Papers
Azimuthal vorticity gradient in the formative stages of vortex breakdown
- M. KUROSAKA, C. B. CAIN, S. SRIGRAROM, J. D. WIMER, D. DABIRI, W. F. JOHNSON, J. C. HATCHER, B. R. THOMPSON, M. KIKUCHI, K. HIRANO, T. YUGÉ, T. HONDA
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- 15 November 2006, pp. 1-28
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This paper is motivated by an observation: in the nascent state of vortex breakdown before it develops into a full-grown radial expansion, an initially straight vortex core first swells, and does so even in a straight pipe for no apparent reason. Although this initial swelling may be explained in many ways according to the perspectives chosen, we offer our own interpretation framed solely within vorticity dynamics: the radial swelling as well as the subsequent growth are induced by the azimuthal vorticity gradient decreasing downstream. The negative azimuthal vorticity gradient first appears at start-up and moves eventually into the region where the circulation reaches its steady-state value. The vorticity gradient can become negative without necessarily being accompanied by a sign-switch of the azimuthal vorticity itself.
The key point – that the negative azimuthal vorticity gradient induces initial radial swelling and its growth – is demonstrated in two analyses. First, a kinematic analysis results in an equation for the radial velocity where the azimuthal vorticity gradient appears as a source term. Its solution shows, in general and explicitly, that the negative azimuthal vorticity gradient does induce radially outward velocity. Two heuristic examples serve to illustrate this point further. In the second analysis, using the equation of motion in the streamline coordinates, the negative azimuthal vorticity gradient is shown to diverge the meridional streamlines radially. A numerical simulation using a modified vortex filament method not only corroborates this role of the azimuthal vorticity gradient in initiating and promoting the radial expansion, but also adds details to track the formation process. Both analyses and simulation support our interpretation that the initial radial swelling and its subsequent growth are induced by the negative azimuthal vorticity gradient.
Bounds on double-diffusive convection
- NEIL J. BALMFORTH, SHILPA A. GHADGE, ATICHART KETTAPUN, SHREYAS D. MANDRE
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- 15 November 2006, pp. 29-50
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We consider double-diffusive convection between two parallel plates and compute bounds on the flux of the unstably stratified species using the background method. The bound on the heat flux for Rayleigh–Bénard convection also serves as a bound on the double-diffusive problem (with the thermal Rayleigh number equal to that of the unstably stratified component). In order to incorporate a dependence of the bound on the stably stratified component, an additional constraint must be included, like that used by Joseph (Stability of Fluid Motion, 1976, Springer) to improve the energy stability analysis of this system. Our bound extends Joseph's result beyond his energy stability boundary. At large Rayleigh number, the bound is found to behave like $R_T^{1/2}$ for fixed ratio $R_S/R_T$, where $R_T$ and $R_S$ are the Rayleigh numbers of the unstably and stably stratified components, respectively.
Force measurements on rising bubbles
- WOODROW L. SHEW, SEBASTIEN PONCET, JEAN-FRANÇOIS PINTON
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- 15 November 2006, pp. 51-60
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The dynamics of millimetre-sized air bubbles rising through still water are investigated using ultrasound velocimetry combined with high-speed video. From measurements of speed and three-dimensional trajectories we calculate time-resolved precise drag and lift forces on the bubble, which give rise to planar zigzag and three-dimensional spiralling motion. Temporal correlations of forces and the oscillatory bubble motions, particularly the lift force, emphasize the importance of the wake vortices found in previous studies.
Lock-release gravity currents and dam-break flows
- ANDREW J. HOGG
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- 15 November 2006, pp. 61-87
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Gravity current and dam-break flows, resulting from the instantaneous release of fluid initially at rest behind a lock gate, are modelled theoretically using the shallow water equations. By analysing the motion in the hodograph plane, the governing equations become linear and hence it is possible to integrate them analytically from lock-release initial conditions. This approach provides many advantages: not only are numerical computations obviated, but the analysis clearly reveals how the nature of the ensuing flow depends on the Froude number, $\hbox{\it Fr}$, at the front of the current. It is also demonstrated that the motion comprises uniform and simple wave regions within which both or one of the characteristic variables are constant, respectively, in addition to complex wave regions within which both characteristic variables vary. These solutions reveal phenomena that have not previously been reported for gravity current flow. Specifically, when $\hbox{\it Fr}\,{>}\,2$, the height and velocity fields become discontinuous at late times at an interior point within the current. Conversely, when $\hbox{\it Fr}\,{<}\,2$, there is a wave-like disturbance that propagates along the length of the current, being reflected successively between the rear wall of the lock and the front of the flow.
The fall of a viscous thread onto a moving surface: a ‘fluid-mechanical sewing machine’
- S. CHIU-WEBSTER, J. R. LISTER
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- 15 November 2006, pp. 89-111
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A viscous thread falling onto a steadily moving horizontal belt shows a surprisingly complex range of behaviour in experiments. Low belt speeds produce coiling, as might be expected from the behaviour of a thread falling onto a stationary surface. High belt speeds produce a steady thread, whose shape is predicted well by theory developed to describe a stretching viscous catenary with surface tension and inertia. Intermediate belt speeds show several novel modes of oscillation, which lay down a wide variety of patterns on the belt. The patterns include meanders, side kicks, slanted loops, braiding, figures-of-eight, Ws, and also period-doubled versions of figures-of-eight, meanders and coiling. The experimental boundary between steady and unsteady behaviour occurs at a slightly lower belt speed than the loss of the steady solution for a stretching catenary.
Scattering of flexural–gravity waves at the boundaries between three floating sheets with applications
- T. D. WILLIAMS, V. A. SQUIRE
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- 15 November 2006, pp. 113-140
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A theoretical model is reported that describes wave propagation between three floating Euler–Bernoulli thin elastic sheets extending, respectively, from $-\infty$ to 0, from 0 to $l$, and from $l$ to $\infty$, with properties, e.g. thickness, that can be specified independently. The sheets are assumed either to be welded together or to have free edges separating them. Two methods of solution are employed – the Wiener–Hopf technique and residue calculus, which allows the theoretical development to be verified at various points along the way. The model generalizes the considerable body of published work concerned with wave propagation into and out of floating ice sheets, and across features contained therein such as cracks, open or refrozen leads and embedded icebergs. It can also be applied to breakwaters, very large floating structures and vessels in a seaway. After validation, results are presented showing (a) the details of how the reflection coefficient depends on the geometry of the configuration being modelled; and (b) how a wave energy spectrum evolves as it propagates in a marginal ice zone composed of a large number of identical (coherent) or randomly specified (incoherent) sea-ice plates, as commonly observed in the polar or subpolar oceans.
Low-dimensional modelling of a confined three-dimensional wake flow
- M. BUFFONI, S. CAMARRI, A. IOLLO, M. V. SALVETTI
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- 15 November 2006, pp. 141-150
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The laminar flow past a square cylinder symmetrically placed between two parallel walls is considered. A classical vortex wake is shed from the cylinder, but three-dimensional instabilities are present and they develop in complicated flow patterns. The possibility of extracting an accurate low-order model of this flow is explored.
High-Reynolds-number steady flow in a collapsible channel
- J. C. GUNERATNE, T. J. PEDLEY
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- 15 November 2006, pp. 151-184
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We have studied steady flow in a two-dimensional channel in which a section of one wall has been replaced by an elastic membrane under dimensionless longitudinal tension $T$ but possessing no bending stiffness. The dimensionless upstream transmural pressure takes a value $P_{\hbox{\it\scriptsize ext}}$, the membrane section is assumed to be long compared with the channel width and its deformation is assumed to remain within the viscous boundary layers. Standard high-Reynolds-number asymptotic methods are applied to arrive at a coupled boundary-layer-membrane problem. A non-zero cross-stream pressure gradient, leading to flow perturbations upstream of the membrane, is included in the analysis.
Linearization of the boundary-layer problem yields firstly an analytic solution at non-zero $P_{\hbox{\it\scriptsize ext}}$ and asymptotically high $T$. This takes the form of an expansion in $T^{-1}$ for which the membrane shape and the flow decouple at each order. Extension of this solution branch to smaller values of the tension suggests a singularity at finite tension, where the deformation of the membrane becomes very large. Secondly, when the upstream transmural pressure is zero the trivial solution is valid for all values of the tension. However, we also obtain eigensolutions where the membrane tension plays the role of eigenvalue. There are thus non-trivial solutions of the problem at these particular values of the tension.
The nonlinear coupled boundary-layer–membrane problem is then solved numerically. A finite-difference, Keller-box, marching scheme is used, together with a shooting algorithm to satisfy the boundary condition at the downstream end of the membrane. This reveals a variety of different solutions, showing the relation between the two cases captured by the linearized analysis and demonstrating the existence of parameter ranges for which no solutions exist under the specified constraints. Such parameter ranges appear not to exist if the downstream, rather than the upstream, transmural pressure is held constant.
The relation to our results of solutions obtained by solving the two-dimensional Navier–Stokes equations directly is discussed. Reasonable agreement between parameters is obtained, once allowance is made for the finite Reynolds number and membrane length in those computations.
A combined direct numerical simulation–particle image velocimetry study of the turbulent near wake
- S. DONG, G. E. KARNIADAKIS, A. EKMEKCI, D. ROCKWELL
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- 15 November 2006, pp. 185-207
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We investigate the near wake of a cylinder at values of Reynolds number corresponding to the onset and development of shear-layer instabilities. By combining quantitative experimental imaging (particle image velocimetry, PIV) and direct numerical simulations at $Re \,{=}\, 3900/4000$ and 10000, we show that the flow structure is notably altered. At higher Reynolds number, the lengths of both the wake bubble and the separating shear layer decrease substantially. Corresponding patterns of velocity fluctuations and Reynolds stress contract towards the base of the cylinder. The elevated values of Reynolds stress at upstream locations in the separated layer indicate earlier onset of shear-layer transition. These features are intimately associated with the details of the shear-layer instability, which leads to small-scale vortices. The simulated signatures of the shear-layer vortices are characterized by a broadband peak at $Re \,{=}\, 3900$ and a broadband high spectral-density ‘plateau’ at $Re \,{=}\, 10\,000$ in the power spectra. The shear-layer frequencies from the present direct numerical simulations study agree well with previous experimentally measured values, and follow the power law suggested by other workers.
Direct numerical simulation of flow and heat transfer in a turbine cascade with incoming wakes
- JAN G. WISSINK, WOLFGANG RODI
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- 15 November 2006, pp. 209-247
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Direct numerical simulations (DNS) of flow in a turbine cascade with heat transfer have been performed. The set-up of the simulations was chosen in close accordance with previous experiments. Three of the experimental situations were simulated: one without free-stream turbulence and two with periodically incoming wakes of different frequency and with different levels of background fluctuation. Hence, the calculations allow us to study the influence of impinging wakes and background fluctuations on the development of the boundary layers and the local Nusselt number along the surfaces of the heated blade. Along the suction side, the pressure gradient is first favourable and then turns adverse near the trailing edge and the boundary layer remains laminar for the case without free-stream turbulence with the Nusselt number showing the typical decay from the leading to the trailing edge. With periodic wakes and background turbulence, transition occurs when the pressure gradient turns adverse, but intermittency persists so that the boundary layer is not fully turbulent when the trailing edge is reached. In this region, the heat transfer is increased significantly by an amount comparable to that found in the experiments. In the pre-transitional region with favourable pressure gradient, the flow acceleration stretches the free-stream vortices, aligning their axis with the flow direction, thereby forming streamwise vortical structures. These increase the laminar heat transfer in this region by 20–30%, which is, however, much less than observed in the experiments. On the pressure side, the pressure gradient is favourable along the entire blade so that the boundary layer remains laminar. Here, the wakes, through their impingement, also generate streamwise vortical structures which, because of the low convection speed on this side, have a very long lifetime compared to the structures along the suction side. Also these structures increase the laminar heat transfer by about 30%, which for the case with the highest wake frequency is again much less than in the experiments. The simulated average level of fluctuations in the laminar parts of the boundary layers is comparable or even higher than that in the experiments so that it seems likely that a difference in the spectral contents causes the discrepancies. The wake turbulence entering the calculation domain corresponds to that in far wakes with relatively small-scale structures, whereas in the experiments the wakes most probably still carried some large-scale fluctuations of the size of the wake width, which have been found to be more effective in increasing laminar heat transfer.
Weakly nonlinear internal gravity wavepackets
- BRUCE R. SUTHERLAND
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- 15 November 2006, pp. 249-258
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Horizontally periodic, vertically localized internal wavepackets evolve nonlinearly due only to interactions between the waves and their wave-induced mean flow. The corresponding weakly nonlinear equation that describes the evolution of the amplitude envelope before the onset of parametric subharmonic instability is examined. The results are compared with fully nonlinear numerical simulations and are shown to lie in excellent agreement for over 15 buoyancy periods. Analysis of the equation shows that the evolution is modulationally unstable if the wave frequency exceeds that of waves with the fastest vertical group speed and if the amplitude is sufficiently large. Waves that move close to the fastest vertical group speed are unstable even if their relative amplitude is a tiny fraction of the inverse relative vertical extent of the wavepacket. At late times in the evolution of an unstable wavepacket third-order dispersion terms become non-negligible and act in conjunction with weakly nonlinear effects to retard the vertical advance of the wavepacket as a whole.
Hierarchical structures in a turbulent free shear flow
- XIAO-QIN JIANG, HAO GONG, JIAN-KUN LIU, MING-DE ZHOU, ZHEN-SU SHE
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- 15 November 2006, pp. 259-286
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We have conducted a detailed analysis of scaling for longitudinal and transverse velocity structure functions in a turbulent free shear flow. The free shear flow is generated via a mixing layer under varying conditions of upstream flow disturbances. Two velocity components are simultaneously measured with a pair of cross-wires at two spanwise locations, with varying positions of the second cross-wire, which allows us to study the statistics of two longitudinal and four transverse velocity increments. Spectra, probability density functions of the velocity increments, and scaling exponents are measured and discussed in relation to flow structures such as streamwise and spanwise vortices. Scaling exponents of the velocity structure functions are interpreted in the phenomenological framework of the hierarchical structure (HS) model of She & Leveque (Phys. Rev. Lett. vol. 72, 1994, p. 336). One HS parameter ($\beta$) specifying similarity between weak and strong vortices is shown to be universal for all structure functions, and another HS parameter ($\gamma$) related to the singularity index of the so-called most intermittent structures shows strong dependence on flow structures. The strongest intermittency occurs in the form of streamwise vortices. The results confirm that coherent small-scale flow structures are responsible for intermittency effects and anomalous scaling, and a complete set of measurements of longitudinal and transverse velocity variations are required to derive flow structural information.
On the model coefficients for the standard and the variational multi-scale Smagorinsky model
- JOHAN MEYERS, PIERRE SAGAUT
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- 15 November 2006, pp. 287-319
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A theoretical analysis is presented on the behaviour of the model coefficients for the well-known Smagorinsky model and two variational multi-scale (VMS) variants of the Smagorinsky model. The dependency on two important parameters is addressed, i.e. the ratio of the LES-filter width $\varDelta$ and the Kolmogorov scale $\eta$ on the one hand, and the ratio of the integral length scale $L$ and the LES-filter width $\varDelta$ on the other hand. First of all, it is demonstrated that the model coefficients vary strongly with $\varDelta/\eta$. By evaluating the model coefficients as functions of the subgrid activity $s$ (which expresses the relative contribution of the subgrid-scale model in the total dissipation, and corresponds to a nonlinear transformation of $\varDelta/\eta$), we show that a classical Lilly–Smagorinsky model overestimates the dissipation, even in cases where the dissipation of the subgrid-scale model is dominant. Therefore, generic and easy-to-use modifications to the different models are proposed, which provide close approximations to the models employing ‘exact’ coefficients. For the standard Smagorinsky model, this modified model corresponds to approximating the eddy viscosity $\nu_t$ as $\nu_t\,{=}\,(\nu_{\mbox{\textit{\scriptsize Lilly}}}^2\,{+}\,\nu^2)^{1/2} -\nu$, with $\nu_{\mbox{\textit{\scriptsize Lilly}}}$ the turbulent viscosity obtained by employing Lilly's classical Smagorinsky constant and $\nu$ the laminar viscosity. Similar easy-to-use relations are presented for the variational multi-scale Smagorinsky models. Next to the $\varDelta/\eta$ dependence of the model coefficients, the $L/\varDelta$ behaviour is also elaborated. Although a strong dependence on $L/\varDelta$ is observed for low values of the ratio, we do not advocate the use of $L/\varDelta$-dependent model coefficients. Rather, the asymptotic $L/\varDelta$ independence and the speed of asymptotic convergence are used as a tool to compare the quality of subgrid-scale models (e.g. $L/\varDelta \,{>}\, 10$ is a minimum order of magnitude for the small–small VMS model), and differences are observed between the standard Smagorinsky model and its two VMS variants. Finally, for the VMS models, the influence of the shape of the high-pass filter, used in the variational multi-scale formulation, is investigated. We observed that smooth high-pass filters result in more robust VMS Smagorinsky models.
Flow-driven opening of a valvular leaflet
- G. PEDRIZZETTI, F. DOMENICHINI
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- 15 November 2006, pp. 321-330
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The understanding of valvular opening is a central issue in cardiac flows, whose analysis is often prohibited by the unavailability of (in vivo) data about tissue properties. Asymptotic or approximate representations of fluid–structure interaction are thus sought. The dynamics of an accelerated stream, in a two-dimensional channel initially closed by a rigid inertialess movable leaflet, is studied as a simple model problem aimed at demonstrating the main phenomena contributing to the fluid–structure interaction. The problem is solved by the coupled numerical solution of equations for the flow and solid. The results show that the leaflet initially opens in a no-shedding regime, driven by fluid mass conservation and a predictable dynamics. Then the leaflet motion jumps, after the saturation of a very rapid intermediate vortex-shedding phase, to the asymptotic slower regime with a stable self-similar wake structure.
Travelling circular waves in axisymmetric rotating convection
- J. M. LOPEZ, A. RUBIO, F. MARQUES
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- 15 November 2006, pp. 331-348
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Rayleigh–Bénard convection in a finite rotating cylinder of moderate aspect ratio (radius four times the depth) is investigated numerically for a fluid of Prandtl number equal to 7 (corresponding essentially to water). We consider the effects of rotation from both the Coriolis force and the centrifugal force and find that the centrifugal force plays a significant dynamic role. In this initial study, we restrict the computations to the axisymmetric subspace in which the convection patterns near onset consist of steady concentric circular cells, the so-called target patterns, which have been studied and observed experimentally under different conditions by a number of investigators. As the convection is driven far enough beyond onset, the steady cellular patterns give way to time-periodic states in which the target patterns travel radially inward. We have identified two such travelling modes, primarily distinguished by one having alternating warm and cold plumes forming at the cylinder sidewall and then propagating radially inward to quench alternately cold and warm plumes on the axis. The other mode always has a cold plume descending on the sidewall and the adjacent warm plume periodically splits into two, with the innermost of the split pair travelling radially inward. The first of these modes is found when the centrifugal force is weak and the second for stronger centrifugal force. The large-scale meridional circulation driven by the centrifugal buoyancy is seen to favour having a cold plume descending on the sidewall, accounting for the switch to the second travelling mode.
Hysteresis and non-uniqueness in the speed of the onset of instability in curtain coating
- J. O. MARSTON, S. P. DECENT, M. J. H. SIMMONS
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- 15 November 2006, pp. 349-363
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The maximum speed of stable coating is determined experimentally by the onset of air entrainment in curtain coating onto a pre-wetted surface over a broad range of dimensionless parameters (Reynolds number: $0.14\,{<}\,\hbox{\it Re}\,{<}\,33.02$; capillary number: $0.19\,{<}\,\hbox{\it Ca}\,{<}\,25.07$). We show not only that the substrate speed at the onset of instability is flow-rate dependent for a pre-wetted surface, but that the speed at which instability occurs is not always unique for a given flow rate.
Particle-size segregation and diffusive remixing in shallow granular avalanches
- J. M. N. T. GRAY, V. A. CHUGUNOV
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- 15 November 2006, pp. 365-398
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Segregation and mixing of dissimilar grains is a problem in many industrial and pharmaceutical processes, as well as in hazardous geophysical flows, where the size-distribution can have a major impact on the local rheology and the overall run-out. In this paper, a simple binary mixture theory is used to formulate a model for particle-size segregation and diffusive remixing of large and small particles in shallow gravity-driven free-surface flows. This builds on a recent theory for the process of kinetic sieving, which is the dominant mechanism for segregation in granular avalanches provided the density-ratio and the size-ratio of the particles are not too large. The resulting nonlinear parabolic segregation–remixing equation reduces to a quasi-linear hyperbolic equation in the no-remixing limit. It assumes that the bulk velocity is incompressible and that the bulk pressure is lithostatic, making it compatible with most theories used to compute the motion of shallow granular free-surface flows. In steady-state, the segregation–remixing equation reduces to a logistic type equation and the ‘S’-shaped solutions are in very good agreement with existing particle dynamics simulations for both size and density segregation. Laterally uniform time-dependent solutions are constructed by mapping the segregation–remixing equation to Burgers equation and using the Cole–Hopf transformation to linearize the problem. It is then shown how solutions for arbitrary initial conditions can be constructed using standard methods. Three examples are investigated in which the initial concentration is (i) homogeneous, (ii) reverse graded with the coarse grains above the fines, and, (iii) normally graded with the fines above the coarse grains. Time-dependent two-dimensional solutions are also constructed for plug-flow in a semi-infinite chute.
Euclidean invariance and weak-equilibrium condition for the algebraic Reynolds stress model
- FUJIHIRO HAMBA
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- 15 November 2006, pp. 399-408
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Taking into account the frame-invariance of a model expression under arbitrarily rotating transformations, Weis & Hutter (J. Fluid Mech. vol. 476, 2003, p. 63) proposed a Euclidean-objective weak-equilibrium condition for the algebraic Reynolds stress model (ARSM). However, Gatski & Wallin (J. Fluid Mech. vol. 518, 2004, p. 147) pointed out that the weak-equilibrium condition proposed is not correct in actual rotating flows such as a rotating channel flow and showed that a non-objective weak-equilibrium condition extended to curved and rotating flows should be assumed. The frame-invariance is an important issue not only for the ARSM but also for general nonlinear eddy-viscosity models. By introducing the corotational derivative of the Reynolds stress, the transport equation for the Reynolds stress can be written to be frame-invariant. It is shown that a frame-invariant expression is desirable as a general model by comparing the error of model expressions in different rotating frames. The extended weak-equilibrium condition of Gatski & Wallin is examined to show that it is in reality objective and it does not contradict a frame-invariant model expression for the Reynolds stress.
Non-Oberbeck–Boussinesq effects in strongly turbulent Rayleigh–Bénard convection
- GUENTER AHLERS, ERIC BROWN, FRANCISCO FONTENELE ARAUJO, DENIS FUNFSCHILLING, SIEGFRIED GROSSMANN, DETLEF LOHSE
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- 15 November 2006, pp. 409-445
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Non-Oberbeck–Boussinesq (NOB) effects on the Nusselt number $Nu$ and Reynolds number $\hbox{\it Re}$ in strongly turbulent Rayleigh–Bénard (RB) convection in liquids were investigated both experimentally and theoretically. In the experiments the heat current, the temperature difference, and the temperature at the horizontal midplane were measured. Three cells of different heights $L$, all filled with water and all with aspect ratio $\Gamma$ close to 1, were used. For each $L$, about 1.5 decades in $Ra$ were covered, together spanning the range $10^8 \,{\le}\, Ra \,{\le}\, 10^{11}$. For the largest temperature difference between the bottom and top plates, $\Delta \,{=}\, 40$K, the kinematic viscosity and the thermal expansion coefficient, owing to their temperature dependence, varied by more than a factor of 2. The Oberbeck–Boussinesq (OB) approximation of temperature-independent material parameters thus was no longer valid. The ratio $\chi$ of the temperature drops across the bottom and top thermal boundary layers became as small as $\chi\,{=}\,0.83$, which may be compared with the ratio $\chi \,{=}\, 1$ in the OB case. Nevertheless, the Nusselt number $Nu$ was found to be only slightly smaller (by at most 1.4%) than in the next larger cell with the same Rayleigh number, where the material parameters were still nearly height independent. The Reynolds numbers in the OB and NOB case agreed with each other within the experimental resolution of about 2%, showing that NOB effects for this parameter were small as well. Thus $Nu$ and $\hbox{\it Re}$ are rather insensitive against even significant deviations from OB conditions. Theoretically, we first account for the robustness of $Nu$ with respect to NOB corrections: the NOB effects in the top boundary layer cancel those which arise in the bottom boundary layer as long as they are linear in the temperature difference $\Delta$. The net effects on $Nu$ are proportional to $ \Delta^2$ and thus increase only slowly and still remain minor despite drastic material-parameter changes. We then extend the Prandtl–Blasius boundary-layer theory to NOB Rayleigh–Bénard flow with temperature-dependent viscosity and thermal diffusivity. This allows calculation of the shift in the bulk temperature, the temperature drops across the boundary layers, and the ratio $\chi$ without the introduction of any fitting parameter. The calculated quantities are in very good agreement with experiment. When in addition we use the experimental finding that for water the sum of the top and bottom thermal boundary-layer widths (based on the slopes of the temperature profiles at the plates) remains unchanged under NOB effects within the experimental resolution, the theory also gives the measured small Nusselt-number reduction for the NOB case. In addition, it predicts an increase by about 0.5% of the Reynolds number, which is also consistent with the experimental data. By studying theoretically hypothetical liquids for which only one of the material parameters is temperature dependent, we are able to shed further light on the origin of NOB corrections in water: while the NOB deviation of $\chi$ from its OB value $\chi \,{=}\, 1$ mainly originates from the temperature dependence of the viscosity, the NOB correction of the Nusselt number primarily originates from the temperature dependence of the thermal diffusivity. Finally, we give predictions from our theory for the NOB corrections if glycerol were used as the operating liquid.
A unified theory of balance in the extratropics
- SIMAL SAUJANI, THEODORE G. SHEPHERD
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- 15 November 2006, pp. 447-464
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Many physical systems exhibit dynamics with vastly different time scales. Often the different motions interact only weakly and the slow dynamics is naturally constrained to a subspace of phase space, in the vicinity of a slow manifold. In geophysical fluid dynamics this reduction in phase space is called balance. Classically, balance is understood by way of the Rossby number $R$ or the Froude number $F$; either $R \,{\ll}\, 1$ or $F \,{\ll}\, 1$.
We examined the shallow-water equations and Boussinesq equations on an $f$-plane and determined a dimensionless parameter $\epsilon$, small values of which imply a time-scale separation. In terms of $R$ and $F$, \[\epsilon = \frac{RF}{\sqrt{R^2 + F^2}}.\] We then developed a unified theory of (extratropical) balance based on $\epsilon$ that includes all cases of small $R$ and/or small $F$. The leading-order systems are ensured to be Hamiltonian and turn out to be governed by the quasi-geostrophic potential-vorticity equation. However, the height field is not necessarily in geostrophic balance, so the leading-order dynamics are more general than in quasi-geostrophy. Thus the quasi-geostrophic potential-vorticity equation (as distinct from the quasi-geostrophic dynamics) is valid more generally than its traditional derivation would suggest. In the case of the Boussinesq equations, we have found that balanced dynamics generally implies hydrostatic balance without any assumption on the aspect ratio; only when the Froude number is not small and it is the Rossby number that guarantees a time-scale separation must we impose the requirement of a small aspect ratio to ensure hydrostatic balance.