9.2 Enthalpy Damping 焓阻尼[cfd-9-2]

在某些情形下，总焓 $$H$$ （式(2.12)）在整个流场中为常数。例如，在无热源和外力的情况下，由欧拉方程支配的外部流动就会出现这种情形。我们可以利用这一事实来减少计算量。en

In certain cases, the total enthalpy $$H$$ (Eq. (2.12)) is constant in the whole flow field. This situation occurs, for example, for external flows governed by the Euler equations in absence of heat sources and external forces. We can take advantage of this fact in order to reduce the computational effort.

第一种可能的做法，是在流动域内规定总焓的值。这样一来，便可以从欧拉方程(2.45)中省去能量方程，从而节省内存和CPU时间。en

The first possibility would be to prescribe the value of the total enthalpy in the flow domain. In consequence, the energy equation can be omitted from the Euler equations (2.45). This saves memory and CPU time.

另一种方法——所谓的焓阻尼（enthalpy damping）——由Jameson为求解位势流方程而提出[3]。它利用总焓 $$H$$ 与其自由流值 $H_\infty$ 之差来定义一个附加的强迫项。据此，向定常态的收敛可以得到显著加速。把焓阻尼应用于欧拉方程(2.45)，得到如下修改形式[4]en

A different methodology - the so-called enthalpy damping - was suggested by Jameson for the solution of the potential flow equation [3]. It employs the difference between the total enthalpy $$H$$ and its freestream value $H_\infty$ to define an additional forcing term. With this, the convergence to the steady state can be considerably accelerated. The application of the enthalpy damping to the Euler equations (2.45) results in the following modification [4]

\frac{\partial}{\partial t}\int_{\Omega}\vec{W}\,d\Omega + \oint_{\partial\Omega}\vec{F}_c\,dS = \int_{\Omega}\left(\vec{Q} - \vec{Q}_{ED}\right)d\Omega\,, \tag{9.1}

其中强迫项 $\vec{Q}_{ED}$ 由下式给出en

where the forcing term $\vec{Q}_{ED}$ is given by

\vec{Q}_{ED} = \vartheta \begin{bmatrix} \rho\left(H - H_\infty\right)\\ \rho u\left(H - H_\infty\right)\\ \rho v\left(H - H_\infty\right)\\ \rho w\left(H - H_\infty\right)\\ \rho\left(H - H_\infty\right) \end{bmatrix}\,. \tag{9.2}

阻尼因子 $\vartheta$ 是一个小常数，必须凭经验确定。空间离散格式，特别是人工耗散，必须按这样的方式实现：使 $H = H_\infty$ 成为离散化方程的一个有效解。这样，加入源项 $\vec{Q}_{ED}$ 就不会改变最终的定常态。en

The damping factor $\vartheta$ is a small constant, which has to determined empirically. The spatial discretisation scheme, and particularly the artificial dissipation has to be implemented in such a way that $H = H_\infty$ is a valid solution of the discretised equations. Then, the addition of the source term $\vec{Q}_{ED}$ does not alter the final steady state.

焓阻尼作为附加的步骤，在每次更新流动解之后执行。例如，对于显式 $$m$$ 级时间推进格式（6.1.1或6.1.2小节），阻尼步骤为en

The enthalpy damping is conducted as an additional step after each update of the flow solution. For example, in the case of the explicit $$m$$ -stage time-stepping scheme (Subsection 6.1.1 or 6.1.2), the damping step reads

\vec{W}_I^{\,n+1} = \frac{1}{1 + \vartheta\left(H_I^{(m)} - H_\infty\right)}\,\vec{W}_I^{(m)} \tag{9.3}

能量方程除外，它变为en

with the exception of the energy equation which becomes

\left(\rho E\right)_I^{n+1} = \frac{1}{1 + \vartheta}\left[\left(\rho E\right)_I^{(m)} - \vartheta p_I^{(m)}\right]\,. \tag{9.4}

在式(9.3)中， $\vec{W}_I^{(m)}$ 表示 $$m$$ 级显式时间推进格式的最终解。文献[1]中针对绕NACA 0012翼型跨声速流动（ $M_\infty = 0.8$ 、 $\alpha = 1.25^{\circ}$ ）所做的数值实验证实，到达定常态所需的时间步数可以减少约一半。en

In Eq. (9.3), $\vec{W}_I^{(m)}$ denotes the final solution of the $$m$$ -stage explicit time-stepping scheme. Numerical experiments in Ref. [1], which were conducted for a transonic flow past the NACA 0012 airfoil ( $M_\infty = 0.8$ , $\alpha = 1.25^{\circ}$ ), confirmed that the number of time steps to reach the steady state can be reduced by about factor of two.