Vertical End Walls Research Articles

Modelling the impact of a dam‐break wave on a vertical wall

SummaryDam‐Break waves arise from the sudden release of the water stored behind a wall, and they cause destruction and losses on nearby areas. Characterisation of the impact force on obstacles is very important for risk assessment. In this work, we performed a series of lock‐release experiments to evaluate the wave‐induced impact force on a vertical end wall situated some distance from the lock. Laboratory data were used to evaluate the performance of several types of numerical models to predict the wave‐induced force on the vertical end wall. The selected models were as follows: the hydrostatic 1D De Saint‐Venant model, the multi‐layer hydrostatic model and the Navier–Stokes two‐phase flow. For the latter two models, both 2D (in the ‐ plane) and 3D simulations were performed. The results show that, independently of the hydrostatic assumption, the 2‐D simulations generate nonphysical oscillations which are not present in the measured temporal histories of the pressure and force values. By contrast, the predictions of both 3D numerical models better reproduce the temporal variation of the pressure forces on the end wall. The present study also shows that from a practical point of view, the 1D De Saint‐Venant model predicts with sufficient accuracy the impact time and the magnitude of the peak value of the impact force. Therefore, if the use of a fully 3D models is not a feasible option, in terms of a trade‐off between computation cost and accuracy, the use of a simple hydrostatic 1D De Saint‐Venant model is preferable to a 2D model.

Redefined interface error, 2D verification and validation for pure solid-gallium phase change modeling by enthalpy-porosity methodology

For simulating phase change process of pure solid gallium, previously published literature contains some key problems, including the inappropriate definition & computation for interface errors, and the unsatisfactory numerical verifications & validations. Herein, the 2D verifications and 2D validations for the enthalpy-porosity modeling of pure gallium melting starting from a vertical end wall have been reported comprehensively. The well-known and classic experimental data presented in literature are used to verify and validate the model. The groundbreaking contributions in this work are that: i) The definition and computational method of the interface-position errors between the numerical and experimental results are revisited and proven carefully; ii) The numerical schemes as well as major parameters of 2D modeling gallium melting are selected with overall interface error <9%. However, these schemes and parameters are not appropriate for the solidification modeling of gallium due to the irregular and non-reproducible interface shapes caused by highly anisotropic properties of solid crystal gallium as experimentally reported in literature; iii) The 2D validations of enthalpy-porosity modeling of gallium melting are investigated in detail, and the maximum interface error is within the limits of ±12%. Moreover, the simulated liquid volume fractions and dimensionless heat transfer coefficients are well correlated in terms of related criteria parameters, and they are compared with correlations available in literature. It is concluded that to solidly validate the modeling of pure metal melting, both the global parameters (e.g., volume-averaged liquid fraction) and the local parameters (e.g., interface position) need to be considered together.

Force action of discontinuous waves on a vertical wall

This paper presents the results of an experimental study of the dynamic action exerted on the vertical end wall of a rectangular channel by the wave propagating in the tailwater region after total dam break. It is shown that the results of calculations using the first shallow-water approximation differ from experimental data by not more than 5%.

Normal modes of a rectangular tank with corrugated bottom

We study some effects of regular bottom corrugations on water waves in a long rectangular tank with vertical endwalls and open top. In particular, we consider motions which are normal modes of oscillation in such a tank. Attention is focused on the modes whose internodal spacing, in the absence of corrugations, would be near the wavelength of the corrugations. In these cases, the perturbation of the eigenfunctions (though not of their frequencies) can be significant, e.g. the amplitude of the eigenfunction can be greater by a factor of ten or more near one end of the tank than at the other end. This is due to a cooperative effect of the corrugations, called Bragg resonance. We first study these effects using an asymptotic theory, which assumes that the bottom corrugations are of small amplitude and that the motions are slowly varying everywhere. We then present an exact theory, utilizing continued fractions. This allows us to deal with the rapidly varying components of the flow. The exact theory confirms the essential correctness of the asymptotic results for the slowly varying aspects of the motions. The rapidly varying parts (evanescent waves) are, however, needed to satisfy accurately the true boundary conditions, hence of importance to the flow near the endwalls.

2D Large-Eddy Simulation of Lock-Exchange Gravity Current Flows at High Grashof Numbers

Lock-exchange gravity current flows produced by the instantaneous release of a heavy fluid are investigated by means of high-resolution two-dimensional (2D) large-eddy simulation (LES) using a dynamic subgrid scale model for Grashof numbers (Gr) up to 1012 . The model is validated using 2D direct numerical simulation results for the classical lock-exchange gravity current flow in an infinite channel with no-slip walls. Then the model is applied to study lock-exchange flows for the case in which the heavier (lock) fluid is initially situated in between a vertical end wall and the lock barrier. Three cases with different aspect ratios of the initial lock fluid are considered for Grashof numbers of the order of 109 – 1010 . It is found that 2D LES is able to capture most of the physics observed in experiments including the evolution of the head, as well as the formation of coherent billow structures at the apex of the head due to growth of the Kelvin-Helmholtz (KH) instabilities. However, due to the fact tha...

The influence of double-diffusive processes on the melting of ice in the Arctic Ocean: laboratory analogue experiments and their interpretation

This study has been motivated by two oceanographic observations: an increased rate of melting of sea ice in the Arctic Ocean, and the advance of an anomalously warm tongue of Atlantic water across the Arctic below the halocline over the last few decades. A series of laboratory experiments has been carried out in order to explore the physical principles underlying these phenomena, and the possibility that the extra heating at depth is responsible for the enhanced melting rate. A tank was filled with salt solution having various constant vertical density gradients. A block of ice one third of the length of the tank was floated on the surface at one end, and the rest of the surface and the walls of the tank were insulated. When no extra heat was supplied the melting rate (loss of weight of the ice in 1 h) systematically decreased as the stratification was changed from homogeneous fluid to increasingly large density gradients, while keeping the salinity of the solution in contact with the ice constant. An analogue of the intruding Atlantic water was produced by heating the lower portion of the vertical end wall at the end of the tank opposite to the ice end, keeping its temperature constant, and using the same range of salinity gradients as in the unheated experiments. Again the melting rate decreased as the density gradient was increased, but for low gradients it was larger than that in the unheated experiments. Above a certain intermediate gradient there was no significant difference in melting rate between the unheated and heated runs. The melting data were supplemented by photographs and vertical temperature and salinity profiles. The upward transfer of heat from the body of the fluid to melt the ice was clearly double-diffusive: overturning layers, separated by ‘diffusive’ interfaces, were visible on shadowgraphs, and the thickness of the layers decreased as the density gradient increased. The mean thickness of the layers through the depth of the tank also systematically decreased as the density gradient increased. With weak gradients an extra heat flux to the ice came from the intruding heated layer, but at large gradients this tongue of warm water at depth did not add to the flux near the surface. Though they were obtained in a simple, arbitrary and fixed geometry, we believe that the results of these experiments can be used as the basis for a better physical understanding of the melting rates of ice in the Arctic under various conditions.

Observations of inertial waves in a rectangular basin with one sloping boundary

Inertial waves in a homogeneous rotating fluid travel along rays that are inclined with respect to the rotation axis. The angle of inclination depends solely on the ratio of the wave frequency and twice the angular frequency. Because of this fixed angle, the waves can become focused when reflected at a sloping wall. In an infinitely long channel with a sloping wall, the repeated action of focusing may lead to the approach to a limit cycle, the so-called wave attractor, where the energy is concentrated. This effect is studied in the laboratory in a rectangular tank with one sloping wall, placed excentrically on a rotating table. The waves are excited by modulation of the background rotation. Several frequency ratios are used to study different wave attractors and one standing wave. The observations consist mainly of particle image velocimetry data in horizontal and vertical cross-sections in one half of the basin. The attractors are observed in the vertical cross-sections. They show continuous phase propagation, which distinguishes them from the standing wave where the phase changes at the same time over the whole cross-section. However, particle motion of inertial waves is three-dimensional and the actual basin is not an ideal twodimensional channel but is of finite length. This implies that the waves must adapt to the vertical endwalls, although a prediction of the nature of these adaptations and the structure of the three-dimensional wave field is at present lacking. For critical waves, whose rays are parallel to the slope, clear three-dimensional behaviour is observed. The location of most intense motion along this critical slope attractor changes in the horizontal direction and horizontal phase propagation is observed, with a wavelength between 1/5 and 1/4 of the basin length. For the other attractors there is little evidence of phase propagation in the horizontal direction. The motion along the attractor is however stronger near the vertical endwalls for attractors with wave rays of slopes close to 1 or larger. The standing wave and the other attractors are more clearly visible near 1/5 of the tank length.

Continuous Separation of Fines from Fibers in a Wedge-Shaped Vessel

The continuous removal of fines from a pulp suspension can be achieved by a particle elutriation process using, for example, a wedge-shaped vessel with an inclined tube at one vertical end wall that serves as an inlet for the suspension. Near the other vertical end, the suspension flow is split into a bottom stream containing the recovered pulp and a top stream containing the separated fines. The effect of operating conditions on both the hydrodynamic behavior and the separation efficiency was investigated. There are two critical limits for the operating flow rates of the feed suspension: below a minimum flow rate, fibers settle near the bottom exit and above a maximum flow rate, fibers escape in the top stream. These limiting flow rates do not show a strong dependence on operating conditions, except for a maximum inlet consistency above which the pulp bed expands into the outlet section, thus preventing fractionation. The particle concentration in the top stream was measured with a spectrophotometer, from which the separation efficiency was determined as a function of the inlet and outlet flow rates and the inlet consistency. The separation efficiency increases with increasing split ratio (i.e., the ratio of top to bottom streams) and with decreasing consistency of the feed suspension. Fiber loss in the top stream is about 1% of the fibers entering the vessel.

Experimental insights on the geometry and kinematics of fold-and-thrust belts above weak, viscous evaporitic décollement

This paper discusses the differences in mechanical properties and kinematics between fold-and-thrust belts detaching on evaporitic décollements and those detaching on stronger detachments. Physical experiments are described that model shortening of a thick brittle cover overlying a weak, viscous décollement to gain a better understanding of these differences. We tested the influence of (1) the décollement layer thickness and (2) the presence of a deformable backstop on the hinterland side and of a décollement pinch-out on the foreland side. Because of the very low shear strength of the viscous décollement, folds and thrusts did not propagate according to a piggy-back sequence but by centripetal, back-and-forth propagation. Additional shortening was accommodated by growth of all existing structures. Fold symmetry and thrust vergence varied between experiments. Models confined between two rigid, vertical endwalls always deformed by symmetric folding and thrusting. In models having a deformable backstop and a foreland décollement pinch-out, forethrusting initially dominated, folds were asymmetric, and fault blocks rotated. In models having a thick décollement layer, folds kept growing asymmetrically. Diapirism also depended on the initial décollement geometry. Diapirs formed only in models having a deformable backstop and were restricted to the hanging wall of a fault-related fold. Finally, the implications of such kinematics on the thermal history are discussed as well as the likelihood of hydrocarbon maturation and preservation.

Reflection of a Dam–Break Wave at a Vertical Wall. Numerical Modeling and Experiment

Results of the experimental study and numerical modeling of the reflection of a dam–break wave at the vertical end wall of a channel are given. A wave forms with distance from a partition creating the initial level difference of the liquid. It is shown that a numerical calculation based on the Zheleznyak—Pelinovskii nonlinear dispersion model satisfactorily describes the height of the splash–up, the amplitude of reflected waves, and the wave velocity in front of the wall for smooth and dam–break waves. It is also shown that, for smooth and weakly breaking (without significant entrainment of air) incoming waves, the experimental values of the height of the splash–up at the wall agree well with relevant experimental and calculated data for solitary waves.

Convection in a shallow rectangular cavity due to internal heat generation

Steady two-dimensional convective motions generated in a shallow cavity by uniformly distributed internal heat sources are analysed for the case where the vertical end walls are isothermal and the horizontal boundaries are adiabatic. Rayleigh numbers are considered for which the flow and temperature fields in the cavity are influenced by nonlinear effects. Solutions are obtained using the method of matched asymptotic expansions.

Parallel computation for natural convection

Parallel computation for two-dimensional convective flows in cavities with adiabatic horizontal boundaries and driven by differential heating of the two vertical end walls are investigated using the Intel Paragon, Intel Touchstone Delta, Cray T3D and IBM SP2. The numerical scheme, including a parallel multigrid solver, and domain decomposition techniques for parallel computing are discussed in detail. Performance comparisons are made for the different parallel systems, and numerical results using various numbers of processors are discussed. © 1997 John Wiley & Sons, Ltd.

NATURAL CONVECTION OF WATER NEAR THE DENSITY EXTREMUM FOR A WIDE RANGE OF RAYLEIGH NUMBERS

A time-dependent penalty finite element model was used to investigate natural convection of pure water in a rectangular enclosure with vertical end walls differentially heated near the temperature of maximum density. Attention was focused on the main features of the flow at high Rayleigh numbers not considered previously (Ra = 105−108). Water in a rectangular enclosure for the aspect ratio of 1.25 was initially kept at 4°C. The enclosure was then suddenly heated and cooled on the opposing vertical walls, i.e., 8°C and 0°C. The Rayleigh number was varied by the size of the enclosure. In the Rayleigh number range considered here, steady state was reached within a time of the order of 2tf, where tf is the time to steady state suggested by Patterson and Imberger [1]. The steady state flow field and temperature structure were symmetric at about the enclosure's midpoint, and a stable sinking jet was formed in the interior of the enclosure as a result of density inversion. The velocity profile near the vertical ...

Natural convection flow and heat transfer within a rectangular enclosure containing a vertical porous layer

An experimental and theoretical study of natural convection flow and heat transfer within a rectangular enclosure partially filled with an anisotropic porous medium is conducted as a step to understand the effects of the mushy region flow characteristics on the interacting flows of the melt pool and the mushy zone of solidifying alloys. The test cavity is filled with a porous medium for half the lateral distance to the cold wall and is heated and cooled at the vertical end walls by imposing uniform but different temperatures. The solid matrices of the porous media are constructed with perforated plates, and measured flow characteristics are used in the predictions. Flow visualization in the porous region is made possible by the latticed structure of the porous matrices and electrolysis dye generation. The effect of anisotropic flow characteristics of the porous medium on the flow and heat transfer is modeled mathematically using volume-averaged conservation equations. The mathematical model was validated by comparing the predicted velocity and temperature fields with the visualized streamlines and measured temperatures. To investigate the effect of anisotropic permeability for wider ranges of dimensionless parameters, numerical experiments were conducted. The importance of incorporating the mathematical models which account for the anisotropic characteristics of the mushy zone is emphasized.

Experimental and numerical analyses of the natural convection of water through its density maximum in a rectangular enclosure

An experimental and numerical investigation is presented concerning the transient and steady-state natural convection of water through its density maximum in a rectangular enclosure with the vertical end walls held at different temperatures. The water in the enclosure is initially at a uniform temperature above the density maximum at 3.98°C, and then the temperature of the cold wall is suddenly reduced to and maintained at 0°C. Experimental flow patterns are presented for 5 and 8°C initial and hot wall temperatures. A pH indicator technique is used to visualize the convective flow patterns in the enclosure. Numerical solutions of the convective flow patterns are presented for transient and steady-state natural convection, and are shown to be in good agreement with experimental flow patterns. The effect of the density inversion on the transient convective flow patterns was dominant at a 5°C initial and hot wall temperature. Steady-state results for an 8°C initial and hot wall temperature show an asymmetric convective flow pattern with the dominant flow cell being above the density inversion.

Numerical study of thermal convection in shallow cavities with conducting boundaries

Two-dimensional convective flows in shallow cavities with conducting horizontal boundaries and driven by differential heating of the two vertical end walls, are studied numerically over a range of Rayleigh numbers and Prandtl numbers. As the Rayleigh number increases, nonlinearity first affects the flow structure in the turning regions near the ends of the cavity. These “end-zone problems” have been investigated by a combined computational and analytical approach. Numerical solutions are found using a Dufort-Frankel-Multigrid method, and appear to be in good agreement with stability analysis.

Thermocapillary convection in two immiscible liquid layers with free surface

Thermocapillary convection is studied in two immiscible liquid layers with one free surface, one liquid/liquid interface, and differential heating applied parallel to the interfaces. An analytical solution is introduced for infinite horizontal layers. The defining parameter for the flow pattern is λ, the ratio of the temperature coefficient of the interfacial tension to that of the surface tension. Four different flow patterns exist under zero gravity conditions. ‘‘Halt’’ conditions which halt the fluid motion in the lower encapsulated liquid layer have been found. A numerical experiment is carried out to study effects of vertical end walls on the double layer convection in a two-dimensional cavity. The halt condition obtained from the analytical study is found to be valid in the limit of small Reynolds numbers. The flow in the encapsulated liquid layer can be suppressed substantially.

Effect of density extremum on the solidification of water on a vertical wall of a rectangular cavity

Experiments were performed in a two-dimensional rectangular cavity to study the transient flow in an initially isothermal and motionless fluid due to a step decrease in temperature on one of the two vertical end walls. In the experiments water was used as the phase-change medium, with the cold-wall temperature maintained below the freezing temperature. The opposite vertical wall was kept at the initial temperature, greater than the temperature where the density extremum occurs. The growth of ice and the transient flow in the cavity were visualized with the aid of a tracer technique to examine the effect of density inversion. The temperature field was continuously recorded by an array of thermocouples. It was found that the density inversion of water strongly influences both the growth of ice and the convective flow in the liquid region of the test cavity.

Transient Natural Convection in Enclosures at High Rayleigh Number

Transient natural convection at Rayleigh numbers of 1010 was studied experimentally in two enclosures of aspect ratio 1/3, one a 1:5.5 physical scale model containing the dense refrigerant gas R114, and the second a full-scale room containing air. In one type of transient the vertical endwall temperature was suddenly changed, while in a second type of transient the isothermal, well-mixed gas was suddenly exposed to hot and cold vertical endwalls. The experiments indicated that the dominant time constant was a convective one. Comparisons between the scale model and full scale show that R114 gas can simulate the transient behavior of air-filled enclosures.

Convection in a rotating cavity: modelling ocean circulation

Large-scale oceanic flows, ranging from large estuaries to basin-scale abyssal circulation, can be driven by imposed lateral density gradients and are significantly influenced by the Earth's rotation. Some essential features of such flows have been incorporated into a laboratory model by differentially heating and cooling the vertical endwalls of a shallow rectangular cavity, rotating about a vertical axis through its centre. In this paper, the flow observed after this system had been allowed to develop to a steady mean state is described. Measurements were made of the velocity and temperature fields, as well as the variability in temperature field associated with eddy motions. The temperature data were also used to calculate the potential vorticity distribution within the cavity. These results indicate that the heated and cooled end-walls acted not only as continuous sources of buoyancy, but also of anomalous potential vorticity. It is shown that the potential vorticity served as an indicator and tracer of the laboratory flow. The structure and location of boundary currents were largely determined by nonlinear potential vorticity dynamics which resulted in a mean circulation consisting of two basin-scale counter-rotating gyres. This flow differed markedly from the initial boundary currents set up by Kelvin waves.