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Review

Recent advances on the numerical modelling of turbulent flows

C.D. Argyropoulos a,⇑, N.C. Markatos b,c,⇑

a Department of Chemical Engineering, Imperial College London, South Kensington Campus, London SW7 2AZ, UK b Computational Fluid Dynamics Unit, School of Chemical Engineering, National Technical University of Athens, 9 Iroon Polytechniou Str., Zografou Campus,

15780 Athens, Greecec Metropolitan College, School of Engineering, 74 Sorou Str., Marousi, Athens 15125, Greece

a r t i c l e i n f o

Article history:

Received 15 February 2013

Received in revised form 9 June 2014

Accepted 7 July 2014

Available online 14 July 2014

Keywords:

Turbulence modelling

DNS

LES

URANS

DES

Reynolds stress models

a b s t r a c t

This paper reviews the problems and successes of computing turbulent flow. Most of the

flow phenomena that are important to modern technology involve turbulence. The review

is concerned with methods for turbulent flow computer predictions and their applications,

and describes several of them. These computational methods are aimed at simulating

either as much detail of the turbulent motion as possible by current computer power or,

more commonly, its overall effect on the mean-flow behaviour. The methods are still being

developed and some of the most recent concepts involved are discussed.

Some success has been achieved with two-equation models for relatively simple hydro-

dynamic phenomena; indeed, routine design work has been undertaken during the last

three decades in several applications of engineering practise, for which extensive studies

have optimised these models.

Failures are still common for many applications particularly those that involve strong

curvature, intermittency, strong buoyancy influences, low-Reynolds-number effects, rapidcompression or expansion, strong swirl, and kinetically-influenced chemical reaction. New

conceptual developments are needed in these areas, probably along the lines of actually

calculating the principal manifestation of turbulence, e.g. intermittency. A start has been

made in this direction in the form of ‘multi-fluid’ models, and full simulations.

The turbulence modelling approaches presented here are, Reynolds-Averaged

Navier–Stokes (RANS), two-fluid models, Very Large Eddy Simulation (VLES), Unsteady

Reynolds-Averaged Navier–Stokes (URANS), Detached Eddy Simulation (DES) and some

interesting, relatively recent, hybrid LES/RANS techniques.

A large number of relatively recent studies are considered, together with reference to the

numerical experiments existing on the subject.

The authors hope that they provide the interested reader with most of the appropriate

sources of turbulence modelling, exhibiting either as much detail as it is possible, by means

of bibliography, or illustrating some of the most recent developments on the numerical

modelling of turbulent flows. Thus, the potential user has the appropriate information,

for him to select the suitable turbulence model for his own case of interest.

2014 Elsevier Inc. All rights reserved.

http://dx.doi.org/10.1016/j.apm.2014.07.001

0307-904X/ 2014 Elsevier Inc. All rights reserved.

⇑ Current address: Metropolitan College, School of Engineering, 74 Sorou Str., Marousi, Athens 15125, Greece (N.C. Markatos). Tel./fax: +30 210 7723126.

E-mail addresses: [email protected] (C.D. Argyropoulos), [email protected] (N.C. Markatos).

URL: http://www.amc.edu.gr (N.C. Markatos).

Applied Mathematical Modelling 39 (2015) 693–732

Contents lists available at ScienceDirect

Applied Mathematical Modelling

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a te / a p m

http://dx.doi.org/10.1016/j.apm.2014.07.001mailto:[email protected]:[email protected]://www.amc.edu.gr/http://dx.doi.org/10.1016/j.apm.2014.07.001http://www.sciencedirect.com/science/journal/0307904Xhttp://www.elsevier.com/locate/apmhttp://www.elsevier.com/locate/apmhttp://www.sciencedirect.com/science/journal/0307904Xhttp://dx.doi.org/10.1016/j.apm.2014.07.001http://www.amc.edu.gr/mailto:[email protected]:[email protected]://dx.doi.org/10.1016/j.apm.2014.07.001http://-/?-http://crossmark.crossref.org/dialog/?doi=10.1016/j.apm.2014.07.001&domain=pdf


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Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694

2. Computer modelling of turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695

2.1. The differential equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695

2.2. Direct Numerical Simulation (DNS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 696

2.3. Reynolds-Averaged Navier Stokes (RANS) models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 697

2.3.1. Physical concepts of turbulence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 697

2.3.2. The equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6982.3.3. Zero-equation or algebraic models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 699

2.3.4. Half-equation models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 700

2.3.5. One-equation models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 700

2.3.6. Two-equation models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 701

2.3.6.1 The k–e model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7012.3.6.2 Modified k–e model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7012.3.6.3 The k–x model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7022.3.6.4 More recent two-equation models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 702

2.3.6.5 Low Reynolds number modifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 704

2.3.7. Non-Linear Eddy Viscosity Models (NLEVM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 705

2.3.8. Recent advances in eddy viscosity modelling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 706

2.4. Differential Second-Moment (DSM) and Algebraic Stress Models (ASM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 707

2.5. Two-fluid models of turbulence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 709

2.6. Large Eddy Simulation (LES). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7102.6.1. Validation of the LES approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 713

2.7. Monotone Integrated LES (MILES) and Implicit LES (ILES) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 714

2.8. Unsteady Reynolds-Averaged Navier–Stokes (URANS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 714

2.9. Very LES (VLES) and Detached-Eddy Simulation (DES). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 715

2.10. Hybrid RANS/LES strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 715

3. Applications of DNS and LES to flows in pipes and flows with a free surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 715

3.1. DNS of turbulent pipe flows. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 716

3.2. DNS of turbulent free-surface flows. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 719

3.3. LES of turbulent pipe flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 721

3.4. LES of turbulent free-surface flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 724

4. Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 725

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 726

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 726

1. Introduction

Turbulence is the most complicated kind of fluid motion, making even its precise definition difficult. Literature contains

many definitions as, for example, that included in Markatos [1]: ‘‘A fluid motion is described as turbulent if it is three-dimen-

sional, rotational, intermittent, highly disordered, diffusive and dissipative’’.

Turbulence is a three-dimensional, time-dependent, nonlinear phenomenon. Its modelling is very attractive, as it saves

huge amounts of money, by avoiding the need to build and test prototypes, and as it transforms technologies by allowing

improved understanding of turbulence. This is particularly true in industrial flows which, apart from the complexities of

turbulence, involve also very complicated geometries and several design parameters, requiring optimisation [2]. Thus,

shape design is one of the most important drivers for the use of simulation approaches in fluid-engineering industry.

Examples refer to the drag of an aircraft or ship, propulsive efficiency of aeroengines or propellers, turbomachinery, chem-

ical process engineering, among others. In comparison to experiments, Computational Modelling offers a competitive

advantage if it is able to guide the analyst to a better design.

Computer programs now exist which are capable of solving three-dimensional, time-dependent Navier–Stokes (NS) equa-

tions, within practical computer resources. The reason that we do not make direct computer simulations of turbulence is that

turbulence is dissipated, and momentum exchanged by small-scale fluctuations [3]. The crucial difference between visuali-

sations of laminar and turbulent flows is the appearance of eddying motions of a wide range of length scales in turbulent

flows [4,5].

A typical flow domain of 0.1 m by 0.1 m with a high Reynolds number turbulent flow might contain eddies down to 10–

100lm size. We would need computing meshes of 109 up to 1012 points to be able to describe processes at all length scales.The fastest events take place with a frequency of the order of 10 kHz, so we would need to discretise time into steps of about

100ls. We have estimated that the direct simulation of a turbulent channel flow at a Reynolds number of 800,000 requires acomputer which is half a million times faster than a current generation supercomputer. This estimate is analogous to the one

made by Speziale in 1991 [6], who stated that direct simulation of a turbulent pipe flow at a Reynolds number 500,000

required a computer 10 million times faster than the CRAY supercomputer of that time.

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With present day computing power it has only recently started to become possible to track the dynamics of eddies in

relatively simple flows at transitional Reynolds number. The computing requirements for the direct solution of the time-

dependent Navier–Stokes equations for fully turbulent practical flows at high Reynolds numbers are truly phenomenal

and must await major developments in computer hardware, possibly those based on quantum computing.

Meanwhile, engineers need computational procedures which can supply adequate information about the turbulent

processes, but which avoid the need to predict the effect of each and every eddy in the flow [7]. Therefore, in quantitative

work one is obliged to use turbulence models based on using averaged NS equations and, in addition, a set of equations

that supposedly express the relations between terms appearing in the NS equations [8–10]. It must be realised that most

of the available models pay no respect to the actual physical modes of turbulence (eddies, velocity patterns, high-vorticity

regions, large structures that stretch and engulf...) and, therefore, obscure the physical processes they purport to represent.

Flow visualisation experiments [11–16] confirm this point and demonstrate the difficulty of precise definition and

modelling. It is therefore hardly surprising that the actual physics of turbulence are nowhere to be seen in the available

models; simply because nobody can see as yet how mathematics can be employed to represent them in the models. It is,

however, also true that the engineering community has fortuitously often obtained very useful results by using relatively

simple models, such as those described in Section 2.3 below, results that would have required much more man-time and

experimental cost to obtain in their absence. Therefore, cautiously exercised and interpreted the turbulence models can be

valuable tools in research and design despite their physical deficiencies.

The purpose of the present effort is to provide a comprehensive review of the available turbulence modelling techniques.

The relevant material is certainly too much to be reviewed in a single paper. For this reason the authors confine attention to

what they consider the better established or more promising models. No disrespect is therefore implied for the models that

are scarcely – or not at all – mentioned. Extensive use has been made of the published literature on the topic and of earlier

reviews [17–20,11,21–40].

In addition, ERCOFTAC (European Research Community On Flow, Turbulence And Combustion) organises workshops and

special courses on best practise guidelines for CFD users. The Special Interest Group (SIG) 15 of ERCOFTAC is devoted to tur-

bulence modelling, and provides the appropriate data (e.g. experimental, DNS, highly-resolved LES databases) for the veri-

fication and validation of turbulence models, thus promoting their use for fundamental research and for industrial

applications [41].

Turbulent heat and mass transport are not explicitly covered in this review; the interested reader is directed to the review

by Launder [9]. Multi-phase phenomena are also not explicitly covered, apart from presenting the general differential equa-

tions and some necessary, to the authors’ mind, discussion on considerable work done for free-surface flows.

The review concludes with a summary of the advantages and disadvantages of the various turbulence models, in an

attempt to assist the potential user in choosing the most suitable model for his particular problem.

In the remainder of this review paper: Section 2 illustrates all of the available techniques for predicting turbulent flows;

in Sections 3, a literature survey is presented for applications using the LES and DNS technique; finally in Section 4, conclu-sions and some recommendations for future research are outlined.

2. Computer modelling of turbulence

2.1. The differential equations

The motion of a fluid in three dimensions is described by a system of partial differential equations that represent math-

ematical statements of the conservation laws of physics (mass, momentum, energy and concentration conservation). The

momentum conservation equations are called the Navier–Stokes equations. In what follows the ‘‘Eulerian’’ equations gov-

erning the dynamics and heat/mass transfer of a turbulent fluid are given, in Cartesian tensor notation, using the

repeated-suffix summation convention. The equations are presented in the most general form of multi-phase flows

[42–54], as the single-phase ones are easily derived by just setting the volume fraction, r i equal to unity.

A convenient assumption for deriving these equations is based on the concepts of time- and space-averaging; it is thatmore than one phase can exist at the same location at the same time [46,54]. Then, any small volume of the domain of inter-

est can be imagined as containing, at any particular time, a volume fraction r i of the ith phase. As a consequence, if there are n

phases in total,

Xni¼1

r i ¼ 1: ð1Þ

When flow properties are to be computed over finite time intervals, a suitable averaging over space and time must be carried

out. Following the above notion, that treats each phase as a continuum in the domain of interest, we can derive the following

balance equations:

Conservation of phase mass:

@

@ t ðqir i

Þ þdiv

ðqir i

~V iÞ ¼

_mi;

ð2

Þ

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where qi is the density, ~V i is the velocity vector, r i is the volume fraction of phase i; _mi is the mass per unit volume enteringthe phase i, from all sources per unit time, and div is the divergence operator (i.e. the limit of the outflow divided by the

volume as the volume tends to zero).

Summation of (2) over all phases leads to the ‘‘over-all’’ mass-conservation equation:

Xni¼1

@

@ t ðqir iÞ þ divðqir i~V iÞ

¼ 0; ð3Þ

which has of course a zero on the right-hand side.Conservation of phase momentum:

@

@ t ðqir iuikÞ þ divðqir i~V iuikÞ ¼ r ið~k grad p þ BikÞ þ F ik þ lik; ð4Þ

where: uik is the velocity component in the direction k of phase i; p is pressure, assumed to be shared between the phases;~k

is a unit vector in the k-direction; Bik is the k-direction body force per unit volume of phase i; F ik is the friction force exerted

on phase i by viscous action within that phase; and l ik is the momentum transfer to phase i from interactions with other

phases occupying the same space.

Conservation of phase energy:

@

@ t ðr iðqihi pÞÞ þ divðqir i~V ihiÞ ¼ r iQ i þ H i þ J i; ð5Þ

where: h i is stagnation enthalpy of phase i per unit mass (i.e. the thermodynamic enthalpy plus the kinetic energy of thephase plus any potential energy); Q i is the heat transfer to phase i per unit volume; H i is heat transfer within the same phase,

e.g. by thermal conduction and viscous action; and J i is the effect of interactions with other phases.

Conservation of species-in-phase mass:

@

@ t ðqir imilÞ þ divðqir i~V imilÞ ¼ divðr iCil grad milÞ þ r iRil þ _miM il; ð6Þ

where: mil is the mass fraction of chemical species l present in phase i; Ril is the rate of production of species l, by chemical

reaction, per unit volume of phase i present; Cil is the exchange coefficient of species l (diffusion); and M il is the l-fraction of

the mass crossing the phase boundary, i.e. it represents the effect of interactions with other phases.

All of the above equations can be expressed in a single form as follows:

@

@ t ðqir iuiÞ þ divðqir i~V iuiÞ ¼ divðr iCui grad uiÞ þ _miUi þ r isui total source of ui; ð7Þ

where: ui is any extensive fluid property; the first term on the right-hand side expresses the whole of that part of the sourceterm which can be so expressed, with Cui being the exchange coefficient for ui. sui is the source/sink term for ui, per unitphase volume; and _miUi represents the contribution to the total source of any interactions between the phases, such as any

phase change (withUi being the value of ui in the material crossing the phase boundary, during phase change). Distributionof effects between _miUi and sui is sometimes arbitrary, reflecting modelling convenience. For single-phase situations, the

above equations are valid by setting the r’s to unity.

For turbulent flow, averaging over times which are large compared with the fluctuation time leads to similar equations

for time-average values of ui with fluctuating-velocity effects usually represented by enlargement of Cui. More details on theabove concepts and equations may be found in [1].

2.2. Direct Numerical Simulation (DNS)

Solutions of turbulent flow problems (Eqs. (1)–(4)) can be obtained by using various analytical or numerical approaches,

with different level of accuracy in each case. Among the latter approaches, the Direct Numerical Simulation (DNS) has made a

significant contribution in turbulence research over the last decades [21], as it involves the numerical solution of the above

full three-dimensional, time-dependent Navier–Stokes equations without the need of any turbulence model. DNS is indeed

useful for the investigation of turbulence mechanisms, the improvement and development of turbulence models and for

assessing two-point closure theories.

Until the 1970’s the DNS approach was impossible to be used due to computer systems with insufficient memory and

speed to accommodate the required resolution needed for the small-scale turbulence effects. The first attempts for the inves-

tigation of homogeneous turbulence with DNS originated at the National Center for Atmospheric Research (NCAR) by Lilly

[55] and Orszag and Patterson [56] for 2-D and 3-D dimensional simulations, respectively. Rogallo [57] investigated the

effects of mean shear, rotational and irrotational strain on turbulence, based on the extension of Orszag and Patterson algo-

rithm. Kim et al. [58], Moser et al. [59], Abe et al. [60], Del Alamo et al. [61] and Hoyas and Jimenez [62], among others, per-

formed DNS for the investigation of wall turbulence for channel flows at1 Res = 180, 395, 640, 1900 and 2003.

1 Reynolds number based on the friction velocity us and the channel half width.


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There have also been extensive investigations of DNS in turbulent boundary layers at2 Reh = 700 [63], 1410 [64], 2500 [65],

2900 [66], 4060 [67] and 6650 [68,69]. More complex problems in wall-bounded turbulent flows (e.g. square duct, homoge-

neous isotropic turbulence, heat transfer, turbulence control and wavy boundary) have also been studied by DNS. The interested

reader may also consider the review papers by Moin and Mahesh [21], Kasagi and Shikazono [20] and Ishihara et al. [23]. The

most recent review concerning almost all the aspects of DNS (e.g. wall-bounded turbulence, turbulence control, bluff body tur-

bulence, turbulent flow structures and high performance computing) may be found in the work of Alfonsi [70].

The current largest scientific DNS was performed by Lee et al. [71] for turbulent channel flow at Res = 5200 and 3.5 times

more degrees of freedom than the DNS (40963 grid points) obtained by Kaneda et al. [72] and Kaneda and Ishihara [73] on

the Earth Simulator in Japan. The maximum Reynolds number obtained was approximately 1200 (Taylor microscale) which

is similar to the current capabilities obtained by laboratory experiments.

The absence of a turbulence model implies that the simulation is obtained by numerically solving over all the spatial and

temporal scales of turbulence, and its accuracy, therefore, is unrivalled by other methods. However, the DNS of high-Rey-

nolds number flows poses overwhelming demands on present-day available computing resources (speed and storage). It

is, therefore, necessary that DNS satisfies the following two constraints, according to Rogallo and Moin [17] and Kasagi

and Shikazono [20]:

(1) The dimensions of the computational domain must be large enough to comprise the largest turbulence scales.

(2) Grid resolution must be fine enough to capture the dissipation length scale, which is known as the Kolmogorov micro-

scale, g = (v3/e)1/4, where e is the average rate of dissipation of turbulence kinetic energy per unit mass, and v is thekinematic viscosity of the fluid.

As a result, the required number of grid points for a given DNS is dependent on the Kolmogorov micro-scale and Kolmogo-

rov micro-timescale (s = (v/e)1/2) of the flow. The higher the Reynolds number, the finer the mesh should be. Hence, thecell size in each direction of the computational domain should decrease with Re3/4 and the time step should decrease withRe1/2 [74,3]. It is worth mentioning that the DNS time step is always smaller than the Kolmogorov micro-timescale in orderto maintain the algorithm’s numerical stability [75].

The required resolution for DNS in the directions parallel to the wall, according to the work of Kim et al. [58], isD x+ = 8 and

D z + = 4, where D x+ is the streamwise and D z + is the spanwise grid spacing, respectively. In wall-normal directions, a rule of

thumb is to place at least three grid points below y+ = 1 (non-dimensional distance from the wall to the first grid point) and at

least 10 grid points for y+ < 10, while in the outer region such as the pipe/channel centre line a value of D y+ = 10 must beused.

Even with modern super-computers, the applicability of DNS is limited to flows of low to moderate Reynolds numbers.

Despite this current limitation, DNS is an effective and very useful tool for turbulence research leading to satisfactory results,

and used for testing simpler turbulence models, but it is still not practical for industrial or general engineering applications.

Among other benefits, DNS has contributed remarkably to testing conventional models and ideas and therefore to the devel-

opment of turbulence theory, in many ways which are summarized briefly below [20,74]. Furthermore, DNS data are impor-

tant for the development and improvement of turbulence models, due to the ability of DNS to provide the appropriate

turbulence statistics, including pressure and all spatial derivatives.

Important dimensionless numbers such as Reynolds and Prandtl can be varied in DNS, a fact of significant importance for

the derivation of a turbulence model with wide applicability. DNS is also suitable for studying a virtual flow which may occur

in reality now or in the near future. The latter advantage is important for the study of a dynamical turbulence phenomenon

[76] and for the evaluation of turbulence control methodologies [77].

Another important issue about DNS is the validation of the obtained results. According to Sandham [75] and Coleman and

Sandberg [78] the following are the criteria for such a validation: (a) validation of the obtained numerical data against

analytical solutions, experimental data and different numerical codes; (b) parametric studies with different grid resolutions,

domain sizes and time steps; (c) the time step (Dt ) should be comparable with the Kolmogorov time-scale and the grid

spacing, D xi, with Kolmogorov micro-scale, while the ratios of Dt /s and D xi/s should be of order unity; (d) evaluation of the statistical quantities budgets.

2.3. Reynolds-Averaged Navier Stokes (RANS) models

For the purpose of introducing the concepts of turbulent flow modelling we restrict attention to single-phase, incom-

pressible flow with constant laminar viscosity. The introduction of two-phase considerations, variable density and viscosity

are nowadays relatively easy tasks in modern solution algorithms. Only a generic presentation of turbulence modelling is

attempted here, for the sake of clarity. Details on the manner in which turbulence models properly couple into multi-phase

flow solvers may be found in literature (for example [79,80]).

2.3.1. Physical concepts of turbulence

Before discussing the turbulence models a very brief description of some concepts is provided. The main characteristic of

turbulence is the transfer of energy to smaller spatial scales across a continuous wave-number spectrum, i.e. a 3D, nonlinear

2 Reynolds number based on momentum thickness h and free-stream velocity.


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process. A useful concept for discussing the main mechanisms of turbulence is that of an ‘eddy’ [81,82,3]. An eddy can be

thought of as a typical turbulence pattern, covering a range of wave- lengths, large and small eddies co-existing in the same

volume of fluid. The actual modes of turbulence are eddies and high-vorticity regions [4,3]. By analogy with molecular vis-

cosity, which is a property of the fluid, turbulence is often described by eddy viscosity as a local property of the fluid; the

corresponding mixing length in eddy viscosity models is treated in an analogous manner to the molecular mean-free path

derived from the kinetic theory of gases. This description is based on erroneous physical concepts but has proved useful in

the quantitative prediction of simple turbulent flows [7].

The eddies can be considered as a tangle of vortex elements (or lines) that are stretched in a preferred direction by mean

flow and in a random direction by one another. This mechanism, the so-called ‘vortex stretching’, ultimately leads to the

breaking down of large eddies into smaller ones. This process takes the form of an ‘energy cascade’. Since eddies of compa-

rable size can only exchange energy with one another [9], the kinetic energy from the mean motion is extracted from the

largest eddies [3]. This energy is then transferred to neighbouring eddies of smaller scales continuing to smaller and smaller

scales (larger and larger velocity gradients), the smallest scale being reached when the eddies lose energy by the direct action

of viscous stresses which finally convert it into internal thermal energy on the smallest-sized eddies [82]. It is important to

note that viscosity does not play any role in the stretching process nor does it determine the amount of dissipated energy; it

only determines the smallest scale at which dissipation takes place. It is the large eddies (comparable with the linear dimen-

sions of the flow domain), characterising the large-scale motion, that determine the rate at which the mean-flow kinetic

energy is fed into turbulent motion, and can be passed on to smaller scales and be finally dissipated. The larger eddies

are thus mainly responsible for the transport of momentum and heat, and hence need to be properly simulated in a turbu-

lence model. Because of direct interaction with the mean flow, the large-scale motion depends strongly on the boundary

conditions of the problem under consideration.

An increase in Reynolds number increases the width of the spectrum, i.e. the difference between the largest eddies (asso-

ciated with low-frequency fluctuations) and the smallest eddies (associated with high-frequency fluctuations). This suggests

that at high Reynolds numbers the turbulent motion can be well approximated by a three-level procedure, namely, a mean

motion, a large-scale motion and a small-scale motion [83].

Viscosity does not usually affect the larger-scale eddies which are primarily responsible for turbulent mixing, with the

exception of the ‘viscous sublayer’ very close to a solid surface. Furthermore, the effects of density fluctuations on turbulence

are small if, as in the majority of practical situations, the density fluctuations are small compared to the mean density, the

exception being the effect of temporal fluctuations and spatial gradients of density in a gravitational field. Therefore, one can

usually neglect the direct effect of viscosity and compressibility on turbulence. It is also important to note that it is the fluc-

tuating velocity field that drives the fluctuating scalar field, the effect of the latter on the former usually being negligible.

2.3.2. The equations

Eqs. (8)–(11) below constitute the mathematical representation of fluid flows, under the assumptions that the turbulentfluid is a continuum, Newtonian in nature and that the flow can be described by the Navier–Stokes equations. For turbulent

flows, the latter represent the instantaneous values of the flow properties [1,84,85].

The equations for turbulence fluctuations are obtained by Reynolds de-composition which describes the turbulent motion

as a random variation about a mean value [1]:

/ ¼ /þ /0; ð8Þwhere / is the instantaneous scalar quantity, / its time- mean value and /

0the fluctuating part. The time-average of the fluc-

tuating value is zero /0 ¼ 0, and the mean value / is defined as:

/ð xÞ ¼ limt !1 1Dt

Z t 1þDt t 1

/ð x; t Þdt t 1 Dt t 2; ð9Þ

where t 1 is the time scale of the rapid fluctuations and t 2 the time scale of the slow motion (for time-dependent mean value,

i.e. for non-stationary turbulence). By substituting Eq. (8) into the form of Eqs. (1)–(3) for single-phase, incompressible flowsand then taking the time-mean of the resulting equations, one derives the following continuity and NS equations:

@ ui@ x ¼ 0; ð10Þ

@ ui@ t þ @

@ x jðuiu jÞ ¼ 1q

@ p

@ xiþ m @

2ui@ x j@ x j

@ @ x j

ðu0iu0 jÞ; ð11Þ

where ui is the mean velocity, u0i the fluctuating velocity, q the fluid density and v the kinematic viscosity. Eq. (11) is known

as the Reynolds-Averaged Navier–Stokes (RANS) equation, while the term u0iu0

j is the Reynolds-stress tensor:

sij ¼ u0iu0 j; ð12Þ

which is a symmetric tensor with six independent components. It is observed from Eqs. (10) and (11) that the number of

unknown quantities (pressure, three velocity components and six stresses) is larger than the number of the available


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equations (continuity and Navier–Stokes). As a result, the system of equations is not yet closed and the problem of ‘‘closure’’

is reduced to the modelling of the Reynolds-stresses, in terms of mean-flow quantities.

The most popular approach to resolving this problem of ‘‘closure’’ is the use of the Boussinesq eddy-viscosity approxima-

tion [86]. The latter is based on an analogy between molecular and turbulent motions, in order to correlate Reynolds stresses

to the rate of strain of the mean motion. The turbulence eddies are thought of as parcels of fluid, which like molecules, collide

and exchange momentum, obeying the kinetic theory of gases. Thus, in analogy with the molecular viscous stress, the Rey-

nolds (turbulence) stresses are modelled as follows [1]:

sij ¼ u0iu0 j ¼ 23jdij mt @ ui@ x j

þ @ u j@ xi

; ð13Þ

k ¼ 12

u0iu0

i ¼ 1

2 u021 þ u022 þ u023

; ð14Þwhere k is the turbulence kinetic energy and mt ð¼ lt =qÞ is the turbulence or eddy (kinematic) viscosity which, in contrast tothe molecular (kinematic) viscosity is not constant; and may vary significantly from flow to flow and from point to point [1];

and dij is the Kronecker delta. Substituting Eq. (13) into Eq. (11) leads to:

@ ui@ t þ @


@ p

@ xiþ @

@ x jðmþ mt Þ @

ui@ x j

: ð15Þ

The isotropic part of the Reynolds-stress tensor is absorbed normally into the pressure term as p ¼ p þ 2k3

.

Dimensional analysis dictates that the unknown vt must be proportional to the product of a characteristic velocity V t and

a characteristic length scale Lt . The difference between zero-equation, one-equation and two-equation models, discussedbelow, lies in the way they choose to calculate them [1]. Thus, zero-equation models prescribe both characteristic velocity

and length-scale as algebraic expressions. One-equation models consider as characteristic velocity the square root of the tur-

bulence kinetic energy and prescribe algebraically the length scale, therefore:

vt ¼ C v 1 ffiffiffi

kp

L; ð16Þwhere C v 1 is a dimensionless constant. Two-equation models, such as k–e and k–x [1], described below in this subsection,use differential equations to compute both the characteristic velocity and length scale and then estimate the value of vt by

the following equations:

vt ¼ C l f l

k2

e ðk—e modelsÞa kx ðk—x modelsÞ

( ; ð17Þ

where f l is a damping function, C l and a are constants, e is the turbulence energy dissipation rate and x the dissipation perunit turbulence kinetic energy.

Recent developments have led to the construction of non-linear eddy viscosity models, aiming at including non-linear

terms of the strain-rate [40]. More details for these models are presented in Section 2.3.7.

The traditional linear-eddy-viscosity RANS models may be divided into the following four main categories [1,87]: (a) alge-

braic (zero-equation) models, (b) half equation models (c) one-equation models and (d) two-equation models.

In the remainder of this section, a number of the better-established and most promising, according to the present authors’

experience, linear and non-linear eddy viscosity models, along with some more recently advanced ones, will be presented and

discussed.

2.3.3. Zero-equation or algebraic models

Zero-equation or algebraic models use partial differential equations only for computing the mean fields, while only alge-

braic expressions for the turbulence quantities [1]. This class of models is the oldest one, it is characterised by simplicity to

implement and has given good results for some applications of engineering relevance. For example, the best known of this

class, Prandtl’s mixing length model [88], is suitable for the prediction of thin-shear-layer flows such as boundary layers, jets,

mixing layers, and wakes. According to Prandtl, [88], in a boundary layer flow the eddy viscosity is given by:

vt ¼ ‘2mix@ u

@ y

; ð18Þ

where ‘mix is the mixing length, that depends upon the type of flow, and is specified algebraically, while y is the direction

normal to the wall. This model is not suitable for predicting flows with recirculation and separation.

More modern variants of this category, following the contribution of Van Driest, Clauser and Klebanoff modifications [86],

are the Cebeci–Smith [89] and Baldwin–Lomax models [90]. These models are characterised by two-layer mixing-length

eddy viscosities, one as an inner and one as an outer layer viscosity. The second model is distinguished from the first because

of the different outer-layer length viscosity equation. Both are suitable for predicting turbulent flows in aerodynamics (e.g.

around airfoils) with similar accuracy, but are unreliable for separated flows. The mathematical formulations of these models

may be found in the textbook by Wilcox [86]. Nowadays, zero-equation models are used rarely and only for getting an initial

prediction of the flow field [91].




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2.3.4. Half-equation models

In 1985 Johnson and King [92] developed a two layer model for the investigation of pressure driven separated flows. The

J–K model has been improved by Johnson [93] and Johnson and Coakley [94] in order to become applicable to compressible

flows as well. In addition, the J–K model has been extended for 3-D dimensional flows by Savill et al. [95]. The mathematical

equations along with comparisons against computational and experimental data may be found in Wilcox [86]. Even though

the J–K model has improved the classical algebraic models for predicting turbulent, transonic separate flows, it still suffers

from the same drawbacks as the Cebeci–Smith and Baldwin–Lomax models.

2.3.5. One-equation models

One-equation models are characterised by formulating one additional transport equation for the computation of a turbu-

lence quantity, usually the turbulence kinetic energy (k). For all of them there is still a need of prescribing a length-scale

distribution (L), which is defined algebraically and is usually based on available experimental data. For elliptic flows, like

recirculating and separated ones, experimental data is generally not available, making it difficult to prescribe algebraically

such a length scale. Therefore, most researchers decided to adopt two- or even more-equation models [1]. One-equation

models were used mainly in the nuclear and aeronautics industries (e.g. aircraft wings, fuselage, nuclear reactors) and the

most well known for aerospace applications are Baldwin and Barth [96] and Spalart and Allmaras [97] models. The Spal-

art–Allmaras was designed and optimised for flows past wings and airfoils and produced very good results. It is also easy

to implement for any type of grid (e.g. structured or unstructured, single-block or multi-block) [40]. However, both models

create enormous diffusion, in particular for regions of 3-D vortical flow [40]. Improvements of the aforementioned models

are presented in the works of Spalart and Shur [98], Dacles-Mariani et al. [99] and Rahman et al. [100], regarding the effects

of curvature, rotation, decrease of diffusion and for near-wall effects. Recent studies with the Spalart–Allmaras model havebeen presented by Karabelas and Markatos [101] and Karabelas [102] for flow over an airfoil and past a flapping multi-ele-

ment airfoil, respectively. Karabelas [102] performed simulations past a plunging multi-element airfoil at Re = 6 105(Fig. 1).

Accurate resolution of such flows still constitutes a great mathematical challenge for RANS modelling. It is well known

that the latter is often inaccurate even in terms of integrated quantities, such as lift and drag coefficients. This is due to

Fig. 1. Turbulence simulations of the flow past a plunging multi-element airfoil at Re = 6 105: Path-lines and pressure distribution at three fixed

geometric angles of attack, soaring flight regime (left), mid-time of the up-stroke (middle) and mid-time of the down-stroke phase (right), reproduced withthe author’s permission [102]. Reprinted by permission of the American Institute of Aeronautics and Astronautics, Inc.


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the fact that in flow regimes past multi-element airfoils, multiple transition processes from laminar to turbulent states and

vice versa could occur. In this study, we mention the one-equation model of Spalart Allmaras because its performance was

found to be superior [103] even over other more complete two-equation models. Further tuning of the latter models to

include transition effects did not increase considerably the accuracy of the simulations. It is worth mentioning, that in

the workshop held by NASA [104], for validating turbulence modelling of the flow past the multi-element airfoil MDA

30P-30N, mainly one-equation models were used in the governing equations.

Recently, Fares and Schroder [105] developed a complete and general one-equation model based on the two-equation k–

x model for predicting general turbulent flows such as wakes, jets, boundary layers and vortex flows. The new model provedmore accurate compared to the Spalart–Allmaras model, especially for jets and vortex flows. More information for zero-,

half- and one-equation models is provided in detail in the review papers by Markatos [1], Alfonsi [106] and in the classic

text by Wilcox [86].

2.3.6. Two-equation models

Two-equation models use, in addition to the mean-flow Navier–Stokes equations, two transport equations for two turbu-

lence properties. The first one is usually that for the turbulence kinetic energy (k) and the second any other from a variety

that includes: the dissipation rate of turbulence kinetic energy (e), the specific dissipation rate (x), the length scale (l), theproduct of k l, the time scale s, the product of k ands, among others [1]. This class of models is the most preferred by indus-try it looks like remaining so for the foreseeable future [85]. Two-equation eddy viscosity models are still the first choice for

general CFD calculations, with the standard k–e model [107] and k–x [108] being the most widely used. There is no partic-ular reason for this preference, but at least those models have been applied so widely, that we know their behaviour

beforehand.

In this section only the k–e and k–x models are presented, as being representative of the two-equation models, alongwith their improvements and some interesting low-Re versions.

2.3.6.1. The k–e modelThe k–e model is by far the most widely used and tested two-equation model, with many improvements incorporated

over the years. The standard k–e model of Launder and Sharma [107] is specified as follows:Kinematic eddy viscosity (vt) equation:

mt ¼ C l k2

e : ð19Þ

Turbulence kinetic energy (k) equation:

@ k

@ t þu j

@ k

@ x j ¼

@

@ x j

ðmþ vtÞ

rk

@ k

@ x j eþ sij@ ui

@ x j

:

ð20

ÞTurbulence dissipation rate (e) equation:

@ e@ t þ u j @ e

@ x j¼ @

@ x j

ðm þ vtÞre

@ e@ x j

þ C e1 e

ksij

@ ui@ x j

C e2 e2

k ; ð21Þ

where rk = 1.0 and re = 1.3 are the Prandtl numbers for k and e, respectively. The remaining model constants are: C l = 0.09,C e1 = 1.44, C e2 = 1.92. The standard k-e model behaves very in predicting turbulent shear flows, in many applications of engi-neering interest. However, this model is unable to predict accurately flows with adverse pressure gradients and extra strains

(e.g. streamline curvature, skewing, rotation [91]). As a result it yields poor results for separated flows, whilst it is rather

difficult to be integrated through the viscous sublayer [86]. Despite the above shortcomings, the k–e model is recommendedfor an at least gross estimation of the flow field and for cases such as combustion, multiphase flows and flows with chemical

reactions [91].

2.3.6.2. Modified k–e modelImprovements and modifications of the standard k–e model are many (for example, for flows with strong buoyancy, [1])

with probably the most important being the realisable k–e model [109] and the Renormalization Group (RNG) k–e model[110].

The first model is based on the satisfaction of the realizability constraints on the normal Reynolds stresses and the

Schwartz inequality for turbulent shear stresses. Beside this, the C l constant of standard k–e model is not anymore aconstant but it is computed in this improved model by an eddy-viscosity equation. Performance is substantially improved

for jets and mixing layers, channels, boundary layers and separated flows compared to the standard k-e model [109]. Theconstants of the realisable k-e model are: C e1 = 1.44, C e2 = 1.9, rk = 1.0 and re = 1.2.

The RNG k–e model [110] is a modification of the classical k–e model with better predictions of the recirculation length inseparating flows. The model is represented by the same equations (19)–(21) of standard k–e model but with a modified coef-ficient, C e2, which is computed by the following equation:

C e2 C e2 þC lg3

ð1

g=g0

Þ1 þ b1g3 ; ð22Þ




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g ¼ Ske ; S ¼

ffiffiffiffiffiffiffiffiffiffiffiffi2S ijS ij

q ; S ¼ 1

2

@ ui@ x j

þ @ u j@ xi

; ð23Þ

where S denotes the mean strain-rate of the flow and S ij the deformation tensor. The model constants are: C l = 0.085,

C e1 = 1.42, C e2 = 1.68, rk = re = 0.72, b = 0.012 and g0 = 4.38. This model gives better results than the standard k–e modelfor separating flows, but fails to predict flows with acceleration [91].

Implementation of standard k–e model and RNG k–e model for pollutant dispersion from large tank-fires [111–114] andstreet canyon flows [115], respectively, have been recently undertaken by the authors. In addition, high Reynolds number

turbulent flow past a rotating cylinder has also been examined by the authors and their colleagues. In Fig. 2, supercriticalstreamline patterns are illustrated compared with laminar ones at Re = 200 and for some common examined cases (same

rotational rate) for flow past a rotating cylinder. More details for the study can be found in the work of Karabelas et al. [116].

2.3.6.3. The k–x modelAnother ‘successful’ model and also widely used is the k–x model. The initial form of the model was proposed by

Kolmogorov in 1942 [117]. An improved version of the model was developed by the Imperial College group under Prof. B.

Spalding [118]. Further development and application of k–x model was performed by many scientists and engineers, butthe most important development was by Wilcox [108]. In the present paper, the most recent version of the model (Wilcox

(2006) k–x model) is presented below [86,108]:Kinematic eddy viscosity (vt) equation:

mt ¼

k

~x; ~x

¼max x;C

lim ffiffiffiffiffiffiffiffiffiffiffiffi2S ijS ijbs ( ); C lim ¼7

8:

ð24Þ

Turbulence kinetic energy (k) equation:

@ k

@ t þ u j @ k

@ x j¼ @

@ x jv þ r k

x

@ k

@ x j

bkxþ sij @

ui@ x j

: ð25Þ

Specific dissipation rate (x) equation:

@ x@ t þ u j @ x

@ x j¼ @

@ x jv þ r k

x

@ x@ x j

bx2 þ rd

x@ k

@ x j

@ x@ x j

þ axk sij

@ ui@ x j

: ð26Þ

The auxiliary relations and closure coefficients of the model are specified as follows:

a ¼ 0:52; b ¼ b0 f b; b0 ¼ 0:0708; b ¼ 0:09; r ¼ 0:5; r ¼ 0:6; rd0 ¼ 0:125; ð27Þ

rd ¼0; @ k

@ x j

@ x@ x j6 0

rd0; @ k@ x j@ x@ x j

> 0

8><>: ; f b ¼ 1 þ 85vx1 þ 100vx ; vx XijX jk

S ki

ðbxÞ3

; Xij ¼ 12 @ ui@ x j @ u j@ xi

; ð28Þ

where C lim is the stress-limiter strength, f b the vortex-stretching function, vx the dimensionless vortex-stretching parameterandXij the mean-rotation tensor. The k–x model is superior to the standard k–e model for several reasons. For instance, itachieves higher accuracy for boundary layers with adverse pressure gradient and can be easily integrated into the viscous

sub-layer without any additional damping functions [86]. In addition, the recent version of Wilcox (2006) k–x model ismuch more accurate for free shear flows and separated flows. The model still suffers from weaknesses when applied to flows

with free-stream boundaries (e.g. jets), according to the review paper by Menter [119].

2.3.6.4. More recent two-equation models

A more advanced turbulence model is the Shear Stress Transport (SST) model by Menter [120]. This model combines theadvantages of k–e and k–x models in predicting aerodynamic flows, and in particular in predicting boundary layers understrong adverse pressure gradients. The model has been validated against many other applications with good results such as

turbomachinery blades, wind turbines, free shear layers, zero pressure gradient and adverse pressure gradient boundary

layers. Recent improvements of the model are an enhanced version for rotation and streamline curvature [121] and the

replacement of the vorticity in the eddy viscosity with the strain rate [119]. The mathematical formulation of the model

is not repeated due to space limitations, but it may be found in the above mentioned references.

Another class of two-equation models is the two-time scale models, with significantly improved results compared to the

k–e model. Hanjalic et al. [122] proposed a multi-scale model in which separate transport equations are solved for theturbulence energy transfer rate across the spectrum. The mathematical formulation of the proposed turbulence model is

as follows [123].

vt

¼ C l

kkP

eP ; ð29Þ


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Fig. 2. Streamline patterns at Re = 200, 5 105

, 106

, 5 106

and rotational rates a = 2, 3, 4, 5, 6, 7 and 8. L1 and L2 stagnation points are apparent for lowrotational rates (laminar flow), while A, B, C and Z are addressed for the super-critical Reynolds numbers [116].



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DkP Dt ¼ @

@ x jðmþ vtÞ @ kP

@ x j

þ P k eP ; ð30Þ

DkT Dt ¼ @

@ x jðv þ vtÞ @ kT

@ x j

þ eP eT ; ð31Þ

DeP

Dt ¼ @

@ x j ðv

þvtÞ@ eP

@ x j þ C P 1eP

kP P k

C P 2e2P

kP þC 0P 1kP

@ ul

@ xm

@ ui

@ x jelmkeijk;

ð32

ÞDeT Dt ¼ @

@ x jðv þ vtÞ @ eT

@ x j

þ C T 1 eP eT

kT C T 2 e

T 2

kT : ð33Þ

The form of Eq. (29) has been obtained from simplifying the mean-Reynolds-stress (MRS) equation by considering normal

Reynolds stresses proportional to k and by taking the time scale for pressure strain to be kP eP . The above mentioned Eqs.

(29)–(33) use the following set of coefficients and functions:

C l ¼ 0:09; C P 1 ¼ 2:2; C 0P 1 ¼ 0:11; C P 2 ¼ 1:8 0:3kP kT 1

kP kT þ 1 ; C T 1 ¼ 1:08

eP eT

; C T 2 ¼ 1:15; ð34Þ

where P k ¼ uiu j @ ui@ x j . Here k p and kI are, respectively, the turbulence kinetic energy in the production and dissipation ranges,P

k is the rate at which turbulence energy is produced (or extracted) from the mean motion, e

p is the rate at which energy is

transferred out of the production range, eT is the rate at which energy is transferred into the dissipation range from the iner-tia range and e is the rate at which turbulence energy is dissipated (i.e. converted into internal energy).

It is worth mentioning that the proposed version of k–e model performs better than the standard (single-scale) k–e modeldue to the fact that e p (rate at which energy is transferred out of the production range) replaces P k (turbulence production) inthe dissipation rate (e) equation, simply because in flows where P k is suddenly switched off, e is not expected to decreaseimmediately. The present model gives better predictions than the single-scale k–e model in plane and round jets [1].

The main advantage of the two-scale k–e model is the combination of modelling the cascade process of turbulence kineticenergy and of solving complex flows such as separating and reattaching flows. Improvements of the model may be achieved

by accounting for the proper empirical coefficients which affect the spectrum shape. Applications of the model for breaking

waves [124], plane synthetic and swirling jet [125], wake-boundary-layer interaction and compressible flow [126], may be

found in the literature. Finally, new models for non-equilibrium flows have also been developed by Klein et al. [127] with

satisfactory results.

2.3.6.5. Low Reynolds number modifications

Most of the above models are applicable for turbulent flows at high- Re numbers, but are inaccurate for the prediction of

the flow in the vicinity of the wall, where viscous forces dominate. In order to treat this shortcoming, many scientists and

engineers have proposed a number of near-wall modifications. These models with near-wall modifications are referred to as

‘‘Low-Reynolds Number’’ (LRN) models. A full list of these models is presented in the text by Wilcox [86] and in the review

paper by Patel et al. [128]. In the present paper two popular LRN models will be presented, the Lam–Bremhorst k–e model[129] and Bredberg et al. k–x model [130]. The mathematical formulation of the first model can be written in the followingboundary layer form:

u@ k

@ xþ t @ k

@ y¼ @

@ y v þ vt

rk

@ k

@ y

þ vt @ u

@ y

2 e; ð35Þ

u@ ~e@ x þ t

@ ~e@ y ¼

@

@ y v þ vtrk

@ ~e@ y

þ C e1 f 1 ~ek vt @ u@ y 2

C e2 f 2~e2

k þ E ; ð36Þ

where the turbulence dissipation (e) is given by the following equation:

e ¼ e0 þ ~e ð37Þand e0 is the value of e at y = 0. The kinematic eddy viscosity is defined as:

vt ¼ C l f lk

2

~e : ð38Þ

The damping functions ð f 1; f 2; f l; e0 and E Þ and closure coefficients for the Lam–Bremhorst k–e model are presented below:

f l ¼ ð1 expð0:0165R yÞÞ2 1 þ20:5

Ret

; f 1 ¼ 1 þ

0:05

f l

!3; f 2 ¼ 1 expðRe2t Þ; Ret ¼

k2

~ev; R y ¼ k

0:5 y

v ;

e0 ¼ 0; E ¼ 0; C e1 ¼ 1:44; C e2 ¼ 1:92; C l ¼ 0:09; rk ¼ 1:0; re ¼ 1:3: ð39Þ


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The development of LRN k–e models improved the original k–e model by making it more compatible with the law of thewall. However, LRN modifications did not improve the problem with strong adverse pressure gradient. More details for the

difficulty of LRN k–e models to predict turbulent flows with pressure gradients may be found in Wilcox [86]. Finally, Patelet al. [128] claim that any improvement in predicting flows with adverse pressure gradient would require modifications to

the original k–e model itself.Another class of LRN models is the k–x models and one of the most popular is the standard k–x model by Wilcox [108].

Extensions and improvements of the model have been proposed by Wilcox [131], Peng et al. [132], Bredberg et al. [130],

among others.

The mathematical equations of Bredberg et al. k–x model are as follows:

@ k

@ t þ @

@ x jðu jkÞ ¼ @

@ x jv þ v t

rk

@ k

@ x j

þ P k C kkx; ð40Þ

@ x@ t þ @

@ x jðu jxÞ ¼ @

@ x jv þ vt

rx

@ x@ x j

þ C x v

kþ vt

k

@ k@ x j

@ x@ x j

þ C x1 xk

P k C x2x2: ð41Þ

The turbulence kinematic viscosity is defined as:

vt ¼ C l f lk

x; ð42Þ

where the damp function f l is given by the following equation:

f l ¼ 0:09 þ 0:91 þ 1

Re3t

! 1 exp Ret

25

2:75( )" #: ð43Þ

Finally, the constants of the model are denoted as:

C l ¼ 1; C k ¼ 0:09; C x ¼ 1:1; C x1 ¼ 0:49; C x2 ¼ 0:071; rk ¼ 1; rx ¼ 1:8: ð44ÞThe model of Bredberg et al. [130] presents improved results against the original Wilcox k–x model, compared to DNS

and experimental data for three different cases (channel flow, backward facing step flow and rib-roughened channel flow).

Recently, an extension of the model to viscoelastic fluids was proposed by Resende et al. [133].

2.3.7. Non-Linear Eddy Viscosity Models (NLEVM)

As mentioned earlier in Section 2.3.2, the Non-Linear Eddy Viscosity Models (NLEVM), may be defined as non-linear

extensions of the eddy-viscosity models in which Eqs. (13) and (15) can be rewritten in a more general form, in order to

include non-linear terms of the strain-rate [40]:

sij ¼ u0iu0 j ¼2

3jdij þ

XN nþ1

anT ðnÞij ; ð45Þ

@ ui@ t þ @


@ p

@ xiþ @

@ x jðmþ mt Þ @

ui@ x j

þ N :S :T :; ð46Þ

where N.S.T are non-linear source terms deriving from Eq. (45).

This class of models has been developed to overcoming the deficiencies of eddy-viscosity models, in particular for two-

equation models. There is a large number of NLEVM in the literature and they may be categorised as quadratic and cubic

models. Popular quadratic models have been proposed by Gatski and Speziale [134] and Shin et al. [135], among others.

The first model is a high-Re k–e model which supports separation in adverse pressure gradient flows. The model of Shin

et al. [135] has shown improved results for backward facing step compared to classical linear eddy viscosity models, butit also suffers with rotational effects especially for channel flow [136].

The mathematical formulation of the cubic model by Craft et al. [137] is selected for presentation here, as a general form

of the category. The anisotropic tensor and turbulence kinetic energy are defined as:

aij ¼ uiu jk 2

3dij and k ¼ 1

2ukuk: ð47Þ

The mathematical formulation of the cubic model is as follows [137]:

aij ¼ vtk

S ij þ c 1 vt~e S ikS kj 1

3S klS kldij

þ c 2 vt~e ðXikS kj þX jkS klÞ þ c 3

vt~e XikX jk 1

3XlkXlkdij

þ c 4 vtk~e2 ðS kiXlj

þ S kjXliÞS kl þ c 5 vtk~e2 XilXlmS mj þ S ilXlmXmj 2

3S lmXmnXnldij

þ c 6 vtk~e2 S ijS klS kl þ c 7

vtk~e2

S ijXklXkl; ð48Þ

where S ij is the mean strain-rate tensor and Xij the mean vorticity tensor:




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S ij ¼ @ ui

@ x jþ @ u j

@ xiXij ¼ @

ui@ x j

@ u j@ xi

: ð49Þ

The empirical coefficients of the model are presented in Table 1, where ~S is the dimensionless strain parameter, ~X the dimen-

sionless vorticity parameter, ~e the homogenous dissipation rate, which are defined as:

~S ¼ k~e

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiS ijS ij=2

q ; ~X ¼ k

~e

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXijXij=2

q ; vt ¼ c l k

2

~e : ð50Þ

The model appears better compared to ordinary linear eddy viscosity models (e.g. for impinging jet flows). The compu-

tational time is approximately 20% more compared to a low-Re k–e model. One drawback of the model is its performance for

convex surfaces, according to Craft et al. [137].Another popular cubic low-Re k–e model was developed by Apsley and Leschziner [138], with its free parameters cali-

brated with data from DNS data for channel flow. The model leads to better results for airfoil and diffuser flows compared

to other linear and nonlinear EVM. For more details for the NLEVM, the interest reader is directed to the review paper by

Hellsten and Wallin [136].

2.3.8. Recent advances in eddy viscosity modelling

It is worth mentioning some recent eddy-viscosity models, such as Durbin’s t2-f model (also known as v2–f ) [139] and f–f model [140]. The t2–f model is based on the elliptic relaxation concept and employs two additional equations, apart from thek and e ones. One for the velocity scale t2 and one for the elliptic relaxation function, f. The main motivation for the devel-opment of this model was the improved modelling in the vicinity of the wall (near-wall turbulence). More applications (e.g.

rotating cylinder, rotating channel flow, axially rotating pipe and square duct) and validation of the model with experimental

and DNS data, may be found in the work of Durbin and Petterson [141].

The f–f model is based on the similar concept of elliptic relaxation, but instead of solving the t2 equation it solves for thevelocity scale ratio f = t2/k [140]. The full equations of the f– f model, which are similar to those of the t2– f model, are [140]:

vt ¼ C lfks; ð51Þ

Dk

Dt ¼ @

@ x jv þ vt

rt

@ k

@ x j

þ P e; ð52Þ

DeDt ¼ @

@ x jv þ vt

re

@ e@ x j

þ ðC e1P C e2eÞ

s ; ð53Þ

L2r2 f f ¼ 1s

c 1 þ C 02P

e

f 2

3

; ð54Þ

Df

Dt ¼ @

@ xkv þ vt

rf

@ f

@ xk

þ f f

kP : ð55Þ

Completeness of the model is achieved by Durbin’s [142] realizability constraints, combined with the lower bounds (Kol-

mogorov time- and length- scale):

s ¼ max min ke;

a ffiffiffi6

p C ljS jf

!;C s

v

e

0:5" #; ð56Þ

L ¼ C L max min k1:5

e ;

k0:5 ffiffiffi

6p

C ljS jf

!; C g

v3

e

0:25" #; ð57Þ

where a6 1 (recommended

a = 0.6 [140]). The coefficients of this model are:

C l = 0.22,

C e1 = 1.4(1 + 0.012/f),

C e2 = 1.9,

c 1 = 0.4, C 02 ¼ 0:65, rk = 1, re = 1.3, rf = 1.2, C s = 6.0, C L = 0.36, C g = 85.

Table 1

The model coefficients [136].

c 1 c 2 c 3 c 4 c 5 c 6 c 7

0:05 f q f l 0:11 f q f l

0:21 f q

~S

f lð~S þ~XÞ=20.8 f c 0 0.5 f c 0.5 f c

C l f l0:667r n 1exp 0:415expð1:3n5=6Þ½ f g

1þ1:8n1:1

ffiffi~ee

p 10:8expð~Rt =30Þ½

1þ0:6 A2 þ0:2 A3:52r

n f

q f

c

1 þ 1 exp½ð2 A2Þ3n o

1 þ 4 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffi

exp ~Rt 20 r r n

ð1þ0:0086g2Þ0:5 r 2n

1þ0:45n2:5


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It should be noted that the f–f model is more stable [141] compared to the t2– f model. Both models are better for com-puting wall-bounded flows compared to the classical low-Re two equation models (e.g. k–x and k–e), but they are still weakagainst DSM (introduced next in Section 2.4) and advanced NLEVMs. The f–f model has been extensively validated with

experimental and DNS data for plane channel, backward-facing step and multiple-impinging jets flows, presenting satisfac-

tory agreement.

Recently, a new robust version of the t2– f model was proposed by Billiard and Laurence [143] with improved numericalstability, known as the BL-t2=k. The model is based on the elliptic blending method of Manceau and Hanjalic [144] and wasvalidated for pressure induced separated flows, as well as buoyancy impairing turbulent flows, with satisfactory results.

More detailed evaluation of the model against other turbulence models and test cases (e.g. 3-D diffuser and swept wing)

is presented in the work of Billiard et al. [145]. In Fig. 3, the mean velocity streamlines for swept wing are compared to data

from Implicit Large Eddy Simulation (LES). Both models capture the leading edge vortex but the secondary vortex region

(red-dashed line) is reproduced by the EBRSM model and secondly from the BL-t2=k model. A detailed review of t2– f modelevolution may be found in the work of Billiard and Laurence [143].

2.4. Differential Second-Moment (DSM) and Algebraic Stress Models (ASM)

A type of turbulence closure models with great expectations to replace the widely used k–e model is the DifferentialSecond-Moment (DSM) or Reynolds Stress (RS) or Mean-Reynolds Stress (MRS) model. DSM presents natural superiority

compared to the two equations turbulence models, as it is physically the more complete model (history, transport and

anisotropy of turbulent stresses are all accounted for). More specifically, DSM closure models explicitly employ transport

equations for the individual Reynolds Stresses, u0iu0 j (as well as for u0 jT 0), each of them representing a separate velocity scale.The transport equation of u0iu0 j for an incompressible fluid, excluding effects of rotation and body force, may be written ingeneral symbolic tensor form as follows:

Lij þ C ij ¼ P ij þ /ij þ Dij eij; ð58Þwhere Lij is the local change in time, C ij the convective transport, P ij the production by mean-flow deformation, /ij the stress

redistribution tensor due to pressure strain, Dij the diffusive transport and eij the viscous dissipation tensor. The Lij, C ij and P ij

terms do not require any modelling and are given by the following equations:

Lij þ C ij ¼ @ @ t þ um @

@ xm

u0iu

0 j; ð59Þ

P ij ¼ u0iu0m@ u j@ xm

þ u0 ju0m@ ui@ xm

: ð60Þ

The remaining terms, /ij, Dij and e ij need to be modelled. The simplest way to model the viscous dissipation tensor is byassuming local isotropy:

eij ¼ 23edij; e ¼ v @ um@ um

@ uk@ uk; ð61Þ

where e

is the turbulence dissipation and dij

the Kronecker unit tensor. The diffusion of turbulence (Dij

) is usually treated by

the popular Daly–Harlow model [146]:

Fig. 3. Mean velocity streamlines over the swept wing surface, highlighting regions of interest [145].


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Dij ¼ @ @ xk

C sk

eu0

lu0m

@ u0iu0 j

@ xl

! with C s ¼ 0:25: ð62Þ

Instead of the popular Daly–Harlow model, there are more advanced models that can be used and have been developed

over the years, such as the models by Magnaudet [147] and Nagano and Tagawa [148]. More details and validation of the

above mentioned models can be found in the review paper by Hanjalic [149].

The radically new feature of the DSM-equation is the pressure-strain ‘redistribution’ term (/ij), which does not appear in

the exact solution of k–e equation. This suggests that the pressure-strain term only serves to redistribute the turbulenceenergy among its components and to reduce the shear stresses, thus tending to make the turbulence more isotropic. The

unknown correlations appearing in the DSM-equation are either determined by a transport equation or else they are

expressed in terms of second-order correlations ðu0iu0 jÞ themselves; the latter procedure, closing the DSM-equation at itsown level, is often referred to as ‘second-order closure’. The redistribution term,/ij , is usually modelled by the Isotropization

of Production (IP) model [150]:

/ij ¼ C 1 u0iu0 j 2

3dijk

ek C 2 P ij dij

3 P kk

with C 1 ¼ 1:8 and C 2 ¼ 0:6: ð63Þ

Improvements of the IP pressure-strain model are the LRR-QI model by Launder et al. [151], the SSG model by Speziale

et al. [152], the CL model by Craft and Launder [153], and LT model by Launder and Tselepidakis [154]. For detailed evalu-

ation of the models, the interest reader is referred to the work of Hanjalic and Jakirlic [155]. DSM closure models predict

more accurate physical phenomena which involve streamline curvature, strong pressure gradients, swirling and system rota-

tion effects [155].

It is important to mention that the initial versions of DSM models could not perform very well in handling the return to

isotropy [1]. They may, however, work well in flows dominated by other effects. In addition, the DSM models do not always

perform better than the two-equation models. For instance, recent numerical simulations in street canyon flows performed

by Koutsourakis et al. [115] indicated that DSM performance was not good enough according to the theoretical expectations,

while the RNG k–e model exhibited better results. The evaluation of the models (DSM, RNG k–e and standard k–e) was donecomparing to different experimental and numerical data (LES). Figs. 4 and 5 present numerical results for the velocity pro-

files compared with experimental data. It is concluded that DSM, at least as it has been applied in that study, is not more

useful than simpler models for practical use. However, both modelling and experimental uncertainties are high, so that extra

attention is required in order to draw any definitive conclusions about the quality of the models.

Main disadvantages of DSM are the difficulty in the modelling of more terms in the turbulence equations and the

increased demand on computer resources. The new generation of DSM closure models have solved most of the above-men-

tioned difficulties but the computational demands are roughly twice as large as those for the two-equation models, for high -

Re number flows using wall functions [155].

It is worth noting the Elliptic Blending DSM by Manceau and Hanjalic [144], which belongs to the category of advanced

DSMs. The model is based on the DSM of Durbin [159] but, instead of resolving equations for the stress components, it adopts

a single elliptic equation [160]. Implementation of EBDSM for predicting impinging jet flows can be found in the work of

Thielen et al. [161].

Another interesting approach is the hybridization of DSM with an eddy viscosity model, recently developed by Basara and

Jakirlic [162]. The model combines the advantages of DSM along with the robustness of k–e model and it is known as HybridTurbulence Model (HTM). It may be used as an initialization model for DSMs in order to stabilize it and reduce the

Fig. 4. Experiment 4: Non-dimensional horizontal velocity profiles at leeward and windward side of a street canyon with aspect ratio H /D = 1 and very

rough walls. Uref is the free stream velocity, equal to 8 m/s. Experimental data are extractedfrom Kovar-Panskus et al. [156]. CFD results with the k-e modelfrom theoriginal paperare also included [115]. ‘‘N. Koutsourakis, J. Bartzis, N. Markatos, Evaluation of Reynolds stress, k-e andRNG k-e turbulence models in

street canyon flows using various experimental datasets, Environmental Fluid Mechanics 12 (2012) 379–403. This is Fig. 6 in the publication in which thematerial was originally published. With kind permission from Springer Science and Business Media.’’


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computational time. The model has been tested for many flows such as flow around a car, axially rotating pipe, 180 turned U

bend, backward-facing step and round jet impinging on a flat plane. The results were better than EVMs and closer to those

obtained by full DSM.Simplified versions of DSMs are the Algebraic Stress Models (ASM) or Explicit Algebraic Reynolds Stress Models (EARSM)

[136], obtained by eliminating the transport terms, using instead those of the kinetic energy equation. EARSM have the rep-

utation of being simple and easy to implement for boundary layer flows, while for elliptic, recirculating flows they are very

unstable. In addition, their performance is dependent on the DSM from which they were derived. This category constitutes

an intermediate-level between DSM and eddy viscosity models. EARSM are characterised by less computational demands

and higher accuracy compared to LEVM.

First attempt for the derivation of EARSM was done by Pope [163] for two-dimensional flows and it was later extended

and refined by Gatski and Speziale [134] and by Jongen and Gatski [164] for 3-D flows. The EARSM are popular for predicting

aeronautical flows, in particular with the model by Wallin and Johansson [165].

Due to space limitations, the interested reader is directed to the recent review papers by Hellsten and Wallin [136] and

Alfonsi [106], for the mathematical equations of EARSM.

2.5. Two-fluid models of turbulence

Undoubtedly, the main idea behind a large number of turbulence models derives from the notion of Boussinesq, who

introduced the idea of an effective viscosity, and of Prandtl who conceived the notion of turbulence mixing phenomena being

very similar to those treated by the dynamical theory of gases. The mathematical background of convectional turbulence

theory reflects only the unstructured diffusion of the molecular-collision process; large structure formation and growth,

and fine-structure creation and stretching, are nowhere to be found.

The above-mentioned facts led Spalding [166], among others, to the development of the ‘two-fluid’ theory, briefly pre-

sented below. The origins of two-fluid model ideas are to be found back in the 1940’s and 1950’s, particularly in the field

of turbulent combustion.

The following two-fluid model results in the prediction of the intermittency and of the conditional flow variables within

the turbulent and non-turbulent zones of the flow. The model was proposed by Spalding [166] and then developed by Malin

[167] for the investigation of intermittency in free turbulent shear flows.

The Spalding model is designed on an analogy between intermittent flows and two-phase flows, rather than on any spe-

cific and rigorous closure of conditional-averaged transport equations. It is supposed that two-fluids share occupancy of the

same space, although not necessarily at the same time, their share of space being measured by the volume fractions. There

are many ways in which the two-fluids can be distinguished. In case of turbulence intermittency computations, it is conve-

nient to define the fluids as ‘turbulent’ and ‘non-turbulent’. The equations governing the motion of the turbulent and non-

turbulent fluids are given in detail by Markatos [1,79].

Improvement and expansion of the model for turbulent combustion was presented by Markatos and Kotsifaki [168]. Fur-

thermore Shen et al. [169] used a two-fluid model to simulate turbulent stratified flows, while Yu et al. [170] established a

modified two-fluid model to simulate the flow and heat transfer characteristics of air curtains in an open vertical display

cabinet. For Rayleigh–Taylor mixing so far only two models have been developed, namely Youngs’ model [171] and two-

structure two-fluid two-turbulent (2SFK) model [172]. Liu et al. [173] applied two-fluid model for predicting the flow in

UV disinfection reactor. Finally, Cao et al. [174] used the model for the design of air curtains for open vertical refrigerated

display cases. The numerical results compared with experimental data present good agreement.

Fig. 5. Experiment 6: Performance of the three turbulence models against experimental data extracted from Li et al. [157] in a street canyon with aspect

ratio H /D = 2. LES results of Li et al. [158] are also included [115]. ‘‘N. Koutsourakis, J. Bartzis, N. Markatos, Evaluation of Reynolds stress, k-e and RNG k-eturbulence models in street canyon flows using various experimental datasets, Environmental Fluid Mechanics 12 (2012) 379–403. This is Fig. 8 in the

publication in which the material was originally published. With kind permission from Springer Science and Business Media.’’




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2.6. Large Eddy Simulation (LES)

Another modelling approach, promising to be more accurate and of wider applicability than RANS and less computation-

ally demanding than DNS, is the Large Eddy Simulation (LES) approach. In LES of turbulence, the important large scales are

fully resolved whilst the small sub-grid scales are modelled. The main advantage of LES compared to RANS models is that in

the former only the small, isotropic turbulent scales are modelled and not the entire spectrum (Fig. 6) as it is the case in the

latter. The LES approach is extremely useful for the investigation of turbulence at high Reynolds numbers, for the develop-

ment and assessment of new turbulence models, and for the prediction of complex flows where other turbulence models

may prove inadequate [176,177].

The first attempts of LES are found in the pioneering works of Smagorinsky [178] and Lilly [179] in meteorology and Dear-

dorff [180] in engineering. Since then, LES has seen a tremendous popularity for the study of turbulent flows. The develop-

ment and testing of LES has concentrated at first on isotropic turbulence by Kraichnan [181] and Chasnov [182] and on

turbulent channel flow by Deardorff [180], Schumann [183], Moin and Kim [184] and Piomelli [185]. The basic steps of

LES method, according to Pope [24] and Berselli et al. [25] are: (a) a filtering operation of the N–S equations; (b) a closure

model for SGS stress tensor; (c) imposition of the appropriate boundary and initial conditions with special care for near wall

modelling; (d) selection of the suitable numerical method for spatial and temporal discretization of N–S equations and (e)

performance of the numerical simulation.

Instead of time-averaging, a spatial filtering approach is adopted in order

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