The equation of similar profiles in boundary layer theory with strong blowing

1966 ◽  
Vol 294 (1437) ◽  
pp. 208-234 ◽  
Keyword(s):  
Boundary Layer ◽  
Skin Friction ◽  
Main Part ◽  
Similarity Solutions ◽  
Boundary Layer Theory ◽  
Main Stream ◽  
Outer Solution ◽  
Boundary Layer Equations ◽  
Displacement Thickness ◽  
Complicated Form

This paper contains a study of the similarity solutions of the boundary layer equations for the case of strong blowing through a porous surface. The main part of the boundary layer is thick and almost inviscid in these conditions, but there is a thin viscous region where the boundary layer merges into the main stream. The asymptotic solutions appropriate to these two regions are matched to one another when the blowing velocity is large. The skin friction is found from the inner solution, which is independent of the outer solution, but the displacement thickness involves both solutions and is of more complicated form.

Jeffery-Hamel boundary-layer flows over curved beds

1988 ◽  
Vol 186 ◽  
pp. 583-597 ◽  
Author(s):  
P. M. Eagles
Keyword(s):  
Boundary Layer ◽  
Flow Field ◽  
Film Flow ◽  
Local Approximation ◽  
Main Stream ◽  
Streamwise Direction ◽  
Boundary Layer Flows ◽  
Boundary Layer Equations ◽  
First Approximation ◽  
Local Value

We find certain exact solutions of Jeffery-Hamel type for the boundary-layer equations for film flow over certain beds. If β is the angle of the bed with the horizontal and S is the arclength these beds have equation sin β = (const.)S−3, and allow a description of flows on concave and convex beds. The velocity profiles are markedly different from the semi-Poiseuille flow on a plane bed.We also find a class of beds in which the Jeffery-Hamel flows appear as a first approximation throughout the flow field, which is infinite in streamwise extent. Since the parameter γ specifying the Jeffery-Hamel flow varies in the streamwise direction this allows a description of flows over curved beds which are slowly varying, as described in the theory, in such a way that the local approximation is that Jeffery-Hamel flow with the local value of γ. This allows the description of flows with separation and reattachment of the main stream in some cases.

scholarly journals Similarity Solutions of the MHD Boundary Layer Flow Past a Constant Wedge within Porous Media

2017 ◽  
Vol 2017 ◽  
pp. 1-11 ◽  
Author(s):  
Ramesh B. Kudenatti ◽  
Shreenivas R. Kirsur ◽  
Achala L. Nargund ◽  
N. M. Bujurke
Keyword(s):  
Boundary Layer ◽  
Porous Medium ◽  
Pressure Gradient ◽  
Adverse Pressure Gradient ◽  
Velocity Profiles ◽  
Similarity Solutions ◽  
Positive Pressure ◽  
Boundary Layer Equations ◽  
Similarity Transformations ◽  
Wide Range

The two-dimensional magnetohydrodynamic flow of a viscous fluid over a constant wedge immersed in a porous medium is studied. The flow is induced by suction/injection and also by the mainstream flow that is assumed to vary in a power-law manner with coordinate distance along the boundary. The governing nonlinear boundary layer equations have been transformed into a third-order nonlinear Falkner-Skan equation through similarity transformations. This equation has been solved analytically for a wide range of parameters involved in the study. Various results for the dimensionless velocity profiles and skin frictions are discussed for the pressure gradient parameter, Hartmann number, permeability parameter, and suction/injection. A far-field asymptotic solution is also obtained which has revealed oscillatory velocity profiles when the flow has an adverse pressure gradient. The results show that, for the positive pressure gradient and mass transfer parameters, the thickness of the boundary layer becomes thin and the flow is directed entirely towards the wedge surface whereas for negative values the solutions have very different characters. Also it is found that MHD effects on the boundary layer are exactly the same as the porous medium in which both reduce the boundary layer thickness.

An Accurate Calculation Method for Two-dimensional Incompressible Laminar Boundary Layers, Including Cases with Regions of Sharp Pressure Gradient

1977 ◽  
Vol 28 (3) ◽  
pp. 149-162 ◽  
Author(s):  
N Curle
Keyword(s):  
Boundary Layer ◽  
Pressure Gradient ◽  
Calculation Method ◽  
Boundary Layer Separation ◽  
Pressure Gradients ◽  
Accurate Calculation ◽  
Boundary Layer Equations ◽  
Laminar Boundary Layers ◽  
Displacement Thickness ◽  
Type Boundary

SummaryThe paper develops and extends the calculation method of Stratford, for flows in which a Blasius type boundary layer reacts to a sharp unfavourable pressure gradient. Whereas even the more general of Stratford’s two formulae for predicting the position of boundary-layer separation is based primarily upon an interpolation between only three exact solutions of the boundary layer equations, the present proposals are based upon nine solutions covering a much wider range of conditions. Four of the solutions are for extremely sharp pressure gradients of the type studied by Stratford, and five are for more modest gradients. The method predicts the position of separation extremely accurately for each of these cases.The method may also be used to predict the detailed distributions of skin friction, displacement thickness and momentum thickness, and does so both simply and accurately.

On the propagation of disturbances in a laminar boundary layer. I

1973 ◽  
Vol 73 (3) ◽  
pp. 493-503 ◽  
Author(s):  
S. N. Brown ◽  
K. Stewartson
Keyword(s):  
Boundary Layer ◽  
Asymptotic Formula ◽  
Boundary Conditions ◽  
Laminar Boundary Layer ◽  
Numerical Results ◽  
Boundary Layer Equations ◽  
Displacement Thickness ◽  
Layer I

An analysis is made of the response of a laminar boundary layer to a perturbation, either in the mainstream, or of the boundary conditions at the wall. The disturbance propagates with the mainstream velocity, and the manner in which it decays at large distances downstream is determined by eigensolutions of the boundary-layer equations. The elucidation of the structure of these eigensolutions requires division of the boundary layer into three regions. Comparison of the asymptotic formula obtained for the displacement thickness is made with the numerical results of Ackerberg and Phillips (1).

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.

The Effects of Small Changes of Prandtl Number and the Viscosity-Temperature Index on Skin Friction

1964 ◽  
Vol 15 (4) ◽  
pp. 392-406 ◽  
Author(s):  
A. D. Young
Keyword(s):  
Boundary Layer ◽  
Prandtl Number ◽  
Pressure Gradient ◽  
Skin Friction ◽  
Laminar Boundary Layer ◽  
Wall Temperature ◽  
External Pressure ◽  
Main Stream ◽  
Pressure Gradients ◽  
Temperature Index

SummaryThe analytic simplifications in boundary-layer analysis that result from the assumptions that the Prandtl number σ and the viscosity-temperature index ω are unity make it desirable to be able to assess the effects of the departures of the actual values of these parameters from unity. In this paper only the effects on skin friction are considered. Formulae of acceptable validity and wide application are first used to produce generalised curves for these effects for given main-stream Mach numbers and wall temperature conditions for the case of zero external pressure gradient for both laminar and turbulent boundary layers (Figs. 1 and 2).A number of calculated results for the laminar boundary layer with favourable and adverse pressure gradients is then analysed (Figs. 3, 4 and 5) and it is shown that these results are consistent with the assumption that, for a given wall temperature, the effects of small changes of σ and ω on skin friction are independent of the external gradient, so that the appropriate curves of Figs. 1 and 2 apply. Where the change of a- is associated with a change of wall temperature (e.g. if the heat transfer is specified as zero) then the interaction between pressure gradient and this temperature change can be significant in its effects on skin friction for the laminar boundary layer and can only be assessed if the effects of changes of wall temperature with constant σ and ω have been separately determined for the pressure distribution considered. It is inferred that in all cases, except with large adverse pressure gradients and imminent separation, the effects of changes of ω and σ for the turbulent boundary layer are reliably predicted by the zero pressure gradient curves of Figs. 1 and 2 and the effect of any associated change of wall temperature can then be reliably inferred from the zero pressure gradient formula (equation (15)) in the absence of more specific calculations covering a range of wall temperatures.

scholarly journals The numerical solution of the asymptotic equations of trailing edge flow

1974 ◽  
Vol 340 (1620) ◽  
pp. 91-111 ◽  
Keyword(s):  
Boundary Layer ◽  
Outer Layer ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Trailing Edge ◽  
Navier Stokes Equations ◽  
Boundary Layer Equations ◽  
Asymptotic Equations ◽  
Displacement Thickness

According to Stewartson (1969, 1974) and to Messiter (1970), the flow near the trailing edge of a flat plate has a limit structure for Reynolds number Re →∞ consisting of three layers over a distance O (Re -3/8 ) from the trailing edge: the inner layer of thickness O ( Re -5/8 ) in which the usual boundary layer equations apply; an intermediate layer of thickness O ( Re -1/2 ) in which simplified inviscid equations hold, and the outer layer of thickness O ( Re -3/8 ) in which the full inviscid equations hold. These asymptotic equations have been solved numerically by means of a Cauchy-integral algorithm for the outer layer and a modified Crank-Nicholson boundary layer program for the displacement-thickness interaction between the layers. Results of the computation compare well with experimental data of Janour and with numerical solutions of the Navier-Stokes equations by Dennis & Chang (1969) and Dennis & Dunwoody (1966).

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