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, 𝜌U ≡ 𝜌̃ 𝜌 ≡ 𝜌 2 u and uiuj uiuj. We will allow for density variation, y / h 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 2 end, averages of the Navier–Stokes equations can be formed, y / h 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 , ′ ′ ′ 𝜕 𝜌̃ 𝜕 𝜌̃ ̃ 𝜕 ̃ 𝜕 𝜇 𝜕 ̃ 𝜕 ̃ 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 𝜕 𝜌 𝜕 𝜌̃ 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 𝜌𝜕 U + 𝜌U 𝜕 U =−𝜕 P + 𝜕 [𝜇(𝜕 U + 𝜕 U )] − 𝝏 (𝝆 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 𝜕 𝜌 𝜕 𝜌 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 the highest level consid- ered in most single point moment closure modeling. Dynamical equations for the Reynolds stress tensor can 3 MODELS WITH SCALAR VARIABLES be derived from the equation for the fluctuating velocity. For simplicity, we consider constant viscosity and solenoidal The purpose of closure models is to formulate a soluble set ⋅ 𝜌′ ≪𝜌 velocity fluctuations, ∇ u = 0, so . After subtracting of equations to predict turbulence statistics. These consist (6) from (5), multiplying by uj, averaging, and adding the of exact equations, such as (6), (7), and (9) plus formulas, result to the same equation with i and j reversed or auxiliary equations, that express unclosed terms as func- tions of their dependent variables. The extra formulas must 2 𝜕 u u + U 𝜕 u u =  + ℘ − 𝜀𝛿 + T contain empiricism, often via a fairly small number of exper- t i j k k i j ij ij 3 ij ij imentally determined model constants, although sometimes 1 𝜕 𝜇𝜕 u u + 𝜌 k( k i j) (7) empirical functions are also introduced to fit experimental curves. Unfortunately, the empiricism is not universal. There is obtained. Definitions and terminology for the various is a degree of freedom in which data are used to set the terms on the right are constants. On the other hand, closure models enable one to predict statistics of the very complex phenomenon of turbu-  ≡ −u u 𝜕 U − u u 𝜕 U production ⎫ lent flow by solving a relatively simple set of equations. If ij j k k i i k k j ⎪ ℘ ≡ 𝜕 𝜕 no empirical constants were required, the implication would ij (−uj ip − ui jp ⎪ be that the closure represented either exact laws of fluid +(2∕3)𝛿 u 𝜕 p)∕𝜌 ⎪ ij k k ⎪ dynamics, or a systematically derived approximation. That 𝜈𝜕 u 𝜕 u 𝛿 𝜀 − 2 k i k j +(2∕3) ij redistribution ⎬ degree of exactitude is impossible in predictive models for 𝜀 ≡ 𝜈 𝜕 u 𝜕 u dissipation ⎪ k j k j ⎪ engineering flows. T ≡ −(𝜕 𝜌 u u u +(2∕3)𝛿 u 𝜕 p)∕𝜌 ⎪ The models discussed in this section have scalar dependent ij k k i j ij k k ⎪ turbulent transport⎭ variables. Their basic purpose is to predict an eddy viscosity. (8) 𝜌′ ≪𝜌 On the assumption that the overbar has been dropped 3.1 The k − 𝜺 model from the mean density.  Aside from , the precise definition of the quantities in (8) In part, k − 𝜀 is popular for historical reasons: it was the is usually of minor importance. Their relevant properties are first two-equation model used in applied CFD. However, it 𝜀 ℘ that is nonnegative, ij is trace free, and Tij is the diver- is not the only model available. Indeed, other models may gence of a third-order tensor (if ui is solenoidal). Obviously, be more accurate, or more computationally robust in many (7) is not a closed equation for the second moment. cases. Gradually, these other models are gaining wider usage. The equation for turbulent kinetic energy per unit mass is This chapter reviews several of them. It is inevitable that one-half of the trace of (7): as the applications of CFD grow, further models will be [ ( ) ] 1 1 developed and existing models will be adapted to particular 𝜕 k + U 𝜕 k =  − 𝜀 − 𝜕 u p + 𝜌u u u + 𝜕 (𝜇𝜕 k) t j j 𝜌 j j 2 j i i i i needs. Closure modeling is an open-ended field that fills a (9) critical need. What is now called the “standard” form for ≡ ≡  𝜀 where k 1∕2uiui. Its rate of production is 1∕2 ij = k − is that developed by Jones and Launder (1972), with 𝜕 𝜀 −uiuk kUi and its rate of dissipation is . values for the empirical constants given by Launder and The primary objective of turbulence modeling is to close Sharma (1974). Although an enormous number of variations the mean flow equation (6). Various formulations have been on the k − 𝜀 model have been proposed, the standard model proposed in the course of time. Here we will only survey contains the essential elements, with the caveat that some methods that add auxiliary transport equations; these are the form of fix is needed near to solid boundaries –as willbe sort used in general purpose computational fluid dynamics discussed later. (CFD) codes. The closure schemes will be categorized into The k − 𝜀 formula for eddy viscosity could be rationalized those based on transport of scalars and those based on as follows. In parallel shear flow, the Reynolds shear stress 4 Turbulence Closure Models for Computational Fluid Dynamics

2 2 is assumed to be related to the velocity gradient by −uv =  = 2𝜈 S S − k𝜕 U − 𝜈 (𝜕 U )2 𝜈 𝜕 T ij ji i i T i i T yU. Suppose that dissipation and production of turbulent 3 3 𝜀   energy are approximately in balance, ≈ , where = 𝜈 | |2 2 𝛁 ⋅ 2𝜈 𝛁 ⋅ 2 𝜕  𝜕 = 2 T S − k U − T ( U) −uiuj iUj (equation 8). In parallel shear flow, =−uv yU. 3 3 𝜀  𝜈 Multiplying ≈ by T gives  𝜈 | |2 For incompressible mean flow, this reduces to = 2 T S . 𝜈 𝜀 𝜈  𝜈 𝜕 2 . 2 𝜀 T ≈ T = T (−uv yU)=(uv) ≈ 0 09k (10) The modeled transport equation for cannot be derived systematically. Essentially it is a dimensionally consistent upon invoking the experimental observation that uv∕k ≈ 0.3 analogy to the above k-equation: in many equilibrium shear layers (Townsend, 1976). Rear- [( ) ] ranging the above, 𝜈 ≈ 0.09k2∕𝜀. This formula is usually C  − C 𝜀 𝜇 T 𝜕 𝜀 𝜕 𝜀 𝜀1 𝜀2 1𝜕 𝜇 T 𝜕 𝜀 written t + Uj j = + 𝜌 j + 𝜎 j (15) C k2 T 𝜀 𝜈 𝜇 T = 𝜀 (11) The time scale T = k∕𝜀 makes it dimensionally consistent. and the standard value of C𝜇 is 0.09. Equation (15) is analogous to (14), except that empirical 𝜎 Alternatively, it can simply be argued by dimensional anal- constants C𝜀1, C𝜀2,and 𝜀 have been added because the ysis that the turbulence correlation time scale is T ∼ k∕𝜀 and 𝜀-equation is just an assumed form. The terms on the right 𝜈 2 side can be referred to as production of dissipation, dissipa- the velocity scale squared is k. Then by the formula, T ∼ u T 𝜈 2 𝜀 tion of dissipation diffusion of dissipation derived by Taylor (1921), T ∼ C𝜇k ∕ . By either rationale, ,and . The empirical one sees that formulas to parameterize turbulent mixing can coefficients are chosen in order to impose certain experi- be evaluated from a model that predicts k and 𝜀. mental constraints. The standard values for the constants are The mean flow (6) is computed with the scalar eddy (Launder and Sharma, 1974) viscosity and the constitutive relation ( ) C𝜇 = 0.09; C𝜀 = 1.44; C𝜀 = 1.92; 𝜎𝜀 = 1.3 (16) 1 2 1 2 − u u = 2𝜈 S − 𝛿 S − k𝛿 (12) i j T ij 3 ij kk 3 ij These constants were chosen some time ago. More recent 𝜕 𝜕 where Sij is the mean rate of strain tensor (1∕2)( iUj + jUi). data suggest that slightly different values might be suit- Substituting the constitutive model (12) closes the mean able, but the standard constants give reasonable engineering flow equation (6). The constitutive equation (12) is alinear results in many situations. It is not likely that minor adjust- stress–strain relation, as for a Newtonian fluid; nonlinear ments would significantly affect the predictive accuracy. models also have been proposed at times (Section 3.7). The C𝜀1 is a critical constant. It controls the growth rate of shear 𝜀 problem addressed by the k − transport model is how to layers: a 7% decrease of C𝜀1 increases the spreading rate by 𝜈 robustly predict T . The formula (11) reduces this to: predict about 25%. This is a bit misleading, because the spreading k 𝜀 the spatial and temporal distribution of and . rate is determined by C𝜀1 − 1, so the leading 1 is irrelevant Equation (9), is the exact, but unclosed, evolution equation and the change in spreading rate is nearly proportionate to for k. To “close” it, the transport and pressure diffusion terms C𝜀1 − 1. The standard value of C𝜀1 was chosen to fit the together are replaced by a gradient transport model: spreading rate of a plane mixing layer. Slightly different ( ) ( ) 𝜇 values would be obtained for other flows. 1 T − 𝜕 𝜌u u u − 𝜕 (u p)≈𝜕 𝜕 k (13) Very briefly, C𝜀2 is determined by the decay exponent j j i i j j j 𝜎 j 2 k −n measured in grid turbulence: if k ∼ t then C𝜀2 = 1 + 1∕n (Typical data would give C ≈ 1.83.) The value of C has This is based on a notion that the third velocity moment 𝜀2 𝜇 been discussed previously. represents random convection of turbulent kinetic energy, k 𝜀 that pressure–velocity correlation diffuses momentum, and The − model has an important closed form solution 𝜕 𝜅 that these can be replaced by gradient transport. The transport √in the log-layer. In this layer yU = u∗∕ y, where u∗ = 𝜏 𝜌 𝜏 𝜅 equation for k, with (13) substituted is ( w∕ ), with w being the wall shear stress and is the [( ) ] VonKarman constant. One finds that (Durbin and Reif, 2010; 𝜇 Wilcox, 1998) 𝜕 𝜕  𝜀 1𝜕 𝜇 T 𝜕 tk + Uj jk = − + j + jk (14) u3 𝜌 𝜎 ∗ ⎫ k  = 𝜀 = 𝜅y ⎪ 𝜈 𝜅 𝜎 ≡ T = u∗y ⎬ (17) The usual value of k is 1. Note that −uiujSij. Then, u2 ⎪ k = √ ∗ ⎭ with (12) C𝜇 Turbulence Closure Models for Computational Fluid Dynamics 5

This solution provides the value of 휎휀 60

C휀2 − C휀1 휎휀 = √ (18) 2 휅 C휇 40 Measurements of 휅 are mostly in the range 0.41 ± 0.02. The T + standard value 휎휀 = 1.3 corresponds to the high end of these ν measurements. 20 νT Although this log-layer solution plays a role in compu- 0.09 k2/ε 휀 tational use of k − , the solution for k does not agree with 0.2 v2k/ε . 2 experiments. Older data suggested that k ∼ 3 3u∗ and hence that C휇 = 0.09 in (17). Experiments by DeGraaff and Eaton 0 (2000) question the correctness of these older data. The 0 150 300 450 600 y+ momentum thickness Reynolds number must be on the 2 order of R휃 ≈ 8000, before a zone of constant k∕u∗ is seen. Figure 2. Exact eddy viscosity compared to the k − 휀 and v2 − f However, that value increases with R휃, which would imply formulas in channel flow at R휏 = 590. Curves were computed from that C휇 declines to values significantly lower than 0.09. DNS data of Moser et al. (1999). While that certainly is of concern, it is not devastating: 휅 the log-layer eddy viscosity is predicted to be u∗y,which agrees quite well with data. Therefore, mean flow predic- tions do not fail. Presumably, the apparent Reynolds number In CFD codes this, or something equivalent, is often used to dependence of C휇 would be counteracted by Reynolds impose the no-slip condition (19). number dependence of other coefficients, with little effect Unfortunately, deriving the correct no-slip boundary condi- on mean flow prediction. tions is not the only issue in near-wall modeling. A second need is to prevent a singularity in the 휀-equation (15). If 휀 → 2 → 3.2 Boundary conditions and near-wall the time scale T = k∕ is used, then T O(y ) as y 0, 휀2 modifications and the right side of√ (15) becomes singular like ∕k.The Kolmogoroff scale, (휈∕휀), is an appropriate time scale near The boundary conditions to the k − 휀 model at a no-slip wall the surface. The formula (Durbin, 1991) are quite natural, but the near-wall behavior of the model is ( √ ) k 휈 not. This is rather a serious issue in RANS computation. A T = max , 6 (22) variety of patches have been proposed in the course of time. 휀 휀 At a no-slip surface u = 0, so k = |u|2∕2 has a quadratic zero. Hence both k and its normal derivative vanish. The can be used. The coefficient of 6 in formula (22) isan natural boundary condition is empirical constant. Several other methods that help avoid a singularity in the 휀 equation are discussed in Patel et al. 휕 k = 0; nk = 0 (19) (1984). But the near-wall failure of standard k − 휀 goes further. 휕 ≡ n̂ ⋅ 𝛁 휈 2 휀 where n is the derivative in the wall-normal direc- The formula T = C휇k ∕ gives an erroneous profile of eddy tion. Equation (19) specifies two conditions on k and none viscosity even if exact values of k(y) and 휀(y) are known. The on 휀. That suffices to solve the coupled k − 휀 system. solid line in Figure 2 is the viscosity constructed from DNS However, in segregated equation algorithms, it is common data (Moser et al., 1999) by evaluating the exact definition practice to convert these into k = 0 and a condition on 휀.As 휈+ 휈 T =−uv∕ dyU. It is compared to curves constructed by the wall is approached, (14) has the limiting behavior substituting the exact k and 휀 into the k − 휀 formula (11). The model formula is seen to be grossly in error below y ≈ 50. 휀 휈휕2 + w = y k (20) Overpredicting the eddy viscosity in this region will greatly overpredict skin friction on the surface. Integrating (20) gives k → A + By + 휀 y2∕2휈, where A and w One device to fix (11) consists of “damping” the viscosity: B are integration constants. By (19) A = B = 0, so the wall to this end, it is replaced by value of dissipation is 휈 2 휀 2 k f휇C휇k w = lim (21) 휈 = (23) y→0 y2 T 휀 6 Turbulence Closure Models for Computational Fluid Dynamics

In the literature the damping function, f휇, has been made is obtained if A휀 = 2C𝓁. The eddy viscosity (11) will not dependent either on ky∕휈 or on k2∕휀휈, depending on the have the right√ damping if (26) is substituted. Doing so would 휈 𝓁 model. For example, Launder and Sharma (1974) used give T = (k) 휀. Therefore, a separate length is used in the [ ] formula −3.4 k2 √ f휇 = exp ; R = (24) 2 t 휀휈 휈 𝓁 (1 + RT ∕50) T = C휇 k 휈 (28) ( √ ) −y k∕휈A휈 The reader might consult Patel et al. (1984) for tabulations 𝓁휈 = C𝓁y 1 − e (29) of various other “low Reynolds number k − 휀” formulations. 3∕4 It is not sufficient to simply damp the eddy viscosity; The log-layer solution (17) then gives C𝓁 = 휅∕C휇 .Given 휀 all low Reynolds number k − models also modify the the VonKarman constant 휅 = 0.41 and that C휇 retains its 휀 -equation in one way or another. Launder and Sharma value 0.09, the only new empirical constant is A휈.Thevalue (1974) replace it with A휈 = 62.5 gives a good fit to skin friction data in zero pres- [( ) ] sure gradient boundary layers. ̃휀 휇 휕 ̃휀 휕 ̃휀  ̃휀 1휕 휇 T 휕 ̃휀 The k − 𝓁 and k − 휀 models are patched at some t + Uj j = (C휀1 − f2C휀2 )+휌 j + 휎 j k 휀 distance from the√ wall. Common practice is to do ( ) −y (k)∕휈A 2 this, where 1 − e 휈 reaches 0.95; this occurs at 휕2U √ + 2휈휈 (25) y = log(20)A휈휈∕ (k(y )). Succinctly, the two-layer T 휕y2 sw sw model solves the k-equation (14) at all points in the flow, but instead of (15) the 휀-equation is represented by The dependent variable ̃휀 is defined as 휀 − 휈|∇k|2∕2k.This is a device that causes ̃휀 to be zero at the wall. Throughout [휀]= (30) the model, including the eddy viscosity formula (23), 휀 is ̃휀 replaced by . The last term on the right side of (25) is The operator  and source  are defined as not coordinate independent, but it is only important near the { ( ) ̂ 휇 surface, so y can be considered the wall normal direction. The 휕 휕 1 휕 휇 T 휕 > t + Uj j − 휌 j + 휎 j y ysw blending function  = 휀 (31) ≤ 1 y ysw . 2 f2 = 1 − 0 3exp (−Rt ) ⎧ C  − C 휀 ⎪ 휀1 휀2 y > y  T sw was also inserted in (25). = ⎨ 3∕2 (32) k ≤ 휀 ⎪ 𝓁 y ysw The benefits of adopting low Reynolds number k − ⎩ 휀 models in practical situations seem limited. Given the confusingly large number of the formulations, their numer- This two-layer formulation has proved effective in practical ical stiffness and their inaccurate predictions in flows with computations (Rodi, 1991) and usually gives better predic- significant pressure gradient, they are not attractive for full tions than wall functions. But it does require a fine grid near RANS computation. to walls, and so is more expensive computationally than wall functions. 3.3 Two-layer models 3.4 Wall functions A viable alternative to near-wall damping is to formulate a simplified model for the wall region, and to patch it ontothe Another method that circumvents the erroneous predictions k − 휀 model away from the wall. The k − 𝓁 formulation has in the near-wall region is to abandon the k − 휀 equations in a been used to this end, in an approach called the two layer zone next to the wall and to impose boundary conditions at k − 휀 model (Chen and Patel, 1998). the top of that zone. Within the zone, the turbulence and mean The k − 𝓁 model uses the k-equation (14), but replaces the velocity are assumed to follow prescribed profiles. This is 휀-equation (15) by the algebraic formula the “wall function” method. Conceptually, the wall function is used in the law-of-the-wall region and the k − 휀 model k3∕2 predicts the flow field farther from the surface. Thetwo 휀 = (26) 𝓁휀 share the logarithmic layer as a common asymptote. They are ( √ ) patched together in that layer. Wall functions are used with −y k∕휈A휀 𝓁휀 = C𝓁y 1 − e (27) models other than k − 휀, as well, to reduce grid requirements. Turbulence Closure Models for Computational Fluid Dynamics 7

Let d be the distance from the wall at which the patching computational stiffness is reduced. Because of this stiffness, 휅 is done.√ At that point the log-layer solution dU∕dy = u∗∕ d, the near-wall region requires a disproportionate number of 2 휀 3 휅 k = u∗∕ (C휇),and = u∗∕ d (see equation 17) is assumed grid points; avoiding it with a wall function reduces grid to be valid. This is an exact solution to the standard k − 휀 requirements. Wall functions therefore are widely used for model in a constant stress layer, so smooth matching is engineering prediction. possible in principle; in practice, wall functions are used Various methods have been proposed to extend the validity ≲ even when they are not mathematically justified, such as in a of wall functions to y+ 40 in the interest of more flexible separating flow. meshing. In principle, the wall function boundary condition

The friction velocity, u∗, is found by solving the implicit can be imposed anywhere in the law-of-the-wall region. For formula ( ) instance, a solution in that near-wall region can be tabu- u du U(d)= ∗ ln ∗ + B (33) lated and used as a boundary condition (Kalitzen et al., 휅 휈 2005). Another approach is called a “scalable” wall function, which largely consists of replacing the non-dimensional wall . with a typical value of B = 5 1. The tangential surface shear distance by max(y+, 11.2) (Vieser et al., 2002). stress is assumed to be parallel to the direction of the mean 휏 휌 2 velocity, and is w = u∗ in magnitude. In a finite volume 휏 3.5 The k − 𝝎 model formulation w provides the momentum flux on the wall face. At a two-dimensional separation point, u∗ = 0. To avoid 2 The eddy viscosity is of the form 휈 = C휇u T, where T is a problems as u∗ changes sign, the boundary conditions can T ≡ correlation time. One role of the 휀-equation is to provide this be√ expressed in terms of a different velocity scale, uk 1∕2 휀 (k (C휇)) . They then assume the form (Launder and scale via T = k∕ . Instead, one might consider combining the Spalding, 1974) k-equation directly with a time-scale equation. It turns out to be more suitable to use a quantity, 휔, that has dimensions 휏 3 of inverse time. Then the eddy viscosity is represented as dU w uk dk = ,휀= , = 0 (34) 휈 k 휔 dn 휌휅u y 휅y dn T = ∕ . k p p The k − 휔 model of Wilcox (1998) can be written as In practice, the conditions (34) are applied at the grid point [( ) ] 1 휇 closest to the solid boundary, y(1); this point should be 휕 휕 휈 | |2 휔 휕 휇 T 휕 tk + Uj jk = 2 T S − C휇k + 휌 j + 휎 jk . 훿 훿 k located above y+ ≈ 40 and below y ≈ 0 2 99, where 99 is [( ) ] 휇 the 99% boundary layer thickness. 휕 휔 휕 휔 | |2 휔2 1휕 휇 T 휕 휔 t + Uj j = 2C휔1 S − C휔2 + j + j The wall function procedure is rationalized by appeal to the 휌 휎휔 two-layer, law-of-the-wall/law-of-the-wake, boundary layer 휈 k structure. The law-of-the-wall is assumed to be of a universal T = 휔 (35) form, unaffected by pressure gradients or flow geometry. 휀 휔 Its large y+ asymptote, (34), is its only connection to the The k-equation is altered only by changing to k .The nonuniversal part of the flow field. Through this, the skin 휔-equation is quite analogous to the 휀-equation. The stan- 휎 휎 friction can respond to external forces. However, in highly dard constants are C휔1 = 5∕9, C휔2 = 3∕40, 휔 = k = 2, and perturbed flows the assumption of a universal wall layer is C휇 = 0.09. not consistent with experiments. Predictions made with wall Figure 3 shows the near-wall behavior of a solution to functions then deteriorate. the k − 휔 model in plane channel flow with R휏 = 590. The As a practical matter, it is sometimes impossible to ensure profile of 휀 = C휇k휔 has been multiplied by 50 for display. that the first point of a computational grid lies in the log-layer, The region next to the wall is critical to heat and momentum if a log-layer exists at all. It is not possible a priori to exchange between the fluid and the surface. In the wall generate a computational mesh that will ensure that the first layer, the k − 휔 predictions of k and 휀 are at odds with the computational node is neither too close nor too far from data. 휀 erroneously goes to zero at the surface and has a the wall. Accurate computation may require a posteriori spurious peak near y+ = 10. The consequence of the spurious modification of the mesh to achieve a suitable y+. peak is that k is excessively dissipated near the wall. While In complex flows, it is likely that the wall function will be these erroneous predictions might at first be disconcerting, used beyond its range of justifiability. On the other hand, wall in fact the underestimation of k and overestimation of 휀 functions can significantly reduce the cost of a CFD analysis. are exactly what are needed to counter the overprediction 휈 2 휀 The steepest gradient of turbulent energy occurs near the of T displayed by the formula C휇k ∕ in Figure 2. Both ≲ 휈 wall (y+ 10). By starting the solution above this region, the of these features contribute to giving T a more reasonable 8 Turbulence Closure Models for Computational Fluid Dynamics

U+ Formula (36) is an unwanted constraint in free shear flow. The limiter is confined to the boundary layer by the blending 15 function

2 10 F2 = tan h(arg ) [ √2 ] 2k+ 2 k 500휈 arg = max , (38) 2 C 휔y 휔y2 5 휇

The blending function (38) is devised so that F2 is nearly 50ε+ 0 unity in most of the boundary layer, dropping to zero near 0 50 100 150 the top and in the free-stream. F2 multiplies the second term y+ in the min function of equation (36). To rectify the spurious free-stream sensitivity of the orig- Figure 3. Channel flow solution to k − 휔. The centerline is at y+= inal k − 휔 model, Menter developed a two zone formulation 590. The curves are solutions to the model. are DNS data for U; that uses k − 휔 near the wall and k − 휀 for the rest of the flow. 휀 are data for 2k;and are for 50 . The switch between these forms is by a smooth interpolation. Now the blending function is distribution. Indeed, the U predictions in Figure 3 agree quite 4 well with data, given that no wall corrections have been made F1 = tan h(arg1) [ ( √ ) to the model. k − 휔 is usable near boundaries, without wall k 500휈 functions or wall damping – that is its remarkable property. arg = min max , , 1 C 휔y 휔 2 Near a no-slip surface equations (35) have the 휇 y ] 휔 2 m limiting√ solution + = 6∕(C휔2y+),andk ∝ y+ with m = 2k휔 . 휔 (39) (1∕2)+ (149∕20)=3 23. This shows that is singular at y2max(∇k ⋅ ∇휔, 10−20) no-slip boundaries. Nonsingular solutions also exist; Wilcox (1998) proposed that wall roughness could be represented This seemingly intricate function is simply an operational by prescribing a finite boundary value for 휔.However,on 휔 휀 device to interpolate between the k − and k − models. F1 smooth walls the singular solution is usually invoked. is devised to be near unity in the inner half of the boundary layer and to decrease through the outer half, dropping to zero 3.5.1 The SST variant slightly inside its top edge. The term 휔 Menter (1992) noted two failings of the basic k − model, ( )[ ] 휈 2 one minor and one major. The former is that it overpredicts 2 T |∇k| ∇k ⋅ ∇휀 S휔 = 휈 + − the level of shear stress in adverse pressure gradient boundary T 휎휔 k 휀 layers. The latter is its spurious sensitivity to free-stream conditions: the spreading rate of free shear flows shows an which arises when the exchange 휔 → 휀∕k is used to trans- 휔 휔 휀 unphysical dependence on the free-stream value of . form the -equation into the -equation, is faded out via F1: 휔 To overcome the shortcomings of the basic k − model, [( ) ] Menter proposed the “shear stress transport” (SST) model. C  − C 휀 휈 휕 휀 휕 휀 휀1 휀2 휕 휈 T 휕 휀 휔 t + Uj j = + j + j + F1S휔 This variant of k − has been found quite effective in T 휎휀 predicting many aeronautical flows. To limit the Reynolds (40) shear stress in strong pressure gradients, Menter introduced the bound [ √ ] Thereby a transition between the 휀 and 휔 equations typically C k 휈 k , 휇 is brought about across the middle of the boundary layer. T = min 휔 | 𝛀| (36) 2 F2 To complete this formulation, the model constants also are interpolated as where ( ) ≡ 1 휕 휕 Ωij iUj − jUi (37) . 2 C휀1 = 1 +(1 − F1)0 44 + F1C휔1 . and |𝛀| is its magnitude. C휀2 = 1 +(1 − F1)0 92 + F1C휔2 Turbulence Closure Models for Computational Fluid Dynamics 9

휀 휔 These provide the k − values when F1 = 0andthek − The eigenvalue bound is invoked by introducing the constant 휎 휎 훼 ≤ ∗ values when F1 = 1. The coefficients k and 휀 are interpo- 1. For compressible flows, S should be replaced by S = lated similarly. S −(1∕3)(∇ ⋅ U)I.Thevalue훼 = 0.6 was selected. A simpler method for avoiding free-stream sensitivity is to add a cross-diffusion term to (35) (Wilcox, 1998, Kok, 2000): 3.6 Eddy viscosity transport models [( ) ] 1 휇 휕 휔 휕 휔 | |2 휔2 휕 휇 T 휕 휔 Two equation models construct an eddy viscosity from t + Uj j = 2C휔1 S − C휔2 + 휌 j + 휎 j 휔 velocity and time scales. It might seem prudent instead 휎 + d max{ 훁 k ⋅ ∇휔, 0} (41) to formulate a transport equation directly for the eddy 휔 viscosity. That idea was initiated by Baldwin and Barth (1990) and improved upon by Spalart and Allmaras (1992) Kok (2000) suggests the values 휎 , 휎 = 1.5, 휎 = 2. d k 휔 to produce the model described here. The Spalart–Allmaras (S–A) model enjoyed initial success in predicting aerody- 3.5.2 The large strain rate anomaly namic flows and is now established for general flows. Assume a priori that an effective viscosity, ̃휈, satisfies a The large strain rate anomaly is that excessive levels of turbu- prototype transport equation lent kinetic energy are predicted by standard two-equation [ ] models in regions of large mean rate of strain. The following 휕 ̃휈 ⋅ 훁 ̃휈  휀 1 훁 휈 ̃휈 훁 ̃휈 | 훁 ̃휈|2 t + U = 휈 − 휈 + (( + ) )+cb2 reviews two ideas to solve this problem. 휎휈 In nondivergent flow the rate of production of turbulent (44) energy is  = 2휈 |S|2. Launder and Kato (1993) attribute Aside from the term multiplying cb2 this is analogous to the t 휀 휔 excessive levels of k to an overestimate of  in pure straining k, ,or equations: the right side consists of production, flow. As a pragmatic device to avoid the problem of spurious destruction, and transport. The cb2 term is added to control stagnation point build-up of turbulent kinetic energy, they the evolution of free shear layers. This equation has a propa- | |2 | || | gating front solution, with the propagation speed depending replaced S with S Ω , where Ωij is defined by (37) and |Ω|2 =Ω Ω . Hence the rate of production is zero in irrota- on cb2. Spalart and Allmaras (1992) argue that the front speed ij ij . tional flow (|Ω| = 0). This is the first idea for avoiding the influences shear layer development, and choose cb2 = 0 622, 휎 large rate of strain anomaly. in conjunction with 휈 = 2∕3, to obtain a good representation The second idea is to invoke a time-scale bound. The eddy of the velocity profiles in wakes and mixing layers. viscosity predicted by scalar equation turbulence models can The cleverness in developing an equation for an effec- ̃휈 휈 be characterized by the form tive viscosity, , rather than the actual eddy viscosity, T , is that a numerical amenity can be added. Baldwin and 휇 휌 2 Barth (1990) proposed to make ̃휈 vary linearly throughout t = C휇 u T (42) the law-of-the-wall layer, in particular, to nearly retain the ̃휈 휅 where u is the velocity scale and T is the turbulence time log-layer dependence = u∗y allthewaytothewall. scale. In k − 휀 and k − 휔 models, u2 = k. T equals k∕휀,in For production, choose the dimensionally consistent form k − 휀, and it equals 1∕휔 in k − 휔. Note that T also appears  | ̃ |  휀 휀 휈 = cb1 2Ω (45) in the source term (C휀1 − C휀2 )∕T of the -equation and that production of turbulent energy can be stated as  = 2C k|S|2T. Spalart and Allmaras selected the constant to be cb1 = 휇 . A bound for the turbulent time scale T can be derived 0 1355. The wall distance d is used for a length scale in the destruction term ( ) (Durbin and Reif, 2010) from the condition that the eigen- ̃휈 2 휀휈 = f c values of the Reynolds stress tensor as estimated by formula w w1 d (12) should be nonnegative: [ ] The function fw is discussed below. The expression cw1 = 휅−2 휎 k 훼 cb1 +(1 + cb2)∕ 휈 can be derived from log-layer scaling. T = min , √ The intriguing part is that ̃휈 is contrived to vary nearly 휀 | | 6C휇 S linearly all the way to the wall, in theory. This is achieved or [ ] by deriving the formula 훼 1 , √ ̃휈2 ̃휈 T = min 휔 (43) |Ω̃ | = |Ω| − + (46) C휇 | | 휈 휈 휅 2 휅 2 6C휇 S ( T + )( y) ( y) 10 Turbulence Closure Models for Computational Fluid Dynamics for the effective vorticity in (45). Next to a wall, the second on system rotation. However, it should be warned that factor on the right cancels |Ω| and the third provides |Ω̃ |, nonlinear constitutive models tend to increase numerical leading to the desired behavior of ̃휈 – in consequence of stiffness. | ̃ | Ω ∝(1∕y). One postulates that uiuj is a tensor function of the rate A formulation with a nearly linear solution near the wall of strain S and the rate of rotation 훀,aswellasofthe is attractive; but the real eddy viscosity is not linear near the scalar quantities k and 휀. What are the constraints on the wall. This is rectified by a nonlinear transformation: let possible functional form? The principle of material frame ( ) indifference would demand that material properties be inde- ̃휈 휈 ̃휈 pendent of the rate of rotation, but turbulent stresses are T = f휈 휈 (47) most definitely affected by rotation. This is because they The transforming function are not material properties but properties of the flow. So the only constraints are that the formula must be tensorally and ( ) ̃휈 (̃휈∕휈)3 dimensionally consistent. Our discussion is further simpli- f휈 = 휈 (̃휈∕휈)3 + 7.13 fied by restricting it to two-dimensional mean flow. Let the stress tensor be a function of two trace-free  훀, was adopted by Spalart and Allmaras (1992). tensors: uiuj = ij( S). By the extended Cayley – Hamilton The function fw in front of the destruction term remains to theorem (Goodbody, 1982), in two dimensions, the most be specified. That function implements a constraint that, far general isotropic function can depend tensorally on I, S, 훀, from the surface, the wall distance should drop out of the and S2, where the last is used in place of the two-dimensional model. The particular form has an arbitrary appearance; it is identity tensor. It can also depend on products of these; but, [ ] it can be shown that the only tensorally independent product 1∕6 65 is S ⋅ 훀 (Pope, 1975). f (r)=g , with g = r + 0.3(r6 − r) w g6 + 64 The most general constitutive model for three ̃휈 velocity-component turbulent stresses in a two-dimensional, and r ≡ (48) incompressible mean flow is provided by Ω(̃ 휅d)2 [ 2 2 Formula (48) was selected to provide accurate agreement −uu = C휏 kTS − kI + kT C휏 (S ⋅ 훀 − 훀 ⋅ S) 1 3 2 with experimental data on skin friction beneath a flat plate ( )] 2 1 2 boundary layer. + C휏 S − trace(S )I (49) 3 3

3.7 Nonlinear constitutive equations where T is a turbulence time scale (e.g., k∕휀). The linear model is C휏 = C휇 and C휏 = 0 = C휏 . In nonlinear models 1 2 3 Eddy viscosity models usually invoke the quasilinear these coefficients can be made functions of the invariants stress – strain rate relation (12). There are obvious short- |S|2T2 and |훀|2T2. One method to derive the functional comings to that formula: for instance, it incorrectly predicts dependence is by solving an equilibrium approximation to that the normal stresses are isotropic in parallel shear flow: one of the RST equations, as discussed in Section 4.3 (Gatski u2 = v2 = w2 =(2∕3)k when U is a function only of y.To and Speziale, 1993). Another method is simply to postulate a large extent one hopes that (12) will adequately repre- forms for the coefficients, instances of which are described sent the shear stresses, as these are often dominant in the by Apsley and Leschziner (1999). mean momentum equation. However, the normal stresses In three dimensions, a number of further terms are needed generally will not be correct, even qualitatively. to obtain the most general tensor representation. Several An elaborate correction to some shortcomings of eddy authors have explored truncated versions of the general viscosity is provided by RST models. Stress transport three-dimensional form. Craft et al. (1999) explored a models will be discussed in Section 4. An intermediate version that includes terms cubic in S, 훀, and their products. level of modeling would seem to be provided by nonlinear, Craft et al. (1999) also discuss methods to accelerate numer- but still algebraic, constitutive formulas. In practice, it ical convergence of the RANS equations with nonlinear remains unclear what level of improved prediction nonlinear constitutive models. These methods are of the variety constitutive models provide. They promise to add physical described in Section 5.1: essentially, an effective viscosity effects that are not accommodated by the linear formula. is extracted and included with the viscous term, and dissi- For instance, the normal stresses in parallel shear flow can pative source terms are treated implicitly in the k and 휀 be unequal and the Reynolds stress can become dependent equations. Turbulence Closure Models for Computational Fluid Dynamics 11

4 SECOND MOMENT TRANSPORT dependent variable, uiuj. What is needed is a tensor function of tensors, ℘ ,휕 ,훿 ,휀 Anisotropy exists in all real flows. In parallel shear flows, ij = Fij(uiuj jUi ij; k ) the dominant anisotropy is the shear stress. Eddy viscosity models are designed to represent shear stress; they are It is common to nondimensionalize uiuj by k and subtract 훿 not designed to represent normal stress anisotropy. Second (2∕3) ij to form a trace-free, anisotropy tensor moment closure (SMC) is based on transport equations for the entire stress tensor. uiuj 2 b ≡ − 훿 (51) The limitations to scalar models stem largely from the ij k 3 ij [ ] turbulence being represented by its kinetic energy, which is k ℘ = 휀 b , 휕 U ,훿 (52) a scalar, and from the eddy viscosity assumption (12). The ij ij ij 휀 j i ij former does not correctly represent turbulence anisotropy; in the present case of homogeneous flow. The functional form the latter assumes an instantaneous equilibrium between the is local in time: in particular,  could be a functional of b (t′), Reynolds stress tensor and the mean rate of strain. A short- ij t′ ≤ t. However, all SMC models currently in use invoke a coming in representing the turbulent velocity fluctuations temporally local redistribution model; all variables in (52) solely by the scalar magnitude k is that sometimes an external are evaluated at the same time t. History effects are present, force acts on one component more strongly than others. but only through the evolution equation (50). For example, stable streamline curvature will suppress the Common practice is to separate ℘ into slow and rapid component directed toward the center of curvature. Another contributions, ℘slow + ℘rapid, and to model their functional shortcoming in the eddy viscosity representation is that it dependence,  slow +  rapid, separately. Terms that do not causes the Reynolds stresses to change instantaneously with depend on 휕 U are referred to as the slow terms of the the mean rate of strain. Disequilibrium should come into j i redistribution model. The rapid terms depend on velocity play in rapidly changing flow conditions. SMC incorporates gradients; they are usually tensorally linear in 휕 U . many of these effects in a natural way because it is based j i on the RST equations (7). For instance, curvature appears in  4.1 Models for the slow part the production tensor ij. The price paid for the increased physical content of the model is that more equations must be solved. Hanjalic (1994) discusses further pros and cons in The slow term is associated with the problem of return to 휕 developing models for the transport of Reynolds stresses. isotropy. To isolate slow terms, consider the case jUi = 0. The essence of SMC modeling can be described by refer- Then there is no directional preference imposed on the turbu- ence to homogeneous turbulence. Our discussion begins lence and hence no driving force toward anisotropy. In that ℘ 휀 , 𝜹 there. Under the condition of homogeneity the exact trans- case (52) becomes ij = ij[b ]. The most commonly port equation (7) takes the form used form for slow redistribution is the Rotta model ℘slow 휀 2 ij =−C1 bij (53) 휕 u u =  + ℘ − 휀훿 (50) t i j ij ij 3 ij 훿 with bij toward 0, or of uiuj toward (2∕3)k ij. For that reason it 휀 휀 ≡ 휀 Note that is the dissipation rate of k so that (1∕2) ii. is called a return to isotropy model. Typical empirical values Indeed, the trace of (50) is two times the k equation. In of C1 are in the range 1.5–2.0. The concept that the slow term > that sense, SMC modeling can be looked on as unfolding produces a tendency toward isotropy demands that C1 1 the k − 휀 model to recover a better representation of stress (Durbin and Reif, 2010). anisotropy. The Rotta model is usually quite effective. However, the The explicit form for the production tensor is stated below Cayley–Hamilton theorem shows that the most general func- (7) to be tional dependence of the slow redistribution model is  =−u u 휕 U − u u 휕 U ( ) ij j k k i i k k j 1 ℘slow =−휀C b + 휀Cn b2 − b2 훿 (54) ij 1 ij 1 ij 3 kk ij which involves the dependent variable uiuj and the mean flow n gradients. The only new unclosed term in (50) is redistri- The coefficients C1 and C1 can be functions of the invari- ℘ 휀 2 3 bution, ij, assuming either that the -equation (15) or the ants IIb =(−1∕2)bkk, IIIb =(1∕3)bkk. Speziale et al. (1991) 휔-equation (41) is retained. invoked this form with the coefficients Modeling involves developing formulas and equations to 0.9 C = 1.7 + and Cn = 1.05 relate the unclosed term to the mean flow gradients and to the 1 휀 1 12 Turbulence Closure Models for Computational Fluid Dynamics

C1 was made a function of production over dissipation to The form (57) was introduced by Launder et al. (1975). They incorporate data measured in shear flows. derived the coefficients While there are grounds for including the Cn term, as a 1 c + 8 8c − 2 practical matter nonlinearity can adversely affect numerical C = and C = 2 11 3 11 convergence when SMC models are used in complex geome- tries. For instance, the SSG model (Speziale et al., 1991) from certain constraints and selected the empirical constant n . suffers from stiffness unless C1 is set to zero; doing so has a c = 0 4. The IP model uses negligible effect on predictions in wall bounded flow. 3 C = and C = 0 2 5 3 4.2 Models for the rapid part Speziale et al. (1991) found that data on shear flow ℘ . . The rapid part of the ij model is defined as the portion anisotropy were fit by C2 = 0 4125 and C3 = 0 2125 (the of the model that explicitly involves the tensor components seemingly large number of decimal places is due to trans- 휕 iUj. A good deal of research on SMC has focused on the forming from their variables to the present). They also added rapid pressure-strain model for homogeneous turbulence. In the term √ 1 general, the rapid contribution to (52) is represented as − C∗ b b kS (58) s 2 ij ji ij ℘rapid 휀 rapid 휕 = = Mijkl lUk (55) ij ij The relative importance of the rapid and slow redistribution The various closures amount to different prescriptions of models depends somewhat on flow conditions. The split the tensor M. Constraints on M can be derived, but there between them for the IP and SSG models is shown in is still considerable freedom in how it is selected. The most Figure 4 for plane channel flow. The specifics vary with common models assume M to be a function of the anisotropy the model, but both rapid and slow components make a tensor b. One way of devising a model is to expand in powers significant contribution throughout the flow. One should of anisotropy, treating coefficients as empirical constants; be warned that the predictions of these models are not ≲ an elaboration is to treat the coefficients as functions of the correct in the region y+ 80. The behavior of Dij near invariants of b – possibly expanding them as well, to be the wall produces quite anomalous predictions by the SSG systematic. redistribution model. Methods for correcting the near-wall When the coefficients are constants and the expansion stops behavior of quasi-homogeneous models are discussed at the linear term in b, the general linear model (GLM) is below. ℘ obtained: Closure of equation (50) is effected by replacing ij by the sum of a rapid and a slow model. Physical processes 2 ℘rapid = k(휕 U + 휕 U ) are captured largely through the production tensor, which ij 5 j i i j ( ) is exact. The redistribution formulae partially counteract 2 + kC b 휕 U + b 휕 U − 훿 b 휕 U production. For instance, in parallel shear flow, turbulent 2 ik k j jk k i 3 ij kl k l ( ) energy is produced in the streamwise component of inten- 휕 휕 2훿 휕 sity and fed to the other components by the redistribution + kC3 bik jUk + bjk iUk − ijbkl kUl (56) 3 model. Special cases of (56) are the LRR (Launder et al., 1975) Effects of imposed rotation illustrate the role of the produc- and the isotropization of production (IP), (Noat et al., 1973) tion tensor. In parallel shear flow, U1(x2), rotating about the models. x3-axis, Pij becomes The formula (56) can be rearranged into other forms that ⎛ 2u u (2Ω−휕 U )−u2휕 U 0⎞ appear in the literature. In terms of the production tensor it ⎜ 1 2 2 1 2 2 1 ⎟ becomes ⎜ + 2Ω(u2 − u2) ⎟ [ ] ( ) [P ]=⎜ 2 1 ⎟ (59) 4 4 2 ij ℘rapid = − (C + C ) kS − C  − 훿  ⎜− u2휕 U + 2Ω(u2 − u2)−4Ωu u 0⎟ ij 5 3 2 3 ij 2 ij 3 ij ⎜ 2 2 1 2 1 1 2 ⎟ ( ) ⎝ 000⎠ 2훿  − C3 Dij − ij (57) 3 Consider a positive imposed angular velocity, Ω > 0. 휕 > after defining Usually u1u2 is negative if 2U1 0. According to (59) > < P22 0 in this case, while P22 0 for negative frame D u u 휕 U u u 휕 U ij =− i k j k − j k i k rotation. This accounts for the stabilizing or destabilizing Turbulence Closure Models for Computational Fluid Dynamics 13

4.3 Equilibrium approximation 50 The concept of moving equilibrium was introduced by Rodi 0 (1976). It assumes that the anisotropy tensor (51) asymptotes at large time to a constant value – albeit uiuj and k remain SSG model time dependent. Then, by substituting equation (50) into −50 Slow

Rapid 11 휕 휕 Slow tuiuj uiuj tk Redistribution (a.u.) Redistribution 휕 Rapid 22 tbij = − = 0 (60) −100 Slow k k2 Rapid 12 Thereby, an implicit formula for the Reynolds stress as a 0 50 100 150 function of the mean velocity gradient is obtained: (a) y+ 2  − 훿 P + ℘ +(P − 휀)b = 0 (61) 50 ij 3 ij ij ij

25 It is assumed that transport equations will be solved for k and 휀. ℘ 0 A model for ij is needed to make (61) concrete. For instance, with the GLM (56) and Rotta (53) models, it IP model −25 Slow becomes Rapid 11 Slow 휀 8 Redistribution (a.u.) Redistribution −50 0 =(1 − C )b − S Rapid 22 1 ij k 15 ij Slow ( ) −75 Rapid 12 2 +(C + C − 1) b S + b S − 훿 b S 2 3 ik kj jk ki 3 ij kl lk 0 50 100 150 +(C2 − C3 − 1)(bikΩkj + bjkΩki) (62) (b) y+

This can be solved for bij to obtain the Reynolds stress Figure 4. Contributions of the rapid and slow redistribution model required by the RANS equations (6). in plane channel flow for the SSG and IP models. A closed form solution exists for equation (62), or for fairly general quasi-linear models, such as SSG. The three-dimensional solution is quite unmanageable (Gatski tendencies of rotation, or swirl. When the rotation has the and Speziale, 1993); however, the two-dimensional solution opposite sign to the shear, it tends to suppress turbulent is a nonlinear constitutive model (Section 3.7) of the form energy. (49) with The role of the redistribution tensor usually is to compen- sate unequal production rates. In the case of (59) it will (−8∕15) g C휏 = 2 1 2||2 2| |2 generate the u3 component of intensity, which is not [1 −(2∕3)]g + 2g W produced directly. Symmetry prevents u1u3 from being C휏 = gC휏 (1 − C + C ) generated in this case. 2 1 2 3 Expression (59) illustrates why SMC can be numer- C휏 =−2gC휏 (1 − C − C ) (63) 3 1 2 3 ically intransigent: the individual stress components are coupled through the production and redistribution where tensors; u u appears in the u2 and u2 equations, u2 1 2 1 2 1 1 g = u2 u u 휀 and 2 appear in the 1 2-equation, and so on. Compu- C1 − 1 + P∕ tational schemes often treat production explicitly and || | | k destruction implicitly. This requires special treatment = S (1 − C2 − C3)휀 of source terms, and, in the present case, simulta- | | |훀| k neous solution of the various components. Usually the W = (1 − C2 + C3)휀 latter is not done; hence, failing to couple the compo- nents numerically can be a source of computational In practice, the two-dimensional constitutive model (49) with stiffness. these coefficients is used in fully three-dimensional flow. The 14 Turbulence Closure Models for Computational Fluid Dynamics complete equilibrium solution requires that (49) be used to must preserve the conservation form and be symmetric in i compute P, which becomes a cubic equation for P in the case and j. of (63). The notion that the third velocity moment represents Speziale and Mac Giolla Mhuiris (1989) showed that two random convection by turbulence is invoked. The Marko- solution branches exist in rotating flow. External forces, such vian assumption, that this can be modeled by gradient as those caused by rotation, can cause one solution branch diffusion, is made in most engineering models. The to disappear at a parameter value. This is referred to as a most common closure is the Daly and Harlow (1970) bifurcation, although the use of this term is a bit unconven- formula, tional. Mathematically the bifurcation explains how SMC ( ) models respond to external forces; conceptually, it describes 2 u 휕 p 휕 u u u k k 휕 C T u u 휕 u u − k k i j − 휌훿 = k( s k l l i j) (64) a transition from growing turbulence to decaying turbulence 3 ij as a stabilizing effect is increased. However, the complete equilibrium solution is not used for predictive purposes; ≡ 휈 Here, the eddy viscosity tensor is CsukulT T . A typical either the nonlinear stress-velocity gradient relation (49), . kl value of the empirical constant is Cs = 0 22 (Launder, or the algebraic system (62) is invoked as a constitutive 1989). Near to a wall the dominant component of the model. gradient is in the wall-normal direction, y. Then (64) Some of the pros and cons of nonlinear constitutive models 휕 2휕 is approximately y(CsTv yuiuj). The dominant eddy are discussed in Apsley and Leschziner (1999). The domi- 휈 2 viscosity is T = CsT v . One influence of a wall is to nant effects arise through the dependence of C휏 on S and 1 suppress v2 relative to the other intensities. With the 훀; in other words, they can be represented by a vari- caveat that the model must be able to predict v2 correctly, able C휇 in (12). The latter is the nature of (36) or (43). (64) proves to be more accurate than a simpler model, in Other C휇 formulas, often motivated by equilibrium anal- yses such as that leading to (63), also have been proposed; which the eddy viscosity scales on the turbulent energy, 휈 = 훿 C Tk. Mellor and Yamada (1982) do so to obtain effects of density Tkl kl s stratification. The closed RST equation with (64) is 2 휕 u u + U 휕 u u = ℘ − 훿 휀 + 휕 (C T u u 휕 u u ) 4.4 Turbulent transport t i j k k i j ij 3 ij k s k l l i j u u 휕 U u u 휕 U 휈 2u u − j k k i − i k k j + ∇ i j There are two critical effects of nonhomogeneity on the (65) mathematical modeling: the turbulent transport terms in the Reynolds stress budget (7) do not vanish and the redistribu- The algebraic formulas in Sections 4.1 and 4.2 can be substi- tion models discussed above must be modified for effects of ℘ ℘rapid ℘slow tuted for ij = ij + ij . This closure can be used to boundaries. Transport terms will be discussed first, and then predict free shear flows; but near to walls the algebraic the more difficult topic of wall effects on the redistribution ℘ formulas for ij can be rather erroneous. model is covered. To skirt the problem, wall function boundary conditions The Reynolds stress budget (7) contains both turbulent and can be applied in the log-layer; in lieu of a suitable model, molecular transport on its right side. For constant density the practical computations often resort to this. The approach is relevant terms are quite similar to the method discussed in connection with ( ) the k − 휀 model. The wall function consists of adding u u to 2 u p i j 휕 u u u k 휈 2u u equation (34) by prescribing ratios u u ∕k. In 2-D a suitable − k k i j − 휌훿 + ∇ i j i j 3 ij set of experimental values is The second term, molecular transport, is closed and needs no { } ⎛ 1.06 −0.32 0 ⎞ modeling. Oftentimes modeling turbulent transport is char- uiuj ⎜ ⎟ acterized as representing u u u as a tensor function of u u . = ⎜ −0.32 0.42 0 ⎟ (66) k i j i j k ⎜ ⎟ This philosophy leads to rather complex formulas because ⎝ 000.52 ⎠ the symmetry in i, j, k should be respected. However, the , term being modeled is only symmetric in i j. The three-fold In 3-D, x1 is the flow direction and x2 is normal to the wall. symmetry is not apparent in the RST equation. Hence, there Wall function boundary conditions for uiuj are harder to is little motive to constrain the model to satisfy the hidden justify than the logarithmic specification for U. For instance, symmetry – especially when it causes a great deal of added the data of DeGraaff and Eaton (2000) show that uiuj∕k is complexity. The inviolable constraints are that the model dependent on Reynolds number. Turbulence Closure Models for Computational Fluid Dynamics 15

4.5 Near-wall modeling as y → 0. The solution to a Reynolds stress model should in principle be consistent with these. However, in practice, In an equilibrium boundary layer, the near-wall region refers it may be sufficient to ensure that −uv and v2 are small 2 ≪ to the zone from the log-layer to the wall. It includes the compared to u when y+ 1. This implements the suppres- viscous dominated region next to the surface and the strongly sion of normal transport in the immediate vicinity of the wall. ≪ inhomogeneous region above it. Thus, it is a region in which The formality y+ 1 can be taken with a grain of salt; the ≲ nonhomogeneity and viscosity play dominant roles. The powers (67) are satisfied by experimental data when y+ 5. 휀 near-wall region is one of “high impedance” to turbulent The limiting behavior of the dissipation rate tensor, ij = transport, in the sense that the wall suppresses the normal 휈 휕 휕 2 ( iuj iuk), is found to be component of turbulence. This means that the layer adjacent to the wall controls skin friction and heat transfer dispropor- 휀 휀 2 휀 11 = O(1); 22 = O(y ); 33 = O(1) tionately, making it critical to engineering applications. It is 휀 휀 휀 also of great interest to turbulence theory because it is the 12 = O(y); 13 = O(1); 23 = O(y) region of high shear and large rates of turbulence production. 휀 → The primary mathematical issues in near-wall modeling are These are derived by considerations such as 12 휈 휕 휕 boundary conditions and nonlocal wall effects on redistribu- 2 ( yu yv)=O(y), using the near-wall behavior of the 휀 tion. The issue of nonlocal influences of the wall presents fluctuation velocity cited above. Note that ij = O(uiuj∕k) → rather a challenge to an analytical model. These wall influ- as y 0. This proves to be a useful observation about ences can have pronounced effects. Various methods have near-wall scaling. been developed to represent nonlocal wall influences. The A consideration of the various contributions to the two discussed in Sections 4.5.2 and 4.5.3 are wall echo and Reynolds stress budget (7) shows that the dominant balance elliptic relaxation. near a surface is between dissipation, molecular diffusion, and the pressure term. The budget reduces simply to

4.5.1 No-slip 휈휕2 휀 y uiuj = ij It might seem that the boundary condition at a no-slip wall if i and j are not 2, and to for the Reynolds stress tensor is simply uiuj = 0. While that is correct, the power of y with which the zero value 휈휕2 휕 휕 휀 is approached is often of importance. Models should be y uiuj = ui jp + uj ip + ij designed such that their near-wall asymptotic behavior is reasonable. We will examine the consequences of the no-slip if i or j equals 2. The first recovers the limit boundary condition on the asymptotic behavior of turbulence 2휈u u u u 휀 statistics near a wall, determining the power of y with which 휀 → i j i j ij ≈ various quantities vary as y → 0. y2 k Let the no-slip wall be the plane y = 0. The no-slip condi- i, j ≠ 휀 y tion is that all components of velocity vanish: u = 0. Even if if 2; the second shows that ij is of the same order in u u 휀 k the wall is moving, all components of the turbulent velocity as i j ∕ , although not exactly equal to it. vanish, provided the wall motion is not random. If the velocity is a smooth function of y, it can be expanded in a 4.5.2 Wall echo Taylor series, 2 The elliptic nature of wall effects was recognized early in ui = ai + biy + ciy … the literature on turbulence modeling and has continued to influence thoughts about how to incorporate nonlocal where ai, bi,andci are functions of x and z. The no-slip condi- influences of boundaries (Launder et al., 1975; Durbin, 1993; tion requires that ai = 0. Thus the tangential components satisfy u = O(y), w = O(y) as y → 0. However, the conti- Manceau et al., 2001). 휕 휕 휕 In the literature on closure modeling the nonlocal effect is nuity equation, yv =− xu − zw shows that b2 = 0 and thus v = O(y2). From these limits of the fluctuating velocity, the often referred to as “pressure reflection” or “pressure echo” Reynolds stresses are found to behave like because it originates with the surface boundary condition imposed on the Poisson equation for the perturbation pres- u2 = O(y2); v2 = O(y4); w2 = O(y2) sure. The boundary condition influences the pressure interior to the fluid, which can be described as a nonlocal, kinematic 3 2 3 uv = O(y ); uw = O(y ); vw = O(y ) (67) effect. 16 Turbulence Closure Models for Computational Fluid Dynamics

The velocity–pressure gradient correlation that appears suitable manner for each homogeneous redistribution model in the redistribution term must be corrected for the wall to which it is applied. For instance, the relative magnitudes ℘ influence. In the wall echo approach, it is taken tobean of the rapid and slow contributions to ij differ between the additive correction IP and SSG models. This demands that wall echo be adapted differently in each instance. Lai and So (1990) present a wall ℘ ℘h ℘w ij = ij + ij (68) echo function for the SSG model.

The ℘h term represents one of the models developed above, ij 4.5.3 Elliptic relaxation and elliptic blending such as the GLM (57) plus the Rotta return to isotropy (53). ℘w The additive wall correction ij is often referred to as the Elliptic relaxation (Durbin, 1993) is a rather different wall echo contribution. It is modeled as a function of the approach to wall effects. It is incorporated by solving an unit wall normal n̂ and of the shortest distance to the wall, d. elliptic equation. Contact with homogeneous redistribution A simple example is the formula models, such as those described in Section 4, is made via the [ ] source term in the elliptic equation. The particular equation 휀 2 L ℘w =−Cw u u n̂ n̂ + u u n̂ n̂ − u u n̂ n̂ 훿 is of the modified Helmoltz form: ij 1 k i m m j j m m i 3 m l m l ij d [ ] ℘h 휀 ℘qh w rapid rapid 2 rapid L + bij − C ℘ n̂ n̂ + ℘ n̂ n̂ − ℘ n̂ n̂ 훿 2 2 ij ij 2 im m j jm m i ml m l ij L ∇ f − f =− ≡ − (70) 3 d ij ij k k (69) On the right side, the superscript qh acknowledges that this is proposed by Gibson and Launder (1978). Here L = k3∕2∕휀 the quasi-homogeneous model. To use this method, the RST ̂ and the ni are components of the unit wall normal vector. equation is rewritten as The factor of L∕d causes this correction to vanish far from uiuj the surface. The idea is that wall effects decay at a distance D u u + 휀 =  + ℘ + 휕 [휈 휕 u u ]+휈∇2u u on the order of the correlation scale of the turbulent eddies. t i j k ij ij k Tkl l i j i j Gibson and Launder (1978) used (69) in conjunction with (71) the IP model for ℘rapid. The model constants Cw = 0.3and 1 f is an intermediate variable, related to the redistribution Cw = 0.18 were suggested. ij 2 tensor by ℘ = kf . The solution to (70) provides the nonho- If the wall normal is the x -direction then the rapid contri- ij ij 2 mogeneous model. The turbulent kinetic energy, k, is used as bution to the 22-component of (69) is ℘ → a factor in order to enforce the correct behavior, ij 0, at 4 L a no-slip boundary (Section 4.5.1). ℘w =− Cw℘rapid 22 2 22 The anisotropic influence of the wall on Reynolds stresses 3 x2 interior to the fluid arises by imposing suitable boundary for the wall normal intensity. In shear flow parallel to conditions on the components of the uiuj − fij system. the wall, energy is redistributed from u2 into v2 so Boundary conditions are described in Section 5.1. The wall ℘rapid > ℘rapid > 22 0. With 22 0 the wall correction is nega- normal now enters only into the wall boundary condition. tive. This is consistent with the idea that blocking suppresses The length scale in (70) is prescribed by analogy to (22) as the wall normal component of intensity. [ ( ) ] However, in a flow toward the wall the velocity hasa 3∕2 휈3 1∕4 k , component V(y). On the stagnation streamline the mean L = max cL 휀 c휂 휀 (72) 휕 2 rate of strain yV will produce v and energy will be redis- ℘rapid < tributed out of this component: 22 0, so the wall correc- In fully turbulent flow it has been found that Kolmogoroff tion, stated above, is positive. It therefore has the erroneous scaling collapses near-wall data quite effectively (Laurence, effect of enhancing the normal component of intensity. Craft 2002; Manceau et al., 2001). Hence the Kolmogoroff scale et al. (1993) proposed a more complex wall echo function to is used for the lower bound in (72). The important feature is correct this fault. that L and T do not vanish at no-slip surfaces. If they vanished The formula (69) illustrates that wall corrections are then the equations would become singular. tensoral operators that act on the Reynolds stress tensor. The elliptic relaxation procedure (70) accepts a homoge- ̂ The ni dependence of these operators has to be adjusted neous redistribution model on its right side and operates to properly damp each component of uiuj. The formula on it with a Helmholtz type of Green’s function, imposing ℘w for the correction function, ij , has to be readjusted in a suitable wall conditions. The net result can be a substantial Turbulence Closure Models for Computational Fluid Dynamics 17 alteration of the near-wall behavior from that of the original contrast, eddy viscosities, as a rule, assist convergence. The redistribution model. The v2 − f and 휁 − f models (Hanjalic tendency for the flow equations to develop chaotic solutions et al., 2005) are scalar, eddy viscosity application of elliptic is overcome by eddy viscous dissipation. relaxation. It is common practice to decouple the turbulence model Elliptic blending invokes the f -equation solver from the mean flow solver. The only communication from the model might be an eddy viscosity that is passed to L2∇2f − f =−1 (73) the mean flow. The mean flow solver would then compute the Navier–Stokes equations with variable viscosity. Most Manceau (2015) uses f 2 to interpolate between the near applied CFD codes incorporate a selection of more than one wall redistribution model and the main flow. The boundary eddy viscosity scheme; isolating the model solution from the conditions are f = 0atthewallandf → 1 at infinity. For mean flow solution simplifies implementation of the various instance, the pressure strain is written as models. It is not just the plethora of models that motivates a segre- ℘ 2℘h 2 ℘w ij = f ij +(1 − f ) ij gated solution. Sometimes different algorithms are used to solve each portion. As a rule, turbulence models are solved ̂ This is a source term in RST equations. A unit vector n = more readily with implicit numerical schemes, while explicit | | ∇f ∕ ∇f replaces the wall normal, which appears in earlier schemes are sometimes preferred for the mean flow (Turner ℘w formulations of ij . and Jenions, 1993). A case in point is provided by the Spalart–Allmaras model: explicit methods are popular for compressible aerodynamic flows; the S–A eddy viscosity 5 REYNOLDS-AVERAGED transport model (Section 3.6) is also popular for aerody- COMPUTATION namics; unfortunately, there has been little success solving S–A with explicit schemes. On the other hand, the S–A Closure models are ultimately meant to be used in CFD equations are readily integrated with alternating direction codes for the purpose of predicting mean flow fields, and implicit (ADI) or other implicit methods. possibly the Reynolds stress components. The only informa- It has been argued that first order, upwind discretization tion that they can provide is these low order statistics. RANS of the convective derivative is acceptable for the turbulence computations do not simulate random eddying, although variables, even though such low accuracy usually is not quite complex mean flows are routinely calculated. acceptable for the mean flow. A rationale is that the turbu- Standard discretization methods (see Finite Difference lence equations are dominated by source and sink terms, Methods, Finite Element Methods) can be applied to turbu- so inaccuracies in the convection term are quantitatively 휀 lent transport equations, such as those for k, ,oruiuj. small. In most cases, this line of reasoning has been veri- The relevant chapters Finite Difference Methods, Finite fied. However, where production is small, or changes are Element Methods should be consulted on matters of numer- rapid, local inaccuracies exist. A common strategy is to first ical analysis. The following section is a brief mention of a compute a preliminary solution with first-order convection, few special topics that bear specifically on solving the closure then to reconverge it with higher order accuracy. In the same models. vein, an eddy viscosity solution is sometimes used to initiate a second moment computation. 5.1 Numerical issues A general rule of thumb is to make dissipation implicit and production explicit. More exactly, treat these source terms so Most practical engineering computations are done with as to improve diagonal dominance. This rule of thumb gener- some variety of eddy viscosity formulation (Section 3). alizes to the evolution equation for any variable 휙, where 휙

Second moment closures (Section 4) promise greater fidelity would be k in the k-equation, or uiuj in a Reynolds stress to turbulence physics (Hanjalic, 1994); however, the compu- equation. In order to distinguish the implicit and explicit tational difficulties they present are manifold. The absence parts, the source term is arranged into the form A − B휙 where of numerically stabilizing, eddy viscous terms, in the mean A, B ≥ 0; that is, the evolution equation is put into the form flow equations, the strong coupling between Reynolds stress 휙 휙 components via production and redistribution, the increased Dt = A − B +··· number of equations, and other computationally adverse properties lead to slow, tenuous convergence. Special where A and B can be functions of the dependent variables. methods that overcome this have been explored; the work of The rule of thumb for treating dissipation and production is Leschziner and Lien (2002) provides many suggestions. By implemented by updating the source term as An − Bn휙n+1. 18 Turbulence Closure Models for Computational Fluid Dynamics

However, the splitting between the explicit contribution A Boundary conditions can be problematic when the k − 휀 and the implicit contribution B휙 is not unique. As an example model is integrated to a no-slip wall. of the leeway, consider the right side of the k-equation:  − 휀 [ ]  휀  k(y) might be rewritten as −( ∕k)k for which A = and B = 휀(0)=2휈 lim (75) 휀∕k. The update is then y→0 y2

| n | is usually invoked, per (20). A ratio evaluated at the first n |휀 | n  − | | k +1 |kn | computational point from the wall is substituted for the limit on the left side. The factor of y−2 can cause stiffness if the absolute value ensures that B ≥ 0. In this same vein, this condition is not implemented implicitly. (If the k and 휀 Spalart and Allmaras (1992) recommend treating the sum of equations are coupled, (19) can be used to avoid the y−2.) production and dissipation implicitly or explicitly depending The singular boundary condition to the k − 휔 model on the sign of its Jacobian. After rewriting P휈 − 휀휈 in (44) (Section 3.5) is exactly in the form (JP − J휀)̃휈, they then construct the [ ] 6휈 source as 휔(0) = lim (76) y→ 2 0 C휔2y n n n (P휈 − 휀휈) − pos[(J휀 − JP) ]Δ̃휈 − pos[휕휈(J휀 − JP)휈 ]Δ̃휈 Wilcox (1998) suggests specifying this as the solution for where pos(x)=(x + |x|)∕2andΔ̃휈 = ̃휈n+1 − ̃휈n. the first few grid points; Menter (1992) recommends using 휔 휈 2 Leschziner and Lien (2002) discuss the treatment of source = 10[6 ∕(C휔2y1)] as the boundary value. The factor of 10 terms in SMCs. Numerical stiffness is often encountered is arbitrary, but Menter states that the results are not sensitive to the precise value. when solving transport models for uiuj. The root cause is the functional coupling between the Reynolds stress components Modifications to turbulence models that are made for introduced by the production and redistribution tensors. For numerical reasons can result in inconsistent predictions. example, the evolution equation for uv contains v2휕 U in the Rumsey (2009) has created a website with solutions for veri- y fying consistent implementation of several popular models. 2 휕 production term, and the equation for v contains uv yU in 2 the redistribution term. When uv and v are solved sepa- 5.2 The assessment of models rately – often called a segregated solution – justice is not done to this intimate coupling. On occasion it has been A good deal of literature deals with assessment of the predic- proposed to solve the full set of components simultane- tive accuracy of models. These endeavors tend to be anec- ously, as a coupled system (Laurence, 2002). However, no dotal and rarely are models dismissed unequivocally. Never- general scheme has been offered for robustly solving RST theless, some remarks will be made on the relative merits of equations. different models. The Reynolds stress appears as a body force if it is treated A large number of variants on basic formulations described explicitly in the mean flow equation (6). But the Reynolds herein can be found. To an extent, that is a reflection of stress is inherently a diffusive term that should provide the degrees of freedom introduced by empirical content. At numerical stability. To recover the diffusive property, eddy times the number of variations on a theme can be over- diffusion can be subtracted from both sides of the mean flow whelming. But that should not distract from the fact that the equation, treating it implicitly on one side and explicitly on number of basic formulations is few: eddy viscosity trans- the other, as in port, two-equation, Reynolds stress transport (SMC). The second two categories include another equation to determine 휕 휕 휕 휈 휕 휕 (n+1) 1휕 tUi + Uj jUi − j[ T ( jUi + iUj)] =−휌 iP a turbulent time scale, via either 휔 or 휀. The equations are of convection–diffusion form, with production and dissipa- 휈 2 휕 (n) 휕 휈 휕 휕 (n) + ∇ Ui − jujui − j[ T ( jUi + iUj)] (74) tion appearing as source terms, and in the case of SMC, as a redistribution term linking the components (Section 4). The procedure (74) will be the most effective if Near-wall modeling can play a disproportionate role because that is a region of reduced turbulent mixing, 2 u u − k훿 + 휈 (휕 U + 휕 U ) and hence it can be rate limiting. It adds further degrees i j 3 ij T j i i j of freedom; nevertheless models again can be classified is made small. In practice, simply using the k − 휀 formula into a few varieties – wall functions, damping functions, 휈 (11) for T in equation (74) adds greatly to the ease of two-layers, elliptic relaxation – and the pros and cons convergence. assessed for each class. Turbulence Closure Models for Computational Fluid Dynamics 19

Factors that tend to distinguish the models are their ability 12 1 2 to predict flow separation, surface heat transfer, or the influ- ) ence of surface curvature and system rotation. Bardina et al. − 1 8 (1997) compared two-equation and eddy viscosity trans- 4 port models, largely on the basis of their ability to predict separation. They favored the SST (equation 36, ff.) and 0 휔 S–A (Section 3.6) models over k − and low Reynolds −4 number k − 휀. S–A was preferred on numerical grounds Axial velocity (m s −8 and because of its satisfactory predictive accuracy. In this −0.1 0 0.1−0.1 0 0.1 study and others, k − 휀 with wall functions was found unre- (a) Radius (m) (b) Radius (m) liable for predicting flow separation; its tendency isto Experiment 3 predict attached flow beyond the point at which it should SMC separate. Scalar model U Iacovides and Raisee (1999) have concluded that heat 1 transfer prediction argues for integration to the wall, rather than resorting to wall functions (Section 3.4). Again, low 2 Reynolds number k − 휀 (equation (25)) was not recom- 3 mended. In this study, RST was favored because it responds to curvature in a natural way. It is safe to say that when −0.1 0 0.1 such effects significantly influence the turbulent intensity, (c) Radius (m) (d) RST formulations are more reliable than scalar formulations. For instance, suppression of turbulent energy in a strongly Figure 5. Flow in a cyclone swirler illustrates the failure of scalar swirling flow can be captured, as illustrated by Figure 5.In models in a flow that is stabilized by swirl. (Reproduced with this geometry, the flow enters at the top, as shown inthe permission from G. Iaccarino. © G. Iaccarino, 2004.) figure, then swirls round toward the closed lower end.The stabilizing centrifugal acceleration suppresses turbulence in the vicinity of r = 0 and permits an upward jet along the axis, provides a more generic, and often more accurate, approach as shown by the experimental data and the second moment but adds expense; suffice it to say that near-wall treatments closure model (Section 4). for SMC are too insecure to warrant a recommendation on Scalar models (Section 3) are insensitive to centrifugal which is most effective. In practice, it is common to use stabilization. In the scalar model case of Figure 5, the fluid wall functions (equation 66), or a variant of the two-layer stays turbulent near the axis and the flow is downward at the approach (Section 3.3). center, with a slow upward flow next to the walls – quite Ooi et al. (2002) found that heat transfer predictions are differently from the experiment. It should be mentioned, strongly dependent on the eddy viscosity distribution very 휈 however, that scalar models can be modified to incorporate near to the surface. In a flat plate boundary layer, T becomes 휈 rotational stabilization (Durbin, 2011). Such approaches are equal to when y+ ≈ 10. If a model predicts that the eddy less reliable than SMC and are used for simplification. viscosity remains higher than molecular viscosity signifi- However, when influences such as swirl or streamline cantly below this, heat transfer will be overpredicted and curvature have a secondary effect, scalar models can be as vice versa.InOoiet al. (2002), both the S–A model and good as, or better than, SMC: swirling flow in a pipe expan- the two-layer k − 휀 model (Section 3.3) severely underpre- sion can be predicted adequately by a scalar model because dicted heat transfer in a ribbed, square duct flow. This could 휈 centrifugal and pressure gradient effects on the mean flow be traced to underprediction of T adjacent to the wall. The are captured by the averaged Navier–Stokes equation, and v2 − f model was found to be more accurate. The ribs in this the direct swirl effect on the turbulence is secondary. geometry cause complex secondary flows and large separa- A wall echo of the form (69) was invoked by Iacovides and tion bubbles. One might hope that the principle here is that Raisee (1999), but with some additional terms and factors. models using length scales that are prescribed functions of Because of the latter, evaluations of this type are of limited wall distance are less flexible than those based on local vari- guidance; those that adhere to widely used forms of the ables. S–A and two-layer models are of the former ilk, while considered models are more suited to advisory tasks. Iaco- v2 − f is of the latter; however, the evidence in favor of the vides and Raisee (1999) concluded that some such form of aforementioned principle is not compelling. wall treatment is preferred to wall functions, but that the Apsley and Leschziner (1999) report on a collaborative formulation in equation (69) is not reliable. Elliptic blending assessment of closure schemes. They consider the gamut 20 Turbulence Closure Models for Computational Fluid Dynamics

from quasilinear eddy viscosity, to nonlinear constitutive instance of modified C휇, was a great improvement over equations (equation 49), to full RST. Their study inferred the native k − 휔 formulation (35). In that case, response little value from nonlinear constitutive formulations, finding to pressure gradients is made more accurate by (in effect) that the linear constitutive relation (12), modified by adding making C휇 depend on the rate of strain. Leading order effects strain-rate dependence to C휇 in (11), constitutes a more of curvature and system rotation can be incorporated by effective formulation. In particular the SST model, as an variable C휇 as well.

1 1

0 0 z / h z / h

−1 −1 0 2 0 2 (a) x/h t = 1/4T (b) t = 1/2T x/h

1 1

0 0 z / h z / h

−1 −1 0 2 0 2 (c) x/h t = 3/4T (d) t = T x/h

Figure 6. Skin friction lines in periodic flow round a surface mounted cube. (Reproduced with permission from Iaccarino et al., 2002. © Elsevier, 2002.)

x/H = 4.33 x/H = 4.56

(a) (b)

x/H = 5.33 x/H = 7.00

(c) (d)

Figure 7. Velocity profiles downstream of the cube, showing the difference between········· steady( ) and time-averaged unsteady (–––) computations. (Reproduced with permission from Iaccarino et al., 2002. © Elsevier, 2002.) Turbulence Closure Models for Computational Fluid Dynamics 21

Generally, Apsley and Leschziner (1999) found k − 휀 Baldwin BS and Barth TJ. A one-equation turbulence transport formulations to be unsatisfactory. There is an emerging senti- model for high Reynolds number wall-bounded flows. NASA ment that, among two-equation formulations, k − 휔 variants TM-102847, 1990. tend to be more satisfactory than k − 휀; but the need for fixes Bardina JE, Huang PG and Coakley TJ. Turbulence modeling valida- of the large strain anomaly and the free-stream sensitivity, tion. AIAA Paper 97–2121, 1997. discussed in Section 3.5, should not be overlooked. Also, the Barre S, Bonnet J-P, Gatski TB and Sandham ND. Compressible, 휔 high speed flow. In Closure Strategies for Turbulent and Tran- singular behavior of as a wall is approached can impose sitional Flows, Launder BE and Sandham ND (eds). Cambridge stringent grid requirements. University Press: Cambridge, 2002; 522–581. Turbulence models represent averaged effects of random Chen C-J and Patel VC. Near-wall turbulence models for complex eddying. This is the ensemble average, defined by flows including separation. AIAA J. 1998; 26:641–648. equation (1). Statistical averaging is not synonymous Craft T, Graham L and Launder BE. Impinging jet studies for with time-averaging; Nevertheless, there is frequently a turbulence model assessment-II: An examination of the perfor- confusion between ensemble and time-averaging. For this mance of four turbulence models. Int. J. Heat Mass Transfer 1993; reason, it has become usual to refer to unsteady RANS 36:2685–2697. when time-dependent flow is computed. A very intriguing Craft T, Iacovides H and Yoon JH. Progress in the use of application of unsteady RANS (a.k.a. URANS) is to bluff non-linear two-equation models in the computation of convec- body flows with periodic vortex shedding. In this casethe tive heat-transfer in impinging and separated flows. Flow, Turbul. Combust. 1999; 63:59–80. turbulence model represents the broad-band component of the frequency spectrum of the velocity field. The shedding Daly BJ and Harlow FH. Transport equations of turbulence. Phys. Fluids 1970; 13:2634–2649. frequency and its harmonics are spikes in the spectrum that DeGraaff DB and Eaton J. Reynolds number scaling of the flat plate deterministic are eddies, such as the vonKarman vortex turbulent boundary layer. J. Fluid Mech. 2000; 422:319–386. street. These are part of the mean flow and hence must be Durbin PA. Near-wall turbulence closure modeling without ‘damping computed explicitly. Often a steady solution can be obtained functions’. Theor. Comput. Fluid Dyn. 1991; 3:1–13. by imposing a symmetry on the calculation. However, Durbin PA. A Reynolds stress model for near-wall turbulence. the steady solution generally underpredicts mixing and J. Fluid Mech. 1993; 249:465–498. overpredicts the length of wakes. Errors associated with Durbin PA. Review: adapting scalar turbulence closure models for steady calculation of unsteady flows are mistakes in the complex turbulent flows. J. Fluids Eng. 2011; 133:061205-1–8. mean flow, and should not be attributed to the turbulence Durbin PA and Pettersson Reif BA. Statistical Theory and Modeling model. for Turbulent Flow (2nd edn). John Wiley & Sons, Ltd: Chichester, Figure 6 shows skin friction lines in periodic flow round a 2010. surface mounted cube (from Iaccarino et al., 2002). The flow Gatski TB and Speziale CG. On explicit algebraic stress models for develops periodic unsteadiness in the immediate lee of the complex turbulent flows. J. Fluid Mech. 1993; 254:59–78. cube. The spiral nodes are a footprint of vortices that arch Gibson MM and Launder BE. Ground effects on pressure fluctu- over the cube. The bands near the top and bottom of each ations in the atmospheric boundary layer. J. Fluid Mech. 1978; panel are caused by vortices that wrap around the front of 86:491–511. the cube. Goodbody AM. Cartesian Tensors. Ellis Horwood: Chichester, 1982. The comparison in Figure 7 illustrates how a steady Hanjalic K. Advanced turbulence closure models: a view of current computation of this flow is inaccurate, while a time average status and future prospects. Int. J. Heat Fluid Flow 1994; of an unsteady computation agrees nicely with data. The 15:178–203. steady calculation shows a wake extending too far down- Hanjalic K, Popovac M and Hadziabdic M. A robust near-wall elliptic stream. It omits the component of mixing produced by the relaxation eddy-viscosity turbulence model for CFD. Int. J. Heat Fluid Flow 2005; 25:1047–1051. unsteady mean flow vortices. Comparison to the unsteady RANS simulation gives an idea of the contribution of mean Iaccarino G, Ooi A, Durbin PA and Behnia M. Reynolds averaged simulation of unsteady separated flow. Int. J. Heat Fluid Flow flow vortices to mixing. 2002; 24:147–156. Iacovides H and Raisee M. Recent progress in the computation of flow and heat transfer in internal cooling passages of turbine blades. Int. J. Heat Fluid Flow 1999; 20:320–328. REFERENCES Jones WP and Launder BE. The prediction of laminarization with a two-equation model. Int. J. Heat Mass Transfer 1972; 15:301–314. Apsley DD and Leschziner MA. Advanced turbulence modelling Kalitzen G, Medic G, Iaccarino G and Durbin PA. Near-wall behavior of separated flow in a diffuser. Flow, Turbul. Combust. 1999; of RANS turbulence models and implications for wall functions. J. 63:81–112. Comput. Phys. 2005; 204:265–291. 22 Turbulence Closure Models for Computational Fluid Dynamics

Kok J. Resolving the dependence on freestream values for the k − 휔 Parneix S, Durbin PA and Behnia M. Computation of 3-D turbulent turbulence model. AIAA J. 2000; 38:1292–1295. boundary layers using the v2 − f model. Flow, Turbul. Combust. Lai YG and So RMC. On near-wall turbulent flow modelling. AIAA 1998; 10:19–46. J. 1990; 34:2291–2298. Patel VC, Rodi W and Scheurer G. Turbulence modeling for Launder BE. Second-moment closure: present … and future. Int. J. near-wall and low Reynolds number flows: a review. AIAA J. 1984; Heat Fluid Flow 1989; 10:282–300. 23:1308–1319. Launder BE and Kato M. Modelling flow-induced oscillations Pope SB. A more general effective-viscosity hypothesis. J. Fluid in turbulent flow around a square cylinder. ASME FED 1993; Mech. 1975; 72:331–340. 157:189–199. Pope SB. Turbulent Flows. Cambridge University Press: Cambridge, Launder BE and Sharma BL. Application of the energy-dissipation 2000. model of turbulence to the calculation of flow near a spinning disk. Rodi W. A new algebraic relation for calculating the Reynolds Lett. Heat Mass Transfer 1974; 1:131–138. stresses. Z. Angew. Math. Mech. 1976; 56:T219–T221. Launder BE and Spalding B. The numerical computation of turbulent Rodi W. Experience using two-layer models combining the k − 휀 flows. Comput. Methods Appl. Mech. Eng. 1974; 3:269–289. model with a one-equation model near the wall. AIAA Paper Launder BE, Reece GJ and Rodi W. Progress in the development 91–0609, 1991. of a Reynolds stress turbulence closure. J. Fluid Mech. 1975; Rumsey C. Consistency, verification and validation of turbu- 68:537–566. lence models for Reynolds-averaged Native-Stokes applica- Laurence D. Applications of RANS Equations to Engineering Prob- tions. 3rd European Conference for Aerospace Sciences Paper lems, Von Karman Institute Lecture Series: Introduction to Turbu- –EUCASS2009-7, 2009. lence Modelling, VKI-LS 1997-03, Rhode Saint Genese, Belgium: Spalart PR and Allmaras SR. A one-equation turbulence model for Von Karman Institute for Fluid Dynamics, 2002. aerodynamic flows. AIAA Paper 92–0439, 1992. Leschziner MA and Lien FS. Numerical aspects of applying Speziale CG and Mac Giolla Mhuiris N. On the prediction of equi- second-moment closure to complex flow. In Closure Strategies librium states in homogeneous turbulence. J. Fluid Mech. 1989; for Turbulent and Transitional Flows, Launder BE and Sandham 209:591–615. ND (eds). Cambridge University Press: Cambridge, 2002; Speziale CG, Sarkar S and Gatski TB. Modeling the pressure-strain 153–187. correlation of turbulence: an invariant dynamical systems Manceau R. Recent progress in development of the elliptic blending approach. J. Fluid Mech. 1991; 227:245–272. Reynolds-stress model. Int. J. Heat Fluid Flow 2015; 51:195–220. Taylor GI. Diffusion by continuous movements. Proc. London Math. Manceau R, Wang M and Laurence D. Inhomogeneity and Soc. 1921; 20:196–212. anisotropy effects on the redistribution term in Reynolds-averaged Townsend AA. The Structure of Turbulent Shear Flow. Cambridge Navier–Stokes modelling. J. Fluid Mech. 2001; 438:307–338. University Press: Cambridge, 1976. Mellor GL and Yamada T. Development of a turbulence closure Turner MG and Jenions IK. An investigation of turbulence modeling model for geophysical fluid problems. Rev. Geophys. Space Phys. in transonic fans including a novel implementation of an implicit 1982; 20:851–875. k − 휀 turbulence model. ASME J. Turbomach. 1993; 115:248–260. Menter FR. Two-equation eddy-viscosity turbulence models for engi- Vieser W, Esch T and Menter FR. Heat transfer predictions using neering applications. AIAA J. 1992; 32:1598–1605. advanced two-equation turbulence models. Technical Memo- Moser RD, Kim J and Mansour NN. Direct numerical simulation randum CFX-VAL10/0602, 2002. of turbulent channel flow up to Re휏 = 590. Phys. Fluids 1999; White FM. Viscous Fluid Flow (2nd edn). McGraw-Hill: New York, 11:943–945. 1991. Noat D, Shavit A and Wolfshtein M. Two-point correlation model Wilcox DC. Turbulence Modeling for CFD (2nd edn). DCW Indus- and the redistribution of Reynolds stresses. Phys. Fluids 1973; tries, Inc.: La Canada CA, 1998. 16:738–743. Wolfshtein M. The velocity and temperature distribution in Ooi A, Iaccarino G, Durbin PA and Behnia M. Reynolds averaged one-dimensional flow with turbulence augmentation and pressure simulation of flow and heat transfer in ribbed ducts. Int. J. Heat gradient. Int. J. Heat Mass Transfer 1969; 12:301–318. Fluid Flow 2002; 23:750–757.