SEMI-IMPLICIT FINITE VOLUME SHALLOW-WATER FLOW AND SOLUTE TRANSPORT SOLVER WITH k–ε TURBULENCE MODEL

A 3D semi-implicit finite volume scheme for shallow- water flow with the hydrostatic pressure assumption has been developed using the σ-co-ordinate system, incorporating a standard k–ε turbulence transport model and variable density solute transport with the Boussinesq approximation for the resulting horizontal pressure gradients. The mesh spacing in the vertical direction varies parabolically to give fine resolution near the bed and free surface to resolve high gradients of velocity, k and ε. In this study, wall functions are used at the bed (defined by the bed roughness) and wind stress at the surface is not considered. Surface elevation gradient terms and vertical diffusion terms are handled implicitly and horizontal diffusion and source terms explicitly, including the Boussinesq pressure gradient term due to the horizontal density gradient. The advection terms are handled in explicit (conservative) form using linear upwind interpolation giving second-order accuracy. A fully coupled solution for the flow field is obtained by substi- tuting for velocity in the depth-integrated continuity equation and solving for surface elevation using a conjugate gradient equation solver. Evaluation of horizontal gradients in the σ-co-ordinate system requires high-order derivatives which can cause spurious flows and this is avoided by obtaining these gradients in real space. In this paper the method is applied to parallel oscillatory (tidal) flow in deep and shallow water and compared with field measurements. It is then applied to current flow about a conical island of small side slope where vortex shedding occurs and velocities are compared with data from the laboratory. Computed concentration distributions are also compared with dye visualization and an example of the influence of temperature on plume dispersion is presented. © 1997 John Wiley & Sons, Ltd.     

Stability of thermal convection between non-uniformly heated plates

The effect of horizontal as well as vertical temperature gradients on the stability of natural convection in a thin horizontal layer of viscous, incompressible fluid is studied on the basis of linear theory. The boundaries are taken to be rigid, perfectly thermally conducting, having prescribed temperatures and the horizontal temperature gradient is assumed to be small. It is found that for Prandtl number greater than 0.13, the critical Rayleigh number is always larger than that for the corresponding Benard problem. The preferred mode of disturbance is stationary and will be a transverse roll (having axes normal to the basic flow) or a longitudinal roll (having axes aligned in the direction of the basic flow) depending on whether the Prandtl number is less or larger than 1.7. Finally, it is shown that the instability is of thermal origin.     

Radiative-convective heat transfer in a flow past an evaporating semitransparent melt film

A conjugate problem of nonstationary radiative-convective heat transfer in a turbulent flow of a mixture of gases with solid particles around a horizontal evaporating semitransparent melt film is numerically solved. The moving film is subjected to intense radiative heating by an external source whose radiation interacts with the gas-particle medium and the film in a bounded spectral range. The temperature fields and velocities in the boundary layer and the film are calculated. The computational results given allow determination of the impact of radiation on heat transfer and film dynamics in the boundary layer-film system.     

Stability of Flow with Viscous Dissipation in a Horizontal Porous Layer with an Open Boundary Having a Prescribed Temperature Gradient

A buoyancy-induced stationary flow with viscous dissipation in a horizontal porous layer is investigated. The lower boundary surface is impermeable and subject to a uniform heat flux. The upper open boundary has a prescribed, linearly varying, temperature distribution. The buoyancy-induced basic velocity profile is parallel and non-uniform. The linear stability of this basic solution is analysed numerically by solving the disturbance equations for oblique rolls arbitrarily oriented with respect to the basic velocity field. The onset conditions of thermal instability are governed by the Rayleigh number associated with the prescribed wall heat flux at the lower boundary, by the horizontal Rayleigh number associated with the imposed temperature gradient on the upper open boundary, and by the Gebhart number associated with the effect of viscous dissipation. The critical value of the Rayleigh number for the onset of the thermal instability is evaluated as a function of the horizontal Rayleigh number and of the Gebhart number. It is shown that the longitudinal rolls, having axis parallel to the basic velocity, are the most unstable in all the cases examined. Moreover, the imposed horizontal temperature gradient tends to stabilise the basic flow, while the viscous dissipation turns out to have a destabilising effect.     

Natural convection due to solar radiation over a non-absorbing plate with and without heat losses

Natural convection over a non-reflecting, non-absorbing, ideally transparent semi-infinite vertical flat plate due to absorption of incident radiation (solar radiation) is considered. The absorbed radiation acts as a distributed source which initiates buoyancy-driven flow and convection in the absorbing layer. The plate when heated by the absorbing fluid loses heat to the surroundings from its external side. Solution of the governing equations of the flow under these circumstances is non-similar because of both the heat source term in the energy equation and the temperature boundary condition at the plate. A local non-similar technique is used to obtain solutions for a wide range of the dimensionless distance along the plate and of the dimensionless loss coefficient to the surroundings. The results show that the temperature distribution has a maximum temperature in the depth of the fluid rather than on the plate. A new definition for a local heat transfer coefficient between the plate and the absorbing fluid is introduced which is based on the local maximum temperature rise in the fluid. A formula to calculate this heat transfer coefficient is given for the anticipated range of the loss coefficient.     

Investigation of the fluid temperature field inside a flat-plate solar collector

An experimental study was conducted to investigate fluid temperature fields inside a flat-plate solar collector tube. The results show the highest fluid temperature at the upper end of the tube which decreased gradually to the lowest value at the bottom end of the tube, whereas, the temperature field in the horizontal plane is symmetric about the centerline. The vertical temperature gradients vary with the axial distance. The local fluid temperature increased nonlinearly along the collector length and its magnitude decreased with an increase in the Reynolds number. The local Rayleigh number increased with the axial distance and at a given location, its magnitude increased with a decrease in the Reynolds number, whereas, the local Nusselt number trends in flat-plate collector tube are in general similar to that in the conventional laminar channel flows. The local fluid temperature increased with an increase in the incident heat flux at a given collector orientation but decreased for the inclined collectors. The results show that over the given Reynolds number range, the fluid in a flat-plate collector tube is stably stratified over most of the fluid cross-sectional domain and the convective currents are suppressed and restricted to a thin layer adjacent to the lower tube wall. The results from the present study provide the physical explanation for the heat transfer enhancement by insert devices. That is, the insert devices disrupt the stably stratified layer and induce mixing which enhances the heat transfer.     

Gravity wave radiation from unsteady rotational flow in an f-plane shallow water system

Spontaneous gravity wave radiation from an unsteady rotational flow is investigated numerically in an f-plane shallow water system. Unlike the classical Rossby adjustment problem, where free development of an initially unbalanced state is investigated, we consider development of a barotropically unstable zonal flow which is initially balanced but maintained by zonal mean forcing. Gravity waves are continuously radiated from a nearly balanced rotational flow region even when the Froude number is so small that balance dynamics is thought to be a good approximation for the full system. The source of gravity waves is discussed by analogy with the theory of aero-acoustic sound wave radiation (the Lighthill theory). It is shown that the source regions correspond to regions of strong rotational flow. The gradual change of rotational flow causes gravity wave radiation. We propose an approximation for these strong sources on the assumption that the dominant flow in the jet region is non-divergent rotational flow. In addition, we calculate the zonally symmetric component of gravity waves far from the source regions, solving the Lighthill equation. Using scaling analyses for perturbations, these gravity waves can be calculated with only one approximated source term that is related to the latitudinal gradient of the fluid depth and the latitudinal mass flux. In spite of its simplicity, this approximation not only explains the physical cause of gravity wave radiation, but gives an amount of source close to that obtained by classical approximation derived from vortical motion.     

Migration of salts in the unsaturated zone caused by heating

Heat-transfer phenomena as well as moisture movement in unsaturated soils due to thermal gradients, have been extensively studied during the last four decades. Less attention has been devoted to the transport and redistribution of solutes caused by heating.Solar radiation, radioactive waste repositories, underground energy storage, buried electric cables and steam pipes, disposal of waste heat from power plants are examples of heat sources in the soil.Soil-water properties, such as surface tension, viscosity, density, as well as the equilibrium composition of phases, depend on temperature. Hence, nonuniform heating of a soil partially saturated by saline water has an effect on such processes as water flow under capillary and gravitational forces, evaporation, condensation and diffusion of vapor and transport and precipitation of salts.A mathematical model is presented for the migration of salts in the vadoze zone in the soil under nonisothermal conditions, taking into account the above-mentioned phenomena. The physical assumptions underlying the model are briefly discussed.The study of a particular case shows that under certain conditions, a heat source may attract dissolved salts, and cause their precipitation in the hot area.     

Stability of two-layer fluid flow

The problem of two-layer convective flow of viscous incompressible fluids in a horizontal channel with solid walls in the presence of evaporation is considered in the Oberbeck–Boussinesq approximation assuming that the interface is an undeformable thermocapillary surface and taking into account the Dufour effect in the upper layer which is a mixture of gas and liquid vapor. The effects of longitudinal temperature gradients at the boundaries of the channel and the thicknesses of the layer on the flow pattern and the evaporation rate are studied under conditions of specified gas flow and the absence of vapor flow on the upper boundary of the channel. It is shown that the long-wavelength asymptotics for the decrement is determined from the flow characteristics, the longwavelength perturbations occurring in the system decay monotonically, and the thermal instability mechanism is not potentially the most dangerous.     

Horizontal mixing layer in shallow water flows

Horizontal-shear thin-layer homogeneous fluid flow in the open channel is considered. A one-dimensional mathematical model of the development and evolution of the horizontal mixing layer is derived within the framework of the three-layer scheme. The steady-state solutions of the equations of motion are constructed and investigated. In particular, supercritical (subcritical)-in-average flow concepts are introduced and the problem of the mixing layer structure is solved. The proposed model is verified on the basis of comparison with a numerical solution of two-dimensional equations of shallow water theory.     

Forced convective flow and heat transfer over a?porous plate in a?Darcy-Forchheimer porous medium in presence of?radiation

A mathematical model is presented for analyzing the boundary layer forced convective flow and heat transfer of an incompressible fluid past a plate embedded in a Darcy-Forchheimer porous medium. Thermal radiation term is considered in the energy equation. The similarity solutions for the problem are obtained and the reduced nonlinear ordinary differential equations are solved numerically. It is noticed that the boundary layer decreases with an increase in the value of inertial parameter and in this case the temperature profile is found to decrease smoothly within the boundary layer. In case of porous plate, fluid velocity increases whereas non-dimensional temperature decreases for increasing values of suction parameter. The rate of heat transfer increases with the increasing values of Prandtl number. The effect of thermal radiation on temperature field is also analyzed.     

Stability of plane-parallel convective fluid flow in a horizontal layer relative to spatial perturbations

The stability of stationary plane-parallel convective flow between horizontal planes along which a constant temperature gradient is given, is investigated relative to spatial perturbations. It is shown that the flow crisis is caused by spiral perturbations in a broad range of Prandtl number values (P > 0.24). Spiral perturbations are developed in unstably stratified fluid layers adjoining the upper and lower layer boundaries, and are of Rayleigh nature.     

Effects of a barrier on temperature structure and mixing in thermally stratified water cooled from above

A Mach-Zehnder interferometer was used to study the unsteady temperature structure in radiation stratified water cooled from above. Temperature distribution measurements in a test cell filled with distilled water provide conclusive evidence that the thermal structure between the air-water interface and the stable region is controlled by buoyancy induced natural convection. The cooling from above produces a complex vertical temperature profile which can be divided into several distinct regimes. Introduction of a thin, rigid transparent (glass) plate into the water before thermal stratification by radiation and cooling confines the natural convection driven flow and reduces the intensity of mixing. As a result, the energy transport from the interior of the water layer to the interface is decreased. However, under the experimental conditions tested use of rigid, horizontal plates introduced in the fluid were not very effective in reducing the transport of heat from the warm interior to the cooler interface.     

Thermocapillary convection in an absorbing-scattering medium subjected to irradiation

The criteria for the onset of thermocapillary convection in a horizontal radiating fluid layer heated by an incident thermal radiative energy source are determined. The fluid layer is an absorbing and isotropically scattering medium confined between a free upper surface and an insulated rigid lower surface. Linear analysis is performed on the continuity, momentum, energy, and approximate radiative equations. The resulting disturbance equations are solved using a numerical optimization technique to obtain the eigenvalues governing the onset of convective motion. The influence of thermal radiation on the critical Marangoni number is examined. Attention is drawn to the physical significance of the heat transfer mode, gravitational force, the scattering effect, and the surface radiative properties. The conditions leading to the onset of convection are presented as functions of the optical thickness, scattering albedo, Planck number, surface emissivities, and transmissivities.     

The onset of convection in a shallow box occupied by a heterogeneous porous medium with constant flux boundaries

The effects of hydrodynamic and thermal heterogeneity, for the case of variation in both the horizontal and vertical directions, on the onset of convection in a horizontal layer of a saturated porous medium uniformly heated from below, are studied analytically using linear stability theory for the case of weak heterogeneity. Attention is focused on the case of constant flux upper and lower boundaries, a case for which the critical horizontal wavenumber is zero, and attention is also concentrated on the case of a shallow layer. It is found that the effect of such heterogeneity on the critical value of the Rayleigh number Ra based on mean properties is of second order if the properties vary in a piecewise constant or linear fashion. The effects of horizontal heterogeneity and vertical heterogeneity are then comparable once the aspect ratio is taken into account, and to a first approximation are independent. The combination of permeability heterogeneity and conductivity heterogeneity can be either stabilizing or destabilizing for the present case.     

Effects of partial slip on boundary layer flow past a permeable exponential stretching sheet in presence of thermal radiation

An analysis is presented to describe the boundary layer flow and heat transfer towards a porous exponential stretching sheet. Velocity and thermal slips are considered instead of no-slip conditions at the boundary. Thermal radiation term is incorporated in the temperature equation. Similarity transformations are used to convert the partial differential equations corresponding to the momentum and heat equations into highly non-linear ordinary differential equations. Numerical solutions of these equations are obtained by shooting method. It is found that the fluid velocity and temperature decrease with increasing slip parameter. Temperature is found to decrease with an increase of thermal slip parameter. Thermal radiation enhances the effective thermal diffusivity and the temperature rises.     

Flux and source term discretization in two‐dimensional shallow water models with porosity on unstructured grids

Two‐dimensional shallow water models with porosity appear as an interesting path for the large‐scale modelling of floodplains with urbanized areas. The porosity accounts for the reduction in storage and in the exchange sections due to the presence of buildings and other structures in the floodplain. The introduction of a porosity into the two‐dimensional shallow water equations leads to modified expressions for the fluxes and source terms. An extra source term appears in the momentum equation. This paper presents a discretization of the modified fluxes using a modified HLL Riemann solver on unstructured grids. The source term arising from the gradients in the topography and in the porosity is treated in an upwind fashion so as to enhance the stability of the solution. The Riemann solver is tested against new analytical solutions with variable porosity. A new formulation is proposed for the macroscopic head loss in urban areas. An application example is presented, where the large scale model with porosity is compared to a refined flow model containing obstacles that represent a schematic urban area. The quality of the results illustrates the potential usefulness of porosity‐based shallow water models for large scale floodplain simulations. Copyright © 2005 John Wiley & Sons, Ltd.     

Nanofluid bioconvection: interaction of microorganisms oxytactic upswimming, nanoparticle distribution, and heating/cooling from below

The aim of this paper is to develop a theory describing the onset of convection instability (called here nanofluid bioconvecion) that is induced by simultaneous effects produced by oxytactic microorganisms, nanoparticles, and vertical temperature variation. The theory is developed for the situation when the nanofluid occupies a shallow horizontal layer of finite depth. The layer is defined as shallow as long as oxygen concentration at the bottom of the layer is above the minimum concentration required for the bacteria to be active (to actively swim up the oxygen gradient). The lower boundary of the layer is assumed rigid, while at the upper boundary both situations when the boundary is rigid or stress free are considered. Physical mechanisms responsible for the slip velocity between the nanoparticles and the base fluid, such as Brownian motion and thermophoresis, are accounted for in the model. A linear instability analysis is performed, and the resulting eigenvalue problem is solved analytically using the Galerkin method.     

Axisymmetric convective boundary layer in a vibrating fluid

A vibrating convective flow around a uniformly heated sphere in weightlessness conditions is studied theoretically for circularly polarized vibrations. It is found that the fluid motion has the form of two jets spreading from the sphere in opposite directions along the symmetry axis, perpendicular to the vibration polarization plane. For large characteristic temperature gradients, the flow becomes self-similar. The equations describing thermovibrational convection in the boundary layer approximation are derived. A class of self-similar solutions for a point heat source is found. The results obtained on the basis of the full equations and in the boundary layer approximation are compared.     

The Effect of Sudden Permeability Changes in Porous Media Filling Box Flows

We report upon experimental and analytical investigations of filling box flows in non-uniform porous media characterized by a sudden change in permeability. The porous medium consists of two layers separated by a horizontal permeability jump and is initially filled with light ambient fluid. A line source located at the top of the upper layer supplies dense contaminated fluid that falls toward the bottom of the domain. Two configurations are studied, i.e., a low-permeability layer on top of a high-permeability layer and vice versa. In the former scenario, the flow dynamics are qualitatively similar to the case of a uniform porous medium. Thus, the analytical formulation of Sahu and Flynn (J Fluid Mech 782:455–478, 2015) can be adopted to compute the parameters of interest, e.g., the plume volume flux. In the latter scenario, the flow dynamics are significantly different from those of the uniform porous medium case; after reaching the permeability jump, some fraction of the dense plume propagates horizontally as a pair of oppositely directed interfacial gravity currents. Meanwhile, the remaining fraction of the plume flows downward into the lower layer where it accumulates along the bottom boundary in the form of a deepening layer of discharged plume fluid. Depending on the permeability ratio of the upper and lower layers and the source conditions, the gravity currents may become temporarily arrested after traveling some finite horizontal length. An analytical prediction for this so-called run-out length is derived, motivated, in part, by the immiscible analysis of Goda and Sato (J Fluid Mech 673:60–79, 2011). Finally, a prediction of the filling box time, consisting of the time required to fill the control volume up to the point of contaminated fluid overflow, is made. These predictions are compared with analog experimental measurements. Generally positive agreement is found when the higher-permeability layer is located below the lower-permeability layer. In the opposite circumstance, the agreement is conditional. If the run-out length of the gravity current is less than the horizontal dimensions of the control volume (or tank in case of the experiments), the agreement is good. By contrast, when the run-out length is large, comparatively poor agreement may be realized: In spite of the higher density of the contaminated fluid, it may occupy the entirety of the upper layer before filling the lower layer.