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In this paper, we propose an upwind compact difference method with fourth-order spatial accuracy and second-order temporal accuracy for solving the streamfunction-velocity formulation of the two-dimensional unsteady incompressible Navier–Stokes equations. The streamfunction and its first-order partial derivatives (velocities) are treated as unknown variables. Three types of compact difference schemes are employed to discretize the first-order partial derivatives of the streamfunction. Specifically, these schemes include the fourth-order symmetric scheme, the fifth-order upwind scheme, and the sixth-order symmetric scheme derived by combining the two parts of the fifth-order upwind scheme. As a result, the fourth-order spatial discretization schemes are established for the Laplacian term, the biharmonic term, and the nonlinear convective term, along with the Crank–Nicolson scheme for the temporal discretization. The unconditional stability characteristic of the scheme for the linear model is proved by discrete von Neumann analysis. Moreover, six numerical experiments involving three test problems with the analytic solutions, and three flow problems including doubly periodic double shear layer, lid-driven cavity flow, and dipole-wall interaction are carried out to demonstrate the accuracy, robustness, and efficiency of the present method. The results indicate that the present method not only has good numerical performance but also exhibits quite efficiency. [ABSTRACT FROM AUTHOR]

The Taylor cone jet is an electrohydrodynamic flow typically induced by applying an external electric field to a liquid within a capillary, commonly utilized in colloidal thrusters. This flow generation involves a complex multiphase and multiphysics process, with stability contingent upon specific operational parameters. The operational window is intrinsically linked to flow rate and applied electric voltage magnitude. High voltages can induce atomization instabilities, resulting in the production of an electrospray. Our study presents initially a numerical investigation into the atomization process of a Taylor cone jet using computational fluid dynamics. Implemented within OpenFOAM, our numerical model utilizes a volume-of-fluid approach coupled with Maxwell's equations to incorporate electric body forces into the incompressible Navier–Stokes equations. We employ the leaky-dielectric model, subjecting the interface between phases to hydrodynamic surface tension and electric stress (Maxwell stress). With this model, we studied the droplet breakup of a heptane liquid jet, for a range of operation of 1.53–7.0 nL s−1 and 2.4–4.5 kV of extraction. First, the developed high-fidelity numerical solution is studied for the jet breakup and acceleration of the droplets. Second, we integrate a machine learning model capable of extending the parametric windows of operation. Additionally, we explore the influence of extractor and acceleration plates on colloidal propulsion systems. This work offers a numerical exploration of the Taylor cone–jet transition and droplet acceleration using novel, numerically accurate approaches. Subsequently, we integrate machine learning models, specifically an artificial neural network and a one-dimensional convolutional neural network, to predict the jet's performance under conditions not previously evaluated by computationally heavy numerical models. Notably, we demonstrate that the convolutional neural network outperforms the artificial neural network for this type of application data, achieving a 2% droplet size prediction accuracy. [ABSTRACT FROM AUTHOR]

In this study, phenomenological observations and the Kreuer interpretation of the origin of viscosity were used to develop a computational method for solving the turbulence problem of incompressible viscous Newtonian fluids based on extended Navier–Stokes (N–S) equations. The shear process in fluid flow was hypothesized to be accompanied by eddy formation, and the effects of eddies on the convection and diffusion were considered. The classical N–S equations were improved to obtain extended N–S equations. The extended equations are closed, and the sources of the velocity fluctuations are explicitly considered to be additional convection and diffusion. The extended equations are compatible with the classical N–S equations; thus, they can describe laminar and turbulent flows in a unified manner. In fluid flow simulations, the equations describing the mean flow quantities could be directly obtained from the extended N–S equations without any additional turbulence models. A numerical investigation was carried out to verify the extended equations by exploring the flow over a cube placed in a channel. The simulation results were compared with both the large eddy simulation and experimental results. [ABSTRACT FROM AUTHOR]

We present a novel machine learning approach to reduce the dimensionality of state variables in stratified turbulent flows governed by the Navier–Stokes equations in the Boussinesq approximation. The aim of the new method is to perform an accurate reconstruction of the temperature and the three-dimensional velocity of geophysical turbulent flows developing non-homogeneities, starting from a low-dimensional representation in latent space, yet conserving important information about non-Gaussian structures captured by high-order moments of distributions. To achieve this goal, we modify the standard convolutional autoencoder (CAE) by implementing a customized loss function that enforces the accuracy of the reconstructed high-order statistical moments. We present results for compression coefficients up to 16, demonstrating how the proposed method is more efficient than a standard CAE in performing dimensionality reduction of simulations of stratified geophysical flows characterized by intermittent phenomena, as observed in the atmosphere and the oceans. [ABSTRACT FROM AUTHOR]

Rogue waves are known to be much more common on jet currents. A possible explanation was put forward in Shrira and Slunyaev ["Nonlinear dynamics of trapped waves on jet currents and rogue waves," Phys. Rev. E 89, 041002(R) (2014)] for the waves trapped on current robust long-lived envelope solitary waves localized in both horizontal directions become possible, such wave patterns cannot exist in the absence of the current. In this work, we investigate interactions between envelope solitons of essentially nonlinear trapped waves by means of the direct numerical simulation of the Euler equations. The solitary waves remain localized in both horizontal directions for hundreds of wave periods. We also demonstrate a high efficiency of the developed analytic nonlinear mode theory for description of the long-lived solitary patterns up to remarkably steep waves. We show robustness of the solitons in the course of interactions and the possibility of extreme wave generation as a result of solitons' collisions. Their collisions are shown to be nearly elastic. These robust solitary waves obtained from the Euler equations without weak nonlinearity assumptions are viewed as a plausible model of rogue waves on jet currents. [ABSTRACT FROM AUTHOR]

Renewable energy converters, such as bio-inspired fluttering foils, are gaining popularity due to their eco-friendly properties. However, the system with multiple objects has received scant attention. Here, we analyze how spacing influences the transverse (one-degree-of-freedom) vortex-induced vibration of two tandem identical elliptic cylinders at a constant Reynolds number by employing a wide range of reduced velocities ( U r ∈ [ 2 , 14 ]) and space ratios ( L ∗ ∈ [ 2 , 6 ]). The incompressible Navier–Stokes equations are solved using the overset mesh method in the OpenFOAM® library. The findings indicate that the wake structure goes through eight distinct wake modes, as well as two gap flow patterns (reattachment and co-shedding). Vibrational responses, force parameters, and flow patterns determine three spacing configurations. At a small spacing ( L ∗ = 2), the upstream cylinder (U C) has the traditional lock-in (the frequency ratio f y / f n ≃ 0.95 –1.05) at the reduced velocity ( U r ≃ 7), and the downstream cylinder (D C) has a narrow lock-in region around U r ≃ 9. However, the U C has a wide soft-lock-in (the synchronization region of f y / f n ≃ 1.15) at high reduced velocities ( U r ≃ 8 –10). Here, the transverse vibrations of both cylinders, but especially the D C , reach relatively high amplitudes. At a moderate spacing ( L ∗ = 3), the U C bears a lock-in zone analogous to a single cylinder with the same mass ratio, while the D C shows a vast soft-lock-in zone ( U r ≃ 8 –14). At a large spacing ( L ∗ = 4 , 5 , and 6), the amplitude of the D C is often larger than that of a single cylinder when it is in the lock-in region. The D C exhibits a peak in amplitude at Ur = 7 and a wake-galloping region for Ur > 12. [ABSTRACT FROM AUTHOR]

The present numerical investigation delves into the intricate interplay between Mach number (Ms), flow characteristics, and vorticity dynamics within a T106A low-pressure turbine (LPT) blade passage. The two-dimensional (2D) compressible Navier–Stokes equations are solved using a high-accuracy, dispersion relation preserving methodology, which is validated against benchmark direct numerical simulations. Four Ms ranging from 0.15 to 0.30 are computed in order to display the intricate response of compressibility on the separation-induced transition process. The emergence and evolution of unsteady separation bubbles along the suction surface of the T106A blade are explored, revealing a growing trend with Ms. The time-averaged boundary layer parameters evaluated along the suction surface display a delayed separation with a smaller streamwise extent with increasing Ms. However, an overall increase in the blade profile loss and a decrease in turbulent mixing are observed with increasing Ms, suggesting a detrimental effect on LPT performance. Applying the compressible enstrophy transport equation (CETE) to the flow in a T106A blade passage reveals that while a linear relationship exists between Ms and certain CETE budget terms, other terms have a nuanced dependency, which paves the way for future investigations into the role of compressibility on enstrophy dynamics. [ABSTRACT FROM AUTHOR]

We present a computational study of free surface flows with rheologically complex interfaces in the film formation region of a slot coater. The equations of motion for incompressible Newtonian liquids in the bulk flow are coupled with the Boussinesq–Scriven constitutive equation for viscous interfaces in the dynamic boundary condition at the liquid-air free surface and solved with a mixed finite element method. We show that the interfacial viscosity plays a major role in the flow dynamics and operating limits of slot coating. We find that the interfacial viscosity makes viscous interfaces generally stiffer than their simple counterparts, affecting both the normal and the tangential stress jumps across the free surface. As a result, the interfacial viscosity counteracts the meniscus retraction and slows down the film flow, increasing the development length over the substrate and changing the topology of the recirculation region in the coating bead. Remarkably, we also find that the interfacial viscosity can substantially broaden the operating boundaries of the coating window associated with the low-flow limit, suggesting that surface-active components can be suitably designed to allow for the stable production of thinner films at higher speeds by tuning interfacial material properties in slot coating applications. [ABSTRACT FROM AUTHOR]

This study aims to investigate numerically the Rayleigh–Bénard Convection using an in-house Fortran 90 code based on the lattice Boltzmann method. The bottom wall is equipped with two hot circular/elliptical sources and the right wall is open. The non-linear coupled differential governing equations are formulated using the lattice Boltzmann equation associated with the Boussinesq approximation. The simulations are conducted for (103 ≤ Ra ≤ 106) and Pr = 0.7 (corresponding to air). The code verification showed a good reliability of the present mesoscopic numerical approach. Several configurations related to the size and shape of the heaters were studied. It was found that elliptically shaped heat sources provide higher heat transfer rates compared to circular sources. [ABSTRACT FROM AUTHOR]

Analysis on natural convective heat transfer in different engineering systems allows optimization of the technical apparatus. For this purpose, numerical simulation of the fluid flow and heat transport within the system is combined with study of entropy generation. The latter is very important considering the Gouy–Stodola theorem of thermodynamics. The present research deals with the mathematical modeling of thermal convection and entropy generation in a right-angled trapezoidal cavity under the influence of sinusoidal vertical wall temperature distribution. Control Oberbeck–Galerkin finite element technique has solved Boussinesq equations formulated using the non-dimensional primitive variables. Analyses of flow structures, thermal and entropy generation patterns for different values of the Rayleigh number, and parameters of non-uniform wall temperature were performed. It was found that a rise in the sinusoidal wall temperature amplitude increases the average Nusselt and Bejan numbers and average entropy generation. Moreover, growth in the non-uniform wall temperature wave number decreases the energy transport strength and Bejan number. [ABSTRACT FROM AUTHOR]

Generating a digital twin of any complex system requires modeling and computational approaches that are efficient, accurate, and modular. Traditional reduced order modeling techniques are targeted at only the first two, but the novel nonintrusive approach presented in this study is an attempt at taking all three into account effectively compared to their traditional counterparts. Based on dimensionality reduction using proper orthogonal decomposition (POD), we introduce a long short-term memory neural network architecture together with a principal interval decomposition (PID) framework as an enabler to account for localized modal deformation. As an effective partitioning tool for breaking the Kolmogorov barrier, our PID framework, therefore, can be considered a key element in the accurate reduced order modeling of convective flows. Our applications for convection-dominated systems governed by Burgers, Navier-Stokes, and Boussinesq equations demonstrate that the proposed approach yields significantly more accurate predictions than the POD-Galerkin method and could be a key enabler toward near real-time predictions of unsteady flows. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a modular deep neural network (DNN) framework for data-driven reduced order modeling of dynamical systems relevant to fluid flows. We propose various DNN architectures which numerically predict evolution of dynamical systems by learning from either using discrete state or slope information of the system. Our approach has been demonstrated using both residual formula and backward difference scheme formulas. However, it can be easily generalized into many different numerical schemes as well. We give a demonstration of our framework for three examples: (i) Kraichnan-Orszag system, an illustrative coupled nonlinear ordinary differential equation, (ii) Lorenz system exhibiting chaotic behavior, and (iii) a nonintrusive model order reduction framework for the two-dimensional Boussinesq equations with a differentially heated cavity flow setup at various Rayleigh numbers. Using only snapshots of state variables at discrete time instances, our data-driven approach can be considered truly nonintrusive since any prior information about the underlying governing equations is not required for generating the reduced order model. Our a posteriori analysis shows that the proposed data-driven approach is remarkably accurate and can be used as a robust predictive tool for nonintrusive model order reduction of complex fluid flows. [ABSTRACT FROM AUTHOR]

We consider a two-dimensional fluid exposed to Kolmogorov's forcing cos(ny) and heated from above. The stabilizing effects of temperature are taken into account using the Boussinesq approximation. The fluid with no temperature stratification has been widely studied and, although relying on strong simplifications, it is considered an important tool for the theoretical and experimental study of transition to turbulence. In this paper, we are interested in the set of transitions leading the temperature stratified fluid from the laminar solution [ U ∝ cos (n y) , 0 , T ∝ y] to more complex states until the onset of chaotic states. We will consider Reynolds numbers 0 < Re ≤ 30, while the Richardson numbers shall be kept in the regime of weak stratifications (Ri ≤ 5 × 10−3). We shall first review the non-stratified Kolmogorov flow and find a new period-tripling bifurcation as the precursor of chaotic states. Introducing the stabilizing temperature gradient, we shall observe that higher Re are required to trigger instabilities. More importantly, we shall see new states and phenomena: the newly discovered period-tripling bifurcation is supercritical or subcritical according to Ri; more period-tripling and doubling bifurcations may depart from this new state; strong enough stratifications trigger new regions of chaotic solutions and, on the drifting solution branch, non-chaotic bursting solutions. [ABSTRACT FROM AUTHOR]

Nonhydrostatic effects on convectively forced mesoscale flows are theoretically investigated using a linearized, two-dimensional, steady-state, nonrotating, Boussinesq airflow system with prescribed convective forcing. The nondimensionalized airflow system contains the nonhydrostaticity factor β = U/Na, where U is the basic-state wind speed, N is the basic-state buoyancy frequency, and a is the half-width of the convective forcing. In an inviscid-limit system, the solution for vertical velocity is classified into the propagating mode (k ≤ β−1, where k is the nondimensional horizontal wavenumber) and the evanescent mode (k > β−1). As β increases, an alternating wavy pattern of updrafts and downdrafts appears downstream of the convective forcing with a nondimensional horizontal wavelength of 2πβ corresponding to the nondimensional critical horizontal wavenumber kc = β−1. The momentum flux analysis shows that the alternating updrafts and downdrafts are almost horizontally propagating gravity waves of the propagating mode whose k is slightly smaller than kc and that these gravity waves strengthen the momentum flux above the convective forcing. In a viscous system, the solution for vertical velocity has propagating and decaying components simultaneously that cannot be explicitly separated. Here, the propagating mode and two evanescent modes are defined by comparing the magnitudes of the nondimensional vertical wavenumber and decay rate. For large viscous coefficients, the k-range of the propagating mode becomes narrow and the alternating updrafts and downdrafts are dissipated. As β increases, the propagating mode, which strengthens the momentum flux above the convective forcing, is effectively dissipated even with a small viscous coefficient. [ABSTRACT FROM AUTHOR]

In this paper, we first revisit the celebrated Boussinesq approximation in stratified flows. Using scaling arguments we show that when the background shear is weak, the Boussinesq approximation yields either (i) A t ≪ O (1) or (ii) F r c 2 ≪ O (1) , where At is the ratio of density variation to the mean density and Frc is the ratio of the phase speed to the long wave speed. The second clause implies that, in contrast to the commonly accepted notion, a flow with large density variations can also be Boussinesq. Indeed, we show that deep water surface gravity waves are Boussinesq, while shallow water surface gravity waves are not. However, in the presence of moderate/strong shear, the Boussinesq approximation implies the conventionally accepted A t ≪ O (1). To understand the inertial effects of density variation, our second objective is to explore various non-Boussinesq shear flows and study different kinds of stably propagating waves that can be present at an interface between two fluids of different background densities and vorticities. Furthermore, three kinds of density interfaces—neutral, stable, and unstable—embedded in a background shear layer are investigated. Instabilities ensuing from these configurations, which include Kelvin-Helmholtz, Holmboe, Rayleigh-Taylor, and triangular-jet, are studied in terms of resonant wave interactions. The effects of density stratification and the shear on the stability of each of these flow configurations are explored. Some of the results, e.g., the destabilizing role of density stratification, stabilizing role of shear, etc., are apparently counter-intuitive, but physical explanations are possible if the instabilities are interpreted from wave interaction perspective. [ABSTRACT FROM AUTHOR]

In this paper, a recently developed direct numerical simulation technique, the Finite Element Fictitious Boundary Method (FEM-FBM) [K. Walayat et al., "An efficient multi-grid finite element fictitious boundary method for particulate flows with thermal convection," Int. J. Heat Mass Transfer 126, 452–465 (2018)], is used to simulate sedimentation of an elliptic particle with thermal convection. The momentum and temperature flow fields are coupled with the aid of Boussinesq approximation. The thermal and momentum interactions between solid and fluid phases are handled by using the fictitious boundary method (FBM). The continuity, momentum, and energy equations are solved on a fixed Eulerian mesh which is independent of flow features by using a multi-grid finite element scheme. Two validation tests are conducted to show the accuracy of the present method, and then the effects of thermal properties of fluid on the sedimentation of an elliptic particle are studied. It is demonstrated that the dynamics of hot elliptic particle sedimentation depend on the thermal diffusivity and thermal expansion of the fluid. A comparative study of the forces and torque acting on the hot, cold, and isothermal particle is reported. Moreover, different sedimentation modes of hot and cold elliptic particles are identified in an infinitely long channel. The mechanism of transitions of particle settling modes from tumbling to inclined and then to the horizontal mode is discovered. Also, we discovered a new sedimentation mode of the hot elliptic particle in cold fluid, i.e., the vertical mode. Furthermore, buoyancy effects for the catalyst particle are studied at different initial orientations. [ABSTRACT FROM AUTHOR]

We investigate the oscillatory convective instability in a liquid layer with a deformable free surface. We provide linear stability analysis within a non-Boussinesq approach as our aim is to examine the influence of deformability of the free surface on the Rayleigh–Bénard–Marangoni instability. Within this approach, fluid is assumed to be isothermally incompressible; the density variations are accounted for in the continuity equation and in the buoyancy and inertial terms of the momentum equations. The numerical results show significant differences in stability behavior as compared to results obtained using the Boussinesq approach. Moreover, a novel oscillatory mode of instability is revealed for the zero Marangoni number and zero gravity. This result could not be obtained within the framework of the conventional Boussinesq approximation. Thorough investigation of the novel oscillatory mode let us propose the mechanism of this mode. It is connected with the capillary wave that enforced with thermal expansion of fluid. Weakly nonlinear analysis shows that supercritical branching of standing rolls is possible. [ABSTRACT FROM AUTHOR]

Criticality plays a central role in the study of reductions and stability of hydrodynamical systems. At critical points, it is often the case that nonlinear reductions with dispersion arise to govern solution behavior. By considering when such models become bidirectional and lose their initial dispersive properties, it will be shown that higher order dispersive models may be supported in hydrodynamical systems. Precisely, this equation is a two-way Boussinesq equation with sixth order dispersion. The case of two layered shallow water is considered to illustrate this, and it is reasoned why such an environment is natural for such a system to emerge. Further, it is demonstrated that the regions in the parameter space for nontrivial flow, which admit this reduction, are vast and in fact form a continuum. The reduced model is then numerically simulated to illustrate how the two-way and higher dispersive properties suggest more exotic families of solitary wave solutions can emerge in stratified flows. [ABSTRACT FROM AUTHOR]

We present a combined theoretical-experimental investigation of particle-driven gravity currents advancing in circular cross section channels in the high-Reynolds number Boussinesq regime; the ambient fluid is either homogeneous or linearly stratified. The predictions of the theoretical model are compared with experiments performed in lock-release configuration; experiments were performed with conditions of both full-depth and partial-depth locks. Two different particles were used for the turbidity current, and the full range 0 ≤ S ≤ 1 of the stratification parameter was explored (S = 0 corresponds to the homogeneous case and S = 1 when the density of the ambient fluid and of the current are equal at the bottom). In addition, a few saline gravity currents were tested for comparison. The results show good agreement for the full-depth configuration, with the initial depth of the current in the lock being equal to the depth of the ambient fluid. The agreement is less good for the partial-depth cases and is improved by the introduction of a simple adjustment coefficient for the Froude number at the front of the current and accounting for dissipation. The general parameter dependencies and behaviour of the current, although influenced by many factors (e.g., mixing and internal waves), are well predicted by the relatively simple model. [ABSTRACT FROM AUTHOR]

The heuristic Fermi-Neumann box model is used to study the non-Boussinesq lock exchange flows in an infinite horizontal channel. This allowed us to find the asymptotic growth laws for intrusion tongues at a late stage of their development. It is shown that these laws are essentially different. The classical scenario of the linear growth x1 ∞ t is supported only by the lower (heavy) intrusion tongue. The same is not true for the upper (light) intrusion tongue that obeys the law x2 ∞ t=ln t. [ABSTRACT FROM AUTHOR]

Numerical experiments have been carried out using a nonhydrostatic and non-Boussinesq regional oceanic circulation model to investigate the nonlinear processes generated by supercritical tidal flow in shallow straits. Our approach relies on idealized direct numerical simulations inspired by oceanic observations. By analyzing a large set of simulations, a regime diagram is proposed for the nonlinear processes generated in the lee of these straits. The results show that the topography shape of the strait plays a crucial role in the formation of internal solitary waves (ISWs) and in the occurrence of local breaking events. Both of these nonlinear processes are important turbulence producing phenomena. The topographic control, observed in mode 1 ISW formation in previous studies [Y. Dossmann, F. Auclair, and A. Paci, "Topographically induced internal solitary waves in a pycnocline: Primary generation and topographic control," Phys. Fluids 25, 066601 (2013) and Y. Dossmann et al., "Topographically induced internal solitary waves in a pycnocline: Ultrasonic probes and stereo-correlation measurements," Phys. Fluids 26, 056601 (2014)], is clearly reproducible for mode-2 ISW above shallow straits. Strong plunging breaking events are observed above "narrow" straits (straits with a width less than mode 1 wavelength) when the fluid velocity exceeds the local mode 1 wave speed. These results are a step towards future works on vertical mixing quantification and localization around complex strait areas. [ABSTRACT FROM AUTHOR]

The present study is focused on Lapwood convection in isotropic porous media saturated with non- Newtonian shear thinning fluid. The non-Newtonian rheological behavior of the fluid is modeled using the general viscosity model of Carreau-Yasuda. The convection configuration consists of a shallow porous cavity with a finite aspect ratio and subject to a vertical constant heat flux, whereas the vertical walls are maintained impermeable and adiabatic. An approximate analytical solution is developed on the basis of the parallel flow assumption, and numerical solutions are obtained by solving the full governing equations. The Darcy model with the Boussinesq approximation and energy transport equations are solved numerically using a finite difference method. The results are obtained in terms of the Nusselt number and the flow fields as functions of the governing parameters. A good agreement is obtained between the analytical approximation and the numerical solution of the full governing equations. The effects of the rheological parameters of the Carreau-Yasuda fluid and Rayleigh number on the onset of subcritical convection thresholds are demonstrated. Regardless of the aspect ratio of the enclosure and thermal boundary condition type, the subcritical convective flows are seen to occur below the onset of stationary convection. Correlations are proposed to estimate the subcritical Rayleigh number for the onset of finite amplitude convection as a function of the fluid rheological parameters. Linear stability of the convective motion, predicted by the parallel flow approximation, is studied, and the onset of Hopf bifurcation, from steady convective flowto oscillatory behavior, is found to depend strongly on the rheological parameters. In general, Hopf bifurcation is triggered earlier as the fluid becomes more and more shear-thinning. [ABSTRACT FROM AUTHOR]

The present work extends the semi-integrated method proposed by Antuono and Brocchini ["Beyond Boussinesq-type equations: Semi-integrated models for coastal dynamics," Phys. Fluids 25(1), 016603 (2013)], which comprises a subset of depth-averaged equations (similar to Boussinesq-like models) and a Poisson equation that accounts for vertical dynamics. Here, the subset of depth-averaged equations has been reshaped in a conservative-like form and both the Poisson equation formulations proposed by Antuono and Brocchini ["Beyond Boussinesq-type equations: Semi-integrated models for coastal dynamics," Phys. Fluids 25(1), 016603 (2013)] are investigated: the former uses the vertical velocity component (formulation A) and the latter a specific depth semi-averaged variable, Υ (formulation B). Our analyses reveal that formulation A is prone to instabilities as wave nonlinearity increases. On the contrary, formulation B allows an accurate, robust numerical implementation. Test cases derived from the scientific literature on Boussinesq-type models--i.e., solitary and Stokes wave analytical solutions for linear dispersion and nonlinear evolution and experimental data for shoaling properties--are used to assess the proposed solution strategy. It is found that the present method gives reliable predictions of wave propagation in shallow to intermediate waters, in terms of both semi-averaged variables and conservation properties. [ABSTRACT FROM AUTHOR]

The dielectrophoresis of a Newtonian uncharged drop in the presence of an axisymmetric nonuniform DC electric field is studied analytically. The present study is focused on the effects of interfacial viscosities on the dielectrophoretic motion and shape deformation of an isolated suspended drop. The interfacial viscosities generate surface-excess viscous stress which is modeled as a two-dimensional Newtonian fluid which obeys the Boussinesq-Scriven constitutive law with constant values of interfacial tension, interfacial shear, and dilatational viscosities. In the regime of small drop deformation, we have obtained analytical solution for the drop velocity and deformed shape by neglecting surface charge convection and fluid inertia. Our study demonstrates that the drop velocity is independent of the interfacial shear viscosity, while the interfacial dilatational viscosity strongly affects the drop velocity. The interfacial viscous effects always retard the dielectrophoretic motion of a perfectly conducting/dielectric drop. Notably, the interfacial viscous effects can retard or augment the dielectrophoretic motion of a leaky dielectric drop depending on the electrohydrodynamic properties. The shape deformation of a leaky dielectric drop is found to decrease (or increase) due to interfacial shear (or dilatational) viscosity. [ABSTRACT FROM AUTHOR]

Planar, Boussinesq, compositional gravity currents formed by the release of a fixed volume of heavier fluid from a closed lock and advancing on an inclined no-slip bottom surface in a reservoir with a horizontal free surface are investigated based on 3-D large eddy simulation. The initial region containing the lock fluid has a rectangular shape. Simulations are conducted for bottom slope angles, ϴ, between 0° and 60°. The running length of the inclined bottom was sufficiently long to allow a detailed study of the evolution, front dynamics and structure of the current during the latter stages of the deceleration phase (front velocity reduces with time). Results show that currents advancing over inclined surfaces with ϴ > 10° are characterized by the formation of an intensified mixed vortex (IMV) at the back of the head. The IMV forms faster and its coherence, circulation, and size increase monotonically with increasing bottom slope angle. The paper discusses how the buoyancy in the head varies with varying bottom slope angle and with time. In particular, for ϴ ≥ 30°, the current reaches a regime where the total buoyancy of the head and IMV is close to a constant and the value of this constant increases with increasing ϴ. During this regime, the head mainly loses buoyancy to the IMV. For ϴ ≥ 40°, a close to linear decay of the head buoyancy with time is observed during the later stages of this regime. Simulation results show that, while for relatively small bottom slope angles most of the sediment is entrained beneath the head, for ϴ > 20° the IMV has a much larger capacity to entrain the sediment compared to the head region past the initial stages of the propagation of the current. This means that sediment entrainment patterns of currents propagating over highly inclined surfaces are qualitatively very different from the widely studied case of currents propagating over horizontal surfaces. The paper also discusses the different regimes observed in the temporal evolution of the front velocity and the applicability of theoretical models derived based on the data obtained for relatively small bottom slope angles and a relatively short evolution of the current to describe the evolution of currents propagating over large bottom slope angles and/or at large times after the start of the deceleration phase. While it is found that mixing increases monotonically with increasing ϴ, the largest total kinetic energy for a given front position is observed for ϴ = 30°-40°. Results also show that the largest magnitude of the bed friction velocity is induced for ϴ = 30°-40°, which means that the currents with the largest capacity to entrain sediment are those with the largest rate of increase of total kinetic energy with the propagation distance. [ABSTRACT FROM AUTHOR]

Nonlinear evolution of pancake-like vortices in a uniformly rotating and stratified fluid is studied using a 3D Boussinesq numerical model at large Rossby numbers. After the initial stage of viscous decay, the simulations reveal exponential growth of toroidal circulation cells (aka Taylor vortices) at the peripheral annulus with a negative Rayleigh discriminant. At the nonlinear stage, these thin cells redistribute the angular momentum and density differently at the levels of radial outflow and inflow. Resulting layering, with a vertical stacking of sharp variations in velocity and density, enhances small-scale mixing and energy decay. Characteristic detectable stretching patterns are produced in the density field. The circulation patterns, induced by centrifugal instability, tend to homogenize the angular momentum in the vicinity of the unstable region. We demonstrate that the peak intensity of the cells and the vortex energy decay are dramatically reduced by the earth's rotation due to conservation of total absolute angular momentum. The results have important implications for better understanding the fate of pancake vortices and physical mechanisms of energy transfer in stratified fluids. [ABSTRACT FROM AUTHOR]

We present an experimental investigation, supported by a theoretical model, of the motion of lock-release, constant inflow, and time varying inflow gravity currents (GCs) into a linearly stratified ambient fluid at large Reynolds number. The aim is the experimental validation of a simple model able to predict the slumping phase front speed and the asymptotic self-similar front speed for rectangular and circular cross section channels. The first investigated system is of Boussinesq type with the dense current (salt water dyed with aniline) released in a circular channel of 19 cm diameter and 400 cm long (605 cm in the inflow experiments), half-filled of linearly stratified ambient fluid (salt water with varying salt concentration). The second system has the same components but with a channel of rectangular cross section of 14 cm width, 11 cm ambient fluid depth, and 504 cm length. The density stratification of the ambient fluid was obtained with a computer controlled set of pumps and of mixing tanks. For the experiments with inflow, a multi-pipes drainage system was set at the opposite end with respect to the inflow section, computer controlled to avoid the selective withdrawal. The numerous experiments (28 for circular cross section, lock release; 26 for circular and 14 for rectangular cross section, constant inflow (fluid volume ∝ tα, with α = 1); 6 for circular cross section, linearly increasing inflow (α = 2)), with several combination of the stratification parameter (0 < S < 1) confirm the theory within ≈30% (≈40% for a single series of experiments), which is considered a good result in view of the various underlying simplifications and approximations. The results on the front speed of the GCs are discussed in the presence of the internal waves, which have a celerity given by a theoretical and experimentally tested model for the rectangular but not for the circular cross section. The theoretical analysis of internal waves in circular cross sections has been extended and experimentally validated. [ABSTRACT FROM AUTHOR]

This paper presents three-dimensional direct numerical simulations of laboratoryscale double-diffusive gravity currents. Flow is governed by the incompressible Navier-Stokes equations under the Boussinesq approximation, with salinity and temperature coupled to the equations of motion using a nonlinear approximation to the UNESCO equation of state. The effects of vertical boundary conditions and current volume are examined, with focus on flow pattern development, current propagation speed, three-dimensionalization, dissipation, and stirring and mixing. It was observed that no-slip boundaries cause the gravity current head to take the standard lobe-and-cleft shape and encourage both a greater degree and an earlier onset of three-dimensionalization when compared to what occurs in the case of a free-slip boundary. Additionally, numerical simulations with no-slip boundary conditions experience greater viscous dissipation, stirring, and mixing when compared to similar configurations using free-slip conditions. [ABSTRACT FROM AUTHOR]

A steady pseudo-plane ideal flow (PIF) model is derived from the 3D Euler equations under Boussinesq approximation. The model is solved analytically to yield high-degree polynomial exact solutions. Unlike quadratic flows, the cubic and quartic solutions display reduced geometry in the form of straightline jet, circular vortex, and multipolar strain field. The high-order circular-vortex solutions are vertically aligned and even the non-aligned multipolar strain-field solutions display vertical concentricity. Such geometry reduction is explained by an analytical theorem stating that only straightline jet and circular vortex have functional solutions to the PIF model. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to highlight the existence of simple algebraic expressions linking the second order moments of velocity and concentration in Rayleigh-Taylor turbulence, in the Boussinesq limit. Focusing on the concentration variance, these relations allow to underline the influence of mixing on the remaining second order correlations, as well as on the growth rate of the mixing zone. [ABSTRACT FROM AUTHOR]

Double criticality and its nonlinear implications are considered for stratified N-layer shallow water flows with N = 1, 2, 3. Double criticality arises when the linearization of the steady problem about a uniform flow has a double zero eigenvalue. We find that there are two types of double criticality: non-semisimple (one eigenvector and one generalized eigenvector) and semi-simple (two independent eigenvectors). Using a multiple scales argument, dictated by the type of singularity, it is shown that the weakly nonlinear problem near double criticality is governed by a two-way Boussinesq equation (non-semisimple case) and a coupled Kortewegde Vries equation (semisimple case). Parameter values and reduced equations are constructed for the examples of two-layer and three-layer stratified shallow water hydrodynamics. [ABSTRACT FROM AUTHOR]

We report upon a laboratory experimental study of dense full- and partial-depth locks released gravity currents propagating through a two-layer ambient and over sinusoidal topography of amplitude A and wavelength λ. Attention is restricted to a Boussinesq flow regime and the initial (or slumping) stage of motion. Because of the presence of the topography, the lower layer depth and channel depth vary in the downstream direction by as much as ±52% and ±40%, respectively. Despite such large variations, the instantaneous gravity current front speed typically changes by only a small amount from its average value, Ū. We conclude that the reason for this unexpected observation is a counterbalancing between the contraction/expansion of the channel on one hand and along-slope variations of the buoyancy force on the other hand. Compared to the case of gravity current flow over a horizontal surface, the topography has a retarding effect on Ū we characterize the variation of Ū with the following parameters: A, A/λ, the interface height, the height of gravity current fluid inside the lock, and the layer densities as characterized by S ≡ (ρ1 - ρ2)/(ρ0 - ρ2) where ρ0 is the gravity current density, ρ1 is the lower ambient layer density, and ρ2 is the upper ambient layer density. The topography also alters the structure of the gravity current head by inducing large-scaleKelvin-Helmholtz (K-H) vortices directly downstream of regions of significant shear. These vortices are transient flow features; after formation and saturation, they relax at which point gravity current fluid sloshes back and forth in the topographic troughs. A simple geometric model is presented that prescribes the minimum number of topographic peaks a gravity current fluid will overcome. As in the horizontal bottom case, the forward advance of the gravity current can excite a downstream-propagating interfacial disturbance, which is more prominent for larger S. We identify when the gravity current front will travel faster (supercritical flow) or slower (subcritical flow) than the interfacial disturbance based on S and A/λ. Unlike in the horizontal-bottom case, however, we typically also observe interfacial oscillations far ahead of the gravity current front. These have wavelength approximately equal to λ and are due to spatial variations in the speed of the ambient return flow. [ABSTRACT FROM AUTHOR]

Effects of the baroclinic torque on wave propagation normally neglected under the Boussinesq approximation is investigated here, with a special focus on the associated consequences for the mechanistic interpretation of shear instability arising from the interaction between a pair of vorticity-propagating waves. To illustrate and elucidate the physical effects that modify wave propagation, we consider three examples of increasing complexity: wave propagation supported by a uniform background flow; wave propagation supported on a piecewise-linear basic state possessing one jump; and an instability problem of a piecewise-linear basic state possessing two jumps, which supports the possibility of shear instability. We find that the non-Boussinesq effects introduce a preference for the direction of wave propagation that depends on the sign of the shear in the region where waves are supported. This in turn affects phase-locking of waves that is crucial for the mechanistic interpretation for shear instability, and is seen here to have an inherent tendency for stabilisation. [ABSTRACT FROM AUTHOR]

Steady but generally unstable solutions of the 2D Boussinesq equations are obtained for no-slip boundary conditions and Prandtl number 7. The primary solution that bifurcates from the conduction state at Rayleigh number Ra 1708 has been calculated up to Ra 5.106 and its Nusselt number is Nu ~ 0.143 Ra0.28 with a delicate spiral structure in the temperature field. Another solution that maximizes Nu over the horizontal wavenumber has been calculated up to Ra = 109 and scales as Nu ~ 0.115 Ra0.31 for 107 < Ra = 109, quite similar to 3D turbulent data that show Nu ~ 0.105 Ra0.31 in that range. The optimum solution is a simple yet multi-scale coherent solution whose horizontal wavenumber scales as 0.133 Ra0.217. That solution is unstable to larger scale perturbations and in particular to mean shear flows, yet it appears to be relevant as a backbone for turbulent solutions, possibly setting the scale, strength, and spacing of elemental plumes. [ABSTRACT FROM AUTHOR]

The focus of this paper is to establish the effect of source Reynolds number, Re0, on the mean rise heights of Boussinesq miscible fountains and thereby the range of Re0 for which rise heights are independent of Re0. We present results from experiments on aqueous-saline fountains spanning source Reynolds numbers of 15 ≤ Re0 ≤ 4000 and source Froude numbers of 0.3 ≤ Fr0 ≤ 40. We provide threshold values for the source Reynolds number, i.e., values above which fountain rise height may be regarded as independent of Re0. For Reynolds numbers immediately beneath these threshold values, the fountain rise height increases as Re0 decreases""in agreement with Lin and Armfield ["The Reynolds and Prandtl number dependence of weak fountains," Comput. Mech. 31, 379""389 (2003)]. At even lower Re0, we show that the rise heights reach a maximum and that further decreases in Re0 decrease the rise height attained""in agreement with Phillipe et al., ["Penetration of a negatively buoyant jet in a miscible liquid," Phys. Fluids 17, 053601 (2005)]. As such, we resolve an apparent inconsistency within the existing fountain literature. [ABSTRACT FROM AUTHOR]

We study the effects of a stable density stratification on the turbulent dynamics of thin fluid layers forced at intermediate scales. By means of a set of high-resolution numerical simulations, performed within the Boussinesq approximation, we investigate how the stratification and confinement affect the mechanisms of kinetic and potential energy transfer. The detailed analysis of the statistics of the energy-dissipation rates and energy-exchange rates and of the spectral fluxes of potential and kinetic energy shows that stratification provides a new channel for the energy transfer towards small scales which reduces the large-scale flux of kinetic energy. We also discuss the role of vortex stretching and enstrophy flux in the transfer of kinetic energy into potential energy at small scales. [ABSTRACT FROM AUTHOR]

In order to understand the characteristics of Rayleigh-Bénard convection of cold water near its density maximum in a cubical cavity with different thermal boundaries on the sidewalls, a series of direct numerical simulations were carried out by using the control volume approach. Flow pattern structures and their bifurcation series were detailedly analyzed through the bifurcation diagram, and the heat transfer ability of each branch was discussed. Results showed that both the flow pattern and heat transfer characteristic of Rayleigh-Bénard convection of cold water have many notable differences as compared with those of common fluids satisfying the Boussinesq approximation; the flow behavior depends both qualitatively and quantitatively on the thermal boundary conditions on the sidewalls. Furthermore, there are multiple flow pattern coexistence and hysteresis phenomenon during the flow pattern transition. [ABSTRACT FROM AUTHOR]

Natural convection fluid flow in air-filled tall tilted cavities is studied numerically with a direct projection method applied on the unsteady Boussinesq approximation in primitive variables. The study is focused on the so called cat's eyes and multiple disjoint cells as the aspect ratio A and the angle of inclination 0 of the cavity vary. Results have already been reported with primitive and stream function-vorticity variables. The former are validated with the latter ones, which in turn were validated through mesh size and time-step independence studies. The new results complemented with the previous ones lead to find out the fluid motion and heat transfer invariant properties of this thermal phenomenon, which is the novelty here. [ABSTRACT FROM AUTHOR]

In this paper, we present numerical simulations of rapidly rotating Rayleigh-Benard convection in the Boussinesq approximation with stress-free boundary conditions. At moderately low Rossby number and large Rayleigh number, we show that a largescale depth-invariant flow is formed, reminiscent of the condensate state observed in two-dimensional flows. We show that the large-scale circulation shares many similarities with the so-called vortex, or slow-mode, of forced rotating turbulence. Our investigations show that at a fixed rotation rate the large-scale vortex is only observed for a finite range of Rayleigh numbers, as the quasi-two-dimensional nature of the flow disappears at very high Rayleigh numbers. We observe slow vortex merging events and find a non-local inverse cascade of energy in addition to the regular direct cascade associated with fast small-scale turbulent motions. Finally, we show that cyclonic structures are dominant in the small-scale turbulent flow and this symmetry breaking persists in the large-scale vortex motion. [ABSTRACT FROM AUTHOR]

We investigate properties of convective solutions of the Boussinesq thermal convection in a moderately rotating spherical shell allowing the respective rotation of the inner and outer spheres due to the viscous torque of the fluid. The ratio of the inner and outer radii of the spheres, the Prandtl number, and the Taylor number are fixed to 0.4, 1, and 500², respectively. The Rayleigh number is varied from 2.6 × 104 to 3.4 × 104. In this parameter range, the behaviours of obtained asymptotic convective solutions are almost similar to those in the system whose inner and outer spheres are restricted to rotate with the same constant angular velocity, although the difference is found in the transition process to chaotic solutions. The convective solution changes from an equatorially symmetric quasi-periodic one to an equatorially symmetric chaotic one, and further to an equatorially asymmetric chaotic one, as the Rayleigh number is increased. This is in contrast to the transition in the system whose inner and outer spheres are assumed to rotate with the same constant angular velocity, where the convective solution changes from an equatorially symmetric quasi-periodic one, to an equatorially asymmetric quasi-periodic one, and to equatorially asymmetric chaotic one. The inner sphere rotates in the retrograde direction on average in the parameter range; however, it sometimes undergoes the prograde rotation when the convective solution becomes chaotic. [ABSTRACT FROM AUTHOR]

Rotating Rayleigh-Bénard convection in water is studied in direct numerical simulations, where the temperature dependence of the viscosity, the thermal conductivity, and the density within the buoyancy term is taken into account. In all simulations, the arithmetic mean of the lowest and highest temperature in the system equals 40°C, corresponding to a Prandtl number of Pr = 4.38. In the non-rotational case, the Rayleigh number Ra ranges from 107 to 1.16 × 109 and temperature differences ▵ up to 70 K are considered, whereas in the rotational case the inverse Rossby number range from 0.07 ⩽ 1/Ro⩽ 14.1 is studied for ▵ = 40K with the focus on Ra = 108. The non-Oberbeck-Boussinesq (NOB) effects in water are reflected in an up to 5.5K enhancement of the center temperature and in an up to 5% reduction of the Nusselt number. The top thermal and viscous boundary layer thicknesses increase and the bottom ones decrease, while the sum of the corresponding top and bottom thicknesses remains as in the classical Oberbeck-Boussinesq (OB) case. Rotation applied to NOB thermal convection reduces the central temperature enhancement. Under NOB conditions the top (bottom) thermal and viscous boundary layers become equal for a slightly larger (smaller) inverse Rossby number than in the OB case. Furthermore, for rapid rotation the thermal bottom boundary layers become thicker than the top ones. The Nusselt number normalized by that in the non-rotating case depends similarly on 1/Ro in both, the NOB and the OB cases. The deviation between the Nusselt number under OB and NOB conditions is minimal when the thermal and viscous boundary layers are equal. [ABSTRACT FROM AUTHOR]

We apply numerical and analytic techniques to study the Boussinesq approximation in Rayleigh-Taylor and Richtmyer-Meshkov instabilities. In this approximation, one sets the Atwood number A equal to zero except where it multiplies the acceleration g or velocity-jump ▵ v. While this approximation is generally applied to low-A systems, we show that it can be applied to high-A systems also in certain regimes and to the "bubble" part of the instability, i.e., the penetration depth of the lighter fluid into the heavier fluid. It cannot be applied to the spike. We extend the Boussinesq approximation for incompressible fluids and show that it always overestimates the penetration depth but the error is never more than about 41%. The effect of compressibility is studied by analytic techniques in the linear regime which indicate that compressibility has the opposite effect and the Boussinesq approximation underestimates bubbles by about 14%. We also present direct numerical simulations of two compressible systems which have approximately the same A▵ v: a low-A air/CO2 system shocked at Ms = 1.57, and a high-A air/SF6 system shocked at Ms = 1.24. While the bubbles are approximately equal, the lower-A system has a shorter (less penetrating) spike; however, because its mushrooms are more tightly wound, the low-A system has the larger interface area. [ABSTRACT FROM AUTHOR]

In this paper we study the problem of thermal convection in a laterally heated, finite, horizontal cylinder. We consider cylinders of moderate aspect ratio (height/diameter 2) containing a small Prandtl number fluid (s < 0.026) typical of molten metals and molten semiconductors. We use the Navier-Stokes and energy equations in the Boussinesq approximation to calculate numerically the basic steady states, analyze their linear stability, and compute some nonlinear secondary flows originated from the instabilities. All the calculated flows and the stability analysis are characterized by their symmetry properties. Due to the confined cylindrical geometry, -presence of lateral walls and lids-, all the flows are completely three dimensional even for the basic steady states. In the range of Prandtl numbers studied, we have identified four different types of instabilities, either oscillatory or stationary. The physical mechanisms, shear or buoyancy, of the corresponding flow transitions have been analyzed. As the value of the Prandtl number approaches s = 0.026 the scenario of bifurcations becomes more complicated due to the existence of two different stable basic states originated in a saddle-node bifurcation; a fact that had been overlooked in previous works. [ABSTRACT FROM AUTHOR]

We investigate the stability and bifurcation of Boussinesq thermal convection in a moderately rotating spherical shell, with the inner sphere free to rotate as a solid body due to the viscous torque of the fluid. The ratio of the inner and outer radii of the spheres and the Prandtl number are fixed to 0.4 and 1, respectively. The Taylor number is varied from 522 to 5002 and the Rayleigh number from 1500 to 10 000. In this parameter range, the finite-amplitude traveling wave solutions, which have four-fold symmetry in the azimuthal direction, bifurcate supercritically at the critical points. The inner sphere rotates in the prograde direction due to the viscous torque of the fluid when the rotation rate is small while it rotates in the retrograde direction when the rotation rate is large. However, the stable region of these traveling wave solutions is quantitatively similar to that in the co-rotating system where the inner and outer spheres rotate with the same angular velocity. The structures of convective motions of these solutions such as the radial component of velocity are quantitatively similar to those in the co-rotating system, but the structure of mean zonal flows is effectively changed by the inner sphere rotation. [ABSTRACT FROM AUTHOR]

We report a high-resolution numerical study of two-dimensional (2D) miscible Rayleigh-Taylor (RT) incompressible turbulence with the Boussinesq approximation. An ensemble of 100 independent realizations were performed at small Atwood number and unit Prandtl number with a spatial resolution of 2048 × 8193 grid points. Our main focus is on the temporal evolution and the scaling behavior of global quantities and of small-scale turbulence properties. Our results show that the buoyancy force balances the inertial force at all scales below the integral length scale and thus validate the basic force-balance assumption of the Bolgiano-Obukhov scenario in 2D RT turbulence. It is further found that the Kolmogorov dissipation scale η(t) ∼ t1/8, the kinetic-energy dissipation rate &eh;u(t) ∼ t-1/2, and the thermal dissipation rate &eh;θ(t) ∼ t-1. All of these scaling properties are in excellent agreement with the theoretical predictions of the Chertkov model ['Phenomenology of Rayleigh-Taylor turbulence,' Phys. Rev. Lett. 91, 115001 (2003)]. We further discuss the emergence of intermittency and anomalous scaling for high order moments of velocity and temperature differences. The scaling exponents ξpr of the pth-order temperature structure functions are shown to saturate to ξ∞r≃0.78±0.15 for the highest orders, p ∼ 10. The value of ξ∞r and the order at which saturation occurs are compatible with those of turbulent Rayleigh-Bénard (RB) convection [A. Celani, T. Matsumoto, A. Mazzino, and M. Vergassola, 'Scaling and universality in turbulent convection,' Phys. Rev. Lett. 88, 054503 (2002)], supporting the scenario of universality of buoyancy-driven turbulence with respect to the different boundary conditions characterizing the RT and RB systems. [ABSTRACT FROM AUTHOR]

Vortex-shedding suppression in two-dimensional mixed convective flows past circular and square cylinders is investigated numerically at two supercritical Reynolds numbers, Re = 60 and 100, at a fixed Prandtl (Pr) number of 0.71. The Richardson number (Ri) and free-stream orientation (α) are varied in the range [0, 1.6] and [0, π/2], respectively. The investigations involve the numerical solutions of mass, momentum, and energy equations subject to Boussinesq approximation in generalized curvilinear body-fitted coordinates. The critical Richardson numbers corresponding to the onset of suppression of vortex-shedding are determined for different free-stream orientations using the numerical data and the Stuart-Landau theory. For the case of circular cylinder, the critical Richardson number exhibits a 'cosine-law' with respect to the free-stream orientation, while a non-monotonic trend is observed for the case of the square cylinder. By examining the near critical steady flow field data, two distinct components of the baroclinic vorticity generation rate are identified that appear to control the shedding suppression laws (relationships between the critical Richardson number and free-stream orientation) in theRi-α parametric space for the circular and the square cylinders. Supported by numerical experiments, the plausible roles of these baroclinic vorticity generation rate components are identified and utilized to theoretically deduce the functional forms of the shedding suppression laws that agree with the laws observed in the numerical experiments. [ABSTRACT FROM AUTHOR]

The influence of surface deformations on the Rayleigh–Bénard–Marangoni instability of a uniform layer of a non-Boussinesq fluid heated from below is investigated. In particular, the stability of the conductive state of a horizontal fluid layer with a deformable surface, a flat isothermal rigid lower boundary, and a convective heat transfer condition at the upper free surface is considered. The fluid is assumed to be isothermally incompressible. In contrast to the Boussinesq approximation, density variations are accounted for in the continuity equation and in the buoyancy and inertial terms of the momentum equations. Two different types of temperature dependence of the density are considered: linear and exponential. The longwave instability is studied analytically, and instability to perturbations with finite wavenumber is examined numerically. It is found that there is a decrease in stability of the system with respect to the onset of longwave Marangoni convection. This result could not be obtained within the framework of the conventional Boussinesq approximation. It is also shown that at Ma = 0 the critical Rayleigh number increases with Ga (the ratio of gravity to viscous forces or Galileo number). At some value of Ga, the Rayleigh–Bénard instability vanishes. This stabilization occurs for each of the density equations of state. At small values of Ga and when deformation of the free surface is important, it is shown that there are significant differences in stability behavior as compared to results obtained using the Boussinesq approximation. [ABSTRACT FROM AUTHOR]

Using a Boussinesq model with improved linear dispersion, we show numerical evidence that bottom non-uniformity can provoke significantly increased probability of freak waves as a wave field propagates into shallower water, in agreement with recent experimental results [K. Trulsen, H. Zeng, and O. Gramstad, 'Laboratory evidence of freak waves provoked by non-uniform bathymetry,' Phys. Fluids 24, 097101 (2012)]. Increased values of skewness, kurtosis, and probability of freak waves can be found on the shallower side of a bottom slope, with a maximum close to the end of the slope. The increased probability of freak waves is typically seen to endure some distance into the shallower domain, before it decreases and reaches a stable value depending on the depth. The maxima of the statistical parameters are observed both in the case where there is a region of constant depth after the slope, and in the case where the uphill slope is immediately followed by a downhill slope. In the case that waves propagate over a slope from shallower to deeper water, however, we do not find any increase in freak wave occurrence. [ABSTRACT FROM AUTHOR]

In addition to the conservation of mass and horizontal momentum, existing analytical models of gravity currents traditionally require an assumption about the conservation or loss of energy along specific streamlines for closure. Here, we show that the front velocity of gravity currents can be predicted as a function of their height from mass and momentum balances alone by considering the evolution of interfacial vorticity. This approach does not require information on the pressure field and therefore avoids the need for the energy conservation arguments invoked by earlier models. Predictions by the new theory are shown to be in close agreement with results from numerical simulations. We also discuss the influence of downstream mixing on the front velocity predicted by this theory. [ABSTRACT FROM AUTHOR]