论文标题
速度场的客观变形部分
The Objective Deformation Component of a Velocity Field
论文作者
论文摘要
对于在有限的，固定的空间域上定义的任意速度字段$ \ mathbf {v} $，我们找到了最接近的刚体速度字段$ \ mathbf {v} _ {rb} _ {rb} $ to $ \ mathbf {v} $ in $ l^2 $ norm。所得的变形速度组件，$ \ mathbf {v} _ {d} = \ MathBf {v- \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {v}} _ {rb} $，事实证明是框架 - 内差和物理上可观察的。具体而言，如果$ \ mathbf {q} _ {\ text {rb}}（t）$是旋转张量，描述了最接近的刚性车身框架的运动，则$ \ m athbf {v} $被视为$ \ mathbf {q} _ {\ text {rb}}}^{t} \ mathbf {v} _ {d} $由该帧中的观察者。结果，当从变形速度组件$ \ mathbf {v} _ {d} $计算时，流动的动量，能量，涡度，胚胎和螺旋度都会变得框架。
For an arbitrary velocity field $\mathbf{v}$ defined on a finite, fixed spatial domain, we find the closest rigid-body velocity field $\mathbf{v}_{RB}$ to $\mathbf{v}$ in the $L^2$ norm. The resulting deformation velocity component, $\mathbf{v}_{d}=\mathbf{v-\mathbf{v}}_{RB}$, turns out to be frame-indifferent and physically observable. Specifically, if $\mathbf{Q}_{\text{RB}}(t)$ is the rotation tensor describing the motion of the closest rigid body frame, then $\mathbf{v}$ is seen as $\mathbf{Q}_{\text{RB}}^{T}\mathbf{v}_{d}$ by an observer in that frame. As a consequence, the momentum, energy, vorticity, enstrophy, and helicity of the flow all become frame-indifferent when computed from the deformation velocity component $\mathbf{v}_{d}$.
