Forces on bodies moving unsteadily in rapidly compressed flows

Abstract

The inviscid compressible flow generated by a rigid body of volume ${\cal V}$ moving unsteadily with a velocity ${\bm U}$ in a rapidly compressed homentropic flow is considered. The fluid is compressed isentropically at a rate ${\bm \nabla}\,{\bm \cdot}\,{\bm v}_0$ uniformly over a scale much larger than the size of the body and the body moves slowly enough that the Mach number $M$ is low. The flow is initially irrotational and remains so during compression. The perturbation to the flow generated by the body moving unsteadily is non-divergent within an evolving region ${\cal D}$ of distance $\int_0^t c_1\,{\rm d}t$ from the body, where $c_1$ is the speed of sound. Within ${\cal D}$, the flow is dominated by a source of strength $({\bm \nabla} \,{\bm \cdot}\, {\bm v}_0){\cal V}$ and a dipolar contribution which is independent of the rate of compression, while outside ${\cal D}$, compressional waves propagate away from the body. When the body is much smaller than the characteristic distance $\|({\bm \nabla}{\bm v}_0)|_{{\bm x}_0}\|/\|({\bm \nabla} {\bm \nabla} {\bm v}_0 )|_{{\bm x}_0}\|$ and the size of the region ${\cal D}$, the separation of length scales enables the force on the body to be calculated analytically from the momentum flux far from the body (but within the region ${\cal D}$). The contribution to the total force arising from fluid compression is $\rho(t) ({\bm \nabla} \,{\bm \cdot}\, {\bm v}_0) {\cal V} ({\bm U}-{\bm v}_0)\,{\bm \cdot}\, \boldsymbol{\alpha} $, where ${\bm v}_0$ is the velocity field in the absence of the particles and $\boldsymbol{\alpha}$ is the virtual inertia tensor. Thus a body experiences a drag (thrust) force during fluid compression (expansion) because the density of the fluid displaced forward by the body increases (decreases) with time. The analysis indicates that the sum of the compressional and added-mass force is equal to the rate of decrease of fluid impulse ${\bm P} = \rho(t){\cal V}({\bm U}-{\bm v}_0)\,{\bm \cdot}\, \boldsymbol{\alpha}$. Thus the concept of fluid impulse naturally extends to the class of flows where the fluid density changes with time, but is spatially uniform.

These new results are applied to consider the inviscid dynamics of a rigid sphere and cylinder projected into a uniformly compressed or expanded fluid. When the fluid rapidly expands, a rigid body ultimately moves with a constant velocity because the total force, which is proportional to the density of the fluid, tends rapidly to zero. When the body moves perpendicular to the axis of compression, it slows down and stops when the density of the fluid is comparable to the density of the body. However, a body moving parallel to the axis of compression is accelerated by pressure gradients which are proportional to fluid density and increases in time.