Boundary-layer flow self-similar

Similarity Variables The physical meaning of the term similarity relates to internal similitude, or self-similitude. Thus, similar solutions in boundary-layer flow over a horizontal flat plate are those for which the horizontal component of velocity u has the property that two velocity profiles located at different coordinates x differ only by a scale factor. The mathematical interpretation of the term similarity is a transformation of variables carried out so that a reduction in the number of independent variables is achieved. There are essentially two methods for finding similarity variables, "separation of variables (not the classical concept) and the use of "continuous transformation groups. The basic theory is available in Ames (1965). 

The continuity equations for mass, x-direction momentum, chemical species and energy in the plane, stationary, laminar boundary layer flow have already been given as Eqs. (7.1) to (7.4). The stream function ij/, by means of which the mass continuity equation is automatically satisfied, is defined by Eqs. (7.5). Following the approaches of Lees (1956), Fay and Riddell (1958), and Chung (1965), self-similar solutions in the stagnation region are obtained via transformations from (x, y) co-ordinates to the two new variables... 

So far, we have been talking about the stability of zero pressure gradient flows. It is possible to extend the studies to include flows with pressure gradient using quasi-parallel flow assumption. To study the effects in a systematic manner, one can also use the equilibrium solution provided by the self-similar velocity profiles of the Falkner-Skan family. These similarity profiles are for wedge flows, whose external velocity distribution is of the form, 11 = k x . This family of similarity flow is characterized by the Hartree parameter jSh = 2 1 the shape factor, H =. Some typical non-dimensional flow profiles of this family are plotted against non-dimensional wall-normal co-ordinate in Fig. 2.7. The wall-normal distance is normalized by the boundary layer thickness of the shear layer. 

Near the point where the two streams first meet the chemical reaction rate is small and a self-similar frozen-flow solution for Yp applies. This frozen solution has been used as the first term in a series expansion [62] or as the first approximation in an iterative approach [64]. An integral method also has been developed [62], in which ordinary differential equations are solved for the streamwise evolution of parameters that characterize profile shapes. The problem also is well suited for application of activation-energy asymptotics, as may be seen by analogy with [65]. The boundary-layer approximation fails in the downstream region of flame spreading unless the burning velocity is small compared with u it may also fail near the point where the temperature bulge develops because of the rapid onset of heat release there,... 

In the book [117], some data are given on the hydrodynamic characteristics of bodies of various shapes these data mainly pertain to the region of precrisis self-similarity. The influence of roughness of the cylinder surface and the turbulence level of the incoming flow on the drag coefficient is discussed in [522]. In [211], the relationship between hydrodynamic flow characteristics in turbulent boundary layers and the longitudinal pressure gradient is studied. Analysis of the transition to turbulence in the wake of circular cylinders is presented in [333]. 

The concept of a self-similar solution is well known to the student familiar with the development of a boundary layer along a flat plate where the velocity profile remains the same when distance from the wall is scaled with the boundary layer thickness that varies along the direction of flow. Similarly, diffusion profiles in semi-infinite media are known to be self-similar when distance is scaled with respect to the square root of diffusion time. 

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