2.2.1 The Continuity Equation 连续方程[cfd-2-2-1]

如果把注意力限定在单相流体上，质量守恒定律表达的是这样一个事实：在这样的流体系统中，质量既不能被创造，也不能消失。连续方程中也没有扩散通量的贡献，因为对静止流体而言，质量的任何变化都意味着流体粒子的位移。en

If we restrict our attention to single-phase fluids, the law of mass conservation expresses the fact that mass cannot be created in such a fluid system, nor it can disappear. There is also no diffusive flux contribution to the continuity equation, since for a fluid at rest, any variation of mass would imply a displacement of the fluid particles.

为了导出连续方程，考虑如图2.1所示的固定在空间中的有限控制体模型。在控制面上的某一点处，流动速度为 $\vec{v}$ ，单位法向量为 $\vec{n}$ ， $$dS$$ 表示一个微元面积。此情形下的守恒量是密度 $\rho$ 。对有限体积 $\Omega$ 内部总质量的时间变化率，我们有en

In order to derive the continuity equation, consider the model of a finite control volume fixed in space, as sketched in Fig. 2.1. At a point on the control surface, the flow velocity is $\vec{v}$ , the unit normal vector is $\vec{n}$ , and $$dS$$ denotes an elemental surface area. The conserved quantity in this case is the density $\rho$ . For the time rate of change of the total mass inside the finite volume $\Omega$ we have

\frac{\partial}{\partial t}\int_{\Omega}\rho\,d\Omega. \tag{1}

流体通过某个固定在空间中的表面的质量流量，等于（密度）×（表面面积）×（垂直于表面的速度分量）的乘积。因此，对流通过每个面元 $$dS$$ 的贡献便为en

The mass flow of a fluid through some surface fixed in space equals to the product of (density) × (surface area) × (velocity component perpendicular to the surface). Therefore, the contribution from the convective flux across each surface element $$dS$$ becomes

\rho\left(\vec{v}\cdot\vec{n}\right)dS. \tag{2}

由于在对流中 $\vec{n}$ 总是指向控制体外，当乘积 $\left(\vec{v}\cdot\vec{n}\right)$ 为负时我们称之为流入（inflow），为正时则称为流出（outflow），此时质量离开控制体。en

Since by convection $\vec{n}$ always points out of the control volume, we speak of inflow if the product $\left(\vec{v}\cdot\vec{n}\right)$ is negative, and of outflow if it is positive and hence the mass leaves the control volume.

如上所述，此时不存在任何体积源或面源。于是，考虑到方程(2.1)的一般形式，我们可以写出en

As stated above, there are no volume or surface sources present. Thus, by taking into account the general formulation of Eq. (2.1), we can write

\frac{\partial}{\partial t}\int_{\Omega}\rho\,d\Omega + \oint_{\partial\Omega}\rho\left(\vec{v}\cdot\vec{n}\right)dS = 0. \tag{2.3}

这就是连续方程的积分形式——质量守恒定律。en

This represents the integral form of the continuity equation - the conservation law of mass.