Shallow Fluid Layer Research Articles

Experimental characterization of the 3D dynamics of a laminar shallow vortex dipole

Experimental results on the dynamics of a vortex dipole evolving in a shallow fluid layer are presented. In particular, the generation of a spanwise vortex at the front of the dipole is observed in agreement with previous experiments at larger Reynolds numbers. The results show that this secondary vortex is of comparable strength to the dipole. The present physical analysis suggests that the origin of this structure involves the stretching induced by the dipole of the boundary-layer vorticity generated by the dipole’s advection over the no-slip bottom.

The early stages of shallow flows in an inclined flume

The motion of an initially quiescent shallow layer of fluid within an impulsively tilted flume is modelled using the nonlinear shallow water equations. Analytical solutions for the two-dimensional flow are constructed using the method of characteristics and, in regions where neither of the characteristic variables is constant, by adopting hodograph variables and using the Riemann construction for the solution. These solutions reveal that the motion is strongly influenced by the impermeable endwalls of the flume. They show that discontinuous solutions emerge after some period following the initiation of the flow and that for sufficiently long flumes there is a moving interface between wetted and dry regions. Using the hodograph variables we are able to track the evolution of the flow analytically. After the discontinuities develop, we also calculate the velocity and height fields by using jump conditions to express conservation of mass and momentum across the shock and thus we show how the hydraulic jump moves within the domain and how its magnitude grows. In addition to providing the behaviour of the flow in this physical scenario, this unsteady solution also provides an important test case for numerical algorithms designed to integrate the shallow water equations.

Instability in gravity-driven flow over uneven surfaces

We consider the gravity-driven laminar flow of a shallow fluid layer down an uneven incline with the principal objective of investigating the effect of bottom topography and surface tension on the stability of the flow. The equations of motion are approximations to the Navier–Stokes equations which exploit the assumed relative shallowness of the fluid layer. Included in these equations are diffusive terms that are second order relative to the shallowness parameter. These terms, while small in magnitude, represent an important dependence of the flow dynamics on the variation in bottom topography and play a significant role in theoretically capturing important aspects of the flow. Some of the second-order terms include normal shear contributions, while others lead to a nonhydrostatic pressure distribution. The explicit dependence on the cross-stream coordinate is eliminated from the equations of motion by means of a weighted residual approach. The resulting mathematical formulation constitutes an extension of the modified integral-boundary-layer equations proposed by Ruyer-Quil and Manneville [Eur. Phys. J. B 15, 357 (2000)] for flows over even surfaces to flows over variable topography. A linear stability analysis of the steady flow is carried out by taking advantage of Floquet–Bloch theory. A numerical scheme is devised for solving the nonlinear governing equations and is used to calculate the evolution of the perturbed equilibrium flow. The simulations are used to confirm the analytical predictions and to investigate the interfacial wave structure. The bottom profile considered in this investigation corresponds to periodic undulations characterized by measures of wavelength and amplitude. Conclusions are drawn on the combined effect of bottom topography and surface tension.

On the onset of natural convection in differentially heated shallow fluid layers with internal heat generation

This paper reports the results of an analytical and numerical investigation to determine the effect of internal heat generation on the onset of convection, in a differentially heated shallow fluid layer. The case with the bottom plate at a temperature higher than the top plate mimics the classical Rayleigh Benard convection. However, internal heat generation adds a new dimension to the problem. Linear stability analysis is first carried out for the case of an infinitely wide cavity. The effect of aspect ratio on the onset of convection is studied by solving the full Navier–Stokes equations and the equation of energy and observing the temperature contours. A bisection algorithm is used for an accurate prediction of the onset. The numerical results are used to plot the stability curves for eight different aspect ratios. A general correlation is developed to determine the onset of convection in a differentially heated cavity for various aspect ratios. For an aspect ratio of 10, it is seen that the cavity approaches the limit of an infinite cavity. Analytical results obtained by using linear stability analysis agree very well with the “full” CFD simulations, for the above aspect ratio.

Two-Dimensional Navier–Stokes Turbulence in Bounded Domains

In this review we will discuss recent experimental and numerical results of quasi-two-dimensional decaying and forced Navier–Stokes turbulence in bounded domains. We will give a concise overview of developments in two-dimensional turbulence research, with emphasis on the progress made during the past 10 years. The scope of this review concerns the self-organization of two-dimensional Navier–Stokes turbulence, the quasi-stationary final states in domains with no-slip boundaries, the role of the lateral no-slip walls on two-dimensional turbulence, and their role on the possible destabilization of domain-sized vortices. The overview of the laboratory experiments on quasi-two-dimensional turbulence is restricted to include only those carried out in thin electromagnetically forced shallow fluid layers and in stratified fluids. The effects of the quasi-two-dimensional character of the turbulence in the laboratory experiments will be discussed briefly. As a supplement, the main results from numerical simulations of forced and decaying two-dimensional turbulence in rectangular and circular domains, thus explicitly taking into account the lateral sidewalls, will be summarized and compared with the experimental observations.

Laboratory Modeling of Geophysical Vortices

Investigators have modeled oceanic and atmospheric vortices in the laboratory in a number of different ways, employing background rotation, density effects, and geometrical confinement. In this article, we address barotropic vortices in a rotating fluid, emphasizing generation techniques, instability issues, and topography effects in particular. We then review work on vortices in shallow fluid layers, including topography effects on vortices in coastal areas and the role of vortices in tidal exchange between two connected basins.

The three-dimensional structure of an electromagnetically generated dipolar vortex in a shallow fluid layer

Many experiments have been performed in electromagnetically driven shallow fluid layers to study quasi-two-dimensional (Q2D) turbulence, the shallowness of the layer commonly is assumed to ensure Q2D dynamics. In this paper, however, we demonstrate that shallow fluid flows exhibit complex three-dimensional (3D) structures. For this purpose we study one of the elementary vortex structures in Q2D turbulence, the dipolar vortex, in a shallow fluid layer. The flow evolution is studied both experimentally and by numerical simulations. Experimentally, stereoscopic particle image velocimetry is used to measure instantaneously all three components of the velocity field in a horizontal plane, and 3D numerical simulations provide the full 3D velocity and vorticity fields over the entire flow domain. It is found that significant and complex 3D structures and vertical motions occur throughout the flow evolution, i.e., during and after the forcing phase. We conclude that the bottom friction is not the main mechanism leading to three-dimensionality of the flow but rather the impermeability of the boundaries. It is further shown that free-surface deformations, i.e., gravity waves, are of minor importance too as a mechanism to generate 3D motion. Furthermore, it is demonstrated that the observations are not due to three-dimensionality introduced by the forcing mechanism but intrinsically due to the flow dynamics itself. The flow evolution is analyzed with respect to its quasi-two-dimensionality by adopting the ratio of “horizontal” to “vertical” kinetic energies, the normalized horizontal divergence, and a measure of the relaxation to a Poiseuille-like profile. An important observation is that, although the relative magnitude of the vertical velocity as compared to the horizontal flow components decreases for decreasing fluid depth, the vertical profile of the horizontal flow relaxes rather slowly to a Poiseuille-like profile, i.e., not faster than the bottom friction time scale.

Dipole-wall collision in a shallow fluid

Recent experiments on a freely evolving dipolar vortex in a homogeneous shallow fluid layer have clearly shown the importance of vertical secondary flows on top of the primary horizontal motion. The present contribution focuses on the interaction of such a dipolar vortex with a sidewall. Accurate measurements of the three velocity components in a single horizontal plane have been performed using the Stereoscopic Particle Image Velocimetry (SPIV) technique. The experimental results, supported by numerical simulations, indicate that the complex vertical structure of a shallow-layer dipole becomes even more complex during the collision process. The observed growth of the kinetic energy associated with enhanced vertical motion pinpoints the strong discrepancies between vortex-wall interactions in shallow fluid layers and in purely two-dimensional wall-bounded turbulence.

A Mathematical And Numerical Study Of Roll Waves

In this paper we analyze the gravity-driven laminar flow of a shallow fluid layer down an uneven incline with the principal objective of investigating the effect of bottom topography on the instability of the flow. The equations of motion are approximations to the Navier-Stokes equations which exploit the assumed relative shallowness of the fluid layer and fast laminar flow. The explicit dependence on the cross-stream coordinate is eliminated by depth-integrating the equations of motion. A linear stability analysis of the steady flow is carried out by taking advantage of Floquet-Bloch theory. A numerical scheme is devised for solving the nonlinear governing equations and is used to calculate the evolution of the perturbed steady flow. The results are used to confirm the analytical predictions and to investigate the structure of roll waves. The effect of bottom topography for cases where the bottom profile corresponds to periodic undulations is discussed.

Reflecting tidal wave beams and local generation of solitary waves in the ocean thermocline

It is generally accepted that ocean internal solitary waves can arise from the interaction of the barotropic tide with the continental shelf, which generates an internal tide that in turn steepens and forms solitary waves as it propagates shorewards. Some field observations, however, reveal large-amplitude internal solitary waves in deep water, hundreds of kilometres away from the continental shelf, suggesting an alternative generation mechanism: tidal flow over steep topography forces a propagating beam of internal tidal wave energy which impacts the thermocline at a considerable distance from the forcing site and gives rise to internal solitary waves there. Motivated by this possibility, a simple nonlinear long-wave model is proposed for the interaction of a tidal wave beam with the thermocline and the ensuing local generation of solitary waves. The thermocline is modelled as a density jump across the interface of a shallow homogeneous fluid layer on top of a deep uniformly stratified fluid, and a finite-amplitude propagating internal wave beam of tidal frequency in the lower fluid is assumed to be incident and reflected at the interface. The induced weakly nonlinear long-wave disturbance on the interface is governed in the far field by an integral-differential equation which accounts for nonlinear and dispersive effects as well as energy loss owing to radiation into the lower fluid. Depending on the strength of the thermocline and the intensity of the incident beam, nonlinear wave steepening can overcome radiation damping so a series of solitary waves may arise in the thermocline. Sample numerical solutions of the governing evolution equation suggest that this mechanism is quite robust for typical oceanic conditions.

Nonlinear resonant characteristics of shallow fluid layers

Results are presented from an experimental investigation of the motion of a shallow layer of water in a square tank that was oscillated horizontally with small amplitude at frequencies close to the natural frequency of the layer. The aim was to assess the validity of certain theoretical predictions relating to nonlinear resonance in shallow layers based on asymptotic analysis. These concern the hysteretic nature of the resonance profile and the existence of non-trivial oscillatory components associated with shock-like discontinuities in the derived solution for the free surface configuration. The experimental results support the theoretical predictions and show a coincidence in parameter space between the regions of hysteresis and oscillatory transition.

Influence of initial conditions on decaying two-dimensional turbulence

A numerical study of freely decaying two-dimensional turbulence is presented to show how the time evolution of characteristic flow quantities is influenced by the initial conditions. The numerical method adopted is a standard two-dimensional (2D) Fourier pseudospectral algorithm with Newtonian viscosity. Vortex statistics are extracted using a vortex census method. Several characteristic initial vorticity distributions analogous to those employed in previous laboratory experiments are considered. Some of the initial vorticity distributions have in common a dominant subset of vortices. Reliable statistics are obtained for each characteristic distribution by ensemble averaging. For the dominant subset, the time evolutions of the global enstrophy and the number density, respectively, are found to collapse confirming the self-similarity of 2D turbulence, one of the starting points for the scaling theory proposed by Carnevale et al. [Phys. Rev. Lett. 66, 2735 (1991)]. The relationship between the relevant scaling exponents as predicted by the scaling theory is not confirmed, however, and thus seems questionable within the considered parameter range. Furthermore, power-law exponents for both the number density and the global enstrophy are found to be affected by the initial number density and the initial vortex size distribution. Our results thus suggest that for experiments in shallow fluid layers, any agreement with a universal scaling exponent seems coincidental.

Wave analysis of different splitting methods in the linearised shallow water equations

The shallow water equations describe motions in a shallow, incompressible and nonviscous fluid layer on the rotating Earth. Due to their relative simplicity, they are widely used for testing and analysing new numerical methods developed for weather prediction models. In this paper, we apply different operator splittings, based on directional decomposition, to the linearised form of the shallow water equations obtained by the method of small perturbations. This system has three types of harmonic wave solutions with known dispersion relations and phase velocities. We investigate how the application of operator splitting modifies these important characteristics, and compare the performance of different splitting methods from this point of view.

Stability analysis of thermo-bioconvection in suspensions of gravitactic microorganisms in a fluid layer

In this paper, we investigate the effect of heating or cooling from below on the stability of a suspension of motile gravitactic microorganisms in a shallow fluid layer. The linear perturbation theory is used to obtain the stability diagram and the critical conditions for the onset of convection. It is found that the thermo-effects may either stabilize or destabilize the suspension, and decrease or increase the wavelength of the bioconvective pattern.

Investigation of the onset of bioconvection in a suspension of oxytactic microorganisms subjected to high-frequency vertical vibration

This paper investigates the effect of vertical vibration on the stability of a dilute suspension of oxytactic microorganisms in a shallow horizontal fluid layer. For the case of high-frequency vibration, an averaging method is utilized to derive the equations describing the mean flow by decomposing the solutions of governing equations into two components: one that varies slowly with time, and a second that varies rapidly with time. Linear stability analysis is used to investigate the stability of the obtained averaged equations. It is predicted that high-frequency, low-amplitude vertical vibration has a stabilizing effect on a suspension of oxytactic microorganisms confined in a shallow horizontal layer.

Forcing function control of Faraday wave instabilities in viscous shallow fluids

We investigate the relationship between the linear surface wave instabilities of a shallow viscous fluid layer and the shape of the periodic, parametric-forcing function (describing the vertical acceleration of the fluid container) that excites them. We find numerically that the envelope of the resonance tongues can only develop multiple minima when the forcing function has more than two local extrema per cycle. With this insight, we construct a multi-frequency forcing function that generates at onset a nontrivial harmonic instability which is distinct from a subharmonic response to any of its frequency components. We measure the corresponding surface patterns experimentally and verify that small changes in the forcing waveform cause a transition, through a bicritical point, from the predicted harmonic short-wavelength pattern to a much larger standard subharmonic pattern. Using a formulation valid in the lubrication regime (thin viscous fluid layer) and a Wentzel-Kramers-Brillouin (WKB) method to find its analytic solutions, we explore the origin of the observed relation between the forcing function shape and the resonance tongue structure. In particular, we show that for square and triangular forcing functions the envelope of these tongues has only one minimum, as in the usual sinusoidal case.

Flow of aerosol in a 3D alveolated bend: experimental measurements by Particle Image Velocimetry (PIV) and Particle Tracking Velocimetry (PTV)

Recent experiments on a freely evolving dipolar vortex in a homogeneous shallow fluid layer have clearly shown the importance of vertical secondary flows on top of the primary horizontal motion. The present contribution focuses on the interaction of such a dipolar vortex with a sidewall. Accurate measurements of the three velocity components in a single horizontal plane have been performed using the Stereoscopic Particle Image Velocimetry (SPIV) technique. The experimental results, supported by numerical simulations, indicate that the complex vertical structure of a shallow-layer dipole becomes even more complex during the collision process. The observed growth of the kinetic energy associated with enhanced vertical motion pinpoints the strong discrepancies between vortex-wall interactions in shallow fluid layers and in purely two-dimensional wall-bounded turbulence.

Investigation of the onset of thermo-bioconvection in a suspension of oxytactic microorganisms in a shallow fluid layer heated from below

This paper investigates the combined effect of density stratification due to oxytactic upswimming and heating from below on the stability of a suspension of motile oxytactic microorganisms in a shallow fluid layer. Different from traditional bioconvection, thermo-bioconvection has two destabilizing mechanisms that contribute to creating the unstable density stratification. This problem may be relevant to a number of geophysical applications, such as the investigation of the dynamics of some species of thermophiles (heat loving microorganisms) living in hot springs. By performing a linear stability analysis, we obtained a correlation between the critical value of the bioconvection Rayleigh number and the traditional, “thermal” Rayleigh number. It is established that heating from below makes the system more unstable and helps the development of bioconvection.

310 内部発熱対流における温度場の可視化と解析(S42-1 流れの可視化実験解析(1),S42 流れの可視化実験解析)

Natural convection in a shallow fluid layer induced by internal heat generation was investigated experimentally. A horizontal temperature field at each Rayleigh number was visualized by Thermo-chromic Liquid Crystal (TLC), and then the obtained images were converted to the temperature field by a calibration curve, which correlates between color information extracted from the image and temperature. Individual cells were extracted from the converted temperature field and a variation of area of the cells with respect to the Rayleigh number represented a cell dilatation. Variations of statistical values, standard deviation of the temperature field and mean size of a cell, with Rayleigh number increases showed that there might be a gradual transition of the convection.

Jupiter's and Saturn's convectively driven banded jets in the laboratory

The banded patterns of cloud and wind are among the most striking features of the atmospheres of Jupiter and Saturn, but their dynamical origin remains poorly understood. Most approaches towards understanding zonation so far (also in the terrestrial oceans) have used highly idealized models to show that it might originate from dynamical anisotropy in a shallow turbulent fluid layer due to the planetary β‐effect. Here we report the results of laboratory experiments, conducted on a 14‐m diameter turntable, which quantitatively confirm that multiple zonal jets may indeed be generated and maintained by this mechanism in the presence of deep convection and a topographic β‐effect. At the very small values of Ekman number (≤2 × 10−5) and large local Reynolds numbers (≥2000, based on jet scales) achieved, the kinetic energy spectra suggest the presence of both energy‐cascading and enstrophy‐cascading inertial ranges in addition to the zonation near twice the Rhines wave number.