Heat Transfer Through Laminar Boundary Layers on Semi-Infinite Cylinders of Arbitrary Cross Section

Abstract

Summary This paper shows how to calculate the rate of heat transfer through a laminar boundary layer on a semi-infinite cylinder of arbitrary cross section. The cylinder is placed in a stream of incompressible fluid, the flow at infinity being parallel to the generators, and is maintained at a uniform temperature. A series solution for small downstream distances and an asymptotic formula for large downstream distances are given. To cover the intermediate range an approximate Pohlhausen solution is obtained; a correction of the error involved in the Pohlhausen solution is suggested which, it is believed, will lead to final errors of at most 2 percent. The calculations are applied to elliptic cylinders, and illustrate the effect on the local rate of heat transfer of varying the ratio of the major and minor axes of cross section, the length of perimeter being held fixed. der as those given by Seban and Bond1 for the circular cylinder. A straightforward extension of Cooke's analysis will show that a similar result applies to the Seban and Bond series for heat flux, per unit length. An asymptotic formula valid for large downstream stations may be obtained by considering the work of Batchelor5 or Hasimoto.6 These authors use Ray- leigh's method in which they find first the fluid motion generated by suddenly moving an infinite cylinder parallel to its length: the fluid and cylinder are initially at rest, and at all times / after the start the cylinder velocity is maintained at a constant value U. An approximation for the skin friction on a semi- infinite cylinder placed in a stream of velocity U, at a distance x downstream of its leading edge, is then in­ ferred from the Rayleigh solution by interpreting t as x/U. This approximate result improves as x becomes large and is asymptoticall y correct: it shows that at large distances downstream the skin friction on the arbitrary cylinder is the same as that on an equivalent'' circular cylinder of radius a, where a is such that the cross section of either cylinder may be mapped con- formally into the cross section of the other while leav­ ing the region at infinity unchanged. We shall first ob­ tain the above result by a slightly more direct method, dealing at once with the nonlinear equation for the steady flow past an arbitrary cylinder. It will become clear from this that at large downstream distances the rate of heat transfer from the arbitrary cylinder is the same as that from the equivalent circular cylinder; the exact asymptotic series provided by Bourne and Davies3 for the circular cylinder may therefore be applied here. Just how many terms of this series may be used for any given (arbitrary) cylinder is open to conjecture, but we shall present evidence to suggest that using three terms leads to errors of not more than 2 percent when applied to elliptic cylinders. In order to provide a Pohlhausen solution of the temperature boundary-layer equation it is necessary first to know the velocity distribution in the boundary layer. An approximate Pohlhausen profile has been provided by Cooke 4 and Varley.7 The form of solu­ tion given by Cooke is not easily employed in this con­ text, but by using Varley's form a solution of the tem­ perature boundary-layer equation can be provided. Taking Cartesian coordinate axes x% y, and z, with x