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.

On Interaction Between Falling Bodies and the Surrounding Fluid

“…On the other hand, suspension flow studies tend to be concerned with sparser grain flows where the interstitial fluid is important for the grain dynamics [13,22,30,55,56], such as in the study of aeolian or fluvial transport [54]. These studies address interesting issues such as entrainment, which are potentially relevant to some body-fluid interactions of present concern.…”

Grain transport mechanics in shallow overland flow

“…By introducing kinematic flow concepts, Eames et al (2004) derived an expression for the Eulerian mean velocity (U Ł ) of the liquid passing through an array of bodies. Their framework was developed by replacing the obstacles with a distributed drag force, while ignoring the inviscid blocking by the rigid bodies.…”

“…where C m is the additive mass coefficient for the particle matrix (Eames et al, 2004). For near spherical shapes, C m is about 0Ð5.…”

Fluid Mechanical Aspects of the Gas-Lift Technique

“…This effect is attributed to the hydrodynamic interactions and to the reference liquid velocity seen by the bubbles, or interstitial velocity, which is different from the time average liquid velocity (Kowe et al 1988). Recently, Eames et al (2004) studied the velocity through and around a group of bodies and extended this interstitial velocity concept developed by Kowe et al (1988). It was shown that, for low values of the void fraction , the hindering function could be expressed as h( who analyzed binary bubble interactions and obtained h( ) = 1 − C , where the constant C was ranging from 1.43 to 1.56 depending on the truncation used in the calculations.…”