Solution For Channel Flow Research Articles

Analytical solution of channel flow with downstream control

Analytical solution of channel flow with downstream control

Application of a slow-zone model to contaminant dispersion in laminar shear flows

A slow-zone model which was developed for dispersion in two-dimensional turbulent open channel flow is here applied to contaminant dispersion in laminar shear flow in a circular pipe or between parallel plates. In this model the flow is divided into two zones—a fast zone in the middle of the flow and a slow zone near the wall. The resulting pair of coupled linear dispersion equations is solved by Fourier transformation and the Fourier integrals are inverted approximately. The significance of the results is that explicit analytic expressions are obtained for the fast zone, slow zone and cross-sectional average contaminant concentrations. These expressions are valid for all longitudinal positions and all time following contaminant discharge. The results show that laminar dispersion can be described in two stages. In the first stage, corresponding to small times after contaminant discharge, the mean contaminant distribution ( c) consists of a leading Gaussian (mostly in the fast zone), followed by a trailing Gaussian (mostly in the slow zone), and the peak values decay exponentially with time. In the second stage (larger times) c gradually approaches a single Gaussian which travels with the mean speed in accordance with the Taylor model. These theoretical results are shown to be in reasonable agreement with previously calculated exact (numerical) solutions for laminar pipe or channel flows.

Steady viscous flows by compact differences in boundary-fitted coordinates

Fourth-order compact differencing is applied to the steady solution of two-dimensional viscous incompressible flows at moderate Reynolds number. The physical region where the fluid flow occurs is mapped onto a rectangle by means of the boundary-fitted coordinates transformation method. The design of the general numerical algorithm rests upon discretization of velocity components and pressure field at the same grid points. The validity of this procedure is assessed by the investigation of a Stokes test problem. Fourth-order numerical results are compared to the analytical solution, second-order results and Chebyshev tau approximation. It is shown that fourth-order differences provide good precision, particularly when the ability of generating irregular meshes is fully exploited. Standard problems as Poiseuille and Couette flows, and the square cavity problem are solved. The fourth-order results on coarse mesh compare favourably with other techniques such as finite element method and second-order differences. A global convergence analysis was performed. On a two-dimensional problem with smooth boundary conditions, one observes a rate of convergence of order four for the velocity components and of order three for the pressure. For the square cavity problem with the two corner singularities, the rates of convergence are decreased by almost an order of magnitude. The solution of a plane constricted channel flow enhances the overall improvement in accuracy, gained by the treatment of the geometrically complex region through the mapping technique.

The calculation of eigenvalues for the stationary perturbation of poiseuille flow

The problem considered is that of two-dimensional viscous flow in a straight channel. The decay of a stationary perturbation from the Poiseuille flow in both the upstream and downstream directions is sought. A differential eigenvalue equation resembling the Orr-Sommerfeld equation is solved using a spectral method for values of the Reynolds number R between 0 and 2000. Although the decay downstream has received considerable previous attention, the situation upstream has not been studied in detail. It is found that for R greater than about 250, the upstream disturbances decay more rapidly by an order of magnitude greater than the downstream disturbances. The eigen-functions are used to obtain an asymptotic solution for two-dimensional channel flow both upstream and downstream and comparison is made with a previous numerical solution for flow in a channel with a step change in width.

Effects of external circuit on MHD channel flow of a rarefied gas

An exact solution of MHD channel flow of a rarefied gas has been carried out on taking into account the external circuit. Closed form solutions have been derived for velocity, amplitude, current density, magnetic field, normalised flow rate and the skin-friction. The effects of λ (rarefaction parameter), M (Hartmann number), V T ∗ (terminal voltage), I ∗ (total current) have been discussed on the velocity, current density, magnetic field, flow rate and the skin-friction.

Variable-viscosity flows in channels with high heat generation

This paper presents a similarity solution for plane channel flow of a very viscous fluid, whose viscosity is exponentially dependent upon temperature, when heat generation is very large. A dimensionless formulation of the problem involves two length scales (the depthhand lengthl, respectively, of the channel), one velocity scale (the mean velocityVof the fluid along the channel), the thermal conductivityk, thermal diffusivitykand viscosityVof the fluid, and the temperature coefficientbof the viscosity. From these, two important dimensionless groups arise, the Graetz number (Gz = Vh2/kl) and the Nahme–Griffith number (G= μV2b/k). In the case of steady flow withG−1[Lt ]Gz−1[Lt ] 1 a thin thermal boundary layer of thickness proportional toGz−½arises at each wall with an even thinner shear layer, detached from the wall and embedded in the thermal boundary layer, of thickness proportional toGz−½(lnG)−1, coinciding with the region of maximum temperature (lnG)/b. The similarity variable is (Pe½y/x½) wherePeis the Péclet number (Vh/k) andyandxare measured away from and along (either) boundary wall. The analogous unsteady uniform flow solution is also given.

Channel and Tube Flows with Surface Mass Transfer and Velocity Slip

An analysis of internal flows with surface mass transfer is performed in which account is taken of velocity slip at the surface of the porous bounding wall. The solutions for channel and tube flows reveal a variety of interesting effects of velocity slip, which depend upon whether fluid is injected or withdrawn at the bounding wall and upon the sign of the streamwise pressure gradient. For example, relative to the situation in which there is no slip, the streamwise pressure gradient may either increase or decrease in the presence of slip and may even undergo a change of sign.

Nonuniform conductivity MHD channel flow solution by B. G. Galerkin method.

The steady, fully developed MHD channel flow of a viscous, electrically conducting fluid is investigated under applied magnetic and electric fields. The conductivity is assumed to be a function of the space variable across the channel, and the Hall effect is neglected. The application of the B. G. Galerkin method is formulated for the general problem, and numerical results are presented for various conductivity profiles under different loading conditions. It is shown that the variational method has distinct computational advantages over the classical mathematical solutions. The results verify that the flow is strongly affected by the variations in the conductivity, and the assumption of uniformity in the fluid may lead to considerable errors in the analysis of the actual physical problem.

The stability of a family of Jeffery–Hamel solutions for divergent channel flow

A set of Jeffery–Hamel profiles (for radial, viscous, incompressible flow) have been shown by Fraenkel (1962, 1963) to approximate to profiles in certain two-dimensional divergent channels. The stability of a family of these profiles is investigated by a numerical solution of the Orr-Sommerfeld problem. Neutralstability curves are calculated in the (R,k)-planes (where R is the Reynolds number of the basic flow and k is the wave-number of the disturbance), and fairly low critical Reynolds numbers are found. For those profiles that have regions of reversed flow, negative wave velocities are found on the lower branch of the neutral curve, and also it is found that Rk tends to a finite limit as R → ∞ on the lower branch. These unexpected results are further discussed and verified by independent methods. The relation of the calculations to some experiments of Patterson (1934, 1935) is discussed.

Viscous Fluid Flow Under the Influence of a Resonant Acoustic Field

This paper presents an analytical solution for laminar channel flow under the influence of a resonant acoustic field. The solution predicts that standing vortices exist in the flow and that the size of these vortices, the ratio of the width of the vortex to the half channel width, is dependent upon the parameter M¯/M02 where M is the average Mach number of the through flow and M0 is the acoustic Mach number which is based upon the maximum amplitude of the time dependent velocity. The existence of the standing vortices was confirmed experimentally by a visualization study of tube flow. Typical flow fields are shown photographically. Finally the threshold data of Eastwood for convection heat transfer under the influence of a resonant acoustic field are correlated. It was shown that sound affects the local heat-transfer rate if M¯/M02 is less than 63.

Hall current and inlet disturbances in constant area channel flows

The problem considered is that of obtaining the fluid flow fields introduced within an electromagnetic accelerator, by the off-axis of the Lorentz force created when a component of the electric current exists. In addition, the growth or decay of inlet disturbances is investigated. In this paper a perturbation solution for the fluid properties is obtained. The reference flow about which the perturbation is taken is a modified channel-flow solution which becomes exact as the ratio of channel height to channel length approaches zero. Several example solutions are computed, all but one of which lead to closed form expressions for the dependent variables in terms of the velocity of the imposed channel flow. The solutions indicate that for values of the Hall parameter near or greater than unity, the validity of the channel-flow solutions will be questioned in some cases of high electromagnetic loading, unless the channel width is kept very small. The results for the behavior of inlet disturbances show that in all cases considered such disturbances grow to some extent downstream.

An Approximate Solution for Laminar Channel Flow of a Conducting Fluid Under Transverse Magnetic Field

An Approximate Solution for Laminar Channel Flow of a Conducting Fluid Under Transverse Magnetic Field

Nonuniformities in Magnetogasdynamic Channel Flow

On the Entropy Change Through a Shock Layer JAMES SERRIN AND Y. C. WHANG 990 T QUASI-ONE-DIMENSIONAL magnetogasdynamic channel flow equations have been presented for the case of mutually perpendicular applied electric and magnetic fields. As it was later pointed out, the flow variables-—e.g., pressure, velocity, current, magnetic field, electromagnetic body force, etc.—are to be taken as average values, and the need for Maxwell's equations arises only in evaluating the induced part of the magnetic field by means of the Biot-Savart law. This induced field is then subtracted from the total field used in the channel-flow solution, in order to ascertain the field to be applied at each position. Following the indicated calculation, the resulting expressions for the x and z components of the induced field are seen to exhibit a dependence not only on position, but on channel height, a, width, b, and length, L. The notation is that of Ref. 2 with x indicating the flow direction.

Magnetohydrodynamic Channel Flow under Time-Dependent Pressure Gradient

Hartmann's work on magnetohydrodynamic channel flow is extended to include the case with a time-dependent pressure gradient. Cases treated are: (a) suddenly applied pressure gradients which are periodic in time, and (b) pressure gradients which, with respect to time, are step and delta functions. The channel walls are assumed to be perfectly conductive. As a limiting case, the solution for hydrodynamic channel flow with a time-dependent pressure gradient is also given. The nature of viscous and magnetic drag on the wall is discussed. By use of the convolution integral and super-position principle, solutions can be obtained for any arbitrary time-dependent pressure gradient. Figures show the physical consequence of the solutions.

Shock Response to Downstream Disturbances

Shock Response to Upstream Disturbances, Roy Gundersen. 226 Variational Analysis of Ablation, M. A. Biot and H. Daughaday 227 Shock-Wave Profiles of Sharp Wedges in Helium Flow at M = 28.9, Robert H. Johnson 229 Roll Damping of a Fleet Ballistic-Missile Submarine, Frank J. Regan 230 Dusty-Gas Flow Through an Oblique Shock Wave, Kinge Okauchi 230 Constant-Electric-Field and Constant-Magnetic-Field Magnetogasdynamic Channel Flow, Gordon C. Oates 231 Effect of Porosity on the Two-Dimensional Supersonic Sail, E. A. Boyd 232 Response of a Schuler-Tuned Inertial Guidance System to Random Errors, James T. Ephgrave 233 A Method of Obtaining an Approximate Solution of the Falkner and Skan Boundary-Layer Problem, Eleanor M. James 234 The Use of the Howarth Transformation in Turbulent, Compressible Flow, William T. Snyder 235 Special Solutions to the Equations of Motion for Maneuvering Entry, W. Scott Jackson 236 An Experimental Test of Compressibility Transformations for Turbulent Boundary Layers, William Squire 237 Blunt-Cone Pressure Distributions at Hypersonic Mach Numbers, Andrew F. Burke and James T. Curtis 237 The Evaluation of Response of Aeroelastic Systems to Transient and Random Turbulence by Kernel-Function Techniques, Arthur K. Cross 239 Pressure Variation of Strong-Detonation Limits, K. M. Foreman 240 A Uniqueness Theorem for a Nonlinear Maxwell Body, Jerome L. Sackman 240 On the Hodograph Method in Magnetogasdynamics, Milomir M. Stanisic 242 Historical Note on the 1.5 Factor of Safety for Aircraft Structures, F. R. Shanley 243 A Magnetofluid-Dynamic Kutta-Joukowsky Condition, E. Cumberbatch, L. Sarason, and H. Weitzner 244 A Technique for Determining the Nozzle-Flow Properties of Air in an Equilibrium, Nonequilibrium, or Frozen State, Lovick 0. Hayman, Jr. and Roger B. Stewart 245 A Note on the Design of a Two-Dimensional Contracting Channel, Y. S. Nanjunda Swamy 246 An Approximate Solution for Laminar Channel Flow of a Conducting Fluid under Transverse Magnetic Field, Fujihiko Sakao 246 Direct Beam Stiffness-Matrix Calculations Including Shear Effects, Claus J. Meissner 247 A LINEARIZED ANALYSIS, based on small area variations, > 2 is utilized to determine the perturbation of an initially plane and uniform shock wave propagating through a tube of varying cross-sectional area into an initially uniform and isentropic state, which is subjected to a nonisentropic perturbation at some section, x = a, say. (The case of conditions prescribed at t = 0 would be treated similarly.) The shock is perturbed by the area variations and further by the perturbed flow upstream so that the flow behind the shock is nonisentropic. By applying the conditions on x = a to the general solution for the nonisentropic perturbation of an initially uniform state, the perturbed flow in front of the shock is completely determined. By writing the shock relations in perturbation form, a formula for the shock-velocity perturbation is derived by superposition of the perturbations due to the area variations and upstream perturbation. I t is found that the shock-velocity perturbation is singular for sonic flow just behind the shock with the singularity contained in the terms arising from the upstream perturbation. The results obtained may be applied to the case of an initially stationary shock obtained by setting the original shock-velocity equal to zero. When the equations which govern the one-dimensional, unsteady flow of an inviscid, ideal compressible gas in a tube of variable cross-sectional area are linearized in the neighborhood of a known (isentropic) solution, the general solution for an initially uniform state is: