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

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.

Numerical studies of the laminar boundary layer for Mach numbers up to 15

1969 ◽  
Vol 36 (2) ◽  
pp. 347-366 ◽  
Author(s):  
Henry A. Fitzhugh
Keyword(s):  
Boundary Layer ◽  
Skin Friction Coefficient ◽  
Order Effects ◽  
Approximate Methods ◽  
First Order ◽  
Boundary Layer Equations ◽  
Numerical Studies ◽  
Thickness Skin ◽  
Displacement Thickness ◽  
Exact Computer

A comprehensive set of exact solutions to the first-order boundary-layer equations has been computed using the finite difference computer programme of Sells, with and without wall cooling. The effects of Prandtl number, wall cooling and Mach number on separation point location were studied. Values of displacement thickness, skin friction coefficient and Stanton number are displayed graphically for the supersonic flow over a circular concave arc, for a subsonic cooled cylinder and for the case of a linearly retarded velocity distribution. The influence of pressure gradient on recovery factor was studied. Velocity and temperature profiles are shown for four cold wall cases. The exact computer results show the errors in many of the more approximate methods available for the case whereUe=U∞(1 -X/L). The importance of second-order effects and the applicability of a first-order solution are discussed briefly.

Asymptotic defect boundary layer theory applied to thermochemical non-equilibrium hypersonic flows

1997 ◽  
Vol 339 ◽  
pp. 213-238 ◽  
Author(s):  
S. SÉROR ◽  
D. E. ZEITOUN ◽  
J.-Ph. BRAZIER ◽  
E. SCHALL
Keyword(s):  
Boundary Layer ◽  
Heat Flux ◽  
Wind Tunnel ◽  
Boundary Layer Theory ◽  
Hypersonic Flows ◽  
Navier Stokes ◽  
First Order ◽  
Boundary Layer Equations ◽  
Defect Boundary ◽  
Non Equilibrium

Viscous flow computations are required to predict the heat flux or the viscous drag on an hypersonic re-entry vehicle. When real gas effects are included, Navier–Stokes computations are very expensive, whereas the use of standard boundary layer approximations does not correctly account for the ‘entropy layer swallowing’ phenomenon. The purpose of this paper is to present an extension of a new boundary layer theory, called the ‘defect approach’, to two-dimensional hypersonic flows including chemical and vibrational non-equilibrium phenomena. This method ensures a smooth matching of the boundary layer with the inviscid solution in hypersonic flows with strong entropy gradients. A new set of first-order boundary layer equations has been derived, using a defect formulation in the viscous region together with a matched asymptotic expansions technique. These equations and the associated transport coefficient models as well as thermochemical models have been implemented. The prediction of the flow field around the blunt-cone wind tunnel model ELECTRE with non-equilibrium free-stream conditions has been done by solving first the inviscid flow equations and then the first-order defect boundary layer equations. The numerical simulations of the boundary layer flow were performed with catalytic and non-catalytic conditions for the chemistry and the vibrational mode. The comparison with Navier–Stokes computations shows good agreement. The wall heat flux predictions are compared to experimental measurements carried out during the MSTP campaign in the ONERA F4 wind tunnel facility. The defect approach improves the skin friction prediction in comparison with a classical boundary layer computation.

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.

Nonlinear Theory for Flexural Motions of Thin Elastic Plate, Part 3: Numerical Evaluation of Boundary Layer Solutions

1982 ◽  
Vol 49 (2) ◽  
pp. 409-416
Author(s):  
N. Sugimoto
Keyword(s):  
Boundary Layer ◽  
Stress Singularity ◽  
Free Edge ◽  
Thin Elastic Plate ◽  
Normal Stresses ◽  
Interior Region ◽  
First Order ◽  
Plausible Values ◽  
Bending Stresses ◽  
Order Stress

The boundary layer solutions previoulsy obtained in Part 2 of this series for the cases of the built-in edge and the free edge are evaluated numerically. For the built-in edge, a characteristic penetration depth of the boundary layer toward the interior region is given by 0.13 εh, εh being the normalized thickness of the plate, while for the free edge, it is given by 0.32 εh. Thus the boundary layer for the free edge penetrates more deeply toward the interior region than that for the built-in edge. The first-order stress distribution in each boundary layer is displayed. For the built-in edge, the stress singularity appears on the edge. It is shown that, in the boundary layer, the shearing and normal stresses become comparable with the bending stresses. Similarly for the free edge, the shearing stress also becomes comparable with the twisting stress. It should be remarked that, in the boundary layer, the shearing or the normal stress plays a primarily important role as the bending or the twisting stress. But the former decays toward the interior region and remains higher order than the latter. Finally owing to these numerical results, the coefficients involved in the “reduced” boundary conditions for the built-in edge are evaluated for the various plausible values of Poisson’s ratio.

Numerical studies of the hypersonic strong interaction boundary-layer equations

AIAA Journal ◽  
1971 ◽  
Vol 9 (10) ◽  
pp. 2058-2060 ◽  
Author(s):  
M. J. WERLE ◽  
S. F. WORNOM
Keyword(s):  
Boundary Layer ◽  
Strong Interaction ◽  
Boundary Layer Equations ◽  
Numerical Studies
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