420.
ON THE SUGGESTED ANALOGY BETWEEN THE CONDUCTION OF HEAT AND MOMENTUM DURING THE TURBULENT MOTION OF A FLUID.
[Advisory Committee for Aeronautics, T. 941, 1917.]
THE idea that the passage of heat from solids to liquids moving past them is governed by the same principles as apply in virtue of viscosity to the passage of momentum, originated with Reynolds (Manchester Proc., 1874); and it has been further developed by Stanton (Phil. Trans., Vol. cxc. p. 67, 1897; Tech. Rep. Adv. Committee, 1912-13, p. 45) and Lanchester (same Report, p. 40). Both these writers express some doubt as to the exactitude of the analogy, or at any rate of the proofs which have been given of it. The object of the present note is to show definitely that the analogy is not complete.
The problem which is the simplest, and presumably the most favourable to the analogy, is that of fluid enclosed between two parallel plane solid surfaces. One of these surfaces at y = 0 is supposed to be fixed, while the other at y = 1 moves in the direction of x in its own plane with unit velocity. If the motion of the fluid is in plane strata, as would happen if the viscosity were high enough, the velocity u in permanent regime of any stratum y is represented by y simply. And by definition, if the viscosity be unity, the tangential traction per unit area on the bounding planes is also unity.
Let us now suppose that the fixed surface is maintained at temperature 0, and bhe moving surface at temperature 1. So long as the motion is stratified, the flow of heat is the same as if the fluid were at rest, and the temperature (ff) at any stratum y has the same value y as has u. If the conductivity is unity, the passage of heat per unit area and unit time is also unity. In -this case, the analogy under examination is seen to be complete. The question is—will it still hold when the motion becomes turbulent ? It appears that the identity in the values of 0 and u then fails.
The equations for the motion of the fluid when there are no impressed forces are
Du 1 dp _.
-----=---------— 4- i/V2 U
Dt 'ntained constant, small alterations 8/3, 87, Sg are incurred. Neglecting the small variations of ft, 7, g when multiplied by a2 and higher powers of a, we get