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International Journal of Heat and Fluid Flow 27 (2006) 994–1009

Turbulent flow around a wall-mounted cube: A directnumerical simulation

Alexander Yakhot a,*, Heping Liu a,1, Nikolay Nikitin b

a The Pearlstone Center for Aeronautical Engineering Studies, Department of Mechanical Engineering,

Ben-Gurion University of the Negev, Beersheva Beersheva 84105, Israelb Institute of Mechanics, Moscow State University, 1 Michurinski Prospekt, 119899 Moscow, Russia

Received 24 August 2004; received in revised form 29 December 2005; accepted 3 February 2006Available online 12 May 2006

Abstract

An immersed-boundary method was employed to perform a direct numerical simulation (DNS) of flow around a wall-mounted cubein a fully developed turbulent channel for a Reynolds number Re = 5610, based on the bulk velocity and the channel height. Instanta-neous results of the DNS of a plain channel flow were used as a fully developed inflow condition for the main channel. The results con-firm the unsteadiness of the considered flow caused by the unstable interaction of a horseshoe vortex formed in front of the cube and onboth its sides with an arch-type vortex behind the cube. The time-averaged data of the turbulence mean-square intensities, Reynoldsshear stresses, kinetic energy and dissipation rate are presented. The negative turbulence production is predicted in the region in frontof the cube where the main horseshoe vortex originates.� 2006 Elsevier Inc. All rights reserved.

Keywords: Turbulence; Direct numerical simulation; Negative turbulence production; Immersed-boundary method

1. Introduction

Flow and heat transfer in a channel with wall-mountedcubes represent a general engineering configuration that isrelevant in many applications. Owing to simple geometrybut with complex vortical structures and generic flow phe-nomena associated with a turbulent flow, this configurationhas attracted increasing attention from researchers and hasbeen used for the bench-marking purposes to validate tur-bulent models and numerical methods.

Before the 1990s, measurements of three-dimensionalflows around wall-mounted cubical obstacles were unavail-able. Instead any understanding of the major global fea-tures of flow patterns were based on flow visualizationstudies. The first published measurements of the turbulent

0142-727X/$ - see front matter � 2006 Elsevier Inc. All rights reserved.

doi:10.1016/j.ijheatfluidflow.2006.02.026

* Corresponding author.E-mail address: yakhot@bgu.ac.il (A. Yakhot).

1 Present address: Central Iron and Steel Research Institute, 76 XueyuanNanLu, Beijing 100081, People’s Republic of China.

velocity field, energy balance and heat transfer around asurface-mounted cube begun to appear in the 1990s. Sev-eral experimental works have been performed on a flowaround a single wall-mounted cube in a developed turbu-lent channel flow. Results showed that this flow is charac-terized by the appearance of a horseshoe-type vortex at thewindward face, an arc-shaped vortex in the wake of thecube, flow separation at the top and side face of the cubeand vortex shedding. The flow features and experimentaldata for time-averaged flow quantities have been well doc-umented in Martinuzzi and Tropea (1993) and Hussein andMartinuzzi (1996). Nakamura et al. (2001) experimentallyinvestigated a fluid flow and local heat transfer around acube mounted on a wall of a plane channel. The turbulentboundary layer thickness was about two times higher thanthe cube height. Meinders et al. (1999) experimental studywas also performed in a developing turbulent channel flow.

The ability to perform these measurement aroused greatinterest in numerical simulation of these flows. Mostnumerical studies were performed by using the Reynolds-

Nomenclature

D mean grid spacing, (DxDyDz)1/3

d boundary layer thicknessE dissipation rate of the turbulence energygK Kolmogorov’s length scale, ðm3=EÞ1=4

K turbulence kinetic energy, 0:5ðu02 þ v02 þ w02Þm kinematic viscosityu0iu0j Reynolds stresses, i, j = x, y, z

q densitysw wall shear stressu velocity vectoru02; v02;w02 turbulence mean-square intensitiesf frequencyH height of the channelh height of the cubep kinematic (divided by density) pressure

PK production rate of turbulence kinetic energy,u0iu0jSij

Reh Reynolds number, Umh/mRem Reynolds number, UmH/mSij rate of strain tensor, 0.5(oUi/oxj + oUj/oxi), i,

j = x, y, z

St Strouhal number, fh/Um

u 0, v 0, w 0 fluctuating velocity components in x-, y- and z-directions

U, V, W mean velocity components in x-, y- and z-direc-tions

Um mean velocityus friction velocity,

ffiffiffiffiffiffiffiffiffiffisw=q

px, y, z steamwise, wall-normal and spanwise directions

A. Yakhot et al. / Int. J. Heat and Fluid Flow 27 (2006) 994–1009 995

averaged Navier–Stokes (RANS) method with differentturbulent models (Iaccarino et al., 2003; Lakehal and Rodi,1997) and large-eddy simulations (LES) (Krajnovic andDavidson, 1999, 2001; Niceno et al., 2002; Rodi et al.,1997; Shah and Ferziger, 1997). For the flow consideredin the present study, RANS approach could not reproducethe details of the complex fluid structure near the wall, e.g.the converging–diverging horseshoe vortex and separationof the boundary layer in front of the cube, nor the separa-tion length behind the cube. It is commonly believed thatthis inability is due to the disregard of unsteady effects suchas vortex shedding. In contrast, global flow features andcharacteristics predicted by LES and more recently byunsteady RANS-based modeling (Iaccarino et al., 2003)showed good agreement with experimental data.

A numerical study of the fundamental hydrodynamiceffects in complex geometries is a challenging task for dis-cretization of Navier–Stokes equations in the vicinity ofcomplex geometry boundaries. Implementing boundary-fit-ted, structured or unstructured grids in finite element meth-ods help to deal with this problem, however the numericalalgorithms designed for such grids are usually inefficientin comparison to those designed for simple rectangularmeshes. This disadvantage is particularly pronounced forsimulating non-steady incompressible flows when the Pois-son equation for the pressure has to be solved at each timestep. Iterative methods for complex meshes have low con-vergence rates, especially for fine grids. On the other hand,very efficient and stable algorithms have been developed forsolving Navier–Stokes equations in rectangular domains. Inparticular, these algorithms use fast direct methods for solv-ing the Poisson’s equation for the pressure (Swarztrauber,1974). The evident simplicity and other advantages of suchsolvers led to the development of approaches for formulat-ing complex geometry flows on simple rectangular domains.

One such approach is based on the immersed-boundary(IB) method originally introduced to reduce the simulation

of complex geometry flows to that defined on simple (rect-angular) domains. This can be illustrated if we consider aflow of an incompressible fluid around an obstacle X (Sis its boundary) placed into a rectangular domain (P).To impose the no-slip condition on an obstacle surface S

(which becomes an internal surface for the rectangulardomain wherein the problem is formulated), a source termf (an artificial body force) is added to the Navier–Stokesequations. The purpose of the forcing term is to imposethe no-slip boundary condition on the xS-points whichdefine the immersed boundary S.

Immersed-boundary-based methods are considered tobe a powerful tool for simulating complex flows. Differentimmersed-boundary methods can be found in Balaras(2004), Fadlun et al. (2000), Kim et al. (2001), Tseng andFerziger (2003). IB-based approaches differ by the methodused to introduce an artificial force into the governingequations. Fadlun et al. (2000) and Kim et al. (2001) devel-oped the idea of ‘‘direct forcing’’ for implementing finite-volume methods on a staggered grid. Kim et al. (2001)contributed two basic approaches for introducing directforcing when using immersed-boundary methods. Onewas a numerically stable interpolation procedure for evalu-ating the forcing term, and the other introducing a masssource/sink to enhance the solution’s accuracy.

The main advantages of IB methods are that they arebased on relatively simple numerical codes and highly effec-tive algorithms, resulting in considerable reduction ofrequired computing resources. The main disadvantage,however, is the difficulty in resolving local regions withextreme variations of flow characteristics. These are espe-cially pronounced for high Reynolds number flows. More-over, in order to impose the boundary conditions,numerical algorithms require that the node velocity valuesbe interpolated onto the boundary points because theboundary S does not coincide with the gridpoints of a rect-angular mesh. Finally, due to the time-stepping algorithms

996 A. Yakhot et al. / Int. J. Heat and Fluid Flow 27 (2006) 994–1009

used, for example, in ‘‘direct forcing’’ IB methods, the no-slip boundary condition is imposed with O(Dt2) accuracy.Therefore, implementation of IB methods to simulate tur-bulent flows requires careful monitoring to avoid possiblecontamination of numerical results arising from inaccurateboundary conditions.

All of the aforementioned numerical studies used indi-rect methods to test the various modeling approaches.However, it is often difficult to judge the adequacy of a tur-bulence model due to uncertainties of the verifiability ofavailable experimental data. Due to the vast, continuousrange of excited scales of motion, which must be correctlyresolved by numerical simulation, the DNS approach is themost accurate but, unfortunately, it is also the most expen-sive. Any DNS simulation, especially in complex geometry,is very time consuming and has extensive storage require-ments. At present, the application of the DNS to realisticflows of engineering importance is restricted by relativelylow Reynolds numbers. The objective of DNS not neces-sarily for reproducing real-life flows, but performing a con-trolled study that allows better insight into flow physics inorder to develop turbulence modeling approaches. This isthe first paper to present DNS-based simulations carriedout with inlet boundary conditions which are physicallyrealistic and without the use of turbulence models. Ourmajor concern was to perform the DNS of a case thatyields reliable results. The aims of this work were to testthe recent trend to employ immersed-boundary methodsfor simulating turbulent flows and to examine the feasibil-ity of using DNS for complex geometry flows. We presentthe time-averaged data of the following turbulence statis-tics: mean-square intensities, Reynolds shear stress, kineticenergy and dissipation rate. We draw attention to theoccurrence of negative turbulence kinetic energy produc-tion in front of the cube that is relevant for turbulenceLES/RANS modeling.

2 Gridpoints were added near the front and top cube’s faces and near thechannel base.

2. Numerical method

The flow of an incompressible fluid around an obstacleX placed into a rectangular domain (P) is governed bythe Navier–Stokes and incompressibility equations withthe no-slip boundary condition. The basic idea of IB meth-ods is to describe a problem, defined in P � X, by solvingthe governing equations inside an entire rectangular Pwithout an obstacle, which allows using simple rectangularmeshes. To impose boundary conditions on an obstaclesurface S (which becomes an internal surface for the rect-angular domain where the problem is formulated), weadd a body force f term to the governing equations:

ou

ot¼ �ðurÞu�rp þ mr2uþ f ð1Þ

where u is the velocity, p is the kinematic (divided by den-sity) pressure and m is the kinematic viscosity.

The numerical solution to the system of equation (1) wasobtained by using the IB approach suggested by Kim et al.

(2001). The only difference is that instead of the time-advancing scheme used in Kim et al. (2001), we employedanother third-order Runge–Kutta algorithm suggested inNikitin (1996). This time-treatment results in a calculationprocedure allowing numerical accuracy and time-stepcontrol.

3. Numerical simulation: problem formulation

3.1. Computational domain, spatial and temporal resolution

A configuration of a channel with a wall-mountedcube is shown in Fig. 1. The computational domain is14h · 3h · 6.4h in the streamwise (x), normal to the channelwalls (y) and spanwise (z) directions, respectively, where h

is the cube’s height. The computational domain inlet andoutlet are located at x = �3h and x = 11h, respectively,and the cube is located between 0 6 x 6 h, 0 6 y 6 h,�0.5h 6 z 6 0.5h. A computational grid is set of 181 ·121 · 256 gridpoints in the x, y and z directions, respec-tively. A non-uniform mesh was used in the streamwiseand wall-normal directions with gridpoints clustering nearthe channel and cube walls. The finest grid spacing is oforder Dxmin = 0.024h near the cube’s front and rear wallsand Dymin = 0.006h near the channel and cube’s top walls.

The computational domain also contains an entrance

channel, which is not shown in Fig. 1. To generate a fullydeveloped turbulent flow to be used as the inlet conditionto the channel with a cube, DNS was performed in anentrance channel measuring 9h · 3h · 6.4h. A computa-tional grid in the entrance channel measured 64 · 121 ·256 gridpoints in the x, y and z directions, respectively.The total number of 7,589,120 gridpoints were used. Forthe grid refinement tests,2 the number of gridpoints was11,009,280. The computations have been performed onCRAY SV1 Supercomputer, maximum memory used�115 MW, CPU-time is 70 s/time step.

A uniform mesh of Dx = 0.141h was used in the stream-wise direction. In the wall-normal direction, the gridpointswere clustered near the channel walls with Dymin = 0.006h.For both the entrance and main channels, in the spanwisez-direction, the gridpoints were equally spaced withDz = 0.025h. The computational grid data are summarizedin Table 1. The subscript ‘‘+’’ denotes that a quantity isnormalized by the wall units: the characteristic length,ls = m/us, and velocity, us = (sw/q)1/2, computed for thefully developed flow in the entrance channel. For isotropicand homogenous turbulence, the Kolmogorov’s length

scale, gK ¼ ðm3=EÞ1=4 (E ¼ mou0ioxj

ou0ioxj

is the mean dissipation

rate of the turbulence energy) is the smallest scale to be spa-tially resolved. The flow considered in this paper exhibitsstrong anisotropy. However, an estimation of the smallest

X

Y

Z

3h

3h

h

14h

6.4h

3.2h

h

h

Fig. 1. Geometry of a cube mounted on a wall.

Table 1Computational domain and grid data

Domain Main channel Entrance channel

Lx · Ly · Lz 14h · 3h · 6.4h 9h · 3h · 6.4h

Nx · Ny · Nz 181 · 121 · 256 64 · 121 · 256Cube grid 40 · 51 · 40Dx+ 3.1–40.7 19.51Dy+ 0.76–6.46 0.76–6.46Dz+ 3.46 3.46

A. Yakhot et al. / Int. J. Heat and Fluid Flow 27 (2006) 994–1009 997

scale, ld, to be spatially resolved by numerical simulatingthe anisotropic turbulence is beyond the scope of thispaper. It is believed that the mean grid widthD = (DxDyDz)

1/3 is O(gK) and is typically greater than gK.DNS-based results reported in the literature show goodagreement with experimental data even though the Kol-mogorov’s length scale was not resolved, e.g. D/gK =c > 1 (Moin and Mahesh, 1998). Moser and Moin (1987)note that in a curved channel, most of the dissipationoccurs at scales greater than 15gK. For simulations of thewall-bounded turbulence in simple geometries (channel,pipe, boundary layer), gK reaches the smallest values inthe near-wall regions where the dissipation rate, E, isexpected to be maximal. For complex geometry simula-

Table 2Comparison of computational resolution around the cube with KolmogorovD = (DxDyDz)

1/3, gK ¼ ðm3=EÞ1=4

c = D/gK c-range Cells with 3 < c < 4 (%) Cells with 4 <

cx 0.5–14.0 16.3 9.5cy 0.2–7.2 7.3 2.7cz 0.5–9.6 16.0 4.2c 0.4–5.9 13.7 4.2

tions, gK is unknown a priori. As an a posteriori test, wecomputed Kolmogorov’s length scale. Table 2 summarizesthe data of the spatial resolution in the near-cube region. Inour simulations, we found that 2.0 < c,cx,cy,cz < 5.0 in theregions of complex vortical flows of the horseshoe, lateraland top vortices. The relatively high (>5) values of cy

and cz were detected in the vicinity of the sharp edges ofthe cube. The high values of cx > 6 were found in the recov-ery region, downstream to the rear vortex reattachmentpoint. Note that ad hoc computing the smallest scales thatneed to be resolved as ld � gK leads to an underestimating

of ld. This is because computing the dissipation rate from

E ¼ mou0ioxj

ou0ioxj

results in overestimating E since all the scales

are accounted for, including those which are stronglyanisotropic and do not in fact lead to energy dissipation.

Cartesian coordinates with a staggered grid wasemployed for spatial finite-difference discretization. The sec-ond-order central difference scheme was used for the convec-tion and viscous terms of the Navier–Stokes equations. Theimposing no-slip boundary conditions and incompressibilitywere combined into a single time-step procedure. A periodicboundary condition was imposed in the spanwise direction,the no-slip boundary condition was used at all rigid wallsand the Neumann condition was used for pressure at all

’s length scale, 1.5 < x/h < 6.0, 0 < y/h < 2.0, �1.0 < z/h < 1.0, D = cgK,

c < 5 (%) Cells with 5 < c < 6 (%) Cells with c > 6 (%)

3.8 9.10.6 0.040.4 0.075.2 –

998 A. Yakhot et al. / Int. J. Heat and Fluid Flow 27 (2006) 994–1009

boundaries. At the downstream outlet boundary, convectivevelocity conditions were imposed.

3.2. Inflow condition: fully developed turbulent channel flow

Numerical simulation of spatially developing turbulentflows requires specification of time-dependent inflow condi-tions. In cases with flow separation, free shear layers, lam-inar-turbulent transitional flows, the downstream flow ishighly dependent on the conditions specified at the inlet.For studying flows around bluff bodies placed into a devel-oping boundary layer, a possible approach is to employ amean velocity profile and a level of turbulent kinetic energy(if those are available from the experiment). Then, a smalltime-dependent perturbation can be imposed on the inflowto obtain the desired statistics. Evidently, this implies thatthe inflow satisfies the Navier–Stokes equations, whichmay require excessively long streamwise domains for thedeveloping boundary layer simulations. Imposing experi-mentally-obtained inflow conditions are not within thescope of this study due to their insuperable complexity.For example, in Meinders et al. (1999), the flow around acube consisted of a developing turbulent boundary layerflow near the channel wall where the cube was placed (d/h > 1, d is the boundary layer thickness) and an almostlaminar boundary layer at the opposite wall with the coreflow remaining uniform. In the present study, the oncom-ing flow is a fully developed turbulent channel flow. Weintroduce an entrance (upstream) domain which is muchless costly. In this domain, the inflow is computed usingperiodic boundary conditions. These simulations wererun simultaneously with the main computation procedurein order to specify the fully developed inflow condition.The dimensions and grid data of the entrance channelare summarized in Table 1; the first gridpoint was aty+ = 0.76. The Reynolds number based on the bulk veloc-ity and the channel height is Rem = 5610, identical to thebenchmark simulations of Kim et al. (1987). The calculatedroot-mean-square velocity fluctuations and the Reynoldsshear stress profiles were compared with the results of

Fig. 2. Time-averaged streamlines and press

Kim et al. (1987) showing very good agreement betweenthe results.

4. Results and discussion

All flow parameters presented in this section are non-dimensionalized, unless they are explicitly written in thedimensional form; the cube’s height, h, and the mean veloc-ity Um are characteristic length and velocity, respectively.

A detailed experimental investigation by Meinders et al.(1999) was performed for Reynolds numbers rangingbetween 2750 < Reh < 4970, based on the bulk velocityand the cube height. The flow visualization results werefound to be independent of Reh in the range of2000 < Reh < 7000. The present computations have beenperformed at the relatively low Reynolds number ofReh = 1870. The experiments were carried out in a two-dimensional channel with a ratio of H/h = 3.3, where H

is the channel height. The case of H/h = 2 served as a testcase at the Rottach–Egern large-eddy simulation (LES)workshop for two Reynolds numbers Reh = 3000 and40,000. The LES calculations showed that the flow behav-ior at the two Reynolds number is very similar (Rodi et al.,1997).

4.1. Flow pattern

Past experiments have been performed for both rela-tively low and moderate Reynolds number flows. Afterthe first experimental study carried out by Martinuzziand Tropea (1993) at Reh = 40,000, a series of works werepublished by Meinders and co-authors (Meinders et al.,1998, 1999; Meinders and Hanjalic, 1999, 2002). Visualiza-tion and measurements indicated that the flow possessesvery complex features. The flow separates in front of thecube and at its front corners on the top (roof) and sidewalls. The main vortex forms a horseshoe-type vortexaround the cube and interacts with the main separationregion behind the cube. Up to four separation regions areinstantaneously disclosed.

ure distribution on the symmetry plane.

Fig. 3. Time-averaged streamlines in the symmetry plane obtained byMartinuzzi (see Rodi et al. (1997)).

A. Yakhot et al. / Int. J. Heat and Fluid Flow 27 (2006) 994–1009 999

Fig. 2 shows the time-averaged streamlines and pressuredistribution (the background color) at the symmetry plane.The experimental outline of time-averaged streamlinesobtained by Martinuzzi (see Rodi et al. (1997)) at a highReynolds number (Reh = 40,000) is depicted in Fig. 3. Itcan be seen that, despite the different Reynolds numbers,the general flow features and the overall prediction of thefront, top and rear vortex modes (marked F, T and R)are quite similar, although some differences are visible. Inour simulations for lower Reynolds numbers, there aretwo small counter-clockwise secondary vortices found inthe bottom corner part of the front face (marked M) andthe rear face (marked N); the horseshoe-type vortex,formed by the separation region in the front and bentaround the cube, is initiated more upstream (marked XF1)and the predicted reattachment length (marked XR1) ofthe primary vortex behind the cube is shorter. Moreover,the flow reattaches on the cube’s top surface. These pre-dicted features differ from the high Reynolds number caseof Martinuzzi and Tropea (1993), but they are in goodagreement with the experimental study of Meinders et al.(1999) carried out in a developing turbulent channel flowat a low Reynolds number. In their experiment, Meinderset al. (1999) also observed flow reattachment on the topsurface and the shorter wake recirculation region. In the

Fig. 4. The near-wall (y/h = 0.003) and near-top-surface (Dy/h = 0.

present case, the windward separation point (marked Sb)is located much further downstream (1.21h) than in theirexperiment (about 1.4h). This difference may be causedby different inflow conditions, namely, the fully developedturbulent channel flow used by us and the developing flowin Meinders et al. (1999). The predicted by us value of thefront separation region XF1 = 1.21h lies between the valuesof 1.0h and 1.4h measured in Martinuzzi and Tropea (1993)and Meinders et al. (1999), respectively. The front stagna-tion point XF2 = 0.65h, where the flow is spread radially,and the rear stagnation point XR2 = 0.15h, where anupward flow is formed, can also be seen from Fig. 2. Thesevalues agree with those found by Nakamura et al. (2001)for a wall-mounted cube in a turbulent boundary layer.

Fig. 4 depicts the near-wall (y/h = 0.003) and near-top-surface (Dy/h = 0.003) distribution of the time-averagedstreamlines and pressure. All marked points in this figurecorrespond to those used in the oil-film visualization inMeinders et al. (1999) (not shown here). All flow featuresdetected in Meinders et al. (1999) can be seen also fromour results. The oncoming flow separates at the saddlepoint (marked Sb) located approximately 1.21 cube heightin the front of the cube, where streamlines extend down-stream along the channel floor past the sides of the cube.Just ahead of the cube, the flow separation line (markedSa) is deflected downstream along the leading side facesto form two legs of a horseshoe vortex (marked A). Theconverging–diverging streamlines inside the horseshoe vor-tex show traces of this vortex and delineate its extent ofseparation. Fig. 4 shows that the flow separations fromthe cube’s side and top faces form a complex vortical struc-ture consisting of two side vortices (marked E) and the topvortex (marked by T in Fig. 2). Behind the cube, the flowreattachment from the side flow and the shear layer formedover the cube lead to an arc-shaped vortex tube. A pair ofcounter-rotating flows (marked B) close to the base wall

003) distribution of the time-averaged streamlines and pressure.

1000 A. Yakhot et al. / Int. J. Heat and Fluid Flow 27 (2006) 994–1009

behind the cube can be easily observed. The areas of rotat-ing flow show the footprints of the arch vortex. In the rearof the cube, the shear layer separated at the top leadingedge reattaches at approximately 1.5h cube height down-stream to the trailing face (marked C). Fig. 4 shows alsothe near-top cube’s surface projected onto the channelfloor. This imprint clearly shows two counter-rotating vor-tices (marked D), which were also observed in the oil-filmresults of Meinders et al. (1999).

The correlation between the time-averaged pressure dis-tribution and flow structure can be seen in Figs. 2 and 4. Ahigh pressure region is found in front where the oncomingflow impinges upon the cube. The maximum value of thepressure is observed on the cube front face where the front

X

Z

2 30

0.8

0.9

0.6

0.7

0.4

0.5

0.2

0.1

0.3

1

X

Z

2 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

X

Z

2 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

X

Z

2 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Fig. 5. The time-averaged streamlines in the xz-plane for diffe

stagnation point appears. In the vicinity of the downstreamcorners, the top and side vortices separation yields the min-imum pressure.

The time-averaged streamlines in the xz-plane for differ-ent y/h levels are shown in Fig. 5a–d for one half a channel.The side recirculation region extends upward along the lat-eral faces. We can see that substantial alteration of its sizewas not detected. On the other hand, the arc-type vorticalstructure in the wake, gradually grows towards the channelfloor. Fig. 5a shows the imprint of the horseshoe vortex aty/h = 0.1, but the imprint of the horseshoe vortex at y/h = 0.25 is hardly detected. Meinders et al. (1999) reportedthat the horseshoe vortex extends up to approximately halfof the cube height. To estimate the horseshoe vortex size, in

4 5

(a)

(b)

(c)

(d)

4 5

4 5

4 5

rent y/h levels; y/h = 0.1 (a), 0.25 (b), 0.5 (c) and 0.75 (d).

A. Yakhot et al. / Int. J. Heat and Fluid Flow 27 (2006) 994–1009 1001

Fig. 6 we show the front view of the time-averaged stream-wise vorticity, Xx, contour plots for different streamwise x/h-locations in the vicinity of the cube. Only the contoursassociated with the horseshoe vortex are ad hoc shown inFig. 6. The converging–diverging nature of the horse-shoe-type vortex tube can be observed and its cross-streamsize increases with the increase of x/h but does not exceed0.35h. The horseshoe vortex is detached from the cube, itsdistance from the side faces (zh) and its height (yh) increase.The increase rate is more intensive from xh = 3.11 toxh = 3.50 than from xh = 3.50 to xh = 3.91. At x/h = 3.91, close to the rear face, the vortex tube extendsuntil about zh = 0.79h and yh = 0.34h.

Fig. 7 depicts the position of the horseshoe vortex in theplane normal to the main flow direction as obtained both inexperiments and in our simulations. The Reynolds numberin the simulations Reh = 5610 differs from that in the exper-iments, Reh = 40,000. Moreover, we recall that our DNSwas carried out in a channel with a ratio of H/h = 3 whileH/h = 2 was used in the experiments of Hussein and Mar-

Fig. 6. Time-averaged streamwise vorticity, Xx, contour plots for differ

tinuzzi (1996). Nonetheless, the location of the horseshoevortex is quite similar. To explain this circumstance werefer to Castro et al. (1977) who found that global flowcharacteristics (e.g. the location of the horseshoe legs, thesize of the wake recirculation zone) depend on the bound-ary layer thickness of the oncoming flow. It appears thatthe fully developed inlet conditions used in our simulationsand experiments led to quite satisfactory agreement in pre-dicting the location of the horseshoe vortex. Krajnovic andDavidson (2001) performed LES where the Reynolds num-ber was the same as in the experiments (Reh = 40,000),however the inlet velocity profile was not fully developed.They, in contrast, found significant difference in the loca-tion of the predicted and experimentally observed horse-shoe vortex.

Fig. 8 shows the selected time-averaged streamlines. Themain vortical structures are clearly seen and are very simi-lar to the ‘‘artist’s impression’’ sketch of the flow structurepresented in Meinders et al. (1999). Fig. 9 demonstratesthat there is good agreement between the time-averaged

ent streamwise x/h-locations: x/h = 3.11 (a), 3.50 (b) and 3.91 (c).

Fig. 8. Time-averaged volume streamlines around a cube.

1002 A. Yakhot et al. / Int. J. Heat and Fluid Flow 27 (2006) 994–1009

streamwise velocity profile normalized by the bulk velocityand the experimental results of Meinders et al. (1999) onthe symmetry plane. The consistently larger magnitude ofthe velocity away from the channel wall may be a resultof the smaller channel-height-to-cube-height ratio H/h = 3 which were used in our calculations instead of 3.3as used in Meinders et al. (1999). The quantitative discrep-ancies in negative mean velocities close to the cube’s frontand rear faces are related to the aforementioned differencesin the horseshoe/arch vortical structures. In their experi-ments, Martinuzzi and Tropea (1993), Hussein and Mar-tinuzzi (1996) and Meinders et al. (1999) detecteddominant fluctuation frequencies sideways behind thecube. It is now well-established that the origin of thisunsteadiness is the interaction between the horseshoe vor-tex and the arch vortex that develops behind the cube.According to the kinematic principles formulated by Huntet al. (1978), this horseshoe vortex is sidetracked aroundthe edges of the cube into the downstream flow. Fig. 4shows the imprint of the arch vortex as it detaches fromthe shear layers shed from the upstream edges of the cube.The legs of the arch vortex lean on a pair of vortices orig-

Fig. 7. Time-averaged velocity field in the plane X/h = 4.75; (a) present,(b) Hussein and Martinuzzi (1996).

inating on the channel wall (marked by a focus point ‘‘B’’in Fig. 4). This focus point illustrates that the vortexentrains part of the fluid carried downstream by the horse-shoe vortex. Our DNS-based visualization confirmed theconclusion derived by Hunt et al. (1978), that the entrainedpart of the side fluid whirls upward, where it is shed intothe downstream flow. Unsteady RANS simulationsrecently performed by Iaccarino et al. (2003) agree withthe horseshoe-arch vortices interaction scenario describedby Shah and Ferziger (1997). They showed that the vortexpair forming the arch vortex base alternatively intensifiesby acquiring vorticity from the flow passing sideways.Owing to this mechanism, the arch vortex legs change theirlocation, which leads to a side-to-side oscillation. To con-firm this, a phase-averaging procedure of DNS resultsshould be applied, which has not been carried out duringthis study.

In our simulations, the dominant characteristic fre-quency in the rear wake was obtained using FFT analysisof the power spectral density (PSD) of the computed span-wise velocity. The spectrum was obtained from the datacollected during about 100 vortex shedding cycles at thepoint in the symmetry plane z = 0 with the coordinatesx/h = 6.0, y/h = 0.5 (see Fig. 3). Fig. 10 shows PSD ofthe squared spanwise velocity, where a dominant frequencyof 0.104 is clearly seen. This value corresponds to theStrouhal number St = fh/Um = 0.104 (based on the cubeheight h and the bulk velocity Um) and is consistent withthe experimental Strouhal numbers of 0.095 and 0.109reported in Meinders et al. (1999) and Meinders and Hanj-alic (1999), respectively.

4.2. Turbulence statistics

The governing equations were integrated in time until astatistically steady-state was reached. Then the mean flowand turbulence time-averaged statistical quantities wereobtained by further time-advancing and averaging. Theturbulence mean-square intensities (the normal Reynolds

X

U/U

m

0 2 4 6 8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

y/h=0.10y/h=0.31y/h=0.50y/h=0.69y/h=0.90Exp.y/h=0.1Exp.y/h=0.3Exp.y/h=0.5Exp.y/h=0.7Exp.y/h=0.9

Fig. 9. Time-averaged streamwise velocity in the symmetry plane; symbols: Meinders et al. (1999).

A. Yakhot et al. / Int. J. Heat and Fluid Flow 27 (2006) 994–1009 1003

stresses) u02, v02 and w02, kinetic energy K ¼ 0:5ðu02 þ v02þw02Þ, energy dissipation rate E ¼ m

ou0ioxj

ou0ioxj

, Reynolds stresses

ðu0iu0jÞ, and production ðP K ¼ u0iu0jSijÞ rate of turbulence

kinetic energy were computed using DNS data. The accu-mulating turbulence statistics and the time-averaged char-acteristics were performed over about 100 and 30 vortexshedding cycles for the mean-square intensities and sec-ond-order moments, respectively.

4.2.1. Turbulence mean-square intensities, kinetic energy,

dissipation rate

The contours of the mean-square intensities and kineticenergy dissipation rate in the symmetry plane are shown inFigs. 11a–c and 14, respectively. Our results qualitativelyreproduce the results reported in Meinders and Hanjalic

0.01 0.1 1 10Frequency

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

PS

D o

f a

span

wis

e ve

loci

ty

Fig. 10. Power spectrum of the spanwise velocity in the rear wake.

(1999). The maxima in the streamwise velocity fluctuationsare found in the horseshoe vortex (see the location X = x/h � 2.7 in Figs. 4 and 11a) and in the top vortices regionswhere the shear layers separate and reattach. Fig. 11cdepicts the spanwise mean-square intensities. The maximaof the spanwise fluctuations in front of the cube occur inthe recirculation region and in the vicinity of the front facewhere the oncoming flow impinges the cube. The difference

in the distribution of u02 and v02, w02 mean-square intensitiescomputed above the cube and behind it deserves somecomment. First, the distribution of v02 and w02 depicts cer-tain qualitative and quantitative similarities. Second, from

Fig. 11a, the turbulence streamwise mean-square intensities

u02, generated in the shear layer separated at the leading topedge, reach a maximum in the recirculation region at thetop and gradually weaken downstream. Fig. 11b and creveal the two maxima of the normal and spanwisemean-square intensities v02 and w02, respectively. Asidefrom the local maximum that occurs in the recirculationregion on the top, another occurs approximately atX = x/h = 5.2, Y = y/h = 0.75. This phenomenon can beattributed to the enhanced mixing in that region (seeFig. 2) and was also found experimentally by Husseinand Martinuzzi (1996).

Fig. 12 shows the time-averaged turbulent kinetic energyK ¼ 0:5ðu02 þ v02 þ w02Þ in the xz-plane for y/h = 0.1, 0.25,0.50 and 0.75 levels. An imprint in Fig. 12(a) indicatingthe enhancement of turbulence production inside the horse-shoe-type vortex is clearly seen. It should be noted that inthe side recirculation region, the distribution of the kineticenergy is quasi-two-dimensional (y-independent). FromFig. 12 and from the time-averaged streamlines shown inFig. 5, one may assume that the side recirculation regionis confined into a coherent quasi-two-dimensional vortex.The distribution of the dissipation rate presented inFig. 13 lends support to this assumption showing very weakdependency in the vertical y-direction in the lateral regions.

0.015

0.040.08

0.03

0.04

0.060.140.220.015

0.015

0.015

0.03

0.06

0.08

0.03

X

Y

2 2.5 3 3.5 4 4.5 5 5.5 6 6.50

0.5

1

1.5

2(a)

0.04

0.05

0.01

0.04 0.030.02

0.01

0.03

0.02

0.05

0.070.09

0.020.08

0.01

X

Y

2 2.5 3 3.5 4 4.5 5 5.5 6 6.50

0.5

1

1.5

2(b)

0.007

0.02

0.05

0.060.07

0.04

0.030.

020.01

0.070.08

0.01

0.0050.007

X

Y

2 2.5 3 3.5 4 4.5 5 5.5 6 6.50

0.5

1

1.5

2(c)

Fig. 11. Mean-square intensities in the symmetry plane Z = 0 plane; (a) u02, (b) v02, (c) w02.

1004 A. Yakhot et al. / Int. J. Heat and Fluid Flow 27 (2006) 994–1009

4.2.2. Reynolds shear stress

Fig. 15 shows Reynolds stresses u0v0 in the symmetryplane Z = 0. The contours depict the Reynolds stressenhancement in the recirculation region over the cube’stop face. The local maxima of u0v0 clearly observed in frontof the cube are discussed below. We performed a numericalexperiment in which, for the U and V mean velocity fieldsobtained from DNS, the Reynolds stresses are calculatedfrom the Bussinesq approximation �u0v0 ¼ mtðoU

oy þ oVoxÞ.

For mt ¼ 0:09K2=E, the modeled Reynolds stresses are

shown in Fig. 16. Figs. 15 and 16 depict a surprising qual-itative and quantitative similarity in the recirculationregion behind the cube. In front of the cube and on thetop, the difference is by a factor of about 2 and 3, respec-tively, due to overestimated turbulent viscosity.

4.2.3. Negative turbulence production

For complex flows around wall-mounted bluff bodies,one expects that in certain regions (such as those of stagna-tion) turbulent kinetic energy will exhibit a substantial

0.03

0.060.080.090.08

0.060.03

0.050.090.11

0.130.16

0.05

X

Z

2 2.5 3 3.5 4 4.5 5 5.5 6 6.50

0.5

1

1.5

2

(d)

0.08

0.050.06

0.04

0.090.130.150.18

0.040.05

0.06

0.11

0.07

5

0.065

X

Z

2 2.5 3 3.5 4 4.5 5 5.5 6 6.50

0.5

1

1.5

2

(c)

0.03

0.05

0.080.09

0.03

0.110.130.17

0.05 0.06

0.03

0.03

0.06

0.15

0.05

X

Z

2 2.5 3 3.5 4 4.5 5 5.5 6 6.50

0.5

1

1.5

2

(b)

0.04

0.050.06

0.050.06

0.08

0.09

0.04

0.080.130.17

0.04

0.11

0.03

0.030.07

X

Z

2 3 4 5 60

0.5

1

1.5

2

(a)

Fig. 12. Contour plots of the turbulent kinetic energy in the xz plane; y/h = 0.10 (a), 0.25 (b), 0.50 (c) and 0.75 (d).

A. Yakhot et al. / Int. J. Heat and Fluid Flow 27 (2006) 994–1009 1005

decay. Therefore, special concern was given to detect thoseregions where turbulence production is negative. Negativeproduction points to states that are not in equilibrium

and indicates the reverse energy transport from small tolarge scales. This latter phenomenon is an intrinsic featureof essentially anisotropic flows. For such flows, the energy

Fig. 13. Contour plots of the dissipation rate E in the yz-plane, x/h = 3.5.

1006 A. Yakhot et al. / Int. J. Heat and Fluid Flow 27 (2006) 994–1009

of small scales tends to accumulate into large-scale coher-ent vortices that dominate the flow. In subgrid scale mod-eling, reverse transport is known as ‘‘backscattering’’ and isof importance for developing LES approaches (Piomelliet al., 1991). Krajnovic and Davidson (1999) predicted that

Fig. 14. Contour plots of the dissipation

-0.005

-0.05-0.07

-0.04

X

Y

2 3 40

0.5

1

1.5

2

Fig. 15. Reynolds stress u0v0 in

the strongest ‘‘backscattering’’ of energy takes placenear the front vertical edges of a cube. In Fig. 17, we showthe iso-surfaces and contours of negative productionobtained in front of the cube in our calculations. To esti-mate the individual contributions of the Reynolds stressesto production, we rewrite PK in the following form:

P K ¼1

2ðP uu þ P vv þ P wwÞ ð2Þ

P uu ¼ �2 u02oUoxþ u0v0

oUoyþ u0w0

oUoz

� �ð3Þ

P vv ¼ �2 u0v0oVoxþ v02

oVoyþ v0w0

oVoz

� �ð4Þ

P ww ¼ �2 u0w0oWoxþ v0w0

oWoyþ w02

oWoz

� �ð5Þ

The contours of negative values of PK, Puu, Pvv and Pww areshown in Fig. 18a–d. In this region, the oncoming flow im-pinges on the wall, the flow rapidly decelerates and oU/ox

is negative. The absolute value of u02oU=ox is much greaterthan u0v0oU=oy and u0w0oU=oz. As a result, Puu is positive,as can be seen in Fig. 18b where only contours of negativePuu are shown. On the other hand, owing to the incom-pressibility constraint, negative oU/ox yields that oV/

rate E in the symmetry Z = 0 plane.

-0.01

-0.0

2

-0.01

-0.0

3

-0.005

-0.005

-0.025-0.02

5 6 7

the symmetry plane Z = 0.

-0.015 -0.01

-0.005

-0.02

-0.02

-0.03

-0.06-0.1-0.2

-0.005-0.025

-0.005-0.01

-0.01

-0.005

-0.15

-0.1

X

Y

2 3 4 5 6 70

0.5

1

1.5

2

Fig. 16. The modeled Reynolds stresses u0v0 in the symmetry Z = 0 plane; �u0v0 ¼ mtðoUoy þ oV

oxÞ; mt ¼ 0:09K2=E, K ¼ 0:5ðu02 þ v02 þ w02Þ.

Fig. 17. (a) Iso-surfaces of PK = �0.03 and (b) contours of negative kinetic energy production in the symmetry Z = 0 plane.

A. Yakhot et al. / Int. J. Heat and Fluid Flow 27 (2006) 994–1009 1007

oy > 0 and oW/oz > 0. This explains why Pvv and Pww be-come negative near the front wall (see Fig. 18c and d).However, from Fig. 18a, the maximum negative value ofPK = �0.07 appears just ahead of the cube, near the chan-nel wall, at X = x/h = 2.82, Y = y/h = 0.035. This point isin the vicinity of the saddle point marked by ‘‘A’’ inFig. 4. The situation here is quiet different from that justdescribed. Immediately in front of the cube, below the stag-nation streamline marked by ‘‘M’’ in Fig. 2, the flow turnsdownward and oV/oy becomes negative near the channelwall. In turn, oU/ox becomes positive. Furthermore, theterms u0v0oU=oy and u0w0oU=oz are relevant only withinthe shear layer, which means that P uu � �2u02oU=ox < 0.Fig. 18b shows that mainly the Puu contributes to PK

becoming negative. To explain why Pww is negative, wemust turn to Figs. 4, 2 and 8, which depict, with notableagreement, the flow features predicted analytically by Huntet al. (1978) and observed later experimentally by Martin-uzzi and Tropea (1993), Hussein and Martinuzzi (1996)and Meinders et al. (1999). In the region, marked by ‘‘A’’in Fig. 4, the separation streamline forms a saddle point(Sa in Fig. 4) and the flow is sidetracked forming a horse-shoe vortex (Fig. 8). This sideward detachment of a spiral

flow implies positive oW/oz, which results in negative Pww.Positive oU/ox and oW/oz imply negative oV/oy and, as aresult, positive Pvv (not shown in Fig. 18c, where only thecontours of negative Pvv are shown).

The probability distribution of the u 0 and w 0 velocityfluctuations at x/h = 2.82, y/h = 0.04, z/h = 0 show Gauss-ian distribution. The probability distribution of the verticalfluctuation v 0 is non-Gaussian with Kurtosis Ku = 10instead of the Gaussian value of 3. The origin of the highKurtosis can be related to events associated with large-scale instability that appear in this region as the result ofnegative turbulence production. Based on the above obser-vations, we suggest that flow dynamics in front of the cubeis strongly affected by negative energy production.

5. Concluding remarks

Direct numerical simulations were performed on flowaround a wall-mounted cube in a fully developed channelflow by using an immersed-boundary method. To minimizeany uncertainties related to imposed boundary conditions,we simulated the case of a wall-mounted cube in a fullydeveloped channel flow. Instantaneous results of the

Fig. 18. Contour plots of negative production in front of the cube in the symmetry z = 0 plane; (a) PK, (b) Puu, (c) Pvv, (d) Pww.

1008 A. Yakhot et al. / Int. J. Heat and Fluid Flow 27 (2006) 994–1009

DNS of a plain channel flow were used as a fully developedinflow condition for the main channel. The Reynolds num-ber, based on the bulk velocity and the channel height, was5610. The results confirmed the unsteadiness of the consid-ered flow caused by unstable interaction of a horseshoevortex formed in front of the cube and on both its sideswith an arch-type vortex behind the cube. This unsteadi-ness leads to the vortex shedding phenomenon. The domi-nant characteristic frequency in the rear wake was detectedusing FFT analysis of the computed spanwise velocity andagrees with that found experimentally. The time-averagedglobal flow pattern and characteristics (a horseshoe vortex,an arch-type vortex behind the cube, the length of the recir-culation regions) were found to be in good agreement withexperimental results.

We presented DNS-based data of the following turbu-lence statistics: mean-square intensities, Reynolds shearstresses, kinetic energy and dissipation rate. We reportedresults on negative turbulence production obtained in theregion in front of the cube (where the main horseshoe vor-tex originates). The negative production points to statesthat are not in equilibrium and indicates the reverse energytransport from small to large scales which is an intrinsicfeature of essentially anisotropic flows. The occurrence ofnegative production in front of the cube can partiallyexplain the failure of some LES/RANS simulations to pre-dict flows around a wall-mounted cube.

Immersed-boundary-based approaches differ by themethods used to introduce an artificial force into the gov-erning equations. The direct forcing method that we usedintroduces an error of O(Dt2) in the no-slip boundary con-dition. Implementation of immersed-boundary methods tosimulate turbulent flows requires careful monitoring toavoid possible contamination of numerical results arisingfrom inaccurate boundary conditions. One may expect thatan error in imposing boundary conditions is unavoidablytranslated into erroneous predictions. In general, our com-putations are in good agreement with the experimentalfindings.

In conclusion, our results lend support to the recenttrend to employ immersed-boundary methods formulatedon rectangular meshes as a tool for simulating complex tur-bulent flows.

Acknowledgements

This work was supported by the Israel Science Founda-tion Grant 159/02 and in part by the CEAR of the HebrewUniversity of Jerusalem. The third author (NN) was alsopartially supported by the Russian Foundation for BasicResearch under Grant 02-01-00492. The first author (AY)wishes to express his appreciation to Profs. S. Sukorianskyand V. Yakhot for valuable discussions. Our special grati-tude goes to the IUCC – the Inter-University Computation

A. Yakhot et al. / Int. J. Heat and Fluid Flow 27 (2006) 994–1009 1009

Center, Tel Aviv, where the numerical simulations havebeen carried out on Cray SV1 supercomputer.

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