Large eddy simulation of free-shear flows using an algebraic model

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Computers & Fluids 36 (2007) 1384–1397

Large eddy simulation of free-shear flows using an algebraic model

S. Bhushan *,1, Z.U.A. Warsi 2

Mississippi State University, MS 39762, USA

Received 30 April 2005; received in revised form 30 August 2006; accepted 30 December 2006Available online 14 April 2007

Abstract

Large eddy simulation of the free-shear flows has been performed using an algebraic subgrid stress model. The test cases consideredare the temporal planar jet and mixing layer. The results are compared with that of the Smagorinsky, dynamic Smagorinsky, mixed mod-els and the DNS data. For the mean flow quantities none of the models perform exceptionally better over the others. However, both thealgebraic and mixed models predict better statistics for the turbulent quantities than the linear models. The results also demonstrate thecapability of the algebraic model in depicting backscatter of energy explicitly, which is consistent in both the cases considered.� 2007 Elsevier Ltd. All rights reserved.

1. Introduction

Free-shear turbulence is observed in many engineeringapplications, for example in environmental flows. So farthe simulation of these complex problems has not beenpossible using the direct numerical simulation (DNS).However, large eddy simulation (LES) provides an attrac-tive alternative since it is numerically inexpensive and capa-ble of capturing the details of turbulent scales of motion.The prototypical free-shear flow cases studied in details,both experimentally and using DNS, are the planar jets[1,2], and mixing layers [3–5]. These well understood flowsform good validation cases for the subgrid stress (SGS)model. The purpose of this paper is to present the capabil-ity of an algebraic model in depicting the turbulent struc-ture of these flows.

In LES, the turbulent flow variables are decomposedinto resolved ðuÞ and subgrid scale components using suit-able filter functions [6,7]. The incompressible LES are thusgoverned by the filtered Navier–Stokes equations,

0045-7930/$ - see front matter � 2007 Elsevier Ltd. All rights reserved.

doi:10.1016/j.compfluid.2006.12.007

* Corresponding author. Tel.: +1 319 335 5580; fax: +1 319 335 5238.E-mail address: [email protected] (S. Bhushan).

1 Presently: Assistant Research Scientist, University of Iowa.2 Professor Emeritus, Department of Aerospace Engineering.

div u ¼ 0

o

otuþ ðu � gradÞu ¼ � 1

qgrad p þ vdivðgrad uÞ � div s

ð1Þ

where s is the SGS tensor which needs to be modeled. It hasbeen identified that the main effect of SGS is to transfer en-ergy between resolved and subgrid scales of motion. Theseenergy transfers occur bi-directionally leading to both for-ward and backscatter [8]. Backscatter of energy plays a sig-nificant role in determining the turbulent structure of theflow, and must be captured accurately by a SGS model[9,10].

The most commonly used model for SGS is the purelydissipative Smagorinsky model (SM). One of the disadvan-tages of this model is that the model coefficient needs to beadjusted from one case to another. The dynamic modelcoefficient evaluation approach (DSM) [11] helps to over-come this disadvantage. DSM is also reported to be capa-ble of capturing backscatter, but only in an averaged sense[8]. It is expected that a SGS model should predict back-scatter phenomenon explicitly and thus must be modeledindependently [12]. The eddy viscosity models have oftenbeen coupled with stochastic modeling of the backscatterenergy transfer [13]. However, the stochastic approach

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S. Bhushan, Z.U.A. Warsi / Computers & Fluids 36 (2007) 1384–1397 1385

disregards the organized vortical structure of backscatterphenomenon [14] and does not provide a desired determin-istic model.

Recently, the scale-similarity model has received muchattention and is found to depict turbulent structure accu-rately [15,5]. The scale-similar terms have been exploredin various other formulations such as the gradient model,and also has close ties with the deconvolution models[16,23]. Usually these models are combined with eddy-viscosity term to obtain the mixed model (MM) [17,5].Analysis shows that the scale-similarity terms are essen-tially the modified Leonard’s stress term, which on vol-ume averaging leads to net forward scatter of energy[18,23]. The MM coefficients computed in a dynamicframework may lead to negative dissipation locally,thereby providing backscatter naturally [10,5]. However,they are also found to be the source of numerical instabil-ity (cf. [19] for fresh perspective on this issue) and thusbackscatter is avoided by clipping [15,20,21]. It is expectedthat the net SGS dissipation should be locally positiveand the model coefficients that fail to satisfy this criteriashould be investigated.

More important source of backscatter turns out to bethe non-local interactions [22] which are due to a com-pletely separate mechanism than the scale-similar model(cf. [23] for details). Although deconvolution and gradientmodels have been shown to provide backscatter locally[16,10], they do not provide backscatter as an independentenergy transfer mechanism, as expected from a SGS model[12]. One of the current research directions in SGS model-ing is towards obtaining terms able to account for back-scatter independently [15,24]. Formulation of such termsintroduces unknown model coefficients, often computeddynamically. Such approach makes LES computationmore expensive and susceptible to numerical instability[15].

This paper uses an algebraic model (AM), obtaineddirectly from the second-order closure of the constitutiveequations of SGS [23]. The model has an additional termover the MM which accounts for backscatter of energyexplicitly. A similar term is also present in LANS-a model,which was obtained from a completely different frame-work. Recently Geurts and Holm [21] acknowledge thebackscatter capability of this term too. The AM has beenused in the simulation of channel flow in the previous study[23] where it showed better performance than the DSM.The AM has constant model coefficients which were com-puted based on isotropic turbulence theory [23]. The samemodel coefficients are used in the study of free-shear flowhere.

In the following section the AM is described and theenergy transfer capability of the model terms are presentedfor the isotropic decaying turbulence. In Section 3 thenumerical method used in the present study is describedin details. Sections 4 and 5 present the results of the planarjet and mixing layer simulations respectively. Finally con-clusions are drawn in Section 6.

2. Algebraic model

The transport equation for the subgrid stresses can beobtained directly from the Navier–Stokes equation, whichinvolves several correlation terms. Introducing the general-ized central moments for the subgrid scales of motion thesecorrelations can be expressed as diffusion, dissipation,production and pressure–strain correlation terms (cf. [32],Section 6.35 for details). These correlation terms can bemodeled following the Reynolds’ Averaged Navier–Stokes(RANS) approach, and using generalized subgrid momentsinstead. Similarly, the transport equation of the subgridscale kinetic energy can be obtained. These two equationsalong with the structural equilibrium assumption yieldsthe algebraic equation for subgrid stresses. Substitutingthe modeling of the correlation terms, in the algebraicequation, constitutive equation for the sub-grid stressesare obtained, which are solved using deductive iteration.The first order closure of the equations give the linearmodel, i.e., the Smagorinsky model, whereas the secondorder approximation gives the AM (cf. [23] for details),

s ¼ 2

3KsgsI� 2C2

s D2½2bD : bD�1=2 bD

þ a1D2 Du � ðruÞT � 1

3ðru : ruÞI

� �� a2D

2 ðruÞT � ru� 1

3ðru : ruÞI

� �ð2Þ

where D is the filter width and for conciseness ($) is used inplace of grad. Interested readers are referred to the text ofWarsi [32] for component form of the model. The secondterm on the RHS of Eq. (2) is the (Smagorinsky) eddy vis-cosity term and the third term is the Taylor’s expansion ofthe modified Leonard’s stress (related to the scale-similaror deconvolution model [17,16,10]). As is evident the AMprovides an additional term (fourth term) over the MM.Role of these model terms, in the energy transfer mecha-nism, can be studied analytically, by the closure of subgridstresses with the rate-of-strain tensor ðbDÞ, in the canonicalframework (homogeneous isotropic turbulence with infiniteinertial subrange). Analysis suggests that the second term ispurely dissipative in nature, whereas (for positive modelcoefficient) fourth term provides backscatter of energy[23]. This additional term (forth term in Eq. 2) is also pres-ent in the LANS-a model [21] where its backscatter capabil-ity has also been recognized in the turbulent mixing layersimulation. Moreover, it must be noted that such termcannot be obtained from the Taylor’s expansion of scale-similar terms or from the deconvolution method, butappears naturally in the algebraic approach. The AM thusdoes not involve any heuristic modeling of backscatter ofenergy.

The three unknown model coefficients appearing in themodel are computed to satisfy the dissipation criteria, suchthat in the canonical framework the eddy viscosity term pro-vides dissipation equal to the forward scatter (cf). Further,

1386 S. Bhushan, Z.U.A. Warsi / Computers & Fluids 36 (2007) 1384–1397

coefficients of third and fourth term are adjusted such as toaccount for the Leonard’s dissipation (cL) and backscatter(cb) respectively. These energy transfer coefficients wereestimated analytically for the canonical case as presentedin Table 2 [23]. The model coefficients thus obtained are pre-sented in Table 1. The model coefficient of MM can also beobtained similarly such that eddy-viscosity term accountsfor both forward scatter and backscatter.

In the previous study [23], dissipative characteristics ofthe model has been tested for the decaying isotropic turbu-lence (at Re = 1346) and forced isotropic turbulent flows.The AM results were compared with the linear (Smagorin-sky) model and mixed model. The results showed that themodel terms depict the energy transfer consistent with theanalytic theory and performs better than other models inpredicting the Kolmogrov’s spectra. The model was furtherapplied to the wall bounded, i.e., channel flow, over a rangeof Reynolds number. In this case the model results werecompared with the dynamic and constant coefficientSmagorinsky models, and also with the DNS data. Resultsshow that all the models capture the mean flow character-istics well, but algebraic model predicted better results forthe turbulent intensity profiles and higher order statisticsof the turbulent flow. The main advantage of the modelis its capability to capture explicitly the subgrid energybackscatter, which was also demonstrated. In this paperthe model has been applied to the free shear flow cases,in order to further validate the backscatter capability ofthe AM. An additional simulation of the decaying isotropicturbulence case has also been performed below, which is ata higher Reynolds number than the previous case.

Numerical simulation of isotropic decaying turbulenceprovides an ideal platform to study the capability of thealgebraic model in depicting the subgrid scale energy trans-fer mechanism. The governing equations for the simula-tions are the filtered Navier–Stokes equation, Eq. (1),which are solved using pseudo-spectral method with 3/2rule to remove dealiasing error. The simulation is per-formed, to match the experimental results of Kang et al.[25], on a cubic box of side 2p discretized into 643 grid(after dealiasing). This case corresponds to the Reynoldsnumber of 2.472 · 104. The initial conditions are obtainedusing the energy spectra at x/M = 20 (of the experiment)with random phases constructed following Rogallo [26].The simulation using SM are run assuming sharp spectralfilter and the spectra is filtered a posteriori using Gaussianfilter to compare with the MM and AM simulations. TheAM and MM are based on Gaussian filter, thus filtered

Table 1Coefficients of subgrid stress models computed based on dissipationcriteria

C2s a1 a2

Smagorinsky model 0.03 – –Mixed model 0.0162 1

12 –Algebraic model 0.0213 1

12 0.03364

energy spectrum is used for initialization. The filter widthfor these simulation is chosen such that the filter transferfunction at cut-off is G(kc) = 0.8 [27].

All the models considered provide accurate amount ofdissipation, however the AM predicts the spectra betterin the intermediate wavenumber range as shown inFig. 1a. The results show that the fourth term in the model(2) provides backscatter explicitly, although less than thatof canonical case, as shown in Table 2. The transfer spectravia the model terms, Fig. 1b, show that the backscatteroccurs to all the resolved scales and decreases near cut-off. The peak of the backscatter and Leonard’s dissipationare almost at the same location, whereas the eddy viscositydissipation drops sharply near the cut-off. The energytransfer via the Leonard’s term is same in both the MMand AM. The eddy term in the MM leads to an amountof dissipation equal to the (cf � cb: as obtained from theAM), which is in accordance with the assumptionsinvolved in the MM. For the AM, the transfer coefficientsalthough vary from the canonical case, but are in accept-able limits. These results suggest that the AM depicts thesubgrid scale energy transfers as expected.

Fig. 1. (a) Energy spectra compared with experimental result. (—)Experimental [25]; (s) SM; (h) MM; (+) AM. (b) Transfer spectral ofSGS model Eq. (2) terms. (—) 2nd term (eddy viscosity); (- - -) third term(Leonard’s) ; (s) forth term (backscatter). Location corresponding tox/M = 48 of experimental results. Abscissa is wavenumber.

Table 2Energy transfer coefficients by the algebraic model terms in numericalsimulations

cf cL cb

Canonical case 0.736 0.44 0.176Decaying turbulence 0.81 0.315 0.125Jet flow 0.855 0.245 0.1Mixing layer 0.854 0.245 0.099


3. Numerical method

The temporally evolving free-shear flows considered inthis paper involves periodic streamwise (x1) and spanwise(x3) directions and inhomogeneous normal direction (x2)[28]. The filtered Navier–Stokes Eq. (1) are solved usingpseudo-spectral method employing Fourier series in thehomogeneous directions and Chebyshev polynomial alongthe inhomogeneous direction. The momentum equationsare solved using a fractional step method which involvessplitting the equations in three steps [29]. The first stepmarches the convective term using second-order AdamsBashforth method i.e.,

uð1Þ ¼ uðNÞ þ dt3

2ðuðNÞ � gradÞuðNÞ � 1

2ðuðN�1Þ � gradÞuðN�1Þ

� �ð3Þ

where the superscript (1) implies the level of the fractionalstep, (N) the time iteration level and dt is the time step size.The computation of the convective term involves convolu-tion in wavenumber space, which is computed in physicalspace using 3/2 dealiasing rule. The second step is the pres-sure correction step which imposes incompressibility. Theequations to be solved in this step are:

uð2Þ ¼ uð1Þ � dt1

2grad p ð4Þ

along with the continuity equation

div uð2Þ ¼ 0 ð5Þ

The solution of these equations can be easily obtained inspectral space, which on taking the Fourier transformreduce to (cf. [29]),

o2�uð2Þ2

ox22

� ðj21 þ j2

3Þ�uð2Þ2 ¼ �ðj2

1 þ j23Þ�uð1Þ2 � i j1

o�uð1Þ1

ox2

þ j3o�uð1Þ3

ox2

" #ð6Þ

where the superscript ð�Þ has been eliminated for conve-nience. The superscript ð��Þ denote the corresponding twodimensional Fourier components ðFÞ of the quantities, ob-tained by Fast Fourier transform along homogeneousdirections i.e.,

Fx1Fx3

u2ðx1; x2; x3Þ ¼ �u2ðj1; x2; j3Þ

After solving Eq. (6) the pressure correction can be ob-tained as,

dt�pq¼ � 1

j21 þ j2

3

iðj1�uð1Þ1 þ j3�u

ð1Þ3 Þ þ

o�uð2Þ2

ox2

" #However, the error induced in solving the above equationis more than solving the Poisson’s equation [30] for thepressure

o2�p

ox22

� ðj21 þ j2

3Þ�p ¼ �qdt

iðj1�uð1Þ1 þ j3�u

ð1Þ3 Þ þ

o�uð1Þ2

ox2

" #ð7Þ

We choose to solve the Poisson’s equation with zero pres-sure fluctuation as a boundary condition. Computation ofthe other velocity components are computed as follows:

�uð2Þ1 ¼ �uð1Þ1 � ij1dt�pq

ð8Þ

�uð2Þ3 ¼ �uð1Þ3 � ij3dt�pq

ð9Þ

The third step incorporates the viscous and subgridstresses,

dtRe

o2�uðNþ1Þ

l

ox22

� dtReðj2

1 þ j23Þ þ 1

� ��uðNþ1Þ

l

¼ ��uð2Þl � dt iðj1�sl1 þ j3�sl3Þ þo�sl2

ox2

� �ð10Þ

where l ¼ 1; 2; 3 and stresses �sij are computed based on thevelocities �uð2Þl .

The flow considered here involves quiescent flow in thenormal direction thus the infinite dimension is mappedonto the Chebyshev Gauss-Lobatto points (n) using alge-braic mapping [29], as

x2 ¼Ln

ð1� n4Þ1=4

The solution of Poisson’s equations (6), (7) and (10) with theabove mapping are carried out using collocation diagonal-ization method as described in Appendix. The eigenvectorsand eigenvalues required for the calculations (as describedin Appendix) are computed apriori using MATLAB [31].

3.1. Simulation parameters

The simulations here are performed using the SM,DSM, MM and AM models discussed in the previous sec-tion. Simulation is also performed without any modelreferred to as NM. This case gives an idea about the effectof SGS modeling. For the SM, coefficient Cs = 0.1 is used,which is usually recommended [5]. The AM and MM sim-ulation are performed using model coefficients as in Table1. The filter width (D) is computed based on grid size as(D1D3dn)1/3, which leads to an inhomogeneous filter width.Authors acknowledge that such a definition of D contra-dicts the assumptions involved in deriving the governingequations [7]. However, results based on the presentapproach will form the basis of future improvements ofthe model. The DSM coefficient is computed every time

Fig. 2. Streamwise mean velocity profile along normal direction at U0t/d0 = 100 for the jet flow: (a) coarse grid; (b) fine grid. (—) DNS [28]; (- - -)NM; (+) SM; (s) DSM; (h) MM; (n) AM.


step following Lilly’s [11] approach. In these calculations,secondary filtering is performed only along the homoge-neous directions (thus does not involve inhomogeneityissue) using Simpson’s rule [28], where secondary filterwidth is assumed to be twice the grid scale.

The domain size considered in the streamwise and span-wise directions are:

L1 ¼16pa

and L3 ¼16pb

where parameters a and b are chosen as prescribed in DNSsimulations [28,4]. The initial mean velocity fields are pre-scribed as the laminar profile,

Jet flow: U 1ðx2Þ ¼U 0

2tanh 6:25 1þ x2

d0

� �� þ tanh 6:25 1� x2

d0

� �� Mixing layer: U 1ðx2Þ ¼ U 0 tanh

x2

d0

� �

where d0 is the initial half jet width or the momentumthickness for the mixing layer simulation. The mean veloc-ity field is superimposed with perturbation specified as[28,4],

u0ðxÞ ¼ r � ½/ðxÞIðx2Þ�where /(x) are the isotropic random phase fluctuations ob-tained following Rogallo [26] assuming an energy spectrum

EðjÞ ¼ K

1þ j2K2

� �where K is chosen as 0.48 d0 The factor I(x2) is used to nor-malize the fluctuation along the inhomogeneous directionsuch as to obtain the realistic perturbation. Based onRMS fluctuation database it is specified as [28],

Iðx2Þ ¼ e �0:2x22=d

20ð Þ

Finally, the fluctuations are scaled such that the turbulentenergy is of the order (10�6) of the mean flow. Further,the derivative computation using Chebyshev’s polynomialare accurate for smooth functions, but for non-smoothfunctions (as prescribed here) produces spurious oscilla-tions [30], at the boundary. Thus for numerical stabilityl/4th of the higher wavenumber modes are forced to bezero after every time step. Similar methodology was alsoadopted by Ribault et al. [20].

In this paper the performance of the SGS models in thefully developed turbulent state are studied. The results arecompared with the DNS values of Ansari [28,4] at the par-ticular instant of time. The LES results have been averagedalong the homogeneous streamwise and spanwise direc-tions to obtain mean values and no time averaging is per-formed. The streamwise fluctuating velocity field iscomputed based on such mean values.

4. Turbulent planar jet flow

The simulation for temporally evolving planar jet flow isperformed for the Reynolds number (Re=) 1319 (based on

centerline velocity and jet half width), which correspondsto the DNS case of Ansari [28]. The initial jet half width(d0 = 1) and centerline velocity (U0 = 1) are chosen. Thedomain size is prescribed by specifying the length parame-ters L = 6, a = 2.65 and b = 2.65 as used in the DNS sim-ulation. The simulations are performed on two gridresolutions 32 · 49 · 32 and 32 · 65 · 32 along streamwise,normal and spanwise directions respectively. Henceforth,these simulations are referred to as the coarse and fine gridcases respectively.

The transition of the flow from the laminar to fullydeveloped turbulent state is monitored from the evolutionof the turbulent kinetic energy. The turbulent kineticenergy grew in the transition zone to a quasi-steady valuein the turbulent zone. The vorticity contours show the for-mation of Kelvin–Helmholtz structures at the two jet edges(figures not shown). These vortices start breaking andmerging with each other leading to a fully developed smallscale turbulent structure. The development of the jet fromthe laminar to turbulent region depends on the subgrid


model, as expected the DSM adjusts to the transition betterthan constant coefficient models. Post-processing of theresults in between tU0/d0 = 95 and 105 show the self-simi-lar behavior of the flow, as expected (for which figure is notpresented). The main emphasis of the paper is to presentand compare the turbulent characteristics of the flow usingLES, with the DNS data. Thus, the results presented hereare at time tU0/d0 = 100, which is the same time at whichDNS data were reported [28].

4.1. Mean flow

All the model simulations overestimated jet half width(about �2%) except for DSM which correlated well withDNS value in the fine grid case. However, on the coarse gridcase the nature was reversed where DSM overestimates itsvalue by about 1.5%. The plot of mean velocity profile,Fig. 2, shows that all the models underpredict the meanstreamwise velocity. On the coarse grid SM performs thebest followed by DSM, AM, MM and NM. On the fine gridthe NM case is close to the DNS value whereas, the addi-

Fig. 3. RMS profile of streamwise velocity fluctuation u01 along normaldirection: (a) coarse grid; (b) fine grid. Same keys as in Fig. 2.

tional dissipation from the SGS models leads to an under-prediction. DSM results are close to that of the NM case,this is because the SGS dissipation by DSM is much smallerthan that of the other models (discussed later).

4.2. Turbulent statistics

The profiles of the RMS velocity fluctuations are pre-sented in Figs. 3–5. As seen the NM case performs poorlyfor both the grid resolution and thus improvement in theseresults are expected from the SGS model. On the coarsegrid DSM shows less turbulence intensities for all the threevelocity components. SM performance is similar to that ofthe DSM for streamwise and normal component butunderpredict the spanwise component considerably. AMpredicts higher value of the turbulence intensities thanthe DSM and SM especially for the normal and spanwisecomponents. The intensities predicted by the MM are com-parable with that of the AM but shows more asymmetrythan the other models. The finer grid case shows wide var-iation in the LES results. The DSM results are no better

Fig. 4. RMS profile of normal velocity fluctuation u02 along normaldirection: (a) coarse grid; (b) fine grid. Same keys as in Fig. 2.

Fig. 5. RMS profile of spanwise velocity fluctuation u03 along normaldirection: (a) coarse grid; (b) fine grid. Same keys as in Fig. 2.

Fig. 6. Reynolds shear stress, hu01u02i, profile: (a) coarse grid; (b) fine grid.Same keys as in Fig. 2.


than the NM case. The intensities are even smaller thanthat of the NM, especially in the normal component. Forthe streamwise component SM results are closer to DNS,but fails to capture the profile in the other directions.AM overpredicts the turbulence intensities for all the veloc-ity components. MM predicts their values comparable tothat of the AM, but AM captures the profile better thanother models.

Fig. 6 shows the Reynolds shear stress profile. On thecoarse grid all the models perform similarly and underesti-mate the stress. Closer inspection reveals that the stress ismost in MM followed by AM, SM and DSM. Surprisingly,NM predicts stress better than all the models at least onone side of the shear layer. Fine grid case shows much bet-ter results where the NM case underestimates the stress themost. AM shows much better agreement with DNS thanother models. DSM predict the least stress comparable toNM. The MM results are of the order of the AM, whereasSM has an intermediate behavior. Comparison of the SGSshear stress (Fig. 7) shows that AM provides the moststress followed by MM, SM and DSM.

4.3. Higher order statistics

More information about the turbulent structure of theflow can be obtained from the velocity skewness S(/)and flatness (kurtosis) F(/) defined as

Sðu0iÞ ¼hu03i ihu02i i

3=2and F ðu0iÞ ¼

hu04i ihu02i i

2

where repeated indices do not imply summation andu0i ði ¼ 1; 2; 3Þ are the fluctuating velocity components.Since the DNS values are not available for these higher or-der statistics only qualitative analysis is performed basedon the probability distribution function (PDF) observedfor such flow. The experimental results [3,6] show thatthe PDF has a Gaussian profile at the center of the jetand diverges considerably at the edge of the flow. All themodel simulation show a similar behavior as seen inFigs. 8 and 9. Skewness and flatness values are close to 0and 3 respectively at the center of the jet, which correspond

Fig. 7. Subgrid shear stress (s12) profile. Same keys as in Fig. 2.

Fig. 8. Skewness profile for: (a) u01; (b) u02; (c) u03. Same keys as in Fig. 2.


to the Gaussian distribution. Divergence in skewness is notas large as flatness, in accordance with experimental results[6]. The skewness profile of the streamwise velocity fluctu-ation shows symmetry as these velocity fluctuation arealong the same direction on either side of the jet, whereasthe normal velocity skewness is antisymmetic because thefluid entrainment is in opposite directions at the upperand lower edges of the jet. Spanwise direction is the homo-geneous direction thus the velocity fluctuation in thesedirection do not show any structure but remains close tozero throughout. The AM shows most divergence followedby MM, SM and DSM. NM case although it shows similarnature but does not show much variation. Similar nature isobserved for the flatness profiles. The NM case fails to pre-dict divergence for the streamwise and spanwise velocitycomponents.

Fig. 9. Flatness profile for: (a) u01; (b) u02; (c) u03. Same keys as in Fig. 2.

Fig. 11. Energy transfer coefficients profile in jet flow. (—) Coarse grid;(- - -) fine grid. (No symbol) cf; (s) cL; (h) cb.


4.4. SGS dissipation

Fig. 10a compares the dissipative characteristics of allthe models considered. As is evident DSM is least dissipa-tive with wide fluctuations in the dissipation spectra, owingto the variation of the model coefficient. SM is slightlymore dissipative than the DSM. However, both MM andAM are most dissipative about 2.5 times more thanDSM. The energy dissipation by the AM model termsare presented in Fig. 10b, which shows the capability ofthe AM in predicting backscatter explicitly via fourth termin Eq. (2). As seen in Fig. 11, throughout the jet width theenergy transfer coefficients from the model terms are nearlyconstant for both the grid resolution. The amount of back-scatter obtained by the AM is much lower (than 40%) asreported by Akhavan et al. [1]. It must be noted that theirstudy considered the backscatter via the near-local oracross cut-off interactions, which has not been consideredexplicitly in the AM [23]. However, the backscatter

Fig. 10. (a) SGS dissipation profile by various models. Same keys as inFig. 2. (b) Forward scatter, backscater and Leonard’s dissipation inAlgebraic model. (—) Second term in Eq. (2); (- - -) fourth term in Eq. (2);(s) third term in Eq. (2).

through the non-local interactions are reported to be about10% (for mixing layer [21]) comparable to the valueobtained here.

5. Turbulent planar mixing layer

The turbulent mixing layer case is simulated for initialReynolds number of Red = 220 (defined based onDU = 2U0 velocity difference between two streams and d0

the initial momentum thickness). For the computationsperformed here U0 = 1 and d0 = 1, thus the quiescent flowvelocities are ±1. Parameters defining the length scale arechosen such that L = 12, a = 0.446 and b = 0.8, which isidentical to the DNS case [4]. The simulation is performedonly on a single grid resolution of 32 · 65 · 32.

As before the simulation was started with the laminarprofile with random phase perturbations imposed over it.The simulation undergoes transition leading to the growthof the momentum thickness which has a slope of 0.003 inthe laminar region and about 0.018 in the turbulent region.The vorticity contour shows the commonly observed roll-up of fundamental instability and successive pairing atthe interface of the opposite moving fluids. The resultsare compared with the DNS results at DUt/d0 = 400, wherethe flow has achieved the fully developed turbulent state.Results presented here are non-dimensionalized by theDU and momentum thickness dm.

5.1. Mean flow quantities

The momentum thickness of the mixing layer is overpre-dicted in all the simulations most by MM and least in NMcase. The self-similar characteristics of the flow were con-firmed in this case too. Instead of going into details ofthe development of the mixing layer we compare only theturbulent characteristics. The plot of the mean streamwiseprofile (Fig. 12) shows that the results from all the model

Fig. 12. Streamwise mean velocity profile normalized by U0 at DUt/d0 = 400 along normal direction normalized by dm for the mixing layer.(—) DNS [4]; (- - -) NM; (+) SM; (s) DSM; (h) MM; (n) AM.


collapse on the DNS data whereas, the NM case deviatesconsiderably.

Fig. 13. RMS profile of fluctuating velocity components (a) u01, (b) u02 and

5.2. Turbulent statistics

Fig. 13 shows the plane averaged RMS values of veloc-ity fluctuations. As seen the NM case deviates the mostfrom the DNS results for all the components. All the mod-els overestimate the streamwise component, most by SMand least by DSM. The MM does not show a definite peakwhereas, AM predicts a better profile. For the normal com-ponent, SM overpredicts the peak whereas DSM is under-predictive. Both MM and AM compare well with DNS,but in this case MM predicts the profile better. The turbu-lent intensity for the spanwise velocity component is over-predicted by all the models most by SM, whereas DSM,MM and AM do not differ much from each other. AMshows intermediate profile of DSM and MM.

The turbulent shear stress profile, Fig. 14, shows thatSM does not correlate well with the DNS data. It overpre-dicts its value considerably and deviates a lot from DNSprofile. The DSM results fail to show a definite peak.The AM also predicts a higher value of the stress, butshows a better profile than the SM. The MM predicts smal-ler value than the AM, but has a wider profile. The NMresults does not agree at all with DNS results. The SGSshear stress profile reveals that it has the same nature asthat of Reynolds stress (also observed in the jet flow) withmaximum stress produced by AM and least by SM.

(c) u03 normalized by DU along normal direction normalized by dm. Samekeys as in Fig. 12.

5.3. Higher order statistics

The skewness and flatness of the velocity fluctuations asdefined earlier are compared for different model simula-tions in Figs. 15 and 16. The skewness of the streamwisevelocity fluctuation shows a nearly linear profile with posi-

tive values in the lower stream and negative values in theupper stream. The change in the sign in the skewness pro-file is due to the opposite direction of the flow entrainmentin the upper and lower streams of flow (as opposed to the

Fig. 14. Shear stress profile (a) Reynolds ð�hu01u02iÞ normalized by DU2

and (b) subgrid stress (�s12). Same keys as in Fig. 12.

Fig. 15. Skewness profile for: (a) u01; (b) u02; (c) u03. Same keys as in Fig. 12.


jet flow case). The skewness is zero at the center of the mix-ing layer which is in accordance with the PDF distribution,which shows Gaussian nature at the mixing layer center.The skewness deviates from the Gaussianity at the edgeof the mixing layer leading to divergence. The normalvelocity component shows a similar nature as that observedin the jet case. This behavior is due to entrainment of thenormal velocity from opposite directions on either side ofthe mixing layer. The variation in the streamwise velocityis most for AM and least for DSM. The NM case barelyshows any structure in the flow. The normal componentvariations are better in DSM than AM. The MM andSM has behavior intermediate of these models. The span-wise component is nearly homogeneous in nature withskewness varying randomly but, stays close to zero. Allthe models show similar magnitude of the scatter.

The flatness profile shows a similar behavior as observedin the jet flow case. Flatness for all the velocity componentsis close to 3 in the mixing layer center and diverges at theedge. The divergence is much larger than the skewness

value as expected. Although all the models show similarnature but the NM case gives a poor variation even inthe mixing layer zone where it underestimates its valuefor the normal component and overestimates for the span-wise component. For the streamwise component it does notshow any divergence at all. As observed there is a signifi-cant asymmetry in the mixing layer which is expected forthe turbulent flow. Overall both DSM and AM show betterbehavior, whereas the variations in SM are the least.

5.4. SGS dissipation

As shown in Fig. 17 both the MM and AM provideequal amount of dissipation about 1.6 times as that ofDSM. The SM is found to be least dissipative and almosthalf of the AM. The large variations in the DSM dissipa-tion is because of the dynamic model coefficients in the

Fig. 16. Flatness profile for: (a) u01; (b) u02; (c) u03. Same keys as in Fig. 12.

Fig. 17. (a) SGS dissipation profile by various models. Same keys as inFig. 12. (b) Forward scatter, backscatter and Leonard’s dissipation inAlgebraic model. (—) Second term in Eq. (2); (- - -) Fourth term in Eq. (2);(s) third term in Eq. (2).


mixing layer. Again, the AM leads to backscatter indepen-dently via the fourth term in the model. The amount ofenergy dissipation via the model terms are presented inFig. 17b. The energy dissipation through these terms arepresented in terms of the transfer coefficient in Table 2.The energy transfer coefficients show nearly constant valuethroughout the mixing layer (figure not presented) and areclose to that obtained from the jet flow case. The amountof backscatter produced by AM is in good agreement withother numerical results (cf. [5,15]).

The computation of the energy transfer coefficient forthe MM shows that the Leonard’s term produces sameamount of dissipation as that in the AM. Thus the eddyviscosity term in the MM accounts for net effect of forwardscatter and backscatter. Same results were obtained both indecaying and jet flow cases. This result impresses on thefact that AM accounts for backscatter, which are not

explicitly accounted in MM, thereby capturing the SGSenergy transfer better.

6. Conclusion

Numerical simulation of free-shear flow has been per-formed using algebraic model. The results of the algebraicmodel is compared with the Smagorinsky, dynamic Smago-rinsky, constant coefficient mixed model and the DNSdata. Two well documented cases have been consideredplanar jet and mixing layer. The results have been com-pared only in the fully developed turbulent region for theturbulent intensities, higher order statistics and dissipativecharacteristics. The model results were also compared withthe no-model case, which shows significant improvement inthe turbulent characteristics of the flow because of the SGSmodeling. Comparison of the mean flow characteristics, asobtained by various models, suggest that none the modelsperform exceptionally well over the other. However, both


the algebraic and mixed models (non-linear models) predictlarger turbulent intensities (implying more energy in theresolved scales of motion) than the dynamic and Smagorin-sky models (linear models). Similar trends were alsoobserved for the higher order statistics. The foremostadvantage of the algebraic model is its backscatter capabil-ity which is in close agreement with the DNS value. Theother models fail to predict the backscatter mechanismexplicitly (cf. [5]), whereas the algebraic model has thecapability to depicts the subgrid scale energy transfer bet-ter. The energy dissipation by the algebraic model termsare presented as energy transfer coefficients which are con-sistent in both the free-shear flow cases.

The constant coefficient algebraic model is based onlyon the first order gradient terms and uses only 3% moreCPU time than the Smagorinsky model. This is muchcheaper than the dynamic model, which uses 12% moretime. These inherent qualities of the algebraic model makesit a suitable candidate for the complex engineering applica-tions. Despite the advantages of the model it also suffersfrom few limitations. Such as, the algebraic model containsthe gradient model terms, which have been shown to be asource of numerical instability. Thus, such instability mayalso occur in the algebraic model. Although no instabilitieswere observed in the flows considered here, but mightoccur in complex engineering problems. The model canbe further improved by adopting the modifications avail-able for the gradient model, which is currently under inves-tigation. Further, the constant coefficient model does notperform well (compared to the dynamic model) in the tran-sition zone which needs to be investigated further.

Appendix

Here we describe the salient points of the methodologyto solve the Poisson’s equation using collocation method[30]. Consider the equation,

o2y

ox2þ Ky ¼ f ðxÞ ð11Þ

with the algebraic mapping

x ¼ gðnÞ; �1 6 n 6 1

The partial derivatives are expressed as,

oyox¼ on

oxoyon

ando

2yox2¼ on

ox

� �2o

2y

on2þ o

2nox2

oyon

In the collocation method the derivatives can be repre-sented in terms of the node values of the variable (cf. [30,pp. 50–52]) such that,

oyon¼ dð1Þij yj;

o2y

on2¼ dð2Þij yj

where dðnÞij is the nth order derivative matrix. Eq. (11) thusreduces to,

QijyðnjÞ þ KdijyðnjÞ ¼ f ðnjÞ ð12Þ

where

Qij ¼onox

� �2

i

dð2Þij þo2nox2

� �i

dð1Þij

Matrix diagonalization is carried out such that,

Q ¼ PKP�1

where P is the matrix of right eigenvector and K is the diag-onal matrix of the corresponding eigenvalues. Substitutingthe above decomposition of Q in Eq. (12) the solution isobtained in three steps

F i ¼ P�1ij f ðnjÞ

Y i ¼1

Ki � KF i

yðniÞ ¼ P ijY j

ð13Þ

The variable n can be mapped back to x to obtain the solu-tion of Eq. (11).

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Large eddy simulation of free-shear flows using an algebraic model

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