On the velocity and temperature distributions in the turbulent wake behind a heated body of revolution

Abstract

Thevelocity distributionin the wake behind a body of revolution calculated by Miss Swain on the momentum transfer theory withlconstant over a cross-section of the wake is not in accordance with experiment.On the generalized vorticity transfer theory the equation for the first approximation to the velocity defectufar downstream isThere are two methods of simplifying this equation on the analogy of the successful simplification of the corresponding equation for the two-dimensional case. One method gives the same equation as if the turbulence were symmetrical; the other gives the equation of the modified vorticity transfer theory.For the former there is no real solution iflis constant over a cross-section; for a real solutionlmust be infinite on the axis of symmetry. The equation was integrated withl∝r−1over a cross-section; this variation oflwas suggested by putting the change in the area enclosed by a circular vortex-line proportional to the area of the section of the wake. The agreement with experiment is fairly good over the middle portion of the wake, but not near the edge, where the theoretical value of ∂u/∂ris not zero. Ifl2∝rp, ∂u/∂rvanishes at the edge of the wake only ifp= − 3. The solution is carried through for this case; it shows only fair agreement with experiment over the middle of the wake, and none at the edge.The calculation on the modified vorticity transfer theory is carried out withlconstant over a section; in this case there is no definite edge to the wake, nor doesufall off sufficiently rapidly to make the momentum integral for the drag converge. (u∝r−2for larger.) Even over the middle of the wake there is a discrepancy with experimental results, rather like that shown by the momentum transfer theory.Thetemperature distributionbehind a heated body was calculated according to each of the foregoing theories.The temperature distribution was also calculated from the experimental velocity distribution, without any assumption being made concerning the kinematic coefficient of turbulent diffusion, which was eliminated between the equations for the velocity and temperature. The vorticity transfer theory with symmetrical turbulence gives an impossible result. The modified vorticity transfer theory gives θ/θmax<u/umaxwhen the experimental velocity distribution is used.