Free convection-radiation interaction from an isothermal plate inclined at a small angle to the horizontal

Summary The interaction of free convection with thermal radiation in boundary layer flow from an inclined isothermal plate is studied numerically. Introducing appropriate transformations the equations governing the flow are expressed in the form of local nonsimilarity equations valid near the leading edge as well as in the downstream region. A group of transformations is also introduced such that the flow near the leading edge and far downstream can be described. Heated upward facing plates with positive and negative inclination angles are investigated. When the inclination is negative the boundary layer separates from the surface and the numerical solutions can be extended downstream past the point of separation. From the present investigation it may be concluded that the position of the separation point moves away from the leading edge with the increase of either of the thermal radiation parameter or the surface temperature parameter of the heated surface.     

The Navier-Stokes solution for laminar flow past a semi-infinite flat plate

Summary A numerical solution joining Carrier and Lin's solution near the leading edge to the boundary layer solution at large distance of the leading edge is presented. The solution is valid for any Reynolds number. Results are given for the skin friction, the integrated skin friction, the displacement thickness, the pressure along the plate and the velocity ahead of the plate. The asymptotic value of the integrated skin friction agrees very well with the exact value. The displacement thickness is already different from zero for small distances ahead of the plate.     

On the nonlinear distortion of waves generated by interacting supersonic boundary layers

Summary In order to obtain uniformly valid solutions which describe the interaction between laminar boundary layers and supersonic flows it is necessary, in general, to account for the nonlinear distortion of the associated wave fields. Results obtained by means of the analytical method of characteristics are applied to typical examples such as the flow over compression ramps, the flow near the trailing edge of a flat plate and shock boundary layer interactions.With 6 FiguresDedicated to Prof. Dr. K. Oswatitsch on the occasion of his 75th birthday.     

Three-dimensional wall-bounded laminar boundary layer with span-wise cross free stream and moving boundary

In the current work, the 3D boundary layers of wall-bounded flow configurations were extended to the situations with span-wise cross moving boundary and free stream. The unsteady boundary layer is also addressed for the Falkner–Skan wedge flow with a span-wise oscillating wall or oscillating free stream. The span-wise secondary boundary layer equation is obtained using similarity transformation technique and solved analytically in terms of the primary stream-wise boundary layer flow solutions. Different fluid motion behaviors are found for these new solutions. It is found that for the span-wise secondary boundary layer flow there is no flow separation for any wall cross moving velocity, which is different from the primary stream-wise boundary layers with a reverse flow. For the unsteady boundary layer with an oscillating wall or free stream, it is seen that the solution is different from the Stokes oscillating plate or free stream problem. The unsteady wall drag increases with the increase in the oscillating frequency and decreases with increasing the primary span-wise free stream magnitude. The velocity overshooting near the wall is also seen for an oscillating free stream for a large oscillating frequency or a lower primary stream-wise free stream magnitude.     

Variable thermal conditions for flooded roofs

Certain properties of the equations of motion and the boundary conditions for a slowly flowing suspension are examined with allowance for sedimentation. The method obtained is applied to the calculation of the sedimentation boundary layer near an inclined plate.     

Mixed convection in wall plume of power-law fluids

Summary The non-similar boundary layer solutions are presented to study the mixed convective flow of a power-law fluid above a vertical adiabatic surface with a steady thermal source at the leading edge. The boundary layer equations contain an important mixed convection parameter. The governing non-similar equations are solved by means of a novel finite difference scheme for several values of the power-law viscosity index, the buoyancy parameter and the non-Newtonian Prandtl number of the fluid. The solutions obtained are uniformly valid over the entire regime of the buoyancy parameter ranging from forced convection to free convection limits. Consideration is given to the case of buoyancy-assisted and buoyancy-opposed plumes.     

Mixed convection boundary layer flow on a vertical surface in a saturated porous medium

Summary The flow of a uniform stream past an impermeable vertical surface embedded in a saturated porous medium and which is supplying heat to the porous medium at a constant rate is considered. The cases when the flow and the buoyancy forces are in the same direction and when they are in opposite direction are discussed. In the former case, the flow develops from mainly forced convection near the leading edge to mainly free convection far downstream. Series solutions are derived in both cases and a numerical solution of the equations is used to describe the flow in the intermediate region. In the latter case, the numerical solution indicates that the flow separates downstream of the leading edge and the nature of the solution near this separation point is discussed.     

Free convective heat transfer at a vertical semiinfinite plate

The effect of the boundary layer at the leading edge on heat transfer near a vertical semiinfinite heated plate is determined by means of matched asymptotic expansions. The criterial relation for air is in good agreement with existing experimental data.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 33, No. 1, pp. 32–39, July, 1977.     

Aerodynamic structure of a flat plate laminar boundary layer with diffusion flame

An aerodynamic structure of a laminar boundary layer over a flat plate with uniform fuel injection from the flat plate and with diffusion flame is investigated numerically. Elliptic type conservation equations are used to take into account the pressure variation within the boundary layer. Velocities and the pressure are solved numerically by SIMPLER algorithm. One step irreversible chemical reaction of methane is assumed. An Arrhenius type chemical reaction rate model is assumed and the pre-exponential factor is varied from 1.0 × 10¹² to 1.0 × 10³⁰ m³/(kg s) as a parameter of the reactivity in order to elucidate the effect of the reactivity on the structure of the boundary layer. When the chemical reaction is very fast, the leading edge of the reaction zone reaches the flat plate. As the chemical reaction rate is decreased with a decrease in the pre- exponential factor, the leading edge of the reaction zone parts from the flat plate and it shifts downstream. The fuel is injected in front of the leading edge of the reaction zone, where the air is dominant, and the oxygen penetrates into the fuel dominant zone through the region between the leading edge and the flat plate. As a consequence, a premixed gas is formed around the leading edge of the reaction zone. The premixed gas seems to react in the region apart from the main visible flame.Part of this work was presented at ICCM 86, Tokyo, Japan, May 25–29, 1986     

Wall-bounded laminar sink flows

Summary The laminar flow near an infinite plane wall perpendicular to a line sink of constant strength is investigated in the limit of large Reynolds numbers. Self-similarity requires that fluid is issuing from the boundary layer. The inviscid flow outside the boundary layer is governed by the Euler equations. A one-parametric set of solutions to the Euler equations with appropriate boundary conditions is given. Uniqueness of the inviscid flow solution is obtained from matching with the boundary layer expansion. The solution of the boundary-layer equations is given both in closed form and numerically. It is found that at the edge of the boundary layer the vorticity decays algebraically.Dedicated to Prof. Dr. Dr. h. c. Franz Ziegler on the occasion of his 60th birthday     

Free convection from a vertical permeable circular cone with non-uniform surface temperature

Summary Laminar free convection from a vertical permeable circular cone maintained at nonuniform surface temperature is considered. Non-similar 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 three methods, namely, (i) a finite difference method, (ii) a series solution method, and (iii) an asymptotic solution method. Solutions are obtained in terms of skin friction, heat transfer, velocity profile and temperature profile for smaller values of Prandtl number and temperature gradient are displayed in both tabular and graphical forms. Finite difference solutions are compared with the solutions obtained by perturbation and asymptotic techniques and found to be in excellent agreement.     

Boundary integral equation method for conductive cracks in two and three-dimensional transversely isotropic piezoelectric media

Using the fundamental solutions and the Somigliana identity of piezoelectric medium, the boundary integral equations are obtained for a conductive planar crack of arbitrary shape in three-dimensional transversely isotropic piezoelectric medium. The singular behaviors near the crack edge are studied by boundary integral equation approach, and the intensity factors are derived in terms of the displacement discontinuity and the electric displacement boundary value sum near the crack edge on crack faces. The boundary integral equations for two dimensional crack problems are deduced as a special case of infinite strip planar crack. Based on the analogy of the obtained boundary integral equations and those for cracks in conventional isotropic elastic material and for contact problem of half-space under the action of a rigid punch, an analysis method is proposed. As an example, the solution to conductive Griffith crack is derived.     

Flow near the leading edge of a rectangular wing of small aspect ratio with applications to the bow of a ship

Summary A local solution is developed for the square-root singularity at the leading edge of a rectangular flat plate of small aspect ratio. The analysis is based on the assumption that the spanwise load distribution is elliptical, and it follows that the two-dimensional lifting-surface integral equation can be reduced to a one-dimensional integral equation of the Wiener-Hopf form. A solution is then obtained, valid near the leading edge, which yields the strength of the square-root singularity. This solution is used to compute the leading-edge suction force, which differs by 15% from the known value based on the total lift and drag expressions. The local solution is applied to study the flow around the stem of a ship's bow, as a result of a yawing motion, using Lighthill's rule to correct for the finite radius of curvature at the bow.Supported by the Office of Naval Research, Contract N00014-67-A-0204-0023, NR062-411, and by the National Science Foundation, Grant GK 10846. Reproduction in whole or in part is permitted for any purpose of the United States Government.     

Thermal stress singularities at tips of a crack in a semi-infinite medium under uniform heat flow

In the framework of plane thermoelastic problems is discussed the thermal stress field near the tips of an arbitrarily inclined crack in an isotropic semi-infinite medium with the thermally insulated edge surface under uniform heat flow. The crack is replaced by continuous distributions of quasi-Volterra dislocations corresponding to line heat sources and edge dislocations, and we obtain a set of simultaneous singular integral equations for dislocation density functions, whose solution is given in the forms of series in terms of Tchebycheff polynomials of the first kind. By means of this method, the thermal stress singularities at the crack tips are estimated exactly and the stress intensity factors can be readily evaluated. Numerical results are given for the particular case where the surface of the inclined crack is maintained at constant temperature and the heat supplied across the surface of the crack vanishes as a whole. The effects of the distance from the crack tip to the edge surface of the semi-infinite medium and the angle of inclination of the crack on the stress intensity factors and the initial direction of crack extension are shown graphically.     

The asymptotic boundary layer on a circular cylinder in axisymmetric micropolar fluid flow

An asymptotic boundary layer analysis is presented using the theory of micropolar fluids due to Eringen. The laminar boundary layer induced on the outside of a long, slender cylinder due to the flow of an incompressible micropolar fluid parallel to the axis of the cylinder is investigated. For reasons of both analytical and practical interests the boundary layer characteristics far down stream from the leading edge are analyzed on the basis of their asymptotic nature. Asymptotic series solutions for the velocity and micro-rotation fields are obtained. An expression for the new micropolar boundary layer thickness is derived. Central to the present investigation is the result that while calculating the skin friction one should take into account the total surface stress effects, not only due to the usual shear stresses but also due to the couple stresses. As a result, it is shown that the micropolar theory does predict a reduction in skin friction as is observed in experiments thus confirming Eringen's well known conjecture.     

Analysis of interactions between shock waves and laminar boundary layers with heat transfer

Summary This paper deals with interactions between shock waves and laminar boundary layers on flat plates with heat transfer. In order to describe this phenomenon the boundary layer is divided into inner viscous layer and outer inviscid layer after Gadd. The boundary layer approximations are assumed to remain valid in the inner layer and the momentum integral equation for the layer is utilized instead of the Pohlhausen's wall condition. In the outer inviscid layer the motion is described by Euler's equation in terms of the isobar coordinates and the deflection angle is determined to match with that at the outer edge of the inner layer. The present theory predicts that self-induced separation does occur for highly cooled wall and yields results in good agreement with the measurement of Lewis et al.With 6 Figures     

A flat plate in a rotating,stratified flow

Summary The boundary layer over a flat plate of semi-infinite extent in a stratified and rotating flow grows forward from the trailing edge, and is characterized by an intrinsic length scaleL, which represents the distance from the trailing edge at which vortex stretching becomes just as important in the boundary layer as baroclinic vorticity production. Near the trailing edge, the layer is essentially the layer in a purely stratified flow; far upstream (manyL), it is an Ekman layer. The boundary layer entrains no fluid, but induces at its edge a transverse velocity component which drives an higher-order streamwise outer flow. If the flow is bounded above and below by horizontal planes, the Wiener-Hopf solution for this outer flow indicates that the disturbance decays rapidly downstream, but the transverse velocity component is non-zero far upstream.     

Surface-breaking crack in an elastic half-space

A semi-infinite crack terminating at the boundary of an elastic half-space is considered. It is assumed that the crack is subjected to a mode-I load applied in a finite region remote from the crack plane, and the boundary of the half-space and the crack surfaces are free of tractions. The problem is formulated in terms of a hypersingular integral equation with respect to the relative crack-face separation defined over the region occupied by the crack. The behaviour of the solution near the corner point where the crack edge intersects the boundary is analysed and results which show the dependence of the stress singularity exponent on the angle of inclination of the crack edge are presented.     

Laminar forced convection in wedge flow with separation

Abstract

A perturbation method is used to study the steady and unsteady laminar boundary layer heat transfer from a wedge with separation for a step‐discontinuity in the surface temperature. The analytic solutions obtained can be used to calculate the steady and unsteady heat transfer rate with arbitrary surface temperature. The effects of the Prandtl number on the temperature distribution and the heat transfer rate are discussed in detail. The solution is valid for large or moderate Prandtl number.     

Bending vibration of a simply supported rectangular plate with a crack parallel to one edge

Natural flexural vibration of a simply supported rectangular plate with a symmetrically located crack parallel to one edge is considered. The problem is analyzed by means of finite Fourier transformation of discontinuous functions. The unknowns of the problem are the discontinuities of the displacement and of the slope across the crack. The singularities of the moments at the tips are “built” in the solution. The conditional equations are obtained by satisfying the boundary conditions at the crack's edge. This requires differentiation of Fourier series representing a discontinuous function. The characteristic equation in form of an infinite determinant is obtained. Numerical data for certain geometries as well as comparison with existing results are included.