EPVS-FMDFfor...

EPVS-FMDF for LES of High-Speed Turbulent Flows

M.B. Nik∗, P. Givi†

University of Pittsburgh, Pittsburgh, PA

C.K. Madnia‡,

State University of New York, Buffalo, NY

and S.B. Pope§

Cornell University, Ithaca, NY

Work is in progress towards development of the “energy-pressure-velocity-scalar filteredmass density function” (EPVS-FMDF) as a new subgrid scale (SGS) model for large eddysimulation (LES) of high-speed turbulent flows. This is an extension of the previouslydeveloped “velocity-scalar filtered mass density function” method in low-speed flows. InEPVS-FMDF, the effects of compressibility are taken into account by including two addi-tional thermodynamic variables: the pressure and the internal energy. The EPVS-FMDFis obtained by solving its transport equation, in which the effect of convection appears ina closed form. The modeled EPVS-FMDF is employed for LES of a temporally developingmixing layer.

I. Introduction

Our portion of work at the Center is concentrated on development of “Generation III models.” Theseare based on advanced large eddy simulation (LES) and assessment of such simulation via direct numericalsimulation (DNS) and laboratory data. The subgrid scale (SGS) closure in LES is primarily based on ourfiltered density function (FDF) model.1, 2 This is the counterpart of the probability density function (PDF)methods in RANS.3

Since its original conception,4, 5 the FDF has become very popular in turbulent combustion research.In its stand-alone form, the FDF must account for the joint statistics of all the relevant physical variables.The most sophisticated FDF closure available to-date is the frequency-velocity-scalar FMDF (FVS-FMDF),6

and a simpler version (VS-FMDF) which does not include the SGS frequency.7, 8 Hydrodynamic closure inincompressible, non-reacting flows has been successfully achieved via the velocity-FDF (V-FDF),9 and theone which has been utilized the most only considers the scalar field (S-FDF and S-FMDF). This is the mostelementary form of FDF when it was first introduced,10, 11 and has experienced widespread usage. Some ofthe subsequent contributions in FDF are in its basic implementation,12–33 fine-tuning of its sub-closures,34–36

and its validation via laboratory experiments.17, 37–41 See Ref.2 for a recent review of the state of progressin FDF.

The objective of the present work is to extend the FDF methodology for LES of high-speed flows. Thisis accomplished by considering the joint “energy-pressure-velocity-scalar filtered mass density function”(EPVS-FMDF). This is work in progress, and when completed it will provide the most comprehensive formof the FDF formulation. With the definition of the EPVS-FMDF, the mathematical framework for itsimplementation in LES is established. A transport equation is developed for the EPVS-FMDF in which

∗Graduate Student, Department of Mechanical Engineering and Materials Science, University of Pittsburgh, Pittsburgh, PA15261, USA, Student Member AIAA.

†James T. MacLeod Professor, Department of Mechanical Engineering and Materials Science, University of Pittsburgh,Pittsburgh, PA 15261, USA, AIAA Fellow.

‡Professor, Department of Mechanical and Aerospace Engineering, State University of New York, Buffalo, NY 14260, USA,AIAA Associate Fellow.

§Sibley College Professor, Sibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca, NY 14853,USA, AIAA Associate Fellow.

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American Institute of Aeronautics and Astronautics

the effect of SGS convection appears in a closed form. The unclosed terms in this equation are modeledin a fashion similar to those in the Reynolds-averaged procedures. A Lagrangian Monte Carlo procedureis developed and implemented for numerical solution of the modeled EPVS-FMDF transport equation.Simulations are conducted of a non-reacting mixing layer. The extension of the formulation to reacting flowsis straightforward, as there would be no additional closure problem associated with SGS chemistry.

II. Formulation

In a multiple species, non-reacting flow, the primary transport variables are the density ρ(x, t), thevelocity vector ui(x, t) (i = 1, 2, 3), the pressure p(x, t), the internal energy e(x, t) and the species massfractions φα (α = 1, . . .Ns). The equations which govern the transport of these variables in space (xi)(i = 1, 2, 3) and time (t) are the continuity, momentum, energy and the scalar transport, all coupled throughthe equation of state.

∂ρ

∂t+∂ρuj∂xj

= 0 (1a)

∂ρui∂t

+∂ρujui∂xj

= −∂p

∂xi+∂τji∂xj

(1b)

∂ρe

∂t+∂ρuje

∂xj= −

∂qj∂xj

+ σij∂ui∂xj

(1c)

∂ρφα∂t

+∂ρujφα∂xj

= −∂Jαj∂xj

(1d)

p = ρR0

WT (1e)

where R0 denotes the universal gas constant and W is the mean molecular weight of the mixture. T denotethe temperature, e is the internal energy and γ =

cpcv

is the specific heat ratio. The viscous stress tensor τij ,the energy flux qj , the species α diffusive mass flux vector J

αj and σij tensor are represented by

σij = τij − pδij (2a)

τij = µ (T )

(

∂ui∂xj

+∂uj∂xi

−2

3

∂uk∂xk

δij

)

(2b)

qj = −λ (T )∂T

∂xj(2c)

Jαj = −ρΓ (T )∂φα∂xj

(2d)

where µ is the fluid dynamic viscosity, λ denotes the thermal diffusivity and Γ is the mass diffusion coefficient.We assume calorically perfect gas in which the specific heats are constants. Large eddy simulation involvesthe spatial filtering operation3

〈Q(x, t)〉ℓ =

∫ +∞

−∞Q(x′, t)G∆l(x

′,x)dx′, (3)

where G∆l1 (x′,x) denotes a filter function, and 〈Q(x, t)〉ℓ denotes the filtered value of the transport variable

Q(x, t). The subscript l1 indicates that 〈Q(x, t)〉ℓ is the first level filter value of the variable Q(x, t).42 In

variable-density flows it is convenient to use the Favre-filtered quantity 〈Q(x, t)〉L = 〈ρQ〉ℓ / 〈ρ〉ℓ. We considera filter function that is spatially and temporally invariant and localized, thus: G∆l1 (x

′,x) ≡ G∆l1 (x′ − x)

with the properties G∆l1 (x) ≥ 0,∫ +∞−∞ G∆l1 (x)dx = 1. The second level spatial filtering operation (used

below) is defined as:

〈〈Q(x, t)〉ℓ〉ℓ2 =

∫ +∞

−∞〈Q(x′, t)〉ℓG∆l2 (x

′,x)dx′, (4)

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where G∆l2 (x′,x) denotes a secondary filter function. Similarly, 〈〈Q(x, t)〉L〉L2 = 〈〈ρQ〉ℓ〉ℓ2 / 〈〈ρ〉ℓ〉ℓ2 .

Applying the first level filtering operation to Eqs. (1) yields:

∂〈ρ〉ℓ∂t

+∂ 〈ρ〉ℓ 〈uj〉L

∂xj= 0 (5a)

∂ 〈ρ〉ℓ 〈ui〉L∂t

+∂ 〈ρ〉ℓ 〈uj〉L 〈ui〉L

∂xj= −

∂ 〈p〉ℓ∂xi

+∂τ̆ij∂xj

−∂ 〈ρ〉ℓ τL(ui, uj)

∂xj(5b)

∂ 〈ρ〉ℓ 〈e〉L∂t

+∂ 〈ρ〉ℓ 〈uj〉L 〈e〉L

∂xj= −

∂q̆j∂xj

−∂ 〈ρ〉ℓ τL(e, uj)

∂xj+ ǫ+ τ̆ij

∂ 〈ui〉L∂xj

−Πd − 〈p〉ℓ∂ 〈ui〉L∂xi

(5c)

∂ 〈ρ〉ℓ 〈φα〉L∂t

+∂ 〈ρ〉ℓ 〈uj〉L 〈φα〉L

∂xj= −

∂J̆αj∂xj

−∂ 〈ρ〉ℓ τL(φα, uj)

∂xj(5d)

In Equation (5), the filtered viscous stress tensor τ̆ij , the filtered energy flux q̆j , the filtered diffusive mass

flux vector J̆αj , the pressure dilatation Πd and the dissipation term ǫ are defined as

τ̆ij = µ (〈T 〉L)

(

∂ 〈ui〉L∂xj

+∂ 〈uj〉L∂xi

−2

3

∂ 〈uk〉L∂xk

δij

)

(6a)

q̆j = −λ (〈T 〉L)∂ 〈T 〉L∂xj

(6b)

J̆αj = −〈ρ〉ℓ Γ (〈T 〉L)∂ 〈φα〉L∂xj

(6c)

Πd =

〈

p∂ui∂xi

〉

ℓ

− 〈p〉ℓ∂ 〈ui〉L∂xi

(6d)

ǫ =

〈

τij∂ui∂xj

〉

ℓ

− τ̆ij∂ 〈ui〉L∂xj

(6e)

and the SGS correlations are defined by:

τL(a, b) = 〈ab〉L − 〈a〉L 〈b〉L (7)

The “energy-pressure-velocity-scalar filtered mass density function” (EPVS-FMDF), denoted by PL, isformally defined as:4

PL (v,ψ, θ,η,x; t) =

∫ +∞

−∞ρ(x′, t)ζ (v,ψ, θ,η;u(x′, t),φ(x′, t), e(x′, t),p(x′, t))G(x′ − x)dx′, (8)

where

ζ (v,ψ, θ,η;u(x, t),φ(x, t), e(x, t),p(x, t)) =

(

3∏

i=1

δ (vi − ui(x, t))

)

×

(

σ=Ns∏

α=1

δ (ψα − φα(x, t))

)

× δ (θ − e(x, t))× δ (η − p(x, t))

(9)

In this equation, δ is the Dirac delta function, and v,ψ, θ,η denote the velocity vector, the scalar array, thesensible internal energy and the pressure in the sample space. The term ζ is the “fine-grained” density,43, 44

and Eq. (8) defines the EPVS-FMDF as the spatially filtered value of the fine-grained density. With the con-dition of a positive filter kernel,45 PL has all of the properties of a mass density function.

44 The “conditionalfiltered value” of the variable Q(x, t) is defined as

〈

Q(x, t)

u(x, t) = v,φ(x, t) = ψ, e(x, t) = θ,p(x, t) = η

〉

ℓ≡〈

Q

v,ψ, θ,η

〉

ℓ=

∫ +∞−∞ ρ(x

′, t)ζ (v,ψ, θ,η;u(x′, t),φ(x′, t), e(x′, t),p(x′, t))G(x′ − x)dx′

PL (v,ψ, θ,η,x; t). (10)

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and the exact transport equation for EPVS-FMDF is

∂PL∂t

+∂vjPL∂xj

=∂

∂vi

(〈

1

ρ

∂p

∂xi

v,ψ, θ,η

〉

ℓ

PL

)

−∂

∂vi

(〈

1

ρ

∂τij∂xj

v,ψ, θ,η

〉

ℓ

PL

)

+∂

∂ψα

(〈

1

ρ

∂Jαj∂xj

v,ψ, θ,η

〉

ℓ

PL

)

+∂

∂θ

(〈

1

ρ

∂qi∂xi

v,ψ, θ,η

〉

ℓ

PL

)

−∂

∂θ

(〈

1

ρτij∂ui∂xj

v,ψ, θ,η

〉

ℓ

PL

)

+∂

∂θ

(〈

1

ρp∂uj∂xj

v,ψ, θ,η

〉

ℓ

PL

)

+ (γ − 1)∂

∂η

(〈

∂qi∂xi

v,ψ, θ,η

〉

ℓ

PL

)

− (γ − 1)∂

∂η

(〈

τij∂ui∂xj

v,ψ, θ,η

〉

ℓ

PL

)

+ γ∂

∂η

(〈

p∂uj∂xj

v,ψ, θ,η

〉

ℓ

PL

)

(11)

In this equation, the effect of convection (the second term on LHS) appears in a closed form. The conditionalterms at the right hand side are the unclosed and require closures. For that, we consider the general diffusionprocess46 given by the system of modeled stochastic differential equations (SDEs) for X+i , U

+i , φ

+α , E

+, P+

which denote the probabilistic representations of position, velocity vector, mass fractions, pressure andenergy variables, respectively. Our proposed model, currently under development, is based on previouscontributions.8–10, 47–50 The proposed Fokker-Planck equation51 for the FDF is: FL(v,ψ, θ,η,x; t), and isof the form:

∂FL∂t

+∂viFL∂xi

=1

〈ρ〉ℓ

∂ 〈p〉ℓ∂xi

∂FL∂vi

−2

〈ρ〉ℓ

∂

∂xj

(

µ∂ 〈ui〉L∂xj

)

∂FL∂vi

−1

〈ρ〉ℓ

∂

∂xj

(

µ∂ 〈uj〉L∂xi

)

∂FL∂vi

+2

3

1

〈ρ〉ℓ

∂

∂xi

(

µ∂ 〈uj〉L∂xj

)

∂FL∂vi

−∂(

Gij(

vj − 〈uj〉L)

FL)

∂vi+

∂

∂xi

(

µ∂(FL/ 〈ρ〉ℓ)

∂xi

)

+∂

∂xi

(

2µ

〈ρ〉ℓ

∂ 〈uj〉L∂xi

∂FL∂vj

)

+µ

〈ρ〉ℓ

∂ 〈uk〉L∂xj

∂ 〈ui〉L∂xj

∂2FL∂vk∂vi

+1

2C0

ǫ

〈ρ〉ℓ

∂2FL∂vi∂vi

+ Cφω∂ ((ψα − 〈φα〉L)FL)

∂ψα+Ceω

γ

∂ ((θ − 〈e〉L)FL)

∂θ−γ − 1

γ

(

ǫ+ τ̆ij∂ 〈ui〉L∂xj

)

∂

∂θ

(

θ

ηFL

)

−γ − 1

γ

∂ (θAFL)

∂θ+γ − 1

γ2∂(

θB2FL)

∂θ−∂ (ηAFL)

∂η

+1

2

(γ − 1)2

γ2∂2(

θ2B2FL)

∂θ∂θ+γ − 1

γ

∂2(

θηB2FL)

∂θ∂η+

1

2

∂2(

η2B2FL)

∂η∂η

(12)

where

Gij =Πd

2k 〈ρ〉ℓ− ω

(

1

2+

3

4C0

)

δij , k =1

2τL (ui, ui) , ǫ = 〈ρ〉ℓ Cǫ

k3/2

∆L, ω =

1

〈ρ〉ℓ

ǫ

k

Πd = CΠ

(

〈

〈p〉ℓ∂ 〈ui〉L∂xi

〉

ℓ2

− 〈〈p〉ℓ〉ℓ1

∂ 〈〈ui〉L〉L2∂xi

)

A = −Ceω

E+(

E+ − 〈e〉L)

+1

E+

(

ǫ

ρ++

1

ρ+τ̆ij∂ 〈ui〉L∂xj

)

− γΠd

τℓ (p, p)

(

P+ − 〈p〉ℓ)

− γ∂ 〈ui〉L∂xi

−γ

γ − 1

1

ρ+E+

(

∂q̆i∂xi

+∂

∂xi

(

µ∂ 〈e〉L∂xi

))

, B = 0 (13)

Here ω is the SGS mixing frequency, ǫ is the dissipation rate, k is the SGS kinetic energy, and ∆L is the LESfilter size. The parameters C0, Cφ, Ce and Cǫ are model constants. The dissipation term ǫ and the pressuredilatation term Πd are modeled.

9, 52

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Numerical solution of the modeled EPVS-FMDF transport equation is obtained by a hybrid finite-difference (FD) / Monte Carlo (MC) procedure.53 The computational domain is discretized on equallyspaced FD grid points and the EPVS-FMDF is represented by an ensemble of MC particles which carryinformation pertaining to the energy, pressure, velocity and the scalar values. This information is updatedvia temporal integration of the modeled SDEs. Statistical information is obtained by considering an ensembleof NE computational particles residing within an ensemble domain of volume VE centered around each of theFD grid points. The FD solver is fourth-order accurate in space and second-order accurate in time.54 All ofthe FD operations are conducted on fixed grid points. The transfer of information from the FD points to theMC particles is accomplished via a linear interpolation. The inverse transfer is accomplished via ensembleaveraging. The FD transport equations include unclosed second order moments which are obtained from theMC. Further details on the hybrid FD-MC can be found in Ref.53

III. Sample Results

Numerical simulation is conducted of a three-dimensional temporally developing mixing layer involvingtransport of a passive scalar variable. Simulations are used to assess the consistency and the overall capabil-ities of the EPVS-FMDF methodology. These predictions are compared with data obtained by DNS of thesame layer configuration. In the representation below, x, y and z denote the streamwise, the cross-stream,and the spanwise directions, respectively. The velocity components along these directions are denoted byu, v and w in the x, y and z directions, respectively. The filtered streamwise velocity and the scalar fields areinitialized with hyperbolic tangent profiles with free-stream conditions as: 〈u〉L = 1, 〈φ〉L = 1 on the top and〈u〉L = −1, 〈φ〉L = 0 on the bottom. The density and temperature fields are initially uniform. The lengthLv is specified such that Lv = 2

NPλu, where NP is the desired number of successive vortex pairings and λuis the wavelength of the most unstable mode corresponding to the mean streamwise velocity profile imposedat the initial time. The flow variables are normalized with respect to the half initial vorticity thickness,

Lr =δv(t=0)

2 , (δv =∆U

|∂〈u〉L/∂y|max

, where 〈u〉L is the Reynolds-averaged value of the filtered streamwise ve-

locity and ∆U is the velocity difference across the layer). The reference velocity is Ur = ∆U/2. The domainis discretized on equally-spaced grid points ∆x = ∆y = ∆z = ∆, with resolutions of 1933 and 653 for DNSand LES, respectively. Simulations are conducted for a range of Reynolds (Re) and Mach (Ma) Numbers.In the results presented below, Re = UrLrν = 50, and Ma =

Ur√γRT

= 0.6. The model parameters are the

same as those suggested by Sheikhi et al.7 No attempt is made to optimize the values of these parameters.In the comparisons, we consider the “resolved” and the SGS components of the Reynolds-averaged mo-

ments.55 The former is denoted by R(a, b) and the latter by τFL (a, b) =〈ρ〉

ℓτL(a,b)

〈ρ〉ℓ

. The MC simulations are

conducted with several sizes of the ensemble domains to generate statistics (Fig. 2). In the results shownhere, VE = (∆x∆y∆z) within which NE ≈ 64. Figure 1 shows the instantaneous iso-surface of the 〈φ〉L fieldat t = 80. By this time, the flow is going through pairings and exhibits strong 3D effects. This is evident bythe formation of large scale spanwise rollers with the presence of secondary structures in streamwise planes.The consistency of the hybrid solver is established by checking the redundancy of the repeated fields asobtained via FD and MC. This is verified for some of the variables in Fig. 3. All of the first order momentsshow excellent agreements with DNS data and are not shown. Several components of the resolved secondorder moments and the SGS terms are presented in Figs. 4- 6. As shown, the EPVS-FMDF predictionscompare well with DNS data.

IV. Summary

The filtered density function (FDF) methodology has proven very effective for large eddy simulation (LES)of turbulent reactive flows.2 All previous contributions in FDF are concentrated on LES of low-speed flows.The objective of the present work is to develop the joint energy-pressure-velocity-scalar filtered mass densityfunction (EPVS-FMDF) methodology for LES of high-speed turbulent flows. The exact transport equationgoverning the evolution of the FDF is derived. It is shown that the effect of subgrid scale (SGS) convectionappears in a closed form. The unclosed terms are modeled in a fashion similar to that in probability densityfunction (PDF) methods. The capability of the EPVS-FMDF is demonstrated by conducting LES of atemporally developing mixing layer. The preliminary comparisons with DNS data are encouraging. Workis in progress on fine-tuning of the EPVS-FMDF sub-closures and applying the methodology for LES of a

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wider class of high-speed flows.

V. Acknowledgements

This research was sponsored by the National Center for Hypersonic Combined Cycle Propulsion GrantFA-9550-09-1-0611. The technical monitors on the grant are Dr. Chiping Li (AFOSR), Dr. Aaron Auslender(NASA) and Dr. Rick Gaffney (NASA). Computational resources are provided by the Extreme Science andEngineering Discovery Environment (XSEDE), which is supported by National Science Foundation underGrant OCI-1053575.

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25Raman, V. and Pitsch, H., “A Consistent LES/Filtered-Density Function Formulation for the Simulation of TurbulentFlames with Detailed Chemistry,” Proc. Combust. Inst., Vol. 31, 2007, pp. 1711–1719.

26Afshari, A., Jaberi, F. A., and Shih, T. I. P., “Large-Eddy Simulations of Turbulent Flows in an Axisymmetric DumpCombustor,” AIAA J., Vol. 46, No. 7, 2008, pp. 1576–1592.

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27Drozda, T. G., Wang, G., Sankaran, V., Mayo, J. R., Oefelein, J. C., and Barlow, R. S., “Scalar Filtered Mass DensityFunctions in Nonpremixed Turbulent Jet Flames,” Combust. Flame, Vol. 155, No. 1-2, 2008, pp. 54–69.

28Pope, S. B., “Self-Conditioned Fields for Large-Eddy Simulations of Turbulent Flows,” J. Fluid. Mech., Vol. 652, 2010,pp. 139–169.

29Yilmaz, S. L., Nik, M. B., Givi, P., and Strakey, P. A., “Scalar Filtered Density Function for Large Eddy Simulation ofa Bunsen Burner,” J. Propul. Power , Vol. 26, No. 1, 2010, pp. 84–93.

30Yilmaz, S. L., Nik, M. B., Sheikhi, M. R. H., Strakey, P. A., and Givi, P., “An Irregularly Portioned Lagrangian MonteCarlo Method for Turbulent Flow Simulation,” J. Sci. Comput., Vol. 47, No. 1, 2011, pp. 109–125.

31Nik, M. B., Yilmaz, S. L., Givi, P., Sheikhi, M. R. H., and Pope, S. B., “Simulation of Sandia Flame D Using Velocity-Scalar Filtered Density Function,” AIAA J., Vol. 48, No. 7, 2010, pp. 1513–1522.

32Nik, M. B., Yilmaz, S. L., Sheikhi, M. R. H., and Givi, P., “Grid Resolution Effects on VSFMDF/LES,” Flow Turbul.Combust., Vol. 85, No. 3-4, 2010, pp. 677–688.

33Ansari, N., Goldin, G. M., Sheikhi, M. R. H., and Givi, P., “Filtered Density Function Simulator on Unstructued Meshes,”J. Comp. Phys., Vol. 230, 2011, pp. 7132–7150.

34Réveillon, J. and Vervisch, L., “Subgrid-Scale Turbulent Micromixing: Dynamic Approach,” AIAA J., Vol. 36, No. 3,1998, pp. 336–341.

35Cha, C. M. and Troullet, P., “A Subgrid-Scale Mixing Model for Large-Eddy Simulations of Turbulent Reacting FlowsUsing the Filtered Density Function,” Phys. Fluids, Vol. 15, No. 6, 2003, pp. 1496–1504.

36Heinz, S., “Unified Turbulence Models for LES and RANS, FDF and PDF Simulations,” Theor. Comp. Fluid Dyn.,Vol. 21, 2007, pp. 99–118.

37Tong, C., “Measurements of Conserved Scalar Filtered Density Function in a Turbulent Jet,” Phys. Fluids, Vol. 13,No. 10, 2001, pp. 2923–2937.

38Wang, D. and Tong, C., “Conditionally Filtered Scalar Dissipation, Scalar Diffusion, and Velocity in a Turbulent Jet,”Phys. Fluids, Vol. 14, No. 7, 2002, pp. 2170–2185.

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41Wang, D. and Tong, C., “Experimental Study of Velocity-Scalar Filtered Joint Density Function for LES of TurbulentCombustion,” Proc. Combust. Inst., Vol. 30, 2005, pp. 567–574.

42Germano, M., “Turbulence: The Filtering Approach,” J. Fluid Mech., Vol. 238, 1992, pp. 325–336.43O’Brien, E. E., “The Probability Density Function (PDF) Approach to Reacting Turbulent Flows,” Turbulent Reacting

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Figure 1. Contour surfaces of the instantaneous 〈φ〉L

field in temporal mixing layer simulations via EPVS-FMDF.

∆

∆E 12

3

Figure 2. Ensemble averaging in MC simulations: 1(∆E = ∆/2), 2(∆E = ∆), 3(∆E = 2∆). Black squares denotethe FD grid points, and the circles denote the MC particles.

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(a)

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

<φ

>m

c

< φ >L

r=0.99982

(b)

(c) (d)

Figure 3. Scatter plots of (a) < u >L, (b) < φ >L , (c) < v >L and (d) < w >L. The thick solid line denotes 45◦.

The parameter r denotes the correlation coefficient.

9 of 12


−20 −10 0 10 200

0.05

0.1

0.15

0.2

R(u

i,u

i)/

2

y

Filtered DNSEPVSFMDF

(a)

−20 −10 0 10 200

0.005

0.01

0.015

0.02

0.025

0.03

τF L(u

i,u

i)/

2

y


(b)

−20 −10 0 10 20−0.08

−0.06

−0.04

−0.02

0

R(u

,v)

y


(c)

−20 −10 0 10 20−8

−6

−4

−2

0x 10−3

τF L(u

,v)

y


(d)

Figure 4. Cross-stream variations of (a) R(ui, ui)/2, (b) τFL (ui, ui)/2 , (c) R(u, v) and (d) τF

L(u, v). The thick solid

line denotes EPVS-FMDF predictions. The circles denote DNS data.

10 of 12


−20 −10 0 10 200

0.02

0.04

0.06

0.08

R(φ

,φ)

y


(a)

−20 −10 0 10 200

0.005

0.01

0.015

τF L(φ

,φ)

y


(b)

−20 −10 0 10 20−0.02

0

0.02

0.04

0.06

0.08

R(u

,φ)

y


(c)

−20 −10 0 10 20−2

0

2

4

6

8x 10−3

τF L(u

,φ)

y


(d)

−20 −10 0 10 20−0.05

−0.04

−0.03

−0.02

−0.01

0

0.01

R(v

,φ)

y


(e)

−20 −10 0 10 20−4

−3

−2

−1

0x 10−3

τF L(v

,φ)

y


(f)

Figure 5. Cross-stream variations of (a) R(φ, φ), (b) τFL(φ, φ) , (c) R(u, φ), (d) τF

L(u, φ), (e) R(v, φ) and (f) τF

L(v, φ).

The thick solid line denotes EPVS-FMDF predictions. The circles denote DNS data.

11 of 12


−20 −10 0 10 200

1

2

x 10−4

R(e

,e)

y


(a)

−20 −10 0 10 200

2

4

6

8x 10−5

τF L(e

,e)

y


(b)

−20 −10 0 10 20−3

−2

−1

0

1

2

3x 10−3

R(u

,e)

y


(c)

−20 −10 0 10 20−4

−2

0

2

4x 10−4

τF L(u

,e)

y


(d)

−20 −10 0 10 20−3

−2

−1

0

1

2x 10−3

R(v

,e)

y


(e)

−20 −10 0 10 20−1.5

−1

−0.5

0

0.5

1

1.5x 10−4

τF L(v

,e)

y


(f)

Figure 6. Cross-stream variations of (a) R(e, e), (b) τFL(e, e) , (c) R(u, e), (d) τF

L(u, e), (e) R(v, e) and (f) τF

L(v, e).

The thick solid line denotes EPVS-FMDF predictions. The circles denote DNS data.

12 of 12


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