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Turbulence Closure Models for Computational Fluid Dynamics Paul A Aerospace Engineering Publications Aerospace Engineering 2-23-2018 Turbulence Closure Models for Computational Fluid Dynamics Paul A. Durbin Iowa State University, [email protected] Follow this and additional works at: https://lib.dr.iastate.edu/aere_pubs Part of the Aerodynamics and Fluid Mechanics Commons, and the Computational Engineering Commons The ompc lete bibliographic information for this item can be found at https://lib.dr.iastate.edu/ aere_pubs/136. For information on how to cite this item, please visit http://lib.dr.iastate.edu/ howtocite.html. This Book Chapter is brought to you for free and open access by the Aerospace Engineering at Iowa State University Digital Repository. It has been accepted for inclusion in Aerospace Engineering Publications by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. Turbulence Closure Models for Computational Fluid Dynamics Abstract The formulation of analytical turbulence closure models for use in computational fluid mechanics is described. The ubjs ect of this chapter is the types of models that are used to predict the statistically averaged flow field – commonly known as Reynolds‐averaged Navier–Stokes equations (RANS) modeling. A variety of models are reviewed: these include two‐equation, eddy viscosity transport, and second moment closures. How they are formulated is not the main theme; it enters the discussion but the models are presented largely at an operational level. Several issues related to numerical implementation are discussed. Relative merits of the various formulations are commented on. Keywords turbulence modeling, RANS, k ‐epsilon model, Reynolds stress transport, second moment closure, turbulence prediction methods Disciplines Aerodynamics and Fluid Mechanics | Aerospace Engineering | Computational Engineering Comments This is a chapter from Durbin, Paul A. "Turbulence Closure Models for Computational Fluid Dynamics." In Encyclopedia of Computational Mechanics, Second Edition, edited by Erwin Stein, Ren de Borst and Thomas J. R. Hughes, 1-22. Chichester: John Wiley & Sons, 2018. DOI: 10.1002/9781119176817.ecm2061. Posted with permission. This book chapter is available at Iowa State University Digital Repository: https://lib.dr.iastate.edu/aere_pubs/136 Turbulence Closure Models for Computational Fluid Dynamics Paul A. Durbin Iowa State University, Ames, IA, USA of turbulent flow produces a reproducible, smooth flow field. 1 Introduction: The Role of Statistical Closure 1 Quantities such as the mean and variance of the velocity 2 Reynolds-Averaged Navier–Stokes Equations 1 provide definite values to be used in engineering analysis. It 3 Models with Scalar Variables 3 would seem that the smoother, averaged flow also is more amenable to computation than the instantaneous, random 4 Second Moment Transport 11 flow. Unfortunately, there are no exact equations governing 5 Reynolds-Averaged Computation 17 this smooth, average field. This is where closure modeling, References 21 the subject of this chapter, comes into play. For historical reasons, the mean velocity is called the Reynolds-averaged velocity and the equations obtained by averaging the Navier–Stokes equations are called the RANS 1 INTRODUCTION: THE ROLE OF (Reynolds-averaged Navier–Stokes) equations. Figure 1 STATISTICAL CLOSURE contrasts an instantaneous, random velocity field, to its ensemble average. The former is a direct numerical simu- Turbulence is the highly irregular flow of fluids. It lation; the latter is its Reynolds average. Statistical closure is governed by the exact, Navier–Stokes, momentum models are meant to predict the field shown in (b), without equations; but it is of little comfort to realize that turbulence having to average over an ensemble of fields shown in is just a particular type of solution to these laws of fluid (a). To accomplish this, new equations must be added to dynamics. Although its full disorderliness can be directly Navier–Stokes so that the average effects of turbulent mixing simulated on the computer (see Turbulence Direct Numer- are represented. These extra, turbulence closure equations, ical Simulation and Large-Eddy Simulation), that is too by necessity, contain empirical coefficients; this is not a expensive computationally and too fine a level of detail shortcoming, it is usually the only prospect one has to for many practical purposes. When the need is for timely obtain predictions of engineering accuracy with manageable prediction of the flow fields in practical devices, statistical computing times. moment closures of the type discussed in this chapter are invariably invoked. There are good reasons why most of the analytical approaches to turbulence prediction resort to statistical 2 REYNOLDS-AVERAGED descriptions. The average over an ensemble of realizations NAVIER–STOKES EQUATIONS Encyclopedia of Computational Mechanics Second Edition, The equations that underlie single point moment closure Edited by Erwin Stein, René de Borst and Thomas J.R. Hughes. models will be briefly summarized. Lengthy discussions of © 2017 John Wiley & Sons, Ltd. their derivation and interpretation can be found in texts on DOI: 10.1002/9781119176817.ecm2061 turbulence (Pope, 2000; Durbin and Reif, 2010). 2 Turbulence Closure Models for Computational Fluid Dynamics 4 where U ≡ ũ. The fluctuation u is defined to be the turbu- 3 lence. For variable density, a convenient formalism consists of the introduction of the density-weighted averages, U ≡ h ̃ ≡ / 2 u and uiuj uiuj. We will allow for density variation, y but we are not primarily concerned with direct effects of 1 compressibility on the turbulence. Ignoring compressibility in the turbulence equations, although not in the mean flow, 0 0 5 10 is generally acceptable well beyond transonic speeds (Barre (a) x/h et al., 2002). The turbulence problem, as presently formulated, is to 4 describe the statistics of the velocity field, without access 3 to realizations of the random flow. It seems sensible to start by attempting to derive equations for statistics. Toward this h / 2 end, averages of the Navier–Stokes equations can be formed, y in hopes of finding equations that govern the mean velocity, 1 the velocity covariances (uiuj), and so on. Regrettably, the 0 averaged equations are unclosed – meaning that each set 0 5 10 of statistical moment equations contains more unknowns (b) x /h than equations. In that sense, they fall short of any ambi- tion to arrive at governing laws for statistics. However, the Figure 1. U contours in a ribbed channel. (a) Instantaneous u field Reynolds-averaged equations give an insight into the factors (DNS) and (b) ensemble average (RANS). (Reproduced from Ikeda and Durbin (2002) courtesy of T. Ikeda. © Ikeda & Durbin, 2002.) that govern the evolution of the mean flow and Reynolds stresses; they also form the basis for the closure models to be discussed subsequently. We introduce the ensemble average of a random function The momentum and continuity equations governing f (x, t) (Newtonian) viscous flow, whether turbulent or laminar, are White (1991) ∑N , 1 , ′ ′ ′ ̃ ̃ ̃ ̃ ̃ ̃ f (x t) = lim fi(x t) or f = f P( f )df (1) t ui + j ujui =− ip + j[ ( iuj + jui)] N→∞ N ∫ i=1 ̃ t + j uj = 0(5) for either discrete samples or a continuous probability density. If the random process is statistically stationary, the If the Reynolds decomposition (4) is substituted into (5) and above can be replaced by a time average the result is averaged, the RANS equations t 1 U + U U =− P + [( U + U )] − ( u u ) f (x) = lim f (x; t′)dt′ (2) t i j j i i j i j j i j j i → ∫ t ∞ t 0 t + j Uj = 0(6) The caveat of statistical stationarity means that the statistics are independent of the time origin; in other words, f (t) and are obtained. The viscous term invokes an assumption that density fluctuations are not significant in viscous regions. f (t + t0) have the same statistical properties for any t0.Iff is statistically homogeneous in direction y, Equations (6) for the mean velocity are equivalent to equations (5) for the total instantaneous velocity, except for 1 y the last, highlighted, term of the momentum equation. This f = lim f (x, z, t; y′)dy′ (3) → ∫ y ∞ y 0 term is a derivative of the Reynolds stress tensor. It comes from the convective derivative, so the Reynolds stresses can be used, the result being independent of the direction of represent the averaged effect of turbulent convection. Any homogeneity, y. understanding of the nature of closure models relies on The Reynolds decomposition of the random variable, ũ,is recognizing that they represent this ensemble averaged into a sum of its mean and a fluctuation, effect, which, for instance, diffuses mean momentum. The mean flow equations (6) are unclosed because they ũ = U + u (4) contain six unknown components of the Reynolds stress Turbulence Closure Models for Computational Fluid Dynamics 3 tensor, uu. These derive from averaging the quadratic nonlin- transport of the Reynolds stress tensor. The first category earity of the Navier–Stokes equations. Any nonlinearity includes single equations for eddy viscosity and pairs of causes moment equations to be unclosed: here, the first equations for turbulent kinetic energy and its dissipation; the moment equation contains second moments; the second second category consists of methods based on the Reynolds moment equation will contain third moments; and so on up stress transport equation (7). the hierarchy. The second moment is
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