Channel Of Finite Depth Research Articles

Surge Hydrodynamic Coefficients Due to a Floating Cylindrical Structure in Presence of a Bottom-Mounted Circular Plate in a Channel of Finite Depth

Surge Hydrodynamic Coefficients Due to a Floating Cylindrical Structure in Presence of a Bottom-Mounted Circular Plate in a Channel of Finite Depth

An algebraic method of solution of a water wave scattering problem involving an asymmetrical trench

The problem of propagation of obliquely incident surface water waves involving an asymmetric rectangular trench in a channel of finite depth is examined for its solution. The problem under consideration leads to multiple series relations involving trigonometric functions. Instead of converting these relations into systems of integral equations, direct algebraic approaches are utilized and the corresponding solutions are obtained approximately, by way of solving the reduced system of over-determined algebraic equations. The system of over-determined algebraic equations are solved with the aid of the well-known algebraic approaches “Least square” (LS) and “Singular value decomposition” (SVD) methods. The presently determined results involving the reflection and transmission coefficients related to the scattering problem under consideration, agree very well, with the known results obtained recently, where integral equations and their approximate numerical solutions were determined. The effect of system of parameters on hydrodynamics quantities such as reflection and transmission coefficients, wave elevation are analysed and shown through tables and graphs. The present algebraic methods appear to be very direct and quick. A similar techniques are found to be applicable to the problem of scattering of surface water waves by any finite number of asymmetrical placed rectangular trenches. The problem of a pair of asymmetrical trenches is under progress. The energy balance relation for the given problem is derived and used to check the accuracy of numerical results. Some important results such as wave elevation profiles, the singularity behaviour of the flow near trench edges are investigated and analyzed through graphs.

Flow from a source above a sloping base

We consider the outflow of water from the peak of a triangular ridge into a channel of finite depth. Solutions are computed for different flow rates and bottom angles. A numerical method is used to compute the flow from the source for small values of flow rate and it is found that there is a maximum flow rate beyond which steady solutions do not seem to exist. Limiting flows are computed for each geometrical configuration. One application of this work is as a model of saline water being returned to the ocean after desalination.

Solution of the problem of propagation of water waves over a pair of asymmetrical rectangular trenches

The problem of propagation of obliquely incident surface water waves over a pair of asymmetrical rectangular trenches in a channel of finite depth is examined for its solution. The mathematical problem is handled for its numerical solution with the aid of a system of singular integral equations of first kind. The resulting integral equations are solved numerically by using suitably designed polynomial approximations of the unknown functions. System of linear algebraic equations is obtained by utilizing the zeros of Chebyshev polynomial of first kind as collocation points and hence the occurrence of ill-conditioned matrix is avoided. The effectiveness of the pair of trenches is studied by analyzing the physical quantities namely the reflection and transmission coefficients. As a special limiting case, the results for a single trench problem are derived and are found to be in excellent agreement with the results available in the literature. The effect of various parameters are analyzed through different graphs for a pair of asymmetrical trenches. The energy balance relation is derived and used to check the accuracy of numerical results.

Large-eddy simulation of stably stratified flow past a rectangular cylinder in a channel of finite depth

Large-eddy simulation is performed to investigate an incompressible stratified flow with a linear temperature gradient past a rectangular cylinder with side ratio b/h=0.6, where b and h are its breadth and depth, respectively. Simulations are performed for Reynolds number Re=8,400 with respect to a series of stratification numbers K. Results show that transition of stratification effect occurs at around K=2 due to the occurrence of second-order stationary internal waves. Both Strouhal number St and drag coefficient CD decrease with stratification number K when K<2 due to suppression of vertical motion, but increase with K when K>2 because of the stationary internal waves. The relationship between the fields of temperature and other flow parameters are studied and a coincidence of temperature field with internal wave is observed at high stratification number. Finally the stratification effect on the vortex street is discussed.

Solitary gravity waves and free surface flows past a point vortex

Nonlinear free surface flows past a disturbance in a channel of finite depth are considered. The fluid is assumed to be incompressible and inviscid and the flow to be two-dimensional, irrotational and supercritical. The disturbance is chosen to be a point vortex. Highly accurate numerical solutions are computed. The basic idea of the numerical approach is first to develop codes to compute solitary waves and then to introduce appropriate modifications to model the point vortex. Previous results are recovered and new solutions are presented.

Three-Dimensional Numerical Simulation of Stably Stratified Flows over a Two-Dimensional Hill

Stably stratified flows over a two-dimensional hill are investigated in a channel of finite depth using a three-dimensional direct numerical simulation (DNS). The present study follows onto our previous two-dimensional DNS studies of stably stratified flows over a hill in a channel of finite depth and provides a more realistic simulation of atmospheric flows than our previous studies. A hill with a constant cross-section in the spanwise (y) direction is placed in a 3-D computational domain. As in the previous 2-D simulations, to avoid the effect of the ground boundary layer that develops upstream of the hill, no-slip conditions are imposed only on the hill surface and the surface downstream of the hill; slip conditions are imposed on the surface upstream of the hill. The simulated 3-D flows are discussed by comparing them to the simulated 2-D flows with a focus on the effect of the stable stratification on the non-periodic separation and reattachment of the flow behind the hill. In neutral (K = 0, where K is a non-dimensional stability parameter) and weakly stable (K = 0.8) conditions, 3-D flows over a hill differ clearly from 2-D flows over a hill mainly because of the three-dimensionality of the flow, that is the development of a spanwise flow component in the 3-D flows. In highly stable conditions (K = 1, 1.3), long-wavelength lee waves develop downstream of the hill in both 2-D and 3-D flows, and the behaviors of the 2-D and 3-D flows are similar in the vicinity of the hill. In other words, the spanwise component of the 3-D flows is strongly suppressed in highly stable conditions, and the flow in the vicinity of the hill becomes approximately two-dimensional in the x and z directions.

On steady non-breaking downstream waves and the wave resistance

In this work we have obtained exact analytical formulae expressing the wave resistance of a two-dimensional body by the parameters of the downstream non-breaking waves. The body moves horizontally at a constant speed $c$in a channel of finite depth $h$. We have analysed the relationships between the parameters of the upstream flow and the downstream waves. Making use of some results by Keady &amp; Norbury (J. Fluid Mech., vol. 70, 1975, pp. 663–671) and Benjamin (J. Fluid Mech., vol. 295, 1995, pp. 337–356), we have rigorously proved that realistic steady free-surface flows with a positive wave resistance exist only if the upstream flow is subcritical, i.e. the Froude number$\mathit{Fr}=c/\sqrt{gh}&lt;1$. For all solutions with downstream waves obtained by a perturbation of a supercritical upstream uniform flow the wave resistance is negative. Applying a numerical technique, we have calculated accurate values of the wave resistance depending on the wavelength, amplitude and mean depth.

Wake angle for surface gravity waves on a finite depth fluid

Linear water wave theory suggests that wave patterns caused by a steadily moving disturbance are contained within a wedge whose half-angle depends on the depth-based Froude number FH. For the problem of flow past an axisymmetric pressure distribution in a finite-depth channel, we report on the apparent angle of the wake, which is the angle of maximum peaks. For moderately deep channels, the dependence of the apparent wake angle on the Froude number is very different to the wedge angle and varies smoothly as FH passes through the critical value FH = 1. For shallow water, the two angles tend to follow each other more closely, which leads to very large apparent wake angles for certain regimes.

Three-dimensional numerical simulation of stably stratified flows over a two-dimensional hill: effect of stratification on the non-periodic separation and reattachment of the flow

Stably stratified flows over a two-dimensional hill are investigated in a channel of finite depth using a three-dimensional direct numerical simulation (DNS). The present study follows on to our previous two-dimensional DNS studies of stably stratified flows over a hill in a channel of finite depth and provides a more realistic simulation of atmospheric flows than our previous studies. A hill with a constant cross-section in the spanwise (y) direction is placed in a 3-D computational domain. As in the previous 2-D simulations, to avoid the effect of the ground boundary layer that develops upstream of the hill, no-slip conditions are imposed only on the hill surface and the surface downstream of the hill; slip conditions are imposed on the surface upstream of the hill. The simulated 3-D flows are discussed by comparing them to the simulated 2-D flows with a focus on the effect of the stable stratification on the non-periodic separation and reattachment of the flow behind the hill. In neutral (K = 0, where K is a non-dimensional stability parameter) and weakly stable (K = 0.8) conditions, 3-D flows over a hill differ clearly from 2-D flows over a hill mainly because of the three-dimensionality of the flow, that is the development of a spanwise flow component in the 3-D flows. In highly stable conditions (K = 1, 1.3), long-wavelength lee waves develop downstream of the hill in both 2-D and 3-D flows, and the behaviors of the 2-D and 3-D flows are similar in the vicinity of the hill. In other words, the spanwise component of the 3-D flows is strongly suppressed in highly stable conditions, and the flow in the vicinity of the hill becomes approximately two-dimensional in the x and z directions.

Quantum finite-depth memory channels: Case study

We analyze the depth of the memory of quantum memory channels generated by a fixed unitary transformation describing the interaction between the principal system and internal degrees of freedom of the process device. We investigate the simplest case of a qubit memory channel with a two-level memory system. In particular, we explicitly characterize all interactions for which the memory depth is finite. We show that the memory effects are either infinite, or they disappear after at most two uses of the channel. Memory channels of finite depth can be to some extent controlled and manipulated by so-called reset sequences. We show that actions separated by the sequences of inputs of the length of the memory depth are independent and constitute memoryless channels.

Low-mode internal tide generation by topography: an experimental and numerical investigation

We analyse the low-mode structure of internal tides generated in laboratory experiments and numerical simulations by a two-dimensional ridge in a channel of finite depth. The height of the ridge is approximately half of the channel depth and the regimes considered span sub- to supercritical topography. For small tidal excursions, of the order of 1% of the topographic width, our results agree well with linear theory. For larger tidal excursions, up to 15% of the topographic width, we find that the scaled mode 1 conversion rate decreases by less than 15%, in spite of nonlinear phenomena that break down the familiar wave-beam structure and generate harmonics and inter-harmonics. Modes two and three, however, are more strongly affected. For this topographic configuration, most of the linear baroclinic energy flux is associated with the mode 1 tide, so our experiments reveal that nonlinear behaviour does not significantly affect the barotropic to baroclinic energy conversion in this regime, which is relevant to large-scale ocean ridges. This may not be the case, however, for smaller scale ridges that generate a response dominated by higher modes.

Water wave interaction with a sphere in a two-layer fluid flowing through a channel of finite depth

Using linear water wave theory, we consider a three-dimensional problem involving the interaction of waves with a sphere in a fluid consisting of two layers with the upper layer and lower layer bounded above and below, respectively, by rigid horizontal walls, which are approximations of the free surface and the bottom surface; these walls can be assumed to constitute a channel. The effects of surface tension at the surface of separation is neglected. For such a situation time-harmonic waves propagate with one wave number only, unlike the case when one of the layers is of infinite depth with the waves propagating with two wave numbers. Method of multipole expansions is used to find the particular solutions for the problems of wave radiation and scattering by a submerged sphere placed in either of the upper or lower layer. The added-mass and damping coefficients for heave and sway motions are derived and plotted against various values of the wave number. Similarly the exciting forces due to heave and sway motions are evaluated and presented graphically. The features of the results find good agreement with previously available results from the point of view of physical interpretation.

Design for Wall Shape Profile of a Millimeter-Scale Air-Heating Channel by Integrating Three-Dimensional CFD Code with Optimization Method

Optimization for the wall shape profile of a millimeter-scale air-heating channel of finite depth has been performed by a computer-aided design approach. The present design approach involves a direct problem solver and a numerical optimizer. A commercial software package is used as the direct problem solver, which is able to provide the numerical predictions of pressure drop and heat transfer performance for the three-dimensional air flows in the channel with an irregular wall shape profile. An objective function is calculated based on the numerical data of average pressure drop and heat flux provided by the direct problem solver. The numerical optimizer is a self-developed program which is used to continuously adjust the channel wall shape profile so as to minimize the objective function in the optimization process. In this article, the interface between the commercial software package and the self-developed optimizer has been established for connection. A scheme incorporating a point-by-point technique with a redistribution method is employed to construct the irregular wall shape profiles. It is found that the search process is robust and is independent of the initial guess. The optimal shapes at different inlet velocities are obtained, and the effect of channel depth on the optimal shape is also investigated.

Binary-mixture convection in connected channels heated from below

Supercritical binary-mixture flows in connected channels of finite depth are studied theoretically and experimentally. It is shown that, in contrast to uniform fluids, in the mixture the convection excitation is “hard” and specific transient flows and oscillation regimes are observed. A mechanism explaining the phenomena observed is proposed and confirmed by a theoretical solution of the problem. The amplitude curves and the channel distributions of the velocity, temperature, and admixture concentration are obtained. These illustrate the competition between the thermodiffusion and thermo-gravitational convection mechanisms.

Variational linear problems in wave-obstacle interaction

We discuss the problem of the steady two-dimensional flow past fixed disturbances in an open channel of finite depth. We consider different types of obstacles: submerged or surface-piercing bodies and localized perturbations of a horizontal botton. By a special variational approach, we prove the unique solvability of the linearized problem for supercritical velocities of the unperturbed flow. We also discuss extensions of the variational method to the limit case of a submerged beam and to subcritical velocities.

Truncation Approximations for Gravity-capillary Free-surface Flows

Gravity-capillary free-surface flows past disturbances in a channel of finite depth are considered. These flows are usually assumed to extend from x = −∞ to x = ∞ where the x-axis is parallel to the bottom. Many numerical schemes truncate this infinite domain to the interval −B < x < A where A and B are large positive numbers. These truncations introduce inaccuracies, especially when the effect of surface tension is included. In this paper numerical methods are presented which remove these inaccuracies. This is achieved by taking into account the contributions from −∞ to −B and from A to ∞. Explicit computations are presented for a semi-circular obstacle at the bottom of the channel.

Standing waves for a two-way model system for water waves

In this paper, we prove the existence of a large family of nontrivial bifurcating standing waves for a model system which describes two-way propagation of water waves in a channel of finite depth or in the near shore zone. In particular, it is shown that, contrary to the classical standing gravity wave problem on a fluid layer of finite depth, the Lyapunov–Schmidt method applies to find the bifurcation equation. The bifurcation set is formed with the discrete union of Whitney's umbrellas in the three-dimensional space formed with 3 parameters representing the time-period and the wave length, and the average of wave amplitude.

Capillary waves past a flat plate in water of finite depth

jjFree-surface flow past a semi-infinite flat plate in a channel of finite depth is considered. The fluid is assumed to be inviscid and incompressible, and the flow to be two-dimensional and irrotational. Surface tension is included in the dynamic boundary condition but the effects of gravity are neglected. It is shown that there is a three-parameter family of solutions with waves in the far field and a discontinuity in slope at the separation point. This family includes as particular cases the solutions previously computed by Osborn & Stump (2001, Phys. Fluids, 13, 616-623) and by Andersson & Vanden-Broeck (1996, Proc. R. Soc., 452, 1985-1997).

A decrease in the effective cross section of a cooling stream caused by hot air penetration into the blowout channel

The phenomenon of penetration of a hot air stream into a channel of finite depth by which a cooling stream is injected was revealed and studied by numerical methods. The penetrating stream decreases the effective cross section of the cooling stream, increases the velocity of this stream, and reduces the near-wall gas screen cooling efficacy.