ARTICLE:

• THE SELF-SIMILAR TURBULENT BOUNDARY LAYER WITH INJECTION  download article

Micha Wolfshtein
Faculty of Aerospace Engineering, Technion - Israel Institute of technology, Haifa, Israel

ABSTRACT

Self-similarity of boundary layers is a very useful phenomenon: It allows easy solutions, better understanding of the boundary layer, and a better organized classification and comprehension of experimental results. The self-similar solutions of incompressible laminar boundary layers have been examined in the early days of boundary layer research and became a very powerful tool for the study of these flows. The situation is different when we consider turbulent boundary layers. A turbulent boundary layer has no self-similar solution because the rate of growth of the inner viscous sub-layer is different from that of the outer part of the turbulent boundary layer. Thus the turbulent boundary layer has an additional independent length scale and does not lend itself easily to similarity analysis. It is possible to show that even if a pressure gradient is present (affecting the rate of growth of both inner and outer layer) the turbulent boundary layer can not posses self-similar solution. The addition of injection or suction from solid walls may ease the situation due to the addition of yet another free parameter.
In a previous paper the possibilities for self-similar solutions of the turbulent incompressible boundary layer under a pressure gradient and with injection were examined. It was shown that a similarity ODE exists under certain constrains. Examination of this equation suggests that a self-similar solution is possible for boundary layers with suction, and for turbulent wall jets.
In the present paper these cases are explored further. The envelope of an approximate domain of existence is presented, in terms of the magnitude of the relevant parameters. Further, the existence of self-similar solutions to the energy equation for incompressible boundary layer is analyzed. The results indicate that the incompressible self-similar thermal boundary layer can exist for isothermal main stream if the wall temperature varies according to a power law in the distance along the wall, provided that the velocity self-similarity exists.
The analysis is carried out by investigation of integral properties of the equations. This technique enables an approximation of the envelope of existence of the solution as well.

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