On the solution of the Prandtl boundary layer equations containing the point of zero skin friction

1982 ◽  
Vol 79 (4) ◽  
pp. 291-304 ◽  
Author(s):  
Ching Shi Liu ◽  
Chun Hian Lee
Keyword(s):  
Boundary Layer ◽  
Skin Friction ◽  
Boundary Layer Equations ◽  
Prandtl Boundary Layer

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.

Potential Systems and Nonlocal Conservation Laws of Prandtl Boundary Layer Equations on the Surface of a Sphere

2017 ◽  
Vol 72 (4) ◽  
pp. 351-357 ◽  
Author(s):  
R. Naz
Keyword(s):  
Boundary Layer ◽  
Conservation Laws ◽  
Linear Combination ◽  
Conservation Law ◽  
Local Conservation ◽  
Boundary Layer Equations ◽  
Potential System ◽  
Nonlocal Conservation Laws ◽  
Prandtl Boundary Layer ◽  
Nonlocally Related Systems

Abstract:The potential systems and nonlocal conservation laws of Prandtl boundary layer equations on the surface of a sphere have been investigated. The multiplier approach yields two local conservation laws for the Prandtl boundary layer equations on the surface of a sphere. Two potential variables ψ and ϕ are introduced corresponding to first and second conservation law. Moreover, another potential variable p is introduced by considering the linear combination of both conservation laws. Two level one potential systems involving a single nonlocal variable ψ or ϕ are constructed. One level two potential system involving both nonlocal variables ψ and ϕ is established. The nonlocal variable p is utilised to derive a spectral potential system. The nonlocal conservation laws of Prandtl boundary layer equations on the surface of a sphere are derived by computing the local conservation laws of its potential systems. The nonlocal conservation laws are utilised to derive the further nonlocally related systems.

scholarly journals ENERGY LOSSES UNDER WAVE ACTION

1970 ◽  
Vol 1 (12) ◽  
pp. 16 ◽  
Author(s):  
P.D. Treloar ◽  
A. Brebner
Keyword(s):  
Boundary Layer ◽  
Energy Dissipation ◽  
Wave Height ◽  
Dissipation Function ◽  
Wave Action ◽  
Energy Losses ◽  
Boundary Layer Equations ◽  
Wave Height Attenuation ◽  
Prandtl Boundary Layer ◽  
Made In

Wave-height attenuation measurements were made in two identical flumes of different widths and the results used to separate bottom energy losses from sidewall energy losses These energy losses, in the form of rates of energy dissipation, were then compared with their theoretical values as calculated by solving the linearized Prandtl boundary layer equations and evaluating the Rayleigh dissipation function Using these results, an adjusted formula for the wave-height attenuation modulus was determined.

scholarly journals Comparison of k-ε Models in Predicting Heat Transfer and Skin Friction Under High Free Stream Turbulence

1996 ◽  
Author(s):  
Ganesh R. Iyer ◽  
Savash Yavuzkurt
Keyword(s):  
Heat Transfer ◽  
Experimental Data ◽  
Boundary Layer ◽  
Skin Friction ◽  
Turbulence Intensity ◽  
Free Stream ◽  
Skin Friction Coefficient ◽  
Empirical Correlations ◽  
Free Stream Turbulence ◽  
Boundary Layer Equations

Calculations of the effects of high free stream turbulence (FST) on heat transfer and skin friction in a flat plate turbulent boundary layer using different k-ε models (Launder-Sharma, K-Y Chien, Lam-Bremhorsi and Jones-Launder) are presented. This study was carried out in order to investigate the prediction capabilities of these models under high FST conditions. In doing so, TEXSTAN, a partial differential equation solver which is based on the ideas of Patankar and Spalding and solves steady-flow boundary layer equations, was used. Firstly, these models were compared as to how they predicted very low FST (≤ 1% turbulence intensity) cases. These baseline cases were tested by comparing predictions with both experimental data and empirical correlations. Then, these models were used in order to determine the effect of high FST (>5% turbulence intensity) on heat transfer and skin friction and compared with experimental data. Predictions for heat transfer and skin friction coefficient for all the turbulence intensities tested by all the models agreed well (within 1–8%) with experimental data. However, all these models predicted poorly the dissipation of turbulent kinetic energy (TKE) in the free stream and TKE profiles. Physical reasoning as to why the aforementioned models differ in their predictions and the probable cause of poor prediction of free-stream TKE and TKE profiles are given.

Dual solutions of the magnetogasdynamic boundary-layer equations

1967 ◽  
Vol 27 (1) ◽  
pp. 145-154 ◽  
Author(s):  
D. B. Ingham
Keyword(s):  
Boundary Layer ◽  
Mach Number ◽  
Skin Friction ◽  
Dual Solutions ◽  
Electrically Conducting ◽  
Current Decay ◽  
Boundary Layer Equations ◽  
Conductivity Parameter ◽  
Aligned Magnetic Field ◽  
Uniform Stream

This paper extends the work of Wilson (1964) to include the effect of compressibility in the recurrence of dual solutions in the flow in the boundary layer on a semi-infinite, thermally insulated, flat plate placed at zero incidence to a uniform stream of electrically-conducting gas with an aligned magnetic field at large distances from the plate. Numerical integration of the boundary-layer equations has been performed for several values of the ratio, β, of the square of the Alfvén speed to the fluid speed in the undisturbed fluid, the conductivity parameter ε = 0·1 and ∞ and the square of the Mach number M2 = 0, 1/2, 1, 2, 2·5, 4 and 5. The effect of compressibility is to increase the value of β for which a solution can exist such that the skin friction at the plate is greater than zero. Dual solutions are seen to occur for non-zero Mach number and all values of ε but no attempt here has been made to explain this phenomenon. An analytic argument indicates that no solutions of the equations exist if the skin friction at the plate is greater than zero and if the vorticity and current decay exponentially, the condition for which is \[ M^2 < 1,\quad \beta > 1/(1-M^2). \] Nothing specific has been proved if this condition is not satisfied.

Boundary-Layer Separation From Downstream Moving Boundaries

1973 ◽  
Vol 40 (2) ◽  
pp. 369-374 ◽  
Author(s):  
D. P. Telionis ◽  
M. J. Werle
Keyword(s):  
Boundary Layer ◽  
Skin Friction ◽  
Adverse Pressure Gradient ◽  
Theoretical Models ◽  
Boundary Layer Separation ◽  
Moving Boundaries ◽  
Unsteady Boundary Layer ◽  
Layer Separation ◽  
Boundary Layer Equations ◽  
Type Singularity

The laminar boundary-layer equations for incompressible flow with a mild adverse pressure gradient were numerically solved for flows over downstream moving boundaries. It was demonstrated that the vanishing of skin friction in this case is not related to separation.2 Indeed the integration proceeds smoothly through a point of vanishing skin friction and further downstream a Goldstein-type singularity appears at a station where all the properties of separation according to the model of Moore, Rott, and Sears are present. It is also numerically demonstrated that the singular behavior is not uniform with n, the distance perpendicular to the wall, but it is initiated at a point away from the wall leaving below a region of nonsingular flow. The foregoing points provide numerical justification of the general theoretical models of unsteady boundary-layer separation suggested by Sears and Telionis.

Oscillatory solutions of the Falkner-Skan equation

1985 ◽  
Vol 397 (1813) ◽  
pp. 415-418 ◽  
Keyword(s):  
Boundary Layer ◽  
Periodic Solution ◽  
Incompressible Flow ◽  
Symbolic Dynamics ◽  
Similarity Solutions ◽  
Oscillatory Solutions ◽  
Smale Horseshoe ◽  
Boundary Layer Equations ◽  
Prandtl Boundary Layer ◽  
Infinitely Many Periodic Solutions

The Falkner-Skan equation for similarity solutions of the Prandtl boundary-layer equations for incompressible flow is analysed for both positive and negative values of the parameter β . For β < — 1 branches of solutions with any number of intervals of overshoot are found analytically, and confirm recent numerical results. For β > 1 we have proved that there is a periodic solution. We conjecture that for β > 2 there are infinitely many periodic solutions and that a form of ‘symbolic dynamics’, of the kind associated with a Smale ‘horseshoe map’ can be constructed. We have shown this rigorously for β close to 2.

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