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 indepen- dent; 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–! 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–! 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 con- stants 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.