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

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 pq 2 , but p + 1/2 pq 2 + 1/2 pq '2. But I do not remember having seen any formal discussion. If V 1 is the total vector velocity, ω 1 1 the total vorticity, p 1 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 v 1 — v 1 X ω 1 = — grad ( p 1 /ρ + Ω + 1/2 v 1 2 ), (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 { p 1 /ρ + Ω + 1/2 (v + v') 2 }.