Unified theory for the solutions of the unsteady thermal boundary-layer equation

1965 ◽  
Vol 61 (3) ◽  
pp. 809-825 ◽  
Author(s):  
G. N. Sarma
Keyword(s):  
Boundary Layer ◽  
Flat Plate ◽  
Thermal Boundary Layer ◽  
Boundary Layer Equation ◽  
Steady Motion ◽  
Unsteady Motion ◽  
Main Stream ◽  
Unified Theory ◽  
Boundary Layer Equations ◽  
Special Cases

AbstractThe unsteady two-dimensional thermal boundary-layer equation linearized as by Lighthill is studied. Two different problems are considered mainly, one in Part I and the other in Part II. Part I deals with the solution when the temperature of the main stream is constant and that of the wall is unsteady and Part II when the temperature of the main stream is constant and the heat transfer from the wall is unsteady. Unified methods are developed from which the results for the stagnation flow and the flow along a flat plate, etc., can be derived as special cases. The results of the unsteady velocity boundary-layer equations analysed by Sarma are used and solutions are obtained in two cases, first, when the main stream is in steady motion and the wall is in an arbitrary motion and secondly when the main stream is in unsteady motion and the wall is at rest. The flat plate problem is considered in detail; the results agree with those given by Lighthill and Moore.

Solutions of unsteady boundary-layer equations

1964 ◽  
Vol 60 (1) ◽  
pp. 137-158 ◽  
Author(s):  
G. N. Sarma
Keyword(s):  
Boundary Layer ◽  
Flat Plate ◽  
Steady Motion ◽  
Unsteady Motion ◽  
Main Stream ◽  
Stagnation Flow ◽  
Boundary Layer Equations ◽  
Special Cases ◽  
Dimensional Boundary ◽  
Unified Method

AbstractThe unsteady two-dimensional boundary-layer equations, linearized as by Lighthill are studied. A unified method is developed, from which the results for the stagnation flow, the flow along a flat plate, the flow in a converging canal, etc., can be derived as special cases. Solutions are obtained in two systems, one when the main stream is in unsteady motion and the wall is at rest and the other when the main stream is in steady motion and the wall is in an arbitrary motion. The stagnation flow has been done by Glauert and generalized by Watson. The flow along a flat plate and the flow in a converging canal are considered in detail.

Steady flow in the laminar boundary layer of a gas

1949 ◽  
Vol 199 (1059) ◽  
pp. 533-558 ◽  
Keyword(s):  
Boundary Layer ◽  
Prandtl Number ◽  
Incompressible Fluid ◽  
Flat Plate ◽  
Horizontal Cylinder ◽  
Vertical Surface ◽  
Main Stream ◽  
Stream Flows ◽  
Boundary Layer Equations ◽  
Plane Jets

If the boundary-layer equations for a gas are transformed by Mises’s transformation, as was done by Kármán & Tsien for the flow along a flat plate of a gas with unit Prandtl number σ, the computation of solutions is simplified, and use may be made of previously computed solutions for an incompressible fluid. For any value of the Prandtl number, and any variation of the viscosity μ with the temperature T , after the method has been applied to flow along a flat plate (a problem otherwise treated by Crocco), the flow near the forward stagnation point of a cylinder is calculated with dissipation neglected, both with the effect of gravity on the flow neglected and with this effect retained for vertical flow past a horizontal cylinder. The approximations involved by the neglect of gravity are considered generally, and the cross-drift is calculated when a horizontal stream flows past a vertical surface. When σ =1, μ∞ T , and the boundary is heat-insulated, it is shown that the boundary-layer equations for a gas may be made identical, whatever be the main stream, with the boundary-layer equations for an incompressible fluid with a certain, determinable, main stream. The method is also applied to free convection at a flat plate (with the heat of dissipation and the variation with altitude of the state of the surrounding fluid neglected) and to laminar flow in plane wakes, but for plane jets the conditions σ =1, μ∞ T , previously imposed by Howarth,are also imposed here in order to obtain simple solutions.

Solution of Similarity Transformation Equations for Boundary Layers Using Spreadsheets

2004 ◽  
Author(s):  
Mohammad H. N. Naraghi
Keyword(s):  
Heat Transfer ◽  
Boundary Layer ◽  
Fluid Mechanics ◽  
Flat Plate ◽  
Thermal Boundary Layer ◽  
Similarity Transformation ◽  
Present Approach ◽  
Published Data ◽  
Boundary Layer Equations ◽  
Transfer Courses

A spreadsheet based solution of the similarity transformation equations of laminar boundary layer equations is presented. In this approach the nonlinear third order differential equations, for both the hydrodynamic and the thermal boundary layer equations, are discretesized using a simple finite difference approach which is suitable for programming spreadsheet cells. This approach was implemented to solve the similarity transform equations for a flat plate (Blasius equations). The thermal boundary layer result was used to obtain the heat transfer correlation for laminar flow over a flat plate in the form of Nu = Nu(Pr,Re). The relative difference between results of the present approach and those of published data are less than 1%. This approach can be easily covered in the undergraduate. Fluid Mechanics and Heat Transfer courses. Also, it can be incorporated in graduate Viscous Fluid Mechanics and Convection Heat Transfer courses. Application of the present approach is not limited to the flat plat boundary layer analysis. It can be used for the solution of a number of similarity transformation equations, including wedge flow problem and natural convection problems that are covered in graduate level courses.

Numerical Study of Boundary Layers With Reverse Wedge Flows Over a Semi-Infinite Flat Plate

2009 ◽  
Vol 77 (2) ◽  
Author(s):  
R. Ahmad ◽  
K. Naeem ◽  
Waqar Ahmed Khan
Keyword(s):  
Boundary Layer ◽  
Flat Plate ◽  
Numerical Study ◽  
Boundary Layer Equation ◽  
Wedge Angle ◽  
Adverse Pressure Gradient ◽  
Problem Solution ◽  
Weighted Residual Method ◽  
Boundary Layer Problem ◽  
Layer Problem

This paper presents the classical approximation scheme to investigate the velocity profile associated with the Falkner–Skan boundary-layer problem. Solution of the boundary-layer equation is obtained for a model problem in which the flow field contains a substantial region of strongly reversed flow. The problem investigates the flow of a viscous liquid past a semi-infinite flat plate against an adverse pressure gradient. Optimized results for the dimensionless velocity profiles of reverse wedge flow are presented graphically for different values of wedge angle parameter β taken from 0≤β≤2.5. Weighted residual method (WRM) is used for determining the solution of nonlinear boundary-layer problem. Finally, for β=0 the results of WRM are compared with the results of homotopy perturbation method.

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.

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.

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