Forces on bodies moving unsteadily in rapidly compressed flows
Forces on bodies moving unsteadily in rapidly compressed flows
复制标题
在快速压缩的流动中不稳定移动的物体上的力
DOI：
10.1017/s0022112004008535
复制
发表时间：
2004
影响因子：
3.7
通讯作者：
J. Hunt
中科院分区：
文献类型：
--
作者：
I. Eames;J. Hunt
考虑由体积 ${\cal V}$ 的刚体在快速压缩的垂直流中以速度 ${\bm U}$ 不稳定移动产生的无粘可压缩流。流体在远大于物体尺寸的尺度上以 ${\bm \nabla}\,{\bm \cdot}\,{\bm v}_0$ 的速率均匀地等熵压缩，并且物体移动得足够慢，以致马赫数 $M$ 很低。流动最初是无旋转的，并且在压缩期间保持无旋转。物体不稳定移动产生的对流动的扰动在距物体 $\int_0^t c_1\,{\rm d}t$ 的演化区域 ${\cal D}$ 内是非发散的，其中 $c_1$ 是声速。在${\cal D}$内，流动由强度源$({\bm \nabla} \,{\bm \cdot}\, {\bm v}_0){\cal V}$和与压缩率无关的偶极贡献主导，而在${\cal D}$之外，压缩波远离身体传播。当物体远小于特征距离 $\|({\bm \nabla}{\bm v}_0)|_{{\bm x}_0}\|/\|({\bm \nabla} {\bm \nabla} {\bm v}_0 )|_{{\bm x}_0}\|$ 和区域大小 ${\cal D}$ 时，长度尺度的分离使得作用在物体上的力可以根据远离物体的动量通量（但在 ${\cal D}$ 区域内）进行分析计算。流体压缩产生的总力为 $\rho(t) ({\bm \nabla} \,{\bm \cdot}\, {\bm v}_0) {\cal V} ({\bm U}-{\bm v}_0)\,{\bm \cdot}\, \boldsymbol{\alpha} $，其中 ${\bm v}_0$ 是没有粒子时的速度场， $\boldsymbol{\alpha}$ 是虚拟惯性张量。因此，物体在流体压缩（膨胀）过程中会受到阻力（推力），因为物体向前移动的流体密度随时间增加（减小）。分析表明，压缩力和附加质量力之和等于流体冲量的减小率 ${\bm P} = \rho(t){\cal V}({\bm U}-{\bm v}_0)\,{\bm \cdot}\, \boldsymbol{\alpha}$。因此，流体冲量的概念自然延伸到流体密度随时间变化但空间均匀的流动类别。这些新结果适用于考虑投射到均匀压缩或膨胀流体中的刚性球体和圆柱体的无粘动力学。当流体快速膨胀时，刚体最终以恒定速度移动，因为与流体密度成正比的总力迅速趋于零。当身体垂直于压缩轴移动时，当流体的密度与身体的密度相当时，它会减慢并停止。然而，平行于压缩轴移动的物体会受到压力梯度的加速，压力梯度与流体密度成正比并随时间增加。
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.