Inviscid mean flow through and around groups of bodies

Abstract

General estimates are derived for mean velocities through and around groups or arrays of fixed and moving bodies, in unbounded and bounded domains, which lie within a defined perimeter. Robust kinematic flow concepts are introduced, namely the Eulerian spatial mean velocity $\overline{u}_E$ in the fluid volume between the bodies, the Eulerian flow outside the group, ${\bm u}_E^{(0)}$, and the Lagrangian mean velocity of material surfaces or fluid particles as they pass through the group of bodies ($\overline{u}_L^{(S)}$, $\overline{u}_L^{(P)}$). The Eulerian mean velocity is related to the momentum in the fluid domain, and is mainly influenced by fast moving regions of the flow. The Lagrangian mean velocity weights slowly moving regions of flow and is related to how material sheets deform as they are advected through groups of bodies. When the bodies are well-separated, the interstitial Eulerian and Lagrangian mean velocities ($\overline{u}_E^{(I)}$, $\overline{u}_L^{(I)}$), are defined and calculated in terms of the far-field contributions from the velocity or displacement field within the group of bodies.

In unbounded flow past well-separated bodies situated within a rectangular perimeter, the difference between the Eulerian and Lagrangian mean velocity is negligible (as the void fraction of the bodies, $\alpha\,{\rightarrow}\,0$). Within wide and short rectangular arrays, the Eulerian mean velocity is faster than the free-stream velocity $U$ because most of the incident flow passes through the array and $\overline{u}_E\,{=}\,U(1-\alpha)^{-1}$. Within long and thin rectangular arrays (and other cases where the reflux velocity is negligible), the Eulerian mean velocity, $\overline{u}_E\,{=}\,U(1-(1+C_m)\alpha)/(1-\alpha)$, is slower than the free-stream velocity, because most of the incident flow passes around the array. For a spherical or circular arrays of bodies, the particle Lagrangian mean velocity is $\overline{u}_L^{(P)}\,{=}\,U(1+C_m\alpha)^{-1}$ and differs from $\overline{u}_E$. These calculations are extended to examine the mean and interstitial flow through clouds of bodies in bounded channel flows.

The new concepts are applied to calculate the mean flow and pressure between and outside clouds of bodies, the average velocity of bubbly flows as a function of void fraction, and the tendency of clouds of bubbles to be distorted depending on their shape.