Non-similar Boundary Layer Equations Research Articles

Free convection from a vertical plate with concentrated and distributed thermal sources

An analysis is developed for the laminar free convection from a vertical plate with uniformly distributed wall heat flux and a concentrated line thermal source embedded at the leading edge. We introduce a parameterξ=(1 +QL/Qw)−1=(1 + RaL/Raw)−1 to describe the relative strength of the two thermal sources; and propose a unified buoyancy parameterλ=(σ RaL+σ Raw)1/5 withσ=1/(1 +Pr−1) to properly scale the dependent and independent variables. The variables are so defined that the resulting nonsimilar boundary-layer equations can describe exactly the buoyancy-induced flow from the dual sources with any relative strength to fluids of any Prandtl number from very small values to infinity. These nonsimilar equations are readily reducible to the self-similar equations of an adiabatic wall plume forξ=0, and to those of free convection from uniform flux plate forξ=1. Rigorous finite-difference solutions for fluids of Pr from 0.001 to ∞ are obtained over the entire range ofξ from 0 to 1. The effects of both relative source strength and Prandtl number on the velocity profiles, temperature profiles, and the variations of wall temperature, are clearly illustrated.

Mixed convection on vertical plate for fluids of any prandtl number

A mixed convection parameterζ=(σRa)1/4/(ωRe)1/2, withσ=Pr/(1+Pr) andω=Pr/(1 +Pr)1/2, is proposed to replace the conventional Richardson number, Gr/Re2, for combined forced and free convection flow on an isothermal vertical plate. This parameter can readily be reduced to the controlling parameters for the relative importance of the forced and the free convection,Ra1/4/(Re1/2Pr1/3) forPr ≫ 1, and (RaPr)1/2/(RePr1/2 forPr ≪ 1. Furthermore, new coordinates and dependent variables are properly defined in terms ofζ, so that the transformed nonsimilar boundary-layer equations give numerical solutions that are uniformly valid over the entire range of mixed convection intensity from forced convection limit to free convection limit for fluids of any Prandtl number from 0.001 to 10,000. The effects of mixed convection intensity and the Prandtl number on the velocity profiles, the temperature profiles, the wall friction, and the heat transfer rate are illustrated for both cases of buoyancy assisting and opposing flow conditions.

Laminar premixed flames in mixed-convection thermal plumes

A theoretical investigation of laminar premixed flames stabilized downstream of a line heat source for mixed-convection conditions is described. The analysis involves solution of the nonsimilar boundary layer equations for an ideal gas with a one-step reaction while neglecting radiation. A deflagration wave is always stabilized at some point downstream of the source for these assumptions, but its position often shifts rapidly from near to far from the source over a limited range of reaction parameters. Conditions at this shift can provide a practical estimate of necessary conditions for ignition (a wake/plume-ignition criterion) if it is assumed that flames far from the source are not observed due to transition to turbulence and quenching by surfaces. Increased free stream velocities have the following effects on the limit: the magnitude of flame position shifts is reduced, source strengths are increased, the flame moves closer to the source, and the rate of lateral spread of the flame is increased. Similar to other ignition and flame stability processes, large shifts, which are representative of well-defined limits, tend to disappear as the activation enery of reaction is reduced.

Analysis of ignition by a plane laminar thermal plume

A theoretical investigation of ignition of a premixed combustible gas by a line heat source is described. Laminar flow was examined by numerical solution of the nonsimilar boundary layer equations, for an ideal gas with a one-step reaction and neglecting radiation. A deflagration wave is always present in the plume for these assumptions. However, the flame makes a rapid transition away from the source over a relatively small range of reaction parameters. This transition state provides a practical estimate of limiting ignition conditions-assuming that flames far from the source are not observed due to flow disturbances, transition to turbulence, or quenching by surrounding surfaces. Computations of limiting ignition conditions using this criterion indicate that while plume ignition approximates ignition by finite-sized sources, as the source size becomes small, there are interesting differences between the two cases due to effects of distributed heat addition and surface friction when the source size is finite. Streamwise gradients were appreciable in some circumstances; therefore, further analysis of this problem is suggested-relaxing the boundary layer approximations.

Flow over Plates with Suction through Porous Strips

This paper addresses the steady, incompressible, two-dimensional flow past a flat plate with suction through porous strips. Closed-form solutions for each flow quantity are developed in the context of linearized triple-deck theory using Fourier transforms. To demonstrate the validity of these closed-form solutions, we compare the wall shear stress and pressure coefficients and the streamwise velocity profiles from the linearized theory with those obtained by the numerical integration of both interacting and nonsimilar boundary-layer equations. The agreement between the linearized triple-deck and interacting boundary-layer equations is good; however, the nonsimilar boundary layers, which fail to account for upstream influence, are shown to be in poor agreement with both interacting boundary layers and the linearized triple deck. The linearized closed-form solutions will therefore be very useful in future stability calculations.

Ignition of supercritical fluds during natural convection from a heated vertical surface

The ignition of a combustible liquid near a heated vertical surface was investigated theoretically and experimentally for laminar natural convection conditions. Only pressures greater than the critical pressure were considered, where the combustible is a single-phase fluid. Analysis of ignition involved solution of the nonsimilar boundary layer equations for flow along the surface and in the plume, allowing for the nonideal properties of supercritical fluids. Ignition was prescribed as the surface conditions which separated regimes where decomposition waves either quenched or developed in the plume. Experiments were conducted with ethylene oxide at pressures of 5.50–19.30 MPa. Predictions could be matched to the measurements for supercritical conditions, using global first-order kinetics. The results indicate a slight increase in ignition resistance as pressure is decreased, for first-order kinetics, due to property variations near the critical point. Limited measurements at subcritical conditions indicated that ignition occurs when the heat flux exceeds the critical heat flux for nucleate boiling, at pressures somewhat below the critical pressure, since the minimum surface temperature for ignition is not accessible for conventionally heated surfaces in this region. This change in ignition mechanism is probably responsible for the unusual variation of ignition properties frequently observed near the critical pressure of combustibles.

Görtler instability

Görtler instability for boundary-layer flows over generally curved walls is considered. The full-linearized disturbance equations are obtained in an orthogonal curvilinear coordinate system. A perturbation procedure to account for second-order effects is used to determine the effects of the displacement thickness and the variation of the streamline curvature on the neutral stability of the Blasius flow. The streamwise pressure gradient in the mean flow is accounted for by solving the nonsimilar boundary-layer equations. Growth rates are obtained for the actual mean flow and compared with those for the Blasius flow and the Falkner–Skan flows. The results demonstrate the strong influence of the streamwise pressure gradient and the nonsimilarity of the basic flow on the stability characteristics.

Ignition of a combustible gas near heated vertical surfaces

A theoretical investigation of the ignition of a combustible gas by a heated vertical surface is described. Laminar, natural convection conditions were treated by numerical solution of the nonsimilar boundary layer equations, assuming an ideal gas and a one-step reaction and neglecting radiation. The plume above the surface was also analyzed, so that the ignition criterion could be specified as the least severe surface condition where a deflagration wave develops in the plume. Heated surfaces included both isothermal and constant heat flux boundary conditions. Plume conditions involved free-standing surfaces and surfaces along a wall. In the latter case, wall boundary conditions included adiabatic and isothermal (equal ambient and wall temperatures) wall plumes. Isothermal surfaces required less severe conditions for ignition than constant heat flux surfaces having the same maximum temperature. The required severity of surface conditions for ignition increased as plume conditions varied in this order: adiabatic wall plume, free-standing plume, isothermal wall plume. The frequently used ignition criterion for heated surfaces, which assumes that ignition occurs where the temperature gradient normal to the surface becomes zero, is shown to overestimate the severity of surface conditions required for ignition in some circumstances. Other limitations of this earlier criterion are that it cannot be applied to either constant heat flux or high-temperature surfaces.

Symmetry Plane Viscous Layer on a Sharp Cone

Numerical solutions for the wind and lee symmetry planes on sharp cones at incidence are obtained for laminar boundary layers. Comparisons are made between solutions obtained with the similarity boundary-layer equations, the nonsimilar boundary-layer equations, and the nonsimilar boundary region equations. For moderate incidence angles, boundary-layer theory fails at the leeplane and a boundary region develops. For large incidence, after crossflow reversal has occurred, nonsimilar boundary-layer solutions are obtained at the leeplane but are in poor agreement with experimental data. The boundary region system provides a significant improvement. 16~18 § analyses. With approximations 1 and 2, it has been shownl that the symmetry plane equations uncouple from the interior system so that for weak viscous-inviscid interaction the wind- and leeplane flow histories are apparently independent of the boundary-layer behavior around the cone. Solutions for the leeplane boundary layer therefore can be obtained by direct integration of the uncoupled symmetry plane equations16 (ordinary differential equations), or, alternatively, by a wind- to-leeplane marching procedure715 using the partial dif- ferential system governing the three-dimensional similarity boundary layer. The windward boundary-layer solution, as obtained from the uncoupled system, serves as the initial con- dition for the marching integration. For group 3 problems this uncoupling is not possible as diffusion effects are important not only normal to the surface, as with boundary layers, but also normal to the symmetry or separation planes. These domains, where a boundary layer forms within a boundary layer, are nonsimilar and termed boundary regions. Solutions of the nonsimilar boundary region equations can be obtained only by a streamwise integration procedure. Boundary-layer profiles are not specified at either of the symmetry planes. For unseparated flows, i.e., small to moderate angles of in- cidence, the boundary-layer methods of categories 1 and 2 have failed to provide leeplane solutions for all but the smallest yaw angles. This difficulty is most apparent with the uncoupled symmetry plane studies, as the calculations have failed to converge, 1-6 but even with the complete boundary- layer integration starting at the windplane, anomalous leeplane behavior has been observed.713 Moreover, the similarity symmetry plane equations exhibit nonunique windward plane solutions for all incidence angles. This nonuniqueness prevails at the leeplane in the limited range where solutions have been obtained.

An Approximate Method for the Solution of a Class of Nonsimilar Laminar Boundary Layer Equations

An approximate method is developed for locally nonsimilar laminar boundary layer flows. This method is applicable to several boundary layer velocity problems where the nonsimilarity stems from the freestream velocity distribution and the transverse curvature. The results are compared with those obtained by other methods and, except in the neighborhood of the point of separation, they are in good agreement.

On the solution of local nonsimilar boundary-layer equations for standard and inverse flows

With the recent advances in numerical techniques and with the availability of high speed computers, the numerical solution of the boundary-layer equations does not impose much difficulty. The governing equations for momentum transfer and heat transfer can be solved accurately and efficiently in their partial differential equation form. Despite this capability, however, there are some problems in engineering in which it is still advantageous as well as convenient to obtain the solution of the boundary-layer equations by using approximate methods, such as Pohlhausen's method, ref. (1), local similarity methods (see ref. (2), for example) and local nonsimilarity methods, ref. (3). These methods avoid the complexity of solving partial differential equations and consider the solution of ordinary differential equations. In this note we discuss the solution of the boundary-layer equations by using the local nonsimilarity method of ref. (3). We consider two-dimensional incompressible laminar flows and obtain their solutions for standard and inverse boundary-layer flows by using an efficient and accurate numerical method described in ref. (4). The method can easily be extended to solve the energy equation, ref. (5), since the solution of that equation is independent of the momentum equation and being a linear ordinary differential equation, can be solved by simple integration methods once the solution of the momentum equation is obtained.

Computations of C0 Laser Radiation Absorption in SF, -Air Boundary Layers

Numerical results are presented for the absorption of CO 2 laser radiation in SF 6-air laminar compressible boundary layers. The sonic flow over a flat plate at sea level is considered, wherein SF6 is injected into the boundary layer through the surface, and CO2 laser radiation at 10.6ju is incident on the plate downstream of the injection. Detailed boundary-layer profiles, the variation of radiative intensity within the boundary layer, and the amount of laser radiation penetrating to the surface is studied using a difference-dif ferential solution of the nonsimilar boundary-layer equations. In the results, strong coupling between the boundary-layer flow and the laser radiation is observed. Because of laser heating of the gas, the attenuation of the radiative intensity within the boundary layer is reduced. As a result, the laser intensity reaching the surface increases nonlinearly with the downstream distance. A limited comparison of the present results with existing experimental data yields good agreement These results bear on the protection of surfaces from laser beams, such as in new welding techniques, and in laser-gasdynamic energy conversion schemes.

Hypersonic Rarefied Viscous Interaction Over a Semi-Infinite Flat Plate

The problem of hypersonic rarefied viscous interaction over a semi-infinite flat plate has been solved using Meksyn’s asymptotic method of integrating the full nonsimilar boundary-layer equations. The primary phenomena of “incomplete compression”, velocity slip, and temperature jump are incorporated into the analysis. Results are obtained for induced wall pressure, skin friction, and Stanton number variation along the plate. Detailed results are also obtained for developing velocity profiles and shock location. The pressure plateau within the merged layer region is correctly predicted and the solutions merge far downstream with the strong interaction solutions where rarefied gas effects become negligible.

On the Ignition of a Pre-Mixed Fuel by a Hot Projectile

Abstract A theoretical study of the problem of ignition of a cold combustible mixture by a hot projectile is performed by approximating, firstly, the flow over its surface to a laminar flow of the mixture over a hot plate, and, secondly, the flow in the wake of the projectile to a plane laminar mixing of the cold unreacted mixture with the hot combustion products. Unlike the earlier analyses consisting of successive approximations and/or series expansions, the non-similar boundary layer equations are directly integrated by following a suitable numerical scheme. A characteristic parameter of the problem—the distance at which ignition occurs, which is defined by the presence of a temperature maximum—is obtained for different values of quantities relating to the properties of propane-air mixture, and to the speed as well as the dimensions of the projectile.

Pressure Work Effect on Natural Convection Flow from a Vertical Circular Cone with Suction and Non-Uniform Surface Temperature

Laminar free convection from a vertical circular cone maintained at non-uniform surface temperature with effects of suction and pressure work are considered. Non-similarity solutions for boundary layer equations are found to exist when the surface temperature follows the power law variations with the distance measured from the leading edge. The numerical solutions of the transformed non-similar boundary layer equations are obtained by using a finite difference method with Keller Box scheme. Solutions obtained in terms of skin-friction, local rate of heat transfer, velocity and temperature profiles for the values of Prandtl number, pressure work parameter and temperature gradient are displayed in both graphical and tabular forms.Keywords: Free convection, Viscous dissipation, Boussinesq approximation, Temperature gradient, Pressure work.Journal of Mechanical Engineering Vol.36 Dec. 2006 pp.6-11DOI = 10.3329/jme.v36i0.805

Separation Points of Magneto-hydrodynamic Boundary Layer Flow Along a Vertical Plate with Exponentially Decreasing Free Stream Velocity

The points of separation of magneto-hydrodynamic mixed convection boundary layer flow along a vertical plate have been investigated. The free stream velocity is considered decreasing exponentially in the stream wise direction. The governing boundary layer equations are transformed into a non-dimensional form and the resulting nonlinear system of partial differential equations are reduced to local non-similar boundary layer equations, which are solved numerically by implicit finite difference method known as Keller box scheme. Here we have focused our attention to find the effects of suction, magnetic field and other relevant physical parameters on the position of boundary layer separation. The numerical results are expressed in terms of local shear stress showing the effects of suction, buoyancy, Prandlt number and magnetic field on the shear stress as well as on the points of separation. Keywords: Separation points, magneto-hydrodynamic, mixed convection, boundary layer, suction, finite difference method, Keller box scheme.  doi:10.3329/jname.v5i1.1868Journal of Naval Architecture and Marine Engineering Vol. 5, No. 1 (June, 2008) 11-18.Â

Surface catalysis in dissociated, chemically frozen air boundary layers with variable transport properties

This work is concerned with the effects of surface catalysis and variable transport properties on the structure of chemically frozen, dissociated air boundary layers enveloping flat plates and slender wedges. Our main interest is in nonsimilar hypersonic flows past catalytic segments placed aft of leading edges or stagnation points coated with noncatalytic materials. The problems are solved numerically; the method of solution is shown to be reliable and applicable in general to the solution of nonsimilar boundarylayer equations. Local convective heat transfer and heat release due to catalysis of atoms are computed separately. These results show that variable transport properties and non-similarity of the flow field have a significant influence on the heat flux due to catalysis and suggest possible serious errors in available diagnostic formulas for catalytic gauges.