The behaviour of similar solutions in a compressible boundary layer

1968 ◽  
Vol 34 (2) ◽  
pp. 337-342 ◽  
Author(s):  
J. B. Mcleod ◽  
J. Serrin
Keyword(s):  
Boundary Layer ◽  
Pressure Gradient ◽  
Compressible Boundary Layer ◽  
Rigorous Proof ◽  
Velocity Overshoot ◽  
Boundary Layer Equations ◽  
Mathematical Properties ◽  
Model Fluid ◽  
Similar Solutions

This paper discusses the mathematical properties of similar solutions of the boundary-layer equations in a compressible model fluid, under assumptions first introduced by Stewartson and by Li & Nagamatsu. Assuming a favourable pressure gradient and that backflow is not present, our results include (among other things) a rigorous proof that velocity overshoot occurs in the boundary layer if the wall is heated, and that this is true whether or not suction, blowing or slipping occurs at the wall; while, conversely, velocity overshoot does not occur when the wall is cooled and the amount of slipping at the wall is suitably restricted.

Further Solution of the Corner Boundary-Layer Equations

1973 ◽  
Vol 24 (3) ◽  
pp. 219-226 ◽  
Author(s):  
M Zamir
Keyword(s):  
Boundary Layer ◽  
Pressure Gradient ◽  
Existence And Uniqueness ◽  
Boundary Layer Equation ◽  
Critical Value ◽  
Numerical Accuracy ◽  
Boundary Layer Equations ◽  
The Past ◽  
Similar Solutions ◽  
High Degree

SummarySimilar solutions of an equation governing the flow in the plane of symmetry of a corner boundary layer with favourable pressure gradient are extended to a “critical” value of the pressure gradient for which no solution could be found previously. It is shown that the failure to achieve this result in the past was due to the “singular” nature of this solution rather than to its non-existence as one was tempted to suspect. The existence and uniqueness of this crucial solution are demonstrated and the solution itself is obtained to a high degree of numerical accuracy. Criticism of the theory on which this corner boundary-layer equation is based is discussed in an Appendix.

Compressible Boundary Layer-Inviscid Flow Interactions in Entrance Region of Internal Flows

1967 ◽  
Vol 89 (4) ◽  
pp. 281-288 ◽  
Author(s):  
V. D. Blankenship ◽  
P. M. Chung
Keyword(s):  
Boundary Layer ◽  
Shock Tube ◽  
Inviscid Flow ◽  
Perturbation Parameter ◽  
Entrance Region ◽  
Compressible Boundary Layer ◽  
Internal Flows ◽  
Flow Equations ◽  
Boundary Layer Equations ◽  
Interaction Problem

The coupling between the inviscid flow and the compressible boundary layer in the developing entrance region for internal flows is analyzed by solving the particular inviscid flow-boundary layer interaction problem. The interaction problem is solved by postulating certain series forms of solutions for the inviscid region and the boundary layer. The boundary-layer equations and inviscid-flow equations are perturbed to third order and each generated equation is solved numerically. In order to preserve the universality of each of the perturbed boundary-layer equations, the perturbation parameter is described by an integral equation which is also solved in series form. The final results describing the interaction problem are then constructed for any given conditions by forming the three series to a consistent order of magnitude. This technique of coordinate perturbation is generalized to show how it may be applied to the entrance regions of pipe flows, including mass injection or suction, and also to the laminar boundary layers in shock tube flows. It demonstrates analytically the manner in which the boundary layer and inviscid flow interact and create a streamwise pressure gradient. In particular, the interaction problem which occurs in shock tube flows is solved in detail by the use of this generalized method, as an example.

Natural convection flow far from a horizontal plate

1999 ◽  
Vol 387 ◽  
pp. 227-254 ◽  
Author(s):  
VALOD NOSHADI ◽  
WILHELM SCHNEIDER
Keyword(s):  
Boundary Layer ◽  
Prandtl Number ◽  
Stokes Equations ◽  
Axisymmetric Flow ◽  
The Self ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Boundary Layer Equations ◽  
Self Similar ◽  
Similar Solutions

Plane and axisymmetric (radial), horizontal laminar jet flows, produced by natural convection on a horizontal finite plate acting as a heat dipole, are considered at large distances from the plate. It is shown that physically acceptable self-similar solutions of the boundary-layer equations, which include buoyancy effects, exist in certain Prandtl-number regimes, i.e. 0.5<Pr[les ]1.470588 for plane, and Pr>1 for axisymmetric flow. In the plane flow case, the eigenvalues of the self-similar solutions are independent of the Prandtl number and can be determined from a momentum balance, whereas in the axisymmetric case the eigenvalues depend on the Prandtl number and are to be determined as part of the solution of the eigenvalue problem. For Prandtl numbers equal to, or smaller than, the lower limiting values of 0.5 and 1 for plane and axisymmetric flow, respectively, the far flow field is a non-buoyant jet, for which self-similar solutions of the boundary-layer equations are also provided. Furthermore it is shown that self-similar solutions of the full Navier–Stokes equations for axisymmetric flow, with the velocity varying as 1/r, exist for arbitrary values of the Prandtl number.Comparisons with finite-element solutions of the full Navier–Stokes equations show that the self-similar boundary-layer solutions are asymptotically approached as the plate Grashof number tends to infinity, whereas the self-similar solution to the full Navier–Stokes equations is applicable, for a given value of the Prandtl number, only to one particular, finite value of the Grashof number.In the Appendices second-order boundary-layer solutions are given, and uniformly valid composite expansions are constructed; asymptotic expansions for large values of the lateral coordinate are performed to study the decay of the self-similar boundary-layer flows; and the stability of the jets is investigated using transient numerical solutions of the Navier–Stokes equations.

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