Temperature and Velocity Profiles in the Compressible Laminar Boundary Layer with Arbitrary Distribution of Surface Temperature

Abstract

An analysis is presented which enables the temperature profiles, velocity profiles, heat transfer, and skin friction to be calculated for laminar flow over a two-dimensional or axially symmetric surface without pressure gradient but with an arbitrary analytic distribution of surface temperature. The general theory is applicable to a gas of any Prandtl Number, although the numerical results given herein have been computed for air (Pr = 0.72). The predictions of the theory for the special case of constant surface temperature are compared with the calculations of Crocco. On the basis of this comparison, it is inferred that the present theory enables heat-transfer and skin-friction calculations accurate to within about 5 per cent to be made for flight conditions up to Mach Numbers near 5 and to within about 1 or 2 per cent for supersonic wind-tunnel conditions up to considerably higher Mach Numbers. A particular effort has been made to present the results, which are simple considering their generality, in a form that can be used readily in practical applications. From the mathematical point of view, the theory is applicable to an arbitrary analytic distribution of surface temperature, but in any given practical case it is necessary that the surface-temperature distribution be approximated by a polynomial. The only unknowns in the final equations developed are the coefficients of this polynomial, so that the work involved in applying the theory in any given case depends entirely on the work involved in approximating a given surfacetemperature distribution by a polynomial. An example is worked out in detail which illustrates some of the principal effects of variable surface temperature. I t is shown that both positively infinite and negatively infinite heat-transfer coefficients can occur. The anomaly of infinite and negative heat-transfer coefficients is discussed and attributed to the customary definition of the heat-transfer coefficient, which is shown to be fundamentally inappropriate for flows with variable surface temperature. In the particular example considered, a conventional method for calculating the net heat transferred yields completely incorrect results. A brief qualitative discussion of the possible effects of the heat transfer on flow separation is given. In order to facilitate the use of the results, all of the principal equations developed are collected and summarized in the section entitled Practical Use of Results.