Three-dimensional Laminar Boundary-layer Equations Research Articles

Inviscid instability of a skewed compressible mixing layer

In this paper we study the inviscid instability of a skewed compressible mixing layer between streams of different velocity magnitude and direction. The mean flow is governed by the three-dimensional laminar boundary-layer equations and can be reduced to a sum of a uniform flow and a two-dimensional shear flow. In the stability analysis, the amplification direction is assumed to be normal to the homogeneous direction of the mean flow. The results show that skewing enhances the instability by a factor of three for the incompressible mixing layer with velocity ratio 0.5 and uniform temperature. Under compressible conditions, skewing still increases the maximum amplification rate for a medium convective Mach number, but the enhancement is smaller. A scaling of the skewing effect is introduced which quantitatively explains the linear stability behaviour. Similarly, a suitably defined convective Mach number explains the compressibility effect.

Determination of the magnitude of the separation criterion for a three-dimensional laminar boundary layer

A separation criterion, i.e., a definite relationship between the external flow and the boundary layer parameters [1], can be used to estimate the possibility of the origination of separation of a two-dimensional boundary layer. A functional form of the separation criterion has also been obtained for a three-dimensional boundary layer [2] on the basis of dimensional analysis. As in the case of the two-dimensional boundary layer, locally self-similar solutions can be used to determine the specific magnitude of the separation criterion as a function of the values of the governing parameters. Locally self-similar solutions of the two-dimensional laminar boundary-layer equations have been found at the separation point for a perfect gas with a linear dependence of the coefficient of viscosity on the temperature (Ω=1) and Prandtl number P=1 [3, 4]. The influence of blowing and suction has been studied for this case [5]. Self-similar solutions have been obtained for Ω=1, P=0.723 for the limit case of hypersonic perfect gas flow [6]. Locally self-similar solutions of the three-dimensional laminar boundary-layer equations at the separation point are presented in [7] for a perfect gas with Ω=1, P=1. There are no such computations for Ω≠1, P≠1; however, the results of computing several examples for a two-dimensional flow [8] show that the influence of the real properties of a gas can be significant and should be taken into account. Self-similar solutions of the two- and three-dimensional boundary-layer equations at the separation point are found in this paper for a perfect gas with a power-law dependence of the viscosity coefficient on the enthalpy (Ω=0.5, 0.75, 1.0) for different values of the Prandtl number (P=0.5, 0.7, 1.0) in a broad range of variation of the external stream velocity (v12/2h1* = 0–0.99) and the temperature of the streamlined surface. Magnitudes of the separation criterion for a laminar boundary layer have been obtained on the basis of these data.

Invariant solutions of the three-dimensional laminar boundary-layer equations on developing surfaces

Exact solutions play an important part in boundary-layer theory. The traditional path to obtaining these solutions is to decrease the number of independent variables in the boundary layer equations. The aim of this paper is to describe the structure of the solutions for which such a simplification is possible.

Uniqueness of the self-similar solutions of three-dimensional laminar boundary-layer equations

The equations of the three-dimensional laminar boundary layer on lines of flow outflow and inflow are studied for conical outer flow under the assumption that the Prandtl number and the productρμ are constant. It is shown that in the case of a positive velocity gradient of the secondary flow (α1>0) the additional conditions which result from the physical flow pattern determine a unique solution of the system of boundary-layer equations. For a negative velocity gradient of the secondary flow (α1≤0) these conditions are satisfied by two solutions. An approximate solution is obtained for the boundary layer equations which is in rather good agreement with the numerical integration results. Compressible gas flow in a three-dimensional laminar boundary layer is described by a system of nonlinear differential equations whose solution is not unique for given boundary conditions. Therefore additional conditions resulting from the physical pattern of the gas flow are imposed on the resulting solution. In the solution of problems with a negative pressure gradient these additional conditions are sufficient for a unique selection of the solution of the boundary-layer equations. However, in the case of a positive pressure gradient the solution of the boundary-layer equations satisfying the boundary and additional conditions may not be unique. In particular, in [1] in a study of a three-dimensional laminar boundary layer in the vicinity of the stagnation point it was shown that for $$c = {{\frac{{\partial v_e }}{{\partial y}}} \mathord{\left/ {\vphantom {{\frac{{\partial v_e }}{{\partial y}}} {\frac{{\partial u_e }}{{\partial x}}}}} \right. \kern-\nulldelimiterspace} {\frac{{\partial u_e }}{{\partial x}}}} > 0$$ the solution is unique, while for c<0 there are two solutions. In the present paper we study the question of the uniqueness of the self-similar solution of the three-dimensional laminar boundary-layer equations on lines of flow outflow and inflow for a conical outer flow.

Calculation of self-similar three-dimensional laminar boundary layer equations by a quasilinear method

Calculation of self-similar three-dimensional laminar boundary layer equations by a quasilinear method

Compressible laminar boundary layer on a cone at high angle of attack

An approximate integral method is applied to the three-dimensional laminar boundarylayer equations, including the energy equation for Prandtl number equal to unity, to obtain a solution to the attached boundary layer on slender cones in hypersonic flight at high angles of attack. The velocity and total enthalpy profiles are of the Timman type. The crossflow velocity profile is a two-parameter family allowing for nonzero crossflow at streamline inflection points. Because of the high angles of attack considered, the solution includes the effect of large crossflow velocity on the boundary layer. Using the pressure data available in the literature, results for the heat transfer coefficient, surface shear stress direction, and position of separation are computed. The computations compare favorably with data from the literature.

Laminar film condensation near a stagnation point on a general curved surface

The three-dimensional laminar boundary layer equations are given for laminar film condensation on a general curved surface in quiescent vapour. An example is investigated which relates to the flow in the neighbourhood of a stagnation point. Numerical results are obtained for the similarity solution in the case of condensation of steam at 100°C and atmospheric pressure. Flow and heat transfer characteristics are presented in graphical and tabular form and compared with those obtained by an approximate method.

Solution of a three-dimensional boundary-layer flow with separation.

A method of calculating accurate solutions of the three-dimensional laminar boundarylayer equations has been developed. The method is applied to a problem that exhibits interesting crossflow phenomena, such as a flow reversal driven by crossflow convective terms and crossflow influences with zero freestream crossflow velocity. The method of solution used is an implicit finite difference scheme, and the stability and convergence properties of this scheme seemed to be good. Also, the important question of initial conditions for threedimensional boundary-layer flow is discussed, and a method of obtaining initial conditions is derived.

A method of solving the three-dimensional laminar boundary-layer equations with application to a lifting re-entry body.

A method is proposed in which the three-dimensional laminar compressible boundarylayer equations are reduced to a sequence of two-dimensional equations by means of a systematic perturbation procedure. The perturbation parameter is related to the inviscid streamline curvature which must be small. The equations of lowest (zeroth) order are identical with the small crossflow equations of Hayes. Two alternate finite-difference methods (for physical and transformed variables, respectively) are developed to solve the zeroth- as well as the first-order equations exactly. Local similarity methods are also considered for comparison. A variety of boundary conditions are possible including the case for which the heatand mass-transfer rates are coupled through an energy balance. The methods have been applied to the problem of boundary-laye r flow on a blunted cone at angle of attack. The results show that the crossflow often becomes rather large near the wall. The first-order corrections to the enthalpy and tangential velocity in the boundary layer are nevertheless small for the cases considered.