[American Institute of Aeronautics and Astronautics 6th AIAA Theoretical Fluid Mechanics Conference...

1

American Institute of Aeronautics and Astronautics

Smart Sub-Grid-Scale Models for LES and Hybrid

RANS/LES

Paul Batten1, Uri Goldberg

2, Eric Kang

3 and Sukumar Chakravarthy

4

Metacomp Technologies, Agoura Hills, CA 91301

This paper considers the implementation of sub-grid scale (SGS) models for large eddy

simulation (LES) or hybrid Reynolds-averaged Navier-Stokes (RANS)/LES methods. The

paper describes how to reconcile Smagorinsky-type eddy viscosity SGS models with

monotonically-integrated large eddy simulation (MILES) approaches by using ‘Smart’ SGS

models that understand, and compensate for, the inherent diffusion in the underlying

numerical transport algorithm. An improved ‘burst’ model of synthetic turbulence is also

introduced for applications to inhomogeneous turbulence.

Nomenclature

Cs = sub-grid scale eddy viscosity coefficient

F = flux vector

L = turbulence length scale

LvK = von Karman length scale

N(a,b) = Gaussian normal random variable with mean a, standard deviation b

J = cell volume

R = right eigenvector matrix

Si = side- or face-area vector

Sij = strain-rate tensor

U = vector of conserved or dependent flow variables

V = turbulence velocity scale

aik = Cholesky decomposition of Reynolds-stress tensor

cn = velocity scaling for nth Fourier mode

d j

n = wave vector for nth Fourier mode

ni = normalized side-area vector

ut = tangential velocity component

v i = isotropic fluctuating velocity vector

x j = position vector in Cartesian frame

= mesh length scale

= diagonal matrix of eigenvalues

= eigenvalue or wave-speed

EN = effective numerical viscosity

SGS = sub-grid scale eddy viscosity

L = left centroidal value of

R = right centroidal value of

l = left interpolated face value of

r = right interpolated face value of

= turbulence time scale, L/V

1Principal Scientist, Metacomp Technologies, 28632 Roadside Dr., Agoura Hills, CA 91301, AIAA Senior Member

2Principal Scientist,Metacomp Technologies, 28632 Roadside Dr., Agoura Hills, CA 91301, AIAA Senior Member.

3Scientist, Metacomp Technologies, 28632 Roadside Dr., Agoura Hills, CA 91301.

4President, Metacomp Technologies, 28632 Roadside Dr., Agoura Hills, CA 91301.

6th AIAA Theoretical Fluid Mechanics Conference27 - 30 June 2011, Honolulu, Hawaii

AIAA 2011-3472

Copyright © 2011 by Metacomp Technologies. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

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I. Introduction

HE earliest attempts at large-eddy simulation (LES) (Smagorinsky1, 1963) involved the use of an explicitly-

added diffusion or viscosity-like term which was designed to mimic the mixing effects of the unresolved scales

of turbulence on the larger, resolved, scales. Although the typical implementation of this Boussinesq or eddy

viscosity hypothesis does not account for backscatter or perturbations that exist at fine temporal resolutions, the

eddy viscosity framework remains the most popular SGS modeling choice, largely on account of its inherent

stability.

The coefficient of the SGS eddy viscosity must (from the perspective of dimensional consistency) be of the

following form:

SGS CSVL (1)

in which Cs is a dimensionless constant (or function, possibly determined from local flow conditions and/or a

separate transport equation) and L and V are, respectively, length and velocity scales representative of the local state

of the turbulence. The model is commonly implemented using the strain magnitude, S, and mesh spacing , as

follows:

2S SSGS C (2)

The dimensionless coefficient,

CS , is normally calibrated with respect to decaying isotropic turbulence,

however, the resulting model is not universally applicable and needs modifying for other scenarios, most notably for

wall-bounded flows. More recent efforts have attempted to replace the tunable coefficient with a modeling

hypothesis, allowing the local value of

CS to be computed dynamically (Germano et al.2), although in practice, this

is still insufficient when dealing with the very high-aspect ratio cells which are usually an unavoidable consequence

of modeling high Reynolds-number wall-bounded flows. The velocity scale, (S), in Equation (2) is also not

universally applicable; it suffers from the obvious deficiency that the strain magnitude, S, does not vanish at walls

(whereas the turbulence shear stress does), and therefore devices such as near-wall damping functions or additional

transport equations are often employed. The literature does not even show a clear consensus on the choice of length

scale, or filter width, . The mesh spacing is an obvious candidate when cells are uniform and isotropic, but in other

situations (such as high-aspect ratio, near-wall cells), the appropriate choice of remains somewhat ambiguous and

dependent on the model and/or practitioner. For hybrid RANS/LES modeling3,4

, we favor the Nyquist mesh length

scale, with additional safety factors based on an appropriate Courant condition and local resolved-structure size9:

vKLtzyx ,,,,max2 (3)

In hybrid RANS/LES models,

SGS fulfills the role of the classical RANS eddy viscosity in the near-wall region;

away from walls,

SGS is typically still active, but now plays the role of providing the (usually weaker) level of

mixing for a traditional sub-grid scale model.

II. MILES

The previous discussion highlights the fact that there remains some degree of uncertainty over the correct levels of

mixing that should be implied by Equation (2). This is in sharp contrast to the precise levels of mixing created by

modern higher-order upwind schemes based on characteristics or wave decomposition and mathematical theories

such as Total Variation Diminishing (TVD) or Essentially Non-Oscillatory (ENO) properties. These mathematical

theories are not based on turbulence considerations, but on a desire to apply the lowest possible level of diffusion to

prevent aliasing errors at the Nyquist limit; a goal shared by most LES SGS models of turbulence. This argument

led some researchers to propose an alternative form of sub-grid model, termed Monotonically-Integrated Large Eddy

Simulation5,6

(MILES). The rationale behind the MILES approach is that monotonic transport algorithms, such as

bounded TVD or ENO schemes, already contain the necessary ingredients to drain energy at the cut-off wave

number. For free-shear flows, LES behavior is mostly independent of the SGS model, provided that the complete

LES framework (SGS model plus numerical diffusion) can drain sufficient energy at those Nyquist scales to prevent

aliasing and spurious energy build-up. Since the more advanced modern numerical schemes can optimally minimize

T

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the level of diffusion based on local flow gradients and individual wave propagation velocities, many authors have

preferred to dispense with any additional sub-grid model, at least when dealing with free-shear flows (see for

example, Shur et al.4).

One significant advantage of the MILES approach is that the use of bounded TVD or ENO schemes can guarantee

robustness even in extreme flow regimes. This is a critical factor in a commercial CFD environment, where a broad

category of flows needs to be routinely and reliably simulated by non-expert users. By contrast, central differencing

schemes – which often work well for simpler academic LES cases - can easily fail in more general situations where

the SGS model alone can simply fails to provide adequate damping.

There is, unfortunately, one major difficulty with the MILES approach that has limited its use. An accurate, low-

diffusion, bounded upwind-transport scheme is simply designed to prevent overshoots and undershoots; if there is

no risk of creating local extrema (such as at a grid-aligned shear- or boundary-layer), a properly formulated upwind

or matrix-diffusion scheme will add no mixing at all. This is a problem for a hybrid RANS/LES model, in which at

least some portion of an attached boundary layer must be treated in RANS mode (which typically implies a high

level of diffusion). A full LES of all attached boundary layers might be possible with MILES, but this would be an

expensive overkill when RANS (or hybrid RANS/LES with high-aspect near-wall cells) provide perfectly adequate

predictions at a fraction of the cost. As a result, MILES is generally considered unsuitable for wall-bounded flows.

III. Smarter SGS Models

Distinct from the ongoing research efforts to

improve models for the sub-grid stress terms is

the question of whether an accurate SGS model

can or should be applied in conjunction with

currently popular numerical schemes, such as

TVD or ENO methods. These types of

numerical method are popular in modern CFD

codes due to their inherent monotonicity and

optimally low levels of numerical diffusion.

Assuming that an optimal or ideal local level of

mixing due to sub-grid scale turbulence (the

SGS model diffusion) were known, Fig. 1

shows, schematically, the problem with naively

adding such an SGS model to a bounded

upwind scheme. The SGS model and the

numerical transport scheme are both designed

to create mixing at the Nyquist mesh-scale limit

and since each is blind to the effects of the other, their combined effect

results in excessive levels of diffusion.

This effect is well understood, but circumventing the problem has

proved challenging. For example, one could completely remove the SGS

model diffusion, but this would result in a MILES approach, which would

not be applicable to wall-bounded flows. One could completely remove

the numerical diffusion, but this would result in a very brittle numerical

method, which would not be suitable for general, commercial CFD

applications. Other practitioners have chosen to blend some portion of

upwind flux in order to maintain numerical stability, but in most of these

cases, there is no attempt to re-calibrate the underlying SGS model and all

guarantees of robustness are lost once the TVD or ENO scheme is blended

with any proportion of central-differencing.

In 1992, Toro8 made an interesting proposal to reduce the level of

numerical diffusion imposed by the flux limiters in his upwind scheme,

based on the local level of physical (or, in our case, modeled, turbulent)

diffusion. The Smart SGS approach in this paper in fact takes the reverse

approach, as it turns out to be simpler to quantify the level of diffusion

from the flux of a monotonic advection scheme than it is to maintain

monotonicity whilst accounting for physical or modeled viscous effects. The Smart SGS approach ensures

Figure 1. Over-predicted total diffusion (right) as a result of

numerical diffusion from the upwind scheme plus SGS

model diffusion.

Figure 2. Schematic of boundary-

layer cells in x-aligned shear.

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robustness of the underlying numerical scheme (i.e., positivity preservation in the sense of internal energies, mass

fractions, etc.) by always preserving the numerical diffusion inherent in the bounded upwind transport algorithm.

The objective of the Smart SGS approach is to then establish, face-by-face, an anisotropic diffusion coefficient that

would give rise to the exact same level of mixing as that of the underlying numerical method and to ensure that the

SGS model understands this level of numerical diffusion and removes it before applying its own level of modeled

SGS stresses or diffusive flux.

A. Construction of a Smart SGS Model

The typical SGS model and the typical bounded convection scheme are designed from very different

perspectives and so have rather different sources or triggers for the generation of local (modeled or numerical)

diffusion. For example, in a simple x-aligned shear, a modeled (diffusive) SGS flux of the Smagorinsky type can be

written as:

y

uF SGSv (4)

In the classical Smagorinsky model, the eddy viscosity is typically defined as:

ijijSSGS SSC 22

.

In the present paper, CS is assumed to be a constant value of 0.2, however, the Smart SGS approach is equally

valid if CS is computed from any other functional form or relation.

By contrast, the numerical flux for a classic first-order upwind scheme, such as Roe‟s method, can be written as:

FC 1

2FL FR

R R1

2U R U L

Subtracting this from a central-difference approximation to the flux, leaves just the well-known matrix diffusion

term:

D 1

2FL FR FC

RR1

2UR UL (5)

D implies a separate level of numerical diffusion for each component in the system. For a segregated scalar

equation with normal propagation velocity, , and a (locally) uniform-density flow, the effective numerical diffusion

for an x-aligned shear (such as that illustrated in Figure 2) becomes:

y

yD LR

22

This can be re-written to look like Equation (4), with:

i

ENENvS

J

yF

2,

where J is the average of the two adjacent cell volumes and

Si is the side- or face-area vector.

In higher-order numerical transport schemes, interpolated data are fed to the interface flux model or Riemann

solver from the left and right limits of the cell face and this effect needs to be included. In addition,

S GS and

EN

both represent scalar quantities. For systems of equations, the effect on the shear-component of velocity can be

isolated into one scalar coefficient by considering the level of diffusion for that component of velocity tangential to

the cell face which gives rise to the maximum shear. This leads to:

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D(ut )

2ur

t ul

t x t

2

ur

t ul

t

uR

t uL

t

ut

x t

EN J

2Si

ur

t ul

t

uR

t uL

t

J D(ut )

Si uR

t uL

t

If the momentum flux jump is primarily due to shear (i.e., away from contact surfaces),

EN can be written in

terms of any central-difference and upwind/bounded (

FU ) flux models as:

EN

J 1

2FL

t FR

t FU

t

Si uR

t uL

t , (6)

with the maximum shear component of the velocity and its flux, tu and

F trespectively, defined as:

.

,/,/

ikkii

kkiitkkiit

nnuut

tttuutttFF

Formulation (6) requires a check for division by zero. However, in practice if the solution is smooth it does no harm

to set 0EN . Recognizing that the target SGS diffusion level should never increase (as a result of a negative local

EN , for example), the smart SGS model viscosity is finally defined as:

0,0,maxmax ENSGSSSGS . (7)

If the momentum-flux jump in (6) occurs across a contact surface or shock, it will create a large value for EN .

This, again, does little harm since

EN SGS SSGS 0 and the inherent level of numerical diffusion from the

convection scheme is always retained to maintain positivity.

As a result of (6), the original target or

required level of SGS model diffusion is

achieved through a combination of the

numerical diffusion and a potentially

lowered level of SGS model diffusion

(Figure 3). In free shear flow where

numerical and SGS model levels of

diffusion are comparable, the resulting

scheme behaves more like MILES, i.e., as

if no SGS model were active. Near walls,

the large levels of eddy viscosity implied

by a hybrid RANS/LES model in high-

aspect ratio cells, far exceed levels of

numerical diffusion (the latter may vanish

completely if the near-wall mesh aligns

with the shear), resulting in typical RANS levels of eddy viscosity. Thus the approach is equally applicable to free

shear and wall-bounded flows.

The only significant expense in computing SSGS

is the calculation of a central-difference flux for each

momentum component. However, computing a central-difference flux (even a higher-order variant) is usually much

less expensive than the calculation of the upwind or bounded flux,

FU . Since the numerical diffusion is, in general,

anisotropic, the computation and use of the smart SGS viscosity, SSGS , must be done on a face-by-face basis.

Using SSGS in place of the standard SGS viscosity sensitizes the SGS model to both the mesh and the underlying

numerical method. Specifically, the numerical diffusion will increase (hence lowering SSGS ) with a lower-order

interpolation, or with a more diffusive interface flux model (such as Rusanov or any scalar artificial viscosity

method). Use of a coarser or more skewed mesh, or a mesh whose cell face locally misaligns with the direction of

Figure 3. Target diffusion level achieved using Smart SGS model.

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maximum shear, will again lead to an increase in numerical diffusion and a reduction in SSGS . As a result,

performance gains with the smart SGS approach will be more significant with less accurate numerical schemes, less

accurate interface flux models or poorer-quality meshes and simpler (more diffusive) SGS models. Thus the

approach has the potential to extend the useful working life of second-order upwind methods.

B. An Example - Homogeneous Turbulence

In the following example, a synthetic Fourier method3 was used to generate an initial isotropic, homogeneous

turbulent velocity field. A 643 mesh was used to study the evolution of the complete LES system in terms of the

energy decay, using a traditional central difference scheme with a standard (constant coefficient) Smagorinsky

method (Central-Difference Blend), a second-order upwind TVD scheme with the smart SGS model

(Upwind+Smart SGS), the same upwind TVD scheme on its own (MILES), and the upwind TVD scheme with the

standard SGS model (Upwind+SGS). All schemes used a preconditioning approach to accelerate convergence and

further minimize numerical diffusion. All upwind TVD methods (including MILES) ran easily without user

intervention; pure central differencing is typically unstable when applied to all equations (it permits pressure

checker-boarding) and so the „Central-Difference Blend‟ results presented here were actually a TVD upwind scheme

with 90% central difference blending applied only to the momentum equations (in addition to the applied SGS

model diffusion). Each calculation used dual-time stepping, with inner residuals reduced by two orders of

magnitude at each physical time step.

Results are shown in Figure 4 in terms of the spectral content after 1000 time steps. All solutions actually adhere

fairly closely to the expected -5/3 exponent decay slope in the inertial range and differences among methods appear

modest when plotted on a logarithmic scale. (There is, however, some evidence that the spectral energy content of

the synthetic turbulence needs to be considered in more detail at some future stage.) A more subtle indication of

diffusion levels is shown in Figures 5a-5d, where iso-surfaces of Jeong and Hussain‟s 2vortex-core detection

function10

are plotted for each of the various solutions. The most detailed resolution is noted with the central-

difference scheme, but results from the Smart SGS are broadly comparable, without suffering the uncertain

robustness problems of the central-difference scheme when applied to more demanding cases. The most diffusive

solution is the upwind scheme with the standard SGS model.

Figure 4. Spectral content of decaying isotropic turbulence case - 64

3mesh

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IV. A Synthetic Burst Model for Inhomogeneous Turbulence

In this section we consider a separate, but related, problem for LES and hybrid RANS/LES methods, namely the

creation of realistic stochastic, unsteady velocity-perturbation fields from an initial statistical description of

inhomogeneous turbulence. This is needed to define inlet conditions in an LES involving wall-bounded flow, shear,

or other mean-flow and statistical non-uniformities and is also useful for converting energy between statistical (k or

Reynolds stress and dissipation-rate data) and directly-resolved representations of turbulence kinetic energy within

sub-domains of a hybrid RANS/LES or within an embedded LES; a process that can appropriately be termed „Large

Eddy Stimulation‟3 (LEST).

The creation of artificial or „synthetic‟ turbulent fluctuations is not the only means to providing initial or inlet

data for LES, but it is arguably one of the most convenient. A common alternative is to use a pre-cursor calculation

with periodic boundary conditions and, optionally, a re-scaling if the flow is not completely periodic in nature. The

advantage to using pre-cursor calculations is that all recycled structures are likely to be supported and maintained by

the combination of grid and numerical scheme. However, in a commercial CFD environment, precursor calculations

place a tremendous burden on the user in terms of having to define separate zones for such a calculation,

determining the location and nature of any periodic boundary conditions, potentially defining a rescaling of the

fluctuation amplitudes and correlations to match a particular set of inlet statistics, plus the cost of the precursor

calculation and simultaneous use (or storage) of the massive amounts of inlet data generated from the precursor

simulation. There is, therefore, a very strong motivation to explore alternative, more automatic and more convenient

procedures for generating realistic eddy content for an LES.

Figure 5a. 2 – Central-difference+SGS Figure 5b. 2 – MILES

Figure 5c. 2 –Upwind+SmartSGS Figure 5d. 2 –Upwind+SGS

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A. Kraichnan’s Fourier-Based Synthetic Turbulence

One of the earliest approaches to generating synthetic turbulence was the Fourier model suggested by

Kraichnan11

. His model was later improved by Smirnov et al.12

to include anisotropic turbulence and refined further

by Batten et al.3, by removing the need to compute similarity transformations. The use of a Fourier model to

describe instantaneous turbulent fluctuations is appealing for a number of reasons: It requires minimal

computational storage, minimal user input (information on turbulence scales can be extracted from existing RANS

calculations) and it can easily supply velocity and velocity derivative information at any arbitrary 4-dimensional

(x,y,z,t) coordinate point where turbulent statistics are known. The modified model of Smirnov et al.12

, presented in

reference 3, is given by the following model for the instantaneous velocity vector, iu , as:

kiki vau (8)

N

n

n

j

n

j

n

i

n

j

n

j

n

iji tLxdqtLxdpN

txv1

//ˆ2sin//ˆ2cos2

),( (9)

/,ˆ LVc

Vdd

n

n

j

n

j

)2/1,0(),1,1(),1,0(, NdNN n

i

nn

i

n

i

n

m

n

jijk

n

i

n

m

n

jijk

n

i

n

k

n

k

n

m

n

lml

n dqdpdddduuc ,,2/3

In the above,

aik is the Cholesky decomposition of the local Reynolds stress tensor (which can always be

obtained as long as realizable stress data are provided as input) and L, V and represent local length, velocity and

time-scales, respectively. A given Fourier mode is considered resolvable (and is therefore synthesized) on the local

mesh if

L/dn 2. A long-time integration of

u i u j(n)/2, gives the energy fraction of mode n as:

Nm

m

i

m

i

n

i

n

i

qp

qp

k

nk

,1

22

22

)()(

)()()( (10)

Summation of Equation (10) over all resolvable modes identifies the energy fraction that is removed from the

modeled k and therefore the remaining (sub-grid) component left behind after the synthetic turbulence energy

conversion process. Thus LEST tracks the turbulence energy to ensure that once a portion of k has been synthesized

(converted into resolvable kinetic energy content in terms of iu ), k is depleted by that amount and a subsequent call

to the same LEST function then has no effect since the remaining k is recognized as purely sub-grid scale kinetic

energy.

B. The Numerical Evolution of Turbulent Eddies

Larger, more energetic eddies that are well resolved spatially and temporally are those most likely to survive and

evolve. For a given combination of mesh and numerical method, there will be certain orientations, frequencies and

phases that make a given structure more vulnerable to diffusion. All eddies will evolve and eventually 'die', implying

an ultimate conversion into heat or internal energy. Smaller, weaker eddies are more susceptible to this process,

both through physical diffusive mechanisms and artificial mechanisms, such as diffusion arising from the

combination of mesh and numerical scheme and the effect of the SGS or the turbulence model. The degree to which

numerical diffusion prematurely kills weaker eddies is a function of the numerical scheme, interpolation type,

interface flux or residual redistribution, alignment of the shear with respect to the local cell faces, degree of skewing

or stretching of the mesh, choice of time-step and time-stepping scheme, etc. Any modeling weakness that leads to

the creation of erroneously small uncorrelated eddies is certain to make those structures more susceptible to all

forms of diffusion. In particular, it is well known that random white-noise data are mostly useless for LES initial or

inlet conditions, as the lack of coherent structure causes a rapid extinction of those tiny uncorrelated structures.

The most notable problem with using Fourier models, such as formulation (8,9), for LES inlet conditions is an

occasional too rapid decay of the turbulence energy (and hence shear stress) downstream of an inlet boundary. There

are several reasons why this can happen. Firstly, imposing any form of turbulence energy at scales beyond those of

the combined spatial and temporal Nyquist limits inevitably leads to aliasing at that frequency and a rapid diffusion

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of those components by both the numerics and the SGS model. Filtering, to ensure the required energy is injected

only at resolvable scales, is therefore important but, luckily, rather easy in Fourier space. Secondly, rapid spatial

variations in amplification scaling factors, such as the Cholesky decomposition in Equation (8), can lead to a slight

under-prediction of spatial correlation lengths. Most significantly, statistical inhomogeneities can cause small

variations in the spatial or temporal scalings in (9) which, over very large distances or very large evolution times,

can lead to significant under-prediction of the two-point spatial or temporal correlations.

The problem of the too-rapid inlet turbulence decay was noted by Keating et al.13

, who implemented an energy-

injection process at planes downstream of the inlet in order to help strengthen the strongest eddies (those most likely

to survive). Their approach is quite general and could be used to help develop resolved-eddy content using

potentially any form of LES inlet data. It is also simpler and less cumbersome than the generation of an entire

precursor simulation, however, the positioning of recycling planes and the definition of the associated time-

dependent sources still represent an unwelcome extra burden for the average CFD user. Hence it would be

advantageous to improve the synthetic models of turbulence so as to minimize both the potential for error and the

additional problem set-up required from the end-user.

Figure 6 shows a basic difficulty of trying to synthesize the evolution of data from statistics at two points that are

arbitrarily close in time or space (Δt and Δx can be interchanged on the x-axis), but with a slight shift in frequency

or wavenumber. Here, this has been exaggerated with a 5% shift, but in practice, each point in a statistical RANS

computation will have a slight variation in both turbulence time-scale and length-scale. Two probe points are

considered as evolving using models of the form:

,sin11

N

n

nn

i tat (11)

.05.1sin12

N

n

nn

i tat (12)

Two sample data sets are created by using Equations (11) and (12), with identical amplitude and frequency

coefficients. The 5% rise in frequency in Equation (12) is illustrative of variations in the turbulence time- or length-

scales that would naturally occur in Equation (9) in the presence of inhomogeneous turbulence. Figure 6 shows the

effect rather dramatically, but even for adjacent points with negligibly-small differences in length- or time-scales,

the signals will, after a large enough time or distance, find themselves 180 degrees out of phase and eventually

lacking any meaningful correlation.

C. The Burst Model for Inhomogeneous, Anisotropic Turbulence

One can note in Figure 6 that at small

times the signals from the two probes are

well correlated. For two points separated

by arbitrarily small t or x, this is the

expectation for all time and all space

coordinates, respectively. Yet, physically,

different regions will experience different

sized eddies that will evolve at different

rates. This dichotomy is reconciled by

introducing a degree of intermittency using

a concept of a turbulence „burst‟, which

represents a region of time or space over

which the turbulence is active. Toward the

edge of the burst, a windowing function

ensures that turbulent fluctuations die away.

In addition, each burst is centered on its

own coordinate system for a local Fourier

representation which is then free to operate

with different length-scales and different

frequencies. This concept is similar in spirit to the synthetic eddy method of Jarrinet al.14

, but does not require the

user to position a large number of eddies in computational space – something which, again, places a large burden on

a CFD user.

A burst model for this simple one-dimensional example can be defined as:

Figure 6. Fourier functions, Equations (11) and (12):

Two identical Fourier representations with a 5% frequency shift.

10


.sin1

*

N

n

nn

i

n tawt (13)

The windowing function and local coordinate frame are defined as:

2

,mod,cos **

ll

l

n bbbtt

b

tw

. (14)

This model introduces two new parameters, the

burst duration (in time) or burst length (in space),

lb , and the burst phase, b . In the example shown

in Figure 7, these were chosen randomly for each

mode, thus different wavelengths need not share the

same locations of intermittency. Admittedly, the

selection of these additional parameters does create

extra work, but this can be automated to some

degree. For example, in four dimensions, burst

duration and burst length can be set equal to the

maximum turbulence time-scale and length-scale,

respectively. For more complex problems it may be

beneficial to use a separate definition of burst

duration and (anisotropic) burst lengths for each

burst.

Figure 7 shows the early-time results of the burst

model representation, applied to two functions with

the same 5% frequency shift. There are locations

where the two solutions deviate, but they also have periods where they synchronize again. Crucially, in the limit as

the frequency shift tends to zero, both signals become perfectly correlated for arbitrarily large times. This is

illustrated in Figures 8 and 9 which show, respectively, Fourier functions modified with a 5% and 1% chirp

function, representative of a turbulent field with spatially or temporally increasing wavenumber.

Figure 8 shows some expected deviations, interspersed with periods of strong correlation at a late time when the

original synthetic model has become completely de-correlated. For the given window shown in Figure 9, the signals

show a very strong degree of correlation as the frequencies converge.

The burst model is extended to four dimensions by modifying Equation (9) as follows:

N

n

nnnn

i

nnnnn

i

n

ji tLdqwtLdpwN

txv1

****//2sin//2cos

2),(

(15)

with the windowing function and local coordinates defined as:

n

l

n

n

t

nn

b

d

b

tw

**

cos,cosmin

(16)

Figure 7. Burst model representation of two functions

with a 5% frequency shift.

Figure 8. Synthetic ‘burst’ applied to two chirp

functions and a 5% frequency shift.

Figure 9. Synthetic ‘burst’ model applied to two

chirp functions and a 1% frequency shift.

11


2

,mod*

n

tn

t

n

t

n bbbtt

(17)

2

,mod*

n

ln

l

n

lj

n

j

n bbbxdd

(18)

2/1,01 Nbb t

n

t (19)

2/1,01 Nbb i

n

l (20)

Figures 10 through 13 show lofted strain-rate contours from the synthetic turbulence generation using statistics

from a turbulent channel flow. A two-dimensional RANS solution was initially computed using a second-moment

closure. The Reynolds-stresses and time-scales were then interpolated onto a three-dimensional mesh and

resolvable structures were synthesized by using both the original model (Equations 8,9) and the burst model

(Equations 8,15-20) in which the burst duration and burst length were taken as the maximum turbulence time scale

and maximum turbulence length scale, respectively, and n

tb and n

lb were chosen randomly across the time and

length scales defined for each mode. Solutions were computed both at t=0 and t=100 max .

Figure 10 shows the original synthetic turbulence model at t=0. Three distinct regions can be observed: the very

near-wall region where no structure is created (the higher-aspect ratio cells cannot resolve the small, near-wall

eddies and hence these are not synthesized and the hybrid RANS/LES model remains in RANS mode here); the

outer region of the boundary layer where some of the (still anisotropic) eddies have grown large enough to be

resolved on the (now more isotropic) mesh cells; and the core-flow region, which is dominated by larger, more

isotropic structures. Figure 11 shows the same solution synthesized at a much later time (100 times the maximum

turbulence time-scale), where de-correlation in the core-flow region is particularly apparent. Figures 12 and 13 show

the burst model synthesis at t=0 and t=100 max , respectively. Although the velocity-perturbation field continues to

evolve, it never experiences the break-down to white noise that the original model suffers at large times.

Note that although the de-correlation effect was demonstrated here with a large time shift, exactly the same

effect is seen when shifting through any other dimension (x, y, or z) that contains inhomogeneous statistics.

Figure 10. Synthetic turbulence model, t=0.

Lofted strain rate on side and streamwise boundaries.

Figure 11. Synthetic turbulence model, t=100max.


Figure 12. Synthetic burst model, t=0.


Figure 13. Synthetic burst model, t=100max.


12


V. Conclusion

This paper has introduced the concept of a Smart SGS model as a means to limit the combined effect of diffusion

from the numerical transport scheme and the SGS model. The approach has been applied here only to the

momentum equations within an eddy-viscosity SGS framework, but extension to additional transport equations

(species or thermal fluxes) or Reynolds-stress SGS models is straightforward. An improved Fourier-based model of

synthetic turbulence was also introduced to improve the stochastic, time-dependent representation of

inhomogeneous turbulence.

Acknowledgments

Paul Batten would like to thank Jiwen Liu and Michael Shur for some useful communications on LES sub-grid

models and synthetic turbulence.

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