Application of the method of integral relations to laminar boundary layers in three dimensions

1977 ◽  
Vol 353 (1674) ◽  
pp. 319-347 ◽  
Keyword(s):  
Boundary Layer ◽  
Velocity Component ◽  
Equations Of Motion ◽  
Method Of Lines ◽  
Streamwise Velocity ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Hyperbolic Partial Differential Equations ◽  
Method Of Integral Relations ◽  
Integral Relations

The method of integral relations, which has been used successfully for the solution of a series of boundary layer problems in two dimensions, is extended to apply to three dimensional compressible boundary layer flows with and without separation. The essential steps in reducing the equations of motion to quasi-incompressible form are discussed and an outline of the method of integral relations as applied in three dimensions is presented. Analytical representations of the streamwise shear stress function θ and the cross flow velocity component V are given in terms of the streamwise velocity component U . Different functions are employed for attached and separated flows reflecting the different physical restrictions for each case. The system of hyperbolic partial differential equations obtained from applications of the method are solved by the method of lines. The method is first applied to two model incompressible problems with known solutions, namely parabolic flow over a flat plate and flow over a yawed cylinder. It is then applied to flow past a plate-cylinder combination previously solved by a finite difference method. Finally, it is applied in its compressible formulation to calculate supersonic laminar boundary layer flow over a swept back wedge. This flow field was recently investigated experimentally and the observed data are compared with results calculated by the method.

scholarly journals Application of the Method of Integral Relations to the Calculation of Two-Dimensional Incompressible Turbulent Boundary Layer

1981 ◽  
Vol 48 (4) ◽  
pp. 701-706 ◽  
Author(s):  
W.-S. Yeung ◽  
R.-J. Yang
Keyword(s):  
Experimental Data ◽  
Boundary Layer ◽  
Turbulent Boundary Layer ◽  
Pressure Gradient ◽  
Numerical Solutions ◽  
Experimental Results ◽  
Two Dimensional ◽  
Incompressible Turbulent Boundary Layer ◽  
Method Of Integral Relations ◽  
Integral Relations

The orthonormal version of the Method of Integral Relations (MIR) was applied to solve for a two-dimensional incompressible turbulent boundary layer. The flow was assumed to be nonseparating. Flows with favorable, unfavorable, and zero pressure gradient were considered, and comparisons made with available experimental data. In general, the method predicted very well the experimental results for flows with favorable or zero pressure gradient; for flows with unfavorable pressure gradient, it predicted the experimental data well only up to a certain distance from the initial station. This result is due to the flow not being in equilibrium beyond that distance. Finally, the scheme was shown to be efficient in obtaining numerical solutions.

Asymptotic solution for the perturbed Stokes problem in a bounded domain in two and three dimensions

1999 ◽  
Vol 129 (4) ◽  
pp. 811-824
Author(s):  
Dialla Konate
Keyword(s):  
Boundary Layer ◽  
Bounded Domain ◽  
Stokes Equation ◽  
Stokes Problem ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Potential Vector ◽  
Small Viscosity ◽  
The Given ◽  
Natural Way

We consider the Stokes problem with a small viscosity. When the viscosity goes to zero, the boundary-layer phenomenon can appear. In this case, the solution of the given perturbed Stokes equation cannot be properly approximated by the solution of its limiting equation ‘near’ the boundary Γ of the domain of study, say Ω To overcome this problem, we need to construct a corrector term in the neighbourhood of Γ Lions has studied this problem and has constructed a corrector for the case where Ω is a half space in ℝ2. The case where Ω is an open and bounded domain of ℝ2 or ℝ3, which remained unsolved, is the concern of this paper. The construction of the corrector to the perturbed Stokes equation depends heavily on the geometry of Ω In two dimensions, we construct the corrector in the form of a stream function, while in ℝ3 we construct it in the form of a potential vector. The corrector acts effectively in a neighbourhood of Γ that is the boundary layer. Using similar methods to those of Baranger and Tartar, we define the thickness of the boundary layer in a natural way. In addition, in this paper we study the behaviour of the corrected solution in some Hölder spaces.

TOPOLOGICAL LATTICE MODEL OF ELECTRONS COUPLED TO A CLASSICAL POLARIZATION FIELD

2001 ◽  
Vol 15 (24n25) ◽  
pp. 3336-3343
Author(s):  
W. Schwalm ◽  
S. Crockett ◽  
B. Moritz
Keyword(s):  
Equations Of Motion ◽  
Tight Binding ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Polarization Field ◽  
Confined Geometries ◽  
Dynamical Equations ◽  
Cross Term ◽  
Topological Lattice ◽  
Elastic Distortion

A simple model for an electron in two dimensions interacting with an elastic distortion field shows interesting polaron-like behaviors. The model consists of a single tight-binding electron in a classical polarization field. The distortion potential ϕ and electron amplitude ψ evolve by coupled dynamical equations derived from a Lagrangian with cross term gϕ|ψ|2. The equations of motion for ψ and ϕ are integrated together in time. For large enough g, the electron self-traps and a bound polaron-like object exists with time independent ϕ in the shape of a well and ψ a stationary state in the well. There is a critical value gmin below which the polaron well disappears. Spectral densities obtained by transforming double-time Green functions into [Formula: see text] show mass enhancement and lifetime broadening effects. The model, such as it is, can be extended easily to arbitrary, confined geometries in two or three dimensions.

Numerical Evaluation of Heat Transfer From a Spherical Dimple in a Flat Plate: Development of Appropriate Boundary Conditions

2007 ◽  
Author(s):  
O. Alshroof ◽  
J. Reizes ◽  
V. Timchenko ◽  
E. Leonardi
Keyword(s):  
Heat Transfer ◽  
Boundary Layer ◽  
Boundary Conditions ◽  
Velocity Distribution ◽  
Flat Plate ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Stream Velocity ◽  
Various Boundary Conditions ◽  
Significant Difference

A numerical investigation has been performed as to the feasibility of using spherical indentations in a flat plate for enhancing heat transfer in the laminar regime. An attempt to validate the calculation procedure resulted in a significant difference with previously published results, in which the inlet boundary condition was stated to be the Polhausen distribution. An investigation of the disparity lead to a study of the effects of various boundary conditions on the development of laminar boundary layers on an infinitesimally thin flat plate. In two-dimensions, regions appear in which velocities are greater than the free stream velocity (overshoot), unless the Blasius distribution is used to predict the inlet velocity in both directions. Surprisingly, although regions of overshoot occur when areas upstream and downstream of the plate are included in calculating the flow near the plate, the velocity distribution within the boundary layer is well represented by the Blasius profile for most of the plate. Outside the boundary layer the velocity distribution depends on the position and the length of the plate. Calculations in three dimensions using inlet boundary conditions developed from the two-dimensional study indicate that a single dimple does not enhance heat transfer.

The Three-Dimensional Turbulent Boundary Layer

1960 ◽  
Vol 64 (599) ◽  
pp. 692-694 ◽  
Author(s):  
D. R. Blackman ◽  
P. N. Joubert
Keyword(s):  
Boundary Layer ◽  
Turbulent Boundary Layer ◽  
Velocity Vector ◽  
Three Dimensional ◽  
Universal Function ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Law Of The Wall ◽  
The Law ◽  
Universal Law

An experimental verification of coles’ postulate of the Law of the Wake in three dimensions has been attempted. This hypothesis, which is strongly supported by experimental evidence in two dimensions, purports to represent the velocity vector in a yawed boundary layer by the sum of two vectors, whose magnitudes are found from the universal law of the wall, and a second universal function, the law of the wake. It has been found that the velocity vector can be represented in this way and the wake function required for this is very similar to Coles’ form. Limitations of the experimental arrangement however prohibited the comparison of vector directions with that proposed by Coles.

Toward the Improvement of Education in Mechanics

2005 ◽  
Author(s):  
Hiroshi Tajima
Keyword(s):  
Engineering Education ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Kinematics And Dynamics ◽  
Recent Effort ◽  
Two Dimensional ◽  
The Creation ◽  
Direct Answer

Two catch phrases of my half year lectures of Mechanics, which are given at two Universities, are as follows: “Three dimensions from the beginning”, and “Various methods to derive the equations of motion”. We often do not have enough time to teach kinematics and dynamics from two-dimensional matters first and then proceed to three dimensions. In many cases three-dimensional subjects are considered to be something advanced, or something which two-dimensional methods can be applied to. As a result we often lose chances to teach three-dimensional matters. I think there are few universities that give clear and firm teaching of three dimensional kinematics and dynamics. Many teachers often escape from three-dimensional discussions saying two dimensions are fundamental. I feel that there are very few teachings and discussions in Japan on the methods for deriving the equations of motion. There are many teachers who tell the importance of the equations of motion, but there are few who can discuss various methods to derive them. Discussion and Recognition of various methods not only broaden the application ability but also give clearer understanding in mechanics itself that will connect to the creation of new methodology. My lecture is a direct answer to these points, and my five year experience gives me more confidence in the importance of them. My recent effort is to get more chances to teach it not only within the universities, but also outside of them, getting more sympathetic people. At present I feel that my lecture will surely give some certain effects to the engineering education in mechanics in Japan.

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