An alternative derivation of the Germano identity as the residual of the LES equation
aa r X i v : . [ phy s i c s . f l u - dyn ] S e p An alternative derivation of the Germano identity as the residual of the LESequation
Siavash Toosi , Johan Larsson Department of Mechanical Engineering, University of Maryland, College Park, MD 20742, USA
1. Introduction
The Germano identity [1] and the resulting dynamic procedure [2, 3] to compute subgrid modelcoefficients have been among the most successful and popular developments in large eddy simulation(LES). The original rationale for the dynamic procedure was that the same subgrid model should beapplicable with the same model coefficient at two different coarse-graining levels (or filter levels), whichwas later interpreted as an argument based on scale-invariance (cf. [4]), a property that is expectedof turbulence in the inertial subrange. The rationale based on scale-invariance was first questionedby Jimenez and Moser [5] and later by Pope [6], partly based on the fact that the dynamic procedureworks well at low Reynolds numbers (transitional flow, near-wall behavior, etc.) where scale-invariancedoes not hold. Jimenez and Moser [5] argued that the success of the dynamic procedure is probablydue to the balance between the production of Leonard stresses and the dissipation rate resulting fromthe application of the dynamic procedure. Pope [6], on the other hand, argued that the reason forsuccess is that the dynamic procedure minimizes the sensitivity of the total (resolved plus modeled)Reynolds stresses to the coarse-graining level. This general argument was later used by Meneveau [7]as well. The current consensus understanding of why the dynamic procedure works seems to be acombination of these arguments, the strength of which is that they require no specific assumptionsabout the characteristics of the flow (e.g., whether it satisfies the scale-similarity hypothesis) and tosome degree about its nature (e.g., whether it is turbulent or not).The objective of this Note is to present an alternative derivation of the Germano identity andits error which provides a subtly different argument for why the dynamic procedure works. Whilethe previous arguments rest on recognizing the importance of the Reynolds stress or the dissipationrate, the present derivation instead follows the path of deriving the residual (in the sense of numericalanalysis) of the LES equation. The residual is of central importance in the field of numerical analysissince it is the source of errors; therefore, the present derivation shows the connection between theerror in the Germano identity and the source of error in LES based on the governing equation alone,with no physical insight required. The present derivation does not contradict the prior arguments byJimenez and Moser [5] or by Pope [6] and Meneveau [7] in any way; rather, it is offered here as acomplement to the prior explanations.
Email addresses: [email protected] (Siavash Toosi), [email protected] (Johan Larsson) Current address: Linn´e FLOW Centre, KTH Mechanics, SE-10044 Stockholm, Sweden
Preprint submitted to Elsevier September 11, 2020 . LES equations
The Navier-Stokes equation for an incompressible and constant viscosity fluid is ∂ U i ∂t + ∂ U i U j ∂x j + 1 ρ ∂ P ∂x i − ν ∂ U i ∂x j ∂x j = 0 , or in short notation N ( U i ) = 0, where ρ and ν are the density and viscosity (both assumed constanthere) and U i and P are the exact velocity and pressure fields (corresponding to a perfect directnumerical simulation, DNS). When coarse-grained or implicitly filtered to a resolution length scale (orfilter width) ∆ the equation becomes (assuming that filtering and differentiation commute) ∂ U i ∂t + ∂ U i U j ∂x j + 1 ρ ∂ P ∂x i − ν ∂ U i ∂x j ∂x j + ∂τ exact ij, ∆ ∂x j = 0 , (1)or N exact∆ ( U i ) = 0 where U i and P are coarse-grained representations of the exact fields and τ exact ij, ∆ = U i U j − U i U j is the exact subgrid scale (SGS) stress tensor. An interesting property of τ exact ij is that itsatisfies the Germano identity [1] \ τ exact ij, ∆ − τ exact ij, b ∆ = [ U i U j − b U i b U j , (2)where b · is a test filtering operation of width b ∆ > ∆, b · is the result of consecutive application of filters∆ and b ∆, and τ exact ij, b ∆ = d U i U j − b U i b U j . This identity provides a “self-consistency condition” [7] that alsoapplies to Eqn. (1) at filter levels ∆ and b ∆.Approximating the exact SGS stress tensor using a model leads to the LES equation in differentialform (i.e., without numerical errors), where we intensionally exclude the numerical errors in order toisolate the effect of modeling errors in the equation, and to be faithful to many of the developmentsin the LES literature. The LES equations at two different filter levels ∆ (say, original) and b ∆ (testfiltered) are ∂u i ∂t + ∂u i u j ∂x j + 1 ρ ∂p∂x i − ν ∂ u i ∂x j ∂x j + ∂τ model ij, ∆ ( u k ) ∂x j = 0 , (3) ∂ b v i ∂t + ∂ b v i b v j ∂x j + 1 ρ ∂ b q∂x i − ν ∂ b v i ∂x j ∂x j + ∂τ model ij, b ∆ ( b v k ) ∂x j = 0 , (4)where ( u i , p ) and ( b v i , b q ) are the solutions at the respective filter levels. These equations are referredto as N model∆ ( u i ) = 0 and N model b ∆ ( b v i ) = 0, respectively.The principle of the dynamic procedure [2] is that any approximate model should satisfy, as wellas possible, the Germano identity. It therefore aims to minimize the error G ij = d u i u j − b u i b u j | {z } L ij − (cid:20) \ τ model ij, ∆ ( u k ) − τ model ij, b ∆ ( b u k ) (cid:21)| {z } M ij , (5)in a least squares sense [3]. Here, G ij is the Germano identity error (GIE), L ij is the Leonard orresolved stress, and M ij is the modeled stress [cf. 8].2 . Residual due to modeling and its connection to the modeling error The residual of an inexact equation N approx ( u approx ) = 0 is the misfit when evaluating the inexactequation for the exact solution, i.e., N approx ( u exact ) in this example. The importance of the residualis made clear by the Taylor expansion N approx ( u exact ) ≈ N approx ( u approx ) | {z } =0 + ∂ N approx ∂u ( u approx − u exact ) | {z } error , which shows how the residual is the source of error in the solution for linearized dynamics. Wetherefore want to find the residual of the LES equation (3). This residual is N model∆ ( U i ) for filter level∆ and N model b ∆ ( b U i ) for filter level b ∆, where we must use the coarse-grained representations of the exactfields (clearly the full field U i is consistent only with the DNS equation, not the coarse-grained onescontaining τ ij ). Therefore, we can write R ∆ ≡ N model∆ ( U i ) = ∂ U i ∂t + ∂ U i U j ∂x j + 1 ρ ∂ P ∂x i − ν ∂ U i ∂x j ∂x j | {z } = − ∂τ exact ij, ∆ /∂x j + ∂τ model ij, ∆ ( U k ) ∂x j = ∂∂x j h τ model ij, ∆ ( U k ) − τ exact ij, ∆ i , where Eqn. (1) is used to replace the terms by the divergence of τ exact ij, ∆ . Similarly, we have R b ∆ ≡ N model b ∆ ( b U i ) = ∂∂x j (cid:20) τ model ij, b ∆ ( b U k ) − τ exact ij, b ∆ (cid:21) . (6)The exact solution is unknown and must therefore be approximated. In the area of numerical anal-ysis, the residual is often approximated by evaluating the numerical operators on a finer representationof the solution [cf. 9]. This approach, however, does not work for LES equations with modeling of thediscarded scales, because obtaining an exact (or sufficiently more accurate) LES equation requires amore accurate τ ij , which in turn requires the estimation of u i u j from only the LES solution u i , which isimpossible due to the limited spectral content of the filtered solution and the projection errors [cf. 10].The solution is to use the N model operator (i.e., the direct approach, as in Eqn. 6, to avoid estimationof τ exact ij ) and to compute the residual at a coarser filter level, specifically the test filter level b ∆, suchthat the test filtered solution b u i can be used in place of b U i to compute the approximate residual. Weshould note that approximating b U i by b u i is only done for the purpose of estimating the residual (verysimilar to the use of the numerical solution for estimating the truncation errors in numerical analysis),and is assumed to be a much weaker approximation than saying that b u i is an accurate representationof b U i in general.With this approximation, the residual at the test-filter level b ∆ is R b ∆ ≈ ∂ b u i ∂t + ∂ b u i b u j ∂x j + 1 ρ ∂ b p∂x i − ν ∂ b u i ∂x j ∂x j + ∂τ model ij, b ∆ ( b u k ) ∂x j . (7)3his equation can be directly computed to estimate R b ∆ ; however, quite interestingly, it can be sim-plified by test-filtering the LES equation (3) and subtracting it from Eqn. (7), which yields (assumingthat filtering and differentiation commute) R b ∆ ≈ ∂∂x j (cid:20)b u i b u j + τ model ij, b ∆ ( b u k ) − d u i u j − \ τ model ij, ∆ ( u k ) (cid:21) = ∂∂x j (cid:20)b u i b u j − d u i u j + τ model ij, b ∆ ( b u k ) − \ τ model ij, ∆ ( u k ) (cid:21) = − ∂∂x j [ L ij − M ij ] , (8)where L ij and M ij are the familiar Leonard (resolved) and modeled stress terms from Eqn. (5). Inother words, we have − ∂∂x j [ L ij − M ij ] ≈ R b ∆ = ∂∂x j (cid:20) τ model ij, b ∆ ( b U k ) − τ exact ij, b ∆ (cid:21) . (9)Therefore, the residual R b ∆ of the LES equation at the test-filter level is approximately equal tothe divergence of the error in the Germano identity L ij − M ij , and the tensor L ij − M ij directlyestimates the modeling error τ model ij − τ exact ij . Minimizing this Germano identity error (GIE) thusdirectly minimizes the modeling errors and the residual that is the source of errors in the (test-filtered)LES equation.
4. Concluding remarks
This Note illustrates the close connection between the residual of the test-filtered LES evolutionequation and the error in the Germano identity (the GIE, generally written as L ij − M ij in mosttexts, [cf. 8]). Equation (9) also shows that the GIE approximates the difference between the modeledSGS stress tensor and the exact one given the exact solution. This can explain why the dynamicprocedure is successful at distinguishing between laminar, transitional, and turbulent flows, and whyit is capable of recovering the correct near-wall behavior of the eddy viscosity at the vicinity of solidwalls: the exact SGS stress tensor τ exact ij has all these properties built in [cf. 11, 12], and by minimizingthe difference between τ exact ij and τ model ij , the SGS model should inherit (to the largest degree possiblegiven the chosen model form) those characteristics.The main purpose of this Note is to complement prior interpretations of why the dynamic procedureworks, and to serve as a connection between the fields of LES and numerical analysis. There is a greatbody of work in the numerical analysis literature that utilizes the residual to, for example, produceerror estimates and to optimally adapt the computational grid. The connection between the GIE andthe residual suggests that the dynamic procedure in a sense uses the same residual to improve thesolution by optimally choosing the model parameter(s). The implication of such a connection is thatmany of the more advanced techniques that are currently used in residual minimization (weighting theresidual by the adjoints, for instance) could (and maybe should?) be used in the dynamic procedure aswell. Furthermore, the present derivation implies that one should be minimizing the volume integralof the residual (i.e., the GIE) as the more meaningful and more optimal approach of reducing theerrors (optimally the GIE should be weighted by some adjoint field), as done by Ghosal et al. [13],4nd shows clearly that we should indeed be minimizing the divergence of the GIE rather than erroritself, as done by Morinishi and Vasilyev [14].Finally, the implications of the findings of this Note extend to uncertainty quantification (UQ) andoutput-based grid/filter adaptation in LES, both of which require an estimate of the residual in theequation. In that sense, this Note also complements our prior work [15] on grid adaptation for LES,that used the same quantity (the divergence of the GIE) as its error indicator, but motivated its usefrom a different point-of-view of solution sensitivity. Acknowledgements
This work has been supported by NASA grant 80NSSC18M0148.