A note on the measurement of total head and static pressure in a turbulent stream

Abstract

Abstract 1—If p is the mean static pressure, q the resultant mean velocity, and q' the resultant turbulent velocity, then it is generally held, I think, that a total head tube in a turbulent stream probably measures not p + 1/2 pq2, but p + 1/2 pq2 + 1/2 pq'2. But I do not remember having seen any formal discussion. If V1 is the total vector velocity, ω11 the total vorticity, p1 the total pressure, Ω the potential of the external forces, and ρ the density (assumed constant), then, if viscosity is neglected, the vector equation of motion is ∂/∂t v1 — v1 X ω1 = — grad (p1/ρ + Ω + 1/2 v12), (1) where the cross denotes a vector product. This vector product has no component along a stream-line. The component of the equation along a stream-line is therefore ∂/∂t (q + q') = — ∂/∂s { p1/ρ + Ω + 1/2 (v + v')2}.