Method for calculation of compressible laminar boundary layer with axial pressure gradient and heat transfer
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The time-average characteristics of boundary layers over a flat plate in nearly quasi-steady flow are determined. The plate may be either insulated or isothermal. The time averages are found without specifying the plate velocity explicitly except that it is positive and has an average value.
Measurements of average skin friction of the turbulent boundary layer have been made on a 15deg total included angle cone with foreign gas injection. Measurements of total skin-friction drag were obtained at free-stream Mach numbers of 0.3, 0.7, 3.5, and 4.7 and within a Reynolds number range from 0.9 x 10(exp 6) to 5.9 x 10(exp 6) with injection of helium, air, and Freon-12 (CCl2F2) through the porous wall. Substantial reductions in skin friction are realized with gas injection within the range of Mach numbers of this test. The relative reduction in skin friction is in accordance with theory-that is, the light gases are most effective when compared on a mass flow basis. There is a marked effect of Mach number on the reduction of average skin friction; this effect is not shown by the available theories. Limited transition location measurements indicate that the boundary layer does not fully trip with gas injection but that the transition point approaches a forward limit with increasing injection. The variation of the skin-friction coefficient, for the lower injection rates with natural transition, is dependent on the flow Reynolds number and type of injected gas; and at the high injection rates the skin friction is in fair agreement with the turbulent boundary layer results.
An approximation which reduces the computation of three dimensional, laminar, compressible, boundary layer equations to the problem of solving two dimensional type boundary layer equations is presented. A comparison of this method with a fully three dimensional boundary layer calculation is provided.
The adjoint parabold stability equations (PSE) formulation is used to calculate the boundary layer receptivity to localized surface roughness and suction for compressible boundary layers. Receptivity efficiency functions predicted by the adjoint PSE approach agree well with results based on other nonparallel methods including linearized Navier-Stokes equations for both Tollmien-Schlichting waves and crossflow instability in swept wing boundary layers. The receptivity efficiency function can be regarded as the Green's function to the disturbance amplitude evolution in a nonparallel (growing) boundary layer. Given the Fourier transformed geometry factor distribution along the chordwise direction, the linear disturbance amplitude evolution for a finite size, distributed nonuniformity can be computed by evaluating the integral effects of both disturbance generation and linear amplification. The synergistic approach via the linear adjoint PSE for receptivity and nonlinear PSE for disturbance evolution downstream of the leading edge forms the basis for an integrated transition prediction tool. Eventually, such physics-based, high fidelity prediction methods could simulate the transition process from the disturbance generation through the nonlinear breakdown in a holistic manner.