Large-eddy simulations of ducts with afree surface

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1 J. Fluid Mech. (23), vol. 484, pp c 23 Cambridge Universit Press DOI: 1.117/S Printed in the United Kingdom 223 Large-edd simulations of ducts with afree surface B RICCARDO BROGLIA 1, ANDREA PASCARELLI 1 AND UGO PIOMELLI 2 1 INSEAN, Via di Vallerano 139, 128 Roma, Ital 2 Department of Mechanical Engineering, Universit of Marland, College Park, MD 2742, USA (Received 27 March 22 and in revised form 7 Januar 23) This work studies the momentum and energ transport mechanisms in the corner between a free surface and a solid wall. We perform large-edd simulations of the incompressible full developed turbulent flow in a square duct bounded above b a free-slip wall, for Renolds numbers based on the mean friction velocit and the duct width equal to 36, 6 and 1. The flow in the corner is strongl affected b the advection due to two counter-rotating secondar-flow regions present immediatel below the free surface. Because of the convection of the inner edd, as the free surface is approached, the friction velocit on the sidewall first decreases, then increases again. Asimilar behaviour is observed for the surface-parallel Renolds-stress components, which first decrease and then increase again ver close to the surface. The budgets of the Renolds stresses show a strong reduction of all terms of the dissipation tensor in both the inner and outer near-corner regions. The exhibit a reduction in both production and dissipation towards the free surface. Ver close to the solid boundar, within 15 2 viscous lengths of the sidewall, the turbulent kinetic energ production and the surface-parallel fluctuations rebound in the thin laer adjacent to the free surface. The Renolds-stress anisotrop appears to be the main factor in the generation of the mean secondar flow. The multi-laer structure of the boundar laer near the free surface is also discussed. 1. Introduction Mixed-boundar corner flows occur at the juncture of a no-slip wall and a free surface. The can be found in a variet of practical applications: the flow around hulls of ships is perhaps the most common example of this configuration, which also occurs on the sidewall boundar of an open channel or of a partiall filled container, and in rivers. An important feature of these flows is the interaction of wall turbulence with a free surface. When turbulence interacts with either a free surface or asolid wall, it becomes highl anisotropic. The character of the anisotrop, however, is substantiall different in the two cases. A free surface cannot support mean shear, and restricts motion in the direction normal to the surface onl, while a solid wall forces all the velocit components to vanish at the boundar. The no-slip condition at a solid wall makes turbulence production and dissipation significant there, whereas at a free surface turbulence production is negligible and the dissipation rate is smaller than in the bulk of the flow because of the vanishing of the velocit gradients. Near amixed-boundar corner, the interaction of the eddies generated b these different

2 224 R. Broglia, A. Pascarelli and U. Piomelli mechanisms creates additional phsical complexities (compared either with the free surface or the solid wall alone) that affect the transport of mass, momentum and energ. In order to predict the flow fields in the applications mentioned above, better understanding of their phsics is required. The characteristics of the flow near a solid flat surface are well known and will not be discussed here. Descriptions of the statistical aspects of the flow and of the turbulent eddies can be found in man textbooks and reviews (see, e.g. Pope 2; Robinson 1991). The structure of the wall turbulence, however, is significantl altered in the corner between two solid walls, where the anisotrop of the Renolds stresses generates turbulence-driven secondar flows in the plane perpendicular to the streamwise direction, commonl referred to as secondar flows of Prandtl s second kind. In straight square ducts, the secondar flow is directed from the centre of the duct toward the corners along the corner bisectors. Although the magnitude of these secondar velocities is extremel small (of the order of 2 3% of the bulk streamwise velocit), the distortion of the axial flow can alter the distribution of the friction coefficient (and thus the mean flow). Because of the complex phsics coupled with a simple geometr, these flows have been studied extensivel, both experimentall and numericall. While the experiments (Launder & Ying 1972; Melling & Whitelaw 1976; Gessner, Po & Emer 1979) were the first to quantif the mean flow, numerical calculations (Madabhushi & Vanka 1991; Gavrilakis 1992; Huser & Biringen 1993; Huser, Biringen & Hata 1994; Su & Friedrich 1994) have been instrumental in addressing questions related to turbulent kinetic energ (TKE) production and transfer, as well as to the modifications to the turbulence structure in these configurations. Huser et al. (1994) investigated the details of the Renolds stress budgets and performed quadrant analsis to examine the dominant structures which give rise to the generation of secondar flows. The observed that the dominant ejections contain two streamwise counter-rotating vortices. Modifications to the turbulence structure occur also when turbulence interacts with afree surface, where the relative velocit between the fluid and the surface interface vanishes, the tangential stresses are ero, and the normal stress must balance the ambient pressure. In addition, in the free-surface case, the boundar can deform. If the free surface can be considered flat, the tangential vorticit vanishes at the free surface but not the component normal to the surface, while in the case of a shearfree deformed free surface, the vorticit at the free surface is non-ero and the surface acquires a solid-bod rotation. The non-dimensional groups that appear in these flows are the Renolds number, the Froude number and the Weber number, defined as Fr = u r / gl r and We = ρl r u 2 r/σ s,respectivel, where g is the acceleration due to gravit, σ s the surface tension, and u r and l r are a reference velocit and length. The Froude number, which is related to the deformation of the free surface, is the most important one for the present stud; low Froude numbers correspond to negligible free-surface deformations. Perhaps the main feature of the interaction of turbulence with the free surface is the modification of the inter-component energ transfer, which has been observed b virtuall all the workers who have examined this problem. In the boundar laer adjacent to the free surface, the fluctuations parallel to the surface are essentiall unaltered, while the Renolds-stress component normal to the surface is redistributed into surface-parallel components, which increase with decreasing distance from the free surface. In the case of a free-slip flat surface, a picture of the vortex/surface interactions on the dnamics of turbulence under the free surface emerges from the

3 Large-edd simulations of ducts with a free surface 225 direct numerical simulation (DNS) studies of Perot & Moin (1995), Walker, Leighton & Gara-Rios (1996) and Nagaosa (1999). Acomplete discussion of the multi-laer structure of free-surface viscous flow is not appropriate here (for a more complete review see Shen et al. 1999), but a few significant results should be described. As shown originall b Hunt & Graham (1978), atwo-laer model can describe the interaction between a shear-free boundar and turbulence. The laer through which the kinematic boundar condition is felt is defined as the blockage or source laer, whereas the thin one near the free surface due to the viscous dnamic conditions is called the surface laer. The blockage laer, which is due to the ero normal-velocit condition at the free surface, obtains for an flow with a boundar constraining the normal motion; redistribution of turbulent intensities takes place here. In the surface laer, the velocit derivatives are highl anisotropic, and the values of the surface-parallel vorticit change from outer values to smaller values induced b the boundar conditions. The important scales in the source laer are the integral scales of the turbulence far below the boundar, while the viscous laer is characteried b velocit and length scales determined b its own internal dnamics. When a turbulent shear flow interacts with a free surface, additional complexities appear. Further investigations of the surface laer can be found in the work of Anthon & Willmarth (1992) and Walker (1997), who associated the origin of the current to the near-surface turbulent-stress anisotrop. When the surface deforms, the Renolds-stress anisotrop is smaller and the free-surface fluctuations make a significant contribution (Hong & Walker 2). Turbulent channel flows bounded b a rigid wall below and an open free surface above have been investigated experimentall (Nakagawa & Neu 1981; Komori et al. 1982; Neu & Rodi 1986) and numericall (Lam & Banerjee 1992; Leighton et al. 1981; Handler et al. 1993; Pan & Banerjee 1995). In this configuration, the turbulence produced near the solid wall owing to the high shear can be convected to the surface and interact with it, a situation that occurs, for instance, in rivers. While the investigations cited above approximated the free surface as flat, other studies emploed more realistic approaches (e.g. Komori et al. 1993; Borue, Orsag & Staroselsk 1995; Tsai 1998). However, in the cases considered, the interfacial deformations and the normal velocit at the free surface remained extremel small, so that no significant deviations from the flat-surface case were observed. In all the free-surface/turbulence interaction studies discussed so far, the flow is homogeneous in the spanwise (surface-parallel) direction. In this case, several workers have inferred the quasi-two-dimensionaliation of turbulence structure near the free surface (Sarpkaa & Suthon 1991; Sarpkaa & Neubert 1994; Pan & Banerjee 1995; Kumar, Gupta & Banerjee 1998). Turbulent shear flows bounded b a vertical wall and a horiontal free surface ma differ significantl from those. In the region close to the walls or in other situations where turbulence is being generated, for instance, the edd structure will certainl be markedl three-dimensional, even near a free surface. The existence of mean secondar flows in the corner (similar to those that occur in the closed ducts) adds complexit to the problem. Studies of turbulent flows in the corner formed b a vertical solid wall and a free surface have been performed, among others, b Grega et al. (1995), Longo, Huang & Stern (1998), Sreedhar & Stern (1998), Hsu et al. (2) and Grega, Hsu & Wei (22). Grega et al. (1995) carried out numerical simulations in conjuction with experimental flow visualiation and single-component laser-doppler anemometr (LDA) velocit measurements. The Renolds number of the experimental investigation was Re θ = 115 (based on the momentum thickness θ), while the numerical

4 226 R. Broglia, A. Pascarelli and U. Piomelli simulation was performed at Re θ = 22. In their work, the presence of the mean inner and outer secondar flow regions was highlighted. While the inner secondar region has a parallel in the flow in a corner formed b two solid walls, the outer secondar flow region does not have an analogue in the wall-bounded corner flow. The inner secondar vortex convects fluid from the free surface toward the corner, surrounded b the outer flow, which convects flow from the sidewall to the free surface. In Grega et al. (1995), the inner secondar flow was ver weak, and could not be observed b flow visualiation; it was, however, resolved in the computations. Longo et al. (1998) examined the flow in the corner formed b a towed surface-piercing plate and a free surface at rest. Three-component laser-doppler velocimetr (LDV) measurements for both boundar laer and wake were also used to obtain scaling properties for the anisotrop of the normal Renolds stresses. However, in this case also, the marginal measurement resolution did not permit visualiation of the inner secondar vortex motion. Sreedhar & Stern (1998) carried out large-edd simulations (LES) of compressible temporall developing solid/solid and solid/rigid-lid corner flows. In these simulations, in which the inner secondar flow was resolved, the TKE increased near the rigid-lid boundar and a redistribution of energ from the surface normal components to the other two was also observed in this region. There is some controvers on this issue, since Longo et al. (1998) and Sreedhar & Stern (1998) predict an increase of mean-square streamwise velocit fluctuations as the distance from the free surface decreases, while Grega et al. (1995), in their experiments, measure an increase in this quantit. Similar discrepancies are found in the behaviour of the anisotrop-tensor profiles. As an extension of the work of Grega et al. (1995), Hsu et al. (2) studied the TKE transport mechanisms in the mixed-boundar corner using high-resolution digital particle image velocimetr (DPIV) and LDV measurements. The found that both TKE production and dissipation are dramaticall reduced close the free surface. Some of the conclusions of Longo et al. (1998) and Sreedhar & Stern (1998) also appear to be in conflict with the results obtained b the follow-up experimental stud of the same problem b Hsu et al. (2), suggesting that the dnamics of boundarlaer/free-surface juncture have not been perfectl elucidated. Furthermore, certain issues were not full addressed owing to limitations of experimental investigations, for example the resolution of the flow ver near the free surface. The origin of the inner secondar flow in the context of vorticit transport, the role plaed b the diverging surface current found when studing jets, wakes or boundar laers parallel to afree surface (Walker 1997), the vortex structures in a turbulent mixed-boundar corner, are examples of the questions not et conclusivel investigated. Man of the discrepancies are probabl due to different set-ups in the different studies, as also mentioned b Grega et al. (22). High-resolution DPIV measurements made in the cross-stream plane b Grega et al. (22) using the same experimental apparatus as two earlier works b the authors, pointed out that there is an, as et, undetermined source of streamwise vorticit particularl in the outer secondar flow region close to the free surface. The objective of this work is to address some of the issues mentioned above and resolve some of the discrepancies between previous studies b performing LES of the flow inside a square duct bounded b a gas liquid interface at the top. In particular, we will computationall provide a complete picture of the secondar flows. The features of these flows will be related to the presence of a free-surface current, and inter alia the effects on turbulence statistics of the peculiar boundar-laer structure will be investigated. After having demonstrated the accurac of the numerical tool,

5 Large-edd simulations of ducts with a free surface 227 we will stud the inner secondar flow in the context of the streamwise budget of momentum equation and vorticit transport; then, we will attempt to provide adetailed description of the turbulent energ transport mechanisms in the mixedboundar corner. The use of LES (rather than DNS) allows us to perform calculations at three Renolds numbers, Re τ = 36, 6 and 1 (based on the mean friction velocit and the duct width). The paper is organied as follow: in the next section, the problem formulation, numerical method and subgrid-scale model used are briefl described. Then, the results of a closed-duct calculation will be presented to validate the code. Discussion of the mean flow, Renolds stress and vorticit statistics will follow. Finall, some conclusions will be drawn in the last section. 2. Problem formulation 2.1. Governing equations and numerical method The governing equations for LES are obtained b the application of a spatial filter to the Navier Stokes equations to separate the effects of the (large) resolved scale from the (small) subgrid-scale (SGS) eddies; in the case of an incompressible viscous flow in presence of bod forces, the can be written in the following form: u i t + x j (u j u i )= p x i + 1 Re τ 2 u i τ ji x j + f i, u j x j =, (2.1) where the overbar denotes filtered variables and the effect of the subgrid scales appears through the SGS stresses τ ij = u i u j u i u j ; u j is the fluid velocit component in the j-direction and p is the pressure divided b the constant densit. Equations (2.1) are non-dimensionalied using the mean friction velocit, u τ,andtheductwidth, D, as characteristic velocit and length scales, respectivel. Thus, the Renolds number is defined as Re τ = u τ D/ν, whereν is the kinematic viscosit. The flow is driven b aconstantbod force per unit mass, f 1.Periodic boundar conditions are used in the streamwise directions and no-slip boundar conditions are imposed on walls. The boundar conditions enforced on the top surface of the computational domain are either the no-slip conditions or the no-stress boundar conditions u =, u =, v =, w =, (2.2) v =, w =. (2.3) Despite the limitations of these assumptions, previous numerical studies have shown that a rigid free-slip wall approximation allows us to predict man of the phenomena seen in experiments with waveless interfaces (Lam & Banerjee 1992; Leighton et al. 1981; Handler et al. 1993; Pan & Banerjee 1995). In the present work, the SGS stresses τ ij are parameteried b an edd-viscosit model of the form: τ ij 1 3 δ ij τ kk = 2ν T S ij = 2C 2 S S ij, (2.4) where δ ij is Kronecker s delta, S = (2S ij S ij ) 1/2 is the magnitude of the largescale strain-rate tensor S ij =( u i / x j + u j / x i )/2, and =2( x ) 1/3 is the filter width. Closure of the SGS stresses τ ij is obtained through specification of the

6 228 R. Broglia, A. Pascarelli and U. Piomelli model coefficient C appearing in (2.4). In the present work, the dnamic procedure proposed b Germano et al. (1991) and Lill (1992) is used to determine the eddviscosit coefficient C. Theconstant C is averaged along the homogeneous streamwise direction; this tpe of averaging is effective in removing spurious fluctuations of C which tend to destabilie the calculations, and has been used in several calculations of turbulent flows that are homogeneous in one direction onl (Akselvoll & Moin 1996; Mittal & Moin 1997), with accurate results. The sum of the laminar and edd viscosit is set to ero wherever it becomes negative. The numerical approach emploed for the solution of (2.1) is the fractional step method (Chorin 1967; Kim & Moin 1985). The time-advancement for the intermediate advection-diffusion step is performed in a semi-implicit fashion with a two-substep second-order accurate Runge Kutta algorithm. This is followed b the pressurecorrection step, which enforces the divergence-free condition; the Poisson equation forthe pressure-like function is solved once per time step. All the spatial derivatives are approximated using a second-order-accurate central discretiation on a non-uniform collocated grid. Solution of the Poisson equation is obtained b means of a successivel over-relaxed Gauss Seidel method; a multigrid algorithm of a correction-scheme tpe is used to accelerate the convergence toward the solution. Implicit treatment of the diffusive terms allows a larger time step, which is onl limited b the explicit treatment of the advective and SGS terms. The overall accurac of the method is second order in time and space. Paralleliation of the code is achieved b an effective domain decomposition technique; the domain is divided into a number of subdomains along the streamwise direction. Communications between processors are made b using the message-passing programming and the send, receive and reduce statements of the message-passing librar Message-Passing Interface (MPI). This algorithm is described in detail in Broglia, Di Mascio & Muscari (1999) Geometr and grid parameters The Navier Stokes equations are solved numericall in a rectangular domain of sie 2πD D D in the x-, - and-directions, respectivel. The streamwise length of the domain was chosen based on the two-point velocit correlations computed b Huser & Biringen (1993) to ensure that the domain is large enough to contain the longest structure present in the flow. The phsical domain is discretied using between grid points for the low- Renolds-number simulation, and grid points for the high-renolds-number one. All the discretiations are uniform in the streamwise (x) direction, whereas in the spanwise and normal directions ( and, respectivel) points are clustered towards the walls; in particular, for all the simulations, the first point close to the wall is placed at + or + =.5 andatleast 13 points are in the near-wall region ( + or + < 1). Points are also clustered toward the slip wall, with a minimum grid spacing of about 1.5 wall units for the high Renolds numbers and.5 for the lowest one. The problem geometr is sketched in figure 1, and the mesh characteristics are given in table 1. The equations were integrated in time until a statistical stead state was reached (the stead state was determined b monitoring the time histor of the total wall shear stress). After that, data for the statistics were collected for several large-edd turnover times (LETOTs) D/u τ.streamwise homogeneit and smmetr about the plane =.5D were used to increase the sample sie. The sampling interval for each case (also in LETOTs) is reported in table 2. The relevant non-dimensional parameters

7 Large-edd simulations of ducts with a free surface Points in Simulation nx n n x + min max min max + or + < 1 Closed duct, Re τ = Closed duct, Re τ = Open duct, Re τ = Open duct, Re τ = Open duct, Re τ = Table 1. Grid characteristics. Simulation Sampling time t U b U c Computed u τ Re b Duct Re τ = Duct Re τ = Open duct Re τ = Open duct Re τ = Open duct Re τ = Table 2. Averaging and global flow characteristics. U c is the maximum velocit in the duct. Solid wall L = 1 L = 1 Free-slip wall L = 1 L = 1 Lx = 2π Lx = 2π x x Figure 1. Sketch of the problems. are the Renolds number based on the friction velocit u τ and the duct width D, Re τ = u τ D/ν, and the bulk Renolds number defined using the bulk (area-averaged) streamwise velocit U b, Re b = U b D/ν. Intable 2, the Renolds numbers and some global results of the present simulations are shown. The value of u τ obtained from the calculations is within.5% of the nominal value u τ =1.Thisdifference supplies an estimate of the sampling error for the first- and second-order statistics, which was, respectivel,.5% and 1%. In the following discussions, the angular brackets,, denote an average over time and the homogeneous direction, whereas a double prime denotes fluctuation with respect to mean resolved quantit, q = q q ; thus, the resolved quantities are decomposed into mean values and resolved fluctuations: u i = u i + u i, p = p + p, τ ij = τ ij + τ ij. (2.5)

8 23 R. Broglia, A. Pascarelli and U. Piomelli 2 15 u u + = + u + = 2.5 ln ( + ) τ w Figure 2. Duct flow. Mean streamwise velocit profiles along the wall bisector; wall stress distribution along the sidewall., Re τ = 36;, Re τ = 6;, DNS at Re τ = 3 (Gavrilakis 1992);, DNS at Re τ = 6 (Huser & Biringen 1993). u rms v rms.5 w rms (c) Figure 3. Duct flow. Root mean square velocit fluctuations along the wall bisector. u; v; (c) w., Re τ = 36;, Re τ = 6;, DNS at Re τ = 3 (Gavrilakis 1992);, DNS at Re τ = 6 (Huser & Biringen 1993). Thus, the averaged resolved velocit and turbulent stresses are denoted b u i and u i u j, respectivel. 3. Results and discussion 3.1. Validation: closed duct flow To determine the accurac of the present numerical method in configurations similar to the open duct, LES of full developed turbulent flow in a closed square duct were performed and compared with the available DNS data. The current LES solutions at Re τ = 36 and 6 are compared with low-renolds-number DNS solution b Gavrilakis (1992) at Re τ = 3 (Re b = 441) and with those b Huser & Biringen (1993) at Re τ = 6 (Re b =132). Mean turbulent statistics are shown in figures 2 and 3. Here and in the following, unless otherwise stated, all quantities are normalied using the mean u τ and D. The agreement with the DNS data is good. In particular, the overshoot in the logarithmic laer (figure 2a) observed in the DNS is captured quite well. The main topological features of the flow, such as the secondar vortices often observed in the corner region, are also captured well. Gavrilakis (1992) performed a DNS using a second-order finite difference on a staggered grid with up to grid points ( ) and a

9 Large-edd simulations of ducts with a free surface (c) (d) 1..5 (e) 1..5 ( f ) Figure 4. Open duct flow. (c) Mean streamwise velocit contours and secondar velocit vectors. (d) (f )Vorticitcontours., (d) Re τ = 36;, (e) Re τ = 6; (c), (f ) Re τ = 1. box length of l x =1π. Ourcalculation uses approximatel one fifth of the resolution of this DNS (47,45 points are used to discretie a domain that is one fifth the length of that of Gavrilakis 1992). Our grid resolution, on the other hand, is comparable to that of the DNS b Huser & Biringen (1993), which used grid points with a spectral/high-order finite-difference scheme and a time-splitting integration method Open duct: mean velocit Figure 4 shows mean streamwise-velocit contours and cross-stream velocit vectors in the cross-stream (, )-plane for the three Renolds numbers considered here. In the figures, the free surface is at the top, and onl half of the domain is shown. In the lower corner, the flow is ver similar to that observed in closed square ducts: a turbulencedriven secondar flow, consisting of a streamwise counter-rotating vortex pair deforms the isotachs b convecting high-momentum fluid from the central region of the duct towards the corner region along the corner bisector. Contours of the resolved mean streamwise vorticit ω x = w / v / show positive and negative extrema at the vertical and horiontal walls because of the no-slip conditions. In the centre of the larger flow-cells the vorticit attains extreme values locall. In the upper corner, on the other hand, the effects of the free-shear surface become important. The main feature of the flow field near the mixed corner is the existence of an inner and an outer mean secondar flow, as observed in the unbounded corner flow studied b Grega et al. (1995). The inner secondar region at the mixed corner consists of a

10 232 R. Broglia, A. Pascarelli and U. Piomelli 2 u + = + u + = 2.5 ln ( + ) u u τ Figure 5. Open duct flow. Mean streamwise-velocit profiles in wall units along the bottom-wall bisector; frictionvelocit distributions along the sidewall., Re τ = 36;, Re τ = 6;, Re τ = 1;, u τ obtained at Re τ = 6 from the Clauser plot. weak vortex that convects momentum towards the corner itself, along the free-slip boundar, whereas the outer secondar vortex is responsible for transport of lowspeed momentum from the sidewall boundar-laer, towards the free surface. The scale and strength of the outer secondar vortex are larger than those of the other vortices appearing in the cross-plane view. The maximum value of the secondar velocit is about 3% of the streamwise velocit, slightl greater than that observed in the closed duct case. Furthermore, the contours of mean streamwise vorticit show adirect correspondence with the secondar flow. A small increase of the secondarflow intensit with the Renolds number is observable. The flow behaviour near the sidewall in the region around =.5 shows a transition between the corner and the free-surface behaviour, characteried b a weaker counter-clockwise vortical motion. This vortex is prevented from growing to dimensions and strength comparable to the corner edd b the presence of the two surface vortical structures, but its effects on the flow field are important. In particular, it thickens the wall laer in this region, affecting the wall stress distribution and the momentum transport. Such a vortical region was not observed b Grega et al. (1995), suggesting that is driven b the anisotropic turbulence generated b the bottom no-slip wall. In figure 5, the mean streamwise-velocit profiles in logarithmic scale along the bottom-wall bisector ( =.5) are presented for the three Renolds numbers considered. The satisf the logarithmic law ver well, and maintain the log-law behaviour until ver close to the free surface. Compared with the closed-duct case, we do not observe the overshoot of the logarithmic laer intercept. Since the grid resolution for the closed- and open-duct calculations was the same, this can be interpreted as a results of slightl weaker turbulence production near the walls awa from the duct corners, compared with the companion closed-duct flow. The mean velocit profiles exhibit a maximum near the free surface. In the region.8 </D<1 a one of negative mean shear u / is observed (see figure 6), owing to the augmentation of the vertical descending convective transport caused b the outer secondar flow as the free surface is approached. The centre of the duct is the merging point of two opposite diverging currents generated at the sidewalls (Walker 1997). As reported in Shen, Triantafllou & Yue (2) for an unbounded shear flow, this is a first indicator of the boundar-laer structure near a free surface, whose maximum and minimum mark the ends of the two laers at the interface. Secondar flows clearl appear to have a significant effect on the region where these events

11 Large-edd simulations of ducts with a free surface FS d u + /d (c) d u + /d + d u + /d + Figure 6. Open duct flow. Mean shear profiles normalied with u 2 τ /ν. FS + is the distance from the free surface in wall units (the reference u τ is used in this normaliation). Re τ = 36, Re τ = 6, (c) Re τ = 1., + =3;, + = 15;, + = 3;, + = Re τ /2. occur. Note that around the demarcation between the inner and outer secondar region (+ 4) a behaviour similar to that observed in shear-free unbounded flow is recovered, and that the sie of the surface laer exhibits little Renolds-number dependence. In figure 5, the distributions of the mean friction velocit along the sidewall are shown. Grega et al. (1995) and Hsu et al. (2) reported experimental data that show a thickening at the free-surface edge of the turbulent boundar laer on the vertical wall as a result of the free-surface current. Consistent with these results, the friction velocit from the present computations decreases as the free surface is approached. The present computations, however, also show an increase of u τ ver near the surface, again due to the inflow from the inner secondar cell, which convects high-speed momentum toward the sidewall at the free surface. This result is consistent with the numerical results presented in Grega et al. (1995) and indicates a reduction of the boundar-laer thickness in the vicinit of the free surface, which was not observed experimentall. Two causes can be the origin of the discrepanc between numerical and experimental data. One is the use of the Fr = approximation, which is least accurate in the corner region. However, the experiments of Grega et al. (1995) and Hsu et al. (2) were conducted at a low reference Froude number, so that waves and surface deformations are not expected to have a significant effect on the interaction of the free surface with the turbulence-driven secondar flows developed near the corner. The second, and perhaps most significant, factor relates to the method used in the experimental studies to determine the skin friction. Friction velocities were determined experimentall from the mean streamwise-velocit data measured at two different distances from the surface (6 and 58 viscous lengths below the free surface, the latter profile corresponding to the tunnel centre). From each individual mean velocit profile, u τ and the wall location were determined b fitting to the Clauser plot for a canonical turbulent boundar laer. In the corner region, however, the flow deviates significantl from the standard logarithmic behaviour. In figure 7, the velocit profiles and distances from the wall are normalied using the local value of u τ.near the free surface, the slope of the logarithmic region is significantl lower; here, friction velocities determined using the Clauser plot are not accurate. This is shown in figure 5 inwhichthesolid dots represent the friction velocit calculated b requiring that the velocit profile satisf the logarithmic region at + = 1. Note

12 234 R. Broglia, A. Pascarelli and U. Piomelli 2 16 u l l Figure 7. Open duct flow, Re τ = 6. Mean streamwise-velocit profiles in wall units for different distances from the free surface:, =1;, =.98;, =.95;, =.8;, =.6. further that Stern et al. (1995) observe a secondar flow near the free-surface edge of a turbulent boundar laer on a vertical wall. The rotation of the secondar flow was such that the boundar laer was thinner as the free surface was approached. The secondar flow was explained in terms of the anisotrop in the Renolds stresses. Increasing the Renolds number results in a thickening of the sublaer on the lower wall, but has ver little effect on the region close to the free surface Open duct: Renolds stresses Figure 8 shows the cross-stream spatial distribution of the Renolds stresses in the Re τ = 6 case. In the present simulations, the SGS stresses are an order of magnitude smaller than the resolved ones (at least awa from the walls), and the large-scale quantities shown in the following represent the contribution from most of the significant turbulent motions. Near the upper corner, the normal Renolds stresses distributions are significantl different from the lower corner region. The streamwise normal stress u u is nearl two-dimensional along most of the sidewall, and begins to decrease onl ver near the free surface; in the lower corner, b comparison, the distribution of u u begins to be affected b the bottom wall at.2. The other two normal components are more three-dimensional, affected both b the solid surface on the bottom and the free surface on top. At the free surface the w fluctuations vanish, and w w = u w = v w =.Significant cross-plane shear stresses v w acomponent difficult to measure experimentall are concentrated in small areas, especiall near the corners. Unlike the other components, v w, which is closel related to the presence of secondar flow, exhibits a high degree of antismmetr with respect to the sidewall bisector; this is because w = alongthe top surface, so that both top and bottom walls act as sinks of v w,inagreement with the observation of previous studies of open-channel turbulent flows (Komori et al. 1982; Handler et al. 1993). Figures 8 and 9 (in which the corner region is enlarged and the mean secondar streamlines are superposed on the Renolds-stress contours) show a rapid decrease in the vertical component and an increase in the horiontal components of the turbulence as the free surface is approached; moreover, most of the energ of the

13 Large-edd simulations of ducts with a free surface (c) (d) (e) ( f ) Figure 8. Open duct flow, Re τ = 6: spatial distributions of the resolved Renolds-stress components. u u ; v v ;(c) w w ;(d) u v ;(e) u w ;(f ) v w (c) (d) (e) ( f ) Figure 9. Open duct flow, Re τ = 6: spatial distributions of the resolved Renolds-stress components near the mixed-boundar corner superposed on the streamlines of the mean secondar flow. u u ; v v ;(c) w w ;(d) u v ;(e) u w ;(f ) v w. vertical component is transferred to the spanwise component with onl a small increase in the streamwise one. The vortical structure plas an important role in decreasing the levels of w w near the sidewall b convecting fluid from the free surface (where w w =)towardsthewall. The distributions of u v and u w

14 236 R. Broglia, A. Pascarelli and U. Piomelli 2 (c) FS u v u v u v Figure 1. Open duct flow: u v Renolds shear stress. Re τ = 36, Re τ = 6, (c) Re τ = 1., + =3;, + = 15;, + = 3;, + = Re τ /2. show the effect of the free surface and the demarcation between inner and outer secondar flows. Near the free surface u v decreases significantl, with a local maximum at the free surface. This is also shown in figure 1, in which the profiles of u v along constant -lines are shown; the indicate that, across the interface between inner and outer regions, u v values at the free surface are smaller than those 1 2 viscous lengths awa from the free surface. A ver strong correlation can be observed between the vortex location and the area in which the secondar shear stress, u w,islarge.inthis region, u w is of the same order of magnitude as u v :theturbulent transfer of streamwise momentum in the direction normal to the free-surface is as important as the transport in the wall-normal direction. Hsu et al. (2) observed an increase in surface-parallel turbulent components for distances from the sidewall greater than 5 viscous lengths, while near the wall all three components of turbulent motion were observed to decrease to a minimum at the free surface. In the present calculation, we also observe that ver close to the free surface (within 2 viscous lengths of the free surface) the u u and v v stresses rebound in the near-wall region (figure 11). Outside the inner secondar region ( + 4), in agreement with the behaviour in open channel flows (Leighton et al. 1981; Lam & Banerjee 1992; Handler et al. 1993; Pan & Banerjee 1995), the increases in spanwise velocit fluctuations are larger than the increases in the streamwise fluctuations. Figure 11 shows that at the lower wall bisector ( + = Re τ /2), the r.m.s. velocit-fluctuation profiles exhibit trends similar to open channel flows. Indeed, u rms dominates, and near the free surface, turbulence moves towards a quasi-twodimensional state (see Walker et al. 1996). From a mathematical point of view, this behaviour at the centre of the duct is obvious since the Renolds stresses depend on the shear components, most of which are ero at the centre of the duct since smmetr and full developed conditions are assumed. A different behaviour was observed at all Renolds numbers in the region associated with the inner secondar flow: both u u and v v increase near the surface. This issue will be discussed later based on the Renolds-stress budget data. A difference between the results of the present work and those ofhsuet al. (2) is that the recover of u u occurs across almost all the surface, except at the lower-wall bisector where u / is equal to ero and u / values are almost constant in the region.8 < /D < 1. Previous studies of Perot & Moin (1995) and Walker et al. (1996) revealed significant differences between shear-free boundar laers near solid walls and free surfaces, such as the higher tangential Renolds stress and lower dissipation near a free surface. Walker

15 Large-edd simulations of ducts with a free surface 237 (c) FS u rms u rms u rms 3. (d) (e) ( f ) FS v rms v rms v rms (g) (h) (i) FS w rms w rms w rms Figure 11. Open duct flow: r.m.s. velocit fluctuations normalied with the reference friction velocit. + FS is the distance from the free surface in wall units (the reference u τ is used in this normaliation). (c) u rms ; Re τ = 36, Re τ = 6, (c) Re τ = 1. (d) (f ) v rms ; (d) Re τ = 36, (e) Re τ = 6, (f ) Re τ = 1. (g) (i) w rms ;(g) Re τ = 36, (h) Re τ = 6, (i) Re τ = 1., + =3;, + = 15;, + = 3;, + = Re τ /2. et al. (1996) attribute the apparent increase of turbulence kinetic energ near a rigidlid boundar to the reduced dissipation of the tangential velocit fluctuations. The persistence of the surface-attached vortices observed in the numerical stud of Pan & Banerjee (1995) is also explained b this minimum in dissipation, together with the no-stress boundar conditions. Current calculations indicate that for juncture flows the peculiar production behaviour can affect the velocit fluctuations. To understand the mechanisms that contribute to the Renolds stress distributions shown above, we will examine their budgets. The resolved Renolds-stress transport

16 238 R. Broglia, A. Pascarelli and U. Piomelli equations are of the form: = u k u i u j + P ij ɛ ij + D ij + φ ij + ψ ij + T ij, (3.1) x k where the terms on the right-hand side represent (rate of) advection, production, dissipation, viscous diffusion, pressure strain, pressure diffusion, and turbulent diffusion tensors, respectivel. The last six terms in (3.1) are defined as: [( P ij = u ju k u i + u i u x k u ) j + ( τ ik S jk + τ jk S ik )], (3.2) k x k [ 2 u u i j ɛ ij = ( τ ik S jk + τ jk S ik )], (3.3) Re τ x k x k D ij = 1 2 u i u j, (3.4) Re τ x k x ( k ) u φ ij = p i + u j, (3.5) x j x i ψ ij = ( ) p u x j δ ik + p u i δ jk, (3.6) [ k u i u ju k T ij = + ( u x k x jτ ik + u i τ jk )]. (3.7) k The dissipation and turbulent diffusion consist of two parts: a resolved and a subgridscale contribution. To quantif the contribution of the SGS eddies to the terms in the Renolds-stress budgets, we compare the SGS dissipation ɛ sgs = τ ij S ij,thenet large-scale energ drained b the subgrid scales, to the total (molecular + SGS) one, ɛ. Thisisthemostsignificant of the SGS terms that appear in (3.1). The volumeaveraged ɛ sgs is 18%, 19% and 21% of the total volume-averaged dissipation, for the three Renolds numbers considered. Using LES and DNS of mixing laers, Geurts & Fröhlich (22) found that, when this measure of subgrid-activit is less than 3%, the modelling errors are less than 1%. Figure 12 shows the terms in the budget of the turbulent kinetic energ K = u i u i /2. Here and in the following, all terms are normalied b u τ and ν. Advection is negligible everwhere except in the mixed-boundar corner, as will be discussed later, and is not shown in this figure. Along the side and bottom walls the budget resembles that observed in a flat-plate boundar laer: production and dissipation are nearl in balance, except ver near the wall itself, where turbulent and viscous diffusion become important. An enlargement of the corner region, in which the secondar-flow streamlines are superposed on the budget terms and advection is shown in figure 13. The turbulent kinetic energ is dominated b the streamwise stress u u,whose production (shown in figure 14c) includes the main shear components, u /, u /. Theterms in the budgets for the two other normal components (shown in figures 15 and 16, respectivel) are smaller than those in the streamwise one b an order of magnitude, reflecting the absence of a significant shear in the production term. All terms contribute to the budgets of v v and w w. Hsu et al. (2), observed that both TKE production and dissipation are dramaticall reduced in the near-corner region, close to the free surface. Far from the wall, this results in an increase of the surface-parallel fluctuations ver close to the free surface. Close to the sidewall and within 2 viscous lengths of the free surface, all

17 Large-edd simulations of ducts with a free surface (c) (d) (e) ( f ) Figure 12. Open duct flow, Re τ = 6: spatial distributions of the terms in the budget of K = u i u i /2. Turbulent diffusion; pressure diffusion; (c) production; (d) dissipation; (e) pressure strain; (f )viscous diffusion. three components of turbulent motions were reported to decrease to a minimum at the free surface. This reduction was explained b Shen et al. (1999) through two mechanisms: the annihilation of w due to the constraint on the vertical motion at the free surface, and the vanishing of u / caused b the shear-free boundar condition. Our calculations agree onl in part with the experimental findings. While adecrease of the production is observed in the corner region as the free surface is approached, we observe that the production of u u increases near the free surface (figure 14c), but that of v v decreases (figure 15c). Near the inner secondarflow cell, production appears to rebound, and the imbalance between production and dissipation is balanced b advection of TKE awa from the corner, along the sidewall. Among the production terms that contribute to the TKE, u v u / is the dominant one in the corner region. To complete the above picture, in figure 17 the dependence of the TKE budget terms on the Renolds number is examined. The distributions of different terms, non-dimensionalied with the reference u τ and ν, arepresented. The corner does not affect the wall scaling expected near a solid boundar for most of the terms (notabl production and dissipation, the leading ones in this region). Onl advection, which is determined b the interaction between the inner and outer secondar-flow cells, deviates significantl from this scaling. We will now discuss the budgets of the normal Renolds stresses in the corner region (figures 14 16) b first describing how the solid-wall/free-surface juncture alters the mechanism of the intercomponent energ transfer. The production rate,

18 24 R. Broglia, A. Pascarelli and U. Piomelli (c) (d) (e) ( f ) Figure 13. Open duct flow, Re τ = 6: spatial distributions of the terms in the budget of K = u i u i /2 near the mixed-boundar corner superposed on the streamlines of the mean secondar flow. Advection; pressure and turbulent diffusion; (c) production; (d) dissipation; (e) pressure strain; (f )viscous diffusion (c) (d) (e) ( f ) Figure 14. Open duct flow, Re τ = 6: spatial distributions of the terms in the budget of u u near the mixed-boundar corner superposed on the streamlines of the mean secondar flow. Advection; pressure and turbulent diffusion; (c) production; (d) dissipation; (e) pressure strain; ( f )viscous diffusion.

19 Large-edd simulations of ducts with a free surface (c) (d) 1. (e) 1. ( f ) Figure 15. Open duct flow, Re τ = 6: spatial distributions of the terms in the budget of v v near the mixed-boundar corner superposed on the streamlines of the mean secondar flow. Advection; pressure and turbulent diffusion; (c) production; (d) dissipation; (e) pressure strain; ( f )viscous diffusion (c) (d) 1. (e) ( f ) Figure 16. Open duct flow, Re τ = 6: spatial distributions of the terms in the budget of w w near the mixed-boundar corner superposed on the streamlines of the mean secondar flow. Advection; pressure and turbulent diffusion; (c) production; (d) dissipation; (e) pressure strain; ( f )viscous diffusion.

20 242 R. Broglia, A. Pascarelli and U. Piomelli (c) + FS (d) (e) ( f ) + FS Figure 17. Open duct flow, Re τ = 6: spatial distributions of the terms in the budget of K = u i u i /2 near the mixed-boundar corner. Advection; pressure and turbulent diffusion; (c) production; (d) dissipation; (e) pressure strain; ( f ) viscous diffusion., Re τ = 36;, Re τ = 6;, Re τ = 1. P 33,which,together with P 22 and P 23,contains onl secondar shear components (namel, v /, v /, w / and w / ), goes to ero at the free surface and is not directl related to the corresponding Renolds stress distribution w w (figures 9c and 16c). In this region, the distribution of w w is more influenced b the surface-normal contribution of the pressure strain, φ 33 (see figure 16e). The pressure rate-of-strain tensor, φ ij,distributes energ among the components of normal stresses u u, v v and w w, usuall shifting energ from the components of the Renolds stress tensor that receive most of the production b mean shear into the others. Since φ ij involves pressure fluctuations, it represents a term associated with non-local interactions and (in wall-bounded flows) is the most important contribution to the pressure correlation tensor Π ij = φ ij + ψ ij in the buffer and the logarithmic region. Near the sidewall φ ij and ψ ij are opposite-valued and cancel each other (similarl to what happens in plane channel flow (Mansour, Kim & Moin 1988). Toward the middle plane the distribution of φ 33 is qualitativel different from the results of open-channel flow (Handler et al. 1993; Komori et al. 1993). In the corner region, up to the demarcation between inner and outer secondar flow, φ 33 decreases and becomes negative as the free surface is approached, whereas it remains positive elsewhere. This indicates that the pressure strain term, which increases w w throughout nearl the entire flow region, near the inner secondarflow region transfers the surface-normal component of TKE to the other components. At the free surface in the outer secondar region, φ 33 is also a production term, but opposite to ψ 33 ;thecancel each other. Apart from a small negative region ver near the sidewall, on the other hand, φ 22 remains positive in the boundar laer below the free surface (figure 15e), while φ 11,whichisnegative throughout nearl

21 Large-edd simulations of ducts with a free surface (c) φ ij.5 φ ij.5.5 φ ij.5 Figure 18. Profiles of the diagonal components of the pressure-strain correlation tensor for Re τ = 6 at different distances from the sidewall: =.65; =.23; (c) =.5. Lines: open duct. Lines and smbols: closed duct., φ 11 ;, φ 22 ;, φ 33. the entire mixed-corner region, becomes ver slightl positive at the free surface (figure 14e). Note that ψ 11 =,because the flow is homogeneous in the streamwise direction. Noting that φ 22 > φ 11 through most of the corner region we conclude that, outside of the inner secondar region, the dominant energ transfer is from the surface-normal component to the spanwise one, in agreement with previous studies of boundar laers adjacent to the free surface (Handler et al. 1993; Perot & Moin 1995; Walker et al. 1996). In the inner region, however, a different behaviour can be observed. This is illustrated in figure 18, in which profiles of the diagonal components of the pressure rate-of-strain tensor, φ 11, φ 22 and φ 33,attwo-locations are compared with the corresponding terms from the closed-duct simulation. At =.65 (figure 18a), φ 33 causes a loss of w w immediatel below the free surface (in the open duct) or the wall (in the closed duct). At the solid solid corner, we can distinguish two laers: one, next to the solid wall, in which the w w stress is distributed both to u u and v v through a negative φ 33 and increased φ 11 and φ 22 ;this is similar to the splatting effect that occurs in the vicinit of a solid boundar (Moin & Kim 1982). For.95 <<.98, however, the decrease of w w results mostl in an increase of v v ;thiseffect, in which onl one Renolds-stress component is increased, while another is decreased, is called a corner effect b Huser et al. (1994). Near the mixed juncture there is little evidence of the corner effect: the spanwise component φ 22 is positive and increases approaching the interface; the streamwise component φ 11 also becomes slightl positive in the vicinit of the free surface. Thus, in this region the vertical component of TKE is distributed both to u u and v v.furtherawa from the sidewall, for =.23, onl the splatting effect is present both in the solid and in the mixed corners. In the present problem, all the off-diagonal Renolds stresses are non-ero. Their budgets in the corner region are shown in figures The contours of u v and u w,whichareresponsible, respectivel, for the transport of streamwise momentum in sidewall and free-surface normal directions, are correlated with those of the production terms P 12 and P 13,inwhichaconnection between primar and secondar shear stresses is present, and the terms containing products of main strain components and normal stresses are dominant. The distribution of v v mainl affects the

22 244 R. Broglia, A. Pascarelli and U. Piomelli (c) (d) (e) ( f ) Figure 19. Open duct flow, Re τ = 6: spatial distributions of the terms in the budget of u v near the mixed-boundar corner superposed on the streamlines of the mean secondar flow. Advection; pressure and turbulent diffusion; (c) production; (d) dissipation; (e) pressure strain; ( f )viscous diffusion (c) (d) (e) ( f ) Figure 2. Open duct flow, Re τ = 6: spatial distributions of the terms in the budget of u w near the mixed-boundar corner superposed on the streamlines of the mean secondar flow. Advection; pressure and turbulent diffusion; (c) production; (d) dissipation; (e) pressure strain; ( f )viscous diffusion.

23 Large-edd simulations of ducts with a free surface (c) (d) (e) ( f ) Figure 21. Open duct flow, Re τ = 6: spatial distributions of the terms in the budget of v w near the mixed-boundar corner superposed on the streamlines of the mean secondar flow. Advection; pressure and turbulent diffusion; (c) production; (d) dissipation; (e) pressure strain; ( f )viscous diffusion. production of u v through P 12 v v u /, while the variation of w w is asignificant mechanism for the u w generation, since P 13 w w u /. At the mixed corner, u / monotonicall increases approaching the sidewall, but the production of u v shows a small reduction owing to the changes in the distributions of v v in this region along the free surface. The pressure strain term φ 12 appears as a gain in the u v budget throughout nearl the entire corner region, having a local maximum near the core of the inner cell. The production of u w changes sign at the demarcation between inner and outer secondar flow and becomes negative in regions associated with the inner secondar cell, since u /, ero at the free surface, attains a local maximum immediatel below the interface due to the secondar motions. Term φ 13 presents a more complicated pattern. The negative extreme values of u v and u w in the core of the inner secondar region affect directl the production of u u, P 11 2( u v u / + u w u / ), which increases at the inner secondar flow cell, and remains positive across the surface. As previousl stressed, in the central part of the duct u / goes to ero; this reduces the corresponding term in P 11.Theremaining term in P 11 is also reduced close to the free surface because of the suppression of u / b the shear-free boundar condition. Furthermore, in the budget for u u the pressure strain term φ 11 appears as a slightl positive quantit at the free surface (figure 18c)andhence it ma be thought of as pseudo-production term as far as the streamwise fluctuations are concerned. The gained energ is transported b the diffusion terms and lost b dissipation. In the open-channel flow, the pressure strain term φ 33 decreases and becomes negative near the surface. The term φ 33 and the dissipation term ɛ 33 nearl balance across almost the entire depth of the channel, but near the free surface and near the wall φ 33 balances with ψ 33,thepressure diffusion term. In the open-channel flow, P 22 and P 33 are identicall ero, i.e. in the budgets for

24 246 R. Broglia, A. Pascarelli and U. Piomelli v v and w w,nodirect production term exists. Here, at the lower wall bisector, P 22 2 v v v / and P 33 2 w w w /. While the latter goes to ero at the interface, the former is a positive quantit. This difference balances with a negative φ 22 (see figure 18c). Since Π kk =fromcontinuit, an increase in φ 22 can be expected to result in a decrease in the magnitude of φ 33.Thebudget for w w confirms this line of reasoning. It should be noted here that, as is the case in solid solid corners (Huser et al. 1994), the turbulence dissipation-rate tensor contributes significantl to the Renolds stress balance, both near the walls and far from them. Figure 2 shows high-dissipation events in the high-turbulence production area. The cross-plane Renolds stress, v w (which is identicall ero in twodimensional turbulent shear flows) together with the anisotrop of the normal stresses in the (, )-plane, controls the production of the mean streamwise vorticit, whose complete balance is deferred to the next paragraph. In the present configuration, the Renolds stress production rates that contain the secondar shear stress, v w,are weak. Previous studies on three-dimensional duct flows have shown that generation of v w results from two mechanisms, one associated with the gradients of secondar velocities, the other with the distortion of primar velocit gradients (Perkins 197). The former mechanism seems more effective that the latter (Demuren & Rodi 1984). Along the sidewall there is a positive production of v w caused b the secondar Renolds stresses; this P 23 balances the negative contribution from ɛ 23.Alongthe free surface, on the other hand, the magnitudes of all terms on the right-hand side of the v w balance equation, although small, appear to be of equal magnitude Open duct: vorticit As outlined above, in an open square duct, mean three-dimensionalit results in nonero mean streamwise vorticit ω x = w / v /. Theorigin of the mean secondar flow can be linked to the same mechanisms as those in a closed square duct, since in both cases the anisotrop of the turbulence stress is the driving force that generates the secondar flows. The relationship between the Renolds-stress anisotrop and the secondar flow is illustrated b the transport equation for ω x.in astatisticall stead full developed turbulent flow, in which streamwise gradients are identicall ero, the quasi-inviscid stretching and deflection (skew-induced) generation term cancels and this equation reads v ω x + w ω x 1 ( ) 2 ω x + 2 ω x Re }{{} τ 2 }{{ 2 } I II ( ) ( 2 v 2 + τ w τ } {{ } III ) v w + τ 2 2 } 2 {{} IV =. (3.8) Term I represents advection of the mean streamwise vorticit b mean secondar velocities; term II is the diffusion of vorticit due to viscosit; III and IV are production terms (resolved + SGS) due to the gradient of the difference in the normal Renolds stresses ( v 2 + τ w 2 + τ ), and to the difference in the gradient of the secondar shear Renolds stress ( v w + τ ), respectivel. It has been argued b other investigators (see, for example, Perkins 197) that, for full developed flows, the imbalance between gradients of normal stresses is primaril responsible for the production of secondar flow vorticit, whereas the shear-stress contribution, if not negligible, acts like a transport term in (3.8). For a more thorough

25 Large-edd simulations of ducts with a free surface Figure 22. Open duct flow, Re τ = 6: spatial distribution of the resolved anisotrop term ( v v w w ). Twent-one equispaced contours between ±1. discussion of the associated flow phenomena, see Bradshaw (1987). In the present problem, the mean streamwise vorticit is directl tied to the secondar velocities and the main mechanisms acting in (3.8) pla a role in the origin of the secondar flows. In the context of vorticit transport, the turbulent secondar-flow generation in the mixed-corner flow presents a totall different spatial distribution from that in the solid solid corner, because of the significant differences in the distributions and magnitudes of the turbulent stresses. Figure 22 displas contours of the resolved anisotrop of the normal Renolds stresses, v v w w. Together with the cross-plane resolved Renolds stress v w (shown in figure 8f ), this term contributes to the production and transport of mean streamwise vorticit. The anisotropic term reaches a minimum along the side wall, where the turbulent fluctuations in the direction normal to the sidewall are suppressed b the boundar conditions. At both corners, the contours show two regions of opposite sign, which are ultimatel the cause of the generation of ω x. The gradient of the anisotrop term near the mixed corner, however, is higher than that at the solid solid corner. Figure 23 shows the spatial distributions of the four terms of equation (3.8) for the Re τ = 6 calculation. The two-dimensional fields of the different terms of (3.8) are superimposed on filled contours of the mean streamwise vorticit. The contribution of the SGS stresses was found to be negligible compared with the leading terms in (3.8). As alread pointed out, in the corner formed b the solid wall and the free surface, ω x has positive/negative extreme values in correspondence with the inner/outer secondar flows. In general, advection represents the smallest contribution to the budget (figure 23a), attaining both its extrema inside the inner secondar-flow cell. Advection is negative at the juncture, and reaches a positive maximum on the free surface, between the inner and outer secondar-flow cells, with a marginal impact on the local ω x distribution. At the solid solid corner (not shown), both Renolds normal and shear stress contributions are of the same order of magnitude, of opposite sign and dominant with respect to advection. At the mixed corner, on the other hand, the normal-stress contribution is dominant among the production terms, and positive peak values of term III are alwas larger than negative values of term IV. Furthermore,

26 248 R. Broglia, A. Pascarelli and U. Piomelli (c) (d) Figure 23. Open duct flow, Re τ = 6. Distributions of the resolved mean streamwise vorticit and terms of the governing balance equation. Contour increments are 1, and dotted lines denote negative contours. Advection; viscous diffusion; (c) production due to normal stress; (d) production due to shear stress. the net positive turbulence contribution overcomes the negative viscous diffusion in the same region, giving rise to a positive streamwise vorticit at the inner secondar flow. As expected, diffusion (figure 23b) issignificant both close to the corners and ver close to the sidewall, but decreases rapidl awa from the wall. At the free surface, viscous diffusion acts as a source term in correspondence of the core of the outer secondar flow. At the outer secondar flow, the balance is mainl between terms II and III. The streamwise vorticit distribution along the sidewall, above the viscous sublaer, in the region.6 </D<.9, is due to the sum of the resolved parts of terms III and IV, which represent the main turbulent contributions to (3.8). In LES, the vorticit fluctuations are less well-resolved than the velocit fluctuations, since the unresolved scales contribute more to the former than to the latter. In the present calculation, however, the grid resolution is fine enough that even the SGS contribution to the vorticit is fairl small. An indication of this is the observation that the SGS contribution to the total dissipation is between 18% and 21% for all the cases examined. Thus, we believe that at least qualitative conclusions can be drawn b examining the r.m.s. vorticit fluctuations obtained from the present simulation. The surface-parallel components of the r.m.s. vorticit fluctuation, ωx rms and ω rms,decrease inside a ver thin laer below the free surface and vanish at the free surface because of the local shear-free condition. The anisotropic vorticit laer is much thinner than the anisotropic velocit laer. In figure 1, the vertical profiles of the fluctuating velocit components showed the dependence on the Renolds number of the thickness of the blockage laer. At