Contributions to the theory of the Pitot-tube displacement effect

In an earlier paper the uniform shear flow past a sphere was studied, by investigating how vortex lines are deformed by the ‘primary flow’ (flow in the absence of shear), and deducing the ‘secondary’ vorticity field (first approximation for small shear). In another paper the image system associated with each element of secondary vorticity was found, whence the Biot-Savart law can be used to determine the secondary flow field by integration. The integration is here carried out for the ‘downwash’ (secondary flow component perpendicular to the undisturbed flow, down the velocity gradient) on the dividing streamline. Difficulties due to the infinite domain of integration and singularities of the integrand are overcome by selecting variables of integration carefully and using known analytical properties of the secondary vorticity. From the computation of downwash is inferred the first approximation (for small shear A) to the ‘displacement’ δ (displacement of the dividing streamline, up the velocity gradient, far upstream of the sphere). If U is the upstream flow velocity and a the radius of the sphere, the computed value of lim (Uδ/Aa²) is 0·9. $A \rightarrow 0$ Details of the calculation show that the secondary trailing vorticity is not an important contributor to the displacement, The downwash is due almost entirely to vorticity upstream of the sphere (Hall's earlier simplified theory gave good results, e.g., 1.24 instead of 0.9, because it concentrated on the effect of local vorticity in producing downwash); further, this produces displacement principally through its image vorticity. The relation between theories for a sphere and experimental results on Pitot tubes (beginning with Young & Maas 1936) is discussed. Theoretical evidence on tertiary- and quartary-flow effects is used here in the light of recent work which renders the successive-approximation sequence uniformly valid at infinity. The conclusion is that the theories, taken together, are not inconsistent with the experimental evidence that (i) at values of the ‘shear parameter’ Aa/U at which the displacement is measurable the ratio δ/a seems to have asymptoted to an approximately constant value, and (ii) displacement is greatly reduced in supersonic flow (Johannesen & Mair 1952) or when ‘sharp-lipped’ tubes are used (Livesey 1956).