On unsteady laminar boundary layers

1960 ◽  
Vol 9 (2) ◽  
pp. 300-304 ◽  
Author(s):  
H. A. Hassan
Keyword(s):  
Boundary Layer ◽  
Exact Solution ◽  
Boundary Layers ◽  
Laminar Boundary Layer ◽  
Two Dimensions ◽  
Universal Functions ◽  
Boundary Layer Equations ◽  
Laminar Boundary Layers ◽  
Pressure Distributions

A transformation is introduced which, for a class of outer pressure distributions, reduces the unsteady incompressible laminar boundary-layer equations in two dimensions to an equation in which the time does not appear explicitly. A formally exact solution of the resulting equation is then presented in the form of a series and it is shown that the solution can be expressed in terms of universal functions.

Approximate Methods of Solving the Laminar Boundary-Layer Equations

1962 ◽  
Vol 13 (3) ◽  
pp. 285-290 ◽  
Author(s):  
R. M. Terrill
Keyword(s):  
Boundary Layer ◽  
Circular Cylinder ◽  
Skin Friction ◽  
Boundary Layers ◽  
Laminar Boundary Layer ◽  
Numerical Solutions ◽  
Approximate Methods ◽  
Boundary Layer Equations ◽  
Laminar Boundary Layers ◽  
Remarkable Agreement

SummaryCurie and Skan have modified the approximate methods of Thwaites and Stratford to predict separation properties of laminar boundary layers for flow over an impermeable surface. The work of Curie and Skan has been extended by Curle to include the estimation of laminar skin friction for the whole flow. The purpose of the following note is to compare the approximate methods of Curie and Skan and Curle with the numerical results given by the author for flow past a circular cylinder. It is found that there is remarkable agreement between these approximate methods and the exact numerical solutions. This indicates that these methods can be used widely, both on account of their simplicity and their accuracy.

scholarly journals The laminar boundary-layer equations of bodies of revolution. Motion of a sphere

1948 ◽  
Vol 194 (1037) ◽  
pp. 218-228 ◽  
Keyword(s):  
Boundary Layer ◽  
Boundary Layers ◽  
Laminar Boundary Layer ◽  
Surface Friction ◽  
Momentum Equation ◽  
Bodies Of Revolution ◽  
Boundary Layer Equations ◽  
Laminar Boundary Layers ◽  
Parabolic Distribution ◽  
Forward Part

In a previous paper (Meksyn 1948) the author gave a method of calculating the velocity distribution in a laminar boundary layer on cylindrical bodies. The aim of the present paper is to extend the method to the case of laminar boundary layers on bodies of revolution. The problem has been treated, so far, only in a few papers (Goldstein 1938, §§51, 52, 61). Millikan (1932) has derived boundary-layer equations and Kármán’s momentum equation for bodies of revolution, and, assuming a parabolic distribution for the velocity u in the laminar part, and u ~ y 1/η in the turbulent part (where y is the distance from the surface), he has applied the momentum equation to airship-line bodies. Fediaersky (1934) derived independently the momentum equation for bodies of revolution and applied it to airship-like bodies by assuming u ~ u n , where n is arbitrary. Tomotika (1935) applied the momentum equation to the evaluation of various boundary-layer quantities for bodies of revolution, in particular for a sphere. Fage (1936) used Tomotika’s results to evaluate the surface friction at the forward part of a sphere. In the present paper the general boundary-layer equations are derived for axially symmetrical motions of bodies of revolution; the equations are partially solved for a sphere, and the results compared with Fage’s (1936) measurements.

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.

Similar and Asymptotic Solutions of the Incompressible Laminar Boundary Layer Equations with Suction

1967 ◽  
Vol 18 (2) ◽  
pp. 103-120 ◽  
Author(s):  
M. Zamir ◽  
A. D. Young
Keyword(s):  
Boundary Layer ◽  
Exact Solution ◽  
Asymptotic Solution ◽  
Laminar Boundary Layer ◽  
Incompressible Flow ◽  
Suction Parameter ◽  
Suction Velocity ◽  
Boundary Layer Equations ◽  
Wide Range ◽  
Similar Solutions

SummarySimilar solutions of the boundary layer equations for incompressible flow with external velocity u1 ∞ xm and suction velocity υw ∞ x(m-1)/2 are obtained for negative values of m, in the range −0-1 to −0-9, and a wide range of suction quantities.The results are used, in combination with, existing solutions for positive m, to provide a guide to the ranges of m and suction parameter [(υw/u1√x] for which a general form of the classical asymptotic solution can be regarded as a good approximation to the exact solution.It is shown that the values of both m and suction parameter are generally important in this comparison, but for values of the latter greater than about 8 the approximation is a very good one for all values of m considered. For m≃−0·14 the approximation is good (i.e. the error is less than about 1 per cent) down to values of the suction parameter as low as 1·0.

Unsteady Laminar Boundary Layers Over an Arbitrary Cylinder With Heat Transfer in an Incompressible Flow

1959 ◽  
Vol 26 (2) ◽  
pp. 171-178
Author(s):  
Kwang-Tzu Yang
Keyword(s):  
Heat Transfer ◽  
Boundary Layers ◽  
Arbitrary Shape ◽  
Detailed Calculation ◽  
Energy Integral ◽  
Universal Functions ◽  
Unsteady Problem ◽  
Boundary Layer Equations ◽  
Heated Cylinder ◽  
Laminar Boundary Layers

Abstract A method is presented for calculating the development of momentum and thermal laminar boundary layers on a heated cylinder of arbitrary shape when the cylinder moves in an incompressible fluid at rest with an unsteady velocity. This analysis is based on solutions to the unsteady momentum and energy-integral equations in conjunction with a set of universal functions, derived from exact solutions to the boundary-layer equations for a specific unsteady problem. These universal functions are given in tabulated form. Those associated with the energy-integral equation are calculated with a Prandtl number of 0.7. The reliability and limitation of these functions are indicated and discussed in the light of several simple problems of which solutions are available. A detailed calculation procedure for the general unsteady problem is given and then followed by a numerical example.

Approximate Methods for Predicting Separation Properties of Laminar Boundary Layers

1957 ◽  
Vol 8 (3) ◽  
pp. 257-268 ◽  
Author(s):  
N. Curle ◽  
S. W. Skan
Keyword(s):  
Boundary Layer ◽  
Boundary Layers ◽  
Laminar Boundary Layer ◽  
Boundary Layer Flow ◽  
Approximate Methods ◽  
Laminar Boundary Layers ◽  
Layer Flow

SummarySome new solutions for steady incompressible laminar boundary layer flow, obtained by Gortler, have been used to test the accuracy of two methods which are commonly used to predict separation. A modification of Stratford's criterion for separation is given in this paper and is probably the most accurate and the simplest of all methods at present in use. Modified numerical functions are also given for Thwaites's method of predicting the main characteristics of the boundary layer over the whole surface, which improve the accuracy of the method.

Evaluation of the Separation Properties of Laminar Boundary Layers

1968 ◽  
Vol 19 (3) ◽  
pp. 235-242 ◽  
Author(s):  
C. Y. Liu ◽  
V. A. Sandborn
Keyword(s):  
Boundary Layer ◽  
Velocity Distribution ◽  
Boundary Layers ◽  
Laminar Boundary Layer ◽  
Previous History ◽  
Laminar Boundary Layers ◽  
Detailed Evaluation ◽  
History Of

SummaryDetailed evaluation of the laminar boundary layer parameters and velocity distribution at separation is given. The analysis takes into account the definite variation in separation profiles observed both theoretically and experimentally. The strong dependency of the shape of the separation profile on the previous history of the boundary development is demonstrated.

Unsteady Laminar Boundary Layers in an Incompressible Stagnation Flow

1958 ◽  
Vol 25 (4) ◽  
pp. 421-427
Author(s):  
Kwang-Tzu Yang
Keyword(s):  
Boundary Layer ◽  
Exact Solution ◽  
Approximate Solution ◽  
Flow Velocity ◽  
Boundary Layers ◽  
Flow Parameter ◽  
Stagnation Flow ◽  
Laminar Boundary Layers ◽  
Velocity Changes ◽  
Special Case

Abstract The present study deals with unsteady laminar boundary layers in the immediate vicinity of the stagnation point of a heated blunt-nosed cylinder in an incompressible flow with unsteady velocity. An exact solution is presented for the special case of a flow velocity varying inversely with a linear function of time, together with calculated boundary-layer characteristics for different values of a flow parameter. Based on the results of this exact solution, an approximate method of solutions is proposed for a more general problem where the flow velocity changes arbitrarily with time. The results of five examples are shown and discussed in the light of other available solutions. Finally, the limitation of this approximate solution is pointed out and a possible remedy indicated.

The Departure from Equilibrium of Turbulent Boundary Layers

1968 ◽  
Vol 19 (1) ◽  
pp. 1-19 ◽  
Author(s):  
H. McDonald
Keyword(s):  
Boundary Layer ◽  
Shear Stress ◽  
Kinetic Energy ◽  
Stress Distribution ◽  
Boundary Layers ◽  
Turbulent Boundary Layers ◽  
Two Dimensional ◽  
Pressure Distributions ◽  
Momentum Integral ◽  
State Examination

SummaryRecently two authors, Nash and Goldberg, have suggested, intuitively, that the rate at which the shear stress distribution in an incompressible, two-dimensional, turbulent boundary layer would return to its equilibrium value is directly proportional to the extent of the departure from the equilibrium state. Examination of the behaviour of the integral properties of the boundary layer supports this hypothesis. In the present paper a relationship similar to the suggestion of Nash and Goldberg is derived from the local balance of the kinetic energy of the turbulence. Coupling this simple derived relationship to the boundary layer momentum and moment-of-momentum integral equations results in quite accurate predictions of the behaviour of non-equilibrium turbulent boundary layers in arbitrary adverse (given) pressure distributions.

Laminar boundary layers behind detonation waves

1983 ◽  
Vol 387 (1793) ◽  
pp. 331-349 ◽  
Keyword(s):  
Heat Transfer ◽  
Boundary Layer ◽  
Boundary Layers ◽  
Flow Analysis ◽  
Time Dependent ◽  
Detonation Waves ◽  
Planar Geometry ◽  
Laminar Boundary Layers ◽  
Spherical Detonation ◽  
Layer Flow

New solutions are presented for non-stationary boundary layers induced by planar, cylindrical and spherical Chapman-Jouguet (C-J) detonation waves. The numerical results show that the Prandtl number ( Pr ) has a very significant influence on the boundary-layer-flow structure. A comparison with available time-dependent heat-transfer measurements in a planar geometry in a 2H 2 + O 2 mixture shows much better agreement with the present analysis than has been obtained previously by others. This lends confidence to the new results on boundary layers induced by cylindrical and spherical detonation waves. Only the spherical-flow analysis is given here in detail for brevity.

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