Approximate Solutions of the Unsteady Boundary-Layer Equations

1976 ◽  
Vol 43 (3) ◽  
pp. 396-398 ◽  
Author(s):  
A. Bianchini ◽  
L. de Socio ◽  
A. Pozzi
Keyword(s):  
Boundary Layer ◽  
Linear Partial Differential Equation ◽  
Approximate Solutions ◽  
Scale Function ◽  
Unsteady Boundary Layer ◽  
First Order ◽  
Boundary Layer Equations ◽  
Approximate Integral ◽  
Similar Solutions ◽  
Approximate Integral Method

An approximate integral method is proposed for solving the unsteady laminar boundary-layer equations. The essence of the method is to assume a similar solution for the velocity profile even in those situations where similar solutions do not exist, and then to find a suitable scale function for the similarity variable. The scale function is the solution to a simple first-order linear partial differential equation. This solution is determined in closed form. Satisfactory results were obtained in comparison with more sophisticated and time-consuming procedures.

Second Order Boundary Layer Solutions on a Curved Surface

1972 ◽  
Vol 94 (3) ◽  
pp. 649-654 ◽  
Author(s):  
W. F. Van Tassell ◽  
D. B. Taulbee
Keyword(s):  
Heat Transfer ◽  
Boundary Layer ◽  
Order Effect ◽  
Second Order ◽  
Two Dimensional ◽  
First Order ◽  
Boundary Layer Equations ◽  
Similar Solutions ◽  
Nonsimilar Solutions ◽  
Different Altitudes

Solutions of the second order longitudinal curvature boundary layer equations near the stagnation point of a two-dimensional circular cylinder are presented. Four cases corresponding to 1 first order locally similar solutions, 2 first order nonsimilar solutions, 3 second order locally similar solutions, and 4 second order nonsimilar solutions are considered. For each of the four cases, results for four different altitudes are given. The only second order effect considered is longitudinal curvature. Based on the numerical results, it is concluded that similarity and curvature assumptions can alter the skin friction calculations significantly. The heat transfer calculations are much less sensitive to the various assumptions, at least for the cases studied in this paper.

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.

First-Order Boundary-Layer Equations

1981 ◽  
pp. 10-18
Author(s):  
Ernst Heinrich Hirschel ◽  
Wilhelm Kordulla
Keyword(s):  
Boundary Layer ◽  
First Order ◽  
Boundary Layer Equations

Correlated incompressible and compressible boundary layers

1949 ◽  
Vol 200 (1060) ◽  
pp. 84-100 ◽  
Keyword(s):  
Boundary Layer ◽  
Exact Solution ◽  
Boundary Layers ◽  
Absolute Temperature ◽  
Stream Velocity ◽  
Main Stream ◽  
Boundary Layer Equations ◽  
Compressible Boundary Layers ◽  
Incompressible Boundary ◽  
Similar Solutions

The boundary-layer equations for a compressible fluid are transformed into those for an incompressible fluid, assuming that the boundary is thermally insulating, that the viscosity is proportional to the absolute temperature, and that the Prandtl number is unity. Various results in the theory of incompressible boundary layers are then taken over into the compressible theory. In particular, the existence of ‘similar’ solutions is proved, and Howarth’s method for retarded flows is applied to determine the point of separation for a uniformly retarded main stream velocity. A comparison with an exact solution is used to show that this method gives a closer approximation than does Pohlhausen’s.

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