The Large Eddy Simulation Capability of Reynolds-Averaged Navier-Stokes Equations: Analytical Results

Physics of Fluids LETTER scitation.org/journal/phf

The large eddy simulation capability of Reynolds-averaged Navier-Stokes equations: Analytical results

Cite as: Phys. Fluids 31, 021702 (2019); doi: 10.1063/1.5085435 Submitted: 12 December 2018 • Accepted: 17 January 2019 • Published Online: 4 February 2019

Stefan Heinza)

AFFILIATIONS Department of Mathematics, University of Wyoming, Laramie, Wyoming 82071-0333, USA

a)Author to whom correspondence should be addressed: [email protected]

ABSTRACT A highly attractive idea to overcome shortcomings of both Reynolds-averaged Navier-Stokes (RANS) and large eddy simulation (LES) equations is the implementation of LES capability in RANS models. However, this approach faces questions regarding (i) the measurement of resolved motion and equivalence of various equations having the same resolution, (ii) the continuous variation of resolved and modeled motion under grid variations, and (iii) the explanation of how resolved motion and scaling variables in LES depend on the grid. Corresponding analytical results (addressing the resolution measurement, equivalence of equations, and resolution control) are reported, and grid effects (including the scaling of computational cost) are discussed. Published under license by AIP Publishing. https://doi.org/10.1063/1.5085435

The need for a significant cost reduction of large eddy Q1. Is there a mathematical explanation of (i) how the simulation (LES), in particular for wall-bounded turbulent amount of resolved and modeled motion has to be flows, initiated a relevant research topic that is under inves- measured and (ii) how various turbulence models can tigation over decades:1–12 the hybridization of Reynolds- produce the same resolution? averaged Navier-Stokes (RANS) and LES. An ideal expectation Q2. Is there mathematical proof that many RANS equations for the performance of hybrid RANS-LES methods is their (i) can produce resolved motions so that (ii) modeled optimal performance on each affordable grid up to high motions are continuously in balance with the actual Reynolds numbers: depending on the resolution supported resolved motion? by the grid, the amount of modeled motion smoothly Q3. Which ideas of how the grid (the filter width ∆) affects increases if the grid becomes coarser. In this regard, a (i) the resolution and (ii) LES scale variables (the LES major issue of popular hybrid RANS-LES methods is the bal- length scale L) are supported by mathematical-physical ance of modeled and resolved turbulent motions: modeled principles? and resolved motions cannot be considered to be independent; the modeled motion has to respond to the amount Regarding Q1, a very important and non-trivial ques- of resolved motion to ensure an appropriate total motion tion is how the resolution of LES can be evaluated,14 and (given by the sum of resolved and modeled motions). The hybrid RANS-LES models need to get information about the latter issue is not just an academic problem. For example, actual resolution to enable an appropriate generation of it is known to be the reason for an incorrect reflection modeled motion. A closely related question is how different of wall physics (the logarithmic law of the wall cannot be turbulence models can be designed that have the same res- obtained13). olution. Regarding Q2, the view that RANS equations can From a more general viewpoint, there are several ques- produce resolved motions under certain conditions is sup- tions with respect to the mathematical foundations of hybrid ported by both applications and theoretical arguments.15 RANS-LES and LES methods: By focusing on criteria for the equivalence of various hybrid

Phys. Fluids 31, 021702 (2019); doi: 10.1063/1.5085435 31, 021702-1 Published under license by AIP Publishing Physics of Fluids LETTER scitation.org/journal/phf

methods, Friess et al.10 have recently presented a technique Mathematically, the appropriate way to address the cal- to analyze relations between model coefficients of RANS culation of α∗ is to consider variations of model parame- equations and variables that reflect the amount of modeled ters and implied variations of model variables: the question motion [like the modeled-to-total turbulent kinetic energy is which model parameters satisfy variation equations implied ratio k+ = k/ktot, where ktot = k + kres is the sum of modeled by turbulence models. Technical material in this regard is pre- (k) and resolved (kres) energies]. However, these relations were sented in the first four paragraphs of the last six paragraphs presented under rather restrictive conditions (see below), and of this letter (beginning with Sections IA-ID, respectively). The they change depending on the assumptions made. So there is analysis shown in Section IA leads to the conclusions that 2 ∗ certainly the question of what proves, in general, the ability R = L+ in α = 1 + R(α − 1), which is clearly different from 4–6,12 of RANS equations to produce resolved motions. More specif- R = k+ applied in PANS-PITM approaches (see the corre- ically, a hybrid model has to respond to the actual resolution sponding discussion below). The analysis in Section IB shows with the proper amount of modeled motion to have a balanced that there exists a corresponding detached eddy simulation total motion—the question is how this can be accomplished. (DES) k– model, which is (resolution-wise) equivalent to the Regarding Q3, a valid question is about how the distribution PANS-PITM-type k– model equations (1). The analysis shown of LES and RANS modes changes with the grid. One version in Section IC leads to the conclusion that there exists a PANS- of asking this is to consider how k+ depends on ∆+ = ∆/Ltot. PITM-type k–ω model, which is (resolution-wise) equivalent Here, Ltot = L + Lres is the total (RANS) length scale consisting to the PANS-PITM-type k– model (1). The analysis in Section of modeled and resolved contributions, L and Lres, respectively ID shows that there exists a corresponding DES k–ω model, (similar to ktot = k + kres). A different version of this question is which is (resolution-wise) equivalent to Eq. (1). Within the to ask how the corresponding length scale ratio L+ = L/Ltot framework considered [the equations considered and condi- 10 depends on ∆+. This question is equivalent to the question of tion of Friess et al. that the energy partition variation (δk/k, how the grid affects variables in LES that provide scale infor- δ/) over the domain is uniform16], the results reported in mation, like L. In particular, there is the question of how L Sections IA-ID are exact analytical implications. It is worth scales with ∆ if ∆ is not small. noting that the analysis of Friess et al.10 provides the tech- Let us illustrate the specific problem considered for nical framework for obtaining the results shown in Sections incompressible flow in terms of the popular k– model for IA-ID, but the latter results were not reported so far (Friess the turbulent kinetic energy k and its dissipation rate (other et al.10 focused on another question, the equivalence of hybrid turbulence models will be considered below), methods: see the discussion of Q2 above). Let us use the Sections IA-ID results to present observa- 2 Dk D P ∗ tions regarding Q1 considered above. The characterization of = P − + Dk, = C − α + D. (1) Dt Dt 1 k the degree of flow resolution is very relevant and rather dif- ficult. Current concepts for the evaluation of the resolution Here, D/Dt = ∂/∂t + Uk∂/∂xk refers to the Lagrangian time derivative, which involves the component U of the mean of simulations are based on comparisons of the suitability of k 14 velocity. The sum convention is applied throughout this paper, parameters that appear to be appropriate. No theory was presented so far providing guidance regarding an optimal res- t is time, and xk is the kth space coordinate. No explicit expres- olution measure. A relevant result of the analysis presented sion will be used below for the production of turbulent kinetic 2 2 here is the mathematical identification of L as a resolution energy P. A usual parametrization is given by P = νtS . Here, + = 1/2 measure regarding the models considered. Without involving S (2SikSki) is the characteristic shear rate which involves 2 Eq. (16), L+ was shown to provide information about the grid- the rate of strain Sik = (∂Ui/∂xk + ∂Uk/∂xi)/2. The turbulent 2 induced resolution to the model equations and to control the viscosity is given by νt = Cµ k /, where Cµ is a constant. In ∗ model’s response to the amount of actual resolution. It is of addition, we have α = α with α = C2 /C1 . C1 and C2 are constants having standard values C = 1.44 and C = 1.92. The interest that the use of L+ as a resolution measure agrees with 1 2 14 turbulent transport terms are given by Davidson’s analysis of resolution measures, showing that the consideration of velocity correlation functions (the resolu- ∂ νt ∂k ∂ νt ∂ tion of integral length scales) represents the most promising Dk = ν + , D = ν + . (2) ∂xj " σk ∂xj # ∂xj " σ ∂xj # approach. L+ (the length scale ratio of modeled LES and RANS motions) may be considered as a Knudsen number: it sepa- Here, ν refers to the constant molecular viscosity, and σk and rates LES and RANS regimes similar to the Knudsen number σ are constants. used in fluid dynamics to separate statistical mechanics and The inclusion of resolving motions in the RANS equa- continuum mechanics regimes. With respect to the hybridiza- tions (1) is possible by considering a variable α∗ instead of tion of RANS and LES, it needs a reduction of scale variables the constant α. The problem is to determine α∗ such that a (length or time scales) to enable resolving simulations. This desired resolution can be realized. The particular challenge is can be performed in a variety of ways, e.g., by replacing the 15 to ensure the functioning of this mechanism for varying res- dissipation time τ = k/ in the turbulent viscosity by τk+. 2 olution levels. Popular solutions to this problem are offered In this regard, the advantage of identifying L+ as the con- by partially averaged Navier-Stokes (PANS)6 and partially trol parameter is the exclusion of empirical solutions to this 4,5,12 2 integrated transport modeling (PITM): both approaches problem. The use of L+ includes + = / tot variations: there ∗ assume α = 1 + R(α − 1), where R = k+. is no need to disregard them as often performed in PANS

Phys. Fluids 31, 021702 (2019); doi: 10.1063/1.5085435 31, 021702-2 Published under license by AIP Publishing Physics of Fluids LETTER scitation.org/journal/phf

10 2/3 modeling or to include such effects via approximations. A k+ = L+ if + = 1 is assumed) and the methods described L2 2 2/3 2 remarkable finding is that + was identified as a resolution here (using L+). The difference between L+ and L+ is illus- measure for several (k– and k–ω) turbulence models and 2/3 trated in Fig. 1. It may be seen that the use of L+ significantly for different (PANS-PITM-type and DES-type) hybridization overestimates the amount of modeled motion. To accomplish schemes. In addition to providing support for the suitability of the same resolution, a much finer grid has to be applied in 2 considering L+, this means that there are equivalent formula- PANS-PITM methods. This can be expressed by the equality tions of several turbulence models that can produce the same 2/3 2 condition L+ (∆C+/F) = L+(∆C+), where F is the grid refine- resolution. ment factor needed in PANS-PITM. By involving the disper- Next, let us use the Sections IA-ID results to present sion relation, the evaluation of the equality condition provides observations regarding Q2 considered above. The results pre- F = [1 + 3∆3 (1 + ∆3 )]1/3/∆2 . By using the dispersion relation, sented above prove explicitly, first, that many RANS equations C+ C+ C+ F can be also written F = κ1/3/L2, where κ = 1 + L3(1 + L3). can produce resolved motions (i.e., L ≤ 1) and, second (via + + + + Here, 1 ≤ κ1/3 ≤ 31/3 = 1.44, i.e., F scales first of all with L−2. the fact that variation equations are satisfied), that modeled + F becomes huge for small L , i.e., PANS-PITM approaches pro- and resolved motions can be kept in balance. The essential + duce in the high-cost LES regime huge additional computa- requirement for both properties is that model coefficients tional cost. Another viewpoint implies the same. A reasonable are chosen in consistency with corresponding variation equa- definition of computational cost reads n = 1 + N : N is the tions. Compared to α∗ = 1 + L2(α − 1) used in the k– model + + + + number of grid points in simulations and 1 + N normalizes n presented here, what are actually the consequences of using + + such that n = 1 for RANS. By using Eq. (16), we find n = L−3. On α∗ = 1 + k (α − 1) applied in PANS-PITM methods? The com- + + + + the other hand, PANS-PITM approaches that need a finer grid bination of α∗ = 1 + k (α − 1) with the variation equation (6) + (∆ /F instead of ∆ in n = 1+∆−3) imply n = L−9. Hence, PANS- implies + + + C+ + + PITM methods have a scaling that is six orders of magnitude δk δ higher than the scaling of methods presented here. N − γ(Nk − 1 − N ) = 2 + γ(1 − Nk ) , (3) k In summary, this paper reports analytical results regarding the measurement of resolution, equivalence of hybrid γ D α − k where = k/[( 1) + ]. For independent variations, the methods, and resolution control in hybrid methods. These only way to satisfy this equation is to neglect variations results are exact within the framework considered [the equa- N and, consequently, to set = 0. Then, the first term requires tions considered and condition of Friess et al.10 that the γ D = 0, which means that k = 0. Thus, PANS-PITM methods energy partition variation (δk/k, δ/) over the domain is are (in addition to the neglect of variations) equivalent to uniform16]. In particular, the following answers to the ques- the consideration of homogeneous turbulence. With respect tions Q1-Q3 considered above were obtained. The question of k ω to the – model, we find a corresponding result as the con- how the amount of resolved (modeled) motion has to be mea- β∗ k β − sequence of the PANS-PITM setting = 1 + +( 1). As a sured is debated over decades.14 Here, a mathematical expla- consequence of PANS-PITM concepts, hybrid models involv- nation was provided showing that the amount of modeled ing + , 1 effects and variations of Cµ in the turbulent viscosity motion is given by L+. This conclusion was obtained for several need to account for these effects approximately, and they are 10 turbulence models, meaning also that various turbulence different depending on the effects considered. On the other models can produce the same resolution. Strict explanations hand, there is no need for considering such different models by using the concepts presented here. Technical material about the grid influence on the resolution, which will be used in the following paragraph to address Q3 considered above, is presented in Section II (the paragraph before the last paragraph of this letter). The relation- = ∆ / ∆3 1/3 ∆ ship L+ C+ (1 + C+) between L+ and + considered in Section II was already presented before based on spectral arguments.12,17 However, given its relevance, the purpose of Section II is to provide independent (probabilistic) evidence for = ∆ / ∆3 1/3 the validity of L+ C+ (1+ C+) for inhomogeneous turbulent flows. Let us use now the Section II results to present observations regarding Q3 considered above. The relationship = ∆ / ∆3 1/3 L+ C+ (1 + C+) , denoted as dispersion relation below, represents a simple interpolation between limits. Evidence for its suitability was provided here by taking reference to probabilistic arguments, in particular, the realizability and statis- tically most-likely (SML) structure of the related probability 2 density function (PDF) f = dL+/d∆C+. The dispersion relation 2/3 2 FIG. 1. L+ , L+, and L+ determined by Eq. (16). The dashed line shows ∆C+. The enables a better insight into the differences between PANS- 2 inset shows the PDF f = dL+/d∆C+. PITM approaches (using k+ to reflect the resolution, we have

Phys. Fluids 31, 021702 (2019); doi: 10.1063/1.5085435 31, 021702-3 Published under license by AIP Publishing Physics of Fluids LETTER scitation.org/journal/phf

of why RANS equations can be resolving and how modeled and N = 2 ﬁnally implies

resolved motions can be kept in balance were not presented ∗ so far. Evidence for that was provided here subject to the δα δk δ δL ∗ = 3 − 2 = 2 , (7) condition that model coefficients can correctly satisfy varia- α − 1 k L tion equations implied by the RANS equations. Various turbu- where L = k3/2/ is the turbulence model length scale. The lence models can act equivalently if their model coefficients integration of Eq. (7) from the RANS state to a state with are chosen consistently. Both PANS and PITM approaches are α∗ a certain level of resolved motion results in ∫ dx/(x − 1) known to produce imbalances in this regard, e.g., regarding α = 2 ∫ L dy/y. The evaluation of the latter expression implies the imposed and the actual resolution.16 The reason for such Ltot imbalances was explained by the fact that PANS-PITM con- α∗ = 1 + R(α − 1), (8) cepts apply a relationship between model coefficients and res- 2 olution parameters that applies to homogeneous flows and where R = L+. needs the neglect of variations. So far, there is no agree- Section IB. It is of interest to look at other versions of ment about how the grid affects the resolution in hybrid setting up hybrid models within the k– model framework. methods and scale variables in LES like L.12,18–20 The proba- According to the DES concept,10,12 we may consider bilistic interpretation presented here supports the validity of 2 ∆ / ∆3 1/3 Dk D P L+ = C+ (1 + C ) . On this basis, it was shown that the use of − − + = P ψα + Dk, = C1 α + D. (9) PITM-PANS concepts has a computational cost scaling with L+ Dt Dt k −9 that is six orders of magnitude higher (n+ = L+ ) than the cost −3 Variations of resolved and modeled motions can be included scaling of methods presented here (n+ = L ), which matters, + here via the function ψα, which has to be determined. In cor- in particular, in the high-cost LES regime. Regarding the scal- 3 1/3 respondence to the analysis presented before, we neglect the ing of LES, L = C+∆/(1 + ∆ ) explains the scaling of the LES C+ LHS of both equations and use P = ψα − Dk in the equation length scale L with ∆. −1 2 so that 0 = C k (ψα − D / − α) + D . The comparison with 1 k Section IA. We apply variational analysis to the k– model Eq. (5) reveals the equivalence of models, provided we have equations (1) to find out which α∗ is needed to enable a ∗ desired resolution. To prepare the following developments, α = 1 + α − ψα. (10) we consider the variations (denoted by δ) of diffusion terms. By following, basically, the analysis of Friess et al.,10 we find Section IC. It is also of interest to look at corresponding δ / = α3/α − δ / = α2 − implications for other two-equation turbulence models, like Dk Dk k 1 and D D k 1. The relevant condition applied here is that variations of αk = 1 + δk/k and α = 1 the k–ω model, + δ/ in space can be neglected, which is equivalent to look- Dk Dω P ing for model coefficient variations that produce an uniform = P − + D , = C ω2 − β∗ + D , (11) Dt k Dt ω1 ω variation of the energy partition over the domain: see the jus- 10 16 2 tification provided by Friess et al. and Manceau et al. In the which involves ω = /k. Here, we have P = νtS with νt = Cµ k/ω 3 ∗ first order of approximation, the relations δDk/Dk = α /α − 1 k and β = β with β = Cω2 /(CkCω1 ). The diffusion term in δ / = α2 − ω ∂ ν ν σ ∂ω ∂ ∂ and D D k 1 read the equation reads Dω = [( + t/ ω ) / xj]/ xj. The constants involved have the values Cω 1 = 0.49, Cω 2 = 0.072, δD δk δ δD δk k = N − N = N Ck = 0.09, Cω = 1.1, σω = 1.8. Usually, a cross diffusion term k k , , (4) −1 Dk k D k Dωc = Cω k (ν + νt)[∂k/∂xj][∂ω/∂xj] is added to Eq. (11). How- ever, it does only affect the model behavior near boundaries (it where N = 3, N = 1, and N = 2. The noticeable difference k k acts like a damping function). Thus, it will be neglected for the to the corresponding expressions obtained by Friess et al.10 is following analysis. The hybridization of the RANS equations (11) the inclusion of variations. By following the analysis of Friess ∗ requires to determine a variable β to enable a certain resolu- et al.,10 we neglect (with respect to the variational analysis tion. As performed with respect to the k– models, we neglect performed here) the left-hand side (LHS) of both equations in both LHS’s in Eqs. (11) and use P = − Dk in the ω equation to the k– model (1), which is equivalent to assuming that both 2 ∗ obtain 0 = Cω ω (1 − D / − β ) + Dω . The variational analysis of k and are in equilibrium along mean streamlines. By using 1 k this equation can be performed by following the correspond- P = − Dk in the equation, we obtain ing analysis of k– models. We find δDω /Dω = αk − 1. In the 2 first order of approximation, we obtain δDω /Dω = Nω δk/k, Dk ∗ 0 = C 1 − − α + D . (5) where N = 1. On this basis, we find 1 k ω

∗ ∗ ∗ δk δ The function α can be found by considering the variation of δβ = (β − 1) (2 + Nω ) − 2 − Tβ , (12) Eq. (5), " k # ∗ ∗ δk δ −1 − − − δα = (α − 1) (1 + N ) − 2 − Tα, (6) where Tβ = Dk[(1 Nk )δ/ + (Nk 2 Nω )δk/k]. Com- " k # 2 bined with Nk = 3, Nk = 1, Nω = 1, and R = L+, the latter relation −1 implies where Tα = Dk[(1 − Nk )δ/ + (Nk − 1 − N )δk/k]. Here, ∗ Eq. (5) was used to replace D . The use of Nk = 3, Nk = 1, and β = 1 + R(β − 1). (13)

Phys. Fluids 31, 021702 (2019); doi: 10.1063/1.5085435 31, 021702-4 Published under license by AIP Publishing Physics of Fluids LETTER scitation.org/journal/phf

Section ID. In correspondence to the DES version of the a probability perspective, this CDF should be realizable, i.e., k– model, we may consider implied by an existing PDF. In particular, this PDF should have existing moments, at least a mean and variance (the existence Dk Dω P − 2 − = P ψβ + Dk, = Cω1 ω β + Dω , (14) of finite moments of such a PDF is not ensured in general: Dt Dt 2 4 for example, the PDF implied by an earlier L+ model does where ψβ needs to be determined. We neglect both LHS’s and not have existing moments). The PDF considered is given by − 2 2 ∆ 1/p − 1+1/p use P = ψβ Dk in the ω equation to obtain 0 = Cω1 ω (ψβ f(t) = dL+/d C+ = 2t (1 t) (see the inset in Fig. 1). Here, − / − β ω ≤ ≤ − = ∆p −1 > Dk ) + Dω . The comparison with the corresponding k– 0 t 1 is defined via 1 t (1 + C+) . For p 1, the PDF 2 ∗ equation 0 = Cω ω (1 − Dk/ − β ) + Dω reveals the consistency −1 1 1/p 2/p−1 1 f(t[∆C+]) has an existing mean, mf = 2p ∫0[t/(1 − t)] t dt, of models, provided we have > σ2 = −1 1 / and for p 2, it has an existing variance, f 2p ∫0[t (1 ∗ − β = 1 + β − ψβ . (15) − 2/p 2/p 1 − 2 σ2 t)] t dt mf . The evaluation of mf and f results in mf −1 − σ2 = −1 / − / − 2 Section II. Let us consider the calculation of L+, which = 2p B(3/p, 1 1/p) and f 2p B(4 p, 1 2 p) mf , respec- is needed in the equations presented above. L+ can be pre- 1 A−1 B−1 tively. B(A, B) = ∫0 t (1−t) dt is the beta function: see Ref. 22. scribed or directly calculated during the simulation. There A further analysis reveals that this PDF has theoretical sup- are versions of performing this (using relations of L+ to k+, port; it represents a SML PDF that maximizes the correspond- + = / tot, and M+, where M+ refers to the modeled-to- ing entropy on the probability space considered.23 The setting total ratio of the Reynolds stress considered in a coordinate- p = 3 (which implies a PDF mean mf = 1) can be justified as 20 invariant manner). For example, L+ can be determined via follows. According to Eq. (16), the modeled volume ratio of tur- / its relationship to M+ combined with a prescribed value of bulence elements reads L3 = 1/(1 + N p 3)3/p, where N = ∆−3 20 + + + C+ M+. Nevertheless, there is the question of how L+ is related characterizes the number of grid points in three-dimensional to the grid, which is relevant because of both practical and / − simulations (∆ = ∆/L ). This implies dL3/dN = −N p 3 1L3+p. theoretical reasons. There are questions regarding the setup + tot + + + + To exclude a divergence of dL3/dN , e.g., for large N , we have of simulations, e.g., what approximately is k for a certain grid + + + + to require that dL3/dN is independent of N . The latter is the and what is the change of the energy ratio if other grids are + + + case for p = 3. used. More importantly, the question of how L+ is related to ∆+ is relevant to the theoretical setup of LES and hybrid meth- The author would like to acknowledge partial support ods. For LES, there is the problem to explain the scaling of the through NASA’s NRA research opportunities in aeronau- model length scale L with ∆, see Refs. 15 and 20. With respect tics program (Grant No. NNX12AJ71A with Dr. P. Balakumar to hybrid RANS-LES, there is the question of how the mode as technical officer) and the National Science Foundation distribution changes with the grid. A significant number of (DMS–CDS&E-MSS, Grant No. 1622488 with Dr. Y. Zeng relationships between L+ and ∆+ are in use, see the reviews as technical officer). Substantial support from the Hanse- in Refs. 12, 18, and 19. Only a few suggestions for solving this Wissenschaftskolleg (Delmenhorst, Germany, Technical Mon- problem were presented as being supported by theory.4,5,17 itor: W. Stenzel) is gratefully acknowledged. I am very thankful Written in terms of L+, the recent version of this development, to the referees for their helpful suggestions for improvements. which involves any number p, reads17

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Phys. Fluids 31, 021702 (2019); doi: 10.1063/1.5085435 31, 021702-6 Published under license by AIP Publishing