2.4.2 Formulation for a Real Gas 真实气体的形式[cfd-2-4-2]

当必须处理真实气体（real gas）时，问题变得更加复杂。原因在于，除流体动力学外，现在还必须对热力学过程和化学反应进行建模。真实气体流动的例子有：燃烧的模拟、再入飞行器的高超声速绕流，以及汽轮机内的流动。原则上，可以采取两种不同的方法来求解这一问题。第一种方法适用于气体处于化学平衡和热力学平衡的情形。这意味着存在唯一的状态方程，此时控制方程(2.19)保持不变，只是压力、温度、黏度等的数值要通过曲线拟合从查询表中插值得到[15]-[17]。但在实际中，气体更多是处于化学和/或热力学非平衡状态，必须相应地加以处理。en

The matter becomes more complicated when one has to deal with a real gas. The reason is that now we have to model a thermodynamic process and chemical reactions in addition to the fluid dynamics. Examples for a real gas flow are the simulation of combustion, the hypersonic flow past a re-entry vehicle, or the flow in a steam turbine. In principle, two different methods can be pursued to solve the problem. The first methodology is applicable in cases, where the gas is in chemical and in thermodynamical equilibrium. This implies that there is a unique equation of state. Then, the governing equations (2.19) remain unchanged. Only the values of pressure, temperature, viscosity, etc. are interpolated from lookup tables using curve fits [15]-[17]. But in practice, the gas is more often in chemical and/or thermodynamical non-equilibrium and has to be treated correspondingly.

为便于说明，考虑由 $$N$$ 种不同组分组成的气体混合物。对于有限的Damköhler数——其定义为流动滞留时间与化学反应时间之比——必须在模型中纳入有限速率化学。该模型需要描述化学反应引起的组分的生成与消亡。下面我们进一步假定，流体动力学和化学反应的时间尺度与空间尺度都远大于热力学的相应尺度。于是，我们假定气体处于热力学平衡但化学非平衡状态。为了模拟这种气体混合物的行为，必须为 $$N$$ 种组分补充 $$(N-1)$$ 个附加输运方程，从而扩展Navier-Stokes方程[18]-[23]。这样，我们得到形式上与方程(2.19)相同的方程组，但守恒变量向量 $\vec{W}$ 、通量向量 $\vec{F}_c$ 和 $\vec{F}_v$ 以及源项 $\vec{Q}$ 都被 $$(N-1)$$ 个组分方程所扩展。回顾表达式(2.20)至(2.25)，守恒变量向量现在为en

Let us for illustration consider a gas mixture consisting of $$N$$ different species. For a finite Damköhler number, which is defined as the ratio of flow-residence time to chemical-reaction time, we have to include finite-rate chemistry into our model. It has to describe the generation/destruction of species due to chemical reactions. In what follows, we will furthermore assume that the temporal and the spatial scales of fluid dynamics and of chemical reactions are much larger compared to those of thermodynamics. Thus, we suppose the gas is thermodynamically in equilibrium but chemically in non-equilibrium. In order to simulate the behaviour of such a gas mixture, the Navier-Stokes equations have to be augmented by $$(N-1)$$ additional transport equations for the $$N$$ species [18]-[23]. Hence, we obtain formally the same system like Eq. (2.19), but now with the vectors of the conservative variables $\vec{W}$ , the flux vectors $\vec{F}_c$ and $\vec{F}_v$ , as well as with the source term $\vec{Q}$ extended by $$(N-1)$$ species equations. Recalling the expressions (2.20) to (2.25), the vector of the conservative variables reads now

\vec{W} = \begin{bmatrix} \rho\\ \rho u\\ \rho v\\ \rho w\\ \rho E\\ \rho Y_1\\ \vdots\\ \rho Y_{N-1} \end{bmatrix}. \tag{2.32}

对流通量向量和黏性通量向量则变换为en

The convective and the viscous flux vectors transform into

\vec{F}_c = \begin{bmatrix} \rho V\\ \rho uV + n_x p\\ \rho vV + n_y p\\ \rho wV + n_z p\\ \rho HV\\ \rho Y_1 V\\ \vdots\\ \rho Y_{N-1} V \end{bmatrix}, \quad \vec{F}_v = \begin{bmatrix} 0\\ n_x\tau_{xx} + n_y\tau_{xy} + n_z\tau_{xz}\\ n_x\tau_{yx} + n_y\tau_{yy} + n_z\tau_{yz}\\ n_x\tau_{zx} + n_y\tau_{zy} + n_z\tau_{zz}\\ n_x\Theta_x + n_y\Theta_y + n_z\Theta_z\\ n_x\Phi_{x,1} + n_y\Phi_{y,1} + n_z\Phi_{z,1}\\ \vdots\\ n_x\Phi_{x,N-1} + n_y\Phi_{y,N-1} + n_z\Phi_{z,N-1} \end{bmatrix}, \tag{2.33}

其中en

where

\begin{aligned} \Theta_x &= u\tau_{xx} + v\tau_{xy} + w\tau_{xz} + k\frac{\partial T}{\partial x} + \rho\sum_{m=1}^{N} h_m D_m\frac{\partial Y_m}{\partial x}\\ \Theta_y &= u\tau_{yx} + v\tau_{yy} + w\tau_{yz} + k\frac{\partial T}{\partial y} + \rho\sum_{m=1}^{N} h_m D_m\frac{\partial Y_m}{\partial y}\\ \Theta_z &= u\tau_{zx} + v\tau_{zy} + w\tau_{zz} + k\frac{\partial T}{\partial z} + \rho\sum_{m=1}^{N} h_m D_m\frac{\partial Y_m}{\partial z}\\ \Phi_{x,m} &= \rho D_m\frac{\partial Y_m}{\partial x}\\ \Phi_{y,m} &= \rho D_m\frac{\partial Y_m}{\partial y}\\ \Phi_{z,m} &= \rho D_m\frac{\partial Y_m}{\partial z}. \end{aligned} \tag{2.34}

最后，源项现在变为en

Finally, the source term now becomes

\vec{Q} = \begin{bmatrix} 0\\ \rho f_{e,x}\\ \rho f_{e,y}\\ \rho f_{e,z}\\ \rho\vec{f}_e\cdot\vec{v} + \dot{q}_h\\ \dot{s}_1\\ \vdots\\ \dot{s}_{N-1} \end{bmatrix}. \tag{2.35}

在上述表达式（方程(2.32)–(2.35)）中， $$Y_m$$ 表示质量分数， $$h_m$$ 为焓， $$D_m$$ 为组分 $$m$$ 的有效二元扩散系数。此外， $\dot{s}_m$ 是化学反应引起的组分 $$m$$ 的变化率。注意，混合物的总密度 $\rho$ 等于各组分密度 $\rho Y_m$ 之和。因此，由于总密度被视为独立变量，只剩下 $$(N-1)$$ 个独立的密度 $\rho Y_m$ 。其余的质量分数 $$Y_N$$ 由下式求得en

In the above expressions Eqs. (2.32)-(2.35), $$Y_m$$ denotes the mass fraction, $$h_m$$ the enthalpy, and $$D_m$$ the effective binary diffusivity of species $$m$$ , respectively. Furthermore, $\dot{s}_m$ is the rate of change of species $$m$$ due to chemical reactions. Note that the total density $\rho$ of the mixture is equal to the sum of the densities of the species $\rho Y_m$ . Therefore, since the total density is regarded as an independent quantity, there are only $$(N-1)$$ independent densities $\rho Y_m$ left. The remaining mass fraction $$Y_N$$ is obtained from

Y_N = 1 - \sum_{m=1}^{N-1} Y_m. \tag{2.36}

为了得到压力 $$p$$ 的表达式，首先假定各组分的行为如同理想气体，即en

In order to find an expression for the pressure $$p$$ , we first assume that the individual species behave like ideal gases, i.e.,

p_m = \rho Y_m\frac{R_u}{W_m}\,T, \tag{2.37}

其中 $$R_u$$ 为普适气体常数， $$W_m$$ 为分子量。结合Dalton定律en

with $$R_u$$ denoting the universal gas constant and $$W_m$$ being the molecular weight, respectively. Together with Dalton's law,

p = \sum_{m=1}^{N} p_m, \tag{2.38}

可以写出en

we can write

p = \rho R_u T\sum_{m=1}^{N}\frac{Y_m}{W_m}. \tag{2.39}

值得注意的是，由于气体处于热力学平衡，所有组分具有相同的温度 $$T$$ 。温度必须由下式迭代求出[21], [24]en

It is important to notice that because the gas is in thermodynamical equilibrium, all species possess the same temperature $$T$$ . The temperature has to be calculated iteratively from the expression [21], [24]

e = \sum_{m=1}^{N}\left[Y_m\left(h^{0}_{f,m} + \int_{T_{ref}}^{T} c_{p,m}\,dT\right)\right] - \frac{p}{\rho}. \tag{2.40}

气体混合物的内能 $$e$$ 由方程(2.6)得到。方程(2.40)中的 $h^{0}_{f,m}$ 、 $c_{p,m}$ 和 $T_{ref}$ 分别表示第 $$m$$ 种组分的生成热、定压比热和参考温度。上述物理量以及各组分的焓、热导率 $$k$$ 和动力黏性 $\mu$ 的数值，都由曲线拟合确定[19], [21], [23]。en

The internal energy of the gas mixture $$e$$ is obtained from Eq. (2.6). The quantities $h^{0}_{f,m}$ , $c_{p,m}$ , and $T_{ref}$ in Eq. (2.40) denote the heat of formation, the specific heat at constant pressure, and the reference temperature of the $$m$$ -th species, respectively. Values of the above quantities as well as of the thermal conductivity $$k$$ and of the dynamic viscosity $\mu$ of the species are determined from curve fits [19], [21], [23].

最后一个尚待建模的部分是方程(2.35)中的化学源项 $\dot{s}_m$ 。对于包含 $$N$$ 种组分、由 $$N_R$$ 个基元反应构成的反应集，其速率方程可以写成如下一般形式en

The last part, which remains to be modelled, is the chemical source term $\dot{s}_m$ in Eq. (2.35). The rate equations for a set of $$N_R$$ elementary reactions involving $$N$$ species can be written in the general form

\sum_{m=1}^{N}\nu'_{lm}C_m \;\overset{K_{fl}}{\underset{K_{bl}}{\rightleftharpoons}}\; \sum_{m=1}^{N}\nu''_{lm}C_m \quad\text{for}\quad l = 1, 2, \ldots, N_R. \tag{2.41}

在方程(2.41)中， $\nu'_{lm}$ 和 $\nu''_{lm}$ 分别是组分 $$m$$ 在第 $$l$$ 个正反应和逆反应中的化学计量系数。此外， $$C_m$$ 表示组分 $$m$$ 的摩尔浓度（ $C_m = \rho Y_m/W_m$ ），而 $K_{fl}$ 和 $K_{bl}$ 分别表示第 $$l$$ 个反应步骤的正反应和逆反应速率常数。它们由经验Arrhenius公式给出en

In the above Eq. (2.41), $\nu'_{lm}$ and $\nu''_{lm}$ are the stoichiometric coefficients for species $$m$$ in the $$l$$ -th forward and backward reaction, respectively. Furthermore, $$C_m$$ stands for the molar concentration of species $$m$$ ( $C_m = \rho Y_m/W_m$ ), and finally $K_{fl}$ and $K_{bl}$ , respectively, denote the forward and the backward reaction rate constants for the $$l$$ -th reaction step. They are given by the empirical Arrhenius formulae

\begin{aligned} K_f &= A_f\,T^{B_f}\exp(-E_f/R_u T)\\ K_b &= A_b\,T^{B_b}\exp(-E_b/R_u T), \end{aligned} \tag{2.42}

其中 $$A_f$$ 和 $$A_b$$ 为Arrhenius系数， $$E_f$$ 和 $$E_b$$ 为活化能， $$B_f$$ 和 $$B_b$$ 为常数。第 $$l$$ 个反应引起的组分 $$m$$ 摩尔浓度的变化率为en

where $$A_f$$ and $$A_b$$ are the Arrhenius coefficients, $$E_f$$ and $$E_b$$ represent the activation energies, and $$B_f$$ as well as $$B_b$$ are constants, respectively. The rate of change of molar concentration of species $$m$$ by the $$l$$ -th reaction is given by

\dot{C}_{lm} = \left(\nu''_{lm} - \nu'_{lm}\right)\left(K_{fl}\prod_{n=1}^{N} C_n^{\nu'_{ln}} - K_{bl}\prod_{n=1}^{N} C_n^{\nu''_{ln}}\right). \tag{2.43}

于是，结合方程(2.43)，可以由下式计算组分 $$m$$ 的总变化率en

Hence, together with Eq. (2.43) we can calculate the total rate of change of species $$m$$ from

\dot{s}_m = W_m\sum_{l=1}^{N_R}\dot{C}_{lm}. \tag{2.44}

更多细节可参见上文引用的文献。关于化学反应流动控制方程的详细综述，包括通量的Jacobian矩阵及其特征值，也可参见[24]。en

More details can be found in the references cited above. A detailed overview of the equations governing a chemically reacting flow, together with the Jacobian matrices of the fluxes and their eigenvalues, can also be found in [24].

真实气体的另一个实用例子是蒸汽的模拟，或者更有挑战性的、叶轮机械应用中湿蒸汽的模拟[25]-[32]。在后一种情形中，蒸汽与水滴混合，即所谓的多相流（multiphase flow）；此时既可以求解一组附加的输运方程，也可以沿若干条流线追踪水滴。这类模拟在现代汽轮机叶栅设计中有着十分重要的应用。例如，对涡轮叶片绕流的分析有助于理解凝结引起的超临界激波以及流动不稳定性的产生，后者会使叶片装置承受额外的动态载荷并导致效率损失。en

Another practical example of real gas is the simulation of steam or, which is more demanding, of wet steam in turbomachinery applications [25]-[32]. In the later case, where the steam is mixed with water droplets, so that we speak of multiphase flow, it is either possible to solve an additional set of transport equations, or to trace the water droplets along a number of streamlines. These simulations have very important applications in the design of modern steam turbine cascades. The analysis of flow past turbine blades can for instance help to understand the occurrence of supercritical shocks by condensation and of flow instabilities, responsible for an additional dynamic load on the bladings and resulting in a loss of the efficiency.