2.2.3 The Energy Equation 能量方程[cfd-2-2-3]

我们在推导能量方程时所依据的基本原理是热力学第一定律。把它应用于图2.1所示的控制体，可以表述为：体积内总能量随时间的任何变化，都是由作用在该体积上的力的做功速率以及流入该体积的净热流引起的。流体单位质量的总能量 $$E$$ ，由其单位质量内能 $$e$$ 加上单位质量动能 $|\vec{v}|^{2}/2$ 而得到。因此，总能量可以写为en

The underlying principle that we will apply in the derivation of the energy equation, is the first law of thermodynamics. Applied to the control volume displayed in Fig. 2.1, it states that any changes in time of the total energy inside the volume are caused by the rate of work of forces acting on the volume and by the net heat flux into it. The total energy per unit mass $$E$$ of a fluid is obtained by adding its internal energy per unit mass, $$e$$ , to its kinetic energy per unit mass $|\vec{v}|^{2}/2$ . Thus, we can write for the total energy

E = e + \frac{|\vec{v}|^{2}}{2} = e + \frac{u^{2}+v^{2}+w^{2}}{2}. \tag{2.6}

此情形下的守恒量是单位体积的总能量，即 $\rho E$ 。它在体积 $\Omega$ 内随时间的变化可表示为en

The conserved quantity is in this case the total energy per unit volume, i.e., $\rho E$ . Its variation in time within the volume $\Omega$ can be expressed as

\frac{\partial}{\partial t}\int_{\Omega}\rho E\,d\Omega. \tag{2}

按照推导一般守恒定律（方程(2.1)）时的讨论，我们可以直接写出对流通量的贡献为en

Following the discussion in course of the derivation of the general conservation law (Eq. (2.1)), we can readily specify the contribution of the convective flux as

-\oint_{\partial\Omega}\rho E\left(\vec{v}\cdot\vec{n}\right)dS. \tag{3}

与连续方程和动量方程不同，这里出现了扩散通量。如前所述，它与单位质量守恒量的梯度成正比（Fick定律）。由于扩散通量 $\vec{F}_D$ 是针对静止流体定义的，只有内能起作用，于是得到en

In contrast to the continuity and the momentum equation, there is now a diffusive flux. As we already stated, it is proportional to the gradient of the conserved quantity per unit mass (Fick's law). Since the diffusive flux $\vec{F}_D$ is defined for a fluid at rest, only the internal energy becomes effective and we obtain

\vec{F}_D = -\gamma\rho\kappa\,\nabla e. \tag{2.7}

上式中， $\gamma = c_p/c_v$ 为比热系数之比， $\kappa$ 为热扩散率系数（thermal diffusivity coefficient）。扩散通量代表进入控制体热流的一部分，即由分子热传导引起的热扩散——由温度梯度导致的传热。因此，方程(2.7)通常写成Fourier热传导定律的形式，即en

In the above, $\gamma = c_p/c_v$ is the ratio of specific heat coefficients, and $\kappa$ denotes the thermal diffusivity coefficient. The diffusion flux represents one part of the heat flux into the control volume, namely the diffusion of heat due to molecular thermal conduction - heat transfer due to temperature gradients. Therefore, Equation (2.7) is in general written in the form of Fourier's law of heat conduction, i.e.,

\vec{F}_D = -k\,\nabla T \tag{2.8}

其中 $$k$$ 为热导率系数（thermal conductivity coefficient）， $$T$$ 为绝对静温。en

with $$k$$ standing for the thermal conductivity coefficient and $$T$$ for the absolute static temperature.

进入有限控制体的净热流的另一部分，是由辐射的吸收或发射、或化学反应引起的体积加热。我们把热源——单位质量传热的时间变化率——记作 $\dot{q}_h$ 。它与为动量方程引入的体力 $\vec{f}_e$ 的做功速率一起，构成完整的体积源en

The other part of the net heat flux into the finite control volume consists of volumetric heating due to the absorption or emission of radiation, or due to chemical reactions. We will denote the heat sources - the time rate of heat transfer per unit mass - as $\dot{q}_h$ . Together with the rate of work done by the body forces $\vec{f}_e$ , which we have introduced for the momentum equation, it completes the volume sources

Q_V = \rho\,\vec{f}_e\cdot\vec{v} + \dot{q}_h. \tag{2.9}

能量守恒中尚待确定的最后一项贡献是面源 $$Q_S$$ 。它对应于压力以及切向和法向应力对流体微元做功的时间变化率（见图2.2），即en

The last contribution to the conservation of energy, which we have yet to determine, are the surface sources $$Q_S$$ . They correspond to the time rate of work done by the pressure as well as the shear and normal stresses on the fluid element (see Fig. 2.2), i.e.,

\vec{Q}_S = -p\,\vec{v} + \overline{\overline{\tau}}\cdot\vec{v}. \tag{2.10}

把上述所有贡献和各项整理起来，便得到能量守恒方程的表达式en

Sorting now all the above contributions and terms, we obtain for the energy conservation equation the expression

\begin{aligned} \frac{\partial}{\partial t}\int_{\Omega}\rho E\,d\Omega + \oint_{\partial\Omega}\rho E\left(\vec{v}\cdot\vec{n}\right)dS = \oint_{\partial\Omega}k\left(\nabla T\cdot\vec{n}\right)dS\\ + \int_{\Omega}\left(\rho\vec{f}_e\cdot\vec{v} + \dot{q}_h\right)d\Omega - \oint_{\partial\Omega}p\left(\vec{v}\cdot\vec{n}\right)dS + \oint_{\partial\Omega}\left(\overline{\overline{\tau}}\cdot\vec{v}\right)\cdot\vec{n}\,dS. \end{aligned} \tag{2.11}

能量方程(2.11)通常写成略有不同的形式。为此，我们将利用总焓、总能量与压力之间的如下一般关系en

The energy equation (2.11) is usually written in a slightly different form. For that purpose, we will utilise the following general relation between the total enthalpy, the total energy and the pressure

H = h + \frac{|\vec{v}|^{2}}{2} = E + \frac{p}{\rho}. \tag{2.12}

当我们在能量守恒定律(2.11)中把对流项（ $\rho E\vec{v}$ ）与压力项（ $p\vec{v}$ ）合并，并应用公式(2.12)时，最终可把能量方程写成en

When we now gather the convective ( $\rho E\vec{v}$ ) and the pressure term ( $p\vec{v}$ ) in the energy conservation law (2.11), and apply the formula (2.12), we can finally write the energy equation in the form

\begin{aligned} \frac{\partial}{\partial t}\int_{\Omega}\rho E\,d\Omega + \oint_{\partial\Omega}\rho H\left(\vec{v}\cdot\vec{n}\right)dS = \oint_{\partial\Omega}k\left(\nabla T\cdot\vec{n}\right)dS\\ + \int_{\Omega}\left(\rho\vec{f}_e\cdot\vec{v} + \dot{q}_h\right)d\Omega + \oint_{\partial\Omega}\left(\overline{\overline{\tau}}\cdot\vec{v}\right)\cdot\vec{n}\,dS. \end{aligned} \tag{2.13}

至此，我们已经导出了三个守恒定律的积分形式：质量守恒(2.3)、动量守恒(2.5)和能量守恒(2.13)。下一节中，我们将更详细地给出法向应力和切向应力的表述。en

Herewith, we have derived integral formulations of the three conservation laws: the conservation of mass (2.3), of momentum (2.5), and of energy (2.13). In the next section, we shall work out the formulation of the normal and the shear stresses in more detail.