filter

)(2

)( 2 wideklkl

ijijwideS g

MM

MLC

ijijij SSSSM~ˆ

~ˆ

2~ˆ

jijiij uuuuL

~ˆ

~ˆ

wide test filter

• Consider replacing test filter with filter defined as the successive applications of the original test filter and a second test filter, .

)ˆ,,(ˆ

yxG )~ˆ,,(~

ˆ

yxG

)~

,,(~ yxG

• The residual stress generated at the -scale is~ˆ

jijir

ij uuuuR~ˆ

~ˆ)(

• The dynamic procedure leads to:

filter width ratiosquared

)~ˆ,,(~

ˆ

yxG

Comments on modeled equations

• The models depend on the ratios and

• Accurate determination of these parameters requires characterization of , and

• In practice is set by the choice of numerical method, thus all three filters are typically poorly characterized.

),,( yxG )ˆ,,(ˆ

yxG

2ˆ

G

.~ˆ

2

).~ˆ,,(~

ˆ

yxG

Discrete test filters• Approximation of a filter operation using quadrature rules

leads to discrete filters

• For example, 2-pt quadrature approximation of a box filtered function leads to:

dyyfxyG )()ˆ,,( 0ˆ 03/23/13/13/2

ˆ)(4

1fffff

y0x 3/1x3/1x3/2x1x 3/2x 1x

h

)2/(1 h

vertexquadrature point

h

Standard and wide discrete filters

• We will be using the following 4 filters:

2/32/12/12/30 8

1

8

3

8

3

8

1~ˆ fffff

)(16

1)(

16

3)(

16

3)(

16

1~ˆ

3/53/43/23/13/13/23/43/50 fffffffff

)(4

1ˆ3/23/13/13/20 fffff

)(2

1ˆ2/12/10 fff filter S1 (standard with rule 1)

filter W2 (wide with rule 2)

filter S2 (standard with rule 2)

filter W1 (wide with rule 1)

Transfer functions for filter S1 on (a) triangles and (b) quads.

(a)

(b)

• Can help find test filter widths:

• Transfer function = Fourier transform of filter kernel ;ˆ

*rk

2/1222zyxr kkkk

*rk = avg. radial wavenumber for a specified value of the transfer function

Decaying isotropic turbulence behind a bar grid

isotropic far field

• In our LES, the larger scales are resolved while thesmaller subgrid-scales are modeled

bar grid

Decaying isotropic turbulence

• In decaying isotropic turbulence the mean flow is zero, motions decay in time due to a lack of kinetic energy production to balance viscous dissipation. Scales have no directional orientation.

• Initial conditions are obtained from experiments of Comte-Bellot and Corrsin (1963). Data exists at two non-dimensional time stations: t42 and t98.

• Domain is a periodic box split by 33 equidistant vertices in each direction.

• We use the Streamline Upwind/Petrov-Galerkin (SUPG) method w/ linear basis and a second order accurate time integrator as described in Whiting and Jansen (2001).

3)2(

Effect of filter width ratio on energy spectraof isotropic turbulence

)(

• We examine the effect of changing the filter width ratio, in on dynamic model results of decaying isotropic turbulence

),(stdf

Filter width ratio assumption based on test filter widths

• Recall the dynamic model coeffs.: and , where

• Two sets of simulations are performed on hexes: (a1) (b1) . Two sets are also performed on tets: (a2) (b2)

• For each set, four simulations are performed: (1) std. filter S1, (2) wide filter W1, (3) std. filter S2, and (4) wide filter W2.

• Test filter widths and are computed based on transfer functions of the standard or wide test filter used.

)(stdf )(wideg

22ˆˆ

h

22 ~ˆ

~ˆ

h

,680.0544.0

h/ h/~

,0.1.588.0

Simulations on hexes

0.53.14

21.5

51.6

0.47.12

9.15

5.11

680.0 680.0

544.0 544.0

22ˆˆ

h

22 ~ˆ

~ˆ

h

Simulations on tets

17.70.21

0.47.11

51.84.22

21.55.12

0.1 0.1

558.0558.0

22ˆˆ

h

22 ~ˆ

~ˆ

h

Dynamic filter width ratio formulation

• Previous results suggest

• Recall stresses and modeled as

• Consider the following identity:

• Least squares minimization between modeled and resolved expressions for the identity above leads to:

)()( stdwide fg

jijir

ij uuuuT ˆˆ)( jijir

ij uuuuR~ˆ

~ˆ)(

ijSdr

ij SSCT ˆˆˆ2 22)( ijSdr

ij SSCR~ˆ

~ˆ

~ˆ2 22)( and

(1)

jijidr

ijdr

ijd

ij uuuuTRQ~ˆ

~ˆˆˆ)()()(

Dynamic filter width ratio formulation (DFWR2)

)/()(2

)ˆ( 2 qqNN

NQC

klkl

ijijS

jijiij uuuuQ~ˆ

~ˆˆˆ ijijij SSSSN

~ˆ

~ˆˆˆ

2

2

2

ˆ

~ˆ

ˆ

~ˆ

• Dividing by results in)/()ˆ( 2 qCS )()( 2 stdstdS fC

)(

)/(

stdf

q (2)

• Recall equation (1): )()( stdwide fg

• Eqns. (1) and (2) can be solved for leading to its dynamic determination and thus a parameter-free model, DFWR2.

Comments on models• The filter width ratio parameter in the classic dynamic

model is not well-characterized.

• Results show sensitivity to filter width ratio. Its accurate determination is important.

• DFWR2 computes the filter width ratio dynamically without parameters.

• DFWR1 is derived similarly to DFWR2. DFWR1 is not parameter-free as it requires the ratio between the widths of the two test filters used.

• DFWR1 and DFWR2 account for implicit filtering characteristic of the numerical method.

DFWR on isotropic turbulence on hexes

Evolution of filter width ratio,

DFWR on isotropic turbulence on tets

Evolution of filter width ratio,

Wall-modeled turbulent channel flow

• Channel geometry:

• Reynolds # based on friction velocity,

• Periodicity in the x- and z-directions. Shear stress boundary condition at the walls (at is obtained via a near-wall model.

• 33 vertices in x, 31 in y and 33 in z. Near-wall features are not well resolved. Vertices are equidistant in each direction. Mesh is made of hexes.

z

y

x

xLzL

4xL)3/4(zL

2800Re180Re

BUu

:u

2

)y

Wall modeled turbulent channel flow

• We compare DFWR2 (w/ S1 and W1) to 2 classic dynamic models: 1) w/ standard filter S1 and 2) w/ wide filter W1.

• For the classic models, test filter widths are based on the filters’ second moment and filter width ratios are taken as:

• No input parameters required for DFWR2.

• Some results are presented in wall units:

31ˆˆ 22

h

91

~ˆ

~ˆ

22

h

uuu /

uhy

y))/(1( 2

1) 2)

Wall force

Expected mean force = 0.435

Mean streamwise (x-) velocity

Direct Numerical Simulation (DNS): Kim, Moin, and Moser (1987)

Dynamic model coefficient

2,,22 )( DFWRwidestdSC

Filter widthratio predicted

by DFWR2

Part II

Physical and numerical energy dissipation

Stabilized FEM Formulation (SUPG)

• See Taylor et.al. (1998), and Hauke and Hughes (1998).

• For our studies:

0),;,( PuqwB ii

dxuqPuuwuwPuqwB iiijijjijitiiii })({),;,( ,*

,,

dxuwLqLuLuw jjiiCiMiijjiMji

e

nel

e

})({ ,,,,~1

dsquPuuw nininnii

h

})({ *

;)( ,*

,,, jijijjitii PuuuL ijijTjiji

Mggcugutc 2

22

1 )()/2(

1

ijT S)(2* ijTij S)(2*

161c

641c

strong SUPG (numerical dissipation)

weak SUPG (numerical dissipation)

SUPG tensor

Numerical and physical dissipations

• Dynamic model (physical) subgrid-scale (SGS) dissipation:

• Numerical dissipation due to SUPG stabilization:

• strong SUPG and weak SUPG.

• Model with standard filter S1: strong (SGS) model Model with wide filter W1: weak (SGS) model

• Wall resolved turbulent channel simulations are performed. 65 vertices in y (normal to walls) are stretched such that there are more vertices near the walls. There are 33 equidistant vertices in x and in y.

• Periodicity in x and z. No-slip velocity at walls. Based on channel half width:

ijijTSGS SS 2

ijijjiMSUPG SLuLu )(

161c 641c

2800Re180Re B

SUPG tensor

Wall forces

Expected mean force = 0.435

Dissipations and eddy viscosity

A modified dynamic model

Model coefficients

SUPG correction:SUPGSGSSGS *

ijijjiijijT SLuLuSS )(2

Corrected SGS diss.: ijijSijijTSGS SSSCSS *22** )(22

Corrected dynamic model: 3

322

*22)()(

)(S

SLuLuSCC

ijijjiMS

S

Strong model w/ and w/out SUPG correction

Expected mean force = 0.435

Strong model w/ and w/out SUPG correction

Weak model w/ and w/out SUPG correction

Expected mean force = 0.435

Weak model w/ and w/out SUPG correction

Other statistics (for strong SUPG cases)

Observations

• SUPG correction allows the dynamic model to properly adjust to the presence of SUPG dissipation.

• When SUPG diss. is of the same order as SGS (dynamic model) diss., SUPG correction has a strong impact.

• The top performer is the weak model with SUPG correction. The worst performer is the strong model without SUPG correction.

• Although not shown, SUPG correction was applied to DFWR2 and helped improve results.

• DFWR2 with SUPG correction is of similar quality as the weak model with SUPG correction.

Summary and future work

• The models proposed here account for the implicit filter characteristic (DFWR1,2) and the dissipative nature nature (SUPG correction) of the numerical method.

• Two main tenets underlie the new models. Dynamic subgrid-scale models should be independent of

• This work has laid the foundation for improved physical subgrid-scale modeling taking into account stabilization and for improved stabilization techniques taking into account physical modeling.

1) the test filter (DFWR2) 2) a change in the numerical method brought about by stabilization

(SUPG correction)

The Navier-Stokes equations

• Continuity:

• Momentum: d

ijjij

jii

xx

p

x

uu

t

u )(1

0

i

i

xu

ijd

ij S 2)(

i

j

j

iij x

u

xu

S21

viscous stress

Transfer functions in multi-dimensions

• Transfer functions can help us determine filter widths:

• is the average radial wavenumber for a specified iso-surface of the transfer function.

• Examples: Filter S1 on regularly connected (a) triangles and (b) quads.

*ˆ

rk

*rk

2/1222zyxr kkkk

quadrature point at centroid

vertex

h

filter support

Ratio between model coefficients)(/)( stdwide fg

Steps in large-eddy simulation of turbulent flows

• The N-S equations are filtered with an arbitrary homogenous kernel .

• Filtering generates a subgrid-scale (SGS) or residual stress which is modeled.

• The modeled filtered N-S eqns. are solved numerically to represent the resolved motions.

),,( yxGPrimary filter kernel