andr és e. tejada-martínez thesis advisor : kenneth e. jansen
DESCRIPTION
Dynamic subgrid-scale modeling in large-eddy simulation of turbulent flows with a stabilized finite element method. Andr és E. Tejada-Martínez Thesis advisor : Kenneth E. Jansen Department of Mechanical, Aerospace, & Nuclear Engineering Rensselaer Polytechnic Institute. Outline. - PowerPoint PPT PresentationTRANSCRIPT
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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
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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
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Large-eddy simulation (LES)
large eddies resolved in
LES
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• Subgrid-scales not resolved in LES, and must be modeled
Large-eddy simulation (LES)
• In direct numerical simulation (DNS) all scales are resolved
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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
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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
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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
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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
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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
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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
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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
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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
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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)
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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
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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
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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(
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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
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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
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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
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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
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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)()()(
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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.
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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.
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DFWR on isotropic turbulence on hexes
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Evolution of filter width ratio,
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DFWR on isotropic turbulence on tets
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Evolution of filter width ratio,
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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
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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)
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Wall force
Expected mean force = 0.435
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Mean streamwise (x-) velocity
Direct Numerical Simulation (DNS): Kim, Moin, and Moser (1987)
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Dynamic model coefficient
2,,22 )( DFWRwidestdSC
Filter widthratio predicted
by DFWR2
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Part II
Physical and numerical energy dissipation
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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
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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
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Wall forces
Expected mean force = 0.435
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Dissipations and eddy viscosity
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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
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Strong model w/ and w/out SUPG correction
Expected mean force = 0.435
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Strong model w/ and w/out SUPG correction
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Weak model w/ and w/out SUPG correction
Expected mean force = 0.435
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Weak model w/ and w/out SUPG correction
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Other statistics (for strong SUPG cases)
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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.
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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)
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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
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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
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Ratio between model coefficients)(/)( stdwide fg
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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