ERCOFTAC Series

Michel StanislasJavier JimenezIvan Marusic Editors

Progress in Wall Turbulence 2Understanding and Modelling

Progress in Wall Turbulence 2

More information about this series at http://www.springer.com/series/5934

ERCOFTAC SERIES

VOLUME 23

Series Editor

Bernard GeurtsFaculty of Mathematical Sciences, University of Twente, Enschede,

The Netherlands

Aims and Scope of the Series

ERCOFTAC (European Research Community on Flow, Turbulence and Com-bustion) was founded as an international association with scientific objectives in1988. ERCOFTAC strongly promotes joint efforts of European research institutesand industries that are active in the field of flow, turbulence and combustion, inorder to enhance the exchange of technical and scientific information on funda-mental and applied research and design. Each year, ERCOFTAC organizes severalmeetings in the form of workshops, conferences and summerschools, whereERCOFTAC members and other researchers meet and exchange information.

The ERCOFTAC Series will publish the proceedings of ERCOFTAC meetings,which cover all aspects of fluid mechanics. The series will comprise proceedings ofconferences and workshops, and of textbooks presenting the material taught atsummerschools.

The series covers the entire domain of fluid mechanics, which includes physicalmodelling, computational fluid dynamics including grid generation and turbulencemodelling, measuring-techniques, flow visualization as applied to industrial flows,aerodynamics, combustion, geophysical and environmental flows, hydraulics,multiphase flows, non-Newtonian flows, astrophysical flows, laminar, turbulent andtransitional flows.

Michel Stanislas • Javier JimenezIvan MarusicEditors

Progress in WallTurbulence 2Understanding and Modelling

123

EditorsMichel StanislasLML UMR 8107Ecole Centrale de LilleVilleneuve d’AscqFrance

Javier JimenezE.T.S. Ingenieros AeronauticosPolytechnic University of MadridMadridSpain

Ivan MarusicMechanical EngineeringUniversity of MelbourneParkville, VICAustralia

ISSN 1382-4309 ISSN 2215-1826 (electronic)ERCOFTAC SeriesISBN 978-3-319-20387-4 ISBN 978-3-319-20388-1 (eBook)DOI 10.1007/978-3-319-20388-1

Library of Congress Control Number: 2015945138

Springer Cham Heidelberg New York Dordrecht London© Springer International Publishing Switzerland 2016This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or partof the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations,recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmissionor information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilarmethodology now known or hereafter developed.The use of general descriptive names, registered names, trademarks, service marks, etc. in thispublication does not imply, even in the absence of a specific statement, that such names are exempt fromthe relevant protective laws and regulations and therefore free for general use.The publisher, the authors and the editors are safe to assume that the advice and information in thisbook are believed to be true and accurate at the date of publication. Neither the publisher nor theauthors or the editors give a warranty, express or implied, with respect to the material contained herein orfor any errors or omissions that may have been made.

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Springer International Publishing AG Switzerland is part of Springer Science+Business Media(www.springer.com)

Preface

This book brings together contributions from participants of the secondWALLTURB workshop on “Understanding and modelling of wall turbulence” heldin Lille (France) from June 18 to 20, 2014.

This workshop follows the inaugural workshop organized in 2009 by theWALLTURB EC project, and aimed to assess the progress made in the field ofnear-wall turbulence in the 5 years separating the two workshops.

The workshop assembled 60 participants from all over the world, with 6 invitedlecturers and 39 contributions, and provided an opportunity to review the recentprogress in theoretical, experimental and numerical approaches to wall turbulence.

This book gathers papers from most of the contributors to the workshop. It isaimed as being a milestone in the research field, thanks to the high level of invitedspeakers and the involvement of the contributors.

Lille Michel StanislasMadrid Javier JimenezMelbourne Ivan MarusicAugust 2009

v

Contents

Part I Invited Lectures

On the Size of the Eddies in the Outer Turbulent Wall Layer:Evidence from Velocity Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3Sergio Pirozzoli

Sensitized-RANS Modelling of Turbulence: Resolving TurbulenceUnsteadiness by a (Near-Wall) Reynolds Stress Model. . . . . . . . . . . . . 17Suad Jakirlić and Robert Maduta

Coherent Structures in Wall-Bounded Turbulence . . . . . . . . . . . . . . . 37Javier Jiménez and Adrián Lozano-Durán

Attached Eddies and High-Order Statistics . . . . . . . . . . . . . . . . . . . . . 47Ivan Marusic and James D. Woodcock

Part II Papers

DNS of Turbulent Boundary Layers in the Quasi-LaminarizationProcess . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63Guillermo Araya, Luciano Castillo and Fazle Hussain

Numerical ABL Wind Tunnel Simulations with Direct Modelingof Roughness Elements Through Immersed Boundary ConditionMethod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73Bruno Lopez, Gabriel Usera, Gabriel Narancio, Mariana Mendina,Maritn Draper and Jose Cataldo

vii

Three-Dimensional Nature of 2D Hairpin Packet Signaturesin a DNS of a Turbulent Boundary Layer . . . . . . . . . . . . . . . . . . . . . 83S. Rahgozar and Y. Maciel

Wall Pressure Signature in Compressible TurbulentBoundary Layers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93N.A. Buchmann, Y.C. Kücükosman, K. Ehrenfried and C.J. Kähler

Three-Dimensional Structure of Pressure–Velocity Correlationsin a Turbulent Boundary Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103Yoshitsugu Naka, Michel Stanislas, Jean-Marc Foucaut,Sebastien Coudert and Jean-Philippe Laval

Computation of Complex Terrain Turbulent Flows Using HybridAlgebraic Structure-Based Models (ASBM) and LES . . . . . . . . . . . . . 115C. Panagiotou, S.C. Kassinos and D. Grigoriadis

Computation of High Reynolds Number Equilibriumand Nonequilibrium Turbulent Wall-Bounded FlowsUsing a Nested LES Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125Yifeng Tang and Rayhaneh Akhavan

An Attempt to Describe Reynolds Stresses of TurbulentBoundary Layer Subjected to Pressure Gradient. . . . . . . . . . . . . . . . . 137Artur Dróżdż and Witold Elsner

The Temporal Coherence of Prograde and Retrograde SpanwiseVortices in Zero-Pressure Gradient Turbulent Boundary Layers . . . . . 147Callum Atkinson, Vassili Kitsios and Soria

Boundary Layer Vorticity and the Rise of “Hairpins” . . . . . . . . . . . . . 159Peter S. Bernard

On the Extension of Polymer Molecules in Turbulent ViscoelasticFlows: Statistical and Tensor Investigation . . . . . . . . . . . . . . . . . . . . . 171Anselmo Soeiro Pereira, Ramon Silva Martins, Gilmar Mompean,Laurent Thais and Roney Leon Thompson

Velocity of Line Plumes on the Hot Plate in TurbulentNatural Convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181Vipin Koothur and Baburaj A. Puthenveettil

viii Contents

LES of a Converging–Diverging Channel Performed with theImmersed Boundary Method and a High-Order CompactDiscretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191Mariusz Ksiezyk and Artur Tyliszczak

On Minimum Aspect Ratio for Experimental Duct Flow Facilities . . . . 201Ricardo Vinuesa, Eduard Bartrons, Daniel Chiu, Jean-Daniel Rüedi,Philipp Schlatter, Aleksandr Obabko and Hassan M. Nagib

Riblets Induced Drag Reduction on a Spatially DevelopingTurbulent Boundary Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213Amaury Bannier, Eric Garnier and Pierre Sagaut

Characterization of Pipe-Flow Turbulence and Mass Transferin a Curved Swirling Flow Behind an Orifice . . . . . . . . . . . . . . . . . . . 225N. Fujisawa, R. Watanabe, T. Yamagata and N. Kanatani

Turbulent Structure of a Concentric Annular Flow . . . . . . . . . . . . . . . 237Sina Ghaemi, Majid Bizhani and Ergun Kuru

Reconstruction of Wall Shear-Stress Fluctuationsin a Shallow Tidal River . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247Romain Mathis, Ivan Marusic, Olivier Cabrit,Nicole L. Jones and Gregory N. Ivey

Analysis of Vortices Generation Process in Turbulent BoundarySubjected to Pressure Gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259Artur Dróżdż and Witold Elsner

Experimental Investigation of a Turbulent BoundaryLayer Subject to an Adverse Pressure Gradient at Rehup to 10000 Using Large-Scale and Long-RangeMicroscopic Particle Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271Tobias Knopp, Nicolas A. Buchmann, Daniel Schanz, ChristianCierpka, Rainer Hain, Andreas Schröder and Christian J. Kähler

The Structure of APG Turbulent Boundary Layers . . . . . . . . . . . . . . 283Ayse G. Gungor, Yvan Maciel and Mark P. Simens

Adverse Pressure Gradients and Curvature Effectsin Turbulent Channel Flows. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295A.B. de Jesus, L.A.C.A. Schiavo, J.L. Azevedo and J.-P. Laval

Contents ix

On the Response of a Separating Turbulent BoundaryLayer to High Amplitude Excitation . . . . . . . . . . . . . . . . . . . . . . . . . . 307Vitali Palei and Avi Seifert

Statistical and Temporal Characterization of TurbulentRayleigh-Bénard Convection Boundary Layers UsingTime-Resolved PIV Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . 317Christian E. Willert, Ronald du Puits and Christian Resagk

Large-Scale Organization of a Near-Wall TurbulentBoundary Layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335R. Dekou, J.-M. Foucaut, S. Roux and M. Stanislas

Near-Wall Study of a Turbulent Boundary LayerUsing High-Speed Tomo-PIV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347Fabio J.W.A. Martins, Jean-Marc Foucaut,Luis F.A. Azevedo and Michel Stanislas

The Effects of Superhydrophobic Surfaces on SkinFriction Drag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357Hyunwook Park and John Kim

Structure and Dynamics of Turbulence in Super-HydrophobicChannel Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367Amirreza Rastegari and Rayhaneh Akhavan

Spectral Assessment of the Turbulent Convection Velocityin a Spatially Developing Flat Plate Turbulent BoundaryLayer at Reynolds Number Reh= 13 000 . . . . . . . . . . . . . . . . . . . . . . . 379Nicolas Renard, Sébastien Deck and Pierre Sagaut

Statistics of Single Self-sustaining Attached Eddyin a Turbulent Channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391Yongyun Hwang

Scaling the Internal Boundary Layer . . . . . . . . . . . . . . . . . . . . . . . . . 399Fanxiao Meng, Donald J. Bergstrom and Bing-Chen Wang

3D Spatial Correlation Tensor from an L-Shaped SPIVExperiment in the Near Wall Region . . . . . . . . . . . . . . . . . . . . . . . . . 405Jean-Marc Foucaut, Christophe Cuvier, Sebastien Coudertand Michel Stanislas

x Contents

On Objective and Non-objective Kinematic FlowClassification Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419Ramon S. Martins, Anselmo S. Pereira, Gilmar Mompean,Laurent Thais and Roney L. Thompson

Quantification of the Full Dissipation Tensor froman L-Shaped SPIV Experiment in the Near Wall Region . . . . . . . . . . . 429Jean-Marc Foucaut, Christophe Cuvier, Michel Stanislasand William K. George

Contents xi

Part IInvited Lectures

On the Size of the Eddies in the OuterTurbulent Wall Layer: Evidencefrom Velocity Spectra

Sergio Pirozzoli

Abstract The scaling of size of the momentum-bearing eddies in wall-parallelplanes in the outer part of turbulent wall layers is analyzed, by examining spectra ofthe fluctuating velocities taken from direct numerical simulations and experiments.For all flows under scrutiny the normalized spectra highlight growth of the eddiessize with the wall distance. The results indicate the capability of a modified mixinglength (Pirozzoli, J Fluid Mech 702:521–532, 2012 [1]) of accounting with greaterprecision for the wall-normal variation of the size of the eddies bearing streamwisemomentum. This observation can be explained by assuming that outer layer momen-tum streaks (superstructures) spread under the collective action of the other eddies,which impart a (nearly) uniform eddy diffusivity throughout the outer wall layer.

Keywords Wall-bounded flows · Turbulence · Coherent structures

1 Introduction

This is the follow-up of a previous study [1] (hereafter referred to as P12), in whichthe issue of the proper scaling of the size of eddies (broadly defined as regions whichretain some degree of coherence in turbulent flow)was addressed. Themain results ofthat study were that, in the outer part of turbulent wall layers (including channels andboundary layers, also in the compressible regime) the spanwise size of the eddies, asmeasured through the integral correlation length scale, is controlled by a modifiedmixing length, defined as

�1/2

δ∼

(∂u+

∂η

)−1/2

, (1)

where η = y/δ, u+ = u/u∗τ , y is the distance from the wall, u∗

τ = (τw/ρ)1/2 is theeffective friction velocity at a given off-wall location (at the wall, u∗

τ coincides with

S. Pirozzoli (B)Dipartimento di Ingegneria Meccanica E Aerospaziale, Università di Roma‘La Sapienza’, Rome, Italye-mail: sergio.pirozzoli@uniroma1.it

© Springer International Publishing Switzerland 2016M. Stanislas et al. (eds.), Progress in Wall Turbulence 2,ERCOFTAC Series 23, DOI 10.1007/978-3-319-20388-1_1

3

4 S. Pirozzoli

the conventional definition of the friction velocity, uτ ), τw is the wall shear stress,ρ is the mean local density, δ is the thickness of the wall layer, and u = ρu/ρ (whichcoincides with u in strictly incompressible flow). The scaling given in Eq. (1) (here-after referred to as the modified mixing length) should be contrasted with Prandtl’sclassical mixing length, namely �m ∼ y, and with its proposed generalization [2]

�m

δ∼

(∂u+

∂η

)−1

, (2)

which reduces to Prandtl’s scaling in the presence of a sizeable log layer. As shownby Mizuno and Jiménez [2], Eq. (2) provides a better prediction for the growth ofthe eddies with the wall distance than the wall distance itself, and hence it will behereafter used instead of the classical Prandtl’s scaling.

In this paper, we address the issue of the proper scaling of the outer layer eddiesin greater detail than in P12 by: (i) scrutinizing the spectral densities of the velocitycomponents, to separately establish the growth of the various scales of motion withthewall distance; (ii) including additional numerical and experimental data, to furtherverify the robustness of the observations.

2 Numerical and Experimental Data

The numerical database used for the analysis includes direct numerical simulations(DNS) of compressible boundary layer flows and incompressible channel flows ofthe Couette–Poiseuille family. The boundary layer data cover the range of frictionReynolds numbers, Reτ = 250 − 4000 (with Reτ = δ/δv , where δv = νw/uτ isthe inner length scale, and δ is the boundary layer thickness), and free-stream Machnumber M∞ ≤ 4. Details on the numerical implementation and mesh resolution areprovided in earlier studies [3, 4], and will not be repeated here. The channel flowdatabase includes DNS performed in a rectangular channel (whose half-height is h),and it includes simulations with stationary walls (corresponding to pure turbulentPoiseuille flow, cases P, PH, PHH), and simulations with with one stationary (S) andone moving (M) wall, the controlling parameters being the bulk Reynolds numberand the ratio of the shear stress at the two walls γ = τM/τS (γ = −1 for Poiseuilleflow). Specifically, flow case P1 corresponds to a Poiseuille-like flow with reducedshear; flow cases C1 and C are Couette-like flows, the latter being very close to pureCouette flow. Details on the channel flow simulations can be retrieved in the originalreferences [5, 6]. Two Poiseuille flow cases from the DNS database of del Álamoet al. [7] were also included (labeled as TO1, TO2). It is noteworthy that the M2HHand the PHH simulations represent the upper limit of Reynolds number currentlyreached in DNS of boundary layers and channel flows, and they both exhibit slightlyless than a decade of quasi-logarithmic variation of the mean velocity [4, 6], thussatisfying the requirements [8, 9] for being representative of high-Reynolds-numberwall turbulence.

On the Size of the Eddies in the Outer Turbulent … 5

Table 1 Flow properties for numerical turbulent boundary layers (left) and Couette–Poiseuilleflows (right)

Flow case M∞ Reτ Reθ Flow case γ Reτ M Reτ S

M03 0.3 432 1098 P −1 284 284

M2L 2 251 1122 P1 −0.24 128 261

M2 2 508 2434 C1 0.27 130 250

M2H 2 1116 6046 C 0.98 242 245

M2HH 2 3940 18787 PH −1 541 541

M3 3 504 3867 PHH −1 4080 4080

M4 4 506 5824 TO1 −1 991 991

TO2 −1 2000 2000

In the table on the left, Reθ = ρ∞ u∞θ/μ∞, Reτ = δ/δv , where θ is the momentum thickness.In the table on the right, γ = τM/τS , Reτ M,S = uτ M,Sh/ν, uτ M,S = (|τM,S |/ρ)1/2, where thesubscripts M and S refer to the moving and the stationary wall, respectively

In the following, δ is stipulated to be the 99% velocity thickness for boundarylayers, even though other choices are possible. In the channel flow cases the outerlength scale is defined to be the distance from the stationary wall to the point wherethe mean shear becomes zero, if any (hence, δ = h for flow cases P, PH, PHH, andTOx). The main parameters for the DNS hereafter reported are shown in Table1.

As will be made clearer in the analysis, verification of the scaling formulas (1) and(2) requires access to the spectral densities of the velocity components in the spanwiseand/or the streamwise direction at several wall distances. For that purpose, we haveconsidered the boundary layer experimental database of Hutchins et al. [10] whichcovers frictionReynolds numbers in the rangeReτ = 500−2300, andwhich includestwo-point correlations of streamwise velocity fluctuations in the spanwise directionat several off-wall stations. For the sake of preciseness, we point out that the values ofthe friction Reynolds number quoted here are slightly lower than those reported in theoriginal reference, because of the different definition of the boundary layer thickness,which was estimated in experiments through a Coles law-of-the-wake fit of the meanvelocity profile. We also consider the pipe flow experimental database by Bailey andSmits [11], which covers a single Reynolds number,Reτ = 3400, andwhich includesspanwise and streamwise correlations of u at five off-wall stations [11]. The readeris referred to the original references for details on the experimental setup.

3 Spectral Densities of Velocity Fluctuations

The power spectral densities associated with the u, v,w velocity fluctuations (respec-tively, the streamwise, wall-normal, and spanwise components) have been analyzedfor all flows presented in Sect. 2, at various wall distances, limited to y/δ ≥ 0.1.Here and in the following ‘+’ superscripts will be used to denote quantities made

6 S. Pirozzoli

nondimensional with respect to uτ and δv . The attention is mainly focused on thespanwise spectra, but information on the scaling of the streamwise spectra is alsoprovided. For the sake of the analysis we define the power spectral densities of thegeneric property ϕ in the i th direction, Ei

ϕ , in such a way that

ϕ′2 =∫ ∞

0Ei

ϕ(ki ) dki , (3)

where ki is the Fourier wavenumber in the i th direction. To account for the effect ofturbulence kinetic energy variation across the wall layer, we consider the normalizedenergy spectra, defined as

E iϕ(ki ) = Ei

ϕ(ki )/ϕ′2. (4)

As shown in standard textbooks, the integral length scale ofϕ in a given homogeneousdirection, defined as the integral of the corresponding two-point correlation function

�iϕ =

∫ ∞

−∞Cϕϕ( xi ) d xi , (5)

is linked with the corresponding normalized spectra through the relation

�iϕ = lim

ki →0π E i

ϕ(ki ). (6)

Equation (6) shows that the integral length scale is closely connected with the asymp-totic behavior of the power spectrum at the largest scales.

The typical organization of the spanwise velocity spectra is shown in Fig. 1, forsome boundary layer DNS at y/δ = 0.3. Note that, for the sake of representation,wavelengths (λz = 2π/kz) are shown on the horizontal axis, rather than the corre-sponding wavenumbers, here scaled by the boundary layer thickness. The spectra ofu and v (those ofw are similar to the former) are both bump shaped, and they exhibit

10-2 10-1 100 10110-3

10-2

10-1

100

λz/δ

kzE

u(k

z)

10-2 10-1 100 10110-3

10-2

10-1

100

λz/δ

kzE

v(k

z)

(a) (b)

Fig. 1 Premultiplied spanwise spectral densities of u (a) and v (b) at y/δ = 0.3 for flow casesM2L(dashes); M2 (dash-dot); M2H (dash-dot-dot); M2HH (solid). The straight diagonal lines mark ak−5/3

z behavior

On the Size of the Eddies in the Outer Turbulent … 7

a k−5/3z power-law behavior in the small-scale range, which reflects the establishment

of a locally isotropic energy cascade [12], and which becomes evident at the highestReτ .

It is the main goal of the forthcoming discussion to verify whether this variationcan be compensated through a suitable scale transformation. For that purpose, in thefollowing we will inspect the premultiplied normalized energy spectra as a functionof the wavelength (λi = 2π/ki ), scaled by a suitable mixing length (either �m or�1/2), in semi-log scale. This type of representation provides hints on the possibleuniversality of the distributions, while retaining the illustrative advantage that equalareas underneath the graphs correspond to equal energies.

3.1 DNS Data

We first consider the spanwise velocity spectra extracted from DNS of boundarylayer and channel flows. The data are shown at several off-wall stations, namelyy/δ = 0.1 i, i = 1, . . . , 7, thus covering the whole outer layer, well into the wakeregion. Only the spectra of u and v are shown in the following for space limitations,those ofw being similar to those of u. The u spectra scaled in classical mixing lengthunits, shown in the left column of Fig. 2 for selected flow cases, have a reasonabledegree of universality. This observation was also made by Mizuno and Jiménez [2],and taken in support for the validity of the classical mixing length scaling, in thegeneralized form given in Eq. (2). However, careful inspection of the figures showsconsistent shift of the scaled spectra to shorter wavelengths as the wall distanceincreases (from the dotted curves to the solid curves). This discrepancy is moreevident at the large scales (i.e., the right end of the spectra), which explains the poorcollapse of the integral length scales with �m observed in P12. The spectra of v scaledwith �m , shown in the right column of Fig. 2, have a slightly different behavior.Whilethe same shift of the normalized spectra of the small scales is found as for u, in thiscase the trend at the largest resolved scales appears to be nearly insensitive to y, andthis is again consistent with the success of �m in parameterizing the integral lengthscales of v observed in P12.

The u and v spanwise spectra scaled with the modified mixing length are shownin Fig. 3. Of course, the qualitative nature of the graphs does not change from Fig. 2,since a y-dependent shift of the horizontal axis is involved. However, all the u-spectraexhibit approximate collapse with the wall distance in this case. In fact, the spectralenergy density of both the small and the large scales becomes nearly y-independent,and scatter is mainly found in the peak amplitude associated with what we definedas energy-containing eddies. Greater scatter is observed in the DNS at the higherReynolds number (M2HH and PHH), which is likely caused by slow convergence ofthe outer layer spectra. It is noteworthy that in boundary layers the spectral peak isplaced at λz ≈ 3�1/2, and at λz ≈ 4�1/2 in channel flows. Hence, in agreement withprevious observations [13], eddies in channels are found to be somewhat larger than

8 S. Pirozzoli

10-2 10-1 100 101 1020

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10-2 10-1 100 101 102

10-2 10-1 100 101 102 10-2 10-1 100 101 102

10-2 10-1 100 101 102 10-2 10-1 100 101 102

10-2 10-1 100 101 102 10-2 10-1 100 101 102

(a) (b)

(c) (d)

(e) (f)

(g) (h)

Fig. 2 DNS data: premultiplied spanwise spectral densities of u (left column) and v (right column)in classical mixing length scaling (�m , as defined in Eq. (2)), for flow cases M2H (a-b), M2HH(c-d), TO2 (e-f), PHH (g-h). The data are shown from y/δ = 0.1 (dots) to y/δ = 0.7 (solid lines),in intervals of 0.1δ

On the Size of the Eddies in the Outer Turbulent … 9

0

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0.6

0.7

0.8

kzE

v(k

z)

λz 1/2

10-2 10-1 100 101 102 10-2 10-1 100 101 102

10-2 10-1 100 101 102 10-2 10-1 100 101 102

10-2 10-1 100 101 10210-2 10-1 100 101 102

10-2 10-1 100 101 102 10-2 10-1 100 101 102

(a) (b)

(c) (d)

(e) (f)

(g) (h)

Fig. 3 DNS data: premultiplied spanwise spectral densities of u (left column) and v (right column)in modified mixing length scaling (�1/2, as defined in Eq. (1)), for flow cases M2H (a-b), M2HH(c-d), TO2 (e-f), PHH (g-h). The data are shown from y/δ = 0.1 (dots) to y/δ = 0.7 (solid lines),in intervals of 0.1δ

10 S. Pirozzoli

in boundary layers. Comparing the v spectra in Fig. 3 with those in Fig. 2, onewill seethat both �m and �1/2 perform similarly in removing the y dependence for all the flowcases. However, the �1/2 scaling seems to be more effective in parameterizing thesmall eddies,whereas the large eddies are probablymore accurately parameterized by�m . The position of the spectral peak is also less sensitive to y in the �1/2 scaling, withλz ≈ 1 − 2�1/2 for boundary layers, and λz ≈ 2�1/2 for channels. Normalization ofthe spectrawith respect to thewall distancewas also attempted as a further possibility,but it was found to yield poorer data collapse than both �1/2 and �m .

Streamwise spectra of u and v are shown in Fig. 4, limited to the TO2 case [7],scaled with respect to δ, �m , and �1/2. In this case, δ scaling yields the best collapseof the u spectra across the whole range of scales, which implies that the longitudinalsize of the u-bearing eddies does not depend on the wall distance to leading order,

0

0.05

0.1

0.15

0.2

0.25

kxE

u(k

x)

λx/δ

0

0.1

0.2

0.3

0.4kxE

v( k

x)

λx/δ

10-2 10-1 100 101 102 1030

0.05

0.1

0.15

0.2

0.25

kxE

u(k

x)

λx m

0

0.1

0.2

0.3

0.4

kxE

v(k

x)

λx m

0

0.05

0.1

0.15

0.2

0.25

kxE

u(k

x)

λx 1/2

0

0.1

0.2

0.3

0.4

kxE

v(k

x)

λx 1/2

10-2 10-1 100 101 102 10-2 10-1 100 101 102

10-2 10-1 100 101 102 103

10-2 10-1 100 101 102 103 10-2 10-1 100 101 102 103

(a) (b)

(c) (d)

(e) (f)

Fig. 4 Premultiplied streamwise spectral densities of u (left column) and v (right column) for flowcase TO2. Wavelengths are scaled with respect to δ (a-b), �m (c-d), and �1/2 (e-f). The data areshown from y/δ = 0.1 (dots) to y/δ = 0.7 (solid lines), in intervals of 0.1δ

On the Size of the Eddies in the Outer Turbulent … 11

and the energy-containing eddies are about 4− 5 δ long. As pointed out by Hutchinsand Marusic [9], the spectral peak of u in the outer layer is the imprint of large-scale momentum streaks, which have a typical meandering pattern in the streamwisedirection. Thus, streamwise spectra do not necessarily convey exhaustive informationon their actual length, and indeed visual inspection supports lengths in excess of20δ [9]. As a consequence, the observed invariance of the δ-scaled u spectra with yis probably to bemore correctly interpreted as evidence that the turns of themeandershave a spacing that does not depend on y. The streamwise spectra of v (right columnof Fig. 4) appear again to be very accurately parameterized by �1/2, both at the smalland at the energy-containing scales. However, the large-scale end of the spectra seemsto scale on δ, and the longitudinal integral scale of v is nearly y-independent.

3.2 Experimental Data

The boundary layer data from Hutchins et al. [10] are considered first. The span-wise spectra of u, which we estimated by Fourier transforming the correspondingtwo-point velocity correlations, shown in Fig. 5, further support the notion that themodifiedmixing length is significantly more accurate in accounting for the growth of

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

kzE

u(k

z)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

kzE

u(k

z)

λz m

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

λz 1/2

10-2 10-1 100 101 102 10-2 10-1 100 101 102

10-2 10-1 100 101 102 10-2 10-1 100 101 102

(a) (b)

(c) (d)

Fig. 5 Boundary layer experiments [10]: premultiplied spanwise spectral densities of u in classicalmixing length scaling (left column) and in modified mixing length scaling (right column) at Reτ =546 (a-b); 2287 (c-d). The data are shown from y/δ = 0.1 (dots) to y/δ = 0.7 (solid lines), inintervals of 0.1δ

12 S. Pirozzoli

the whole range of u-bearing eddies with y. Comparison with the numerical bound-ary layer spectra of u presented in Fig. 3 shows very close similarities, including theposition of the spectral peak, which is very nearly constant (λz ≈ 3�1/2) across therange of wall distances and Reynolds numbers.

We next consider the pipe flow data from Bailey and Smits [11], shown in Fig. 6.The spanwise spectra (left column) exhibit some energy pileup at the smallest scales,caused by lack of smoothness of the spatial correlations. Nevertheless, we refrainedfrom using any smoothing procedure, and we present the raw data. With this caveat,the spectra at the large-scale end appear to have little sensitivity on y, when scaledwith �1/2, with a peak at λz ≈ 3 − 4�1/2. On the other hand, the streamwise spectra

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

kzE

u(k

z)

λz/δ10-2 10-1 100 101 102 1030

0.05

0.1

0.15

0.2

0.25

λx/δ

kxE

u(k

x)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

kzE

u(k

z)

λz m

0

0.05

0.1

0.15

0.2

0.25

λx m

kxE

u(k

x)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

kzE

u(k

z)

λz 1/2

0

0.05

0.1

0.15

0.2

0.25

kxE

u(k

x)

λx 1/2

10-2 10-1 100 101

10-1 100 101 102

10-1 100 101 102

10-2 10-1 100 101 102 103

10-2 10-1 100 101 102 103

(a) (b)

(c) (d)

(e) (f)

Fig. 6 Pipe flow experiments [11]: premultiplied spanwise (left column) and streamwise (rightcolumn) power spectral densities of u in δ scaling (a-b), classical mixing length scaling (c-d), andmodified mixing length scaling (e-f). The data are shown from y/δ = 0.1 (dashed lines), up toy/δ = 0.5 (solid lines), in intervals of 0.1δ

On the Size of the Eddies in the Outer Turbulent … 13

(right column) in the large-scale range appear to be insensitive to y (and thus toscale on δ), whereas the small scales are perhaps equally well parameterized by�1/2. The apparent insensitivity of the streamwise velocity spectra to y was alsonoticed by Bailey and Smits [11], who identified two separate peaks in the spectrum,corresponding to large-scale and very-large-scale motions, whose size varies weaklywith y. They found that this dependence can be further minimized by using the sameconvection velocity at all off-wall stations when applying Taylor’s hypothesis, andinterpreted this fact with the argument that spectral peaks at different wall-normalpositions correspond to the same coherent motions.

4 Conclusions

The key finding is certainly that the typical length scale ofmost eddies inwall-parallelplanes is proportional to �1/2, rather than to �m . Although the two alternatives arenot vastly different, we believe that the consistent improvement in the universalityof the scaled spectra in the �1/2 normalization, observed for a variety of flow cases(only a part of which has been shown), is a sufficient proof of the validity of the newscaling. In this respect, as also observed by Chernyshenko and Baig [14], we arguethat the growth of eddies in wall-parallel planes must be controlled by the inter-play between the local mean shear and diffusion. Turbulent diffusion in wall-parallelplanes in mainly associated with u and w velocity fluctuations. These “sloshing”motions (inactive motions, in Townsend’s terminology, [15]) are not constrained bythe impermeability condition at the wall, and hence their typical length scale is notbound to be proportional to the wall distance. This is not the case for turbulent trans-port in the vertical direction, which is controlled bywall-normal velocity fluctuations(active motions, according to Townsend), whose typical length scale is bound to beproportional (at least sufficiently close to the wall) to the wall distance. This obser-vation points to the relevance of an eddy viscosity based on the friction velocityand on the total wall layer thickness in controlling turbulent diffusion processes inwall-parallel planes,

νt I ∼ u∗τ δ, (7)

where the subscript I denotes the contribution of inactive motions, to discriminatefrom the classical eddy viscosity. Note that the effective friction velocity (u∗

τ ) mayvary across the wall layer in compressible flow, owing to mean density variations.However, since density variations are concentrated in the wall proximity, νt I isapproximately constant in the outer layer. Next, based on this observation, we assumethat the growth of eddies in wall-parallel planes is controlled by turbulent diffusion.Borrowing arguments used by del Álamo et al. [7] we note that, on a time scale t ,diffusion would yield eddies with size � ∼ (νt I t)1/2. If the relevant eddy viscosityfor diffusion is (nearly) constant across the wall layer, as given in Eq. (7), it appearsthat the size of the eddies should be the same at all off-wall locations. However, dif-

14 S. Pirozzoli

fusion processes at a given wall distance are limited in time by the disruptive effectof the mean shear, whose typical time scale is

τS =(

∂ u

∂y

)−1

. (8)

It is then reasonable to conclude that the typical size of eddies resulting from time-limited wall-parallel diffusion processes is

� ∼ (νt I τS)1/2 ∼ (u∗

τ δ)1/2 (

∂ u

∂y

)−1/2

, (9)

which is identical to the mixing length defined in Eq. (1).It is important to state that by no means our arguments imply that the classical

Prandtl’s theory should be discarded. Indeed, our observations do not apply to tur-bulent transport in the wall-normal direction (hence, to the Reynolds shear stress),which is probably still controlled by the classical mixing length, as suggested fromthe analysis of the wall-normal velocity correlations [2]. Hence, it is expected thata signature of Prandtl’s scaling is observed in the wall-parallel spectra of v. This isconsistent with the observation that the largest v-bearing eddies scale on �m , ratherthan on �1/2.

We expect that the present results, while not directly relevant for meanmomentumbalance in wall-bounded flows, may shed some light onto the mechanisms of growthof turbulent eddies in the strongly anisotropic wall layer, and perhaps to build deter-ministic models for the outer layer eddies, in the same line of thought as Townsend’sconical eddies [15]. Also, our results might be relevant for the study of the dispersionof contaminants/pollutants in shear flow. Investigation of the two-dimensional spec-tra in wall-parallel planes and of the vertical velocity correlations, currently ongoing,might lead to fuller characterization of the shape and size of the typical outer layercoherent structures.

Acknowledgments I acknowledge that some results in this paper have been achieved using thePRACE Research Infrastructure resource JUGENE based at the Forschungszentrum Jülich (FZJ) inJülich, Germany. Thanks are also due to Sean C.C. Bailey, Alexander J. Smits, Nicholas Hutchins,and Ivan Marusic, for providing experimental data and other useful information.

References

1. S. Pirozzoli, J. Fluid Mech. 702, 521–532 (2012)2. Y. Mizuno, J. Jiménez, Phys. Fluids 23, 085112 (2011)3. S. Pirozzoli, M. Bernardini, J. Fluid Mech. 688, 120–168 (2011)4. S. Pirozzoli, M. Bernardini, Phys. Fluids 25, 021704 (2013)5. S. Pirozzoli, M. Bernardini, P. Orlandi, J. Fluid Mech. 680, 534–563 (2011)6. M. Bernardini, S. Pirozzoli, P. Orlandi, J. Fluid Mech. (2013). Under review

On the Size of the Eddies in the Outer Turbulent … 15

7. J.C. del Álamo, J. Jiménez, P. Zandonade, R.D. Moser, J. Fluid Mech. 500, 135 (2004)8. J. Jiménez, R. Moser, Philos. Trans. R. Soc. Lond. A 365, 715 (2007)9. N. Hutchins, I. Marusic, J. Fluid Mech. 579, 1 (2007)10. N. Hutchins, W.T. Hambleton, I. Marusic, J. Fluid Mech. 541, 21 (2005)11. S. Bailey, A. Smits, J. Fluid Mech. 651, 339 (2010)12. J. Jiménez, Annu. Rev. Fluid Mech. 44, 27 (2012)13. J. Monty, J. Stewart, R. Williams, M. Chong, J. Fluid Mech. 589, 147 (2007)14. S.I. Chernyshenko, M.F. Baig, J. Fluid Mech. 544, 99 (2005)15. A. Townsend, The Structure of Turbulent Shear Flow, 2nd edn. (Cambridge University Press,

Cambridge, 1976)

Sensitized-RANS Modelling of Turbulence:Resolving Turbulence Unsteadinessby a (Near-Wall) Reynolds Stress Model

Suad Jakirlic and Robert Maduta

Abstract A turbulence model designed and calibrated in the steady RANS(Reynolds-Averaged Navier-Stokes) framework has usually been straightforwardlyapplied to an unsteady calculation. It mostly ended up in a steady velocity field in thecase of confined wall-bounded flows; a somewhat better outcome is to be expected inglobally unstable flows, such as bluff body configurations. However, only a weaklyunsteady mean flow can be returned with the level of unsteadiness being by farlower compared to a referent database. The latter outcome motivated the presentwork dealing with an appropriate extension of a near-wall Second-Moment Clo-sure (SMC) RANSmodel towards an instability-sensitive formulation. Accordingly,a Sensitized-RANS (SRANS) model based on a differential, near-wall Reynoldsstress model of turbulence, capable of resolving the turbulence fluctuations to anextent corresponding to the model’s self-balancing between resolved and modelled(unresolved) contributions to the turbulence kinetic energy, is formulated and appliedto several attached and separated wall-bounded configurations—channel and ductflows, external and internal flows separating from sharp-edged and continuous curvedsurfaces. In most cases considered the fluctuating velocity field was obtained startedfrom the steady RANS results. Themodel proposed does not comprise any parameterdepending explicitly on the grid spacing. An additional term in the correspondinglength scale-determining equation providing a selective assessment of its produc-tion, modelled in terms of the von Karman length scale (formulated in terms of thesecond derivative of the velocity field) in line with the SAS (Scale-Adaptive Simula-tion) proposal (Menter and Egorov, Flow Turbul Combust 85:113–138, (2010) [14]),represents here the key parameter.

S. Jakirlic (B) · R. MadutaInstitute of Fluid Mechanics and Aerodynamics / Center of Smart Interfaces, TechnischeUniversität Darmstadt, Alarich-Weiss-Straße 10, 64287 Darmstadt, Germanye-mail: s.jakirlic@sla.tu-darmstadt.de

R. MadutaOutotec GmbH, Ludwig-Erhard-Strasse 21, 61440 Oberursel, Germanye-mail: roberttibor@gmx.li

© Springer International Publishing Switzerland 2016M. Stanislas et al. (eds.), Progress in Wall Turbulence 2,ERCOFTAC Series 23, DOI 10.1007/978-3-319-20388-1_2

17

18 S. Jakirlic and R. Maduta

1 Introduction

The work on development of the hybrid RANS/LES (Large-Eddy Simulation) meth-ods and novel Unsteady RANS (URANS) methods (RANSmodel plays here the roleof a sub-scale model) has been greatly intensified in recent years. The relevant meth-ods have been proposed by Spalart et al. ([20], DES—Detached Eddy Simulation;see Spalart, [19] for the DES method upgrades, namely Delayed DES and ImprovedDelayed DES), Menter and Egorov ([14]; SAS—Scale-Adaptive Simulations), Gir-imaji ([7]; PANS—Partially Averaged Navier Stokes; see also Basara et al. [1], forthe PANS method extension to account for the near-wall effects), and Chaouat andSchiestel ([4]; PITM—Partially Integrated Transport Model). The common featureof all these models is an appropriate modification of the scale-determining equationproviding a dissipation rate level which suppresses the turbulence intensity towardsthe subgrid (i.e. sub-scale) level in the regions where large coherent structures witha broader spectrum dominate the flow, allowing in such a way evolution of struc-tural features of the associated turbulence. Whereas an appropriate dissipation levelenhancement in the PANS method (similar is in the case of the PITM method)is achieved by reducing selectively (e.g. in the separated shear layer region) thedestruction term in the model dissipation equation, i.e. its coefficient Cε,2 (e.g. thegrid-spacing-dependent model coefficient function in the PANS method providesappropriate decrease of the standard value Cε,2 = 1.92, prevailing in the near-wallregion, towards a significantly lower value in the separated shear layer of the peri-odic 2D hill flow, see e.g. [3]), an additional production term was introduced intothe ω equation (ω ∝ ε/k—inverse turbulent time scale) in the SAS framework. Thisterm is modelled in terms of the von Karman length scale comprising the secondderivative of the velocity field (∇2U), which is capable of capturing the vortex sizevariability, [14].

The work reported here aims at developing an instability sensitive, anisotropy-resolving Second-Moment Closure (SMC) model. This model scheme, functioningas a ‘sub-scale’ model in the Unsteady Sensitized-RANS (SRANS) framework, rep-resents a differential near-wall Reynolds stress model formulated in conjunctionwith the scale-supplying equation governing the homogeneous part of the inverseturbulent time scale: ωh = εh/k. The model capability to account for the vortexlength and time scales variability was enabled through a selective enhancement ofthe production of the dissipation rate in line with the SAS proposal (Scale-AdaptiveSimulation, Menter and Egorov, [14]) pertinent particularly to the highly unsteadyseparated shear layer region. The predictive performances of the proposed model arechecked by computing series of internal and external, two-dimensional and three-dimensional flows in channels, ducts and past bluff bodies including separation fromsharp-edged and continuous curved surfaces in a range of Reynolds numbers.

Sensitized-RANS Modelling of Turbulence: Resolving Turbulence … 19

2 Computational Method

The following section briefly outlines the computational model proposed. It isfollowed by description of the numerical method and associated details.

2.1 Computational Model

The equation governing the homogeneous part of the total viscous dissipation rate,εh = ε−0.5ν∂2k/(∂x j∂x j ), modelled in term-by-termmanner by Jakirlic and Han-jalic [8] represents the starting point for the present development. The RSM-basedωh-equation following directly from the εh-equation (here, instead of originally usedGeneral-Gradient-Diffusion-Hypothesis (GGDH) for the turbulent diffusion mod-elling, the Simple GDH with diffusion coefficient modelled in terms of turbulenceviscosity was applied; thereby, no difference between the Prandtl-Schmidt numberscorresponding to the quantities k and εh was made; σk = σε = σω = 1.1 is adoptedfinally) by using well-known relationship

Dωh

Dt= 1

k

Dεh

Dt− εh

k2Dk

Dt(1)

with

Dk

Dt= ∂

∂xk

[(1

2ν + νt

σk

)∂k

∂xk

]+ Pk − εh and

Dεh

Dt= ∂

∂xk

[(1

2ν + νt

σε

)∂εh

∂xk

]+Cε,1

εh

kPk −Cε,2

εhεh

k+2Cε,3ννt

∂2Ui

∂x j∂xl

∂2Ui

∂x j∂xl

reads:

Dωh

Dt= ∂

∂xk

[(1

2ν + νt

σω

)∂ωh

∂xk

]+ Cω,1

ωh

kPk − Cω,2ω

2h

+2

k

(Ccr,1

1

2ν + Ccr,2

νt

σω

)∂ωh

∂xk

∂k

∂xk+ 2

kCω,3ννt

∂2Ui

∂x j∂xl

∂2Ui

∂x j∂xl(2)

with Pk = −ui u j∂Ui/∂x j representing production of the kinetic energy of tur-bulence and coefficients Cω,1 = Cε,1 − 1 = 0.44, Cω,2 = Cε,2 − 1 = 0.8 andCω,3 = 1.0 taking their standard values. The introduction of the ‘correction’ coeffi-cients Ccr,1 = 0.55 and Ccr,2 = 0.275 into the cross-derivative term copes with thecorrection of the near-wall behaviour of the ωh-variable (see [9] for more details).Last term on the right-hand side represents the gradient production term; here,instead of the original formulation (modelled byusing the vorticity transport theorem)

20 S. Jakirlic and R. Maduta

comprising both the mean rate of strain and second derivative of the velocity field asimplified version (pertinent to an eddy-viscosity model) is applied in line with therequest for a practical model usage. The model for turbulent viscosity νt , accountsfor both Reynolds stress anisotropy (being beyond the reach of the eddy-viscositymodel group) and viscosity effects, with characteristic length representing a switchbetween the Kolmogorov length scale and the turbulent length scale.

The latter equation is appropriately extended through the introduction of the SASterm [14] into the ωh-equation:

Dωh,S AS

Dt= Dωh

Dt+ PS AS; PS AS = CRSM,1 max

[P∗

S AS, 0]

P∗S AS = 2.3713κS2

(L

Lvk

)1/2

− 3CRSM,2k max

[(∇ωh)2

ω2h

,(∇k)2

k2

](3)

with L = k1/2/ωh being the turbulent length scale, Lvk = κS/|∇2U | (with∇2U = [∂2Ui/∂x2j × ∂2Ui/∂x2j ]1/2) representing the 3D generalization of the clas-sical boundary layer definition of the von Karman length scale and S the invariant ofthe mean strain tensor (S = √

2Si j Si j ; Si j = 0.5(∂Ui/∂x j + ∂U j/∂xi )). It shouldbe noted that the PSAS term introduced in the ωh-equation has almost identical formas the one being used in the eddy-viscosity-based k −ω SST-SAS model [14]. How-ever, two coefficients, CRSM,1 = 0.004 and CRSM,2 = 8 and exponent of the lengthscales ratio ((1/2) instead of 2)are introduced adjusting its use in the framework ofa Second-Moment Closure model. The natural decay of the homogeneous isotropicturbulence, fully developed channel flows in a range of Reynolds number (withunderlying velocity field following the logarithmic law) and the non-equilibrium2D hill flow at two different Reynolds numbers (ReH = 10,600 and 37,000) havebeen interactively computed in the process of the coefficients calibration. Equa-tion (3) represents a grid-spacing-free formulation. This explicit non-dependence onthe grid-spacing represents certainly an advantage over some hybrid LES/RANSmodels, especially in the case of unstructured grids with arbitrary grid-cell topology.The contours of the PSAS term in the flow over a periodical arrangement of 2D hillsand past a tandem cylinder configuration depicted in Fig. 1 clearly shows that it isactive only in the region of the separated shear layer. In the reminder of the flowdomain, especially in the near-wall regions, its effect vanishes.

The proposed model is solved in conjunction with the Jakirlic and Hanjalic’s [8]Reynolds stress model equation (εh = ωhk, k = ui ui/2, Pi j = −ui uk∂U j/xk −u j uk∂Ui/∂xk); similar to the scale-supplying equation (Eq.2) the GGDH turbulentdiffusion model is replaced by the corresponding SGDH model formulation:

∂ui u j

∂t+ ∂Ukui u j

∂xk= ∂

∂xk

[(1

2ν + νt

σui u j

)∂ui u j

∂xk

]+ Pi j − εh

i j + Φi j + Φwi j (4)

Sensitized-RANS Modelling of Turbulence: Resolving Turbulence … 21

Fig. 1 Contours of the PSAS term (Eq.3) coloured by its magnitude in the 2D hill flow (upper) andflow past a tandem-cylinder (lower)

For more detailed insight into the modelling rationale interested readers arereferred to [9].

2.2 Numerical Method

All computations were performed using the codeOpen-FOAM, an open source Com-putational Fluid Dynamics toolbox (www.opencfd.co.uk/openfoam), utilizing a cell-center-based finite volume method on an unstructured numerical grid and employingthe solution procedure based on the implicit pressure algorithmwith splitting of oper-ators (PISO) for coupling between pressure and velocity fields. SIMPLE procedurewas applied when computing the steady flows using the RANS-RSM model. Theconvective transport was discretized by a scheme blending between the second ordercentral differencing (CDS) and first order upwind (UDS) schemes with γCDS = 0.95and γUDS = 0.05 in most of the cases considered. For the time integration thesecond order three point backward scheme was used. The code is parallelized apply-ing the Message Passing Interface (MPI) technique for communication between theprocessors.