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Similar Solutions for the Compressible Laminar Boundary Layer With Heat Transfer and Pressure Gradient

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Stewartsons transformation is applied to the laminar compressible boundary-layer equations and the requirement of similarity is introduced, resulting in a set of ordinary nonlinear differential equations previously quoted by Stewartson, but unsolved. The requirements of the system are Prandtl number of 1.0, linear viscosity-temperature relation across the boundary layer, an isothermal surface, and the particular distributions of free-stream velocity consistent with similar solutions. This system admits axial pressure gradients of arbitrary magnitude, heat flux normal to the surface, and arbitrary Mach numbers. The system of differential equations is transformed to an integral system, with the velocity ratio as the independent variable. For this system, solutions are found for pressure gradients varying from that causing separation to the infinitely favorable gradient and for wall temperatures from absolute zero to twice the free-stream temperature. Some solutions for separated flows are also presented. For favorable pressure gradients, the solutions are unique. For adverse pressure gradients, where the solutions are not unique, two solutions of the infinite family of possible solutions are identified as essentially viscid at the outer edge of the boundary layer and the remainder essentially inviscid. For the case of favorable pressure gradients with heated walls, the velocity within a portion of the boundary layer is shown to exceed the local external velocity. The variation of a Reynolds analogy parameter, which indicates the ratio of skin friction to heat transfer, is from zero to 7.4 for a surface of temperature twice the free-stream stagnation temperature, and from zero to 2.8 for a surface held at absolute zero where the value 2 applies to a flat plate.

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  • Aerodynamics
  • Fluid Mechanics

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