Free-surface flow past arbitrary topography and an inverse approach for wave-free solutions

An efficient numerical method to compute nonlinear solutions for two-dimensional steady free-surface flow over an arbitrary channel bottom topography is presented. The approach is based on a boundary integral equation technique which is similar to that of Vanden-Broeck's (1996, J. Fluid Mech., 330, 339-347). The typical approach for this problem is to prescribe the shape of the channel bottom topography, with the free-surface being provided as part of the solution. Here we take an inverse approach and prescribe the shape of the free-surface a priori while solving for the corresponding bottom topography. We show how this inverse approach is particularly useful when studying topographies that give rise to wave-free solutions, allowing us to easily classify eleven basic flow types. Finally, the inverse approach is also adapted to calculate a distribution of pressure on the free-surface, given the free-surface shape itself.

An alternative approach to study nonlinear inviscid flow over arbitrary bottom topography

This paper deals with a new approach to study the nonlinear inviscid flow over arbitrary bottom topography. The problem is formulated as a nonlinear boundary value problem which is reduced to a Dirichlet problem using certain transformations. The Dirichlet problem is solved by applying Plemelj-Sokhotski formulae and it is noticed that the solution of the Dirichlet problem depends on the solution of a coupled Fredholm integral equation of the second kind. These integral equations are solved numerically by using a modified method. The free-surface profile which is unknown at the outset is determined. Different kinds of bottom topographies are considered here to study the influence of bottom topography on the free-surface profile. The effects of the Froude number and the arbitrary bottom topography on the free-surface profile are demonstrated in graphical forms for the subcritical flow. Further, the nonlinear results are validated with the results available in the literature and compared with the results obtained by using linear theory. (C) 2015 Elsevier Inc. All rights reserved.

THREE-LAYER FLUID FLOW OVER A SMALL OBSTRUCTION ON THE BOTTOM OF A CHANNEL

Many boundary value problems occur in a natural way while studying fluid flow problems in a channel. The solutions of two such boundary value problems are obtained and analysed in the context of flow problems involving three layers of fluids of different constant densities in a channel, associated with an impermeable bottom that has a small undulation. The top surface of the channel is either bounded by a rigid lid or free to the atmosphere. The fluid in each layer is assumed to be inviscid and incompressible, and the flow is irrotational and two-dimensional. Only waves that are stationary with respect to the bottom profile are considered in this paper. The effect of surface tension is neglected. In the process of obtaining solutions for both the problems, regular perturbation analysis along with a Fourier transform technique is employed to derive the first-order corrections of some important physical quantities. Two types of bottom topography, such as concave and convex, are considered to derive the profiles of the interfaces. We observe that the profiles are oscillatory in nature, representing waves of variable amplitude with distinct wave numbers propagating downstream and with no wave upstream. The observations are presented in tabular and graphical forms.

Impinging jet simulation of stationary downburst flow over topography

A non-translating, long duration thunderstorm downburst has been simulated experimentally and numerically by modelling a spatially stationary steady flow impinging air jet. Velocity profiles were shown to compare well with an upper-bound of velocity measurements reported for full-scale microbursts. Velocity speed-up over a range of topographic features in simulated downburst flow was also tested with comparisons made to previous work in a similar flow, and also boundary layer wind tunnel experiments. It was found that the amplification measured above the crest of topographic features in simulated downburst flow was up to 35% less than that observed in boundary layer flow for all shapes tested. From the computational standpoint we conclude that the Shear Stress Transport (SST) model performs the best from amongst a range of eddy-viscosity and second moment closures tested for modelling the impinging jet flow.

Dynamics of the Indonesian seas circulation. Part I – The influence of bottom topography on temperature and salinity distributions

The influence of bottom topography on the distribution of temperature and salinity in the Indonesian seas region has been studied with a high-resolution model based on the Princeton Ocean Model. One of the distinctive properties of the model is an adequate reproduction of all major topographic features in the region by the model bottom relief. The three major routes of flow of Pacific water through the region have been identified. The western route follows the flow of North Pacific Water through the Sulawesi Sea, Makassar Strait, Flores Sea, and Banda Sea. This is the main branch of the Indonesian Throughflow. The eastern routes follow the flow of South Pacific water through the eastern Indonesian seas. This water enters the region either through the Halmahera Sea or by flowing to the north around Halmahera Island into the Morotai Basin and then into the Maluku Sea. A deep southward flow of South Pacific Water fills the Seram Sea below 1200 m through the Lifamatola Passage. As it enters the Seram Sea, this overflow turns eastward at depths greater than 2000 m, then upwells in the eastern part of the Seram Sea before returning westward at ~1500-2000 m. The flow continues westward across the Seram Sea, spreading to greater depths before entering the Banda Sea at the Buru-Mangole passage. It is this water that shapes the temperature and salinity of the deep Banda Sea. Topographic elevations break the Indonesian seas region down into separate basins. The difference in the distributions of potential temperature, ?, and salinity, S, in adjacent basins is primarily due to specific properties of advection of ? and S across a topographic rise. By and large, the topographic rise blocks deep flow between basins whereas water shallower than the depth of the rise is free to flow between basins. To understand this process, the structure of simulated fields of temperature and salinity has been analyzed. To identify a range of advected ? or S, special sections over the sills with isotherms or isohalines and isotachs of normal velocity have been considered. Following this approach the impact of various topographic rises on the distribution of ? and S has been identified. There are no substantial structural changes of potential temperature and salinity distributions between seasons, though values of some parameters of temperature and salinity distributions, e.g., magnitudes of maxima and minima, can change. It is shown that the main structure of the observed distributions of temperature and salinity is satisfactorily reproduced by the model throughout the entire domain.

The influence of bottom topography on the decay of modeled Agulhas rings

The influence of a large meridional submarine ridge on the decay of Agulhas rings is investigated with a 1 and 2-layer setup of the isopycnic primitive-equation ocean model MICOM. In the single-layer case we show that the SSH decay of the ring is primarily governed by bottom friction and secondly by the radiation of Rossby waves. When a topographic ridge is present, the effect of the ridge on SSH decay and loss of tracer from the ring is negligible. However, the barotropic ring cannot pass the ridge due to energy and vorticity constraints. In the case of a two-layer ring the initial SSH decay is governed by a mixed barotropic–baroclinic instability of the ring. Again, radiation of barotropic Rossby waves is present. When the ring passes the topographic ridge, it shows a small but significant stagnation of SSH decay, agreeing with satellite altimetry observations. This is found to be due to a reduction of the growth rate of the m = 2 instability, to conversions of kinetic energy to the upper layer, and to a decrease in Rossby-wave radiation. The energy transfer is related to the fact that coherent structures in the lower layer cannot pass the steep ridge due to energy constraints. Furthermore, the loss of tracer from the ring through filamentation is less than for a ring moving over a flat bottom, related to a decrease in propagation speed of the ring. We conclude that ridges like the Walvis Ridge tend to stabilize a multi-layer ring and reduce its decay.

Note on similarity solutions for viscous flow over an impermeable and non-linearly (quadratic) stretching sheet

Similarity solutions for flow over an impermeable, non-linearly (quadratic) stretching sheet were studied recently by Raptis and Perdikis (Int. J. Non-linear Mech. 41 (2006) 527–529) using a stream function of the form ψ=αxf(η)+βx2g(η). A fundamental error in their problem formulation is pointed out. On correction, it is shown that similarity solutions do not exist for this choice of ψ

Shear thinning and shear thickening non-Newtonian confined fluid flow over rotating cylinder

In present work, numerical solution is performed to study the confined flow of power-law non Newtonian fluids over a rotating cylinder. The main purpose is to evaluate drag and thermal coefficients as functions of the related governing dimensionless parameters, namely, power-law index (0.5 ≤ n ≤ 1.4), dimensionless rotational velocity (0 ≤ α ≤ 6) and the Reynolds number (100 ≤ Re ≤ 500). Over the range of Reynolds number, the flow is known to be steady. Results denoted that the increment of power law index and rotational velocity increases the drag coefficient due to momentum diffusivity improvement which is responsible for low rate of heat transfer, because the thicker the boundary layer, the lower the heat transfer is implemented.

Analytical Solution of a Walters’ Liquid B Flow Over a Linear Stretching Sheet in a Porous Medium

This chapter represents the analytical solution of two-dimensional linear stretching sheet problem involving a non-Newtonian liquid and suction by (a) invoking the boundary layer approximation and (b) using this result to solve the stretching sheet problem without using boundary layer approximation. The basic boundary layer equations for momentum, which are non-linear partial differential equations, are converted into non-linear ordinary differential equations by means of similarity transformation. The results reveal a new analytical procedure for solving the boundary layer equations arising in a linear stretching sheet problem involving a non-Newtonian liquid (Walters’ liquid B). The present study throws light on the analytical solution of a class of boundary layer equations arising in the stretching sheet problem.

The effect of thermal radiation on the natural convection boundary layer flow over a wavy horizontal surface

In this article, natural convection boundary layer flow is investigated over a semi-infinite horizontal wavy surface. Such an irregular (wavy) surface is used to exchange heat with an external radiating fluid which obeys Rosseland diffusion approximation. The boundary layer equations are cast into dimensionless form by introducing appropriate scaling. Primitive variable formulations (PVF) and stream function formulations (SFF) are independently used to transform the boundary layer equations into convenient form. The equations obtained from the former formulations are integrated numerically via implicit finite difference iterative scheme whereas equations obtained from lateral formulations are simulated through Keller-box scheme. To validate the results, solutions produced by above two methods are compared graphically. The main parameters: thermal radiation parameter and amplitude of the wavy surface are discussed categorically in terms of shear stress and rate of heat transfer. It is found that wavy surface increases heat transfer rate compared to the smooth wall. Thus optimum heat transfer is accomplished when irregular surface is considered. It is also established that high amplitude of the wavy surface in the boundary layer leads to separation of fluid from the plate.

Unsteady flow over a stretching surface with a magnetic field in a rotating fluid

The effect of the magnetic field on the unsteady flow over a stretching surface in a rotating fluid has been studied. The unsteadiness in the flow field is due to the time-dependent variation of the velocity of the stretching surface and the angular velocity of the rotating fluid. The Navier-Stokes equations and the energy equation governing the flow and the heat transfer admit a self-similar solution if the velocity of the stretching surface and the angular velocity of the rotating fluid vary inversely as a linear function of time. The resulting system of ordinary differential equations is solved numerically using a shooting method. The rotation parameter causes flow reversal in the component of the velocity parallel to the strerching surface and the magnetic field tends to prevent or delay the flow reversal. The surface shear stresses dong the stretching surface and in the rotating direction increase with the rotation parameter, but the surface heat transfer decreases. On the other hand, the magnetic field increases the surface shear stress along the stretching surface, but reduces the surface shear stress in the rotating direction and the surface heat transfer. The effect of the unsteady parameter is more pronounced on the velocity profiles in the rotating direction and temperature profiles.

Flow and heat transfer in a stagnation-point flow over a stretching sheet with a magnetic field

The flow and heat transfer problem in the boundary layer induced by a continuous moving surface is important in many manufacturing processes in industry such as the boundary layer along material handling conveyers, the aerodynamic extrusion of plastic sheet, the cooling of an infinite metalic plate in a cooling bath (which may also be electrolyte). Glass blowing, continuous casting and spinning of fibres also involve the flow due to a stretching surface. Sakiadis [1] was the first to study the flow induced by a semi-infinite moving wall in an ambient fluid. On the other hand, Crane [2] first studied the flow over a linearly stretching sheet in an ambient fluid. Subsequently, Crane [3] also investigated the corresponding heat transfer problem. Since then several authors [4-8] have studied various aspects of this problem such as the effects of mass transfer, variable wall temperature, constant heat flux, magnetic field etc. Recently, Andersson [9] has obtained an exact solution of the Navier-Stokes equations for the MHD flow over a linearly stretching sheet in an ambient fluid. Also Chiam [10] has studied the heat transfer with variable thermal conductivity on a stretching sheet when the velocities of the sheet and the free stream are equal.