Some Developments in Turbulence Modeling of Environmental Flows

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-

−65−

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 ∂  ∂ Ui = Fi − + − τ ij  ν  ρ ∂ xi ∂ x j  ∂ x j Dt 

D T q ∂ = + Dt ρ c p ∂x j

 ∂T  − τθ j   α   ∂x j 

(1)

(2)

−66−

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 = ui u 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 − ui u j (we use now the overbar to denote Reynolds averaged quantities) is expressed in terms of the mean rate of strain Sij=0.5(∂Ui /∂xj+∂Ui /∂xj) and (in some models) of the mean vorticity Ωij=0.5(∂Ui /∂xj-∂Ui /∂xj),

1 ui u j − uk uk δ ij = f (ν t , Sij , Ωij ) 3

(3)

where the kinematic eddy viscosity νt 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 t = C µ k 2 / ε , where k = 0.5uk u k is the turbulence kinetic energy and ε is its dissipation rate, obtained from the solution of the model transport equations Dk ∂k ∂k ∂  ∂k  (4) = +U j = (ν + ν t )  + P + G − ε  Dt ∂t ∂x j ∂xk  ∂xk 

Dε ∂ε ∂ε ∂  ∂ε  C P + Cε 3G − Cε 2ε = +U j = +Y (ν + ν t )  + ε 1  τ Dt ∂t ∂x j ∂xk  ∂xk 

−67−

(5)

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 ui u j − kδ ij = −2ν t Sij 3

(6)

and non-linear (NLEVM) where the stress-strain relation (3) takes a form of a non-linear series expansion in terms of Sij and Ωij, and νt 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 = 2 Sij Sij , 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 2 -f model of Durbin [4] (known also as v2-f model), where the eddy viscosity is defined by ν t = CµDυ 2 k / ε and in which, in addition to k and ε, a transport equation is solved for another scalar υ 2 (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 Dui u j / Dt is solved for each component of ui u j  Dui u j ∂U j ∂U i  − Dij =  f i u j + f j ui  −  ui uk + u j uk +φ −ε Dt ∂xk ∂xk  ij ij    

(7)



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 ui u 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-

−68−

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 υ2-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 υ2 and an additional elliptic relaxation concept to sensitize υ2 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 υ2-f model possess some features that impair its computational efficiency. The main problem is with the wall boundary condition f w → −20υ 2ν 2 /(ε y 4 ) 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 υ2 and f equation, but most commercial as well as in-house codes use more convenient segregated solvers. Alternative formulations of the υ2

−69−

and f equations have been proposed which permit fw=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=υ2/k instead of the equation for υ2, 

ν Dζ ζ ∂  = f − P+ ν+ t  Dt k ∂x j  σ ζ 

 ∂ζ     ∂x j   

(8)

in combination with an elliptic relaxation function (here based on the quasi-linear pressure-strain model of Speziale, Sarkar and Gatski (SSG) [12], P  1 2 (9) L2∇2 f − f =  c1 + C2   ζ −  τ ε  3 Here the eddy viscosity is defined as ν t = cµζ kτ , where cµ is different from the conventional Cµ, 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 w = −2νζ / y 2 , 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 f = f '− 2v(∂ζ 1/ 2 / ∂xn )2 (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

α − L2∇2α = 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

φij = (1 − α 2 )φijw + α 2φijh ε ij = (1 − α 2 )

(11)

ui u j 2 ε + α 2εδ ij k 3

(12)

In equation (11), φ ijh 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

ε 1  φijw = −5  ui uk n j nk + u j uk ni nk − uk ul nk nl ( ni n j + δ ij )  k 2  where the unit wall-normal vector is evaluated from n = ∇α / ∇α .

−70−

(13)

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 stan2 3

dard eddy-viscosity stress-strain relation ui u j − kδ ij = −2ν t Sij with ν t = Cµ k 2 / ε , in which Cµ is now a variable computed from

 ∂U i  Cµ =  −ui u j   ∂x j  

 k2 2   S  ,  ε 

S = 2 Sij S ji

(14)

Equation (14) implies the minimum square error between the stress components obtained by EVM and DSM, dE/dCµ=0, d2E/dCµ2>0, where E = ( 2ν t Sij + ui u j ) . 2

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. 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

−71−

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 (530 ). 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

−81−

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 Cµ, 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 Cµ 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 Cµ (Fig. 12b) acts in a similar spirit as the additional random or stochastic forcing, but it is much simpler. Note that Cµ at the interface is evaluated from the instantaneous field, depending on the definition of νt. For example, for a one-equation (k) model with νt=Cµ lµ k1/2 and for the two-equation (k-ε) model with νt=Cµ fµ k2/ε, respectively, Cµ is evaluated from: Cµ =

ν SGS 0.5

lµ k RANS ,int

;

Cµ =

f µ ( k 2 / ε )ν SGS

( f (k / ε )) 2

(28)

2

µ

Figure 12. Variation of (time invariant) Cµ across the flow (top) and histogram of the instantaneous and homogeneously averaged Cµ at the RANS/LES interface (bottom) [30].

Figure 13 shows the mean velocity and turbulent shear stress profiles in a plane channel for Reτ=2000 obtained using a low-Re-number k-ε model with a damped Cµ for three different locations of the interface. The smoothing effect of the instantaneous, instead of locally averaged Cµ 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

−82−

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 Reτ=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(LLES /LRANS), by

−83−

which the model/computer “sees” the grid. This function controls the switching of the characteristic turbulence length scale from LRANS (computed from the RANS model, e.g. LRANS=k3/2/ε if k-ε model is used) to LLES associated with the reference size of the grid cell (e.g. ∆V1/3). 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 ( C P − Cε 2ε ) + D = f ε1 ε Dt τ

(29)

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 νt just as in the DES approach, as illustrated in Fig. 11b. It is noted that this intervention in the LES region diminishes also νt 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 o

Cε 2 = Cε 1 +

Cε 2 − Cε 1

L  1 + β  RANS   LLES 

(30)

2/3

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: Cε 2 = Cε 1 +

0.48

(31)

α

where α=max(1, LRANS/LLES), LRANS=fµ 3/2 and LLES=(∆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 Reτ=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.

−84−

Figure 15 Seamless RANS/LES coupling with the k-ε model

with Dejaon-Schiestel modification of Cε2.

We close this review by presenting a mixed approach in which an elliptic relaxation EVM (k-υ2-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 (32) Dt ∂ x j  ∂ x j   LRANS    LLES 

α = max  1,

and

LRANS =

3/ 2 ktot

(33) ; LLES = 0.8( ∆x∆y ∆z )1/ 3 ε and ktot.=kres+kmod. In other words, the dissipation rate ε is evaluated from its transport equation in the conventional RANS form when LLES>LRANS, and from k and ∆ when LLES1 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 where

ν t = max(ν tRANS ,ν tLES )

(34)

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 νt 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 Reτ=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.

−85−

Figures 16 Eddy viscosity in hybrid RANS ζ-f + dynamic LES model in a plane channel flow at Reτ=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=1x109 (right)

The velocity defect, ∆S ( x , y , ∆z ) =

U ( x , y , ∆z ) − U 0 ( ∆z )

(35)

U 0 ( ∆z )

computed by various models and hybrid RANS/LES using the Dejoan-Schiestel type of seamless model with Cε2 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

−86−

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-

−87−

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 Ψˆ (xi ,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. ~~ Ψˆ Υˆ = Ψ Υ + Ψ Υ + 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 (λv) and thermal (λθ) 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 1016 (Kenjereš and Hanjalić [42]). It is noted that the maximum Ra achievable by DNS is around 108 and by true LES about 109. 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 1012) in accord with the known correlations Nu∝Ra0.3, λv /H∝Ra-1/7 and λθ /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 λv (Ra) and λθ (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=2x1014. 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=109, 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.

−88−

Fig. 19. T-RANS predictions of Nu number (left) and of hydrodynamic (λν) and thermal (λθ) wall layer thickness in R-B convection for ten decades of Ra [42].

LES (2562x128), Ra=109

Hybrid (822x72) Ra=109

T-RANS ((822x72) Ra=109

T-RANS ((822x72) Ra=2x1014

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=109) and z/H=0.075 (Ra=2x1014); 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.

−89−

Figure 21. Long-term averaged kinetic energy, heat flux and temperature variance in the wall vicinity for Ra=109 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 ±10 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 Tg= ±20 C and ±10 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(1017). 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

−90−

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 < Cmax>). 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 ∆Τ=1 T=T(x,y,z,τ) τ) TEMPERATURE CONCENTRATION C=C(x,y,z,τ) τ) Residential

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.)

−91−

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.

−92−

Next we considered the potential and challenges in merging RANS and LES aimed at capturing some spectral dynamics in attached and mildly separated high Re-number flows, which are inaccessible to the conventional RANS or LES. Several approaches have been discussed and the recent method from authors’ group was illustrated in the application to the benchmark wind flow over a mildly sloped realistic hill (Askevein Hill), showing visible improvements. Computer simulations of diurnal dynamics of air circulation and pollutant dispersion over a mezzo-scale real town valley with complex terrain orography were then presented for two cases of a stable inversion layer over the lower atmosphere, illustrating the application of the transient RANS (T-RANS) method to flows dominated by thermal buoyancy. The full scale simulations gave plausible results, thus confirming the numerical efficiency and robustness of the proposed TRANS approach for predicting the mezzo scale environment over complex terrains. ACKNOWLEDGEMENT The authors thank M. Hadziabdic, R. Hagenzieker and M. Popovac for providing their research results and figures.

REFERENCES 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

Hanjalić, K. 2005, Will RANS survive LES? A view of perspectives. ASME J. Fluid Engng., 27, 831-839. Craft, T.J., Launder, B.E., Suga, K., 1996, Development and application of a cubic eddy-viscosity model of turbulence. Int. J. Heat and Fluid Flow, 17, 108-115. Menter, F. 1994, Two-equation eddy-viscosity turbulence model for engineering applications, AIAA Journal, 32, 1598-1605. Durbin, P. 1991, Near-wall turbulence closure modeling without ‘damping functions’, Theor. Comput. Fluid Dyn. 3, 1-13. Hanjalić, K. 1999, Second Moment Turbulence Closures for CFD: Needs and Prospects. Int. J. CFD, 12, 1999, pp. 67 –97. Hanjalić, K. Jakirlić, S., 2002, Second-moment turbulence closure modelling, In B.E. Launder and N. Sandham (eds) Closure strategies for turbulent and transitional flows, Cambridge Univ. Press, 47-101. Durbin, P.A. 1996, On the k-ε stagnation point anomaly. Int. J. Heat Fluid Flow 17, 89-90. Guimet, V. and Laurence, D. 2002, A linearised turbulent production in the k-ε model for engineering applications, In W. Rodi and N. Fueyo (eds) Engineering Turbulence Modelling and Meas.s 5, Elsevier. 157-166. Basara, B. and Jakirlic, S., 2003, A new hybrid modelling strategy for industrial CFD. Int.J.Num. Meth. Fluids, 42, 89-116 Wallin, S. and Johansson, A., 2002, Modelling streamline curvature effects in explicit algebraic Reynolds stress turbulence models. Int. J. Heat Fluid Flow, 23, 721-730. Hanjalić, K., Popovac, M., Hadžiabdić, M., 2004, A robust near-wall elliptic relaxation eddy-viscosity turbulence model for CFD, Int. J. Heat Fluid Flow, 25, 1047-1051. Speziale, C.G., Sarkar, S. Gatski, T.B. 1991, Modelling the pressure strain correlation of turbulence: an invariant dynamic system approach, J. Fluid Mech. 227, 245-272 Manceau, R. Hanjalić, K., 2002, Elliptic blending model: a new near-wall Reynolds-stress turbulence closure, Physics of Fluids 14(2), 744-754. Durbin, P. A Reynolds stress model for near-wall turbulence, 1993, J. Fluid Mech. 249, 465-498. Thielen, L., Jonker, H.J., Hanjalić, K., 2003, Symmetry breaking of flow and heat transfer in multiple impinging jets, Int. J. Heat Fluid Flow, 24, 444-453. Craft, T.J., Gerasimov, A.V., Iacovides, H. and Launder, B.E., 2002, Progress in the generalization of wall functions treatments. Int. J. Heat Fluid Flow, 23,148-160. Craft, T.J., Gant, S.E., Iacovides, H. and Launder, B.E., 2004, A new Wall Function strategy for complex turbulent flows, Numer. Heat Transfer, Part B, 45, 301-317. Popovac, M. and Hanjalić, K. 2005, A combined WF and ItW treatment of wall boundary conditions for turbulent convective heat transfer, 9th UK National Heat Transfer Conf., Manchester 5-6 Sept. 2005. Kader, B.A. 1981, Temperature and concentration profiles in fully developed turbulent boundary layers, Int. J. Heat Mass Transfer, 24(9) 1541-1544.

−93−

20 Ng, E.Y.K., Tan, H.Y., Lim, H.N., and Choi, D. 2002, Near-wall function for turbulence closure models, Comp. Mechanics 29, 178-181. 21 Hagenzieker, R. Numerical simulations of turbulent flows over hills and complex urban areas with dispersion of pollutants, M.Sc. thesis, Delft University of Technology, 2006. 22 Hagenzieker, R., Kenjeres, S. and Hanjalić, K. (submitted for publication) 23 Brown, M.J., Lawson, R.E., DeCroix D.S. and Lee, R.L.: Mean flow and turbulence measurements around 2-D array of buildings in a wind tunnel. Proc. 11th Conf. on the application of air pollution meteorology, Long Beach Ca, USA. 2000. 24 Pavageau, M. and Schatzmann, M.: Wind tunnel measurements of concentration fluctuations in an urban street canyon, Atmospheric environment, 33:3961-3971, 1999. 25 Brown, M.J., Lawson, R.E., DeCroix D.S. and Lee, R.L.: Comparison of centerline velocity measurements obtained around 2D and 3D building arrays in a wind tunnel. Proc. Int. Society of Environmental Hydraulics Conference, Tempe Arizona, USA, 2001. 26 Ketzel, M., Louka, P., Sahm, P., Guilloteau P.E., Sini, J.-F. and Moussiopolous, M. : Intercomparison of numerical urban dispersion models – Pt 2 : street canyon in Hannover, Water, Air and Soil Pollution: Focus, 2:603613, 2002. 27 Liedtke, J., Leitl, B. and Schatzmann, M.: Car exhaust dispersion in a street canyon – wind tunnel data for validating numerical dispersion model, Proc. 2nd East European Conf. on Wind Engineering, Prague 1998. 28 Hanjalić K., 2002, One-point closure models for buoyancy driven turbulent flows. Annual Review Fluid Mechanics, 34, p. 321-348. 29 Kenjereš, S., Gunarjo S. and Hanjalić K. 2005, Contribution to elliptic relaxation modeling of turbulent natural and mixed convection, Int. J. Heat Fluid Flow, 26(4), 569-586. 30 Hanjalić, K., Hadžiabdić, M., Temmerman, L. and Leschziner, M.A., 2004, Merging LES and RANS strategies: zonal or seamless coupling. In R. Friedrich, B. Geurts and O. Métais (Eds) Direct and Large-Eddy Simulations V, Kluwer Publ. 451-464 31 Temmerman, L., Leschziner, M.A., Hanjalić, K.. 2005, A hybrid two-layer URANS-LES approach for largeeddy simulation at high Reynolds numbers, Int. J. Heat Fluid Flow, 26, 173-190. 32 Spalart, P., 2000, Strategies for turbulence modelling and simulations, Int. J. Heat Fluid Flow, 21, 252-263. 33 Nikitin, N.V., Nocoud, F., Wasistho, B., Squires, K.D., Spalart, P.R., 2000, An approach to wall modelling in large-eddy simulations. Phys. Fluids, 12(7), 1629-1632 34 Piomelli, U., Balaras E., Pasinato, H., Squires, H., Spalart, P., The inner-outer layer interface in large-eddy simulations with wall-layer models. Int. J. Heat and Fluid Flow, 24, pp.538-550, 2003. 35 Davidson, L., Dahlstöm, S., Hybrid LES-RANS: An approach to make LES applicable at high Reynolds number, Proc. (CD) CHT-04, Int. Symp. on Advances in Computational Heat Transfer, April 19-25, Norway, ICHMT/Begell House, 2004. 36 Girimaji, S. S., Srinivasan, R., Jeong, E., PANS turbulence model for seamless transition between RANS and LES: fixed-point analysis and preliminary results, Paper No FEDSM2003-45336, Proc. ASME FEDSM'03, 2003. 37 De Langhe, C., Merci, B., Lodefier, K. and Dick, E. 2003, Hybrid RANS-LES modeling with the Renormalization group. In K. Hanjalić et al. (eds) Turbulence, Heat and Mass Transfer 4, Begell House Inc., 697-704. 38 Dejoan, A. Schiestel, R., 2001, Large eddy simulations of non-equilibrium pulsed turbulent flow using transport equations subgrid scale model. Proc. Int. Symp. Turbulent Shear Flow Phenomena 2 (TSFP2), Stockholm, Sweden, 27-29 June 2001. 39 Hadžiabdić, M. 2005, LES, RANS and combined simulation of impinging flows and heat transfer, PhD thesis, Delft University of Technology. 40 Kenjereš, S. and Hanjalić, K., 2006, LES, T-RANS and hybrid simulations of thermal convection at high Ra numbers, Int. J. Heat Fluid Flow, (in press). 41 Taylor, P.A. and Teunissen, H.W.: The Askervein hill project; overview and background data. Boundary- Layer meteorology, 39:15-39, 1987. 42 Kenjereš, S. and Hanjalić, K., 1999, Transient analysis of Rayleigh-Bénard convection with a RANS model. Int. J. Heat Fluid Flow 20(3), 329-340. 43 Kenjereš, S., Hanjalić, K., 2002. Combined effects of terrain orography and thermal stratification on pollutant dispersion in a town valley: a T-RANS simulation, Journal of Turbulence, .3 (26), p. 1-21. 44 Hanjalić, K. and Kenjereš, S. 2005, Dynamic simulation of pollutant dispersion over complex urban terrains: a tool for sustainable development, control and management, Energy, 30, 1481-1497.

−94−