2.4.1 Formulation for a Perfect Gas 完全气体的形式[cfd-2-4-1]

在纯空气动力学中，一般可以合理地假定工作流体表现为量热完全气体（calorically perfect gas），其状态方程的形式为[13], [14]en

In pure aerodynamics, it is generally reasonable to assume that the working fluid behaves like a calorically perfect gas, for which the equation of state assumes the form [13], [14]

p = \rho RT, \tag{2.26}

其中 $$R$$ 为比气体常数。焓由下式给出en

where $$R$$ denotes the specific gas constant. The enthalpy results from

h = c_p T. \tag{2.27}

用守恒变量表示压力会带来方便。为此，需要把联系总焓与总能量的方程(2.12)与状态方程(2.26)结合起来。把表达式(2.27)代入焓，并利用定义en

It is convenient to express the pressure in terms of the conservative variables. For that purpose, we have to combine Equation (2.12), relating the total enthalpy to the total energy, together with the equation of state (2.26). Substituting expression (2.27) for the enthalpy and using the definitions

R = c_p - c_v, \quad \gamma = \frac{c_p}{c_v}, \tag{2.28}

最终得到压力en

we finally obtain for the pressure

p = (\gamma - 1)\,\rho\left[E - \frac{u^{2}+v^{2}+w^{2}}{2}\right]. \tag{2.29}

温度随后借助关系式(2.26)计算。对完全气体而言，动力黏性系数 $\mu$ 强烈依赖于温度，而受压力的影响很弱。常用的公式是所谓的Sutherland公式。对空气的结果为（SI单位）en

The temperature is then calculated with the aid of the relationship Eq. (2.26). The coefficient of the dynamic viscosity $\mu$ is, for a perfect gas, strongly dependent on temperature but only weakly dependent on pressure. The so-called Sutherland formula is frequently used. The result for air is (in SI units)

\mu = \frac{1.45\,T^{3/2}}{T+110}\cdot 10^{-6}, \tag{2.30}

其中温度 $$T$$ 以开尔文（K）为单位。于是，在 $T = 288\,\mathrm{K}$ 时可得 $\mu = 1.78\cdot 10^{-5}\,\mathrm{kg/ms}$ 。气体中热导率系数 $$k$$ 对温度的依赖关系与 $\mu$ 类似；相反，液体中 $$k$$ 几乎为常数。因此，对空气通常采用关系式en

where the temperature $$T$$ is in degree Kelvin (K). Thus, at $T = 288\,\mathrm{K}$ one obtains $\mu = 1.78\cdot 10^{-5}\,\mathrm{kg/ms}$ . The temperature dependence of the thermal conductivity coefficient $$k$$ resembles that of $\mu$ in the case of gases. By contrast, $$k$$ is virtually constant in the case of liquids. For this reason, the relationship

k = c_p\frac{\mu}{Pr} \tag{2.31}

此外，通常还假定Prandtl数 $$Pr$$ 在整个流场中为常数。对空气，Prandtl数取值 $$Pr = 0.72$$ 。en

is generally used for air. In addition, it is commonly assumed that the Prandtl number $$Pr$$ is constant in the entire flow field. For air, the Prandtl number takes the value $$Pr = 0.72$$ .