Some Developments in Turbulence Modeling of Environmental Flows

Transcription

1 Some Developments in Turbulence Modeling of Environmental Flows Kemal Hanjalić and Saša Kenjereš Department of Multi-scale Physics, Delft University of Technology, Lorentzweg 1, Delft, 2628 CJ The Netherlands ABSTRACT: The article presents some recent efforts in the authors group toward accommodating Reynolds-averaged Navier-Stokes (RANS) modeling to computing complex high-reynolds and high Rayleigh number industrial and environmental flows. Considered are some recent developments both in the second-moment and in the eddy-viscosity framework. The treatment of the wall boundary conditions, which has long been the stumbling block in the RANS computational fluid dynamics (CFD), is discussed with focus on some new concepts for robust integration up to the wall (ItW), generalized wall functions (GWF), and a compound wall treatment (CWT) that combines the both concepts. Arguing that LES will not for long be applicable for large-scale industrial computations, we consider the potential of Transient RANS for flows subjected to strong forcing, as well as combing RANS with LES. Several noted controversial issues in hybrid RANS/LES are discussed, supported with some illustrations. Various concepts and improvements are illustrated in several examples of wind- and buoyancy driven environmental flows. KEYWORDS: RANS, Hybrid RANS/LES, Wall Functions, Compound Wall Treatment. 1 INTRODUCTION Computational Fluid Dynamics (CFD) for Wind and Environmental Engineering continues to rely on the conventional Reynolds-averaged Navier-Stokes (RANS) methods. Their simplicity, robustness and computational economy have not yet been seriously challenged by any other turbulence simulation methods, and it is likely that RANS methods, in the present or improved form (or in a combination with e.g. large-eddy simulations, LES), will continue to dominate the industrial and environmental CFD for some time to come. However, the industrial CFD tends to use oversimplified turbulence models, which are incapable of reproducing some key phenomena encountered in wind and environmental engineering such as flow impingement, bluff body wake, unsteady and three-dimensional boundary layer separation, effects of buoyancy, and others. Accurate predictions of the impingement phenomenon and of the locations at which the flow separates and possibly reattaches, is the prerequisite for computing the wind load on buildings. The same can be said for the effects of thermal and mass buoyancy which often dominate flow and pollutant transport in the environment. Validation of turbulence models and various modifications in such (and a number of other) generic flows prior to their application to real-life cases have shown that most of the popular models incorporated in the industrial CFD codes perform poorly, raising the concern over the models applicability to real-life complex flows. On the other hand, a number of relatively simple model modifications, proven to yield significant improvements in generic flows, do not seem to appeal to CFD vendors, nor to the CFD users. Instead, much hope is placed in LES, which have been gaining in popularity and have been regarded by many as the future industrial tool for computing buildings and terrain aerodynamics. LES certainly possess many desirable features and, in principle, it is superior to RANS. It requires less empiricism and provides information about (large-scale) turbulence spectrum. However, this is still a very expensive technique and in the near future not feasi-

2 ble for computing flow over real complex objects at realistic Reynolds (Re) numbers. LES is, however, a very useful tool for studying flow physics, vortical structures and turbulence, albeit in simplified geometries and at lower Re numbers. An important application of LES is to provide the input (unsteady velocity and pressure field) for computing the aerodynamic noise. In anticipation that the advancement in the computer design and further developments in the LES technique will make this approach more and more attractive for industrial computations, LES deserves to be seriously considered as a complementary tool for studying wind engineering. We begin with a brief discussion of some developments in RANS modeling for complex flows. In focus are the eddy-viscosity models, but in view of some recent progress in numerical treatment of advanced RANS models, we also consider some new concepts in the secondmoment closures. We turn then to some new developments in the treatment of the wall boundary conditions that include the integration to the wall (ItW), generalized wall functions (GWF), and a unified, compound wall treatment that combines the both concepts. Next, we consider some improvements in modeling buoyancy effects. The last sections are focused on accommodating and sensitizing RANS to capture some parts of turbulence spectrum. Considered are the transient RANS (T-RANS) which makes possible to capture large-scale vortical structures such as found in flows dominated by thermal convection, as well as blending of RANS and LES into a unified technique that combines the advantage of both methods. The novelties here discussed are illustrated in several generic flows, as well as in several examples of environmental flows. 2 A PERSPECTIVE ON RANS AND LES The one-point turbulence closures for RANS equations have served for over three decades as the mainstay of the industrial CFD. But, the most popular and most widely used linear eddy-viscosity models (LEVM) have serious fundamental deficiencies and cannot be trusted for predicting genuinely new situations of realistic complexity. Various modifications and new modeling concepts have been proposed, ranging from ad hoc remedies, complex non-linear eddy-viscosity approaches (NLEVM) to multi-equation and multi-scale second-moment closures (Reynolds stress/flux, algebraic or differential models, ASM, DSM). In search for better physical justification and expanding the range of model applicability, current research, primarily in academia, seems to be departing from the traditional RANS strategies. Among such new developments we can identify the following: Unsteady RANS (U-RANS) implying time-solution of the conventional RANS for 3D unsteady problems, with or without special treatment of flow unsteadiness Multi-scale RANS (one-point and spectral closures) Transient RANS (based on conditional or ensemble averaging of NS equations with possibly modified RANS model for the subscale (unresolved) motion VLES and hybrid RANS/LES based on partially averaged NS equations, with zonal, seamless or other coupling of the two methods. It is noted that in most of the approaches one deals with the same form of the continuity, momentum and energy equations, of course with different meaning of the variables: D Ui 1 P U i = Fi + ν τij Dt ρ xi x j x j D T q T = + α τ θ j Dt ρcp x j x j (1) (2)

3 where D/Dt= / t+uj / xj stands for the material derivative and <> denotes Reynolds (time or ensemble) averaged quantities in RANS, and filtered quantities in LES, τ ij = uiu j is the turbulent stress and τ θ j is the turbulent scalar flux, either for the whole turbulence spectrum (RANS) or its parts (multi-scale RANS), or for the unresolved motion (VLES, LES), which in all approaches need to be modeled. The identical forms of equation (1) for RANS, VLES and LES make it convenient not only to use the same computational code and similar numerics, but also to combine the two approaches in a hybrid procedure. The present trend in development of industrial models is characterized by the desire to capture some elements of the turbulence spectrum, i.e. to resolve in time and space parts primarily at large scales- of the unsteady turbulent motion. In most methods the focus is on large, dominating eddy structure, (beyond e.g. vortex shedding that can be captured even with the conventional U-RANS) that preserves some coherence and determinism even if the flow is not separated and is steady in the mean. However, because such approaches require a considerable portion of the turbulence spectrum to be modeled (much larger than the conventional subgrid-scale motion in LES), the modeling remains an important issue, which draws to a large extent to the RANS experience and makes use of the RANS rationale. This in turn brings new demands and new constraints on RANS models, providing new incentive for their research. Apart from the developments where RANS models in their original or modified forms will take a new role as subscale models (in contrast to subgrid-scale in LES), the fact remains that despite disappointments, we have seen no decline in the use of the conventional RANS models among industry. It is conjectured (Hanjalić [1]) that this trend will remain for a foreseeable future, more or less in line with the increase in the computing power. In other words, it is to expect that the increase in the computer power will in the near future be used for RANS, aimed at improving spatial resolution and better numerical accuracy by using larger and better designed numerical meshes and more accurate representation of geometry and its boundaries, as well as using more sophisticated models of turbulence and other phenomena. We can also expect more use of U-RANS for 3D computations of complete bluff bodies to capture better unsteady separation effects. Also, visualization and animation, which usually requires large computing power, will be more and more in use as a tool for identifying some global or local flow features that can help in improving design. More discussion on possible future utilization of the increased computing power can be found in Hanjalić [1]. 3 THE RANS MODELS: A BRIEF OVERVIEW OF THE STATUS Most RANS turbulence models can be classified into two major classes. The most widespread are the Eddy-viscosity models (EVM) where the turbulent stress tensor u i u j (we use now the overbar to denote Reynolds averaged quantities) is expressed in terms of the mean rate of strain S ij =0.5( U i / x j + U i / x j ) and (in some models) of the mean vorticity Ω ij =0.5( U i / x j - U i / x j ), 1 uiu j ukuk δij = f ( νt, Sij, Ω ij ) (3) 3 where the kinematic eddy viscosity ν t is usually defined in term of two or more turbulence properties for which separate transport equations are solved. The most popular are the k-ε models with eddy viscosity defined as ν 2 t = Cµ k / ε, where k = 0.5u k uk is the turbulence kinetic energy and ε is its dissipation rate, obtained from the solution of the model transport equations Dk k k k = + U j = ( ν + ν t ) + P + G ε (4) Dt t x j xk xk Dε ε ε ε Cε1P + Cε 3G Cε 2ε = + U j = ( ν + ν t ) + + Y Dt t x j xk xk τ (5)

4 where P and G denote production of k by the mean rate of strain and body force respectively, and τ is the turbulence time scale, defined usually as k/ε for high Re-number turbulence. Other popular class are the k-ω models, with ω=ε/k. The EVMs can further be divided into linear eddy-viscosity models (LEVM), in which the turbulent stress is linearly proportional to the mean rate of strain, i.e. 2 uu i j kδij = 2ν tsij (6) 3 and non-linear (NLEVM) where the stress-strain relation (3) takes a form of a non-linear series expansion in terms of S ij and Ω ij, and ν t takes one of the above definitions, depending on the closure framework. The expansion is often truncated intuitively guided by pragmatism, though recently a more general tensor-representation theorem is often applied, but leading inevitably to a number of additional empirical coefficients. Some of the coefficients can be determined a priori by imposing physical constrains such as coordinate invariance, realizability and material frame indifference in the two-dimensional limit, but additional tuning is unavoidable. Recognizing the limitation of two-equation models, which can provide only one turbulence time or length scale by which to model different turbulent interactions, a third scalar variable has also been introduced in some EVMs. Example of such three-equation models involve the transport equation for the modulus of the mean rate of strain S = 2S ij S ij, or for one of the invariants of the turbulent stress tensor, all in addition to the k and ε equations (Craft et al.[2]). Other concepts have also been proposed, some focusing primarily on improving the near-wall modeling. Arguing that the k-ω model is more convenient and more robust in the near-wall region when integration to the wall is needed, whereas the k-ε model behaves better in free flow regions away from a solid wall, Menter [3] ( SST model ) proposed a blending of the two models using a set of empirical blending functions. A different approach is followed in the υ -f 2 model of Durbin [4] (known also as v2-f model), where the eddy viscosity is defined by D νt = Cµ υ 2 k / ε and in which, in addition to k and ε, a transport equation is solved for another 2 scalar υ (which reduces to the wall-normal turbulent stress component very close to a solid wall), and for an elliptic relaxation function f. The other large model class are the Differential Second-moment Models (DSM), known also as Reynolds-stress models (RSM), in which a modelled ("model") differential transport equation D uiu j / Dt is solved for each component of uiu j Du u U U Dt x x i j j D i ij fiu j f ju = + i uiuk + u ju k + φij ε ij k k Different DSMs have been proposed in the literature, differing mainly in the treatment (linear or nonlinear) of the pressure-strain term, and in the near-wall and viscous modifications. A variety of truncated versions of DSM have also been proposed and tested, known as Algebraic Stress Models (ASMs). These models have been derived by truncating the differential transport equations for uiu j to an algebraic form (thus resembling the NEVMs), aiming at providing a compromise between the simplicity of the algebraic stress-strain formulation and sounder physics inherent in the DSMs. The new generation of advanced DSM closure models (and, to some extent, their truncated algebraic- and non-linear eddy-viscosity variants) offer much better prospects for ensuring the required accuracy for complex flows than the models found in industrial computer codes. The main reason for their expected superiority is the exact treatment of the source terms that physically maintain the turbulence, as well as effects of body forces. In addition, the DSM closure reproduces the evolution of each stress component which gives better prospects for capturing phe- (7)

5 nomena governed by the anisotropy of the stress field, such as streamline curvature, effect of strong pressure gradient, three-dimensionality, as well as for modeling the scale-determining equation, (see e.g. Hanjalic, [5], Hanjalic and Jakirlic [6]). Despite obvious advantages, the DSMs have not yet been widely accepted by industry. The reason is the increased demand on computer resources (more differential equations to be solved), the still large uncertainty in modeling some of the terms in the Re-stress equations (which are absent from eddy viscosity models), and in the evaluation of the unavoidable additional empirical coefficients. But the major deterrent is the stiffness of the equations in the SMC model and numerical problems that may arise, which all require more skill and experience in carrying out the computations. Some simpler remedies, guided by the SMC modeling, but allowing to retain the simplicity and robustness of simple EVMs, are, however worth mentioning. Such are e.g. the constraints (applicable, in principle to all EVMs) to limit the excessive turbulence energy production (a noted weakness of the LEVMs) by imposing the upper time scale bound (Durbin [7]), or directly as in the linear production model (Guimet and Laurence [8]), or the limiter to the growth of the length scale (for some options see, Hanjalić [5]). An interesting approach in this spirit is the Hybrid EVM-SMC model of Basara and Jakirlić [9]. Some of these models are discussed below. Second-moment closures have also served as inspiration for a number of improvements of lowerorder models. Algebraic stress and flux models based on their differential parent equations in implicit or explicit forms - have been found to perform generally better than the non-linear eddy viscosity models that were derived independently, e.g. Wallin and Johansson [10]. The elliptic relaxation EVM of Durbin [4] was also inspired and derived with resource to the model differential stress equation (7). In view of the above discussion, it is fair to say that the DSM models are witnessing their revival and that we shall see in the near future more extensive use of advanced RANS models applied to complex flows. We consider now briefly some recent advancement, aimed at robust application of advanced models to complex flows. Recently reported novelties are too numerous for this brief coverage and we will restrict the discussion to only a few developments originating from the author s group and some collaborate research with industry. 4 SOME RECENT DEVELOPMENTS IN RANS 4.1 Robust Elliptic Relaxation EVMs. The υ 2 -f model of Durbin [4] appeared as an interesting novelty in engineering turbulence modeling, especially for flow regions adjacent to solid walls. By introducing an additional ("wallnormal") velocity scale υ 2 and an additional elliptic relaxation concept to sensitize υ 2 to the inviscid wall blocking effect, the model dispenses with the conventional practice of introducing empirical damping functions. Because of its physical rationale and of its simplicity, it is gaining in popularity and appeal especially among industrial users. Whilst in complex three-dimensional flows, with strong secondary circulation, rotation and swirl, where the evolution of the complete stress field may be essential for proper reproduction of flow features the model remains still inferior to the full second-moment and advanced non-linear eddy viscosity models, it is certainly a much better option than the conventional near-wall k-ε, k-ω and similar models. However, the original υ 2 -f model possess some features that impair its computational efficiency. The main problem is with the wall boundary condition fw 20 υ ν /( ε y ) when y 0, which makes the computations sensitive to the near-wall grid clustering and - contrary to most other near-wall models - does not tolerate too small y + for the first near-wall grid point. The problem can be obviated by solving simultaneously the υ 2 and f equation, but most commercial as well as in-house codes use more convenient segregated solvers. Alternative formulations of the υ 2

6 and f equations have been proposed which permit f w =0, but these usually perform less satisfactory than the original model and require some re-tuning of the coefficients. Recently a version of the eddy-viscosity model based on Durbin's elliptic relaxation concept has been proposed (Hanjalic et al. [11]), which solves a transport equation for the velocity scale ratio ζ=υ 2 /k instead of the equation for υ 2, Dζ ζ ν f P t ζ ν = + + (8) Dt k x j σ ζ x j in combination with an elliptic relaxation function (here based on the quasi-linear pressure-strain model of Speziale, Sarkar and Gatski (SSG) [12], P 2 L f f = c1 C2 ζ τ + ε 3 (9) Here the eddy viscosity is defined as νt = cµ ζ kτ, where c µ is different from the conventional C µ, and τ is the time scale, equal to k/ε away from a wall. Because of a more convenient formulation of the equation for ζ and especially of the wall boundary condition for the elliptic function f 2 / 2 w = νζ y, this model is more robust and less sensitive to nonuniformities and clustering of the computational grid. Alternatively, one can solve equation (9) for a homogeneous function f with zero wall boundary conditions f w =0, and then obtain 1/2 2 f = f ' 2 v( ζ / x n ) (in analogy with Jones-Launder equation for homogeneous dissipation). The computations of flow and heat transfer in a plane channel, behind a backward facing step and in a round impinging jet show in all cases satisfactory agreement with experiments and direct numerical simulations [11]. 4.2 Elliptic-blending DSM. As an example of a robust second-moment closure suitable for complex near-wall flows, we discuss briefly the Elliptic Blending model (EBM) of Manceau and Hanjalic [13]. The model, based on Durbin s [14] DSM, solves equation (7) in conjunction with the ε equation, but instead of solving six elliptic relaxation equations for the functions corresponding to each stress component, a single scalar elliptic equation is solved 2 2 α L α = 1 (10) The pressure strain term and the stress dissipation are modeled by blending the homogeneous (away from the wall) and the near-wall models 2 2 ij (1 ) ij w ij h φ = α φ + α φ (11) uu 2 ε α ε α εδ k 3 2 i j 2 ij = (1 ) + (12) ij In equation (11), φ h ij can be chosen from any known model (we use SSG), whereas the wall model for the pressure strain, satisfying the exact wall limit and stress budget is defined by w ε 1 φ = 5 u u n n + u u n n u u n n n n + k 2 ( δ ) ij i k j k j k i k k l k l i j ij where the unit wall-normal vector is evaluated from n = α / α. (13)

7 An illustration of the EBM performance can be found in Thielen et al. [15] in the computations of flow and heat transfer in a multiple impinging jets configuration. Besides its relevance for cooling, heating and drying in various applications, impinging jets have long served as a generic benchmark for turbulence and heat transfer modeling and are also relevant for industrial aerodynamics. The EBM showed superior predictions. 4.3 Hybridization of DSM with EVM An interesting simplification of the differential second moment closure has been proposed recently by Basara and Jakirlić [9]. In order to utilize some advantages of the DSM and yet to retain the robustness of the standard k-ε model, they proposed to combine the two approaches into a hybrid turbulence model, HTM. In this method they solve the full differential equations for all stress components, but only the shear stress is used for computing the production of the stress component and in the ε equation, whereas the mean momentum equation is closed by the standard eddy-viscosity stress-strain relation is now a variable computed from 2 U i k i j 2, 2 ij ji x j ε Cµ = u u S S = S S 2 i j δ 2 3 ij νt ij u u k = S with νt = Cµ k 2 / ε, in which C µ Equation (14) implies the minimum square error between the stress components obtained by EVM and DSM, de/dcµ=0, d2e/dcµ2>0, where = ( 2ν + ) 2 E S u u. t ij i j The disadvantage of this approach is that one solves the full DSM for all stress component and yet not utilizing the full advantages of the model. However, the authors have demonstrated that the HTM predicts results in a number of test flows which are much closer to the full DSM than to the EVM, but with notably less computational effort than for the full DSM because the model is very robust due to decoupling of the mean momentum and stress equations. In any case, this approach may prove useful for generating the initial solutions before switching to the full DSM, thus making the computations faster and more robust. (14) 4.4 Generalized Wall Functions (GWF) and Compound Wall Treatment (CWT) The treatment for the wall boundary conditions in RANS computations of complex flows has long been the stumbling block in computation of turbulent flows, especially when accurate predictions of wall friction and heat transfer are the main targets. In such situations the usually affordable computational grids are too coarse to permit integration of the governing equations up to the wall (ItW) and the use of the exact wall boundary conditions. The same problem arises in large-eddy-simulations (LES) of high-re-number wall-bounded flows where proper resolving of the near-wall flow regions requires extremely dense grids and excessive computational resources. It has long been known that the more popular wall function approach (WF) that bridges the near-wall viscous layer tolerates much coarser grids, but here the first cell-center ought to lie outside the viscosity affected region, roughly at y + > 30, which is also difficult to ensure in all regions in complex flows. Besides, the conventional WFs are often inadequate for computing complex flows of industrial relevance because they have been derived for simple wall-attached nearequilibrium flows with a number of presumptions that are valid only in such flows. The continuous increase in computing power has resulted among others - in a trend towards using denser computational grids for computing industrial flows. However, because of

8 prohibitive costs, in most cases such grids are still too coarse to satisfy the prerequisites for the ItW. Instead, often the first grid point lies in the buffer layer (5<y + <30), making neither ItW nor WF applicable. Recently, several proposals appeared in the literature aimed at improving and generalizing the wall treatment, e.g. Craft et al. [16],[17]. We consider here a compound wall treatment (CWT) of proposal by Popovac and Hanjalić [18]. The blending is based on a generalization of the expressions for the mean velocity and temperature profiles of Kader [19] that approximate reasonably well the whole wall region of a boundary layer, including its viscous/conductive and turbulent logarithmic layers. While other universal expressions can be found in the literature, the Kader's blending was adopted because it makes it possible to use the same blending functions for almost all relevant flow and turbulence properties in the same manner. It makes the model insensitive to the precise positioning of the first grid point and - within reasonable limits - to the quality of the mesh in the near-wall region. The CWT can be applied in conjunction with any turbulence model that permits integration to the wall with any wall functions. However, it is advisable to use a well-tuned, physically well-justified and robust ItW model, preferably without empirical damping functions, which has been proved to successfully reproduce properties of a minimum set of generic flows exhibiting various non-equilibrium effects (strong pressure gradients, separation, impingement, and others). Here we use the robust elliptic relaxation ζ-f model discussed earlier (Hanjalić et al. [11] for the ItW. Likewise, no compound treatment will lead to success in computing non-equilibrium wall bounded flows with a relatively coarse grid, if the conventional wall functions are used. For that reasons, we also present the newly developed generalized wall functions (GWF) that account for non-equilibrium effects and yet preserve the simple standard form making its implementation into the existing CFD codes very easy and straightforward Generalized wall functions. The wall functions relate the values of the variables in the cell centres with those at the wall through pre-integrated simplified expressions, thus providing indirectly the wall boundary conditions. A compound wall treatment should automatically ensure this. However, the standard wall functions are known to fail in non-equilibrium flows because they have been derived using several assumptions: constant (turbulent) shear stress uv, proportional to the (also constant) turbulence kinetic energy k (so that uv / k = const 0.3), the semi-logarithmic mean velocity distribution, local energy equilibrium (P k =ε), and a linear length-scale variation. In most flows of practical relevance none of these assumptions holds. In order to relax at least in part the above constraints, some authors have included the pressure gradient to modify the conventional WFs. Such are the "nonequilibrium" WFs of Ng et al. [20] (available in FLUENT CFD code). We present here more general wall functions for the velocity and temperature from analytical integration of the simplified momentum and energy equation (for heat transfer), based on a single assumption proposed by Craft et al. [16] that the non-dimensional eddy viscosity varies linearly with the distance from the wall in turbulent region, which was found to hold reasonably well even in nonequilibrium flows. However, we modified the Craft et al. formulation of µ t into form defined by equation (15) and Fig. 1, which permits a straightforward integration and simpler forms of the WFs, compatible with the standard expressions, so that they can be easily implemented in the existing in-house or commercial CFD codes. The integration of the two-dimensional momentum equation for the wall-tangential direction, with the two integration constants determined from the condition of continuity and smoothness of the velocity profile at the edge of the viscous sublayer leads, after rearrangements, to the following expression for the wall shear stress:

9 0 y < y µ = ρκuτ y y ν ν (15) Figure 1 A sketch of the first near-wall cell with the assumed distribution of eddy viscosity Ctg ρ yν ln( y / yν τ w = ρu + (16) κ uτ µ κ uτ where ρ Utg Utg Utg p Ctg = + Utg + V + t x y x (assumed to be constant over the cell and known from the previous time step or iteration) accounts for convection of the wall-tangential velocity U tg and for the pressure gradient and V is the wall-normal velocity. A priori test in a backward-facing step and in an axysimmetric impinging jet confirmed the feasibility of the above assumption (Popovac and Hanjalić [18]). By inserting the value for the viscous sublayer thickness y + ν = 11, which corresponds to the intersection of the linear and semi-logarithmic velocity laws, and rearranging leads to the generalized semi-logarithmic expression for the velocity distribution in fully turbulent wall layer: U ( Ey ) + C tg + 1 = ln where Ψ = 1 κψ κu + + 1/ 4 1/ 2 and U + =U/u τ, y + =yu τ /ν and u τ = τ w /ρ is replaced by Cµ k to avoid singularity when τ w tends to zero, as common in the conventional WFs. As expected, for equilibrium flows, C tg =0 and ψ=1, thus equation (17) reduces to the standard log-law. The influence of the nonequilibrium factor ψ and the performance of equation (17) is shown in Fig. 2, where the velocity profiles, deduced from the experiments in an a priori manner are compared with measurements for a turbulent boundary layer subjected to strong adverse and favorable pressure gradients at very different p + =ν/ρu τ p/ x. Note that the CWT treatment discussed below is applied to recover the complete velocity profile up to the wall. As seen in both figures, the CWT with the above generalized wall functions reproduces the measurements up to y + close to 100, allowing the placement of the first near wall cell anywhere in this region Compound wall treatment (CWT). We consider now a blending of the two approaches into a unified procedure that should provide the wall boundary conditions irrespective of the location of first new-wall grid nodes. The method should reduce to the ItW or WF procedure when the first grid node happens to lie in the regions appropriate for each approach. If the cell-centre lies in the buffer zone, the boundary conditions are again provided in form of wall functions, but modified to account for the viscous/molecular and other (inviscid) wall effects. (17)

10 Figure 2. A priori test of equation (17) in conjunction with the CWT in a boundary layer subjected to adverse (left) and favourable (right) pressure gradient, illustrating the model validity for 0<y + < 100 for both cases. The quantities for which the boundary conditions ought to be specified in the CWT depend, like in the conventional WF approach, on the turbulence model used. The primary variables are the wall shear stress τ w and its relation to the mean velocity, wall heat flux q w and its relation to the mean temperature, production P k and dissipation ε of the turbulence kinetic energy. For the ζ- f model considered above, we also need the wall function for the elliptic function f. We considered the blending of the wall-limiting and fully turbulent properties in the manner of Kader [19], who proposed a single expression for the temperature profile throughout the whole wall boundary layer, and which can also be applied to the velocity profile as a blending of the viscous and the fully turbulent definition of U + ( ) + + Γ 1 + 1/ Γ 0.01y U = y e + ln E y e where Γ = + κ (18) 1 + 5y This expression has the correct limiting behaviour: it reduces to the viscous or the fully turbulent value in their respective regions, and it reproduces the velocity profile for equilibrium wall boundary layer reasonably well throughout the whole region, including the buffer zone. We apply the same blending to other variables in question P Γ ν 1/ Γ te Φ = Φ e + Φ (19) where Φ denotes the variable considered, and indices ν and t denote the viscous (wall-limit) and the fully turbulent asymptotes respectively. Using the generalized (nonequilibrium) expression for the velocity profile (17), we get the compound expression for the wall shear stress 1/ 4 1/ 2 v Γ t 1/ Γ UP Γ κψ c kp U µ P 1/ Γ τ w = τ e + τ e = ν e e w w + (20) y + P ln Ey ( P ) where the subscript P denotes the values in the next-to-the-wall cell centre. The same blending is also applied for the production and dissipation of the turbulence kinetic energy, thus providing the wall functions for the k and ε equations. In the case of production, the flow calculations with sufficiently fine mesh resolution in the near-wall region will reproduce correctly both the turbulent stress and the velocity gradient, yielding a correct profile of P k, When the coarse mesh is used, however, neither the turbulent stress nor the velocity gradient can be obtained correctly. In the standard WF approach the value of P k is imposed by assuming local equilibrium conditions: logarithmic velocity profile and constant shear stress. This gives a simple relation P k = u 3 τ /(κy), but this is correct only in the fully turbulent region in equilibrium flows. Once we have the analytical expression for the velocity distribution across the complete 4 +

11 near-wall region, we can derive an expression for P k by taking U/ y from equation (18) in combination with the near-wall and fully turbulent expressions for the turbulent stress. This, basically, reduces to the blending according to equation (19), 2 2 3/ 4 3/ 2 U kp U c k Γ µ P Pk P = uv = Cµζ p e + e y ε P y y P κψ P 1/ Γ Likewise, we can derive a blended expression for the dissipation rate. Because of a specific and strong variation of ε in the near-wall region, we modified the exponent of the damping function into = y /(1 + 5 y ) Γ ε 3/ 4 3/ 2 Γ 1/ Γ 2ν k c k P Γ P P v e t e e ε µ ε = ε + ε = + e 2 y P κ yp 1/ Γε Note that the above expressions, aimed at using ζ-f model for the wall limit, can also be used in conjunction with any EVM, including the (low-re-number) k-ε or k-ω model, in which case ζ should be replaced by 0.4 f µ, where 0.4 represents the typical near-wall equilibrium value of ζ=υ 2 /k, and f µ is the eddy-viscosity damping function such as in Jones-Launder model. Figure 3 illustrates the performance of the CWT in two generic test flows where the conventional wall functions are known to fail. For this purpose, the grid was chosen with the first point placed in the buffer region at y + between 10 and 25. The results are compared with the ItW with the ζ-f and v2-f models with the first grid point placed at y + between 1 and 3. We show here that the CWT can provide satisfactory predictions of the Nusselt number, which is especially sensitive parameter to the treatment of the wall boundary conditions. (21) (22) Figure 3. Performance of the CWT with the first near-wall grid point at y+=20-25, compared with the integration up to the wall with the ζ-f and v2-f models: Nusselt number distribution in an axisymmetric impinging jet (left) and in flow behind a backward-facing step (right). 4.5 Some illustration of RANS novelties in computation of environmental flows. Various modifications of the standard k-ε model, the elliptic relaxation EVMs and of compound wall treatment discussed in sections 3 and 4 have been tested in several laboratory and full-scale benchmark cases for environmental flows and pollutant spreading (Hagenzieker [21], Hagenzieker et al. [22]). The test cases include two- and three-dimensional idealized street canyons, as well as a real street. We show here some results for each of the three mentioned test flows.

12 The first case is the 2D street canyon, investigated experimentally by Brown et al. [23] in the wind tunnel of the U.S. Environmental Protection Agency (USEPA), Fig. 4. In addition to comparing the flow and turbulence statistics with data of Brown et al., we also considered pollutant spreading released in the corner of the sixth canyon, which were compared with experiments of Pavageau and Schatzmann [24]. Figures 4 and 5 show the effect of some of the simple remedies such as limiting the time scale and using the linearised kinetic energy constraint, which both bring substantial improvements in impinging flows. The effects are best illustrated in Fig. 6 showing excellent agreement between the computed and measured contours of the tracer gas concentration. Similar improvements have been obtained for 3-D idealized street canyon (a matrix of cubical buildings), for which the velocity, turbulence statistics and tracer gas concentration have been measured by Brown et al [25]. Figure 4. Contours of turbulence kinetic energy k in the full computational domain in the Los Alamos 2-D test case [?] with the standard k-ε model, with scale limiter and linearized production (Hagenzieker 2006). Figure 5. Illustration of the scale-limiter and linarised production cut-off over and between forst rows of 2D obstacles in the Los-Alamos test cases (Hagenzieker 2006).

13 Figure 6. Contours of the computed (left) and measured (right) concentration level. Experiemnts The next test case is the Göttinger Strasse in the German town of Hannover, used as a benchmark case for validating CFD codes by the European Research Network TRAPOS, Fig. 7. Figure 7. Göttinger Strasse in Hanover: building configuration and trajectories of neutral particles released at the ground level along the street canyon centerline. Left: west wind; right: east wind. (Hagenzieker 2006) The full-scale experimental data are provided by the Environmental Agency of Lower Saxony, which conducted systematic monitoring of the concentration level of exhaust gases from approximately vehicles per day. Wind tunnel measurements of the flow field in the same but scaled-down configurations for 4 different wind directions [26],[27] have been used for model validations. Some results of the computations with the standard k-ε and several variants that include various modifications outlined in Sections 3 and 4, are shown in Fig. 8 for two wind directions, west and east which denote two opposite directions normal to the street. Comparison of the nondimensional concentration obtained with different models and experiments is shown in Fig. 9, indicating that simple time-scale limiter provides notable improvement in reproducing the experimental data.

14 Figure 8. Computational results with the k-ε model with Durbin time-scale limiter for wst wind (above) and east wind (below). Left: velocity vectors and contours of the vertical velocity in the palne at z=10 m. Right: pollutant concentration at z=3 m. (Hagenzieker, [2006]). 1 Field experiment 2 TASCflow computations 3 Wind tunnel (detailed) 4 Wind tunnel 5 Standard k-ε 6 k-ε (Kato-Launder model) 7 k-ε + Durbin scale limiter Figure 9. Nondimensional pollutant concentration at a monitoring point for two wind directions: comparison of computations with measurements (Hagenzieker [2006]

15 4.6 Modeling effects of thermal buoyancy Turbulent flows driven by thermal or mass buoyancy have long posed challenge for RANS models, especially when the standard isotropic eddy diffusivity is used by which the heat flux is expressed as directly proportional to the mean temperature gradient, i.e. θui = ( νt / σ ) T / xi. As demonstrated by Hanjalić [28], the models fails in generic situations such as Rayleigh-Bénard convection where T/ x i =0 in most fluid domain apart from the very thin wall-adjacent conductive layer. It also fails in vertical side-heated configurations because here the mean temperature gradient in vertical direction is negligible, yet the turbulent heat flux in the vertical direction is responsible for the buoyant source of turbulence kinetic energy G = βgiθui. In order to provide proper heat flux in a general manner that would satisfy various orientations of the heated walls with respect to the gravitational vector, various algebraic expressions have been propose, drawing on the parent differential flux transport equation for θui analogous to (7). It has been shown that the truncation of this transport equation requires to keep at least all three productions term, which all play important role in various flows regions, especially the buoyant source associated with the temperature variance. However, various testing showed that even if all three terms are retained, it was still very difficult o reproduce the wall-normal heat flux for the two generic situations fluid trapped between the two differentially heated infinite walls when oriented horizontally (g parallel to T) and vertically (g perpendicular to T), Fig. 10. Recently, Kenjereš et al. [29] proposed a quasi-linear expression, which satisfies reasonably well both generic cases, which gives sufficient guarantee that the model should perform well in general situations when the mean temperature gradient can take any orientation with respect to the gravitational vector θu C T τ γuu U ξθu ηβg θ C a θu i 2 ' i = θ i j + j + i + θ 1 ij j x j x j where τ=k/ε in the conventional k-ε framework, or τ=υ 2 /ε=ζ when elliptic relaxation EVM concept is used. Together with the introduction of the buoyancy production into in the expression for the turbulent stress tensor (originating from the parent transport equation (7)) 2 2 uu i j = kδij 2ν tsij + Cθτ β giθuj + gjθui gkθukδ j 3 3 and solving the model transport equation for the temperature variance in which the variance dissipation ε θ is expressed via a new expression for the thermal-to-dynamic time scales, 2 1/ 2 τθ θ ε A R = = max 2 ;0.6A τ 2ε θ k 1+ A2θ the model was shown to perform satisfactory in a range of test flows [29]. An illustration of a priori testing of equations (23) and (25) for two generic situations is presented in Fig. 10. It is noted that such an algebraic flux model served also as a subscale model for Transient RANS (T-RANS) which showed a potential to handle large-scale buoyancy driven environmental flows at extreme Rayleigh numbers, as discussed below. (23) (24) (25)

16 Figure 10. A priori testing of different models for the wall-normal turbulent heat flux (top) and of the time scale ration (bottom) for natural convection in a side-heated (left) and in a heated-from-below infinite plane channel, [29], 5 HYBRID RANS/LES It is recalled that the proper resolution of dynamically important scales with LES requires the grid density to increase with Re 0.4 in regions away from a solid wall, but this constraints becomes much more severe in near-wall regions where the grid density should follow Re 1.8. In contrast, a RANS grid requires clustering only in the wall-normal direction, making the grid requirements proportional to ln(re). For realistic engineering and environmental flows an attractive proposition is to combine LES and RANS. Most approaches currently under exploration can be grouped into two categories: Zonal (two-layer) method with a distinct, predefined RANS-LES interface Seamless methods with a single model throughout the flow, which tends to RANS when the wall is approached and to an LES away from it. The strategies of the two approaches are illustrated in Fig. 11. In the zonal approach the conventional coarse-grid LES is applied in one flow region - usually away from a solid wall, and a onepoint RANS model is applied in the other, usually the near-wall region. The switching from one to another field is made at a suitably chosen interface, Fig. 11a. The key problem, especially in complex flows, is to ensure proper matching conditions at the interface. This is usually based on the equality of the total (resolved + modeled) stress or total viscosity ν + ν = ν + ν (26) res res SGS LES t RANS where

17 ν res LES = ' ' ' ' ( δ / 3) i j k k ij u u u u S S ij S ij ij (27) The overbar denotes filtering and <> implies some local smoothing. Because the resolved motion on both sides of the interface should be the same, ν SGS =ν t and the RANS model yields much larger modeled contribution than the LES subgrid-scale model, the RANS model needs to be damped to match LES at the interface. A way to accomplish this is to damp the RANS eddy viscosity either by damping the coefficient C µ (Hanjalić et al. [30], Temmerman et al. [31]), or by decreasing the RANS kinetic energy or by increasing the dissipation rate. Other approaches have also been reported e.g. a parabolic treatment of the near-wall boundary layer with imbedded solutions using a simple damped mixing length model in the near-wall RANS region, or the simultaneous solution of the parabolic momentum equation again with mixing length, with LES in the outer region. Figure 11. Modeled eddy viscosity in hybrid RANS/ LES methods. Left: zonal method with different interface locations;. right: seamless approach. The second approach is based on continuous (non-zonal) simulations using the same model for the unresolved motion in the complete solution domain, which serves as a RANS model in the near-wall region and as a subgrid-scale model in the outer LES region. The seamless switching between one and the other approach is accomplished by changing the length scale: in the nearwall RANS region the distance from the wall is used whereas in the LES region this is replaced by the representative grid size. The most known method in this category is Spalart s Detached Eddy Simulation (DES) [32,33] in which the Spalart-Almaras (S-A) one-equation model for eddy viscosity is used in both regions, Fig. 11b. The DES approach, just like the zonal one, contains a doze of arbitrariness: the interface between the RANS and LES region is determined by the adopted mesh. The switching parameter can of course be adjusted by an empirical coefficient, but the desired criterion is difficult to know in advance in unknown complex flows. The problem is also in a strong deterioration of the predictions when the switching occurs at larger distances from the wall (y + >30 ). It is also noted that S-A model was tuned for external aerodynamic flows and has been shown to perform unsatisfactory in some other flow types. 5.1 Zonal strategies The zonal approach may seem more appealing because outside the wall boundary layers the conventional LES method (with prescribed subgrid-scale or dynamic modeling) is used without any intervention in the subgrid modeling. However, the crucial issues and problems to address are the

18 location and the definition of the interface, the nature of matching conditions especially for flows in complex geometries, and the receptivity of the RANS region to the LES unsteadiness and the RANS feedback into LES region. Even if the RANS model is adjusted to meet the constraint of continuity in the eddy viscosity and other quantities across the interface (e.g. by modification of C µ, as illustrated in Fig. 12a), insufficient level in small-scale activity that RANS feeds into LES across the interface produces in most circumstances non-physical features (a bump) in the velocity profile around the interface. Several proposals have been published for introducing an extra small-scale forcing. Piomelli et al. [34] suggested a stochastic backscatter generated by random numbers with an envelope dependent on the wall distance. Davidson and Dahlström [35] proposed to add turbulent fluctuations, obtained from DNS of a generic boundary layer, to the momentum equation at the LES side of interface. Hanjalić et al. [30] found that by feeding the instantaneous instead of homogeneously averaged value of C µ at the interface (that matches the RANS eddy viscosity with the subgrid-scale viscosity on the LES side) the anomaly diminishes. This suggests that the noisy instantaneous C µ (Fig. 12b) acts in a similar spirit as the additional random or stochastic forcing, but it is much simpler. Note that C µ at the interface is evaluated from the instantaneous field, depending on the definition of ν t. For example, for a one-equation (k) model with ν t =C µ l µ k 1/2 and for the two-equation (k-ε) model with ν t =C µ f µ k 2 /ε, respectively, C µ is evaluated from: C µ ν SGS = ; 0.5 l k µ RANS,int C µ 2 fµ ( k / ε ) ν 2 ( fµ ( k / ε )) SGS = (28) 2 Figure 12. Variation of (time invariant) C µ across the flow (top) and histogram of the instantaneous and homogeneously averaged C µ at the RANS/LES interface (bottom) [30]. Figure 13 shows the mean velocity and turbulent shear stress profiles in a plane channel for Re τ =2000 obtained using a low-re-number k-ε model with a damped C µ for three different locations of the interface. The smoothing effect of the instantaneous, instead of locally averaged C µ is illustrated in Fig. 12b for the interface at y + =280 for which both results are presented. Defining the criteria for the positioning of the interface is another problem. The kink in the velocity profile seems most visible if the interface is placed in the region populated by coherent streaks (centered around y + =60-100). Because of insufficient spanwise grid spacing, the computed streaks are much wider ( superstreaks ) and their distance much larger than in reality, as

19 shown in Fig. 14. Moving the interface closer to the wall would lead to a greater proportion of the small-scale structure being captured, but reproducing faithfully the streak topology would require the grid to be substantially refined, especially in the spanwise direction, thus departing from the main motivation for the hybrid approach. On the other hand, moving the interface further away from the wall leads to the streaky pattern becoming progressively indistinct. Figure 13 Velocity and shear stress in a plane channel at Re τ =2000, obtained by a zonal method with the k-ε model for the near-wall RANS region and LES (Smagorinski sgs) in the outer region, with different locations of the interface [39]. Paradoxically, with the interface placed at a distance sufficiently large to lose the fine nearwall structure, the anomaly in the velocity profile gradually disappears. This finding may sound discomforting on theoretical grounds, but has comforting implications in the simulation of complex flows at very high Re numbers, where the wall boundary layers are in any case so thin that they cannot be resolved in any event. Figure 14. Spanawise vorticity contours indicating superstreaks for various interface locations in zonal RANS/LES 5.2 Seamless methods Under this name we group the methods that use a single (usually a RANS-type) model for the unresolved motion throughout the whole flow. In the wall limit, the model approaches the standard URANS, whereas away from a solid wall, the model provides the subgrid-scale contribution for the supposed LES. The model modification is continuous, so there is no need to predefine an interface. However, most methods of this kind use a grid detecting function f=f(l LES /L RANS ), by

20 which the model/computer sees the grid. This function controls the switching of the characteristic turbulence length scale from L RANS (computed from the RANS model, e.g. L RANS =k 3/2 /ε if k-ε model is used) to L LES associated with the reference size of the grid cell (e.g. V 1/3 ). The scale switching location represents in fact an interface, though in contrast to zonal models, the gradient (and not only the value) of eddy viscosity is continuous. The interface location is not predefined, but established automatically depending on the local grid density and distribution. Although, the best known method of this type, the DES of Spalart [32], employs oneequation S-A model, most other methods reported use a two-equation model, usually the k-ε model. Here the smooth adjustment of the eddy viscosity is accomplished by either decreasing the modeled kinetic energy or increasing its dissipation rate, both beyond the conventional RANS solutions. Two approaches deserve attention. In the first method, the complete source term in the ε-equation is multiplied by a grid-detecting function f( /L): Dε Dt ( C P C ε ) ε1 ε 2 = f + D (29) ε τ where τ is the time scale. In the near-wall region L will prevail and in the outer region becomes the characteristic length scale forcing ε to increase and thus reducing ν t just as in the DES approach, as illustrated in Fig. 11b. It is noted that this intervention in the LES region diminishes also ν t in the RANS region thus ensuring its continuous and smooth variation across the flow. This intervention on the complete source term in the ε-equation is also in the essence of the Partially averaged Navier-Stokes (PANS) model of Girimaji [36] and of the Renormalization-Groupbased method of De Langhe [37]. Another approach is to affect only the sink term in the ε (or other scale-providing variable) equation. An example is the method based on the subgrid-scale model of Dejoan and Schiestel [38], derived from a multi-scale split-spectrum k-ε model. Here the value of ε is modified by adapting the coefficient in the sink term C o C C ε 2 ε 1 = C + ε 2 ε 1 2 / 3 L 1 + β L RANS LES where β is an adjustable coefficient. This model was further tested in plane channel flows at a range of Re numbers, but its application to impinging jets posed some problems, Hadžiabdić [39]. Recently, Kenjereš and Hanjalić [40] made further simplifications proposed an alternative to (30), which was tested successfully in Rayleigh-Bénard convection and in a wind flow over a hills: 0.48 C = C + (31) ε 2 ε 1 α where α=max(1, L RANS /L LES ), L RANS =f µ <k> 3/2 and L LES =( V) 1/3. The common feature of most seamless approaches is that they are usually void of the kink in the velocity profiles noted in most zonal models. However, the testing of seamless models apart from DES - has so far been reported only for simple flows. Fig. 15 shows some results for a plane channel at Re τ =2000 obtained with the Dejoan-Schiestel model for two values of β, illustrating its effect, [39]. The seamless models are suited for employing more advanced RANS models, which may be needed for complex wall-bounded flows at high Re numbers. In such flows, the RANS/LES interface, despite its vague definition, my lie far away from the wall, thus putting a high burden on the RANS model. This is especially the case for flows containing impingement and separation regions, where the conventional two-equation models such as the k-ε may not be adequate. (30)

21 Figure 15 Seamless RANS/LES coupling with the k-ε model with Dejaon-Schiestel modification of C ε2. We close this review by presenting a mixed approach in which an elliptic relaxation EVM (k-υ 2 -ε-f or k-ζ-ε-f) model for RANS is coupled with the dynamic LES in a manner similar to the seamless methods discussed above (Hadžiabdić [39]). In this approach, the dissipation rate in the k-equation is multiplied by a grid-detection function in term of the RANS and LES length scale ratio, i.e. Dk k = ( ν + vt ) + P αε Dt x j x (32) j L where α = max 1, RANS k 1/3 and L = tot RANS ; LLES = 0.8( x y z ) (33) LLES ε and k tot.=k res +k mod. In other words, the dissipation rate ε is evaluated from its transport equation in the conventional RANS form when L LES >L RANS, and from k and when L LES <L RANS so that the sink in the k-equation is αε=max(ε, k 3/2 / ). Hence, for α 1, the standard RANS model is in play, and for α>1 we should have a one-equation RANS model with a damped viscosity. Another switching is then imposed, this time to the (dynamic) LES using the criterion RANS LES ν = max( ν, ν ) (34) t t t Because at the location where α=1 the eddy-viscosity switching constraint (5) is still not satisfied, a buffer zone is automatically established up to the position where constraint (6) is activated. Here the RANS is still in play but with an automatic adjustment (through α > 1) in the RANS eddy viscosity towards the LES sgs viscosity. This ensures a continuous damping of ν t until the dynamic sgs eddy viscosity is reached when the computations switch to the true dynamic LES, Fig. 16. In the examples shown below the buffer zone extends up to α 1.5, covering only a few cells for the typical RANS and coarse-les grids used here. Figure 16 (right) shows the velocity profile in a plane channel obtained from hybrid computations with the above model for Re τ =20,000 with a 64x90x32 mesh. Note that this grid is by two orders of magnitude smaller than required for properly resolved LES. The results are very satisfactory, though the true test must await justification in more complex flows. 3/2

22 Figures 16 Eddy viscosity in hybrid RANS ζ-f + dynamic LES model in a plane channel flow at Re τ =2000 (left) and the velocity profile for Ret=20.000, [39] 5.3 Some illustration of hybrid RANS/LES for Environmental Flows We show here some results of hybrid RANS/LES used in parallel with some steady RANS models for the computation of flow over 116 m high, moderately sloped Askervein Hill, situated near the coast of South Uist in the Outer Hebrides, U.K., for which extensive field experimental data have been provided by Taylor and Teunissen [41], Fig. 17. Figure 17. Top view of the Askevein Hill with measurements lines A, AA and B and two towers located at the map center (CP) and hilltop (HT) (left); A 3-D view of the hill with solution domain (4125x4125x700m) and mesh (176x176x36), Re=1x10 9 (right) The velocity defect, U ( x, y, z) U 0( z) S ( x, y, z ) = U ( z) 0 computed by various models and hybrid RANS/LES using the Dejoan-Schiestel type of seamless model with C ε2 computed from equation (31), is compared with experiments in Fig. 18. Figure shows horizontal distribution of S at 10 m above the ground along lines AA and A, and vertical (35)

23 profiles at positions CP and HT (for notations see Fig. 17). In both cases, the hybrid approach produced best agreement with experiments. Figure 18. Comparison of computed and measured velocity. Left: horizontal distribution of S at 10 m above the surface along AA (top) and A (bottom). Right: vertical distribution of S at CP (top) and at HT (bottom), [ 21]. 6 T-RANS BASED VLES Many flows are dominated by large-scale coherent structures that can have a (semi)deterministic character. Common examples are vortex shedding from bluff bodies or convective structures in thermal convection. In such flows, the stochastic turbulence often behaves as a passive scalar and has little influence on large scale structures. It may thus suffice to resolve in space and time only the very large, semi-deterministic structure (very large eddy simulations, VLES), which essentially governs the momentum and heat transfer. The name VLES implies a form of LES (not necessarily based on grid-size filtering) with a cut-off filter at much lower wave number, or simply solving ensemble averaged equations. The basic rationale behind VLES is: resolve only very large, coherent or deterministic structures and model the rest! The principle relies on decomposing the total turbulence spectrum into the ensemble-averaged deterministic part and the residue of the ensemble-averaged operation, representing stochastic turbulence, [40, 42]. Because a signifi-

24 cant portion of turbulence spectrum needs to be modeled, the modeling of the unresolved ( subscale ) motion requires more sophisticated approach than used for common LES subgrid-scale models, opening thus a new niche for the RANS modeling. The RANS model can be one of the standard versions (in which case we talk about URANS), or it can be modified to model only the incoherent random fluctuations, while the large scales are resolved. The solution of the resolved part of the spectrum can follow the traditional LES practice using grid size as a basis for defining the filter (in which case we recover a seamless hybrid RANS/LES method), or solve ensemble- or conditionally averaged Navier-Stokes equations with a modified RANS model that is not based on the grid size, as in the Transient RANS (T-RANS). We present here briefly some features and illustrations of the latter approach, and demonstrate its application to turbulent flows subjected to thermal buoyancy. It is recalled that in such flows, an instantaneous field can be decomposed into unsteady ensemble-averaged (organized) motion and random (incoherent) fluctuations, so that the instantaneous flow property Ψ ˆ (x i,t) can be written as sum of the time-mean, deterministic and random part. The ensemble averaged (mean plus deterministic) quantities are fully resolved by solving in time and space the momentum and energy equations just as in LES, whereas the unresolved contribution is modeled using RANS models for instantaneous stress and scalar flux. The total long-term averaged second moments consist of the resolved (deterministic) and incoherent (random) part which are assumed not to interact, i.e. ΨΥ ˆ ˆ ~ ~ = ΨΥ + ΨΥ + ϕγ = Ψ Υ + ϕγ Both parts are expected to be of the same order of magnitude, with the modeled contribution prevailing in the near-wall regions where the deterministic motion is weak. The dominance of the modeled contribution in the near-wall region emphasizes the importance of the RANS model which needs to be well tuned to capture near-wall behavior of turbulent stress and scalar flux. We illustrate the potential of T-RANS in the example of Rayleigh-Bénard convection at extreme Rayleigh numbers, which are inaccessible to either conventional LES (or DNS) or to classic RANS. Here we use the algebraic subscale flux model (23) and the corresponding algebraic stress model in which all variables are evaluated as time dependent. Extensive testing of the RANS subscale model in a number of confined natural convection cases provides confidence in its performance close to walls. Outside the wall layers, the role of the model fades away because the dominant large-scale quasi-deterministic roll structures are fully resolved in time and space. Figure 19 shows T-RANS computations of Nusselt number and of the hydrodynamic (λ v ) and thermal (λ θ ) wall layer thickness (defined by peak positions of the turbulent kinetic energy and temperature variance respectively) as a function of Rayleigh number over ten decades, up to (Kenjereš and Hanjalić [42]). It is noted that the maximum Ra achievable by DNS is around 10 8 and by true LES about The T-RANS computations agree very well with the available DNS for low Ra numbers as well as with the experiments for low and moderate Ra (up to ) in accord with the known correlations Nu Ra 0.3, λ v /H Ra -1/7 and λ θ /H Ra -1/3. For higher Ra numbers the T-RANS shows clearly an increase in the exponent of Ra in accord with Kraichnan s asymptotic theory (n 0.5 for Ra ) and recent experiments. This change in Nu-Ra slope is reflected in the change of the slopes of λ v (Ra) and λ θ (Ra) curves. The capability of T-RANS for capturing the instantaneous structures is illustrated in Fig. 20, where instantaneous streaklines are presented for the central and a near-wall plane for Ra=2x A comparison of the long-term averaged second-moments in the vicinity of the top wall, obtained by hybrid RANS/LES and T-RANS for Ra=10 9, is shown in Fig. 21, displaying for each of the two methods the resolved and modeled contributions. The T-RANS method proved subsequently to be very suitable for real engineering and environmental flows and transport phenomena at mezzo scales, e.g. for predicting diurnal change in air flow and pollutant dispersion.

25 Fig. 19. T-RANS predictions of Nu number (left) and of hydrodynamic (λ ν ) and thermal (λ θ ) wall layer thickness in R-B convection for ten decades of Ra [42]. LES (256 2 x128), Hybrid (82 2 x72) T-RANS ((82 2 x72) T-RANS ((82 2 x72) Ra=10 9 Ra=10 9 Ra=10 9 Ra=2x10 14 Figure 20. Trajectories of massless particles of an instantaneous velocity field portraying the resolved flow structures with a well-resolved LES, Hybrid RANS/LES and T-RANS. Top: the central horizontal plane z/h=0.5. Bottom: inside the thermal boundary layer z/h=10-3 (Ra=10 9 ) and z/h=0.075 (Ra=2x10 14 ); Pr=0.71, [40] 6.1 Illustration of T-RANS of Environmental Flows As an illustration of the application of T-RANS to environmental flows, we present some results of diurnal variation of air movement and pollutant dispersion over a medium-sized town located in a valley during windless winter days when the lower atmosphere is capped by an inversion layer preventing any escape of pollutants. The air movement and pollutant dispersion are governed primarily by the day ground heating and night cooling and by the terrain configuration. The simulated domain covers an area of 12x10x2.5 km, Fig. 22, filled by a numerical mesh of the averaged cell size of 100 m in each direction, but clustered towards the ground.

26 Figure 21. Long-term averaged kinetic energy, heat flux and temperature variance in the wall vicinity for Ra=10 9 compute by fine-resolved LES, Hybrid RANS/LES and T-RANS, showing contributions by the subscale model and the resolved (deterministic) structures, [40]. While any realistic conditions can be imposed, in absence of any field data a hypothetical scenario of a diurnal cycle was adopted by which the ground over the complete solution domain is uniformly heated and cooled over a cycle in a sinusoidal manner with the diurnal and nocturnal temperature amplitudes of ±1 0 C. On top of this time-dependent but spatially uniform ground temperature variation, we superimposed additional ground heating and cooling over the two heat and emission islands, representing distinct residential and industrial zones, over which a sinusoidal variation was imposed, but here both in time (over a diurnal cycle) and in space, with the temperature extrema of T g = ±2 0 C and ±1 0 C respectively at the centre of each of these two zones, Fig. 22. The two zones are also represented by different pollutant emission (C=50% and 100% respectively) during the day, and zero emission during the night. Two consecutive diurnal cycles were simulated (0-24h, day (I) and day (II)) with a time step of 150 sec. Two different situations with respect to the imposed thermal stratification were analyzed. The imposed vertical profile of the potential temperature of dry air was assumed uniform in the lower atmosphere, and linear in the upper layer with an increment of T/ z=4 K/km. The base of the inversion layer (the switch from the uniform to the linear temperature) is located at z/h=2/3 ( 1600m from the valley deepest point) for the first case ("weak stratification"), and at z/h=1/3 ( 800m) for the second case ("strong stratification"). The domain height (H) and the characteristic initial temperature gradients give very high values of Rayleigh number, i.e. O(10 17 ). As an illustration of the potential of the T-RANS approach to capture the instantaneous convective movement under the effect of stable stratification and complex terrain orography, Figs. 23 shows in parallel the instantaneous trajectories of massless fluid particles in a vertical plane over a valley cross-section for two time instants: at noon and midnight of the first day cycle, both for two stratification conditions. The upper figures correspond to active heating periods for the first-day cycle and the lower figures to the nocturnal cooling period at the end of the first-day cycle. It is obvious that the capping inversion layer acts as a kind of a barrier for the vertical convective movement, suppressing the plume penetration and mixing, especially in the case of strong stratification. Distinctive roll structures created during the initial stage of diurnal cycle lose their strength and identity during the stabilizing effect of the nocturnal cooling. It is noted that even a very small undulation of the terrain surface has a great impact on the formation and orientation of the convective rolls. As the time progresses, the interactions between these large structures becomes more intensive, resulting in vigorous motion which is especially

27 noticeable over the urban and industrial areas with elevated ground temperature. The strongest vertical deformations of the capping inversion layer occur above these areas. During the nocturnal periods, the stable stratification suppresses the convective motion and the associated mixing, Fig. 23 and the characteristic roll structures gradually disappear. They are replaced by weak inertial motion which further decays with time. In all figures the effect of the terrain orography on the convective structure is remarkable, what is clearly visible in the vertical cross-sections at locations where there are no local heat sources, i.e. outside the urban and industrial areas, i.e. at the hill slopes. Here, the strongest deformation of the inversion layer are observed above the highest hill peaks. It is interesting to note that a distinct roll structure can still be observed during the nocturnal period, Fig. 23, albeit of low intensity for the strong stratification case. More details can be found in [43, 44]. The above discussed velocity field governs the pollutant dispersion, which is here considered as a passive scalar. Because the air in the valley is trapped by the inversion layer, if the critical situation persist over several days or longer, the emitted pollutant accumulates in the town and can reach critical proportions which may require some measures to regulate the use of fossil fuels. Fig. 24 shows a realistic view of the pollutant concentration front at two selected time instants, close to the end of the first-day cycle (left) and at the end of the second-day cycle (right), both for the weak stratification case. The pollutant cloud is superimposed on the true satellite picture of the town conopy, with its relief and visible residential and green areas. The grey surface corresponds to the pollutant concentration of (0.01 < C max >). It is presented here as an example of a particular visualisation method that can imitate the real smoke appearance with the intensity of its transparentness corresponding to the selected intensity of the concentration. Fig. 24 clearly shows that significant areas of the town, especially on the hill slopes, has not yet been affected by the pollutant dispersion during the two days, thanks to the specific local terrain orography. Such information can be very useful in deciding on location of objects of special interest, e.g. hospitals, schools. Also the simulation can be used for studying the effects of location of various industries and other pollutant emitters. Τ=2 TEMPERATURE Τ=1 T=T(x,y,z,τ) CONCENTRATION Residential C=C(x,y,z,τ) Industrial y Figure 22. Terrain orography, computational domain and mesh for mezzo-scale flow over a town in a mountain valley (left), and the ground temperature and pollutant emission scenarios (shown only in one space dimension) over the residential and industrial zones. (Note that the same scenario applies in other horizontal direction and in time, with maxima at noon and minima at midnight.)

28 Figure 23 Instantaneous fluid trajectories showing the velocity fields over a valley cross section at noon (left) and midnight (right) during the first day cycle for weak (top) and strong stratification (bottom). Figure 24. Time evolution of the pollution front (isosurface of small concentration value) for weak stratification at two time instants. Left: approximately at the end of the first-day cycle; right: at the end of the second-day cycle. 7 CONCLUSION Several novel developments in RANS and combined RANS/LES methods have been presented, aimed at improving accuracy, reliability and robustness of computations of complex flows in wind and environmental engineering. First, two new versions of the near-wall RANS models based on the elliptic relaxation concept have been presented for robust integration up to the wall (ItW), one of the eddy-viscosity type (the ζ-f model) and one at the second-moment closure level (Elliptic Blending Model, EBM). Realizing, however, that for large-scale complex industrial and environmental flows it may still be challenging to use sufficiently fine computational meshes to satisfy the ItW requirements, we presented in parallel some new developments in generalizing wall functions approach, as well as their combination with ItW models labeled as Compound Wall Treatment (CWT). The CWT was shown to provide satisfactory wall boundary conditions irrespective of whether the first near-wall grid node is placed within the viscous sublayer, in the fully turbulent wall region, or in the buffer zone in between. The performance of the concept was illustrated in several examples of generic flows, as well as in computations of laboratory and fullscale examples of flow and pollutant dispersion over block of buildings.