Introduction to classical large eddy simulation (LES) of turbulent flows Andrés E. Tejada-Martínez...
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Transcript of 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:
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:
jijijiji uuuuuuuu
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:
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(
Residual (subgrid-scale (SGS)) stress
isoij
dijij
kkij
kkkkijjijidij uuuuuuuu
)(3
1)(
kk
kkkkijisoij uuuu
)(3
1
• Decompose the SGS stress as
• Note that in general: kkijijdij AAA )3/1(
Residual (subgrid-scale (SGS)) stress
isoij
dijij
kkij
kkkkijjijidij uuuuuuuu
)(3
1)(
kk
kkkkijisoij uuuu
)(3
1
• Decompose the SGS stress as
deviatoric (trace-free) component isotropic component
• Note that in general: kkijijdij AAA )3/1(
Residual (subgrid-scale (SGS)) stress
isoij
dijij
kkij
kkkkijjijidij uuuuuuuu
)(3
1)(
kk
kkkkijisoij uuuu
)(3
1
• Decompose the SGS stress as
• 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
• Note that in general: kkijijdij AAA )3/1(
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
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
resolved (large) scales sub-grid (residual) scales
scales ininertial range
dissipative scales
spectrumbased on
spectrum based on u
u
)(u )(u
Sketch of energy spectrum in LES )(u
most energetic scales usually resolved in RANSS and general ocean circulation simulations
)/(/ 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 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
Smagorinsky SGS model
• Recall that the SGS stress and density buoyancy flux must bemodeled or approximated
Smagorinsky (1967) model: ijTd
jijidij Suuuu 2)(
||)( 2 SCST
i
j
j
iij x
u
x
uS
2
1ijij SSS 2||
Smagorinsky coefficient
Both are trace-free
Smagorinsky SGS model
• Recall that the SGS stress and density buoyancy flux must bemodeled or approximated
Smagorinsky (1967) model: ijTd
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
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
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
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
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
j
dij
j
i
ij
ij
i
x
T
x
u
x
P
x
uu
t
u
2
2 ˆ
Re
1ˆˆˆ
ˆ
ijSd
jijidij SSCuuuu ||)(2)( 2
• With this new filter, the filtered momentum equation becomes:
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
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
j
dij
j
i
ij
ij
i
x
T
x
u
x
P
x
uu
t
u
2
2 ˆ
Re
1ˆˆˆ
ˆ
ijSd
jijidij SSCuuuu ||)(2)( 2
• With this new filter, the filtered momentum equation becomes:
ijSd
jijid
ij SSCuuuuT ˆ|ˆ|)ˆ(2)ˆˆ( 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
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
j
dij
j
i
ij
ij
i
x
T
x
u
x
P
x
uu
t
u
2
2 ˆ
Re
1ˆˆˆ
ˆ
ijSd
jijidij SSCuuuu ||)(2)( 2
• With this new filter, the filtered momentum equation becomes:
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
Dynamic Smagorinsky model
• Consider the following tensor proposed by Germano : dij
dij
dij TL
djiji
djiji
djiji
dij uuuuuuuuuuuuL )ˆˆ()()ˆˆ( (resolved)
Dynamic Smagorinsky model
• Consider the following tensor proposed by Germano : dij
dij
dij TL
djiji
djiji
djiji
dij uuuuuuuuuuuuL )ˆˆ()()ˆˆ( (resolved)
ijSijSdij SSCSSCL ||)(2ˆ|ˆ|)ˆ(2 22 (modeled)
Dynamic Smagorinsky model
• Consider the following tensor proposed by Germano : dij
dij
dij TL
djiji
djiji
djiji
dij uuuuuuuuuuuuL )ˆˆ()()ˆˆ( (resolved)
ijSijSdij SSCSSCL ||)(2ˆ|ˆ|)ˆ(2 22 (modeled)
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)
Dynamic Smagorinsky model
• Consider the following tensor proposed by Germano : dij
dij
dij TL
djiji
djiji
djiji
dij uuuuuuuuuuuuL )ˆˆ()()ˆˆ( (resolved)
ijSijSdij SSCSSCL ||)(2ˆ|ˆ|)ˆ(2 22 (modeled)
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
Dynamic Smagorinsky model
u
)/(/ hOk
Unresolved, subgrid scales Resolved scales
Sketch of spectra
ln(E(k))
ln(k))
Spectrum based on
Dynamic Smagorinsky model
uu
)/(/ hOk
Spectrum based on
Unresolved, subgrid scales Resolved scales
Sketch of spectra
ln(E(k))
ln(k))
Spectrum based on
Dynamic Smagorinsky model
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
• Recall the classical Reynolds decomposition and let
• In LES:
Basic relationships
uuu
yxt ,,
uuu
• Recall the classical Reynolds decomposition and let
• In LES:
Basic relationships
turbulence intensities (not subgrid components)
uuu
2uurms
wu
yxt ,,
uuu
2 rms
• Recall the classical Reynolds decomposition and let
• 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
Stratification effects
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
Stratification effects
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(
Langmuir cells in wind-driven channelz
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(
Langmuir cells in wind-driven channelz
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
• 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(
Langmuir cells in wind-driven channelz
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
• 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