1. Introduction to classical large eddy simulation (LES) of turbulent flows
Andrés E. Tejada-Martínez
Center for Coastal Physical Oceanography
Old Dominion University

2. Outline - introduction to spatial filters
Part I: Theory
Part II: Computations
- introduction to spatial filters
- equations governing the large eddies (filtered (LES) equations)
- importance of numerical discretization (i.e. the numerical solver)
- subgrid-scale (SGS) models/approximations in the filtered equations
LES of isotropic turbulence and unstratified/stratified channel flows
LES of Langmuir turbulence
- animations (flow over an airfoil, flow over a cavity)

3. 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

4. Large-eddy simulation
large eddies resolved in LES

5. Large-eddy simulation
Residual scales not resolved in LES, and must be modeled
Residual scales are modeled under assumption of isotropy

6. 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
1/2h
1/h
box
hat y y
x-h x
x+h
x-h x
x+h
residual (small) component
large component

7. Filtering in real space
Scales <
Note that in general except for the sharp cutoff filter
Filtering attenuates or removes (depending on the shape of the filter
kernel) scales on the order of the filter width:

8. Filter kernels in real space
The width of G(r) may be defined with respect to the box filter as
See Turbulent Flows by S.B. Pope for functional forms of kernels

9. 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:
For the sharp cutoff filter but in general this is not true

10. Sketch of filtered energy spectra
Both filters leave
low wavenumber
content untouched
ln(E(k))
un-filtered spectrum
In the low wavenumber range
filtered spectrum
using cutoff filter
filtered spectrum
using box filter
un-filtered spectrum
In the high wave-
number range
ln(k))
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

11. 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.
No. of grid points

12. Unfiltered equations

13. Unfiltered equations

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

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

16. Filtered momentum equation
Filter the momentum eqn. with an arbitrary homogenous filter of width
Homogeneity of filter allows commutation with differentiation:
leads to
is an unknown stress accounting for the effect of the filtered-out small
scales on the large scales governed by the filtered equation

17. Residual (subgrid-scale (SGS)) stress
Note that in general:
Decompose the SGS stress as

18. Residual (subgrid-scale (SGS)) stress
Note that in general:
Decompose the SGS stress as

19. Residual (subgrid-scale (SGS)) stress
Note that in general:
Decompose the SGS stress as
deviatoric (trace-free) component
isotropic component

20. Residual (subgrid-scale (SGS)) stress
Note that in general:
Decompose the SGS stress as
deviatoric (trace-free) component
isotropic component
This decomposition leads to
The modified filtered pressure contains the isotropic part of the SGS stress

21. Filtered equations

22. Filtered equations
SGS stress
SGS stress:

23. Filtered equations SGS stress: SGS density flux: SGS stress
(obtained in same way as the SGS stress)

24. Comments on the filtered equations
The filtered equations are numerically solved for the filtered variables
describing the large scales
The SGS stress and SGS density flux present closure problems and
must be modeled or approximated in terms of filtered variables only
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.

25. Sketch of energy spectrum in LES
most energetic scales usually resolved in RANSS
and general ocean circulation simulations
ln(E(k)) 3
spectrum -5
based on
dissipative scales
ln(k))

26. Sketch of energy spectrum in LES
most energetic scales usually resolved in RANSS
and general ocean circulation simulations
ln(E(k)) 3
spectrum -5
based on
dissipative scales
spectrum
based on
scales in
inertial range
ln(k))

27. Sketch of energy spectrum in LES
most energetic scales usually resolved in RANSS
and general ocean circulation simulations
ln(E(k)) 3
spectrum -5
based on
dissipative scales
spectrum
based on
scales in
inertial range
resolved (large) scales
sub-grid (residual) scales
ln(k))
We choose the grid size, h, to fall within the inertial range to facilitate SGS modeling

28. Role of discretization in LES
Both A and B do a good job
representing low wavenumber
spectrum of
ln(E(k))
Spectrum of
Spectrum of obtained
with discretization A
Spectrum of obtained
with discretization B
ln(k))
Discretization A behaves more like a sharp cutoff filter, while B behaves
more like a box filter
Ideally we would aim for a discretization like A

29. Smagorinsky SGS model
Recall that the SGS stress and density buoyancy flux must be
modeled or approximated

30. Smagorinsky SGS model
Recall that the SGS stress and density buoyancy flux must be
modeled or approximated
Both are trace-free
Smagorinsky (1967) model:

31. Smagorinsky SGS model
Recall that the SGS stress and density buoyancy flux must be
modeled or approximated
Both are trace-free
Smagorinsky (1967) model:
Smagorinsky coefficient

32. Smagorinsky SGS model
Recall that the SGS stress and density buoyancy flux must be
modeled or approximated
Both are trace-free
Smagorinsky (1967) model:
Smagorinsky coefficient
Analogously:

33. The eddy (turbulent) viscosity
The turbulent viscosity has units Because we are working
with the smallest resolved scales, we can set
In LES the SGS range starts at the inertial range, thus we may invoke
Kolmogorov’s 2nd hypothesis:
Statistics of scales of size, say, within the inertial range have a universal
form uniquely determined by the rate of energy transfer,
And we may have
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:

34. Difficulties with the Smagorinsky model
For isotropic turbulence, Lilly (1967) showed that
Major difficulty:
The constant coefficient allows for a non-vanishing turbulent viscosity
at 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 expression
for the Smagorinsky coefficient

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

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

37. Dynamic Smagorinsky model
Recall filtering the N-S equations with an homogeneous filter of width
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
With this new filter, the filtered momentum equation becomes:

38. Dynamic Smagorinsky model
Recall filtering the N-S equations with an homogeneous filter of width
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
With this new filter, the filtered momentum equation becomes:

39. Dynamic Smagorinsky model
Recall filtering the N-S equations with an homogeneous filter of width
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
With this new filter, the filtered momentum equation becomes:
Scale invariance: Both and are in the inertial range, thus

40. Dynamic Smagorinsky model
Consider the following tensor proposed by Germano :

41. Dynamic Smagorinsky model
Consider the following tensor proposed by Germano :
(resolved)

42. Dynamic Smagorinsky model
Consider the following tensor proposed by Germano :
(resolved)
(modeled)

43. Dynamic Smagorinsky model
Consider the following tensor proposed by Germano :
(resolved)
(modeled)
Minimization of the difference between these two with respect to Cs leads to:
- Averaging in statistically
homogenous direction(s)

44. Dynamic Smagorinsky model
Consider the following tensor proposed by Germano :
(resolved)
(modeled)
Minimization of the difference between these two with respect to Cs leads to:
- Averaging in statistically
homogenous direction(s)
Explicit application of test filter (denoted by a “hat”) is required, unlike 1st filter

45. Dynamic Smagorinsky model
Sketch of spectra
ln(E(k))
Unresolved, subgrid scales
Resolved scales
Spectrum based on
ln(k))

46. Dynamic Smagorinsky model
Sketch of spectra
ln(E(k))
Unresolved, subgrid scales
Resolved scales
Spectrum based on
Spectrum based on
ln(k))

47. Dynamic Smagorinsky model
Sketch of spectra
Subtest scales
ln(E(k))
Unresolved, subgrid scales
Resolved scales
Spectrum based on
Spectrum based on
ln(k))
By applying the test filter, the Germano formulation samples the field between
the subgrid scales and the subtest scales in order to obtain the model coeff.

48. Dynamic mixed model Recall and
true SGS stress
subgrid component
Inserting the former into the latter leads to
+ subgrid-scale terms
The subgrid-scale terms can be approximated via the Smagorinsky model
A dynamic coefficient in the Smagorinky model can be derived here as well
This mixed approach leads to a modeled SGS stress better correlated with
the true SGS stress. Both approaches lead to good correlation with true
SGS energy dissipation

49. LES methodology used in computations
SGS stress
SGS density flux
SGS stress model:
SGS density flux model:
Model coefficients in SGS models are computed dynamically as described

50. Numerical scheme used in computations
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
- restriction to periodic boundaries in x and y
- allows Dirichlet and Neumann boundaries in z
1. de-aliased in x and y
2. filtered in z with a high order implicit filter

51. LES of decaying isotropic turbulence
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.

52. Decaying isotropic turbulence behind a bar grid
isotropic far field

53. Effect of sub-grid scale (SGS, residual) model
3-D energy spectra

54. Geometry and grid for LES of channel
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 z x y

55. Body force
Flow is driven by a body force. There are 2 ways to determine this force:
1. Force control to achieve desired z x

56. Body force
Flow is driven by a body force. There are 2 ways to determine this force:
1. Force control to achieve desired z x
2. Mass control to achieve desired
- Body force is dynamically adjusted towards desired bulk velocity

57. Basic relationships
Recall the classical Reynolds decomposition and let

58. Basic relationships
Recall the classical Reynolds decomposition and let
In LES:

59. Basic relationships
Recall the classical Reynolds decomposition and let
turbulence intensities
(not subgrid components)
In LES:

60. Basic relationships
Recall the classical Reynolds decomposition and let
turbulence intensities
(not subgrid components)
In LES:
We want to study:
resolved Reynolds shear stress

61. Wall forces in unstratified channel flow
Theoretical mean force = 0.435
DNS mean force = 0.426

62. Mean velocity in unstratified channel

63. Variances in unstratified channel flow

64. Shear stresses in unstratified channel

65. 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

66. 1-D spectra in unstratified channel
Results from Najjar and
Tafti, Physics of Fluids,
1996
Ref. 34 is DNS data used
earlier from Kim, Moin &
Moser, J. Fluid Mech.,
1987
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.

67. Stratification effects

68. Stratification effects
strength of stratification
Ri based on bulk velocity
Ri based on friction velocity

69. Stratification effects
strength of stratification
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

70. Wall Forces
Theoretical mean force = 0.435
DNS mean force = 0.426

71. High/low speed streaks near wally x
Fully turbulent
Laminarized

72. Mean velocity

73. Instantaneous streamwise velocity contoursz
Ri = 0 x
Ri = 60

74. Root mean squares of fluctuations

75. Shear stress and turbulence structure

76. Nusselt numbers
Nusselt No. = measure of wall mass transport due to turbulence

77. Density statistics

78. Instantaneous density contoursz
Ri = 0 x
Ri = 60

79. Mixing Efficiency

80. 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

81. Langmuir cells in wind-driven channelz x
no-slip wall

82. Langmuir cells in wind-driven channelz x
no-slip wall
Surface stress is applied such that
Craik-Leibovich (C-L) vortex forcing is added to the filtered momentum
equation to account for Langmuir cells

83. Langmuir cells in wind-driven channelz x
no-slip wall
Surface stress is applied such that
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
- Stokes drift vel. in vortex forcing

84. Langmuir cells in wind-driven channelz x
no-slip wall
Surface stress is applied such that
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
- Stokes drift vel. in vortex forcing
- Simulations with
and
use (33x33x65)-grid with z-stretching
- Simulation with
uses (33x33x97)-grid with z-stretching

85. Mean velocity
Langmuir cells tend to homogenize the upper water column

86. Variances
Langmuir cells increase horizontal fluctuations in the upper water column

87. Time-averaged x-component of vorticity
z/h
y/h
Counter rotating cells are seen in the case with C-L vortex forcing,

88. Engineering applications of LES
LES of flow around an airfoil
Simulation by
Kenneth Jansen, at
Rensselaer Polytechnic
Institute
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