Unsteady Rarefied Gas Flow with Shock Wave in a Channel

The two-dimensional time-dependent problem of rarefied gas flow in a plane channel, formed by parallel plates of finite length and closed at one end, is solved on the basis of the kinetic S-model. The flow develops as a result of rupture of a diaphragm which separates the gas at rest in the channel and the gas at rest in a reservoir of infinite volume. The effect of gas deceleration at the channel walls under the conditions of diffuse molecular reflection from the channel walls and end face is studied. Decay of a shock wave and disappearance of a homogeneous flow zone behind the shock wave is traced for three variants of conditions at the channel inlet: (1) gas enters the channel from a reservoir of infinite length and width (as the basic variant), the simultaneous motion in the reservoir and channel being studied; (2) the high-pressure reservoir represents a usual channel section; and (3) the motion of the gas in the reservoir is not considered at all, instead of this, the boundary conditions of the evaporation-condensation type under the conditions of gas at rest in the reservoir are imposed in the inlet cross-section.     

Unsteady Boundary Layer on a Blunt Body in a Hypersonic Flow of Non-Uniform Dusty Gas

The unsteady heat transfer at the stagnation point on a blunt body traveling at hypersonic velocity through a layer of nonuniform dusty gas with low-inertia particles (not deposited on the body surface) is investigated. Using the matched asymptotic expansion method, the equations of the two-phase unsteady boundary layer near the symmetry axis of the body are derived with account for the polydispersity of the particles. The structure of the unsteady boundary layer and the variation of the friction and heat transfer coefficients at the stagnation point are studied numerically. Layered nonuniformities of the particle concentration and size are considered, the limits of variation of the thermal and mechanicals loads are found, and the effect of the dust polydispersity on the heat transfer is investigated.     

Compressible Flow in a Half-Space with Navier Boundary Conditions

We prove the global existence of weak solutions of the Navier–Stokes equations of compressible flow in a half-space with the boundary condition proposed by Navier: the velocity on the boundary is proportional to the tangential component of the stress. This boundary condition allows for the determination of the scalar function in the Helmholtz decomposition of the acceleration density, which in turn is crucial in obtaining pointwise bounds for the density. Initial data and solutions are small in energy-norm with nonnegative densities having arbitrarily large sup-norm. These results generalize previous results for solutions in the whole space and are the first for solutions in this intermediate regularity class in a region with a boundary.     

Computation of Stokes Flow in a Channel with a Collapsible Segment

The numerical solution of Stokes flow in two-dimensional channel in which a segment of one wall is formed by an elastic membrane under longitudinal tension and the remaining channel boundary is rigid is considered. This model problem is being used to gain an understanding of the complex interactions that occurs between the fluid flow and the wall mechanics when fluid flows through a collapsible tube, examples of which are widespread in physiology. Previous work by Pedley considered a similar system using lubrication theory in which the wall slopes are assumed small. The results showed that as the longitudinal wall tension is reduce, the downstream end of the collapsible segment becomes ever steeper, thus violating the assumptions. Here, lubrication theory is abandoned and a numerical solution of the full governing equations, including the complete expression for wall curvature, is sought using an iterative scheme. The effect of the variation in wall tension due to the fluid shear stresses at the compliant boundary is also included.Results are presented for a range of transmural (internal minus external) pressures and wall tensions. It is found, however, that as the wall tension is reduced, the iterative scheme considered fails to converge. This similar behaviour to that seen by Silliman & Scriven in viscous free-surface flows. Possible reasons for this breakdown together with alternative solution strategies are discussed.     

Darcy Flow in a Wavy Channel Filled with a Porous Medium

Flow in channels bounded by wavy or corrugated walls is of interest in both technological and geological contexts. This paper presents an analytical solution for the steady Darcy flow of an incompressible fluid through a homogeneous, isotropic porous medium filling a channel bounded by symmetric wavy walls. This packed channel may represent an idealized packed fracture, a situation which is of interest as a potential pathway for the leakage of carbon dioxide from a geological sequestration site. The channel walls change from parallel planes, to small amplitude sine waves, to large amplitude nonsinusoidal waves as certain parameters are increased. The direction of gravity is arbitrary. A plot of piezometric head against distance in the direction of mean flow changes from a straight line for parallel planes to a series of steeply sloping sections in the reaches of small aperture alternating with nearly constant sections in the large aperture bulges. Expressions are given for the stream function, specific discharge, piezometric head, and pressure.     

Flow of Particulate-Fluid Suspension in a Channel with Porous Walls

The problem of the dispersed particulate-fluid two-phase flow in a channel with permeable walls under the effect of the Beavers and Joseph slip boundary condition is concerned in this paper. The analytical solution has been derived for the longitude pressure difference, stream functions, and the velocity distribution with the perturbation method based on a small width to length ratio of the channel. The graphical results for pressure, velocity, and stream function are presented and the effects of geometrical coefficients, the slip parameter and the volume fraction density on the pressure variation, the streamline structure and the velocity distribution are evaluated numerically and discussed. It is shown that the sinusoidal channel, accompanied by a higher friction factor, has higher pressure drop than that of the parallel-plate channel under fully developed flow conditions due to the wall-induced curvature effect. The increment of the channel’s width to the length ratio will remarkably increase the flow rate because of the enlargement of the flow area in the channel. At low Reynolds number ranging from 0 to 65, the fluids move forward smoothly following the shape of the channel. Moreover, the slip boundary condition will notably increase the fluid velocity and the decrease of the slip parameter leads to the increment of the velocity magnitude across the channel. The fluid-phase axial velocity decreases with the increment of the volume fraction density.     

Kinematic Characteristics of Fluid Flow in a Channel with a Profiled Bottom

The results of an experimental investigation of fluid flow in a channel with an erodible and profiled rigid bottom are presented. The kinematic parameters of the flow were measured with a laser Doppler anemometer. The experimental data were interpreted using a linear model of potential flow over bottom roughness.     

A Comment on Change of Nusselt Number Sign in a Channel Flow Filled by a Fluid-Saturated Porous Medium with Constant Heat Flux Boundary Conditions

The aim of this Letter is to show that, the Nusselt number sign might be changed without changing of heat transfer direction at the wall of channels, even for flows without viscous dissipation.     

Passive Scalar in a Turbulent Channel Flow with Wall Velocity Disturbances

Direct Numerical Simulations (DNS) of a passive scalar in a turbulent channel flow with a normal velocity disturbance on the lower wall are presented for high and low Reynolds numbers. The aim is to reproduce the complex physics of turbulent rough flows without dealing with the geometric complexity. In addition, isothermal walls that cannot be easily assigned in an experiment, are considered. The paper explains the increase of heat transfer through the changes of the velocity and thermal structures. As in real rough flows, the transpiration produces an isotropization of the turbulence near the wall.     

Darcy-Brinkman Flow Through a Bumpy Channel

The forced flow through a channel with bumpy walls which sandwich a porous medium is studied. The problem models micro-fluidics where, due to the small size of the channel width, the surface roughness of the walls is amplified. The Darcy-Brinkman equation is solved analytically through small perturbations on the ratio of bump amplitude to the half width of the channel. The first- order perturbation solutions give the three-dimensional velocity effects of the bumpiness and the second-order perturbation solutions give the increased resistance due to roughness. The problem depends heavily on the non-dimensional porous medium parameter $k$ which represents the importance of length scale to the square root of permeability. Our solutions reduce to the clear fluid limit when $k$ is zero and to the Darcy limit when $k$ approaches infinity.     

Problem of Optimal Control of the Turbulent Boundary Layer on a Permeable Surface in Supersonic Gas Flow

The problem of constructing the law of distribution of the normal component of the velocity of blowing to the turbulent boundary layer at supersonic flow velocities which ensure the minimum convective heat flow transmitted from the boundary layer to the surface is considered. The power of the control system calculated with regard to Darcy’s law of flow through a porous medium acts as the isoperimetric condition. The problemis solved using the Dorodnitsyn generalized integral relations. The numerical experiments carried out in the case of flow past a sphere showed the effectiveness of the optimal blowing laws as compared with the uniform law, namely, the gain in the minimized functional reaches 31.82%.     

Numerical Simulation of a Turbulent Flow in a Channel with Surface Mounted Cubes

In this paper we report on a fourth-order, spectro-consistent simulation of a complex turbulent flow. A spatial discretization of a convection-diffusion equation is termed spectro-consistent if the spectral properties of the convective and diffusive operators are preserved, i.e. convection skew-symmetric; diffusion symmetric positive definite. We consider a fully developed flow in a channel, where a matrix of cubes is placed at a wall of the channel. The Reynolds number (based on the channel width and the mean bulk velocity) is equal to Re = 13,000. The three-dimensional flow around the surface mounted cubes has served at a test case at the 6th ERCOFTAC/IAHR/COST workshop on refined flow modeling (Delft, June 1997). Here, mean velocity profiles as well as Reynolds stresses at various locations in the channel have been computed without using any turbulence models. The results agree well with the available experimental data.     

Flow of a Weakly Conducting Fluid in a Channel Filled with a Porous Medium

We investigate the fully developed flow in a fluid-saturated porous medium channel with an electrically conducting fluid under the action of a parallel Lorentz force. The Lorentz force varies exponentially in the vertical direction due to low fluid electrical conductivity and the special arrangement of the magnetic and electric fields at the lower plate. Exact analytical solutions are derived for fluid velocity and the results are presented in figures. All these flows are new and are presented for the first time in the literature.     

Two-phase Flow Dynamics at the Interface Between GDL and Gas Distributor Channel Using a Pore-Network Model

Transport in Porous Media - For improved operating conditions of a polymer electrolyte membrane (PEM) fuel cell, a sophisticated water management is crucial. Therefore, it is necessary to...     

Natural Convection Boundary-Layer Flow in a Porous Medium with Temperature-Dependent Boundary Conditions

The natural convection boundary-layer flow on a surface embedded in a fluid-saturated porous medium is discussed in the case when the wall heat flux is related to the wall temperature through a power-law variation. The flow within the porous medium is assumed to be described by Darcy’s law and the Boussinesq approximation is assumed for the density variations. Two cases are discussed, (i) stagnation-point flow and (ii) flow along a vertical surface. The possible steady states are considered first with the governing partial equations reduced to ordinary differential equations by similarity transformations and these latter equations further transformed to previously studied free-convection problems. This identifies values of the exponent N in the power-law wall temperature variation, N = 3/2 for stagnation-point flows and 3/2 ≤ N ≤ 3 for the vertical surface, where similarity solutions do not exist. Time development for stagnation-point flows is seen to depend on N, for N <  3/2 the steady state is approached at large times, for N ≥ 3/2 a singularity develops at finite time leading to thermal runaway. Numerical solutions for the vertical surface, where the temperature-dependent boundary condition becomes more significant as the solution develops, show that, for N < 3/2, the corresponding similarity solution is approached, whereas for N >  3/2 the solution breaks down at a finite distance along the surface.     

Free Fluid Flow Through a Visco-Deformable Porous Medium with a Moving Boundary

One-dimensional Darcy-law flow through a porous matrix representing a high-viscosity liquid is investigated. The flow develops in a region which depends on time due to sedimentation. The problem considered simulates the geological process of sedimentation in a basin. In accordance with geological data, the permeability and viscosity coefficients of the matrix are assumed to depend nonlinearly on the porosity. The asymptotic properties of the flow are described for large times. The agreement between the results of asymptotic and numerical solutions is satisfactory at intermediate times and good at large times under the realistic sedimentary basin conditions. The simplicity of the asymptotic solution obtained makes it possible to vary the problem parameters and determine the porosity, pressure, and velocities for particular geological conditions by means of simple calculations.     

Non-Darcy Mixed Convection in a Vertical Porous Channel with Boundary Conditions of Third Kind

Combined free and forced convection flow in a parallel plate vertical channel filled with porous matrix is analyzed in the fully developed region with boundary conditions of third kind. The flow is modeled using the Brinkman?CForchheimer-extended Darcy equations. The plates exchange heat with an external fluid. Both conditions of equal and different reference temperatures of the external fluid are considered. Governing equations are solved numerically by shooting technique that uses classical explicit Runge?CKutta scheme and Newton?CRaphson method as a correction scheme and analytically using perturbation series method for Darcy model. The velocity field, the temperature field and Nusselt numbers are obtained for governing parameters such as porous parameter, inertia term and perturbation parameter for equal and unequal Biot numbers and are displayed graphically. The dimensionless mean velocity and bulk temperature are also determined. It is found that the numerical solutions agree for small values of the perturbation parameter in the absence of the inertial forces.     

Kinematic Parameters of a Two-Phase Flow in a Rectangular Channel

A kinematic two-phase flow pattern formed in a rectangular channel due to the interaction of a gas flow with an initially stationary or moving water layer is investigated. Using laser diagnostics and hot-wire methods, the velocity distributions in the water and the air are found for a stratified flow regime.     

Boundary Layers and KPP Fronts in a Cellular Flow

We study an eigenvalue problem associated with a reaction-diffusion-advection equation of the KPP type in a cellular flow. We obtain upper and lower bounds on the eigenvalues in the regime of a large flow amplitude A ≪ 1. It follows that the minimal pulsating traveling front speed c _*(A) satisfies the upper and lower bounds C ₁ A ^1/4≦ c _*(A)≦ C ₂ A ^1/4. Physically, the speed enhancement is related to the boundary layer structure of the associated eigenfunction – accordingly, we establish an “averaging along the streamlines” principle for the unique positive eigenfunction.