as approximately constants, mean velocity obeys the semi-logarithmic law and the characteristic turbulence ...... The basic DSM (BDSM) is that of. Launder ...
Closure Models for Incompressible Turbulent Flows
K. Hanjali´c Department of Applied Physics Delft University of Technology Lorentzweg 1, 2628 CJ Delft, The Netherlands
1
Introduction
Turbulence closure models for RANS (Reynolds-averaged-Navier-Stokes) equations (known also as one-point closure methods) are at present the only tool available for the computation of complex turbulent flows of practical relevance. Despite noted deficiencies, they offer still the best means for meeting the industrial needs and reconciling the computational pragmatism with the theoretical rationale. While, admittedly, the RANS turbulence closures have not fulfilled the early expectations, some recent advancements in the second-moment and non linear-eddyviscosity approaches, as well as introducing some new concepts such as elliptic relaxation, offer significant improvement and broaden the range of applicability as compared with the standard liner eddy-viscosity models used by industry. Unfortunately these advances have not found much receptiveness among industrial users. One of the reasons is the unavoidable increase in computational complexity. Discouragement came also from insufficiently tested success claims that were later disapproved after testing the model in other flow types. Numerical handling of complex models has improved and more and more solutions of complex flows by advanced models can be found in the literature. Some CFD vendors offer now the nonlinear eddy-viscosity, improved three-equation and blended linear models, as well as the basic second-moment models, as standard options in their codes. But, perhaps, the most important development is in the awareness that an alternative to RANS modelling for industrial use is still not in the offing. A great variety of turbulence models available in the literature bring some confusion to new and inexperienced users. On the other hand, experienced industrial users are generally aware of the limitations of the basic models that serve as the backbone of most industrial CFD codes, but find it difficult to decide in which direction to search for improvement. For successful application of CFD codes the reliability and hence the quality of the calculations must be ascertained before the results can be used. This is one of the major issues of quality and trust in the prediction methods. The quality depends on both the turbulence model (mathematical equations of an idea to represent turbulence effects on the mean flow), and the numerical method used for solving these equations. With current computer power it seems that we are at, or getting very near to, the stage where the numerical errors can be kept under control. Yet the turbulence models introduce still a high degree of uncertainty. Despite three decades of development and impressive achievements, there are still some physical effects the modelling of which has not yet been accepted as satisfactory among the turbulence research community. Streamline curvature and extra strain rate, unsteadiness, wall proximity, three-dimensionality, flow separation and reversal, swirl, rotation and buoyancy are perhaps the most prominent examples. These effects are present in many turbulent flows. Yet, their modelling still poses difficulties, not only because 1
of insufficient understanding of the physics, but also because of an apparent need to employ mathematical formulations and numerical schemes of greater complexity than had been hitherto used to model simpler, well-behaved flows. This article gives an overview of some of many turbulence closure models available nowadays in the literature. A comprehensive coverage is beyond the scope of the present article. This can be found in a number of books (e.g. J. Piquet 1999, S. Pope 2000, Durbin and Reif 2000, to mention only the most recent ones), or review articles (e.g. Bradshaw 1972, Mellor and Herring 1973, Reynolds 1976, Lumley 1978, Lakshiminarayana 1986, Nallasamy 1987, Rodi 1982, 1986, 1988, Hanjalic 1988, 1994, 1999, Launder 1989, 1990, Leschziner 1989, Speziale 1991, So et al. 1991, and others). We begin with traditional linear eddy-viscosity models trough which the basic modelling rationale and principles are introduced. The focus of the article is, however, on advanced models, primarily on second-moment closures, which offers still the best prospectives for computing complex flows.
1.1
The RANS equations
The RANS approach is based on closing and solving Reynolds-averaged equations for the conservation for mass, momentum, heat and chemical species. For incompressible or mildly compressible fluid (neglecting density fluctuations, but allowing for variable mean density) the RANS equations can be written as • Continuity equation:
∂ρ ∂(ρUi ) + =0 ∂t ∂xi
(1)
• Reynolds-averaged Navier-Stokes equations (’RANS’) ∂P ∂ DUi = Fi − + ρ Dt ∂xi ∂xj
"
µ
• Reynolds-averaged energy equation: DT ρcp = ρcp Dt
2 ∂Uk ∂Ui ∂Uj + − δij ∂xj ∂xi 3 ∂xk
∂T ∂T + Uj ∂t ∂xj
!
∂ = ρ q˙g + ∂xj
!
− ρui uj
∂T λ − ρ cp θui ∂xj
• Reynolds-averaged concentration equation: DC = ρ ρ Dt
∂C ∂C + Uj ∂t ∂xj
!
∂ = ρ r˙ + ∂xj
#
∂C ID − ρcuj ∂xj
!
!
(2)
(3)
(4)
where Ui , T , C and P stand for mean (time-, phase or ensemble averaged) velocity, temperature, concentration and pressure, Fi stands for body forces, ui , θ, c, p and fi denote their fluctuations, q˙g is an internal heat source, r˙ is the reaction rate, µ is dynamic viscosity, λ is thermal conductivity and ID is mass diffusivity. The last terms in equations (2, 3, 4) in the form φuj represent turbulent flux of momentum, heat and species, respectively, i.e. ρ ui uj = τijt is the turbulent stress tensor, ρcp θuj = qit is the turbulent heat flux vector and ρcuj = jit is the turbulent mass flux vector. 2
1.2
The Closure Problem
The Reynolds averaged transport equations contain unknown variables (second moments φuj ) as a consequence of averaging. In order to close the equations set, these variables need to be supplemented before any solution can be obtained. This problem is known as the Turbulence Closure Problem. For this we need additional algebraic or differential relations. A set of mathematical equations which provide unknown variables, is called Turbulence Closure Model. Note that these second moments are always vectors of higher order than the basic variables, i.e. if φ is scalar, φuj is a vector, if φ is a vector φuj is a second order tensor. This complicates the closure problem, because we need to solve many more equations than for a laminar flow of the same complexity. The type (algebraic or differential) and the number of auxiliary equations defines the closure level. Two basic levels of modelling are currently used in computational fluid dynamics and transport processes: Eddy Viscosity/Diffusivity Models (EVM) (known also as first-order models) and Second-Moment Closure Models (SMC) (known also as Reynolds stress/flux models or secondorder models). Each category has a number of variants. The first-order models assume that the turbulent flux of momentum, heat and species are directly related to the mean flow field, i.e. mean velocity, mean temperature and mean concentrations, respectively. In the second-order models the turbulent flux is obtained by solving separate equations for each flux component φuj . If these equations are algebraic, the models are known as Algebraic Second-moment Models (ASM). The most advanced models solve differential transport equations for the second moment, hence the acronym DSM.
1.3
Eddy-viscosity/diffusivity concept
The basic class of turbulence models, Eddy Viscosity/Diffusivity Models (EVM), are based on the Boussinesq (1877) assumption that the turbulent stress tensor can be expressed in terms of the mean rate of strain in the same way as the viscous stress for Newtonian isotropic fluid, except that the coefficient of the molecular viscosity is replaced by eddy viscosity. The same principle applies to other constitutive relations for the flux of species, heat, as well as for other flow properties, i.e.: Turbulent mass flux of n species: (n)t
ji
(n)
= ρ c(n) ui = −IDt
Turbulent heat flux qi = ρ cp θui = −λt Turbulent stress
∂C (n) ∂xi
∂T ∂xi
1 2 τijt = −ρui uj = − ρkδij + 2µt (Sij − Skk δij ) 3 3
(5)
(6)
(7)
1 ∂Ui ∂Uj + ) is the mean rate of strain, k = 21 ui ui is the kinetic energy, µt is the where Sij = ( 2 ∂xj ∂xi (n) eddy (turbulent) viscosity, IDt is the eddy diffusivity for n species, and λt is the eddy thermal 3
conductivity. It is noted that for heat transfer problems the heat flux is often expressed in more convenient form as qi ∂T = θui = −αt (8) ρ cp ∂xi where αt is eddy temperature diffusivity. Further, eddy diffusivity for passive scalars, such as species and heat are often expressed in terms of eddy viscosity, which implies a similarity between the turbulent momentum, heat and species transfer. By analogy with molecular transport, one can define now the turbulent Prandtl number for thermal transport σt =
µt /ρ αt
Similar definition for turbulent Prandtl/Schmidt numbers can be defined for species, but also for other properties, as will be discussed later. Direct analogy between the formulation of molecular and turbulent flux of momentum, heat and species, as implied by Boussinesq EVM approach, makes it possible to express the total flux in terms of effective values (n)
(n)
IDef f = ID(n) + IDt
αef f = α + αt ,
µef f = µ + µt
(9)
Hence, the EVM approach leads to the same form of conservation (transport) equations as for laminar flows, except that the molecular exchange coefficients are replaced by their effective values. This approach is computationally very convenient since the same algorithm and computational code can be used for both laminar and turbulent transport phenomena without having to make any modifications. The problem of closure remains, however, except that now it is reduced to defining the eddy viscosity and diffusivity coefficients. It should be noted that molecular viscosity and diffusivity coefficients are local fluid properties, whereas the turbulent coefficients are flow properties. In flows at high Reynolds numbers the eddy viscosity exceeds the molecular viscosity by orders of magnitude, except very close to a solid wall where the turbulent fluctuations are damped. The ratio of turbulent to molecular diffusivities depends, however, on the molecular Prandtl number: in flows of highly conducting fluids such as liquid metals, where with Pr=O(10−2 ), αt /α can be ≈ 1 whereas µt /µ >> 1. Because the turbulent viscosity and diffusivity are flow (in fact turbulence) properties, the turbulent Prandtl number σt is O(1) irrespective of the type of fluid, reflecting the fact that turbulent transport of heat and mass are controlled by momentum transport, except in case of ’active’ scalar fields (temperature, concentration) as in buoyancy driven flows. 1.3.1
Turbulent viscosity/diffusivity - Classification of Turbulence Models
According to the kinetic theory of gases, the molecular viscosity of a fluid is proportional to the product of the molecular mean free path and the average speed of the molecules. By analogy, the turbulent viscosity can also be expressed as a product of the characteristic turbulence length and velocity scales (the size of eddies transporting momentum and the velocity of this transport). This also follows from dimensional analysis, hence we can write νt =
µt ∝ LU ρ
(10)
where the scales L and U need to be determined. In the early models, both L and U were defined in terms of flow geometry and mean flow properties: L was associated with the 4
characteristic flow dimension and U with mean flow velocity or its gradients (mixing length, algebraic eddy-viscosity models). Because µt represents the transport of momentum by turbulent fluctuations, we may expect that this coefficient will be determined by large-scale eddies, the size of which is close to the characteristic dimension and velocity of the flow itself. This is a basic presumption of algebraic eddy-viscosity models, among which the mixing length model is the simplest and most known in the literature. It is also one of the oldest turbulence models, developed in the beginning of this century. The mixing length model is rarely used nowadays, and only for an initial ’guess’ of the flow field , e.g. for the initiation of the velocity and turbulence field to speed up the computation with a more sophisticated model. Of course, sensible results can be expected only if the mixing length can be defined in terms of flow geometry with a reasonable certainty.
5
2
Basic Eddy Viscosity/Diffusivity Models
An obvious choice for defining U is to use the turbulent kinetic energy k = 12 ui ui (Kolmogorov, 1941, Prandtl 1942), so that U = k 1/2 (11) q
It is noted that k 1/2 = 12 (u21 + u22 + u23 is used as a measure of the averaged turbulence intensity. A transport equation for k can easily be derived. The unknown terms need to be modelled, but the equation remains relatively simple and easy to solve. Defining and providing adequate length scale L is more difficult and uncertain. While all models use k 1/2 as a velocity scale, many variants of models can be found in the literature differing in the choice of L Two basic classes of differential EVMs can be distinguished, depending on how many differential equation need to be solved to provide eddy viscosity νt : • One-equation models (only the differential transport equation for k is solved, whereas L is defined algebraically, usually in terms of flow geometrical parameters, in a similar manner as mixing length); • Two-equation models: in addition to k-equation, another differential transport equation is solved which provides the characteristic turbulence length scale L, either directly, or in combination with k. One-equation models have been popular in some branches of engineering, primarily in aeronautics for computing the flows around aircraft wings, fuselage, and even around the complete aeroplane. The best known one-equation models are Cebeci and Smith (1967), Baldwin and Lomax (1978), Norris and Reynolds (1975) and more recently Spalart and Almares (1997). All these models have been tuned to reproduce external flows (wall boundary layers) such as prevail in aeronautics, but are usually unsuitable and unreliable for predicting any complex internal flows. For flows in complex geometries it is difficult to define L. Besides, the characteristic length scale evolves both in time and space and depends on turbulence and flow ’history’ and boundary conditions, what can not be accounted for by an algebraic specification. In two-equation models the length scale is obtained from a ’scale-providing’ differential transport equation. Although the natural choice would be to derive and model an equation for the length scale L itself, as suggested by Rotta (1951, 1955), this approach was not very successful. Instead, one can derive, model and solve a transport equation for a product involving the length scale and the turbulent kinetic energy, i.e. k p Lq , which will have a standard form: D(k p Lq ) = Production of (k p Lq ) - Destruction of (k p Lq ) + Diffusion of (k p Lq ) Dt Depending on the choice of p and q, the variable can have a different physical meaning. Various combinations have been proposed in the past. There are no theoretical arguments which would favour any form. However, the most popular ’scale-providing’ variable is the rate of turbulent eni ∂ui ergy dissipation ε = ν ∂u ∂xl ∂xl which is the exact sink term in the equation for k (Davidov, 1959, 1960, Hanjali´c and Launder, 1972). Equations for k and ε, together with the eddy-viscosity stress-strain relationship constitute the k-ε turbulence model. Note that the characteristic turbulence length and time scale can be defined in terms of k and ε using dimensional argument, 6
i.e. L = k 3/2 /ε and τ = k/ε. Other popular scale providing variables is ω = ε/k which can be interpreted as a characteristic frequency, r simply as ε normalized with k. Hence, depending on the choice for the scale-providing variable, the eddy viscosity can be defined as µt = Cµ ρk 1/2 L = Cµ ρkτ = Cµ ρ
k2 k = Cµ ρ ε ω
(12)
Note that the definitions of various scale-providing variables may involve a proportionality coefficient,in which case Cµ can have different values for different models.
2.1
The k Equation
This is the fundamental and most important equation in the practice of turbulence modelling. It defines the dynamics (and the budget) of turbulence kinetic energy. Its analysis in some simple generic flows illustrates the major physical mechanisms in a turbulent flow within the frame of the Reynolds averaging approach. The exact k equation is derived by scalar multiplication of the transport equation for the velocity fluctuation Dui /Dt by the velocity fluctuation itself ui , leading to: !2
∂ui ∂Ui ∂k Dk ∂k ∂ = ρ fi ui −ρui uj −µ ≡ρ ρ + ρUj + | {z } Dt ∂t} ∂xj ∂xj ∂xj ∂xj | {z {z } | {z } | | {z } | G L ε C P
∂k µ − ρk ′ uj − pui δij ∂xj {z
D
!
(13)
}
The terms have the following physical meanings: L - local change of k in time C - convective transport of k P - production of k (of turbulence!) due to mean-flow deformation (work of turbulent stresses associated with the mean flow deformation, or transfer of energy from the mean motion to turbulent fluctuations by the action of Reynolds stresses); (note that usually u′i u′j < 0 when ∂Ui /∂xj > 0 so that in most flows P > 0). G is the production (or destruction) of k by body force; ε - dissipation of k into heat by viscosity (Note ε > 0 always!) D - diffusive transport of k (viscous, by fluctuating velocity and by fluctuating pressure, respectively) It should be noted that u′i u′j is obtained from the eddy viscosity model, so that P is determined, whereas ε is supplied from the ε equation. In order to close k equation, it remains only to model the turbulent diffusion by fluctuating velocity and fluctuating pressure. Modelling usually requires to introduce one or more empirical coefficients, which need to be determined from experimental or other (e.g. DNS) data. For this purpose, but also for studying the turbulence physics, we can apply the k (and other) equation to some simple, generic flow situations (or regions of such flows) in which some of the 7
terms are zero or negligible. Several cases can be identified, in which the k-equation reduces to a simple two-term balance: (a) energy decay in homogeneous turbulence:
U1
∂k = −ε dx1
(b) rapid distortion:
Uj
∂k =P dxj
(c) convection-diffusion equilibrium (pure transport):
Uj
∂k =D dxj
(d) local turbulence energy equilibrium:
0=P −ε
(e) diffusion-dissipation equilibrium:
0=D−ε
Of particular importance are the cases (a), (b) and (d), because of their relevance to many industrial and environmental flows. The reduced forms of the turbulence energy equation are regarded as constraints which need to be satisfied by a turbulence model, and serve thus as a basis for defining the empirical coefficients. These cases will be discussed in more detail later, and it suffices here to indicate briefly their relevance. a). In the absence of production (no mean-rate of strain, nor body force) in a homogeneous turbulent flows, the turbulence kinetic energy will freely decay and will ultimately be dissipated by viscosity. This is a natural, irreversible process. In the initial period, while the turbulence Reynolds number is still high, the process occurs by energy cascading towards smaller and smaller eddies, independent of viscosity. At a later stage when the Reynolds number becomes sufficiently small, the decay is governed by viscosity. Experiments show (confirmed by DNS) that both, the initial and final periods of decay can be expressed approximately by exponential law, k ∝ t−n (or ∝ x−n ), with n ≈ 1.2 in the initial period, and n ≈ 2.5 in the final period. This decay law can be used to determine the coefficient associated with the sink term of the scale-providing equation. b). If a turbulent flow is subjected to a sudden mean-rate of strain, the dissipation can be neglected as compared with the imposed turbulence production term. Such is a flow subjected to a sudden contraction, free turbulent flow encountering a solid wall parallel or inclined to the flow direction, or a change in the wall topology, or a boundary layer flow over a stagnant wall encountering transversely moving wall, e.g. turbine rotor. Useful information for modelling and model validation (particularly of second-moment closures) can be obtained by analyzing the response of turbulence to rapid distortion, since the uncertainties in the model ε equation can be ignored. c). At the free edge of a turbulent flow both production and dissipation can be neglected as compared with the convection and diffusion which are roughly in balance. d). The local energy equilibrium prevails in the inner zone (logarithmic region) of a wall boundary layer at zero or weak pressure gradients. Both the convective and diffusive transport are here small in comparison with the production and dissipation which balance each other locally. This energy balance has a number of additional implications: the shear stress can be regarded as approximately constants, mean velocity obeys the semi-logarithmic law and the characteristic turbulence length scale increases linearly with the wall distance. All these features are exploited in tuning the the coefficients in a turbulence model. 8
e). This last case is encountered e.g. in the central zone of a fully developed pipe or channel flow, where the production can be neglected (weak or no mean rate of strain) and convection is zero by definition. Locally dissipated turbulence is supplied by diffusion from other regions in the flow. Although not crucial for most engineering application, satisfying these conditions is often a challenge for a turbulence model.
2.2
The k − ε Model
The k − ε model is most widely used turbulence model, particularly for industrial computations. and has been implemented into most CFD codes. It is numerically robust and has been tested in a broad variety of flows, including heat transfer, combustion, free surface and two-phase flows. Despite numerous shortcomings, which have been discovered over the three decades of use and validation (see later), it is generally accepted that the k − ε model usually yields reasonably realistic predictions of major mean-flow features in most situations. It is particularly recommended for a quick preliminary estimation of the flow field, or in situations where modelling other physical phenomena, such as chemical reactions, combustion, radiation, multi-phase interactions, brings in uncertainties that overweigh those inherent in the k − ε turbulence model. The k − ε model consists of the transport equations for k, and for ε. The k equation has been discussed earlier and we recall it here in symbolic notation: ρ
Dk ≡L+C =P −ε+D Dt
(14)
In the k − ε model ε is supplied from the ε equation. Viscous diffusion is treated in the exact form, but for high Reynolds numbers flows it can be neglected (except close to a solid wall). In order to close the k equation, other terms need to be modelled. 2.2.1
The standard high-Re-number k − ε model
The following modelling practice leads to the standard k − ε model, which is applicable to free and near-equilibrium wall flows at high Reynolds numbers. Mean-Strain Production, P. In eddy-viscosity models ui uj is obtained from the eddy viscosity model (7), so that P is determined, P = −ρui uj
∂Ui 2 = 2µt Sij Sij − (µt Skk + k)Skk ∂xj 3 (15)
Body-Force Production G The body force production (or destruction) needs to be modelled. The approach depends on the character of the body force. The most frequently encountered body force is the gravitation. For the model of the effects of magnetic force (Lorentz) see below. Gravitational production. In the case of gravitational force, this term becomes G = gi ρ′ ui . In 9
most flows the density fluctuations ρ′ can be neglected1 . However, if the mean density stratification due to temperature or concentration variation is substantial (flows driven or affected by thermal or concentration buoyancy) the buoyant source G becomes important. The correlation ρ′ ui can be replaced by the gradient of mean density using the standard eddy-diffusivity ∂ρ , so that the gravitational source becomes expression i.e. −ρ′ ui = σνρt ∂x i G=−
µt ∂ρ ρσρ ∂xi
(16)
Note that σρ ≈ 0.9 is assumed to be the same as for the mean temperature. Alternatively, the density fluctuation ρ′ can be substituted by temperature and concentration fluctuations θ and c, i.e. ρ′ = ραc + ρβθ where α = 1/ρ(∂ρ/∂C)p and β = −1/ρ(∂ρ/∂T )p are concentration and thermal expansion coefficients, respectively. (Note that for multi-component mixture the first term should be replaced by the sum of terms for all components). Hence G = gi ρ(αcui − βθui ). Using eddy-diffusivity expressions for turbulent mass and heat flux (5, 6) the buoyancy production can be written as G = gi (−α
µt ∂T µt ∂C +β ) σC ∂xi σT ∂xi
(17)
Turbulent diffusion It remains to model the turbulent diffusion by fluctuating velocity and fluctuating pressure. Most ˆ = Φ + φ) that −φuj = νt ∂Φ common is the gradient diffusion approach, which assumes (for Φ σφ ∂xj where σφ is the turbulent Prandtl/Schmidt number for property Φ. The pressure transport is usually lumped with fluctuating velocity transport, so that the total turbulent diffusion is expressed as µt ∂k −(ρk ′ uj + puj ) = σk ∂xj where σk ≈ 1 is the Prandtl-Schmidt number for kinetic energy. 2.2.2
Summary of the k − ε model for high-Re-number flows
Hence, the model k-equation is: "
µt ∂k ∂ Dk (µ + ) = P + G − ρε + ρ Dt ∂xj σk ∂xj
#
(18)
In contrast to the k equation, the model ε equation relies much less on exact derivation and more on empiricism. The exact equation for ε can be derived, but all terms except the material derivative (local change and convection) and viscous diffusion need to be modelled. Besides, ε used in turbulence models as a ’scale-providing’ variable represents in fact the energy transfer rate through the spectrum, which is controlled by large energy-containing eddies, whereas the exact ε equation describes the dynamics of small scale turbulence associated with dissipative processes. Only in a spectral equilibrium (when the cascade energy flow from large eddies is 1
except in combusting flows, when the density fluctuations are lumped and jointly modelled with the velocity fluctuations (Favre averaging), or in high-speed supersonic flows when the fluctuating Mach number, based on rms of the local fluctuating velocity is O(1), what corresponds usually to mean flow Mach numbers of O(5) or larger.
10
in balance with dissipation rate) are these two variables the same. For this reason the model ε equation is written here without much details on its derivation. The standard form for high Reynolds number flow is: Dε ∂Uk ε ∂ ρ = (Cε1 P + Cε3 G + Cε4 ρk − Cε2 ε) + Dt ∂xk k ∂xj
µt ∂ε σε ∂xj
!
(19)
It is seen that the model ε equation reflects the composition of the k equation: the source of ε contains the same terms, though each with a different coefficient, but scaled with the turbulence time scale τ = k/ε. One exception is the term with the coefficient Cε4 which takes into account the effect of fluid compression on the source of ε (although P already contains such a term). In this form the ε equation (and the k − ε model as a whole) is very simple and easy to incorporate into a CFD code. However, a close resemblance of k and ε equations indicates also at serious deficiencies and limitation of the model, as the ε equation is too simple to mimic the real physics associated with ε. Various empirical and intuitive modifications have been proposed to extend the validity of the k − ε model, usually in form of an extra source term Sa in ε equation, or bypexpressing the coefficients as functions of nondimensional strain parameter Sk/ε, where S = Sij Sij (e.g. the Renormalization Group Theory RNG of Yakhot and Orszag 1989), or of the ratio of P/ε (Chen and Kim). However, none of modifications proved to yield improvements in all classes of flows. The use of various extension requires a caution and should be limited to flow classes for which the extension has been validated. For example, the enhancement of the effect of irrotational straining (Pope 1978, Hanjali´c and Launder 1980), improves predictions of flows with axial symmetry and with a strong pressure gradient, but is counterproductive in flows with a strong streamline curvature. The RNG modification was showed to improve the prediction of the recirculation length in separating flows, but spoils the prediction of accelerating flows. An additional source term which acts locally to compensate an excessive growth of the length scale, either in terms of the local wall distance (Yap, 1989) or in an invariant form (Hanjali´c and Jakirli´c 1997) improves the predictions of recirculating flows, particularly around separation and reattachment on smooth surfaces. The coefficients have the following standard values:
2.2.3
Cµ
Cε1
Cε2
Cε3
Cε4
σk
σε
σρ
0.09
1.44
1.92
0.8
0.33
1
1.3
0.9
Boundary Conditions: Wall Functions
The high-Re-number and high-Pe-number model equations are not applicable to flows or flow regions when the turbulent Reynolds or Peclet number become small. Because of no-slip conditions at a solid wall, such are all near-wall regions in any flow bounded by walls, irrespective of the bulk Re and Pe numbers. The common way to treat wall boundaries is to avoid the molecular layer and buffer zones adjacent to a wall and to bridge the solutions at the first control cell (assumed to be fully turbulent) with the wall properties. This is achieved by using the ’wall functions’, the set of semi-empirical functions, which have been derived from experimental evidence and similarity arguments.
11
For mean momentum and energy equation the wall functions are based on the assumed logarithmic velocitypand temperature distribution, scaled with the ’inner-wall’ scales: the friction velocity Uτ = τw /ρ and viscosity µ for mean velocity, and wall heat flux and temperature diffusivity for mean temperature. The standard logarithmic ’laws of the wall’ are modified as to relax conditions where τw → 0. For mean velocity the following wall functions are used: U∗ = where
1 ln(Ey ∗ ) κ
(20) 1/2
1/2
U ∗ = Cµ1/4
kp UP Uτ
y ∗ = Cµ1/4
and
ρkP yP µ
(21)
Here κ = 0.41 is von Karman constant, E = 9.8 is the constant in the law-of-the-wall, and index ’P’ denotes the values at the center point of the wall-nearest control cell. For mean temperature similar expressions are used: ∗
T = σT
where
1 ln(E y ∗ ) + Y κ 1/2 ρ cp (TP
T ∗ = Cµ1/4 kP π/4 Y = sin(π/4)
A κ
(22)
− Tw )
(23)
q˙w
1/2
Pr −1 σT
σT Pr
1/4
(24)
where P r is the molecular Prandtl number, σT is the turbulent Prandtl/Schmidt number for mean temperature, A = 26 is van Driest constant. Index ’P’ has the same meaning as in the velocity wall functions and index ’w’ denotes the wall values, i.e. Tw is the wall temperature and q˙w is the wall heat flux. Another expression for Y , proposed by Jayatilaka (1969) is also frequently in use. No wall-function equivalent is used for the k equation. Instead, the k-equation is solved over the first near-wall control cell in the same manner as for all other cells in the solution domain, using the zero-gradient conditions at the wall (zero flux of k), i.e.: ∂k =0 ∂xn
(25)
where xn denotes the coordinate normal to the wall. However, the production term is calculated using the velocity gradient evaluated from the logarithmic velocity law and the shear stress replaced by the wall shear stress, while the normal-stress production is neglected, i.e. τw ∂U = τw 1/4 P ≈ τw (26) 1/2 ∂y Cµ κρk yP P
Also, the dissipation in the k equation is evaluated by similarity arguments, assuming the linear length-scale variation with a distance from the wall: 3/2
εP = Cµ3/4 12
kP κyP
(27)
The ε equation can be treated in a similar manner. However, more common practice is to avoid solving the ε equation in the first cell, but to use the above expression for εP for the first near-wall cell. In cases when the first grid point falls within the viscosity/conductivity affected zone (y ∗ , 11.5), a switch from logarithmic to linear profile is usually practiced. This is, however, not recommended, because the equations are not suited to handle the flow in molecular region. 2.2.4
Near-wall and low-Re-number modifications of k − ε model
High Reynolds number k −ε and other models that require the use of wall functions, cannot deal with any flow where viscosity is important (characterized by low turbulence Reynolds number), such as in wall viscous sublayer and buffer zone or in transitional flows (laminar-to-turbulent and reverse transition). In order to make it possible to integrate equations up to the wall, ”nearwall” modifications are needed to account for viscosity effects, but also for other, nonvisous (kinematic) wall effects, such as wall blocking and pressure reflection. The early (as well as most recent) modifications of linear eddy viscosity models make no distinctions between viscous and nonvisous wall effects and involve relatively simple viscous modifications solely in term of local turbulence Reynolds number - hence the common notion ”low-Re-number” modifications. Whilst such practice is not justified, nor can lead to generally successful models, it has been widely used in industrial applications. The earliest modifications of the k − ε model are those of Jones and Launder (1972) (in further text denoted as JL model). We discuss the JL modification steps, partly for historical reasons, but also partly because this models is still used for computing of simpler flows with low-Re-number effects, and has proved to be more successful in predicting transitional flows than most other models. The first step is to introduce the viscous diffusion terms in the mean momentum equation as well as in the transport equations for turbulence variables k and ε, which are rightly neglected in high-Re-number models. Next we need to account for wall damping of the eddy viscosity, which is usually accomplished by introducing a damping function fµ i.e. µt = Cµ fµ ρ
k2 ε
(28)
where fµ is expressed in terms of turbulence Reynolds number Ret = k 2 /(νε), or some nondimensional distance from the nearest wall, e.g. y + = Uτ y/ν or Rey = k 1/2 y/ν. All these versions of fµ presume that viscosity (i.e. local turbulence Reynolds number) is the major and sole factor in near-wall damping of eddy viscosity, which is not true (see below). The next step in modifying the model is to introduce viscosity effects in some of the terms in the transport equations for turbulence properties. Because the production of k is calculated from the eddy viscosity, no additional modification is required. The same is the case with the sink of k defined by ε, provided that ε is correctly obtained from the solution of its transport equation. Major modifications, are, however, needed in the ε equation (19). Some ideas for these modifications can be inferred from the exact equation for ε, but only as a guide. A major modification is needed in the sink term of the ε equation. Recall that the coefficient Cε2 has been determined from the decay law of isotropic grid turbulence, assumed to obey k ∝ t−n , where n = 1.1. − 1.3 determined from experiments. The application of the k − ε model to such decaying turbulence (production and diffusion terms in both equations set to zero) implies that ε ∝ t−n−1 , yielding Cε2 = (n + 1)/n. The usually adopted value is Cε2 = 1.8 − 1.9. However, the 13
above quoted values for n were measured only during the initial (”inertial”) period of decay. It is evident from the above behaviour of k and ε that the turbulence Reynolds number Ret = k 2 /νε decays as Ret ∝ t1−n . When Ret becomes sufficiently small for viscosity to affect the turbulence decay, turbulence begins to decay much faster and the law of decay changes. We can still assume the same form of the decay law as in the inertial period, but this time experiments suggest that the decay exponent is n ≈ 2.5. If we want to retain the same form of the sink term in the high-Re-number model, in the final period of decay the coefficient Cε2 should reduce to ≈ 1.4. This change of Cε2 can be accommodated by an empirical function, fε which multiplies Cε2 . It is obvious that fε → 1 for high Ret number and 1.4/Cε2 for Ret → 0. The above modification of the sink term in ε equation is still not sufficient to satisfy the wall conditions where k = 0. From Taylor expansion of velocity fluctuations in the near-wall layer in term of wall-normal coordinate y we can express the near-wall value of ε in term of local k, i.e. εw = 2νk/y 2 (or 2ν(∂k 1/2 /∂xn )2 ). Because k ∝ y 2 it follows that εw is finite, which will cause the sink term in ε equation (19) Cε2 fε ε2 /k to approach ∞ when y → 0. A remedy to this is to replace ε2 with ε˜ ε where ε˜ = ε − 2ν(∂k 1/2 /∂xn )2 goes to zero at the wall. The source term in ε equation (19)needs in principle no modifications because viscous damping is accounted for through fµ in production P. However, at low Re numbers, where the model ε equation should represent more closely the true dissipation defined by the exact ε equation (see below) we need to account for some terms that are neglected in high-Re-number model. This is primarily the case with the term involving the second derivatives of the mean velocity (mean vorticity derivatives). A model of such a term has been proposed by Jones and Launder (1972). Accounting for all modifications discussed above leads to the following form of JL low-Re-number model ε equation (with the function fµ of Launder and Sharma 1973): ∂Uk ε ∂ Dε = (Cε1 P + Cε3 G + Cε4 k − Cε2 fε ε) + ρ Dt ∂xk k ∂xj where E = 2ννt and fµ = exp
−3.4 (1 + Ret /50)2
∂ 2 ui ∂xm ∂xn
µt ∂ε σε ∂xj
!2
fε = 1 − 0.3exp(−Re2t )
!
+E
(29)
(30)
(31)
Equations (29) to (31) together with the stress-strain relation (7), definition of eddy viscosity (28) and equation for kinetic energy (18) constitute the Jones-Launder-Sharma low Re-number model. Many other ”low-Re-number” models, both in k − ε and k − ω versions, have been proposed in the literature, which differ mainly in the form of damping functions and parameters used in these functions (Patel, Rodi and Scheuerer, 1985, Bredberg 2002). Some recent models have been found to perform better than the JL However, most models use the local wall distance y + to express the damping functions, what limits their use in more complex geometry and especially in transitional flows.
2.3
Other Two-Equation Eddy-Viscosity Models
In search for better models, a number of other two-equation models have been proposed, differing mainly in the choice of the scale-providing variable. One of the motives was to derive and solve 14
directly transport equation for a characteristic turbulence length or time scale, L or τ . The other motive is to have a variable for which the boundary conditions at a solid wall or free-stream boundary can be defined in exact form, what is not the case with the ε equation. A general scale-providing variable can be expressed in terms of k and ε as a product Ψ = k m εn and an ad hoc transport equation for Ψ can be constructed analogous to the model equation for ε: DΨ ∂Uk Ψ ∂ ρ = (CΨ1 P + CΨ3 G + CΨ4 k − CΨ2 ε) + Dt ∂xk k ∂xj
µt ∂Ψ σΨ ∂xj
!
(32)
It is obvious that one equation can be derived from another and that for homogeneous flows (zero diffusion) they are all the same. However, for inhomogeneous flows, the equations will differ in the diffusion term, and this is what leads to different performance of models using different specification of Ψ (i.e. different values of m and n). If the ε equation is used as a basis, equations for other variables can be derived in form (19), with the coefficients in the source terms taking the values CΨi = m + nCεi
(33)
A summary of various definitions of Ψ, coefficients for the two main source terms,(mean-strain production and viscous destruction) and of wall boundary conditions for Ψ is given in Table below. Model k−ε k−ω k−τ k−L k − kL
Ψ ε ω = ε/k τ = 1/ω = k/ε L = k 3/2 /ε kL = k 5/2 /ε
dimension [m2 s−3 ] [s−1 ] [s] [m] [m3 s−2 ]
ν t /Cµ
m
k 2 /ε k/ω kτ 1/2 k L (kL)/k 1/2
0 -1 1 3/2 5/2
n 1 1 -1 -1 -1
CΨ1
CΨ2
1.44 0.44 -0.44 0.06 1.06
1.92 0.92 -.92 -0.42 0.58
Ψw
2ν
∂k1/2 ∂xj
∞ 0 0 0
2
Despite physical appeal and convenient wall boundary conditions, L, (kL) or τ did not prove to be convenient variables. For example, the direct transformation of ε equation into τ equation transforms the sink term Cε2 ε2 /k into a constant Cτ 2 , which is neither realistic nor practical. The k − ω model of Wilcox et al. (1976, 1988, 1993) has been popular among some CFD users (primarily for aerodynamics applications), because apparently it is more robust and requires practically no modification for near-wall region: the seemingly inconvenient wall boundary condition ωw = ∞ can be easily dealt with by assigning a very large value to ω, or applying a hyperbolic variation ω ∝ 1/y 2 in the first grid point. In fact, using the near-wall polynomial approximation of velocity fluctuations and inner wall scaling (Uτ , ν) leads to the nondimen2 sional expression ω + = 2/y + which agrees very well with DNS data for y + < 5 for several wall flows (wall boundary layer, channel and sink flow) at different Reynolds numbers. However, the major problem arises in specifying the boundary conditions at free boundaries. A non-zero value of ω needs to be prescribed implying a non-zero k, even though the external flow may be non-turbulent. Besides, the solutions seem to be dependent on the adopted boundary value. A combined use of k−ω and k−ε model was proposed by Menter (1994), which uses the ω equation close to a solid wall, but transforms via an empirical blending function into the ε equation away from walls.
15
3
Some Advanced Eddy-Viscosity Models for Complex Flows
3.1
Which Flows are ”Complex”?
Complexity is a subjective notion and no unique criteria can be defined for classifying a turbulent flow as complex. From the point of view of modelling, the notion generally implies a departure from the thin shear layer approximations (Bradshaw et al. 1981), as well as any additional geometrical and physical complexities: • strong pressure gradient (high acceleration and deceleration), pulsation,
• the presence of ”extra” strain rates (in addition to the simple shear ∂U1 /∂x2 ): curvature, lateral divergence, bulk dilatation, swirl.. • body forces (rotational, buoyancy, electromagnetic,) • flow separation and recirculation
• stagnation and streamlines reattachment
• secondary motions (skew-induced and stress-induced) • three-dimensionality effects (cross-flows)
• compressibility effects and flow discontinuities (e.g. shock waves)
• strong property variations, chemical reactions, radiation;
• multi-phases and multi-components.
The phenomena and processes listed above often impose special demands on both mathematical models and numerical methods. These phenomena influence directly the turbulent stresses, (mainly via the stress generation), but also via the turbulence scales. These effects should appear in both the stress- and the scale-providing equations. Some cases require simply the additional exact or modelled equations and terms in the model (e.g. body forces, rotation, compressibility, multi-phase- and multi-component equations). In other cases, the model is expected to account for physical complexities itself (separation and recirculation, secondary motions, turbulence hysteresis, etc.). The ability to capture complexity effects is the basic criteria for judging the success and generality of a flow model. In the next sections we consider some of the ”complexities” listed and compare the ability of the EVM and DSM to capture them. However, we first identify some well known limitations of the basic (linear)two-equation models, discuss some general model requirements and expectation and ten move to discuss the exact equation for the turbulent stress.
3.2
Limitations of Two-Equation Linear Eddy Viscosity Models
Two-equation Eddy Viscosity Models (EVM) and, among them, the k-ε model, are today the most widely employed models for engineering computations. Their major advantage is the simplicity and suitability for an easy incorporation into the existing Navier-Stokes numerical codes. Computation of a variety of flows has shown remarkable agreement with experimental measurements in some simple classes of flows (e.g.2-D attached boundary layers and flows in conduits with pressure gradients, wall suction and blowing, some recirculating flows dominated by pressure etc.). EVM models can serve useful engineering purposes, provided their limitations are known. However, many results are misleading and wrong. 16
Extensive testing and application over the past three decades have revealed a number of shortcomings and deficiencies in linear EVM models such as: • limitation to linear algebraic stress-strain relationship (poor performances wherever the stress transport is important, e.g.: nonequilibrium, fast evolving, separating and buoyant flows ) • insensitivity to the orientation of turbulence structure and stress anisotropy - (poor performances where normal stresses play important role, e.g.: stress-driven secondary flows in noncircular ducts ) • inability to account for extra strain (streamline curvature, skewing, rotation) • poor prediction particularly of flows at strong adverse pressure gradients and in reattachment regions. Other weaknesses are still to be discovered. However, certain facts are already more than obvious: basic EVM with isotropic eddy viscosity are totally inadequate for 3-D flows , as illustrated in Figs. 1 and 2, where the eddy viscosities for two directions are presented for two relatively simple 3-D flows.
Figure 1: Spanwise vs. streamwise eddy viscosity at different depths inside a 3-D wing-body junction boundary layer (⊓ - points inside the line of separation, Davenport and Simpson 1990)
3.3
Figure 2: Eddy-viscosity ratio on selected streamlines in a pressure-driven 3-D boundary layer created by an upstream facing wedge (Anderson and Eaton, 1987)
Requirements and Expectations
The major requirements and expectation of turbulence models can be first summarised as follows: • to mimic faithfully the flow and turbulence physics in a broad range of flow situations • to satisfy mathematical rigor and physical constrains, such as: – tensorial and frame-reference invariance 17
– realizability (impossibility to generate nonrealistic, e.g. negative values of essentially positive quantities, such as turbulence kinetic energy and its components - turbulent normal stresses, temperature concentration and other scalar variances and their dissipation rates), – two-component limit, such as encountered in near-wall area where the wall-normal turbulent stress decays faster than the streamwise and spanwise components – vanishing and infinite turbulence Reynolds and Pecklet number asymptotes, etc. • to be manageable with the relatively simple numerical methods, preferably with available CFD codes (usually design to solve the Navier-Stokes-type of differential equations) • and, in conjunction with the computational methods, to serve as computational tool for predicting new complex flows. So far there is no model available which satisfies all these requirements. However, while the standard EVM with linear stress-strain relationship has not even a remote chance of meeting the major constraints, the second-moment closure offers at least the potential to satisfy most of the requirement. The main advantages lies in the exact treatment of the turbulence production terms, be it by the mean strain or by body forces arising from thermal buoyancy, rotation or other forces. In addition, a solution of a separate transport equation for each component of the turbulent stress enables, in principle, accurate prediction of the turbulent stress field and its anisotropy, which often plays a crucial role in complex flows, either as a major source of turbulence energy such as in the stagnation regions, as a source of secondary motion, or in controlling the dynamics of longitudinal vortices. Accurate prediction of the wall-normal stress component is also important in reproducing the wall phenomena, the wall shear stress, heat and mass transfer. Further, capturing stress anisotropy also enables a more realistic modelling of the scale-determining equation, (dissipation rate or other variable). Before we move to extensive coverage of second-moment closures, we consider some recent advanced eddy-viscosity models that represent an intermediate closure level between the linear eddy-viscosity and second-moment closures. Three classes of such models are especially worth considering because of demonstrated successful performance in a number of wall bounded flows: • Elliptic relaxation concept of Durbin (1991) • Blended k − ω and k − ε model (SST model of Menter 1994) • Non-linear eddy viscosity models The most appealing and probably the most successful of all is the Elliptic relaxation model of Durbin (1991). The major feature is the realistic capturing of inviscid wall blocking effect via an elliptic relaxation function, for which a separate equation is solved. With simple lower bounds on the time and length scale, the model satisfies near-wall limits and thus, together with elliptic relaxation, makes it possible to perform integration of the governing equation up to the wall without using the conventional damping functions. Although still inferior to a full second-moment closure (the near-wall stress anisotropy is only partially reproduced), the elliptic relaxation eddy viscosity model provides a qualitative leap forward in capturing near-wall flow physics at very modest expense in terms of complexity and computer demands. We consider now the original and some recent improved variants of this model. 18
3.4
Elliptic-relaxation eddy-viscosity models
The key element in all conventional near-wall models is the introduction of a damping function in the eddy viscosity definition, usually denoted as fµ , which provides adequate suppression of turbulence very close to a solid wall. This function is invariably defined in terms of local turbulence Reynolds number, defined as Ret = k 2 /(νε), Rey = k 1/2 y/ν or Re+ ≡ y + = Uτ y/ν, thus accounting solely for viscous effect. However, an inspection of the transport equation for the turbulent stress tensor ui uj shows that close to a wall in a fully developed plane channel flow the equation can be truncated to a simple expression for the shear stress uv = −Ckv 2 /ε∂U/∂y suggesting that eddy viscosity should be defined as νt = Cµv kv 2 /ε
(34)
rather than the conventional definition Cµ fµ k 2 /ε, where v 2 is the wall-normal turbulent stress component and Cµ′ is different from Cµ . Comparing the two expression for eddy viscosity yields Cµv v 2 fµ ≈ Cµ k
(35)
Indeed a plot of fµ from equation (35) and of fµ = −uv/[Cµ (k 2 /ε)∂U/∂y] for a plane channel flow using DNS data for all variables on the right hand side of both equations, with Cµ′ ≈ 0.22, shows a striking resemblance except very close to the wall within the viscous sublayer (for y + < 7). Even here the difference is not very large. This shows that the damping function fµ does not represent viscous damping but serves merely as a correction for inadequacy of the conventional eddy viscosity. Major effect of wall damping is of kinematic nature and originates from inviscid blocking of wall-normal velocity and its fluctuations due to wall impermeability. This fact has long been known (e.g. Hanjalic and Launder 1976, Launder Tselepidakis 1988) as confirmed by early second-moment closures which retrieve the correct formulation of eddy viscosity in the near-wall region of a plane channel and equilibrium wall boundary layer. However, it was long considered that the only way to provide v 2 was to solve a full second-moment closure model which provides all stress component. A breakthrough came from Durbin (1991) who proposed to use eddy viscosity defined by equation (34) formulated as νt = Cµv υ 2 τ (36) where τ is the turbulence time scale (= k/ε for high Re-number flows) and υ is a new scalar velocity scale, the square of which reduces to the wall-normal turbulent stress component v 2 close to a solid wall. For υ 2 Durbin proposed a separate transport equation, derived from the parent transport equation for the turbulent stress tensor with simple linear pressure strain models for both the slow and rapid part. In order to keep the scalar character of the equation for υ 2 , all terms involving other stress components, such as P22 , are neglected, what is justified only very close to a wall. To account for the inviscid wall blocking in the pressure strain term, Durbin introduced the concept of elliptic relaxation. 3.4.1
Durbin’s k-υ 2 -f model
As the basis for deriving equation for υ 2 , Durbin used the equation for the wall-normal turbulent stress v 2 = u2 u2 (where ”2” denotes the coordinate normal to the wall) obtained from the general transport equation for the turbulent stress tensor, in which for convenience an extra term nεv 2 /k 19
is added and subtracted, the first being lumped with the pressure strain term and the dissipation ε22 : : Dv 2 v2 = ℘22 + P22 − n ε + D22 (37) Dt k where 2 ∂p v2 −ε22 + n ε ℘22 = − v (38) ρ ∂x2 k {z
|
Π22
}
where n is a parameter introduced to accommodate conveniently the wall boundary condition. In homogeneous flows one can adopt any model of the pressure scrambling term Π22 2 . Using the Rotta’s (1951) linear model for the slow pressure-strain term and the IP model of Naot et al. (1970) model for the rapid term, for homogeneous flows where ε22 ≈ 2/3ε, ℘22 becomes ℘h22
= (1 − C1 )ε
v2 2 − k 3
!
2 v2 P22 − P + (n − 1) ε 3 k
− C2
!
(39)
In order to account for wall blocking effect and flow inhomogeneity, Durbin introduced the elliptic relaxation concept by expressing ℘22 as ℘22 = kf
(40)
where f is obtained from an elliptic equation (for justification, see Durbin 1991) L2 ∇2 f − f = −
℘h22 k
(41)
where L is the characteristic turbulence length scale, equal to k 3/2 /ε in flow regions with high turbulence Re number. It is obvious that in the homogeneous limit f → ℘h22 /k. We can now write the final form of the equations for υ and f based on equations (37) and (41) by neglecting P22 as appropriate for near-wall flow regions "
Dυ 2 νt ∂υ υ2 ∂ (ν + = kf − n ε + ) Dt k ∂xj συ ∂xj c1 L2 ∇2 f − f = τ
υ2 2 − k 3
!
− c2
#
(42)
1 υ2 P + (n − 1) k τ k
(43)
The model is completed with the expression for the eddy viscosity (36) and equations for k and ε, which can be written as "
Dk ∂ νt ∂k = P + G − ρε + (ν + ) Dt ∂xj σk ∂xj "
#
(Cε1 P + Cε3 G + −Cε2 ε) ∂ νt ∂ε Dε = + (ν + ) Dt τ ∂xj σε ∂xj
(44) #
(45)
where τ stands for the turbulence time scale and, for convenience, c1 = C1 − 1 and c2 = 2/3C2 . 2 Note that Πij = Φij + Dij and models usually apply to Φij which, unlike Πij , is traceless for incompressible flow.
20
Because the elliptic relaxation accounts only for the inviscid wall effects, modifications are needed to account for viscosity if the integration is to be performed through the viscous sublayer up to the solid wall. Durbin proposed to achieve this by imposing lower bounds on both the time and length scale in term of Kolmogorov scales, i.e. "
k τ = max , Cτ ε
1/2 #
ν ε
k 3/2 L = CL max , Cη ε
(46)
ν3 ε
!1/4
(47) p
In addition, the coefficient Cε1 in ε equation is expressed as a function Cε1 = 1.4(1 + α k/υ 2 ), α = 0.045. This is in fact a compensation for the deficiency of the simple form of ε equation in the viscous wall region and can be regarded as a substitute for the additional term introduced into the low-Re-number form of ε equation either in terms of second-velocity derivative (Jones and Launder, 1972) or some other form. It resembles an other proposal in form Cε1 = 1.4(1 + αP/ε), which can also be used in the framework of k − υ 2 − f model, though it performs somewhat worse than the expression above. It is noted that equations (46) and (47) can be recast in terms of turbulence Reynolds number, Ret = k 2 /(νε) as τ=
h i k −1/2 max 1, Cτ Ret ε
(48)
h i k 3/2 −3/4 max 1, Cη Ret ε
L = CL
(49)
which associate to the traditional use of Ret to design near-wall viscous damping functions in low-Re-number eddy-viscosity and second-moment closures. The expressions for the time and length scale can be further extended to include the realizability constraints that guarantee non-negative eigenvalues of the turbulent stress tensor and thus always positive normal stress components (Durbin, 1996):
τ = max min
k ak , √ v ε 6Cµ |S|υ 2
!
ν3
, Cτ
k 3/2 k 3/2 L = CL max min , √ v ε 6Cµ |S|υ 2
!
ε
, Cη
!1/2
ν3 ε
where a ≤ 1 (recommended a = 0.6).
(50)
!1/4
(51)
The constraint, especially pertinent to two-equation eddy-viscosity models with P = νt (Sij )2 , was originally aimed at limiting excessive turbulence kinetic energy k in stagnation flow regions. It has been found also beneficial in computing flows with low turbulence level with large time scale τ , where even small P can ”detach” k from ε permitting it to grow uncontrollably. The coefficients in the k-υ 2 -f model are given in the table below. Cµv 0.22
Cε1 p
1.4(1 + 0.045 υ 2 /k)
Cε2
c1
c2
σk
σε
συ
Cτ
CL
Cη
1.92
0.4
0.3
1
1.3
1.0
6.0
0.25
85
It remains to specify the wall boundary conditions. For k, υ 2 and ε, the exact boundary conditions apply as used in other near-wall models, i.e. 21
kw = 0
2 υw =0
2νk y→0 y 2
εw = lim
(52)
The wall boundary condition for f is obtained by enforcing the wall limits in the υ 2 equation to ensure that υ →∝ y 4 as y → 0. For the original model, with n = 1, this constraint yields −20υ 2 ν 2 y→0 εy 4
fw = lim
(53)
This boundary condition can become troublesome and impair numerical stability, especially when the first grid point is too close to a solid wall. This is in contrast to most other nearwall models and contradicts with the need to resolve thin wall layers in some nonequilibrium flows. A remedy to this is solve equations for υ 2 and f in a coupled manner, what is not convenient in most industrial CFD solvers, which usually solve equation set in a segregated procedure. Another way to obviate the stability problem is to adopt n = 6 (this was the reason for introducing the parameter n) which makes it possible to satisfy the wall limiting budget of υ with fw = 0 at the wall. This changes the behaviour of the model and requires some tuning of the coefficients. Durbin recommends for this model version to use CL = 0.23, Cη = 70 and σε = 1.0. The modification of σε seems especially important for recovering the quality of predictions of the original model, but its justification is questionable because σε has a critical value close to 1.05-1.1 (for the commonly adopted values of coefficients in the k and ε equations) around and below which the model exhibits singularities and does not recover the observed behaviour of turbulence scales in some generic homogeneous flows (Umlauf and Burchard, 2003) 3.4.2
The k-ζ-f model
A version of eddy-viscosity model based on Durbin’s elliptic relaxation concept has been proposed by Hanjali´c and Popovac (2004) which solves a transport equation for the velocity scale ratio ζ = υ 2 /k instead of the equation for υ 2 . The motivation behind this development originated from the desire to improve the numerical stability of the scheme, especially when using segregated solvers. The ζ equation can be derived directly from the υ 2 and k equations (42) and (44) The direct transformation yields ζ ∂ Dζ =f− P+ Dt k ∂xk
"
νt ν+ σζ
!
#
∂ζ +X ∂xk
(54)
where the ”cross diffusion” X is a consequence of transformation and can be written in a condensed form as ! νt ∂ζ ∂k 2 ν+ (55) X= τε σζ ∂xk ∂xk The solution of the ζ equation (54) instead of υ 2 should produce the same results. However, from numerical point of view, two advantages can be identified: • instead of ε appearing in υ 2 equation, which is difficult to reproduce correctly in the nearwall layer, ζ equation contains the turbulence kinetic energy production P which is much easier to reproduce accurately if the local turbulent stress and the mean velocity gradient are captured properly - what is the main goal of all models.
22
• because ζ ∝ y 2 when y → 0, the wall boundary condition for ζ deduced from the budged of ζ equation in the limit when the wall is approached, reduce to the balance of only two terms, f and Dζ, with a finite value at the wall, whereas Pk ζ/k varies with y 3 fw =
−2νζ y2
(56)
This is a more convenient and easier reproducible form as compared with (53). In fact the boundary condition for fw has the identical form as that for εw and can be treated jointly in the computational procedure. The mere fact that fw ∝ −y 2 and not to -y 4 brings improved stability of the computational procedure. However, in order to reduce the ζ equation to the simple source-sink-diffusion form, we can omit the term X. This term is not significant, though close to the wall it has some influence. In order to compensate for the omission of X one can re-tune some of the coefficients. To summarize, the recommended k-ζ-f model with eddy viscosity definition νt = cµ ζ
k2 ε
(57)
consist of ζ equation (54) with X = 0, f equation with ζ replacing υ 2 /k and equations for k and ε, (44) and (45). The k-ζ-f model based on quasi-linear pressure strain term. Instead of the simple linear IP model for the rapid part of the pressure strain term in equation (73), we can adopt the more advanced quasi-linear model of Speziale et al. (1991) (see next section) 2 Πij , 2 = −C2′ Paij + C3 kSij + C4 k(aik Sjk + ajk Sik − δij akl Skl ) + C5 k(aik Ωjk + ajk Ωik 3
(58)
which was found to capture better the stress anisotropy in wall boundary layers. Application to the wall normal stress component, with P22 = 0 yields the following form of the f equation in conjunction with the ζ equation (54) (with X = 0): L2 ∇2 f − f =
P 1 c1 + C2′ τ ε
ζ−
2 3
−
C4 P − C5 3 k
(59)
The following coefficient have been optimized with respect to plane channel flow, backwardfacing step and round impinging jet: Cµv
Cε1
Cε2
c1
C2′
C4
C5
σk
σε
σζ
Cτ
CL
Cη
0.22
1.4(1 + 0.012/ζ)
1.92
0.4
0.65
0.625
0.2
1
1.3
1.2
6.0
0.35
85
Some illustrations of the performance of the above defined k − ζ − f model with coefficients specified in Table 3.4.2, are given in Fig. 3, showing the distribution of heat transfer behind a backward facing step and under a normally-impinging round jet. These properties are difficult to predict with the standard linear eddy-viscosity models, and have served as illustrations of benefits with the elliptic relaxation concept. The k − ζ − f model yielded results in good agreement with the original k − v 2 − f model and even some improvements for the impinging jet. 23
0.004
160
0.003
120
Exp. Baughn&Shimizu, 1989 Exp. Baughn et. al., 1991
Nu
St
2
0.002
Exp. Vogel&Eaton, 1985 + k-ζ-f, y max=1.2
0.001
2
+
k-v -f, y
k-v -f, Behnia et. al., 1998 + k-ζ-f, y max=1.6
80
40
=1.2
max
0
0 0
5
10
15
20
0
x/h
1
2
3
4
5
6
r/D
Figure 3: Stanton Number in a backstep flow (left) and Nusselt number in an impinging round jet (right). Comparison of the k-v 2 -f and k-ζ-f models (Popovac, 2004) The k-ζ-f model with fw = 0. One can make further simplifications to satisfy zero wall boundary conditions for fw (in analogy with the original Jones-Launder (1972) formulation of the low-Re-number dissipation equation) by solving equation (3.4.2) but for f˜ with f˜w = 0 and getting f from !2 ∂ζ 1/2 ˜ (60) f = f − 2ν ∂xn which is then used in ζ equation. The second term on the right of equation (60) is just a more general alternative to 2νζ/y 2 , which agrees up to inclusive second term in polynomial expansion of ζ around y = 0, whereas the original term agrees only in the first term.
3.5
SST model of Menter
The k-ω model was claimed to perform better in adverse pressure gradient and to reproduce better flow separation than the k-ε model, and has been favoured by some modellers, especially for aerodynamic flows (e.g. Wilcox 1988, 1993, 1993b, Menter 1994, 1997). A direct transformation of the ω equation into ε equation yields an extra ”cross-term” ∝ νt ∇k∇ω, which seems to sensitize the model to the changes in the pressure gradient. Another advantage of the ω equation is its favourable behaviour very close to a solid wall: the DNS results for plane channel and boundary layers at various (positive and negative) pressure gradients show that the wall limiting 2 expression for ω, nondimensionalised with the inner-wall scales, ω + → 2/y + scales surprisingly well up to y + ≈ 5), in contrast to ε+ . The ω equation was also found to perform well when integrated up to the wall without the need for any extra term to account for low-Re number and wall effects. However, unlike the ε equation, the ω equation was found to be sensitive to the prescribed boundary value of ω at the free edge of turbulent shear layers. In order to utilize the favourable properties of both models, Menter (1994) proposed to combine the two models in such a way that the model reduces to the k-ω close to a solid wall, and to the k-ε model away from the wall. The combination of the two models has been accomplished using a blending function. Whilst the idea sounds plausible, achieving a successful blending requires the use of a number of empirical functions (some including the wall-distance, which is not a desired model feature when dealing with flows in complex geometries). Notwithstanding 24
relative complexity and extensive empiricism, the SST model has been found to perform well in a number of flows including heat transfer, and has been adopted as the baseline EVM in the CFX commercial CFD package. A summary of the complete SST model: "
Dρk µt ∂k ∂ (µ + ) = P˜ + G − β ∗ ρωk + Dt ∂xj σk ∂xj "
#
(61)
#
γ ∂ µt ∂ω 1 ∂k ∂ω γ Dρω = P˜ + G − βρω 2 + ) (µ + + (1 − F1 )2ρσω2 Dt νt νt ∂xj σω ∂xj ω ∂xj ∂xj where: P = −ρui uj
∂Ui ∂xj
and
(62)
P˜ = min(P; c1 ε)
c1 = 10, β ∗ = 0, 09 and all remaining coefficients in equations (61) and (62 are expressed in terms of the blending function F1 : C = F1 C1 + (1 − F1 )C2 where the C1 and C2 stand for the coefficients of the k-ω and k-ε model respectively, for which the following values are recommended: σk1 = 2.0 σk2 = 1.0
σω1 = 2.0 σω2 = 1.168
β1 = 0.075 β2 = 0.0828
with the blending function F1 defined as F1 = tanh(arg14 ) CDkω
γ1 = 0.5532 γ2 = 0.4403
√
k 500ν ; arg1 = min max ∗ β ωy y 2 ω 1 ∂k ∂ω = max 2σω2 : 10−20 ω ∂xj ∂xj
!
4ρσω2 k ; CDkω y 2
!
!
A limiter, similar to those introduced by Durbin (1996), is further introduced, which prevents the excessive growth of the turbulent shear stress. ! √ k 500ν ρ a1 k 2 √ where F2 = tanh(arg2 ) arg2 = max 2 ∗ ; 2 µt = β ωy y ω max(a1 ω : 2S F2 ) where a1 = 0.31.
3.6
Non-linear EVM and Algebraic Re-stress models
There is currently substantial activity in reviving the idea of non-linear eddy viscosity (NEVM) models and their ’relatives’, the algebraic stress models (ASM). While these models offer substantial improvement over liner EVM, it is generally believed that they can serve only as a compromised intermediate tool for some types of flows, and that they will soon give way to physically sounder and potentially more useful differential second-moment (Reynolds-stress) closure models (DSM). The available space will not permit dwelling on these models, and the interested reader is advised to consult the literature, e.g., Speziale 1991, Craft, Launder and Suga (1995, 1996). 25
4
Second-Moment (’Reynolds-stress’) Closures
Differential second-moment (Reynolds-stress) turbulence closure models (DSM) have long been expected to replace the currently popular two-equation k − ε and similar eddy viscosity models (EVM) as the industrial standard for Computational Fluid Dynamics (CFD). Yet, despite almost three decades of development and indisputable progress, only a few commercial CFD vendors offer DSM as a modelling option. Even fewer industrial users recognize the natural superiority of the DSM. These models, used and researched mainly within academic community, are still viewed as a development target rather than as a proven and mature technique for solving complex flow phenomena. Some recent developments demonstrate, however, that DSM have reached the level of maturity and validation at which they can be used to compute complex flows with affordable requirement on computer resources. These models have a great, but yet unexploited, potential. Through a formal analysis and by illustrations in the text that follow, we will demonstrate some obvious advantages. The numerical aspects of using these advanced models are also discussed. That the DSMs have not so far shown an indisputable superiority is for several reasons. First, the equations set contains many more terms than the rudimentary two-equation models. Although the stress production is treated in exact form, there are several other terms in the equations which need to be modelled. One can say that the DSMs have ’more degrees of freedom’ which give more chances of ’going astray’ if wrong or inappropriate formulation is introduced in modelling even a single term; this can annul their inherent higher potential to better capture the physics. A case in point is the pressure-strain (”pressure scrambling”) term which is absent from the kinetic energy equation for incompressible or mildly compressible flows (continuity conditions). This important process in turbulence dynamics, responsible for redistributing energy among the stress components, and, consequently, for stress anisotropy (also the major sink of shear stress at high Reynolds number), is difficult to model and has often in the past been the reason for the failure of DSM to be superior in performance to EVMs. More recent developments, founded on more rigorous analysis (see below), indicate that a better model leads to visible improvement of overall flow properties. A wider application of DSM was discouraged in the past by the high demands on computer resources and the inadequacy of the available numerical methods. The trend to use the available Navier-Stokes computational code to solve the essentially different Reynolds momentum equations (with weak or negligible viscous momentum transport, but strong turbulent momentum transport in terms of stress gradients, supplied from separate transport equations) has often caused serious numerical problems. Numerical difficulties in application of DSMs impeded a wider testing of DSM in more complex flows and has thus far, slowed the research on their improvement. All these problems have contributed to the building of an image of the DSM among some commercial CFD developers as an ’ill-posed’ concept. It should be mentioned that slow progress in the development of DSM was caused also by other factors, such as a lack of information about some terms which need to be modelled (pressure terms, dissipative correlations) and which are intractable to any measuring techniques. Direct Numerical Simulations (DNS), although restricted at present to simple geometries and low Re numbers, now provide much information about flow physics and unknown terms in the transport equations, thus giving a new impetus to the development of DSM and other advanced turbulence models. New incentives for the revival of DSM came also from recent developments in numerical techniques (multigrid acceleration, higher order schemes, fast solvers), and from the advances in hardware: improvement of computer performances, parallel processing, a wider accessibility to powerful mainframe computers, a wide spread of inexpensive workstations and 26
personal computers. Finally, the major impetus comes from the growing awareness among industrial CFD users of the limitations of two-equation and other eddy viscosity models and from the need to model complex flows with higher accuracy and certainty. In this chapter we give an overview of the rationale for employing more advanced models for the computation of complex flows and transport processes, with a focus on Differential Secondmoment (Reynolds stress) closures (DSM). We also discuss reasons for their slow adoption by the CFD community. Physical arguments are briefly given; these illustrate a higher degree of exactness inherent in the second-moment closure approach. The superiority of these models is demonstrated by a series of computational examples, provided by author’s co-workers who used either the same or very similar computational methods and model(s). Examples include several nonequilibrium flows, attached and with separation and reattachment, flow impingement and stagnation, longitudinal vortices, secondary motion, swirl, system rotation. The modelling of molecular effects, both near and away from a solid wall and associated laminar-to-turbulent and reverse transition are also discussed in view of the need for an advanced closure approach particularly when wall phenomena are in focus. Numerical aspects associated with the application of second-moment closure are then discussed, together with current practice used to overcome numerical problems and to reconcile the need for advanced models with unavoidably increasing computational challenge. Several examples related to the automotive industry illustrate the applicability of DSM to real complex flows which have industrial relevance.
4.1
The Rationale
The differential second-moment turbulence closure models (DSM) represent the logical and natural modelling level within the framework of Reynolds-averaged Navier-Stokes, (RANS) approach and are expected to replace the currently popular two-equation k − ε and other eddy viscosity models as the industrial standard for Computational Fluid Dynamics (CFD). These models provide the unknown second-moments (turbulent stress ui uj and turbulent flux of heat and species θuj , cuj ) by solving the model equations for these properties. Hence, instead of modelling directly the second-moments, as implied by eddy-viscosity/diffusivity approach, the modelling task is shifted to unknown higher-order correlation which appear in their differential transport equations. The penalty is that more terms need to be modelled. It is fortunate that these new terms are usually the higher-order statistical moments (third and forth-order correlations), which have a diminishing effect on the mean flow properties. The main advantage of DSM is in the exact treatment of the turbulence production terms, be it by the mean strain or by body forces arising from thermal buoyancy, rotation or other forces. In addition, a solution of a separate transport equation for each component of the turbulent stress enables, in principle, accurate prediction of the turbulent stress field and its anisotropy, which often plays a crucial role in complex flows, either as a major source of turbulence energy such as in the stagnation regions, as a source of secondary motion, or in controlling the dynamics of longitudinal vortices. Accurate prediction of the wall-normal stress component is also important in reproducing the wall phenomena, the wall shear stress, heat and mass transfer. Further, capturing stress anisotropy also enables a more realistic modelling of the scale-determining equation, (dissipation rate or other variable). It should be mentioned that the DSMs do not show always an indisputable superiority over
27
two-equation EVM models. One of the reason is that more terms need to be modelled. While this offer an opportunity to better capture the physics of various turbulence interactions, the advantage may be annulled if some of the terms are modelled wrongly. The use of DSM puts also a higher demand on computing resources, and requires a better skill of the code user, because the model transport equations are not well coupled. However, most of these problems have now been resolved: the advantages of DSM for complex flows have been generally recognized and the numerical difficulties are to a large degree resolved. The demand on computer resources (memory, time) is not excessive (roughly twice as large) as for the two-equations EVM for high Re number flows using wall functions. These advances, together with the growing awareness among industrial CFD users of the limitations of two-equation eddy viscosity models and the need to model complex flows with higher accuracy will lead in the near future to a much wider use of DNS models in CFD. We begin this section by considering first the basic model, and move later to discuss recent trends and advances. Major advantages and inherent potential of the DSMs are then discussed by focussing on some specific features of complex flows, which are usually intractable to standard linear two-equation models.
4.2
4.2.1
The Basic Linear Second-Moment Closure Model for High-Re-number Flows The model equation for ui uj
The exact transport equation for ui uj is
Dui uj ∂ui uj ∂ui uj ∂Uj ∂Ui + Uk = + uj uk = − ui uk + (fi uj + fj ui ) {z } | Dt ∂xk ∂xk ∂xk {z } | {z | ∂t } | {z } Gij Lij
Cij
Pij
∂ui ∂uj + ∂xj ∂xi
p −2Ωk (uj um ǫikm + ui um ǫjkm ) + ρ {z } | Rij
+
∂ ∂ui uj [ν ∂xk ∂xk
| {z } ν Dij
|
|
−
|
ui uj uk {z
t Dij
}
{z
Dij
{z
|
−
2ν |
}
Φij
−
!
p (ui δjk + uj δik ) ] ρ {z
p Dij
}
∂ui ∂uj ∂xk ∂xk {z
εij
}
(63)
}
Terms in boxes must be modelled. Note that Ωk represents the system rotation (angular) velocity, which should be distinguished from the vorticity of a fluid element (”shear vorticity”) ∂Uj i tensor Ωij = 12 ( ∂U ∂xj − ∂xi ) and its dual vector - fluid vorticity ωi = ǫijk Ωkj . The terms have been conveniently denoted so that in further discussion we may refer to symbolic representation of the stress transport equation: p ν t Lij + Cij = Pij + Gij + Rij + Φij − εij + (Dij + Dij + Dij )
where the terms have the following physical meanings (Fig. 4): 28
(64)
Lij - local change in time Cij - convective transport Pij - production by mean flow deformation, Gij - production by body force Rij - production by rotation force Φij - pressure redistribution εij - viscous destruction Dij - diffusive transport. Figure 4: Graphical representation of the physical meanings of terms in ui uj equation (Bradshaw 1978) The modelling of the ui uj and ε equations follows the principles for modelling the k−ε equations, using the characteristic turbulence time scale τ = k/ε and length scales L = k 3/2 /ε, except that ui uj does not need to be modelled. The exception are also the pressure-strain term Φij and the stress dissipation rate εij . The standard modelling practice for the basic model is outlined below. Stress Dissipation. At high Reynolds numbers the large scale motion is unaffected by viscosity, while the fine-scale structure is locally isotropic, i.e. unaffected by the large eddies ∂u ∂u orientation. Consequently, the correlation εij = ν ∂xki ∂xjk - which is associated with smallest eddies - should reduce to zero if i 6= j, while for i = j all three components should be equal. Hence, a common way to model the viscous destruction of stresses for high Re-number flows is: 2 εij = εδij 3
(65)
∂ul ∂ul where ε = ν ∂x and δij is the Kronecker unit tensor. It was mentioned earlier that at k ∂xk high Re numbers ε can be interpreted as the amount of energy given away by large (energy containing) eddies and transferred through the spectrum towards smaller eddies until ultimate dissipation. Hence, although ε represents essentially a viscous process, its value (’dissipation rate’) is governed by large, energy containing eddies and the above assumption is not very appropriate in non homogeneous flow regions such as in the vicinity of a solid wall. Nonetheless, this assumption is widely used in standard DSM models and its deficiency is compensated by the model pressure strain term Φij which accounts for the turbulence anisotropy.
R
Turbulent Diffusion. Dij has a character of divergence so that V Dij dV = 0 over a closed domain bounded by impermeable surface (as follows from Gauss transformation of the volume integral into a surface integral). Hence Dij term is of a transport (diffusive) nature. The most ∂φ popular model is the generalized gradient diffusion (GGD): ϕuk = Cφ τ uk ul ∂x , where τ = k/ε l is the time scale. Application of GDH to the turbulent velocity diffusion of stress yields: t Dij
∂ ∂ (−ui uj uk ) = = ∂xk ∂xk 29
∂ui uj k Cs uk ul ε ∂xl
(66)
A simpler variant is the simple gradient diffusion (SGD) which leads to (Shir 1973). t Dij
k 2 ∂ui uj Cs ε ∂xk
∂ ∂ = (−ui uj uk ) = ∂xk ∂xk
!
(67)
For more advanced models see the next section. The turbulent transport by pressure fluctuations has a different nature (propagation of disturp bances) and none of the gradient transport forms is applicable to modelling Dij . Yet, it is p t common to ”lump” Dij with Dij and to adjust the coefficient Cs . In many flows the pressure transport is smaller than the velocity transport so that this approximation bears no consequences. However, in flows driven by thermal buoyancy (e.g. Rayleigh-Benard convection) this is not the case and such models are not appropriate. Finally, it should be recalled that for high Re-number flows the viscous diffusion is negligible. Pressure-Strain Interaction. Pressure fluctuations act towards scrambling the turbulence structure and redistribute the turbulent stress among components to make turbulence more isotropic. Some insight into the physics and hints for modelling can be gained from the exact Poisson equation for the pressure fluctuations (obtained after differentiating the equation for ul with respect to xl ): ∂fi ∂2p ∂2 ∂Ul ∂um (ρul um − ρul um ) − 2ρ +ρ = − 2 ∂x ∂x ∂x ∂x ∂x ∂xl m m i l l
(68)
which can be integrated to yield p at ~x: 1 p = ρ 4π
Z " V
#
′ ′ ′ ′ ∂2 ′ ′ ′ u′ ) + 2 ∂Ul ∂um − ∂fi dV (x ) (u u − u l m m l ′ ′ ∂xl ∂x′m ∂x′m ∂xl ∂xi |r|
where r = x′ − x. Multiplication with ( expression for Φij : ∂uj ∂ui + ∂xj ∂xi
p Φij = ρ
∂Ul +2 ∂xm |
+
1 4π
|
′
∂um ∂xl
!
′
{z
1 = 4π
∂ui ∂uj + ) (at x!) and averaging yields the exact ∂xj ∂xi
Z
V
′ 2 ∂ ul um
A
∂ui ∂uj + ∂xj ∂xi
1 ∂ ′ p r ∂n′
∂ui ∂uj + ∂xj ∂xi
!
∂uj ∂ui + ∂xj ∂xi
∂xl ∂xm
|
{z
!
−
}
− p′ {z
|
∂fm ∂xm
! }
Φij,1
Φij,2
Z
(69)
!
′
{z
Φij,3
∂ui ∂uj + ∂xj ∂xi
′
∂ui ∂uj dV (x ) + ∂xj ∂xi |x − x′ | !
∂ ∂n′
}
1 dA r
(70)
}
Φw ij
Different terms in (70) can be associated with different physical processes, which can be modelled separately. Most common approach is to split the term into the following parts: w w Φij = Φij,1 + Φij,2 + Φij,3 + Φw ij,1 + Φij,2 + Φij,3
30
(71)
where • Φij,1 - return to isotropy of non-isotropic turbulence, (”slow term”); In the absence of the mean rate of strain Sij and a body force, away from a solid wall, pressure fluctuations will force turbulence to approach an isotropic state. • Φij,2 - ”isotropization” of the process of stress production due to Sij (”rapid term”); Pressure fluctuations will slow down a preferential feeding of turbulence by Sij into a particular component imposed by the active strain rate components. • Φij,3 - ”isotropization” of stress production due to a body force; w w • Φw ij,1 , Φij,2 , Φij,3 - wall blockage (’eddy splatting’) and pressure reflection effect associated with Φij,1 , Φij,2 and Φij,3 respectively. The first process, which is dominant, slows down the isotropizing action of the pressure fluctuations. The pressure reflection acts in fact in opposite way (the pressure wave reflected from a solid surface enhances eddy scrambling), but this effect is smaller in comparison with the wall blockage effect.
Jones and Mussonge (1988) argued that the mean strain rate appears also in the exact transport equation for the slow term Φij,1 and that separate modelling of each part of Φij may not be fully justified. Their ’integral’ model for the complete Φij differs, however, from other models in a small difference in the values of the coefficients (see next section). However, splitting the terms, even if not fully justified, enables to distinguish some physical effects one from another and gives some basis for their modelling. For that reason we follow here the conventional approach The model of Φij,1 (’The slow term’). Based on the idea that the pressure fluctuations tend to diminish turbulence anisotropy, Rotta (1951) proposed a simple linear model by which φij,1 is proportional to the stress anisotropy tensor itself (of course with ’-’ sign). The expression is known as a linear Return-to-Isotropy model: Φij,1 = −C1 εaij = −C1 ε
ui uj 2 − δij k 3
(72)
where C1 must be > 1; the most common value is C1 = 1.8. The models of Φij,2 and Φij,3 (’The rapid terms’). Without going deeper into physics, we can recall at this point that Φij,2 is associated with the mean rate of strain, which is usually the major source of turbulence production. Hence the pressure scrambling action can be expected to extend to the very process of stress production. Following this idea, Naot et al. (1970) proposed a model of Φij,2 analogous to Rotta model of the slow term, known as the ’Isotropisation of Production’ (IP) model: 2 Φij,2 = −C2 (Pij − δij P) 3
(73)
where C2 = 0.6 Analogue approach to the pressure effect on stress generation due to body force leads to: 2 Φij,3 = −C3 (Gij − δij G) 3 31
(74)
where C3 = 0.55 Note that P = 1/2Pkk and G = 1/2Gkk Interdependence of Coefficients. The above listed coefficients C1 , C2 and C3 have been obtained mainly from selected experiments where only one of the process can be isolated (e.g. C1 from the experiment on free return to isotropy of initially strained turbulence, C2 from Rapid Distortion Theory). Of course, the coefficients have been also tuned through subsequent validation in a series of experimentally well documented flows. Values other than those above quoted have also been proposed as more suitable for some classes of flows. However, the validation revealed that a change in one coefficients requires also an adjustment of other(s) in order to reproduce fully the total effect of the pressure-strain term. A useful correlation between C1 and C2 is: C1 ≈ 4.5(1 − C2 ) (75) The most frequently used values are: C1 = 1.8, C2 = 0.6 and C3 = 0.55. Modelling the Wall Effects on Φij Solid walls and free surfaces ”splat” neighbouring eddies, which leads to a larger turbulence anisotropy. Wall impermeability (blocking effect) damps the velocity fluctuations in the wall-normal direction. On the other hand, pressure fluctuations reflecting from bounding surfaces will enhance the pressure scrambling effect. Both these effects are of non-viscous nature and are essentially dependent on the wall distance and the wall topology. The blockage effect is stronger resulting in a slow down the isotropizing action of pressure fluctuations. As a consequence, the stress anisotropy in a near-wall region is higher than in free flows at similar strain rates:
Homogeneous shear flow (Champagne et al.) Near-wall region of a boundary layer
a11 0.30 0.55
a22 -0.18 -0.45
a33 -0.2 -0.11
a12 -0.33 -0.24
Wall damping is expected to affect both the ”slow” and ”rapid” pressure-induced stress-redistribution w processes and we can decompose Φw ij into two terms: Φij,1 ”corrects” the values of the stress components in such a way as to diminish the wall-normal component to the benefit of the streamwiseand spanwise ones, and reduces the shear stress. Likewise, Φw ij,2 modifies the processes of stress production in the near-wall region. The wall correction should attenuate with the distance from the wall: this is usually accounted for by the empirical damping function fw = L/y, where L is the turbulence length scale. Close to a wall L ∝ y so that fw ≈ 1. At a larger wall-distances L ≈ const so that fw → 0 w Based on the above reasoning the models of Φw ij,1 and Φij,2 have been proposed which : most w w frequently employed models for Φij,1 and Φij,2 :
• Shir (1973): w Φw ij,1 = C1
• Gibson & Launder (1978): Φw ij,2
=
(76)
(77)
3 3 ε uk um nk nm δij − ui uk nk nj − uk uj nk ni · fw k 2 2
C2w
3 3 Φkm,2 nk nm δij − Φik,2 nk nj − Φjk,2 nk ni · fw 2 2 32
C1w = 0.5;
where:
C2w = 0.3
fw =
0.4 k 3/2 εxn
,
and xn is the normal distance to the wall and nk is the unit vector of the coordinate normal to the wall. 4.2.2
The Model ε-Equation
Exact equation for ε can not serve much as a basis for modelling except to give some indication on the meanings and importance of various terms. ε appears indeed as the exact sink term in the k equation and needs to be provided to close the model (εij in ui uj equation). However,, what is needed for closing other terms is the characteristic time and length scale of the energycontaining eddies and the equation for energy transfer from these eddies down the spectrum. This energy transfer rate coincides with the dissipation rate only under the conditions of spectral equilibrium. Nevertheless, it is instructive to look at the exact transport equation for ε: Dε ∂ε ∂ε = + Uk Dt |{z} ∂t ∂xk Lε
| {z } Cε
=
−2ν |
+
!
∂Ui ∂xk
{z
}
Pε1 +Pε2
−2ν
|
∂ui ∂uk ∂ul ∂ul + ∂xl ∂xl ∂xi ∂xk
|
∂ui ∂ 2 Ui ∂xl ∂xk ∂xl {z
Pε3
∂ 2 ui ∂ui ∂ui ∂uk ∂fi ∂ui + ν − 2 ν ∂xk ∂xl ∂xl xl ∂xl ∂xk ∂xl {z
}
Pε4
∂ ∂ε (ν ∂xk ∂xk
| {z } Dεν
|
−2νuk
{z
|
}
Gε
− u ε
| {z k } Dεt
{z
Dε
− |
|
{z Y
2ν ∂p ∂uk ) ρ ∂xi ∂xi {z
Dεp
}
2
}
}
(78)
}
The physical meaning of the terms can be inferred from comparison with the transport equation for the square of fluctuating vorticity ωi′2 (’enstrophy’), since for homogeneous turbulence ε ≈ 2νωi′2 (Tennekes and Lumley 1972). At high Reynolds numbers the source terms (Pε4 and Y ) are dominant, while the other production terms can be neglected as smaller (Pε1 + Pε2 and Gε 1/2 by Ret and Pε3 by Ret order of magnitude). Of course, for low-Re-number flows, these terms need to be taken into account. Note that all terms in boxes must be modelled. In the DSM closures the same basic form of model equation for ε is used as in the k − ε model, except that now ui uj is available (and also θui if the second-moment closure level is used also for the thermal field), which has the following implications: • The P and G (production of kinetic energy) in the source term of ε are treated in exact form; • The generalized gradient hypothesis is used to model turbulent diffusion Dεt =
∂ ∂ (−uk ε) = ∂xk ∂xk 33
k ∂ε Cε uk ul ε ∂xl
(79)
Hence, the model equation for ε has the form: Dε ∂ = Dt ∂xk
∂ε k Cε uk ul ε ∂xl
+ (Cε1 P + Cε3 G + Cε4 k
∂Uk ε − Cε2 ε) ∂xk k
(80)
where the coefficients have the same values as in the k − ε model except for the new coefficient Cε = 0.18 (which replaces σε ) 4.2.3
Summary of the Basic DSM model for high Re-numbers in the integral form
Most CFD (Computational Fluid Dynamics) codes use finite volume approach, by which the transport equations are integrated over the elementary control volume (cell) prior to their discretization. It is, therefore useful to give the complete basic differential second-moment closure model for high Reynolds-number flows, integrated over a control volume (with bi denoting local velocity of the control surface):
∂ ∂t
Z
V
ρui uj dV +
Z
A
ρui uj [Uk − bk ] dAk = +
Z
V
∂ ∂t
4.2.4
Z
V
Z A
k µδkl + Cs uk ul ε
∂ui uj dAk ∂xl
(Pij + Gij + Rij + Φij − εij )dV
(81)
Z
Z
∂ε k ρε [Uk − bk ] dAk = µδkl + Cε uk ul ρ εdV + dAk ε ∂x A A l Z ε ε ∂Uk ε2 X + (Cε1 P + Cε3 G + Cε4 ε − Cε2 + Sa ) dV k k ∂xk k V
(82)
Summary of Coefficients
The following values of coefficients are recommended: Cs 0.2
4.3
C1 1.8
C2 0.6
C1w 0.5
C2w 0.3
Cε 0.18
Cε1 1.44
Cε2 1.92
Cε3 1.44
Cε4 0.33
Second-Moment Closure for Scalar Fields for High-Pe-number Flows
The second-moment closure models for scalar fields (thermal, species concentration) follow essentially the same principles as the modelling of the velocity field. This means that the transport equations for the scalar flux hui , θui , cui are modelled starting from their exact parent equations (here h is the fluctuating enthalpy, θ is the fluctuating temperature and c is the fluctuating concentration; for multi-component mixture each species concentration is considered separately, i.e. c is replaced by c(i) . Because the principles are the same, except for the source terms, which usually require no special consideration (nor modelling), we consider here only the equation for the turbulent heat flux θui .
34
4.3.1
The Model Equation for Scalar Flux θui
The exact transport equation for the turbulent heat flux vector for high Peclet numbers (P e = Re.P r) can be derived in the manner analogous to the stress equation: Dθui Dt
= −ui uk |
{z
∂T ∂Ui p ∂θ −θuk −βgi θ2 + ∂xk ∂xk | {z } ρ ∂xi
T Pθi
−(α + ν)
|
}|
{z
U Pθi
}
Gθi
| {z } Φθi
∂ ∂θ ∂ui pθ − (θui uk + δik ) ∂xk ∂xk ∂xk ρ
{z
} |
α −εθi +Dθi
{z
t +D p Dθi θi
(83)
}
The physical meaning of various terms can be inferred by comparison with the Re-stress equation (note that the equation contains three production terms): T - ’thermal’ production (nonuniform temperature field interacting with turbulent stresses) Pθi U - ’mechanical’ production (mean flow deformation interacting with the turbulent heat flux) Pθi
Gθi - gravitational production (gravitation interaction with the fluctuating temperature field). Φθi - pressure–temperature-gradient correlation εθi - molecular destruction Dθi - diffusion transport (were superscripts denote: ′ α′ - molecular, ′ t′ - by turbulent velocity and ′ p′ by pressure fluctuations. All three production terms can be treated in the exact form, but an additional transport equation needs to be provided for the temperature variance θ2 . Other terms need to be modelled. Following the same modelling principles we can express the pressure scrambling term as: Φθi = Φθi,1 + Φθi,2 + Φθi,3 = −Cθ1
θui U − Cθ2 Pθi − Cθ3 Gθi k
(84)
The turbulent diffusion by velocity and pressure fluctuations is modelled by GGD. It should be noted that the viscous diffusion needs also to be modelled, except in the case of Prandtl numbers of O(1). For high Pe numbers εθi can be neglected. Hence, the model equation for the scalar flux (wall effects here omitted) is: Dθui Dt
4.3.2
∂T ∂Ui − (1 − C2θ )θuk − (1 − C3θ )βgi θ2 ∂xk ∂xk " # k ∂θui θui ∂ uk ul −C1θ ε + Cθ k ∂xk ε ∂xl
= −ui uk
(85)
The Model Equation for Scalar Variance θ2
The model equation for θ2 resembles closely the k equation and can be modelled in the same manner. It contains a single production term which can be treated exactly. The turbulent 35
transport is modelled in a usual gradient transport form. The only problem is the sink term
2
∂θ (molecular destruction) εθ = 2α ∂x . A transport equation for εθ can be derived, resembling j the ε equation, except that it has twice as many terms, so that its modelling poses a lot of uncertainty.
The common approximation, based on the assumption that the ratio of the thermal to mechanical time scale τθ /τ = R = const (where τθ = θ2 /2εθ and τ = k/ε) leads to a simple approximation
εθ = ε
θ2 2k
(86)
Hence, the model equation for θ2 is Dθ2 ∂T 1 θ2 ∂ = −2θui − ε + Cθ2 Dt ∂xi 2R 2k ∂xj 4.3.3
k ∂θ2 ui uj ε ∂xi
!
(87)
Summary of Coefficients for Scalar Flux Model
The following values of coefficients can be recommended for the scalar flux model for high Peclet number flows: Cθ2 0.2
Cθ 0.15
C1θ 3.5
C2θ 0.55
C3θ 0.55
R 0.5
4.4
The Algebraic Stress/Flux Models (ASM/AFM)
4.4.1
Algebraic stress models, ASM
A considerable simplification of the equation set can be achieved by eliminating transport terms in individual stress components by that of the kinetic energy. The common approach is to assume so called weak non-equilibrium hypothesis (Rodi, 1976) by which the time and space evolution of the stress anisotropy tensor is equal to zero, i.e.: Daij − Dij aij = 0 Dt The expansion of aij =
(88)
2 ui uj − δij leads to: k 3
Dui uj ui uj − Dij = Dt k
Dk − Dk Dt
=
ui uj (P + G − ε) k
(89)
Each stress component ui uj can now be expressed in terms of an algebraic expression:
2 2 k 2 α1 Pij − Pδij + α2 Gij − Gδij ui uj = δij k + 3 ε 3 3
(90)
where α1 and α2 are functions of P/ε and G/ε (containing also the coefficients from modelled expressions for pressure-strain terms). The ASM have some advantages such as a reduction of computing time in comparison with the full (differential) DSM, give better results than k − ε 36
model where stress anisotropy is strong and important, e.g. secondary flows. However, they are derived from the presumed DSM model and they can at best perform as the parent DSM provided the flow evolution is slow. Major shortcomings are that they really can not replace the DSM particularly when the stress transport and evolution of the stress anisotropy are important, i.e. when the flow ’history’ and development can not be fully accounted for by transport terms in k- and ε-equations. The above ASM is implicit in ui uj . Besides, the functions α have expressions in the denominator, which may become very small or even zero, leading to singularities and numerical instability. In order to overcome the numerical problems, several explicit non-linear ASM and AFM have recently been proposed in the literature (Speciale and Gatski, 1993, Johanson et al. 1996). These models resemble the non-linear EVM. Despite some success, these models suffer from the same deficiencies as all ASMs, as compared with DSM, i.e. inability to fully capture the dynamics of the stress/flux anisotropy. 4.4.2
Algebraic scalar flux models, AFM
The 3D fully-differential stress/flux model consists of 17 transport equations. In complex 3D flows, such a model requires formidable computational resources, especially if near-wall region needs to be fully resolved. There is, therefore, much to be gained if the differential model can be truncated to an algebraic form, especially if the scalar field(s) are to be solved. Applying the weak non-equilibrium hypothesis to the scalar flux (Gibson and Launder 1976, Kenjereˇs 1999), which implies that the anisotropy of the thermal flux θui /(θ2 k)1/2 is approximately constant in space and time, i.e.:
D − Dij Dt
θui 2 (θ k)1/2
!
= 0,
(91)
the sum of the transport terms of θui can be expressed as a function of the transport of the turbulence kinetic energy and temperature variance: Dθui − Dθi = Dt =
θui D 2 1/2 (θ k) − D(θ2 k)1/2 1/2 2 Dt (θ k) "
1 1 θui 2 θ2
Dθ2 − Dθθ Dt
!
1 + k
Dk − Dk Dt
#
(92)
Replacing the total transport of θui (left hand side) by the source terms in equation (85), the total transport of θ2 with the source terms in equation (87) and of k by its source terms, yields the general implicit algebraic expression for the turbulent scalar-flux vector: ∂T ∂Ui + ξθuk + ηgi βθ2 ∂xk ∂xk ε ∂T ∂Ui 1 1 (2θuk −C1θ + + εθ ) + (ui uk + gi βθui + ε) 2 k 2θ ∂xk 2k ∂xk ui uk
θui =
(93)
where, for brevity, ξ = (1 − C2θ ) and η = (1 − C3θ ). On the other hand, full neglect of the transport terms leads to a simpler (‘reduced’) form of the expression (essentially only the numerator of equation (93), Hanjalic 1994) in which the coefficient c1θ needs to be modified:
1 k ∂T ∂Ui θui = − ui uk + ξθuk + ηgi βθ2 C1θ ε ∂xk ∂xk 37
(94)
In simple flows such as forced convection in wall attached flows or natural convection in sideheated vertical channel the components of the turbulent heat flux computed from the complete and reduced algebraic expressions (93) and (94) is small (Dol et al. 1997), the latter being much simpler and more robust, though this conclusion may not apply to more complex flows. It is noted that the neglect of the two last terms involving mean velocity gradient and buoyancy effects (and with adequate modification of the coefficient C1θ ) leads to the anisotropic eddy diffusivity expression for scalar flux, known also as the Generalized Gradient Diffusion Hypothesis (GGDH) (with cθ = 1/C1θ ≈ 0.3. k ∂T θui = −cθ ui uk ε ∂xk
(95)
It is obvious that equation 95 can be further simplified if the stress tensor ui uj is replaced by the its trace 2k (with corresponding adjustment of the index in the temperature gradient vector ∇T , leading the the basic isotropic eddy diffusivity hypothesis, known also as the Simple Gradient Diffusion Hypothesis (SGDH): θui = −c′θ
k 2 ∂T νt ∂T =− ε ∂xi σT ∂xi
(96)
where c′θ = Cµ /σT ≈ 0.08.
4.5
Boundary Conditions
So far only ”High Re-number” variants of RSM and ASM have been developed to a stage to be applicable widely in industrial practice. These models employ the same type of boundary conditions as in the k − ε model. These conditions are summarized below: • Symmetry line (or plane): the turbulent normal stress components are ”even” functions like k and ε, while the shear stress components are ”odd” functions, so that the normal stresses have zero gradients in direction normal to the symmetry line (plane), whereas the shear stresses change the sign and have zero values • At a free surface the same conditions apply as for the k equation • Solid wall: The same standard wall functions approach from the k − ε model is commonly used for defining the boundary conditions for the mean momentum, mean energy and species conservation equations in DSM. The same applies for the ε equation. The DSM require, in addition, to define the boundary conditions for the turbulent stresses. Two approaches are used: the specification of turbulent stresses in the first near-wall control cell, and the integrration of the full transport equations over this cell with help of wall functions. In the first approach the turbulent shear stress is deduced from the local wall shear stress, which is obtained iteratively from the velocity wall function. The normal stress components are expressed as fractions of the kinetic energy. In the integral approach no values of the stress components need to be defined, but the near wall control cell is treated just as any interior cell, except that the values and wall flux in stress equation are zero, and the mean velocity gradient in the production terms is provided from the logarithmic law.
38
5
Advanced Differential Second-Moment Closures
The basic differential second-moment (Reynolds-stress) models have proved to perform better in many flows than any Eddy Viscosity Model. The DSM have reached the level of development which justifies their wider industrial application and they have already been incorporated in some commercial CFD codes. It is likely that they will be more in use in the near future. However, the basic DSM did not meet the early expectations to produce superior prediction in every flow. Over the past decade there has been much activity aimed at improving the basic DSM. All improvement lead necessarily to more complex models which pose additional computational difficulties (numerical instabilities, slower convergence). The model developers often focus on only one or two crucial terms in the ui uj and ε equation and propose more sophisticated expression which better satisfy physical rationale and mathematical constraints (realizability, twocomponent limit, vanishing and infinite Reynolds numbers, etc.). Validation is usually performed in a limited number of test cases that display particular features which are the focus of new development. The resulting complex model is often out of balance with the usually much simpler models of the rest of the terms. We confine our attention here to only a few of advancements, which seem to bring desirable improvement and yet retain the form of the model expressions at a manageable level of sophistication. The focus is on the models of turbulent diffusion and pressure scrambling in the ui uj equation and on some proposal to improve the ε equation.
5.1 5.1.1
Some improvements of the model ui uj equation Turbulent diffusion of ui uj
A tensorially invariant model of Dij can be derived by tensorial expansion of the GDH. Alternatively, a truncation of the model transport equation for triple velocity correlation ui uj uk and retaining only the first order terms, yields (Hanjali´c & Launder, 1972): t Dij =
∂uj uk ∂ ∂uk ui ∂ui uj ∂ ′ k ui ul Cs (−ui uj uk ) = + uj ul + uk ul ∂xk ∂xk ε ∂xl ∂xl ∂xl
(97)
Application of moment generating function leads to still more complex expressions (e.g. Lumley, 1978, Cormac et al. 1978, Magnaudet, 1992). Nagano & Tagawa (1991) proposed a new way to treat triple velocity- and scalar correlations, which brought improvements in near-wall flows. Although shown to perform better than the simplest GDH, more general expressions give rise to a large number of component terms, particularly in non-Cartesian coordinates. Because in many flows with a strong stress production, the turbulent transport is relatively small and a simpler model usually suffices. A simpler expression which satisfies the coordinate-frame invariance and still retains a relatively simple form is the expression proposed by Mellor and Herring (1973) : t Dij
"
2 ∂ ∂ ′ k Cs (−ui uj uk ) = = ∂xk ∂xk ε
39
∂uj uk ∂uk ui ∂ui uj + + ∂xi ∂xj ∂xk
!#
(98)
5.1.2
Pressure-Strain Interaction: The ’slow’ term
The return to isotropy is in fact a non-linear process: a tensorial expansion with Caley Hamilton theorem leads to a quadratic model of the ’slow’term (Lumley 1978, Reynolds 1984, Fu et al. 1987, Speziale, Sarkar & Gatski 1991); 1 Φij,1 = −ε[C1 aij + C1′ (aik ajk − δij A2 )] 3
(99)
where C1 , C1′ are, in general, functions of turbulence Re-number and stress anisotropy invariants A2 = aij aij , A3 = aij ajk aki and ”flatness” parameter A = 1 − 89 (A2 − A3 ). Speciale, Sarkar and Gatski (1991) (SSG) proposed C1′ = −1.05 and this value has been generally accepted in the framework of the complete quadratic pressure-strain model (see below). The UMIST group (Craft and Launder 1991) proposed similar expression (though validated only in free flows): Φij,1 = −C1 ε
1+
C10 C1
1 ′ aij + C1 aik akj − A2 δij 3
(100)
1
with C1 = 3.1(A2 A) 2 , C1′ = 1.2 and C1′ 0 = 1 Earlier, Shih and Lumley (1985) discussed the quadratic expression, but due to the lack of evidence, they discarded the second term and proposed C1 in form of a function dependent on Reynolds number and stress invariants: C1 = 1 + 4.45A ln(1 + 7.8A2 + 6.0A3 )
(101)
The appearance of DNS data for each part of the pressure-strain term makes it possible not only to verify the proposed expression, but also the values of the coefficients. Equation (99) contains two unknowns, C1 and C1′ . Using any pair of experimental data for Φij,1 for two components enables to obtain C1 and C1′ . In fact, because four components of Φij,1 are available in a channel flow, the problem is overdefined and different solutions can emerge for different combinations, if expression (99) is not unique. Such a test in a plane channel flow (Hanjali´c and Jakirli´c, 1997), using the DNS results of Kim et al. (1987) showed that both coefficients, C1 and C1′ vary very strongly across the flow. However, the results for different pairs of Φij,1 collapse indeed into one curve in the region close to a wall for y + < 60 for Rem = 5600, though departing substantially away from the wall, Fig. 5. The data show also that a simple expression C1′ = −A2 C1
(102)
matches well the DNS data in the near-wall region. It should be noted, however, that C1 changes the sign at y + ≈ 12, exhibiting a peak at y + ≈ 6. Most models do not reproduce such a behaviour, but impose a monotonic approach of C1 to zero at the wall. Fig. 5 shows the variation of C1 and C1′ given by different model proposals, discussed above. 5.1.3
Pressure-Strain Interaction: The ’rapid’ term
The basis for modelling of the ’rapid’ term is the general expression Φij,2 =
∂Ul mi (b + bmj li ) ∂xm lj 40
(103)
Figure 5: Variation of C1 and C1′ in a plane channel flow. Symbols: evaluation from pairs of DNS components of Φij,1 . Lines: different models (for acronyms see Table 1) which represents in a symbolic form the Poisson equation (70) after the mean velocity gradient is taken out of the volume integral with presumed local mean flow homogeneity. The modelling task is reduced to expressing the fourth-order tensor bmi lj in terms of available second-order tensors (turbulent stress tensor ui uj and Kronecker unit tensor δij ). A convenient and more general way is to formulate the complete Φij,2 in form of a tensorial expansion series in terms aij , Sij . The complete expression (closed by Caley Hamilton theorem) contains terms up to cubic in aij (Craft and Launder 1996): 2 Φij,2 = −C2′ Paij + C3 kSij + C4 k(aik Sjk + ajk Sik − δij akl Skl ) + C5 (aik Ωjk + ajk Ωik ) 3 −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
+ C6 k (aik akl Sjl + ajk akl Sil − 2akj ali Skl − 3aij akl Skl ) + C7 k (aik akl Ωjl + ajk akl Ωil ) 3 (104) + C8 k[a2mn (aik Ωjk + ajk Ωik ) + ami anj (amk Ωnk + ank Ωmk )] 2
1 where Sij = 2
∂Ui ∂Ui + ∂xj ∂xj
!
1 and Ωij = 2
∂Ui ∂Ui − ∂xj ∂xj
!
The line separates the linear (in aij ) from non-linear terms (though C2′ P aij is in fact quadratic, because P contains aij ). Although non-linear models are claimed to satisfy better the mathematical constraints and physical requirements, the excessive number of terms (and coefficients) make them impractical for industrial applications. Two models, which perform better than the rudimentary IP model and still retain simple form are the linear Quasi-Isotropic model of Launder, Reece and Rodi (1975), denoted as LRR QI, and the quasi-nonlinear model of Speciale, Sarkar and Gatski (1991), denoted as SSG. The models differ in values of coefficients. Besides, the SSG model contains the above mentioned quasi-nonlinear term, which is absent from LLR-QI model, and one of the coefficients is formulated as a function of second stress invariant A2 . A summary of the coefficient for some of the models of Φij known in the literature is given in Table below. It should be mentioned that the LRR-QI model requires the use of wall-echo terms Φw ij defined earlier (Gibson and Launder 1975), whereas the SSG model does not. Apparently the extra quasi-linear term and the function C3 accounts for the stress redistribution modification by a solid wall making the wall-echo terms redundant. While this statement is not fully true 41
Table 1: Summary of coefficients in pressure-strain models
Φij,1
Φij,2
Linear
Quadratic
C1
C1
C2
C3
C4
C5
C6
C7
C8
HL
2.8
0
−0.9
0.8
0.71
0.582
0
0
0
LRR IP
1.8
0
0
0.8
0.6
0.6
0
0
0
LRR QI
1.8
0
0
0.8
0.873
0.655
0
0
0
SSG
1.7
−1.05
0.9
0.625
0.2
0
0
0
CL
1 + 3.1(A2 A)1/2
1.2C1
0
0.8
0.6
0.866
0.2
0.2
1.2
LT
6.3AF 1/2 (1 − f )
0.7C1
0
0.8
0.6
0.866
0.2
0.2
2r
Authors
′
Linear ′
1/2
0.8 − 0.625A2
Quadratic
Cubic
Abbreviations: HL: Hanjali´c and Launder (1972), LRR: Launder, Reece and Rodi (1975), SSG Speziale, Sarkar and Gatski (1991), CL: Craft and Launder (1993), LT: Launder and Tselepidakis (1993).
(LRR-QI+wall echo terms reproduce better the stress anisotropy in a near-wall region, Hadˇzi´c, 1998), the SSG has an appeal as a compromise between the desired accuracy and computational economy: it is more practical than LRR-QI, more accurate than IP and, as a whole, satisfactory for industrial applications. It should be mentioned that the quasi-nonlinear term can be interpreted as an extension of the slow term Φij,1 with the coefficient C1 replaced by a function C1 (1 + C2′ P/ε). The ratio P/ε has been used earlier in several models, even at the two-equation level, to account for departure from local energy equilibrium.
5.1.4
Elliptic-relaxation model of wall effect in second-moment closures
Durbin (1993) extended his elliptic relaxation approach to full second-moment closure by proposing the following elliptic equation for the tensorial damping function fij corresponding to the stress tensor ui uj : Φhij (105) L2 ∇2 fij − fij = − k where Φhij is the ’homogeneous’ (far from a wall) pressure-strain model, for which, in principle, any known model (without wall correction) can be used3 . The function fij is then used to obtain ΦD ij = k fij . With viscous effects accounted for by imposing Kolmogorov scales as lower Note that Durbin used ΦD ij = (Πij − 1/3Πkk δij ) − (εij − ui uj /kε) instead of the conventional pressure-strain p Φij , where Πij = Dij − Φij is the velocity–pressure-gradient correlation, equation (??). 3
42
bounds on both the time and length scales of turbulence (just as in the k-ε-v 2 -f model), the model allows the integration up to a solid wall (for further comments on the implementation of viscous effects, see section 5). A fuller account of the physical basis and the associated analysis for the ER approach, together with examples of applications is provided by Durbin (1991) and in Durbin and Pettersson Reif (2001).
5.2
Elliptic-blending second-moment closure
The ER approach to second-moment closure requires solution up to six elliptic relaxation equations, one for each component of the turbulent stress ui uj . Despite the demonstrated success in reproducing several types of flows, the unavoidable additional computational effort, together with some problems experienced in defining and implementing wall boundary conditions for each fij , have limited wider testing of this model. A significant simplification, while still utilizing the elliptic relaxation concept within the second-moment closure framework, can be achieved by solving a single elliptic equation. It is recalled that elliptic relaxation equation essentially accounts for geometrical effects (wall configuration and topology) and provides a continuous modification of the homogeneous pressure-strain process as the wall is approached to satisfy the wall conditions. Hence, it should be possible to define this transition by a single variable. Manceau and Hanjali´c (2000b) have proposed a ’blending model’ which entails the solution of the elliptic relaxation equation for a blending function α L2 ∇2 α − α = −
1 k
(106)
with boundary conditions kα|w = 0 and kα|∞ → 1. This blending function is then used to provide a transition between the homogeneous (far-from-the-wall) and inhomogeneous (nearwall) pressure-strain model h φ∗ij = (1 − kα)Φw (107) ij + kαΦij and between the isotropic and near-wall nonisotropic stress dissipation rate εij = (1 − Akα)
ui uj 2 ε + Akα εδij k 3
(108)
Here Φhij can be any known homogeneous model of Φij , whereas the inhomogeneous part is defined (satisfying the wall constraints) as Φw ij = −5
ε 1 ui uk nj nk + uj uk ni nk − uk ul nk nl (ni nj − δij ) . k 2
(109)
The unit normal vectors are obtained from ~n =
∇α ||∇α||
(110)
The testing of this model in a plane channel and in flow over a backward-facing step showed very good agreement with the DNS data. Recently, Theilen et al. 2004, simplified further the model to make it more robust in complex flows, by replacing equations 106(107) and (108) by L2 ∇2 α − α = −1
(111)
2 h φ∗ij = (1 − α2 )Φw ij + α Φij
(112)
43
εij = (1 − α2 )
ui uj 2 ε + α2 εδij k 3
(113)
and a modification to the coefficients Cε1 : Cε1 =
0 Cε1
2
s
1 + 0.03(1 − α )
k ui uj ni nj
!
(114)
Application in some complex flows and heat transfer, such as in multiple-impinging jets (Thielen et al. 2004), showed excellent agreement with experiments, see Section 8 and Fig. 25.
5.3
Some Modifications to the ε Equation
The rudimentary form of the ε equation is used in practically all industrial CFD codes to provide the sink term in the kinetic energy equation, and to supply the turbulence scale by which the turbulent diffusion is modelled. This equation is just too simple form for such a task for any turbulent flow away from local energy equilibrium. The inability of the EVM to generate accurately the normal stresses must reflect in the inability to model the production of ε due to normal straining, if these are significant (flow acceleration and deceleration, flow recovery after reattachment, etc.) The use of the second-moment closure with the same ε equation at least obviates this problem, because the normal stress components are well reproduced. However, any extra strain rates or departure from equilibrium may require additional modifications to the ε equation. There have been several proposals to modify and upgrade the simple ε equation, though most of them were aimed at curing a specific deficiency related to a particular flow class. Specific tuning usually resulted in achieving the aim, but in most cases subsequent validation in some other flows produced opposite effects. A case in point is the modification which emerged from the application of the Renormalization Group Theory (RNG). The outcome of this approach is the insertion of an extra term in the ε equation: R=
η(1 − η/η0 ) Pε 1 + βη 3 k
(115)
where η = Sk/ε is the relative strain parameter (in fact the ratio of the turbulence time scale p and of the scale of mean rate of strain) and S = Sij Sji is the strain-rate module. This term was aimed at distinguishing the large from the small strain rates by increasing or decreasing the source of dissipation - depending on whether the strain parameter is larger or smaller, respectively, from what is believed to be a typical value for homogeneous equilibrium shear flow, η0 = 4.8. While the term indeed improved the predictions in the recirculation zone, as well as in the stagnation region, it proved to be harmful in many other flow cases, particularly when the normal straining is dominant. The reason is that the term does not distinguish the sign of the strain Sij , and produces the same effect for the same strain intensity irrespective of its sign, i.e., whether the flow is subjected to acceleration or deceleration, compression or expansion (e.g. in reciprocating engines). Fig. 6 shows the effect of application of RNG term (115) with LRR stress model to flow in an axisymmetric contraction and expansion at roughly the same strain S=62.2 and 86.6 respectively, compared with DNS results of Lee (1985) (Hanjali´c, 1996). The standard model gives poor results, but it is obvious that the RNG modification improves the flow predictions in the expansion, but deteriorates those in the contraction. Hence, the use of such a remedy, particularly by inexperienced users, can yield adverse, insteadqof beneficial, effects. The strain-rate module S and the analogue mean-vorticity module Ω = (Ωij Ωij have 44
Figure 6: Predictions of kinetic energy evolution in an axisymmetric contractions (left) and expansion (right) with the basic LRR model and RNG modifications in the ε equation. been used by some authors to define variable coefficients in the ε equation (primarily Cε1 ), aimed at accounting for nonequilibrium effects and dissipation anisotropy (e.g. Speziale and Gatski (1997). Such modifications suffer from the same deficiency as RNG, because of inability to distinguish the sign of mean strain rate and vorticity. A sounder approach is to use turbulent stress invariants or other turbulence parameters to modify the coefficients, or to define extra source terms. Craft and Launder (1991) proposed to modify Cε2 as Cε2 =
0 Cε2
(116)
1/2
(1 + 0.65A A2 )
0 ). This modification was shown to perform well in several (which also requires modifying Cε1 flows, but within the framework of a nonlinear pressure strain model; its use in connection with conventional models requires additional tuning and validation. This modification has been recently replaced by an additional term in the ε-equation as a complex function of both A and its gradient (Craft and Launder, 1996).
Simpler remedies to improve the predictions of complex industrial flows without having to redefine the rest of the model, are also possible. Such are, e.g., the modifications that involve introducing two extra terms in the standard ε equation. Both terms have local effects only in the flow regions where a remedy is needed, while doing no harm in flows where the conventional dissipation equation serves well: SΩ = Cε5 kΩij Ωij (117) The term SΩ was introduced long ago by Hanjali´c and Launder (1980) to enhance the effects of irrotational straining on the production of ε (note that the coefficient Cε3 = 0.1 while Cε1 is now 2.6, instead of the conventional 1.44!). Although not helpful in dealing with streamline curvature, as originally expected, the term proved to be essential in reproducing flows with very strong adverse and favourable pressure gradients (Hanjali´c et al., 1997). It is recalled that Shimomura’s proposal to account for system rotation has the same form, except that one of the vorticity vectors is replaced by the intrinsic vorticity, as discussed earlier. The second new term Sl is defined as Sl = max
("
1 ∂l Cl ∂xn
2
−1
#
1 ∂l Cl ∂xn
2
)
;0
εε A k
(118)
where l = k 3/2 /ε is the turbulence length scale and Cl = 2.5. This term has been introduced to compensate for excessive growth of the length scale, and it proved beneficial to the improvement 45
of the predictions of reattaching separated flows. The term has indeed a local character, as seen e.g. in a flow behind a backward-facing step, Fig. 7, where the standard IP, QI or SSG models produce anomalous behaviour of the streamline pattern around flow reattachment (Hanjali´c 1996).
Figure 7: Streamline patterns behind a backward-facing step computed respectively with LRRG+WF, SSG+WF and LRRG+WF+Sl models. Contours of the Sl term (below left), Hanjali´c and Jakirli´c 1998)
5.4
Potential of Differential Second-Moment Closures to Model Complex Flow Phenomena
As mentioned earlier, the major advantage of the second-moment closure is that the Reynolds stress ui uj need not be modelled, but is provided from the solution of the model differential transport equation. Other benefits can be deduced from the inspection of the exact transport equation for ui uj . First, it contains several more terms. While this poses a new challenge (a need for modelling) and brings in additional uncertainty (unavoidable new empirical coefficients), it enables the treatment in an exact manner of several important turbulent interactions and, thus far, it also enables the capture of more subtle features of turbulent flows. Other terms can be modelled in a different, more appropriate way than in the EVMs because of the availability of ui uj . Some of these features and terms in the equation are considered below focussing on the potential of DSM, as compared to EVM, to reproduce the physics of the most common ”complexities” in turbulent flows. 5.4.1
Stress Production
The first benefit comes from the possibility to treat the stress generation in the exact form. Turbulent stresses are generated at the expense of mean flow energy by mean flow deformation, Pij , and by body forces (buoyancy, electromagnetic), Gij , and rotation, Rij . Second moments are explicitly present in all generation terms: Pij and Rij contain ui uj and Gij = fi uj is usually replaced by βgi θuj for buoyancy generation, (θ is the fluctuating temperature, or concentration, and β is the corresponding expansion coefficient). The advantage of obtaining second moments 46
from their own transport equations, instead of eddy viscosity/diffusivity model becomes particularly obvious when comparing, e.g., the production of kinetic energy in EVM and DSM in a two-dimensional flow: P
EV M
P DSM
∂U1 2 ∂U2 2 = 2µt + 2µt + µt ∂x1 ∂x2 ∂U1 ∂U1 ∂U2 = −u21 − u22 − u1 u2 ( + ∂x1 ∂x2 ∂x2
∂U1 ∂U2 + ∂x2 ∂x1 ∂U2 ) ∂x1
2
(119) (120)
In thin shear flows (dominated by simple shear ∂U1 /∂x2 ) both expressions give a similar value ∂U2 1 of P, because the effect of normal straining is negligible. In more complex flows, ∂U ∂x1 , ∂x2 , ∂U2 DSM and P EV M are different. ∂x1 ,.. can have significant values and different signs. Hence, P This becomes more evident (and more important) in flows with a complex strain-rate field, i.e. when the ”extra strain rate” originates from streamline curvature, flow skewing, lateral divergence, bulk dilation. Unlike EVM, DSM accounts exactly for stress production by each component of the strain rate. Even a small ”extra strain rate” can have a significant effect on stress production. For example, in a thin shear flow with a mild curvature, such as in a flow ∂U1 2 2 2 ∂U1 2 ∂U2 2 over an airfoil, ∂U ∂x1 ≪ ∂x2 , but u1 ≫ u2 , hence both terms in P12 = −(u2 ∂x2 + u1 ∂x1 ) are of importance (Bradshaw et al. 1981). The problem becomes even more serious in flows fully dominated by normal (dilatational) strain, because the expression for P EV M is always positive and cannot differentiate the sign of the strain rate, i.e., dilatation from compression, or fluid acceleration from deceleration. The exact contribution to P DSM caused by normal straining is the interaction between specific components of turbulent normal stress (positive quantities) with corresponding components of the normal strain-rate that can be either negative or positive. A simple example is a flow in a nozzle and diffusor of the same shape (of equal contraction and expansion ratio), where, depending on the inflow stress anisotropy (for the same k), very different flow development may be expected in two cases (e.g. DNS of Lee and Reynolds, 1985, Hanjali´c 1996). The standard k − ε yields the same results for the same initial level of k and ε for both the compression and expansion. Other, industrially more relevant examples are stagnation regions, flow impingement on a solid surface, boundary layer recovery after reattachment, recirculating flows, the central region in the cylinder of a reciprocating engine, where the flow is subjected to a cyclic compression and expansion, etc. In all these and other cases the computations with EVM k −ε and similar models can never be accurate. 5.4.2
Stress Interaction and Anisotropy
Another important turbulence feature that can be reproduced only by models based on the stress-transport equation, is the stress interaction. In flows with a preferential orientation of the velocity field, the dominant strain rate component feeds energy into selective stress component(s). Pressure fluctuations redistribute a part of the largest stress into other components (also reduces shear stress) making turbulence more isotropic. An illustration of stress interaction and of the role of pressure fluctuations is given in Fig. 8 for a simple thin shear flow, which shows a general flow chart of turbulence energy (stress) components. The exact treatment of the stress generation and a possibility to account for stress-component interactions gives a better prospect for modelling the important turbulence parameter, the stress anisotropy, which governs to a large extent the wall heat and mass transfer. This is particularly 47
Figure 8: A schematic of stress component interactions in a thin shear flow the case in regions with either small or no wall shear, such as around impingement, separation or reattachment, where the transport of mean momentum and heat transfer do not show any correlation or analogy. In these regions, the heat and mass transport are governed by the wall-normal turbulent stress component, and its accurate prediction is a crucial prerequisite for computing accurately the transport phenomena at a solid surface. 5.4.3
Streamline Curvature
Most complex flows involve strong streamline curvature. In principle, we may distinguish between the local streamline curvature irrespective of the shape of flow boundaries (e.g. recirculating region behind a step), and the bulk curvature (e.g. curved channels). The streamline curvature generates an ”extra” strain rate, which can exert a significant effect on stress production. Streamline curvature attenuates the turbulence when the mean flow angular momentum increases with curvature radius (e.g. flow over a convex surface - stabilizing curvature), whereas it amplifies the turbulence in the opposite situation (e.g. over a concave surface - destabilizing curvature). Effects of streamline curvature are directly related to the stress production and stress interactions. Basic EVMs fail to account for curvature effects in most turbulent flows because of their inability to reproduce normal stresses which appear explicitly in the production term Pij . Of course, one can model the effects of streamline curvature with simple models by ad hoc correction. This has been done in k −ε models, e.g., by expressing the coefficients in the sink term of the dissipation equation in terms of the ’curvature Richardson number’ Rit = (k/εR)2 Uθ ∂(RUθ )/∂R, where R is the local radius of the streamline curvature, and Uθ is the resultant mean velocity. Although modification of the scale-determining (ε equation) may be needed anyhow to deal more appropriately with streamline curvature, such a remedy can not compensate the deficiency of the k equation in reproducing the effects of normal stress. Note that Rit changes its sign and magnitude in accordance with the curvature sign, producing thus stabilizing or destabilizing effect. However, DSM captures these effects via exact production terms in the stress equations, which show a selective sensitivity to the streamline curvature. A highly curved shear layer may serve as an example of flow where the streamline curvature is a dominant effect. Computation by Gibson and Rodi (1981) by k − ε and DSM showed that the basic DSM model reproduces the effect much better than the k − ε model. It should be mentioned that the simple algebraic second-moment closures (algebraic stress mod48
els), ASM, perform marginally better than the basic EVM, but predictions can be substantially improved by accounting for extra strain rate ∂U2 /∂x1 (local curvature) or U1 /R (bulk curvature). More complex flows are usually ’contaminated’ by other extra strain rates and it is not easy to distinguish their individual effects on turbulence. Nor is it easy to diagnose a single cause of model failure to reproduce experimental results. Illustrative examples of computations of flows with bulk (longitudinal) curvature are the flows in U-bends and S-bends in circular or square-sectioned pipes, reported by Iacovides & Launder (1985), Anwer et al. (1989), Iacovides et al. 1996, and others. All these computations show the superiority of DSM or ASM over the k − ε model. 5.4.4
System Rotation
The next flow feature which can be better captured by DSM is the system rotation. The bulkflow rotation affects both the mean flow and turbulence by the action of the Coriolis force FiC = −2ρΩj Uk ǫijk . System rotation influences directly the intensity of stress components through stress redistribution. It is also known that rotation affects the turbulence scales. Here, Ωj is the system angular velocity. As mentioned earlier, the stress transport equation contains the exact rotational production term Rij . Because the Coriolis force acts perpendicular to the velocity vector Uk , it has no direct influence on the k-budget (Rii = 0), but redistributes the stress among components and modifies thus far the net production of individual stress components. Rotation also causes a decrease in dissipation, even in isotropic turbulence (DNS by Bardina et al., 1985). The effects of rotation on turbulence can be illustrated in a simple shear flow, such as a plane channel rotating around an axis perpendicular to the main flow direction with an angular velocity vector (”system vorticity”) Ωj (here Ω3 ), Fig. 9. Note that Ωj is aligned with the dual vector of the mean shear vorticity ωk = ǫijk Ωji (here ∂U2 1 ω3 = Ω21 − Ω12 = − ∂U ∂x2 , since ∂x1 = 0). If Ω3 and ω3 (or -Ω12 ) have the same direction, rotation attenuates the turbulence (stabilizing effect, ”suction side”). In contrast, if Ω3 and ω3 have the opposite directions, rotation will amplify the turbulence (destabilizing effect, ”pressure side”). At high rotation rates, the stabilizing effect may completely damp the turbulence at the suction side thus causing local laminarization. This criterion is often expressed in terms of the local Rossby number, defined as the ratio of the mean shear vorticity to the 1 /∂x2 system vorticity Ro = ωl /Ωl (here Ro = − ∂U2Ω ), 3 or its reciprocal S = 1/Ro. The sign of Ro or S indicates whether the effect on turbulence will be amplifying (-) or attenuating (+).
Figure 9: Illustration of system rotation
For a general classification, the bulk rotation number Ro = 2Ω h/Ub is often used, where h is 49
the channel half width and Ub is the bulk velocity. The local Rossby number or Ro and S has served in the past to modify the simple k − ε model by expressing one of the coefficients in ε equation in terms of Ro number. However, the stress transport equation accounts exactly for the effects of rotation through the exact rotational generation term Rij (see equation (63). The table below lists the components of Rij for the plane channel example (note, Ω3 = Ω). ij 11 22 33 12 dU1 1 0 0 −u22 dU Pij −2u1 u2 dx2 dx2 Rij 4Ωu1 u2 −4Ωu1 u2 0 −2Ω(u21 − u22 ) In addition to the exact Rij term in the DSM, system rotation affects also the stress redistribution induced by fluctuating pressure. Hence, the effect of rotation should be accounted for in the model of the pressure-strain term. A simple way to do this is to replace Pij in Φij,2 (see below) by the total stress generation Pij + Rij . In order to ensure the material frame indifference, Pij should be replaced by Pij + 12 Rij (e.g. Launder et al. 1987). So far, no convincing proof has been provided as to which of the two modifications performs better, but both versions have led to substantial improvement of prediction of rotating flows. Because Rii = 0, the basic k − ε model, without modifications, cannot mimic the rotation effects on turbulence. Modifications are introduced usually via additional term(s) in the ε-equation, analogous to the modifications for curvature effects, in terms of rotation Richardson number Rir = −2Ω(k/ε)2 (∂U1 /∂x2 ), or by making the coefficient in eddy-viscosity a function of a rotation parameter. In fact, the ε-equation should, in principle, be modified to account for the effects of rotation on turbulence scale even in conjunction with DSM models. Bardina et al. (1987) suggested an additional term: 1/2 1 Xij Xij (121) −CΩ ε 2 where CΩ = 0.15 and Xij = Ωij + ǫkji Ωk is the ’intrinsic’ mean vorticity tensor4 Shimomura (1993) proposed a slightly different formulation, −CΩ k ωl Ωl , which improved predictions of flow in a rotating plane channel (Jakirli´c et al., 1998). It is interesting to note that this term resembles closely that proposed by Hanjali´c and Launder (1980) for nonrotating flows to enhance the effect of irrotational straining on ε (energy transfer through the spectrum, see below), Cε3 k ωl ωl . This term can be combined with that of Shimomura to yield a joint term which would be effective in both nonrotating and rotating flows, i.e. C kωl (ωl − CΩ /Cε3 Ωl ). Successful DSM computation of rotating channel flows with no modification of ε-equation were reported by Launder et al. (1987) for moderate rotation. These results indicate that the effect of rotation is stronger on the stresses than on the turbulence scale, which illustrates further the advantage of DSM, as compared with EVM. The importance of the integration up to the wall (using a low-Re-number model) in flows where the rotation is sufficiently strong to cause laminarization on the suction side, has been demonstrated by Jakirli´c et al. 1998. 4
or, in terms of dual vectors, −CΩ ε (Xl Xl )1/2 , where Xl = 12 ωl + Ωl
50
5.4.5
Swirl
The swirling flows can be regarded as a special case of fluid rotation with the axis usually aligned with the mean flow direction (longitudinal vortex) so that the Coriolis force is zero. Swirl enhances the turbulent mixing and often induces recirculation. These features are exploited in IC engines and gas-turbine combustors, and for heat transfer enhancement (vortex generators). A common feature of such flows is that the swirl is strong and confined to short cylindrical enclosures, whose length is of the same order as the duct diameter. Another type of confined swirl, usually of weak intensity, is encountered in long tubes, either imposed at the tube entrance, or self-generated by secondary motions as a consequence of upstream multiple tube bends (doubleor S-bends in different planes). A third kind is the swirling motion inside of rotating cylinders or pipes, or their parts, either developing or fully developed. Although all these cases deal with essentially the same phenomenon - rotating fluid in axisymmetric geometries - their predictions pose different challenges. Also different are the free swirling jets which modify substantially the flow characteristics even at a very low swirl intensity. EVM computations employ the swirl-dependent coefficients in the modelled equations, but generally with little success. Basic DSM did not much improve the predictions of a swirling jet. The simple ’Isotropization of Production’ (IP) model of Φij,2 (see below) seems to perform better than the LRR QI (quasi-isotropic) model, (Launder & Morse, 1979). By recognizing that a major deficiency lies in Φij,2 , Fu et al. (1987), proposed the inclusion of convection Cij to ensure material frame indifference, (negligible effects in nonswirling flows): 2 2 Φij,2 = −C2 [(Pij + Rij − P δij ) − (Cij − C δij )] 3 3
(122)
It was found, however, that this modification produces little effect (at least in weak swirls) since the last term in eqn. (122) is smaller than other terms in the expression. Still better predictions are expected with improvements of the scale equation. Several cases of swirling flows reported in the literature show obvious superiority of the DSM model. This is particularly the case of strong swirls in a combustion-chamber type of geometry. Similar improvements are achieved for a swirl in a long pipe. (e.g. Hogg and Leschziner 1989, Jakirli´c 1997). 5.4.6
Mean Pressure Gradient
The transport equations for turbulent stresses and scale properties do not contain mean pressure. However, the mean pressure gradient modifies the mean rate of strain and - depending on the sign - amplifies or attenuates the turbulence. Extreme cases are the laminarization of an originally turbulent flow, when dp/dx ≪ 0 (severe acceleration) and the flow separation when dp/dx ≫ 0 (strong deceleration). Both extreme cases represent a challenge to turbulence modelling. Turbulent flows subjected to periodic variations of pressure gradient or other external conditions (pulsating and oscillating flows) fall into the same category with an additional feature: a hysteresis of turbulence field lagging in phase behind the mean flow perturbations. Basic EVM cannot capture these features. DSM performs generally better, though additional modifications (mainly in the scale equation, see below) are needed. Wall functions are inapplicable for specifying boundary conditions and the integration up to the wall, with appropriate modifications of the model, is essential for reproducing these phenomena accurately. Predictions with a low-Re-number DSM of turbulence evolution and decay in an oscillating flow in a pipe 51
at transitional Re numbers, displaying a visible hysteresis of the stress field, was reported by Hanjali´c et al. (1995). An overview of the performance of DSM in flows with different pressure gradients involving separation is given in Hanjali´c et al. (1999). 5.4.7
Secondary Currents
This term refers usually to a secondary motion with longitudinal, streamwise vorticity ω1 , superimposed on the mean flow in the x1 -direction. Skew-induced (pressure-driven) ω1 (Prandtl’s 1st kind of secondary flows) is essentially an inviscid process, generated by the deflection of existing mean vorticity. Viscous and turbulent stresses cause ω1 to diffuse. Turbulent-stress-induced ω1 (Prandtl’s 2nd kind of secondary flow), is generated by the turbulent stresses due to anisotropy of the normal-stress field. Secondary motions can arise in the form of a ’cross-flow’ such as in 3-D thin shear flows (ω1 ≈ ∂U3 /∂x2 ), or in a form of recirculating ’cross-currents’ such as occur in noncircular ducts (ω1 = ∂U3 /∂x2 − ∂U2 /∂x3 ), Fig. 10. Of course, a secondary current can be imposed on the flow in order to enhance mixing, or heat and mass transfer, such as in the case of vortex generators. Skew-induced secondary velocity can be high, while the stress-induced currents are weak, though still very important for turbulent transport.
Figure 10: Schematics of secondary currents in conduits of noncircular cross-section and wing-tip vortices An illustration of the importance of turbulence in the dynamics of mean-flow vorticity can be envisaged by inspecting the vorticity transport equation, which, e.g., for the ω1 component, reads: Dω1 ∂U1 ∂ω1 ∂U1 ∂U1 + ω2 = ν + ω1 + ω3 Dt ∂xk ∂xk ∂x1 ∂x2 ∂x3 | {z }
vortex stretching
− |
|
∂ 2 u2 u3 ∂ 2 u2 u3 ∂2 + + (u2 − u23 ) ∂x2 ∂x3 2 ∂x22 ∂x23 {z
{z
}
”skew−induced”ω1 (vortex bending)
”stress−induced”generation of Ω1
}
(123)
Skew-induced secondary motion, driven essentially by mean-flow deformation and by the mean pressure field does not require a complex turbulence model. However, the stress-induced motion cannot be handled with EVM and requires a model which can compute individual turbulent stress components, (DSM or ASM). In fully developed flows in ducts of non-circular crosssection the application of ASM is sufficient to capture the stress-induced secondary motion. Illustrations of the prediction of secondary currents in square ducts have been published by Demuren and Rodi (1984), in pipe bends by Anwer et al. (1989), and in U-ducts by Iacovides et al. (1996). 52
5.4.8
Three-dimensionality
Finally, a few remarks must be added concerning the flow three-dimensionality effects. Even a mild three-dimensionality of the mean flow produces significant changes in turbulence structure. In strong cross-flows the effects can be dramatic as, e.g. in the case of unidirectional fluid stream with a superimposed longitudinal vortex, such as produced by a vortex generator to enhance wall heat transfer. The resulting mean velocity profiles may look very skewed, as shown in Fig. 11, which is difficult to reproduce by simple eddy viscosity and similar models. Other examples of relatively simple 3-D boundary layers include wing-body (or blade-rotor) junctions, flows encountering laterally moving walls imposing the transverse shear, such as in stator-rotor assembly in turbomachinery. In fully 3-D separating flows the problem is even more challenging.
Current turbulence models have been developed on the basis of our knowledge of 2-D flows. Plausible extensions to the third dimension do not always yield satisfactory results. This is particularly valid for linear eddy-viscosity models. Even in a simple 3-D boundary layer the eddy viscosity is not isotropic, as discussed earlier, i.e.: u1 u2 u3 u2 6= (124) ∂U1 /∂x2 ∂U3 /∂x2
Figure 11: Schematic of mean velocity profile in flow with a longitudinal vortex
This finding illustrates best that complex flows require turbulence models of a higher order than the EVM. Further illustrations of the inadequacy of the eddy viscosity concept in 3-D flows can be found in Hanjali´c (1994) and elsewhere.
53
6 6.1
Advanced Models for Near-wall and Low-Re-number Flows Wall-functions approach and their deficiency
All industrial CFD codes use the ’wall functions’ to treat the wall boundary conditions. Because the viscosity-affected near-wall region is bridged by placing the first grid point outside the viscous layer in fully turbulent region, high-Reynolds-number turbulence closures can be used for flows at high bulk Reynolds numbers. This simplifies to a great degree both the modelling and computational tasks. By obviating the need to resolve high-gradients very close to the wall, a relatively coarse numerical grid can often suffice for reaching grid-independent solutions. However, the wall functions have been derived on the basis of wall scaling in attached boundary layers in local turbulent energy equilibrium. Their validity in nonequilibrium flows is therefore considered inappropriate, particularly in separating and recirculating flows, around reattachment, at strong pressure gradients and in rotating flows. Fig. 12 shows two examples of deviation of the mean velocity profile from the conventional logarithmic law which serves as a basis for the wall function for the mean velocity. The three profiles on the left correspond to three locations in a flow behind a backward-facing step in the recovery zone downstream from reattachment. Even greater deviation is exhibited in the recirculation zone. Figure on the right shows axial and tangential velocity in a swirl generated by cylinder rotation some time after the rotation was stopped (spin-down). Good agreement with experiments was achieved only with the application of a low-Re-number DSM and integration up to the wall. Various modifications were proposed
Figure 12: Mean velocity profiles at selected locations in the recovery region behind a backwardfacing step flow (Jakirli´c and Hanjali´c 1995) (left) and profiles of axial and tangential velocity in a spin-down flow in an engine cylinder at the beginning of compression (Hanjali´c et al. 1996) (right) to improve and extend the validity of wall functions to non-equilibrium and separating flows, but none of the proposal showed general improvement. The incorporation of pressure gradient is most straightforward and can easily be done by extending the near-wall flow analysis, e.g. Ciofalo and Collins (1989), Kiel and Vieth (1995), Kim and Choudhury (1995) . Such modifications generally lead to some improvement of attached thin shear wall flows with pressure gradient (with convection still being neglected), but its validity is confined only to such situations. More general two-layer approach, based on splitting the wall layer in viscous and nonviscous parts with assumed variation of shear stress and kinetic energy in each layer, was earlier proposed by Chieng and Launder (1980) and Johnson and Launder (1982). The assumed profiles for uv and k enable to integrate separately the stress production and dissipation in the first control volume next to a wall, instead of assumed wall-equilibrium values. However, despite some improvement 54
of wall friction and heat transfer behind a back step and sudden pipe expansion, the approach still has serious deficiencies.
6.2
Models with near-wall and low-Re-number modifications
The integration up to the wall is a more exact alternative to wall functions. This approach requires first the introduction of substantial modification to the turbulence models in order to account for complex wall effects, primarily for viscosity (”low-Re-number models”), but also for nonviscous flow blocking and pressure reflection by a solid wall. This in turn requires a much finer grid resolution in and around the viscosity-affected wall sublayer, and, consequently, increases demands for computation resources, with often formidable requirements on numerical solver to ensure convergent solutions5 . Low-Re-number models are at present available both at the EVM and DSM level, and they are indispensable for predicting the laminar-to-turbulent and reverse transition (at least the forms of transitions which can be handled within the framework of Reynolds averaging approach, such as by-pass transition, revival of inactive background turbulence, turbulence laminarization). A number of proposals for modification of the DSM to account for low Reynolds number- and wall-proximity effects can be found in the literature. These modifications are based on a reference high-Re-number DSM which serves as the asymptotic model to which the modifications should reduce for sufficiently high Reynolds number and at a sufficient distance form a solid wall. Hanjali´c & Launder (1976), Launder & Shima (1989), Hanjali´c & Jakirli´c (1993), Shima (1993), Hanjali´c et al. (1995) base their modifications on the basic DSM with linear pressurestrain models, in which the coefficients are defined as functions of turbulence Reynolds number and invariant turbulence parameters. Earlier models use also the distance from a solid wall. More recent models are based on DNS data and term-by-term modelling, which ensures model realizability, near-wall stress two-component limit, as well as the conditions at the vanishing turbulence Reynolds number. Launder & Tselepidakis (1991), Craft and Launder (1996), Craft (1997) use the cubic pressure-strain model in which the coefficients were determined by imposing, in addition to basic constraints discussed earlier, also the two-component limit. A larger number of coefficients at disposal reduces the need for introducing functional dependence and additional turbulence parameters. A different approach was proposed by Durbin (1991, 1993): elliptic relaxation is used to account for non-viscous blockage effect of a solid wall, whereas a switch of time and length scale from the high-Re-number energy-containing ones to Kolmogorov scales, when the latter become dominant, accounts for viscosity effects. All models require very fine numerical mesh within the viscosity affected wall sub-layer, so that computation becomes time-consuming and impractical for more complex flow cases. Durbin’s elliptic relaxation approach seems to be somewhat less demanding in this respect. In principle, all modifications involve the following: • Inclusion of viscous diffusion in all equations; • Model of εij is provided; 5
A possible compromise, which avoids a need for employing wall function and still remain sensibly within available computer resources is the ”two layer” approach by which a simpler model (say k − ε, one-equation or even mixing length models) is applied within the viscous sublayer, while DSM is used in outside turbulent region (e.g. (Franke & Rodi, 1991).
55
• Additional term is added to the ε-equation (supposedly to model Pε3 ) • Coefficient Cε2 (in some models also C1 , C2 ..) are replaced by functions of turbulence Re number, Ret = k 2 /(νε), nondimensional wall distance and/or other turbulence parameters. 6.2.1
A low-Re-number DSM
An example of low-Re-number second-moment closures (DSM) is the model proposed by Hanjali´c et al (1995) (see also Hanjalic et al. (1997) Jakirli´c (1997), Hadzi´c (1998). This model was used with a reasonable success in a number of 2D and some 3D high- and low -Re-number wall flows including cases of severe acceleration (laminarizing 2D and 3D turbulent boundary layers, some forms of laminar-to-turbulent transition, oscillating flows at transitional Re numbers, rotating and separating flows. Examples will be shown in the next subsection. The model is based on the basic DSM (Section 2) in which the low-Re-number version of ε equation is used and the coefficients in the ui uj equation are expressed as functions of Ret and invariants of the stress and dissipation rate tensors to account for both the viscous and inviscid wall effect, as well as to satisfy the two-componentality and vanishing Reynolds number limits, enabling thus the integration up to the wall. Because viscosity has a scalar character (it dampens all stress components and is independent of the wall distance and its topology), and is only indirectly related to the wall presence via no-slip conditions, its effect can be conveniently accounted for through the turbulence Reynolds number Ret = k 2 /(νε). This should be formulated in a general manner to be applicable both close to a wall and away from it. Inviscid effects are basically dependent on the distance from a solid wall and its orientation, as seen from the Poisson equation for fluctuating pressure (’Stokes term’). This term accounts for the wall blockage and pressure reflection. However, the DNS data for a plane channel show that this term decays fast with the wall distance and becomes insignificant outside the viscous layer. Yet, a notable difference in the stress anisotropy between a homogeneous shear flow and equilibrium wall boundary layer for comparable shear intensity shows that the effect of wall presence permeates much further away from the wall into the log-layer. This indicates at an indirect wall effect through a strong inhomogeneity of the mean shear rate, the fact that is ignored by all available pressure-strain models. In view of above discussion, the use of wall distance through the function fw and wall orientation represented by unit normal vector ni in the adopted models for Φij,w seems reasonable, despite some opposing views in literature. However, these modifications, introduced for and tuned in high-Re-number wall attached flows, can not account for inviscid effects closer to a wall (buffer and viscous layer). It is known that wall impermeability imposes a blockage to fluid velocity and its fluctuations in the normal direction, causing a strong anisotropy of turbulence. This fact has been exploited by Hanjali´c em et al. (1995) by introducing, in addition to Ret , invariants of both the turbulent-stress and dissipation-rate anisotropy, aij = ui uj /k−2/3δij and eij = εij /ε−2/3δij , respectively, A2 , A3 , E2 and E3 , as parameters in the coefficients. This enables to account separately for the wall effect on anisotropy of stress bearing and dissipative scale, shown by the DNS data to be notably different (Hanjali´c et al. 1997, 1999). The sensitivity of stress invariants to pressure gradient is illustrated in Fig. 13, where the Lumley’s two-componentality (’flatness’) parameter A is plotted for boundary layers at zero, favourable and adverse pressure gradients. Also the predictions of A with the here presented low-Re-number second-moment closure model 56
are shown.
Figure 13: Lumley’s two-componentality (’flatness’) parameter for boundary layers at zero, favourable and adverse pressure gradients Based on the above arguments, the following modifications were introduced: The stress transport equation Φij : Linear models are adopted for the slow, rapid and wall terms, equations (72), (73), (76) and (77), in which the coefficients are defined as follows: √ C1 = C + AE 2 C = 2.5AF 1/4 f F = min{0.6; A2 } f = min
(
C2 = 0.8A1/2
Ret 150
3/2
)
;1
"
k 3/2 ; 1.4 fw = min 2.5εxn
C1w = max(1−0.7C; 0.3)
#
C2w = min(A; 0.3)
where 9 A = 1 − (A2 − A3 ) 8
A2 = aij aji
9 E = 1 − (E2 − E3 ) 8
A3 = aij ajk aki
E2 = eij eji
aij =
E3 = eij ejk eki
eij =
2 ui uj − δij k 3 εij 2 − δij ε 3
εij - The stress dissipation rate model: 2 εij = fs ε∗ij + (1 − fs ) δij ε 3
ε [ui uj + (ui uk nj nk + uj uk ni nk + uk ul nk nl ni nj )fd ] ε∗ij = u u k 1 + 3 p q np nq fd
fs = 1 −
2
√
AE 2
k
fd = (1 + 0.1Ret )−1
The ε equation
57
• Equation 80 for ε is modified in the form: Dε Dt
where
∂ε k ∂Uk ε νδkl + Cε uk ul + (Cε1 P + Cε3 G + Cε4 k − Cε2 fε ε˜) ε ∂xl ∂xk k 2 2 k ∂ Ui ∂ U i + SΩ + Sl (125) + Cεν ν uj uk ε ∂xj ∂xl ∂xj ∂xl
=
∂ ∂xk
"
Ret Cε − 1.4 exp − fε = 1 − 2 Cε2 6
2 #
ε˜ = ε − 2ν
and SΩ and Sl have been defined earlier, equations (117) and (118).
58
∂k 1/2 ∂xn
!
7
Second-Moment Closures and Numerical Implications
The illustrations presented in the preceding chapters are only some of the numerous examples, which demonstrate higher potential and overall superiority of advanced turbulence closure models based on differential transport equations for second moment of turbulence fluctuations (Reynolds stress and scalar flux) over standard two-equation models. However, their implementation in general three-dimensional Navier-Stokes solvers has been shown to cause numerical difficulties which at present hinders their wider application to industrial flow computations. In fact, the more advanced and more general the model is, the higher are the demands on computational resources and the more frequent the problems with numerical convergence. Why is it that the seemingly more exact approach, which involves modelling at a higher, more exact level, is associated with increased numerical problems? These and some related issues are addressed from the standpoint of a model developer, not a numericist. One can argue that the change in the physical nature of the governing equations, when implementing different modelling levels, may require a different approach to the discretization, coupling and the numerical solution of the resulting equation set. However, the convergence problem seems to be manageable if some simple remedies are used, as shown below. The origin of the problem lies in the coupling of the mean velocity and turbulent stress field. This is readily seen by comparing the Reynolds-averaged Navier-Stokes equations for incompressible flow, in which the Reynolds stresses ui uj are provided by the Eddy-Viscosity Turbulence Models, with those in the original form in which the Reynolds stresses are supplied from separate transport equations: EVM approach: "
∂ ∂ρUi ∂ρUj Ui ∂Ui ∂P + (µ + µt ) + = Fi − ∂t ∂xj ∂xi ∂xj ∂xj |
DSM approach:
{z
Source
}
|
!#
{z
}
Diffusion
"
∂ ∂P ∂ui uj ∂Ui ∂ρUi ∂ρUj Ui + + = Fi − − µ ∂t ∂xj ∂xi ∂xj ∂xj ∂xj |
{z
Source
}
|
(126)
{z
Diffusion
!#
(127)
}
∂ρUk ui uj ∂ρui uj + = .... ∂t ∂xk Irrespective of the flow Re number, the EVM always ensures the same character of the momentum equation, dominated by the second-order diffusion term. In contrast, the momentum equation with second-moment closure is dominated by the source term supplied from the separate transport equation for ui uj , particularly for high Re-number flows (where the viscous term is negligible) and at a weak pressure gradient. An efficient numerical coupling and solution of the equation set involving the transport equations for ui uj may require a different approach from the conventional Navier-Stokes types of solvers. It should be recalled, however, that several simpler approaches have already been successfully applied to solve complex flows with the second-moment closures. The basic idea behind 59
most current practice is the introduction of an artificial eddy viscosity, which allows the use of the available Navier-Stokes solver. Three possible velocity-stress couplings are shown here for illustration; they are reduced for clarity to a case of a fully developed two-dimensional flow (in conjunction with the standard finite-volume numerical method): ”Interpolation of equations instead of variables” (Obi, Peri´c and Scheuerer, 1989) yields, e.g., the shear-stress term in the momentum equation in the conventional form: Z
−
Ωp
∂ρuv uv n − uv s dΩ ≈ ρ Ωp ∂y δ yp
(128)
The shear-stress equation is rewritten in a form in which the major mean-velocity gradient is subtracted from the production and pressure strain term: Dρuv Dt
= P12 + Φ12 − ε12 + D12 = −Γ12 ρv 2
∂U ′ ′ + P12 + Φ12 − ε12 + D12 ∂y
(129)
where Pij′ and Φ′ij represent the remaining parts of both terms respectively. Now the shear stress in the considered control volume is evaluated from the expression
uv P =
SPuv +
P
AK uv K
AP
−
Γ12 ρv 2 P Un − Us AP δyP
(130)
and the required values at the cell faces, to be supplied to the integrated equation for ui uj (81), are obtained (e.g. for the northern face n) from Figure 14: Control volume notations
−uv n = huv P in = −
|
* |
SPuv +
P
AK uv K
AP
+
+
n
{z
Source
}
{z
*
Γ12 ρv 2 P AP yP
|
+
n
(UN − UP )
{z
Diffusion
Contributions to momentum equation
(131)
}
}
where h·i denote linear interpolation using values at centres of the neighbouring cells around the face ”n” ”Artificial nonisotropic eddy viscosity” derived from the algebraic truncation of stress equations, (Lien and Leschziner, 1993) introduces a form of a tensorial eddy diffusivity which for i 6= j yields µtij = ρ
k u2j 1 − c2 + 1.5 c2 cw 2 (fi + fj ) c1 + 1.5 cw ε 1 (fi + fj ) 60
(132)
and for i = j
(no summation) yields µtii = ρ
2 2 − 4/3 c2 + 2/3 c2 cw 2 (4 fi + fj ) k ui c1 + 2 c w ε 1 fi
(133)
In this approach the momentum equations become ∂ρUi ∂ρUj Ui + ∂t ∂xj ∂P − ∂xi
"
∂Ui ∂Uj (µ + µij ) + ∂xj ∂xi
−
∂ ∂xj
−
∂ui uj ∂ + µij ∂xj ∂xj
"
!#
∂Ui ∂Uj + ∂xj ∂xi
=
!#m−1
(134)
where m denotes the current iteration. Even simpler ’artificial isotropic eddy diffusivity’ was introduced by other authors (e.g. Basara, 1993), yielding the stress tensor:
ui uj
m
= νt
∂Ui ∂Uj + ∂xj ∂xi
!m
+ ui uj
m−1
− νt
∂Ui ∂Uj + ∂xj ∂xi
!m−1
(135)
which leads to the following form of the mean momentum equation: ∂ρUi ∂ρUj Ui + ∂t ∂xj ∂P − ∂xi where µt = ρCµ reached).
"
∂Ui ∂Uj (µ + µt ) + ∂xj ∂xi
"
∂Ui ∂Uj + ∂xj ∂xi
−
∂ ∂xj
−
∂ ui uj + µt ∂xj
!#
=
!#m−1
(136)
k2 retains the conventional k − ε form (irrelevant if the convergent solution is ε
These and other possible approaches, available in literature, have been used successfully to compute a number of complex three-dimensional turbulent flows. While their use requires more effort and computational time than simple EVM and, in some cases, some extra skill in running the computational code if very complex flows are dealt with, there is no doubt that the apparent incongruity between the advanced closure models and the computational methods used in CFD are being reconciled, and that the second-moment closures will in the near-future replace the two-equation models as industrial standards.
61
8
Some Illustration of DSM Performance
Many complex flows contain several ’types’ of strain rate and it is not easy to distinguish their individual effects on turbulence. Moreover, different types of strain or other effects may dominate different regions so that improvements in one region can lead to deterioration in others. Improvements can often be achieved with different remedies and it is not always clear which modifications have a better physical meaning. Some illustrations of such flows are provided in the next two figures.
Figure 15: Predicted mean velocity (a) and turbulence kinetic energy (b) at r/D = 0.5 for an axisymmetric jet impinging on a plane was, obtained with different models (Basara et al. 1997).
Figure 16: Mean velocity profiles (a) (— DSM, .... standard EVM, - - - EVM+RNG); velocity vectors and contours of kinetic energy (DSM) in a subchannel of a tube bank.Symbols: experiments by Simonin and Barcouda (1988) (Hadˇzi´c and Hanjali´c 1994) The first example is a turbulent jet issuing from a circular tube and impinging normally on a plane wall, Fig. 15. The second example is a fully developed flow through a tube bundle, Fig. 16. Both flows are characterized by a strong dilatational strain in the stagnation region and strong streamline curvature, with a complex interaction between the stress components. In both cases the effect of mean pressure is dominant so that the influence of the turbulence model on the predicted mean velocity field is not strong, Fig. 15a. However, the predicted turbulence depends largely on the model applied, as shown in Fig. 15b, where the performance of model variants is shown. The standard k-ε model yields a far too high kinetic energy, because of poor 62
modelling of the normal stress production, as discussed earlier. The next three curves illustrate the effect of the model of the pressure-strain term in DSM. The basic DSM (BDSM) is that of Launder, Reece and Rodi (LRR-IP) model (1975) with Gibson and Launder (1978) wall-echo correction (hence denoted as LRRG), performs much better, though still not satisfactorily owing to inadequacy of the wall-echo term for impinging flows. Hence, better results are obtained by simply omitting the wall-echo term. Further improvement is achieved when using the pressurestrain model of Speziale, Sarkar and Gatski (1991), denoted as SSG, which contains no extra wall-echo term. The application of the cubic model of Craft and Launder (1991) was reported to perform best (not shown here), but at the expense of greater complexity. While the above discussion clearly demonstrates the importance of the pressure-strain model, it would be wrong to conclude that this is the sole cause of unsatisfactory performance. The scale equation (here ε) in the simplest form is clearly inadequate to model complex flows with extra strain rates. The effects of three possible modifications, each involving an extra term, are illustrated by the last three curves in Fig. 15b, showing further possibilities to improve the predictions. Accounting for the effect of irrotational strain (term SΩ ) together with the control of length scale in the near-wall region, brings the results almost in accordance with the experiments. Fig. 16 provides additional illustration: the mean velocity profiles in a tube bank, although not very sensitive to the turbulence model, are improved when EVM (standard k − ε) is replaced by even basic DSM. Fig. 16 also illustrates that improvements in some flows can be achieved also RNG modification of ε equation. The next illustration show examples of swirling flows. A distinction is made between the strong swirls in short cylindrical containers such as combustion chambers, and in long pies (usually with weak swirls). Fig. 17 shows the profiles of axial velocity at two locations in a cylindrical combustion chamber and the velocity evolution at the symmetry axis, computed with standard k − ε model with wall functions, and with a high and low-Re-number DSMs. The latter model show obvious superiority. Similar improvements are achieved for a swirl in a along pipe. Fig. 18 compares the axial and tangential velocities computed by several models with two sets of experimental data. As in the previous example, both the high- and low-Re-number DSM reproduce flow features much better than the two low-Re-number k − ε models (Launder and Sharma and Chien).
Figure 17: Profiles of axial velocity at two locations in a swirling flow in a combustion chamber (left) and the evolution of its value at the symmetry axis (right), computed with standard k-ε+WF model and with a high-and low-Re-number DSM (Jakirli´c 1997) The next example, shown in Fig. 19, is the transitional flow around a round-leading-edge flat plate. Experiments and flow visualization show that in an undisturbed laminar flow a small laminar separation bubble appears shortly after the leading edge. Transition to turbulence 63
Figure 18: Profiles of axial and tangential velocity in two swirl flows in a long pipe (Jakairli´c 1997), computed by low-Re-number k-ε and DSM models, compared with experiments of Kitoh (1991) and Steenberger (1995) occurs at the rear end of the bubble very close to the wall, followed by gradual diffusion of turbulence into the outer flow. Computation with a low-Re-number k − ε model produces the transition and a high turbulence level already in the stagnation region, causing a strong mixing, which prevents the flow separation. The low-Re-number second-moment closure reproduces both the flow pattern with a laminar separation and the transition location which are in close agreement with experimental findings. The computation of both these cases was performed by introducing a very small level of background turbulence in the incoming uniform flow and by applying a block-structured grid with a high resolution in the near wall region. The application of a second-order convection scheme was essential in resolving the near-wall region. Despite the fact that the applied basic LRRG pressure-strain model was found inadequate for impinging flows (see Fig. 15), the low-Re-number modifications, and the dominance of the shear production around the curved leading edge, diminish the effect of the above mentioned model deficiency. An example of more complex configuration is the flow over the NACA 4412 airfoil at maximum lift, with an incidence angle of 13.87o (Coles and Wadcock, 1979), Fig. 20. At high Re numbers (here R = 1.52 × 106 ) the transition length is very short and has little influence on the overall results. However, in contrast to the one- and two-equation models used in aeronautics, the present DSM yields the transition location in accordance with observation without any empirical input (Hanjali´c et al.1999). The predicted mean velocity profiles and turbulence properties are well reproduced by both high- and low-Re number DSM, though the flow pattern around the separation is somewhat different. Note that the separation length is very similar, despite a difference in the shape of the separation bubble. Admittedly, satisfactory predictions of this flow have also been obtained by some two-equation models (e.g. zonal k − ε/k − ω model of Menter, see Guilmineau, 1997), but with a higher degree of empiricism. In any case, this example was presented to demonstrate that the second-moment closure (including its low-Renumber variant with integration up to the wall) may be successfully used to compute complex flow over curved surfaces at very high Re numbers and with strong pressure variation. The above examples indicate that the prediction of wall phenomena (wall shear, heat and mass
64
transfer) in complex flows can at present be achieved only by applying an advanced low-Renumber model with integration up to the wall. However, if wall phenomena are not in focus, it seems that even the classical wall functions in conjunction with a better high-Re-number models (i.e. second-moment closures) can reproduce well the general flow pattern, e.g., recirculation bubbles (see Jakirli´c and Hanjali´c, 1996). This is illustrated in Fig. 21 where the friction factors are shown for a back-step flow and for a sudden pipe expansion. Both cases were computed with the LRRG high Re-number DSM model with wall functions and with the low-Re-number DSM.
Figure 19: Streamlines, kinetic energy contours and profiles of the streamwise turbulence intensity in a transitional flow over a flat plate with round leading edge, computed with a low-Re k-ε (a) and DNS (b) models (Hadˇzi´c and Hanjali´c 1997)
65
Figure 20: Flow around NACA 4412 airfoils at 13.87 deg incidence: streamlines, pressure distribution and velocity profiles computed with k-ε+WF and DSM (SSG)+WF models. Symbols: experiments by Coles and Wadcock 1979) While both models yield very similar flow patterns (almost identical location of reattachment, in good agreement with experiments) the low-Re-number predicts the friction factor much better. It should be noted that the application of advanced low-Re-number models to complex highRe-number industrial 3-D flows, where a nonorthogonal body-fitted grid is needed, may never become a practical option. A middle of the road option may be a ’two-layer’ (zonal) approach in which a simpler model (two-equation or even mixing length) is applied within the viscous sublayer, matched with DSM or another advanced model in the rest of the flow (fully turbulent regime). Yet, such a simplification introduces a serious breach of physical constraints, inconsistent with the model applied in the rest of flow. Simple two- or one-equation low-Re-number models can be designed to account for viscous (scalar) effects, but they can hardly reproduce the wall-topology-dependent nonviscous blocking and the consequent stress anisotropy. This inevitably restricts the generality of this approach to near-equilibrium situations. A better way is to employ algebraic low-Re-number DSM which can be obtained by conventional truncation 66
Figure 21: Friction factor at step-wall in a flow behind a backward-facing step and in a sudden pipe expansion, computed with a high- and low-Re-number DSM. Note different expansion ratios and Reynolds numbers (Jakirli´c and Hanjali´c 1997) of the parent differential low-Re-number model. Such a model seems justified in the near-wall region where the convection and diffusion of the turbulent stress are smaller than the source terms. However, irrespective of whatever the level of modelling is used in the viscosity affected near-wall region, the need for a fine grid still remains if the viscous sublayer is to be resolved. Simple low-Re-number models may be computationally more robust, but demands on computation resources are still extremely high for 3-D flows. Practical flows will, in the foreseeable future, rely on the use of wall functions. Further improvement and generalization of wall functions is currently viewed by industrial CFD community as one of the most urgent tasks. To conclude, we present several examples of real complex flows relevant to automotive application. The first case shows the external flow around an automobile, obtained with the FIRE code, Figure 22, (Basara et al. 1996, 1997).The focus is on the region at the rear window, which notoriously poses the greatest challenge to turbulence models and CFD. The flow was computed using the standard high-Re-number k − ε and SSG Reynolds stress models with about 500.000 grid cells. The examples demonstrate at least two important outcomes of recent efforts to move towards more advanced turbulence closure in industrial computations: first, that the advanced DSM can be successfully applied to complex three-dimensional industrial flows, with affordable computer resources, and, second, that in both cases the DSM models indicate a decided improvement in accuracy, compared with the k − ε model. These illustrations by no means show that all problems have been resolved: many flow details are still unsatisfactory, even when computed with the most advanced models available. However, the mere success in obtaining the results for such flows - without facing much computational difficulty - gives a fresh impetus to efforts towards further improvement of the turbulence models, using sounder physical ground and a more rigorous mathematical derivation. Revisiting some of the conventional options is also encouraged. These new results open up new prospects in the use of advanced closure models in industrial CFD. The next examples comes from IC engine application, and were computed by KIWA code in with the Re-stress model has been incorporated (Yang et al., 1998, 2004). Figure 23 shows the comparison of k − ε and DSM (SSG model) computations of the mean temperature and mean velocity field in a DISC (direct-injected stratified charge) engine chamber at TDC, obtained with KIVA-3 computer code (Yang et al. 1998). Unlike the k − ε model, the DSM produced more realistic predictions, with high-temperature combustion gases confined near the centre of 67
Figure 22: Velocity vectors around the rear of a car body computed with k-ε+WF and DSM (SSG)+WF, and comparison of velocity profiles with measurements at position 4 (Basara et al. 1996) the bowl as a consequence of a squish-induced circulation region with high stress anisotropy. Figure 24 presents velocity vectors at CA of 60 and 120 deg in a four-valve Double-OverheadCamshaft (DOHC) IC engine, obtained by te standard k-ε and with the second-moment closure (DSM) using LRR pressure strain model. Both models are used with wall functions. Due to the lack of reference experimental or other data, no direct verification of the results is made. However, it is evident that the DSM produces much stronger swirl than the k-ε model, which in turn generate more complex vortical pattern (note two distinct vortices at 120 deg in DSM predictions), both in accord with experimental observation (for more details see Yang et al. 1998 and 2004). We close the illustration of the DSM potential with the example of flow and heat transfer in multiple jets, impinging normally on a flat heated surface (Thielen et al. 2004). The configuration is illustrated in Fig. 25 left, showing only one quadrangle. Complex flow pattern involving stagnation region, collision of wall jets on the target surface, upward fountains and recirculation in between, cause a strong stress anisotropy accompanied with strong variation of the wall temperature and heat transfer. Eddy-viscosity models - including the k-v 2 -f - perform badly, whereas the second-moment closure (here Elliptic Blending Model with integration up to the wall, see section 5.1.4) shows very good agreement with experiments both for the velocity and 68
Figure 23: Velocity and temperature field in a bowl of a direct injection stratified-charged (DISC) IC engine at TDC, obtained with k-ε+WF and DSM(SSG)+WF models (Yang et al. 1998)
Figure 24: Velocity vectors in a DOHC IC engine at CA=60 and 120 deg, computed by k−ε+WF and DSM(LRR)+WF models (Yang et al. 2004 temperature field. Figure 25 shows the distribution of heat transfer coefficient along the centrelines of the two jet rows, computed with several turbulence models and two treatments of heat flux: the isotropic (SGDH) and the anisotropic (GGDH) eddy diffusivity models (see section 4.4.2. The latter model gave much better results than SGDH, illustrating the importance of capturing accurately the stress anisotropy: unlike the SGDH, which does not take advantage of the computed full stress field (eddy diffusivity defined in terms of total kinetic energy, section 4.4.2) the still relatively simple GGDH model is sufficient for reproducing the heat flux, provided the stress field is correctly computed.
9
Concluding Remarks • Differential Second-Moment (Reynolds-stress) turbulence models (DSM) are the natural and logical level (within the Reynolds averaging framework) of closing the equations governing turbulent flows. In contrast to Eddy Viscosity Models (EVM), DSM have a sounder physical basis and treat some important turbulence interactions, primarily the stress generation, in its exact form. This allows better reproduction of the evolution of the turbulent stress field and its anisotropy, effects of streamline curvature, flow and system rotation and flow three-dimensionality. 69
x/D=0.0
150 100
Nu 50 Cθ=0.30
0 x/D=4.0
150
Cθ=0.25
100
Nu 50 0 0 k-ε
1 2
v -f
2 EBM-SGDH
3
y/D
4
EBM-GGDH
5 EBM-GGDH
6 Measurement
Figure 25: Nusselt number distribution across the jets centrelines for two locations(Thielen et al. 2004 • The potential of the DSM, although long recognized, has not yet been fully explored nor exploited, mainly due to persisting numerical difficulties, and uncertainties in modelling some terms, such as pressure-scrambling, which do not appear in the two-equation EVM. • Numerical problems, associated with the implementation of advanced turbulence models, and unavoidably increased demands on computing resources still discourage their wider application to the computation of complex flows. • However, over the past few years these problems have been considerably diminished. The DSM models have already been incorporated in some commercial CFD codes and used to solve some very complex flows. It is likely that they will be more widely used in the near future. • The integration up to the wall and a fine resolution of the viscous sublayer cannot be avoided if low-Re-number flows, transition phenomena and accurate wall friction and heat transfer are to be solved. This applies also to multi-component, multi-phase, laminar/turbulent and viscous/inviscid interfaces, as well as around singularities at separation and reattachment. • The need to resolve thin flow regions near walls and interfaces requires adequate model modifications (’low-Re-number’ models), highly non-uniform (and possibly local self-adapting) grid, and a flexible robust solver. For these reasons, integration up to the wall is still not a viable option for complex industrial flows at high-Reynolds numbers. • However, standard wall functions in conjunction with second-moment closure yield reasonable predictions of the flow pattern and pressure distribution, except in the near vicinity of the separation and reattachment, and can be used if wall/interface phenomena are not of primary importance. • Further developments, which will make DSM more appealing, are expected in the near future. In addition to model improvements, new numerical solvers are in the offing 70
specifically targeted at solving the weakly coupled set of transport equations - which will not be burdened by the Navier-Stokes tradition. New wall functions are also needed for complex nonequilibrium flows, which should reproduce more accurately the wall phenomena and yet bridge the viscous sublayer and dispense with the need for a fine grid resolution of the near-wall region. • Still, the present level of development and acquired know-how already permit and, indeed, call for a wider use of advanced models in industrial applications.
Acknowledgement. The author acknowledges the contributions of Dr. B. Basara from AVL LIST GmbH, Graz, Austria; Dr. S. Jakirli´c from Darmstadt University of Technology, Germany, Dr. I. Hadˇzi´c, from the Applied Physics Department of the Delft University of Technology, The Netherlands, and Dr. S.L. Yang from the Michigan Technological University, USA.
71
References Anwer, M., So, R.M.C. and Lai, Y.G. (1989) Perturbation by and recovery from bend curvature of a fully developed pipe flow, Physics of Fluids A 1 (8), 1387-1397. Bardina, J., Ferziger, J.H. and Rogallo, R.S. (1987) Effect of rotation on isotropic turbulence; computation and modelling, J. Fluid Mech., 154, 321-336. Basara, B. (1993) A numerical study into the effects of turbulent flows around full-scale buildings, Ph. D. Thesis, City University, London, UK. Basara, B., Hanjalic, K. and Jakirli´c, S., (1994) (unpublished work) Basara, B., Plimon, A., Bachler, G. and Brandst¨ atter, W. (1996) Calculation of flow around a car body with various turbulence models, Proc. First MIRA International Vehicle Aerodynamic Conference, Birmingham, U.K. Basara, B., Hanjali´c, K. and Jakirli´c, S. (1997) Numerical simulation of turbulent flow in a car compartment with a second-moment turbulence closure, Proc. ASME Fluid Engineering Division Summer Meeting, FEDSM97-3023. Bradshaw, P., Cebeci, T. and Whitelaw, J.H. (1981) Engineering Calculation Mehtods for Turbulent Flows, Academic Press, London-New York Chieng, C.C. and Launder, B.E. (1980) On the calculation of turbulent heat transport downstream from an abrupt pipe expansion, Numerical Heat Transfer 3, 189-207. Ciofalo, M. and Collins, M.W. (1989) The k − ε predictions of heat transfer in turbulent recirculating flows using an improved wall treatment, Numer. Heat Transfer, Pt. B, 3, 21-47. Craft, T.J. and Launder, B.E. (1991) Computation of impinging flows using second-moment closures, Proc. 8th Symp. Turb. Shear Flow, Munich, 8/1-8/5 Craft, T.J. and Launder, B.E. (1996) A Reynolds stress closure designated for complex geometries, Int. J. Heat and Fluid Flow, 17 (3), 245-254. Craft, T.J. Launder, B.E. and Suga K. (1995) A non-linear eddy viscosity model including sensitivity to stress anisotropy, Proc. 10th Symp. on Turbulent Shear Flows Pensylvani State Univ., 23/19-23/25. Craft, T.J. Launder, B.E. and Suga K. (1997) Prediction of turbulent transitional phenomena with a non-linear eddy-viscosity model, Int. J. Heat and Fluid Flow, 18 (1), 15-28. Demuren, A. O. and Rodi, W. (1984) Calculation of turbulence driven secondary motion in non-circular ducts, J. Fluid Mech. 103, 161-182. Dol, H., Hanjali´c K., Kenjereˇs, S. (1997) A comparative assessment of the second-moment differential and algebraic models in turbulent natural convection, Int. J. Heat and Fluid Flow, 18 (1), 4-14. Durbin, P.A. (1991) Near-wall turbulence closure modelling without ’damping functions”, Theoret. Comput. Fluid Dynamics 3, 1-13. Durbin, P.A. (1996) On the k-ε stagnation point anomaly, Int. J. Heat Fluid Flow 17, 89-90. Durbin, P.A. and Pettersson Reif, B.A. (2001) Statistical Theory and Modelling for Turbulent Flows, J. Wiley & Sons Ltd., Chichester, U.K. Fu, S. , Launder, B. E. and Tselepidakis, D. P. (1987) Accommodating the effects of high strain rates in modelling the pressure-strain correlation, Thermofluids report TFD/87/5, UMIST, Manchester. Gerolymos, G.A., Neubauer, J., Sharma, V.C., Vallet, I. (2002)Improved prediction of turbomachinery flows using near-wall Reynolds-stress model, J. Turbomachinery 124, 86-99 Gibson, M.M. and Launder, B. E. (1976) On the calculation of horizontal turbulent free 72
shear flows under gravitational influence, J. HeT Transfer., 98, 81-87. Gibson, M.M. and Launder, B. E. (1978) Ground effects on pressure fluctuations in the atmospheric boundary layer, J. Fluid Mech., 86, 491. Gibson, M.M. and Rodi, W. (1981) A Reynolds-stress closure model of turbulence applied to the calculation of a highly curved mixing layer, J. Fluid Mech. 140, 189-222. Hadˇzi´c, I. (1999), Second-moment closure modelling of transitional and unsteady turbulent flows, Ph.D. thesis, Delft University of Technology, 1999. Hadˇzi´c, I., Hanjali´c, K. and Laurence, D. (2001) Modeling the response of turbulence subjected to cyclic irrotational strain, Physics of Fluids 13/6 1739-1747. Hanjali´c, K. (1994) Advanced turbulence closure models: a view of current status and future prospects. Int. J. Heat and Fluid Flow, 15 (3), 178-203. Hanjali´c, K. (1994) Achievements and limitations in modelling and computation of buoyant turbulent flows and heat transfer. In G. Hewitt (editor): Proc. 10th Int. Heat Transfer Conference, IChemE/Taylor & Francis, Vol. 1, 1-18. Hanjali´c, K. (1999) Second-moment turbulence closures for CFD: Needs and prospects, Int.J. CFD 12 67-97. Hanjali´c, K. (1996) Some resolved and unresolved issues in modelling non-equilibrium and unsteady turbulent flows, Engineering Turbulence Modelling and Experiments 3, ed. W. Rodi and G. Bergeles, Elsevier, 3-18. Hanjali´c, K. and Hadˇzi´c, I. (1996), Modelling the transitional phenomena with statistical turbulence closure models, Transitional Boundary Layers in Aeronautics (Ed. R.A.W.M. Henkes and J.L. van Ingen), North-Holland, Amsterdam, 283-294. Hanjali´c, K. and Hadˇzi´c, I. (1999) Separation-induced transition to turbulence: secondmoment closure modelling, Flow, Turbulence and Combustion 63 157-173. Hanjali´c, K. and Jakirli´c, S. (1998) Contribution towards the second-moment closure modelling of separated turbulent flows, Computers & Fluids 27, 2, 137-156. Hanjali´c, K. and Jakirli´c, S. (2002) Second-Moment Turbulence Closure Modelling, In Closure Strategies for Turbulent and Transitional Flows (Ed. B.E. Launder and N. Sandham), Cambridge Univ. Press,, 47-101. Hanjali´c, K. Jakirli´c S. and Hadˇzi´c I. (1995), Computation of oscillating turbulent flows at transitional Reynolds numbers, Turbulent Shear Flows, 9, Springer, 323-342. Hanjali´c, K. Jakirli´c S. and Hadˇzi´c I.(1997) Expanding the limits of ’equilibrium’ secondmoment turbulence closures, Fluid Dynamics Research. 20, 25-41. Hanjali´c, K., Jakirli´c, S., Tropea, C. and Volkert, J. (1996) Zwischenbericht zum Forschungsvorhaben Tr 194/7-1, LSTM, Univ. of Erlangen Hanjali´c, K. and Launder, B.E. (1972) A Reynolds stress model of turbulence and its application to thin shear flows, J. Fluid Mech. 52,(4), 609-638. Hanjali´c, K. and Launder, B.E. (1976) Contribution towards a Reynolds-stress closure for low-Reynolds number turbulence, J. Fluid Mech. 74,(4), 593-610. Hanjali´c, K. and Launder, B.E. (1980) Sensitizing the dissipation equation to irrotational strains, J. Fluids Engineering 102, 34-40. Hanjali´c, K., Popovac, M. Hadˇziabdi´c, M. (2004) A robust near-wall elliptic relaxation eddyviscosity turbulence model for CFD, submitted for publication to Int. J. Heat Fluid Flow. Iacovides, H. and Launder, B.E. (1985) ASM predictions of turbulent momentum and heat transfer in coils and U-bends, Proc. 4th Int. Conference Numerical Methods Laminar and Turbulent Flow, Swansea, Wales. 73
Iacovides, H., Launder, B.E. and Li, H-Y. (1996) The computation of flow development through stationary and rotating U-ducts of strong curvature, Int. J. Heat and Fluid Flow, 17 (1), 22-33. Jakirli´c, S. (1997) Reynolds-Spannungs-Modellierung komplexer turbulenter Str¨ omungen, Doktor Arbeit, Univerit”at Erlangen-N¨ urnberg, Erlangen, Germany. Jakirli´c, S. and Hanjali´c, K. (1995) A second-moment closure for non-equilibrium and separating high- and low-Re-number flows, Proc. 10th Symp. Turbulent Shear Flows The Pennsylvania State Univ., U.S.A. Jakirli´c, S., Hanjali´c, K., Tropea, C. (2002) Modelling rotating and swirling turbulent flows: A perpetual challenge, AIAA Journal, 40(10), 1984-1996. Jakirli´c, S., Tropea, C. and Hanjali´c, K. (1998) Computations of rotating channel flows with a low-Re-number second-moment closure model, 7th ERCOFTAC/IAHR Workshop on Refined Flow Modelling UMIST, Manchester, 28-29 May, 1998. Jones, W.P. and Mussonge, P. (1988) Phys. Fluids, 31, 3589-3604. Johnson, R. W and Launder, B.E. (1982) Discussion of ”On the calculation of turbulent heat transport downstream from an abrupt pipe expansion”, Numerical Heat Transfer, Technical Note, 5 493-496. Kenjereˇs, S. (1999) Numerical Modelling of Complex Buoyancy-driven Flows, Ph.D. thesis, Delft University of Technology, Delft, The Netherlands. Kiel, R. and Vieth, D. (1995) Experimental and theoretical investigation of the near-wall region ub a turbulent separated and reattached flow, Experimental Thermal and Fluid Sciences 11, 243-256. Kim, S.E. and Choudhury, D. (1995) A near-wall treatment using wall functions sensitized to pressure gradient, In FED Vol. 217, Separated and Complex Flows, ASME 1995, 273-280. Lakshminarayana, B. (1986). Turbulence modelling for comlex shear flows, AIAA J. 24/12, 1900. Launder, B.E. (1990) Phenomenological modelling: present and future. —em Proc. Ehither Turbulence WorkshopJ. Lumley (ed.) Lecture Notes in Physics357, 439. Launder, B.E., Tselepidakis, D.P. and Younis, B.A. (1987) A second-moment closure study of rotating channel flow, J. Fluid Mech. 183 63-75. Launder, B.E., Reece, G. J. and Rodi, W. (1975) Progress in the development of Reynoldsstress turbulence closure, J. Fluid Mech. 68, 537-566. Laurence, D. (1997) Applications of Reynolds Averaged Navier-Stokes equations to engineering problems, Von Karman Institute for Fluid Dynamics, Lecture Series, 1997-03. Leschziner, M. A. (1989) Modelling turbulent recirculating flows by finite-volume methods current status and future directions, Int. J. Heat and Fluid Flow 10/3, 186-202. Lien, F. S. and Leschziner, M. A. (1993) Second-moment modelling of recirculating flow with a non-orthogonal collocated finite-volume algorithm, Turbulent Shear Flows, 8, 205–222. Manceau, R. and Hanjali´c, K. (2000) A new form of the elliptic relaxation equation to account for wall effects in RANS modeling, Physics of Fluids 12/9, 2345-2351. Manceau, R. and Hanjali´c, K. (2002) Elliptic blending model: a new near-wall Reynoldsstress turbulence closure, Physics of Fluids 14/2, 744-754. Menter, F. (1994) Two-equation eddy-viscosity turbulence model for engineering applications, AIAA Journal, 32 1598-1605. Menter, F. (1997) Eddy-viscosity transport equations and their relation to the k-ε model, ASME J. Fluids Engineering, 119 876-884. 74
Menter, F. (2002) Heat transfer predictions using advanced two-equation turbulence models, CFX-VAL10/0902, AEA Technology. Obi, S., Peri´c, M. and Scheurer, G. (1989) A finite-volume calculation procedure for turbulent flows with second-order closure and collocated variable arrangement, Proc. 7th Symp. on Turbulent Shear Flows, 17.4–17.10, Stanford University. Piquet, J. (1999) Turbulent flows: Models and Physics, Springer, Berlin. Pope, S. B. (2000) Turbulent flows, Cambridge Univ. Press, Cambridge, U.K. Rodi, W. (1976) A new algebraic relation for calculating the Reynolds stresses, ZAMM, 56, 219-221. Shimomura, Y. (1993) Turbulence modelling suggested by system rotation, in Near-Wall Turbulent Flows Eds. So et al., Elsevier Science Publishers B.V., 115-123. Speziale, C.G. (1991) Analytical methods for the development of Reynolds-stress closures in turbulence, Ann.Rev. Fluid Mech., 23, 107-157. Speziale, C.G. and Gatski, T.B. (1997) Analysis and modeling anisotropies in the dissipation rate of turbulence, J. Fluid Mech. 344, 155-180. Speziale, C.G., Sarkar S. and Gatski, T.B. (1991) Modelling the pressure-strain correlation of turbulence: an invariant system dynamic approach, J. Fluid Mech. 227, 245-272. Thielen, L., Hanjali´c, K., Jonker, M. and Manceau, R. (2004), Prediction of flow and heat transfer in multiple-impinging jets with an elliptic blending second-moment closure, CD Proc. CHT-04, Int. Symp. on Advances in Computational Heat Transfer, April 19-25, Norway, ICHMT/Begell House Inc. Umlauf, L. and Burchard, H. (2003) A generic length-scale equation for geophysical turbulence models, J. Mar. Res. 60, 699-723. Wenneberger, D. (1995) Entwicklung eines vorhersagefa¨ achigen Berechnungsmodells f¨ ur stark verdrallte Str¨ omungen mit Verbrenung, Doktor Arbeit, Univerit¨ at Erlangen-N¨ urnberg. Wilcox, D.C., (1988) Reassessment of the scale-determining equation for advanced turbulence models. AIAA Journal, 26(11), 1299-1310. Wilcox, D.C., (1993a) Comparison of two-equation turbulence models for boundary layers with pressure gradients. AIAA Journal 31(8), 1414-1421. Wilcox, D.C., (1993b) Turbulence Modelling for CFD. DCW Industries, Inc. La Canada, CA. Wizman, V., Laurence, D., Kanniche, M., Durbin, P. and Demuren, A. (1996) Modelling near-wall effects in second-moment closures by elliptic relaxation, Int. J. Heat and Fluid Flow, 17 (1), 255-266. Yang, S.L., Peschke, B.D. and Hanajli´c, K. (1998) IC-Engine like flow study using a secondmoment closure model, Proc. ASME 1998 Fall Technical Conf., Paper No 98-ICE-141, Vol. 32-2, 115-122. Yang, S.L., Siow, Y.K., Teo, C.Y., Hanajli´c, K.: A Kiwa code with Reynolds-stress model for engine simulation, Energy, (in press)
75
