Reynolds number effects in stratified turbulent wakes

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1 Abstract Reynolds number effects in stratified turbulent wakes Qi Zou 1 and Peter Diamessis 2 1 Department of Applied Matematics and Teoretical Pysics, University of Cambridge 2 Scool of Civil and Environmental Engineering, Cornell University We report large-eddy simulations of te turbulent wake of a towed-spere of diameter D at speed U in a linearly stratified Boussinesq fluid wit buoyancy frequency N. Tese simulations are performed using a spectral-multidomain-penalty-metod-based incompressible Navier-Stokes solver for Re UD/ν {5 10 3, 10 5, } and Fr 2U/(N D) {4, 16, 64}. Increasingly ricer turbulent fine structures are observed in iger-re wakes witin te larger-scale quasi-orizontal vortices at later times, wen it is commonly expected tat buoyancy as suppressed any turbulent motions. Suc penomenology is examined using te (Re, Fr 1 ) pase diagram (Bretouwer et al., 2007). Te body-based wake Re is te parameter wic determines weter te flow can indeed enter te inviscid strongly stratified regime witin wic te buoyancy-driven vertical sear layers can eiter sustain existing turbulence or give rise to secondary searinstabilities and localized turbulent bursts. Te duration of te stratified turbulence regime also scales linearly wit Re posing questions on te structural nature of te associated turbulence and its strengt wit respect to any background motions. 1 Introduction Stratified turbulent wakes are frequently occurring flows in te stably stratified regions of te ocean and te atmospere. Of numerous geopysical and naval applications, suc flows ave motivated extensive researc efforts, e.g., as reviewed by Lin and Pao (1979) and Spedding (2014). In tis paper, we focus on stably stratified towed-spere wakes, a canonical flow involving bot sear and buoyancy forces, wic are described by Reynolds number Re U D/ν and Froude number Fr 2U/(N D), were D is te spere diameter, U is te tow speed, N is te buoyancy frequency, and ν is te kinematic viscosity. Te numerical dataset presented in tis paper is generated troug large-eddy simulations using a spectral-multidomain-penalty-metod-based incompressible Navier-Stokes solver (Diamessis et al., 2005). Tis solver employs Fourier discretization in bot orizontal directions x (streamwise) and y (spanwise), and a Legendre-polynomial-based spectral multidomain sceme in te vertical direction z. Spectral filtering and a penalty sceme ensure te numerical stability of te simulations witout resolving te full spectrum of turbulent motions. Details on te configuration of tese stratified wake simulations can be found in Diamessis et al. (2011) (ereinafter referred to as DSD ). In tis paper, we present preliminary results on te influence of Re on wake turbulence by examining te numerical dataset produced by Zou (2015) at Re {5 10 3, 10 5, } and Fr {4, 16, 64}. Prior simulations by DSD up to Re = 10 5 demonstrate te prolongation of te duration of te non-equilibrium regime (Spedding, 1997) wit increasing Re value due to localized secondary sear instabilities witin te wake, consistent wit te stratified turbulence ypotesis by Lilly (1983) and te pioneering computational study VIII t Int. Symp. on Stratified Flows, San Diego, USA, Aug Sept. 1,

2 by Riley and de Bruyn Kops (2003). Te effects of varying Re in wakes ave also been seen in terms of te internal waves emitted by te wake turbulence (Abdilganie and Diamessis, 2013; Zou and Diamessis, 2016) and te properties of te turbulent/non-turbulent interface (Watanabe et al., 2016). Te dataset recently produced by Zou (2015) brings into play an additional Re data point at Re = As suc it enables an unprecedented systematic investigation of Re effects on te wake turbulence. Hereinafter, eac simulation will be labelled as RaFb, were a = Re/10 3 and b = Fr. Significant progress in stratified turbulence teory as recently been made, as reviewed by Riley and Lindborg (2012). Scaling arguments (Riley et al., 1981; Lilly, 1983; Billant and Comaz, 2001; Bretouwer et al., 2007) based on te Navier-Stokes equations lead to te prediction of a distinct strongly stratified turbulence regime were stratification is strong and viscous effects are weak. An instructive way to identify te potential presence of suc a regime in any given flow is troug a diagram plotted on te (Re, Fr 1 ) pase space, were Re and Fr are appropriately defined Reynolds and Froude numbers based on orizontal turbulent motions. Witin te strongly stratified regime were Re 1 and Fr 1, te formation of buoyancy-driven layers of concentrated sear takes place independently of viscosity wit te layer tickness scaling as U/N, were U is a turbulent orizontal velocity scale. Existing studies on tis regime mainly focus on omogeneous flows (e.g., Riley and de Bruyn Kops (2003); Bretouwer et al. (2007)). Te primary aim of tis paper is to investigate in te stratified turbulent towed-spere wake, a canonical localized inomogeneous turbulent flow, in te context of stratified turbulence teory. Te particular focus lies on caracterizing te dependence of coerent vortical structures on Re, te scaling of various turbulent diagnostics wit Re, te associated persistence time of stratified turbulence witin te wake and te associated trajectories in (Re, Fr 1 ) pase space. 2 Flow structure Figure 1: Contour plots of vertical vorticity ω z at = 150 sampled at te Oxy orizontal mid-plane for simulations at Re {5 10 3, 10 5, } respectively (from left to rigt) and Fr = 4. Spere travels from left to rigt. Te size of te visualization window is 80 3 D in x and 20D in y. Figures 1 and 2 sow contour plots of te vertical, ω z, and spanwise, ω y vorticity on te Oxy and Oxz wake centerplanes in te wake flow in bot orizontal and vertical directions at a later time of = 150. Te ω y snapsots are supplemented wit an earlier time at = 50. More details on te evolution of te wake structure may be found in 7.2 of Zou (2015). At bot of tese times, te R5F4 run as progressed well into te quasi-two-dimensional (Q2D) regime, were quasi-orizontal pancake vortices VIII t Int. Symp. on Stratified Flows, San Diego, USA, Aug Sept. 1,

3 Figure 2: Contour plots of spanwise vorticity ω y at = 50 (left panel) and 150 (rigt panel) sampled at te Oxz vertical mid-plane for simulations at Re {5 10 3, 10 5, } respectively (from top to bottom) and Fr = 4. Spere travels from left to rigt. Te size of te visualization window is 40 3 D in x and 4D in z. dominate te flow and grow troug vortex pairing and vortex diffusion (Spedding, 2002). Vertical transects of ω y sow, at bot times, strongly inclined sear layers wic diffuse even furter over time and remain stable to any perturbations. In contrast, for te two iger Re runs, te tickness of tese layers is markedly reduced. Kelvin-Helmoltz instabilities, resulting in localized secondary patces weak of turbulence as reported by Riley and de Bruyn Kops (2003) and DSD, are visible. Tese secondary events are far more space-filling for te R400F4 case, persisting as late as N t = 150. Te equivalent signature of tese events on te Oxy centerplane is evident in figure 1 were significant fine structure is embedded witin large-scale quasi-orizontal motions in R400F4. In contrast, at R100F4, only a couple patces of turbulent motion remain at tese later times. Overall, tese observations suggest tat stratified turbulence is operative at te two iger Re sown ere: an inverse cascade occurs in te orizontal, driven by pancake vortex pairing, competing against a forward cascade associated wit sear-driven turbulence. 3 Stratified turbulence statistics 3.1 Turbulence Decay Te evolution of te buoyancy Reynolds number Re b is sown in figure 3. Altoug, formally, one defines Re b ε/(νn 2 ), were ε is te turbulence kinetic energy dissipation rate, in tis case te buoyancy Reynolds number is approximated troug Re b Re Fr 2, an approximation valid at least for Re b > O(1) (Hebert and de Bruyn Kops, 2006). Here te orizontal local/turbulent Reynolds and Froude numbers are defined as Re = Ul /ν and Fr = U/(Nl ), were U is taken as te volume-averaged root-mean-square orizontal velocity on te orizontal planes wit te largest vertical sear (DSD) and l is te orizontal integral scale following equation (5.5) of DSD. Re b is a measure of te dynamic range in stratified turbulence, i.e. te scale separation between Ozmidov and Kolmogorov VIII t Int. Symp. on Stratified Flows, San Diego, USA, Aug Sept. 1,

4 scale (Hebert and de Bruyn Kops, 2006). Effectively, wen Re b is sufficiently greater tan unity, a significant range of turbulent scales of motion can operate in a strongly stratified environment wit Fr 1. Te left panel of figure 3 indicates tat tis unity-crossing time is delayed wit increasing Re, wit Re b 1 occurring at 10, 150 and 400 for Re = , 10 5 and Tis observation suggests an increasingly broader and persistent spectrum of turbulent motions witin a stratified wake following te onset of buoyancy control at 2. Rescaling te vertical axis wit 1/Re leads to a roug collapse of all curves (figure 3, rigt panel) wic indicates tat, indeed, te unity crossing time of Re b does scale linearly wit Re. Furter probing te persistence of stratified turbulence, as evidenced by values of Re b 1 during stratified wake evolution, figure 4 sows te time istory of all tree root-meansquare fluctuating velocities in te F4 simulations. At lower Re, w u and v. Driven by later-time stratified turbulence, te magnitude of w increases wit Re remaining as large as 20% of u and 1% of te reference tow speed U up to time Reb Reb/Re Figure 3: Left panel: te buoyancy Reynolds number Re b, approximated by Re Fr 2, as a function of. Rigt panel: Re b rescaled by te wake Reynolds number Re. Re = in red, Re = 10 5 in blue, and Re = in green; Fr = 4, 16 and 64 are in solid, dased and dotted lines respectively. 3.2 Turbulent lengt scales Te time istory of orizontal integral scale l and te volume-averaged vertical Taylor scale l v (defined in Riley and de Bruyn Kops (2003)) is sown in figure 5. Te evolution of l does not seem to be strongly dependent on te wake parameters Re and Fr. l v for te R100 and R400 runs follows te inviscid U/N scaling proposed by Billant and Comaz (2001), indicating tat te buoyancy-generated vertical sear layers are not subject to viscous diffusion and are igly prone to sear instability as sown in figure 2. Note tat te starting time of te window during wic te R100 (blue) and R400 (green) lines collapse, i.e., 10, corresponds to te time wen Fr (see figure 7.14 of Zou (2015)) drops below O(0.1) wile Re remains sufficiently large, providing furter support tat te wake as entered te near-inviscid stratified turbulence flow regime. Te R5 data (red) do not follow te same scaling, as Re is significantly smaller at times 10 and Re b 1. Instead, as suggested by Bretouwer et al. (2007), te low Re data follow te viscous scaling l v /l Re 1/2 (see figure 7.18 of Zou (2015)). VIII t Int. Symp. on Stratified Flows, San Diego, USA, Aug Sept. 1,

5 (u,v,w )/U Figure 4: Time evolution of root-mean-square velocity fluctuations (u, v, w ) wic are normalized by tow speed U for simulations at Fr = 4. Te velocity scales are volume-averaged over L y < y < L y and L z < z < L z, were L y and L z are te alf widt/eigt of te mean wake. Re = in red, Re = 10 5 in blue, and Re = in green; u, v and w are in solid, dased and dotted lines respectively. l/d lv/d lvn/u Figure 5: Time evolution of orizontal and vertical Taylor scales l and l v, bot normalized by spere diameter D, plotted in unit. Te rigt most panel sows l v rescaled wit te inviscid strongly stratified scaling l v U/N (Billant and Comaz, 2001). Line legends are te same as figure Evolution in te pase space Finally, it is instructive to examine te trajectory of a wake across different stratified flow regimes in (Re, Fr 1 ) pase space (Bretouwer et al., 2007). For a fixed value of Fr = 4 (figure 6, left panel), te wake initially operates in te weakly stratified regime wit initially large ε and U and, ence, large Re b and Fr. Te fact tat te trajectory s starting point, at least for te Fr = 4 data does not reside in te classical Kolmogorov turbulence regime, associated wit active turbulence uninibited by buoyancy, may be an artifact of te approximations involved in designing te initial condition wile not explicitly accounting for te spere. Te interested reader is referred to DSD and Zou and Diamessis (2016) for additional discussion. Note tat te initial Re b, measured grapically by te distance of te wake trajectory to its projection onto te diagonal line indicating Re b Re Fr 2 = 1, is set by te wake s Re. As te wake evolves, on one and, Re stays relatively constant. Tis is because te increase in l compensates for te decrease of U (Hebert and de Bruyn Kops, 2006). On te oter and, te increase of l and decrease of VIII t Int. Symp. on Stratified Flows, San Diego, USA, Aug Sept. 1,

6 10 3 Viscosity-affected stratified flow Strongly stratified turbulence /Fr Time Weakly stratified turbulence R5 R100 R400 Classical Kolmogorov turbulence Re 1/Fr F4 F16 F64 Time Re Figure 6: Evolution of wake turbulence in te (Re, Fr 1 ) pase space at Fr = 4 for all tree Re values (left panel) and at Re = for all tree Fr values (rigt panel). Te classification of te flow regimes is after figure 18 of Bretouwer et al. (2007). U result in te decrease of Fr. Re is effectively te factor wic determines weter te flow enters te viscously dominated regime and bypasses te strongly stratified turbulence regime, caracterized by Re 1, Fr 1 and secondary turbulence events, before te wake becomes viscously dominated. Te iger te value of Re, te iger Re b (or Re ) becomes, and te longer te time te wake spends in te strongly stratified regime, as evidenced by comparing te trajectories of R100F4 and R400F4. R5F4 does not enter te strongly stratified regime as small Fr 1 never coexists wit sufficiently large Re b in wake evolution. Fixing Re = , one may focus on te Fr dependence of pase space trajectories (figure 6, rigt panel). Te trajectory sifts to smaller values of Re wit increasing Fr (see figure 7.14 of Zou (2015)). Tis is expected because, wile l /D is weakly Fr-dependent (see figure 5), te velocity scale U scales wit te mean centerline defect velocity U 0 in a Fr-dependent manner (Spedding, 1997; Diamessis et al., 2011; Zou, 2015). Te iger te wake Fr value, te lower te value of U upon onset of buoyancy control, wic is not compensated by te corresponding increase in l /D wit Fr at tis time, and te lower te value of Re. Te question tat ten emerges, similar to te inquiry explored by Spedding (1997), is weter, for a given Re and sufficiently large Re b, tere exists a critical Fr above wic te flow cannot traverse te strongly stratified regime. 4 Discussion Te availability of tree spere-based Re values of , 10 5 and as enabled a first preliminary effort into quantifying te Reynolds number dependence of stratified turbulent wakes wic gives rise to several intriguing questions. Te linear Re scaling of te unity-crossing time of Re b, as approximated by Re b Re Fr 2, poses questions about te persistence of a stratified wake at values typical of ocean engineering applications and geopysical flows. On one and, Re Fr 2 = 1 and not Fr = 1, wic as been VIII t Int. Symp. on Stratified Flows, San Diego, USA, Aug Sept. 1,

7 proposed by lower Re-focused investigations, sould be te most appropriate condition for setting turbulent vertical momentum transport to zero in a wake in predictive selfsimilarity models (Meunier et al., 2006). On te oter and, one wonders about te duration of te initial fraction of te Re Fr 2 > 1 interval during wic te stratified turbulence is distinguisable wit respect to any background motions. To tis end, ow strong does te root-mean-square w ave to be wit respect to te tow speed U and possibly te wake centerline velocity U 0? In addition, ow does te nature of stratified turbulence cange as te fraction of te wake trajectory in te stratified turbulence regime of te (Re, Fr 1 ) pase space becomes larger wit increasing Re? Are te secondary Kelvin-Helmoltz instabilities and resultant turbulence less intermittent during te initial stages of tis regime, corresponding to a igly anisotropic and layered form of spacefilling turbulence? Furter analysis of te existing wake data and stratified omogeneous turbulence datasets (de Bruyn Kops, 2015) will elp answer te above questions as will also additional simulations at even iger Re, possible only wen greater computing power is available. Acknowledgement Te support troug Office of Naval Researc grant N (managed by Dr. R. Joslin) is gratefully acknowledged. Hig performance computing resources were provided troug te US Department of Defence Hig Performance Computing Modernization Program by te Army Engineer Researc and Development Center and te Army Researc Laboratory under Frontier Project FP-CFD-FY (P.I.: Dr. S. de Bruyn Kops). Te work of te first autor was supported, in part, by UK EPSRC Programme Grant EP/K034529/1 entitled Matematical Underpinnings of Stratified Turbulence. References Abdilganie, A. M. and Diamessis, P. J. (2013). Te internal gravity wave field emitted by a stably stratified turbulent wake. J. Fluid Mec., 720: Billant, P. and Comaz, J.-M. (2001). Self-similarity of strongly stratified inviscid flows. Pys. Fluids, 13:1645. Bretouwer, G., Billant, P., Lindborg, E., and Comaz, J.-M. (2007). Scaling analysis and simulation of strongly stratified turbulent flows. J. Fluid Mec., 585: de Bruyn Kops, S. M. (2015). Classical scaling and intermittency in strongly stratified Boussinesq turbulence. J. Fluid Mec., 775: Diamessis, P. J., Domaradzki, J. A., and Hestaven, J. S. (2005). A spectral multidomain penalty metod model for te simulation of ig Reynolds number localized incompressible stratified turbulence. J. Comput. Pys., 202: Diamessis, P. J., Spedding, G. R., and Domaradzki, J. A. (2011). Similarity scaling and vorticity structure in ig-reynolds-number stably stratified turbulent wakes. J. Fluid Mec., 671:52 95, referred to in te text as DSD. Hebert, D. A. and de Bruyn Kops, S. M. (2006). Predicting turbulence in flows wit strong stable stratification. Pys. Fluids, 18: VIII t Int. Symp. on Stratified Flows, San Diego, USA, Aug Sept. 1,

8 Lilly, D. K. (1983). Stratified turbulence and te mesoscale variability of te atmospere. J. Atoms. Sci., 40: Lin, J.-T. and Pao, Y.-H. (1979). Wakes in stratified fluids. Ann. Rev. Fluid Mec., 11: Meunier, P., Diamessis, P. J., and Spedding, G. R. (2006). Self-preservation in stratified momentum wakes. Pys. Fluids, 18: Riley, J. J. and de Bruyn Kops, S. M. (2003). Dynamics of turbulence strongly influenced by buoyancy. Pys. Fluids, 15:2047. Riley, J. J. and Lindborg, E. (2012). Recent progress in stratified turbulence. In Davidson, P. A., Kaneda, Y., and Sreenivasan, K. R., editors, Ten Capters in Turbulence, pages Cambridge University Press. Riley, J. J., Metcalfe, R. W., and Weissman, M. A. (1981). Direct numerical simulations of omogeneous turbulence in density-stratified fluids. AIP Conf. Proc., 76: Spedding, G. R. (1997). Te evolution of initially turbulent bluff-body wakes at ig internal Froude number. J. Fluid Mec., 337: Spedding, G. R. (2002). Vertical structure in stratified wakes wit ig initial Froude number. J. Fluid Mec., 454: Spedding, G. R. (2014). Wake signature detection. Ann. Rev. Fluid Mec., 46: Watanabe, T., Riley, J. J., de Bruyn Kops, S. M., Diamessis, P. J., and Zou, Q. (2016). Turbulent/non-turbulent interfaces in wakes in stably stratified fluids. J. Fluid Mec., 797:R1. Zou, Q. (2015). Reynolds number scaling of a localized stratified turbulent flow and farfield evolution of turbulence-emitted internal waves. PD tesis, Cornell University, Itaca, New York. Zou, Q. and Diamessis, P. J. (2016). Surface manifestation of internal waves emitted by submerged localized stratified turbulence. J. Fluid Mec., 798: VIII t Int. Symp. on Stratified Flows, San Diego, USA, Aug Sept. 1,