University, Jeddah, Kingdom of Saudi Arabia Published online: 08 Apr 2015.

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1 This article was downloaded by: [ ] On: 1 April 215, At: 15:2 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: Registered office: Mortimer House, Mortimer Street, London W1T 3JH, UK Journal of Turbulence Publication details, including instructions for authors and subscription information: Proposed stochastic parameterisations of subgrid turbulence in large eddy simulations of turbulent channel flow V. Kitsios ab, J.A. Sillero c, J.S. Frederiksen b & J. Soria ad a Laboratory For Turbulence Research in Aerospace and Combustion, Department of Mechanical and Aerospace Engineering, Monash University, Clayton, Australia b Oceans and Atmosphere, CSIRO, Aspendale, Australia Click for updates c School of Aeronautics, Universidad Politécnica de Madrid, Madrid, Spain d Department of Aeronautical Engineering, King Abdulaziz University, Jeddah, Kingdom of Saudi Arabia Published online: 8 Apr 215. To cite this article: V. Kitsios, J.A. Sillero, J.S. Frederiksen & J. Soria (215) Proposed stochastic parameterisations of subgrid turbulence in large eddy simulations of turbulent channel flow, Journal of Turbulence, 16:8, , DOI: 1.18/ To link to this article: PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francis makes every effort to ensure the accuracy of all the information (the Content ) contained in the publications on our platform. However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information. Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content. This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. Terms &

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3 Journal of Turbulence, 215 Vol. 16, No. 8, , Proposed stochastic parameterisations of subgrid turbulence in large eddy simulations of turbulent channel flow V. Kitsios a,b,j.a.sillero c,j.s.frederiksen b and J. Soria a,d a Laboratory For Turbulence Research in Aerospace and Combustion, Department of Mechanical and Aerospace Engineering, Monash University, Clayton, Australia; b Oceans and Atmosphere, CSIRO, Aspendale, Australia; c School of Aeronautics, Universidad Politécnica de Madrid, Madrid, Spain; d Department of Aeronautical Engineering, King Abdulaziz University, Jeddah, Kingdom of Saudi Arabia Downloaded by [ ] at 15:2 1 April 215 (Received 1 November 214; accepted 3 March 215) Stochastic and deterministic subgrid parameterisations are developed for the large eddy simulation (LES) of a turbulent channel flow with friction-velocity-based Reynolds number of Re τ = 95 and centreline-based Reynolds number of Re = 2,58. The subgrid model coefficients (eddy viscosities) are determined from the statistics of truncated reference direct numerical simulations (DNSs). The stochastic subgrid model consists of a mean-field shift, a drain eddy viscosity acting on the resolved field and a stochastic backscatter force of variance proportional to the backscatter eddy viscosity. The deterministic variant consists of a eddy viscosity acting on the resolved field, which represents the effect of the drain and backscatter. LES adopting the stochastic and deterministic models is shown to reproduce the time-averaged kiic energy spectra of the DNS within the resolved scales. Keywords: channel flow; large eddy simulation; stochastic subgrid parameterisation 1. Introduction The computational cost of simulating fluid flows at Reynolds numbers of practical engineering and geophysical interest by explicitly resolving all scales of turbulence is currently prohibitive. An alternate approach is to undertake large eddy simulation (LES), where the large-scale eddies are resolved on a computational grid and the interactions between the resolved scales and the unresolved subgrid scales are modelled. Various approaches have been taken to model these subgrid interactions. In its most basic form, the subgrid modelling problem is the determination of the relationship between the subgrid tendency (or equivalently the negative gradient of the subgrid stress tensor) and the resolved field. In the vast majority of studies, such a relationship is proposed, often motivated by physical considerations, which is then developed into a subgrid model. One of the most widely adopted and celebrated models is the empirical Smagorinsky model,[1] inwhichthesubgridstress tensor is related to the local strain rate via a single specified parameter. A stochastic version of this deterministic parameterisation was first proposed by Leith [2]. The dynamic Smagorinsky model [3,4]was thenext major development inthisarea,inwhich thesubgrid stress tensor is again assumed to be proportional to the resolved strain rate, but with the model coefficients calculated from the scales resolved in a test filter at each time step. Corresponding author. vassili.kitsios@monash.edu C 215 Taylor & Francis

4 73 V. Kitsios et al. Downloaded by [ ] at 15:2 1 April 215 In the present study, we take a somewhat different approach, in which no initial physical hypotheses are made, but instead, the subgrid coefficients are determined from the statistics of direct numerical simulations (DNSs). The flow configuration of interest in the present study is turbulent channel flow. In general, it is only possible to represent the statistical effects of the non-linear subgrid interactions[5]; therefore, statistical dynamical closure theory is the natural formulation for developing parameterisations that properly represent the subgrid interactions. The pioneering study in this area was the parameter-free Eulerian direct interaction approximation (DIA) of Kraichnan [6] developed for isotropic homogeneous turbulence, followed by the self-consistent field theory [7]andthelocal energy transfertheory.[8,9]subsequent statistical dynamical subgrid closure models pertaining mainly to three-dimensional homogeneous turbulence include [2,5,1 14]. For turbulent channel flow specifically, the subgrid interactions have been quantified by the energy transfers between the resolved and subgrid scales in [15 19]. In [18], the eddy viscosities linking the energy transfers to the resolved field were also calculated; however, no LES was performed to verify that the eddy viscosities were representative of the interactions. The energy transfers associated with the forward and inverse (or backscatter) energy cascades were represented using the second-order structure function balance equation in [2,21]. These findings lead to the development of a subgrid model consisting of a linear Smagorinsky-type term plus a non-linear tensor to incorporate the backscatter of energy from small to large scales in [22]. Other than the DIA of Kraichnan [6], each of the aforementioned studies has been developed on the basis of an initial physical hypothesis regarding the relationship between the resolved and subgrid scales. In contrast, for homogeneous barotropic (twodimensional) turbulence on a sphere, a self-consistent stochastic subgrid parameterisation based on the DIA was developed, in which no physical assumptions are made.[23] The subgrid coefficients were determined from higher resolution closure calculations, and produced LES that replicated the kiic energy spectra of the benchmark simulation. The stochastic component of the parameterisation represents the chaotic nature of the subgrid interactions.[24,25] Forinhomogeneousturbulentflows,generalexpressionswerederived for the eddy viscosities, stochastic backscatter, mean-field shift, and eddy-topographic force or roughness force based on a quasi-diagonal DIA (QDIA) closure in [26,27]. All of these terms were evaluated for typical barotropic atmospheric flows using the QDIA closure in [28]. The general QDIA closure theory accounts for cross correlations between field variables (e.g., velocity components) and between physical space fields, but has the remarkable property that the eddy viscosity and stochastic backscatter terms are diagonal in spectral space. In other words, the subgrid tendency for a particular wave number can be renormalised to be a function of the resolved fields at only that same wave number. This is not true, however, in physical space. That is, the subgrid tendency at a particular point in space is not dependent upon the resolved fields at only that point in space, but is also dependent upon the surrounding fields. For this reason, the subgrid models in this paper are developed and applied in scale space. Whilst the QDIA closures produce resolution-independent results, they are difficult to implement into a given code as the discretised differential equations need to be reformulated into integro-differential equations involving time lag flow fields. To broaden the applicability, a direct stochastic modelling approach was developed in [29], where the subgrid coefficients are determined from the statistics of a higher resolution reference simulation. The reference simulation is truncated back to a coarser grid and the eddy viscosities are then determined from the subgrid statistics, akin to the approach taken in the self-consistent closure calculations discussed earlier. This approach has been successfully

5 Journal of Turbulence 731 Downloaded by [ ] at 15:2 1 April 215 applied to quasi-geostrophic atmospheric and oceanic simulations with horizontal and vertical shears.[3,31] Inaddition,ifthetruncationsaremadesuchthatthesmallestresolved scales lie within a self-similar inertial range, then the subgrid coefficients are also selfsimilar and may be governed by simple resolution-dependent scaling laws.[32 34] These scaling laws enable the subgrid models to be utilised more widely as they remove the need to generate the subgrid coefficients from a reference simulation for these geophysical flows. In the current study, we develop subgrid models for the LES of smooth-wall turbulent channel flow, with the model coefficients calculated from the statistics of a reference DNS using the approach of Frederiksen and Kepert [29]. In [35], this approach was used to successfully undertake LES for transitionally turbulent channel flow with a frictionvelocity-based Reynolds number of Re τ = 18. Here we determine if the approach is also applicable in the fully turbulent regime, by developing subgrid models and undertaking the LES for a channel flow of Re τ = 95. In Section 2, thedetailsofthednsnavier Stokes solver used for the turbulent channel flow simulations are presented. The approach for determining the subgrid coefficients from the DNS statistics is discussed in Section 3,with the subgrid coefficients themselves presented in Section 4. LESs adopting these subgrid coefficients are compared to the DNS in Section 5. The concluding remarks are made in Section Direct numerical simulation The statistics required to build the forthcoming subgrid models are generated from the DNS of the incompressible isothermal Navier Stokes equations. In physical space, the momentum and continuity equations are given by u a t + u b ua x + 1 p b ρ x ν 2 u a a x b x = f a b,and (1) u a =, (2) xa respectively, with the indices a = 1, 2, 3 and b = 1, 2, 3. The coordinates (x 1, x 2, x 3 ) (x, y, z), where x is the streamwise direction, y the wall-normal direction, and z the spanwise direction, with an associated velocity vector (u 1, u 2, u 3 ) (u, v, w). The force f a represents the boundary conditions, ν is the kinematic viscosity, and ρ the fluid density. The system is non-dimensionalised by the centreline velocity (u ) and channel half-height (h), unless otherwise specified. The general spectral form of the Navier Stokes equations is given by t ua k (t) + Dab (k)ub k (t) = K abc (k, k, k )u b k (t)uc k (t) + f a (k,t), (3) k k where u a k is the spectral component of wave number vector k for the velocity component of index a. Thetermf a(k,t) is the spectral coefficient of f a,thenavier Stokeslinear operator is given by D ab(k), and Kabc (k, k, k )istheinteractioncoefficient.thestructure of Equation (3) holds for any form of scale-space decomposition, with only the specific form of the coefficients differing. The interaction coefficients for three-dimensional Fourier discretisation are detailed in [36]. In the present study, the flow variables are discretised using acollocatedchebyshevdiscretisationiny of polynomial index j, and Fourier discretisation

6 732 V. Kitsios et al. in x and z of respective wave number indices and dealiased using the two-thirds rule. The symmetry properties are such that u a k (t) = ua k (t), where is the complex conjugate operation k (,j,) and k (,j, ). The specific form of D ab(k), K abc (k, k, k ) and f a (k,t)canbededucedfrom[37]. Note, however, for algorithmic efficiency, the code evaluates the non-linear term via grid to spectral transforms as opposed to evaluating the interaction coefficients explicitly. The summations immediately after the equal sign in Equation (3) are over the following set: T = [ k, k T, j N, T T T, j N, T T ], (4) Downloaded by [ ] at 15:2 1 April 215 where the Fourier modes have maximum wave number T, and Chebyshev polynomial of maximum polynomial index N. Thestreamwiseandspanwisewavelengthsareλ x = 2πL x / and λ z = 2πL z /,respectively.forcompleteness,wedefineaverticallengthscale representative of the near-wall region given by λ y = 1 cos (2π/j), which is the distance between the wall and the second turning point of the jth Chebyshev polynomial index. The specific case simulated here is a smooth-wall turbulent channel of friction-velocitybased Reynolds number Re τ u τ h/ν = 95, where the friction velocity u τ = τ wall /ρ, with τ wall the magnitude of the wall shear stress. The associated centreline Reynolds number Re = u h/ν = 2, 58, where u is the centreline velocity. The channel flow is simulated in a rectangular box with extents L x = π, L y = 2, and L z = π/2 in the streamwise, wallnormal and spanwise directions, respectively. The number of collocated grid points in the streamwise (N x )andspanwise(n z )directionsisrelatedtotheidealisedtruncationwave number T = 127 by N x = N z = 3(T + 1) = 384. The grid spacing in the streamwise ( x)and spanwise ( z) directionsisconstant.inviscousunits,thespacingsare x + = xu τ /ν = 7.8 and z + = zu τ /ν = 3.9. The number of Chebyshev collocated grid points in the wall-normal direction is N y = N 1 = 385. These grid points are most densely packed at the wall with the cell spacing ( y) growingasitapproachesthecentreline.inviscous units, the wall-normal cell spacing at the wall is y + wall = y wallu τ /ν =.3 and at the centreline is y + cl = y cl u τ /ν = 7.6. The statistics are accumulated over 39 flow-through times, where one flow-through time is defined as u /L x.themeanandroot-mean-square (rms) profiles illustrated in Figure 1(a) are found to be consistent with the previous DNS of delálamo and Jiménez [38]. An instantaneous flow field is illustrated in Figure 1(b), via iso-surfaces of the discriminant of the velocity gradient tensor in the bottom half of the channel. The iso-surfaces are coloured by streamwise vorticity, where red is positive rotation and blue is negative rotation about the streamwise axis, with the flow from left to right. The subgrid modelling parameterisations are derived from the statistics generated from this DNS. 3. Stochastic subgrid modelling approach For the LES, the wave number set is reduced from the DNS set T and defined as R = [ k, k T R, j N, T R T R T R, j N, T R T R ], (5)

7 Journal of Turbulence 733 u u + u + rms u + rms y + (a) (b) Downloaded by [ ] at 15:2 1 April 215 Figure 1. Characterisation of the bottom half of the Re τ = 95 turbulent channel flow by (a) profiles of the mean (u + )andrms(u + rms ) streamwise velocity in viscous units and (b) instantaneous isosurfaces of the discriminant of the velocity-gradient tensor coloured by streamwise vorticity (red: positive rotation, blue: negative rotation) with flow from left to right. where T R < T is the LES truncation wave number. The subgrid wave number set is then defined as S = T R. NotethatwekeepalloftheChebyshevpolynomials,maintaining all of the vertical scales, and hence consider only the effect of the removal of the horizontal scales. To facilitate the following discussion, we define the three-element column vector u(t) (u k (t),v k (t),w k (t)) T containing the spectral coefficients of each of the velocity components, where the superscript T denotes the transpose operation. The tendency (time derivative) of u is decomposed, such that u t (t) = u R t (t) + us t (t), (6) where u R t is the tendency of the resolved scales with all triadic interactions involving wave numbers within the resolved set, and u S t is the subgrid tendency with at least one wave number in the subgrid set involved in the triadic interactions. Note that as mentioned in the introduction, the subgrid tendency is equivalent to τsgs ab/ xb,wherethesubgridscale stress tensor is given by τsgs ab = ũa u b ũ a ũ b,withthetilderepresentingtheoperation filtering out the subgrid scales. In simulations of smooth-wall turbulence, there are three major classes of subgrid interactions: eddy eddy, between subgrid and resolved eddies; eddy mean field, between subgrid eddies and the resolved mean field; and mean field mean field, between the subgrid and resolved mean field.[36] Instatisticalclosuretheory,thecoefficientsrelatingeachof these classes of subgrid interactions to the associated resolved quantities (fluctuating field, mean field) are referred to as self-energies.[36,39,4] We refer to the following technique as the self-energy subgrid modelling approach, as it is essentially a numerical means to determine the self-energy closure coefficients. We distinguish between these interactions by decomposing the subgrid tendency into its time-averaged (f u S t )andfluctuating(ûs t ) components, such that u S t (t) = f + ûs t (t). (7)

8 734 V. Kitsios et al. The û S t term represents the eddy eddy interactions, and f represents the sum of the eddy mean field and mean field mean field interactions. The means of developing parameterisations for the eddy mean-field and mean-field mean-field coefficients are presented in [34]. For the LES in this particular study, f is extracted from the DNS, and û S t is modelled. The fluctuating component of the subgrid tendency, û S t,isrepresentedbythefollowing stochastic equation: û S t (t) = D d û(t) + ˆf(t), (8) Downloaded by [ ] at 15:2 1 April 215 where û is the fluctuating component of u.thisfunctionalformisconsistentwiththestructure of QDIA closure equations in [26,27], which are determined from the renormalisation of the non-linear statistical equations of motion. In the present case, the drain dissipation, D d,isatime-independent3 3 matrix explicitly capturing the interactions between each of the velocity components. The stochastic backscatter force, ˆf, isatime-dependentthreeelement column vector. The drain dissipation operator is determined by post-multiplying both sides of Equation (8) by û (t ), integrating over the turbulent decorrelation period τ, and ensemble averaging. The expression for D d is based on a generalisation of Gauss s theorem of least squares and given by t +τ t D d = û S +τ 1 t (σ )û (t )dσ û(σ )û (t )dσ, (9) t t where denotes the Hermitian conjugate for vectors and matrices. Here the angled brackets denote ensemble averaging, with each ensemble member determined by shifting the initial time t forward by one time step. For the present flow configuration, D d increases and converges with increasing τ, as previously observed for atmospheric and oceanic flows.[32,33] WepresentD d calculated using a value of τ = 15 t. Thedraindoesnot increase further for larger τ, whichmeansthisdecorrelationtimeissufficienttocapture the memory effects of the subgrid turbulence. The model for ˆf is determined by calculating the non-linear noise covariance matrix F b = F b + F b, where F b = ˆf(t) û (t). Bypost-multiplyingbothsidesofEquation(8) by û (t), and adding the conjugate transpose of Equation (8) pre-multiplied by û(t) yields the following Lyapunov (or balance) equation: ûs t (t)û (t) + û(t)û S t (t) = D d û(t)û (t) û(t)û (t) D d + F b. (1) Given that D d is known, F b can now be calculated. From Equation (1), it is clear that there is a balancing act between the deterministic (D d )andstochastic(f b )componentsof the subgrid model. As D d is dependent upon τ, it is τ that defines this balance. At this point, the formulation is general, and ˆf is coloured noise. For the implementation of the stochastic subgrid parameterisation, ˆf is represented as a white noise process, such that ˆf(t) ˆf (t ) = F b δ(t t ). An eigenvalue decomposition of F b is then used to produce a stochastic model for ˆf,asdetailedin[3]. The fundamental form of subgrid turbulence is a stochastic process, as outlined in Equation (8), comprising of the deterministic drain dissipation and a stochastic backscatter force. However, one can approximate the subgrid interactions by the solely deterministic relationship û S t (t) = D û(t), where D is the dissipation representing the effect of the drain and backscatter. The dissipation is given by

9 Journal of Turbulence 735 D = D d + D b = û S t (t)û (t) û(t)û (t) 1,where (11) D b = F b û(t) û (t) 1 (12) is the backscatter dissipation. From Equation (11), it is clear that D is independent of τ.thedrain,backscatter,anddissipationsarerelatedtotheirrespectiveeddyviscosity forms via D d ν d 2, D b ν b 2, and D ν 2, where 2 is the Laplacian operator. Downloaded by [ ] at 15:2 1 April Subgrid model coefficients We now present the subgrid model coefficients for the turbulent channel flow simulation truncated back to the resolved wave number set R as detailed in Equation (5). The system is truncated back from a maximum DNS wave number of T = 127 to a maximum LES wave number of T R = 63. The resulting subgrid dissipation matrix elements are illustrated in Figure 2. Theyareplottedinthehorizontalwavenumberplane(, ) for the λ + y λ yu τ /ν = 3 vertical scales. Whilst there is some coupling present between the u and w velocity components evident in the matrix elements D uw and D wu,thedomi- nant contribution is from the diagonals (D uu,dvv,dww ). The subgrid tendency for a given velocity component is, therefore, most strongly related to the resolved field of the same velocity component. The diagonal elements are positive throughout the wave number plane, which means energy is being drained out of the resolved scales and sent to the subgrid. The dissipations are also scale selective with the magnitude of the coefficients increasing as they approach the truncation boundaries. This means that the small resolved vortices near truncation have more significant interactions with the subgrid than the larger resolved structures. If the cascade were strictly local in wave number space, as proposed in [41] for isotropic homogeneous turbulence, then one would expect the dissipation to be zero for all wave number pairs except those along the truncation boundary. These results are evidence that the cascade is spectrally non-local, consistent with the findings of [42] forfreeshear flows and [17,18] forwall-boundedflows. The picture becomes somewhat more complicated when inspecting the dissipation of the other vertical scales. For the moment, we concentrate on the D uu term. This element is symmetric about the line = ; therefore, we only present these coefficients in the wave number half-plane (, ). The D uu coefficients are illustrated for λ+ y = 1, λ+ y = 3, λ + y = 2, and λ+ y =.4 infigure 3(i) (l), respectively. The Duu coefficients for λ+ y = 1 and λ + y = 2 have qualitatively the same properties as the λ+ y = 3 coefficients discussed earlier. For λ + y =.4, however, the smallest resolved scales approaching the line = T R = 63 in Figure 3(l) have negative dissipations. This indicates that the subgrid interactions are conspiring to send energy from the subgrid to these wave number pairs, representative of a backscatter of energy. One can determine the stochasticity of the backscatter by decomposing the dissipation into its deterministic and stochastic components. The deterministic drain dissipation Dd uu is illustrated for the same vertical scales in Figure 3(a) (d), with the associated stochastic backscatter dissipation Db uu illustratedinfigure 3(e) (h). Recall by definition, D = D d + D b. These drain dissipations are completely positive and the stochastic backscatter completely negative. This means that the subgrid model represents energy as being sent from the resolved to the subgrid scales in a deterministic manner, and sent from the subgrid to the resolved in a stochastic manner. It is only in Figure 3(l) that

10 736 V. Kitsios et al. Downloaded by [ ] at 15:2 1 April (a) D uu (d) D vu (g) D wu (b) D uv (e) D vv (h) D wv (c) D uw (f) D vw (i) D ww Figure 2. Subgrid dissipation matrix elements illustrated in the horizontal wave number plane (, ) atλ + y = 3, using rectangular truncation with T R = 63: (a) D uu;(b)duv ;(c)duw ;(d)dvu ; (e) D vv ; (f) Dvw ; (g) Dwu ; (h) Dwv ; and (i) Dww.Thecolourbarsinthefinalcolumnareapplicable to all figures the stochastic backscatter overwhelms the drain dissipation, resulting in a negative dissipation for certain wave number pairs. 5. Large eddy simulation We now assess the agreement between the reference DNS and LES using both stochastic and deterministic subgrid models. To serve as a baseline comparison, we also run DNS truncated to the LES resolution with no subgrid model applied, to identify the impact the removed subgrid scales have on the resolved scales. The equations governing the LES are the same as for the DNS, but with u S t (k)addedto the right-hand side of Equation (3), and solved over the wave number set R instead of T. Recall that k (,j,). In the fundamental stochastic representation, u S t (k,t)isgivenby

11 Journal of Turbulence 737 Downloaded by [ ] at 15:2 1 April 215 Reτ = 95 Reτ = 95 Reτ = 95 D uu d D uu b D uu λ + y = 1 λ+ y = 3 λ+ y =2 λ+ y =.4 (a) (b) (c) (d) (e) (i) (f) (j) (g) (k) (h) (l) Figure 3. The upper diagonal component of the subgrid dissipation matrices illustrated in the horizontal wave number half-plane (, ) using rectangular truncation for various wall-normal scales (λ + y ), for an Re τ = 95 truncated at T R = 63: the first row (a) (d) contains the drain coefficients Dd uu ; the second row (e) (h) contains the backscatter coefficients Duu b ; and the third row (i) (l) contains the coefficients D uu. In the first column (a),(e),(i). λ+ y = 1. In the second column (b),(f),(j). λ + y = 3. In the third column (c),(g),(k), λ+ y = 2. In the fourth column (d),(h),(l), λ+ y =.4. All coefficients are symmetric about the line =. The colour bars in the final column are applicable to all figures in that row. the following matrix equation: u S t (k,t) = D d(k) û(k,t) + ˆf(k,t) + f(k), (13) where D d is the drain dissipation acting upon the resolved field, ˆf is the stochastic backscatter force injecting energy into the system, and f is the eddy mean-field force. The dependence upon k has been explicitly included in all of the terms. In the deterministic LES, the drain dissipation and stochastic backscatter are replaced with the dissipation (D ), such that u S t (k,t) = D (k) û(k,t) + f(k), (14) where D is relied upon to both remove and inject energy into the system. In the present simulations, the sum of the negative dissipation elements and the molecular dissipation is in fact positive and effectively increases the Reynolds number at these wave numbers. For situations in which the sum of the subgrid and molecular dissipation is negative, the deterministic subgrid model is not guaranteed to be numerically stable. In this case, it is

12 738 V. Kitsios et al. No Subgrid Model Deterministic Stochastic Downloaded by [ ] at 15:2 1 April 215 E (a) (d) y (b) (e) y (c) (f) y Figure 4. Comparison of the DNS (dotted line) and LES (solid line) on the basis of time-averaged kiic energy spectra (E). The two-dimensional spectrum is compared at y + = 133 for an LES with (a) no subgrid model; (b) the deterministic subgrid model; and (c) the stochastic subgrid model. Contour levels radiating out from the origin are E (1 2.5,1 3,1 3.5,1 4,1 4.5,1 5,1 5.5 ), and the vertical axis in (a) is applicable to (b) and (c). The streamwise wave number summed spectrum in viscous units (E + ) compared at various labelled y + positions to an LES with (d) no subgrid model; (e) the deterministic subgrid model; and (f) the stochastic subgrid model, where the top pair of spectra in each figure has the correct kiic energy, with the remaining spectra shifted down multiples of one decade for clarity. The vertical axis in (d) is applicable to (e) and (f). still possible to produce numerically stable LES if the energy injected at wave numbers of negative dissipation is transferred via the resolved nonlinear interactions to wave numbers of sufficiently positive dissipation, as achieved in previous geophysical simulations.[33,34] Comparisons are made between the DNS and each of the LES variants on the basis of the time-averaged kiic energy spectra (E). In all of the following figures, the DNS is represented by the dotted lines and the LES variants by the solid lines. The spectrum at y + yu τ /ν = 133 is plotted against the streamwise () andspanwise() wavenumbersforthe LES with no subgrid model in Figure 4(a). This figure illustrates a clear accumulation of energy at the smallest resolved scales of motion, distorting the entire spectrum. The now unresolved subgrid interactions would have drained this excess energy out of the resolved scales and into the subgrid, to be eventually damped by viscous dissipation. When adopting either of the subgrid model variants, however, the spectrum is corrected throughout the wave number plane as illustrated for the deterministic model in Figure 4(b) and the stochastic model in Figure 4(c). Comparisons are also made at alternate wall-normal positions on the basis of the time-averaged streamwise wave number summed spectra, in Figure 4(d) (f). Here the kiic energy is presented in viscous units (E + = E/u 2 τ and + = h/l z (ν/u τ )), with

13 Journal of Turbulence 739 the associated y + positions labelled. In each of these figures, the top pair of spectra has the correct kiic energy, with the remaining spectra shifted down multiples of one decade for clarity. Figure 4(d) compares the DNS to an LES with no subgrid model, which again clearly illustrates an accumulation of energy at the tails. As previously observed for the two-dimensional spectra, these one-dimensional spectra are corrected at all wallnormal positions when either the deterministic or stochastic subgrid models are adopted, as illustrated in Figure 4(e) and (f), respectively. The overall agreement between the DNS and LES presented above indicates that the subgrid models capture the appropriate drain and stochastic backscatter injection associated with the subgrid interactions. We also find that the eddy mean-field term (f) is negligible, as there is minimal change to the LES kiic energy spectra when run with f = instead. Downloaded by [ ] at 15:2 1 April Concluding remarks Subgrid parameterisations of turbulent channel flow have been developed by calculating the subgrid coefficients from the statistics of DNS. The stochastic representation of the subgrid interactions consists of a mean-field shift, drain dissipation acting on the resolved field, and a stochastic backscatter noise term of variance proportional to the product of the backscatter dissipation and kiic energy spectrum. The dissipation represents the effect of the two processes, given by the sum of the drain and backscatter. Deterministic and stochastic LESs have been shown to successfully replicate the large-scale DNS statistics. In doing so, we have determined the qualitative properties that more general purpose subgrid models should ideally have. The potential existence of scaling laws identifying how the subgrid coefficients change with Reynolds number and resolution, is as yet undetermined and the subject of the current research. However, in applications to geophysical flows, the subgrid dissipation magnitude increases linearly with resolution.[32,33] Also,the functional form of the dissipation for these flows is such that for a truncation wave number, T R,thedissipationhasasignificant magnitude for scales of wave number greater than T R k E,wherek E is fixed for a given background state and independent of resolution.[32,33]initial results indicate that channel flows have similar properties with regards to both the magnitude and form of the subgrid coefficients. Additionally, for physics-based studies, one can inspect the structure of the subgrid models (validated by LES) to obtain further details on the direction, magnitude, and stochasticity of the energy transfers associated with eddies discriminated on the basis of their scale. Acknowledgements We acknowledge the computational resources provided by the National Computational Infrastructure (NCI) of Australia. Julio Soria gratefully acknowledges the support of an Australian Research Council Discovery Outstanding Researcher Award fellowship. Disclosure statement No potential conflict of interest was reported by the authors.

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