4.3.2 Flux-Vector Splitting Schemes 通量向量分裂格式[cfd-4-3-2]

通量向量分裂方法可以视为最初等的上风格式ï¼因为它们只考虑波的传播方向。通量向量分裂格式把对流通量向量分解为两部分——或按某些特征变量的符号ï¼或分裂为对流部分与压力部分。著名的Van Leer通量向量分裂格式[39]属于基于特征分解的第一类。遵循第二种思路的有较新的方法ï¼如Liou等人的对流上游分裂方法(AUSM)[40]、[41]ï¼以及Jameson的对流上游分裂压力(CUSP)格式[42]、[43]。类似的方法还有Edwards提出的低耗散通量分裂格式(LDFSS)[44]ï¼以及Rossow的基于马赫数的对流压力分裂(MAPS)格式[45]、[46]。en

The flux-vector splitting methods can be viewed as the first level of upwind schemes, since they account only for the direction of wave propagation. The flux-vector splitting schemes decompose the vector of the convective fluxes into two parts - either according to the sign of certain characteristic variables, or into a convective and a pressure part. The well-known Van Leer's flux-vector splitting scheme [39] belongs to the first category based on characteristic decomposition. The second approach is followed by more recent methods like the Advection Upstream Splitting Method (AUSM) of Liou et al. [40], [41], or the Convective Upwind Split Pressure (CUSP) scheme of Jameson [42], [43], respectively. Further similar approaches are the Low-Diffusion Flux-Splitting Scheme (LDFSS) introduced by Edwards [44], or the Mach number-based Advection Pressure Splitting (MAPS) scheme of Rossow [45], [46].

通量向量分裂格式只能在单元中心格式(4.2.1小节)或采用对偶控制体的单元顶点格式(4.2.3小节)的框架下实现。与采用标量人工耗散的中心格式相比ï¼其优点在于数值开销只是适度增加ï¼而激波与边界层的分辨率却好得多。不过ï¼矩阵耗散格式(方程(4.58))也能给出精度相当的结果[37]。由于某些数值上的困难ï¼多位研究者对基本格式(尤其是AUSM)作了大量改进ï¼相关工作至今仍在继续。下面我们将介绍Van Leer、AUSM与CUSP格式的基础ï¼对最重要的改进给出一些提示ï¼并提供相应文献。en

The flux-vector splitting schemes can be implemented only in the framework of the cell-centred scheme (Subsection 4.2.1), or the cell-vertex scheme with dual control volumes (Subsection 4.2.3). Their advantage can be seen in only a moderately increased numerical effort but a much better resolution of shocks and boundary layers, as compared to the central scheme with scalar artificial dissipation. However, the matrix dissipation scheme (Eq. (4.58)), can also produce results of comparable accuracy [37]. Because of certain numerical difficulties, many modifications to the basic schemes (particularly to AUSM) were devised by various researchers and the development still continues. In the following, we shall present the basics of the Van Leer, AUSM and CUSP schemes, give some hints with respect to the most important modifications, and provide references to the corresponding literature.

Van Leer's Scheme Van Leer格式

Van Leer的通量向量分裂格式[39]基于对流通量的特征分解。该方法在贴体网格上的推广见文献[47]、[48]。对流通量被分裂为正、负两部分ï¼即en

Van Leer's flux-vector splitting scheme [39] is based on characteristic decomposition of the convective fluxes. An extension of the approach to body-fitted grids was presented in [47], [48]. The convective flux is split into a positive and a negative part, i.e.,

\vec{F}_c = \vec{F}_c^{+} + \vec{F}_c^{-}\,, \tag{4.60}

分解依据的是垂直于控制体面的马赫数(例如在 $$(I+1/2)$$ 处——见图4.8)en

according to the Mach number normal to the face of the control volume (e.g., at $$(I+1/2)$$ - see Fig. 4.8)

(M_n)_{I+1/2} = \left(\frac{V}{c}\right)_{I+1/2}, \tag{4.61}

其中 $$V$$ 为逆变量速度(2.22), $$c$$ 为声速。在采用对偶控制体的单元顶点格式情形ï¼单元指标须换成节点指标。流动变量 $\rho$ 、 $$u$$ 、 $$v$$ 、 $$w$$ 与 $$p$$ 的值须先按方程(4.19)或方程(4.39)插值到控制体的面上。然后ï¼正通量用左状态计算ï¼负通量用右状态计算。对流马赫数 $(M_n)_{I+1/2}$ 由以下关系得到[39]en

where $$V$$ represents the contravariant velocity (2.22) and $$c$$ the speed of sound, respectively. In the case of the cell-vertex scheme with dual control volumes, the cell indices have to be replaced by node indices. The values of the flow variables $\rho$ , $$u$$ , $$v$$ , $$w$$ , and $$p$$ , respectively, have to be interpolated first to the faces of the control volume correspondingly to Eq. (4.19) or Eq. (4.39). Then, the positive fluxes are computed with the left state and the negative fluxes with the right state. The advection Mach number $(M_n)_{I+1/2}$ is obtained from the relation [39]

(M_n)_{I+1/2} = M^{+}_{L} + M^{-}_{R}\,, \tag{4.62}

其中分裂马赫数定义为en

where the split Mach numbers are defined as

M^{+}_{L} = \begin{cases} M_L & \text{if } M_L \ge +1\\ \frac{1}{4}(M_L + 1)^2 & \text{if } |M_L| < 1\\ 0 & \text{if } M_L \le -1\,, \end{cases} \tag{4.63}

而en

and

M^{-}_{R} = \begin{cases} 0 & \text{if } M_R \ge +1\\ \frac{1}{4}(M_R - 1)^2 & \text{if } |M_R| < 1\\ M_R & \text{if } M_R \le -1\,. \end{cases} \tag{4.64}

马赫数 $$M_L$$ 与 $$M_R$$ 分别用左、右状态计算ï¼即en

The Mach numbers $$M_L$$ and $$M_R$$ are evaluated using the left and right state, respectively, i.e.,

M_L = \frac{V_L}{c_L}\,, \quad M_R = \frac{V_R}{c_R}\,. \tag{4.65}

当 $$|M_n| < 1$$ (亚声速流动)时ï¼正、负通量部分为en

In the case of $$|M_n| < 1$$ (subsonic flow), the positive and the negative flux parts are given by

\vec{F}_c^{\pm} = \begin{bmatrix} f^{\pm}_{\mathrm{mass}}\\ f^{\pm}_{\mathrm{mass}}\left[n_x(-V \pm 2c)/\gamma + u\right]\\ f^{\pm}_{\mathrm{mass}}\left[n_y(-V \pm 2c)/\gamma + v\right]\\ f^{\pm}_{\mathrm{mass}}\left[n_z(-V \pm 2c)/\gamma + w\right]\\ f^{\pm}_{\mathrm{energy}} \end{bmatrix}. \tag{4.66}

质量与能量通量分量定义为en

The mass and energy flux components are defined as

\begin{aligned} f^{+}_{\mathrm{mass}} &= +\rho_L c_L\,\frac{(M_L + 1)^2}{4}\\ f^{-}_{\mathrm{mass}} &= -\rho_R c_R\,\frac{(M_R - 1)^2}{4}\\ f^{\pm}_{\mathrm{energy}} &= f^{\pm}_{\mathrm{mass}}\left\{\frac{\left[(\gamma - 1)V \pm 2c\right]^2}{2(\gamma^2 - 1)} + \frac{u^2 + v^2 + w^2 - V^2}{2}\right\}_{L/R}. \end{aligned} \tag{4.67}

对于超声速流动ï¼即 $|M_n| \ge 1$ ï¼通量按下式计算en

For supersonic flow, i.e., for $|M_n| \ge 1$ , the fluxes are evaluated from

\begin{aligned} \vec{F}_c^{+} = \vec{F}_c\,, \quad \vec{F}_c^{-} = 0 \quad &\text{if } M_n \ge +1\\ \vec{F}_c^{+} = 0\,, \quad \vec{F}_c^{-} = \vec{F}_c \quad &\text{if } M_n \le -1\,. \end{aligned} \tag{4.68}

左、右状态的计算一般遵循MUSCL方法[29]ï¼即方程(4.46)。若流场含有激波等间断ï¼高阶格式 $\hat{\kappa} = -1$ 、 $\hat{\kappa} = 0$ 与 $\hat{\kappa} = 1/3$ 都需要限制器。更多细节见4.3.5小节。en

The evaluation of the left and right state follows generally the MUSCL approach [29], which is given by Eqs. (4.46). The higher order schemes $\hat{\kappa} = -1$ , $\hat{\kappa} = 0$ and $\hat{\kappa} = 1/3$ , respectively, require a limiter if the flow field contains discontinuities like shocks. More details are provided in Subsection 4.3.5.

Van Leer的通量向量分裂格式在欧拉方程情形下表现非常好。但用Navier-Stokes方程进行的若干研究[49]、[50]表明ï¼动量方程与能量方程中的分裂误差会抹平边界层ï¼并导致驻点温度与壁面温度不准确。为此ï¼文献[51]建议对垂直于边界层方向的动量通量作修改;文献[52]对能量通量提出了类似的补救措施。两项修改合在一起可以消除分裂误差ï¼从而显著提高解的精度[53]。en

The flux-vector splitting scheme of Van Leer performs very well in the case of the Euler equations. But several investigations [49], [50], carried out with the Navier-Stokes equations revealed that splitting errors in the momentum and the energy equations smear the boundary layers and also lead to inaccurate stagnation and wall temperatures. A modification to the momentum flux in the direction normal to the boundary layer was therefore suggested in Ref. [51]. A similar remedy for the energy flux was proposed in Ref. [52]. Both modifications together remove the splitting errors, and hence they improve the solution accuracy considerably [53].

AUSM

对流上游分裂方法(AUSM)由Liou与Steffen[40]以及Liou[54]提出。随后经Wada与Liou[55]修改并更名为AUSMD/V。最后,Liou[41]、[56]提出了称为AUSM+的改进版本。en

The Advection Upstream Splitting Method (AUSM) was introduced by Liou and Steffen [40], and Liou [54]. It was subsequently modified by Wada and Liou [55] and renamed as AUSMD/V. Finally, an improved version termed AUSM+ was presented by Liou [41], [56].

该方法的基本思想基于这样的观察:对流通量向量(2.21)由两个物理上截然不同的部分组成ï¼即对流部分与压力部分en

The underlying idea of the approach is based on the observation that the vector of convective fluxes (2.21) consists of two physically distinct parts, namely the convective and the pressure part

\vec{F}_c = V\begin{bmatrix} \rho\\ \rho u\\ \rho v\\ \rho w\\ \rho H \end{bmatrix} + \begin{bmatrix} 0\\ n_x p\\ n_y p\\ n_z p\\ 0 \end{bmatrix}. \tag{4.69}

方程(4.69)中的第一项代表由逆变量速度 $$V$$ 输运的标量量;相比之下ï¼压力项由声波速度支配。现在的想法是:根据 $$V$$ 的符号(即使在亚声速流动中)以纯上风方式离散对流项ï¼即取左状态或右状态之一;而压力项在亚声速情形下同时包含两个状态ï¼只有在超声速流动中才变为完全上风。en

The first term in Eq. (4.69) represents scalar quantities, which are convected by the contravariant velocity $$V$$ . By contrast, the pressure term is governed by the acoustic wave speed. The idea now is to discretise the convective term in purely upwind manner by taking either the left or the right state, depending on the sign of $$V$$ (even for subsonic flow). On the other hand, the pressure term includes both states in the subsonic case. It becomes fully upwind only for a supersonic flow.

遵循文献[40]的基本AUSMï¼我们由方程(4.61)引入对流马赫数 $(M_n)_{I+1/2}$ 。据此ï¼可以把控制体面 $$(I+1/2)$$ 或 $$(i+1/2)$$ 处的对流通量改写为en

Following the basic AUSM from [40], we introduce an advection Mach number $(M_n)_{I+1/2}$ from Eq. (4.61). Herewith, we can recast the convective flux at the face $$(I+1/2)$$ , or $$(i+1/2)$$ of the control volume, respectively, into

(\vec{F}_c)_{I+1/2} = (M_n)_{I+1/2}\begin{bmatrix} \rho c\\ \rho c u\\ \rho c v\\ \rho c w\\ \rho c H \end{bmatrix}_{L/R} + \begin{bmatrix} 0\\ n_x p\\ n_y p\\ n_z p\\ 0 \end{bmatrix}_{I+1/2}, \tag{4.70}

其中en

where

(\bullet)_{L/R} = \begin{cases} (\bullet)_L & \text{if } M_{I+1/2} \ge 0\\ (\bullet)_R & \text{otherwise.} \end{cases} \tag{4.71}

与Van Leer通量向量分裂格式类似ï¼对流马赫数按关系式(4.62)与(4.63)-(4.65)由左、右分裂马赫数之和求出。左、右状态(流动量: $\rho$ 、 $$u$$ 、 $$v$$ 、 $$w$$ 、 $$p$$ 、 $$H$$ )的计算同样依据方程(4.19)或方程(4.39)分别插值到控制体的面上ï¼插值遵循方程(4.46)给出的MUSCL方法[29]。所有高阶MUSCL格式( $\hat{\kappa} = -1$ 、 $\hat{\kappa} = 0$ 与 $\hat{\kappa} = 1/3$ )在流场包含激波等强梯度时都需要限制器。更多细节请参阅4.3.5小节。en

Similar to Van Leer's flux-vector splitting scheme, the advection Mach number is evaluated as a sum of the left and right split Mach numbers according to the relations (4.62) and (4.63)-(4.65). The computation of the left and right state (flow quantities: $\rho$ , $$u$$ , $$v$$ , $$w$$ , $$p$$ , $$H$$ ) is based again on a separate interpolation to the faces of the control volume according to Eq. (4.19) or Eq. (4.39). The interpolation follows the MUSCL methodology [29], as it is given in Eq. (4.46). All higher-order MUSCL schemes ( $\hat{\kappa} = -1$ , $\hat{\kappa} = 0$ , and $\hat{\kappa} = 1/3$ ) require a limiter, if the flow field contains strong gradients like shocks. Please refer to Subsection 4.3.5 for more details.

控制体面 $$(I+1/2)$$ 处的压力由分裂式[40]得到en

The pressure at the face $$(I+1/2)$$ of the control volume is obtained from the splitting [40]

p_{I+1/2} = p^{+}_{L} + p^{-}_{R} \tag{4.72}

其中分裂压力由文献[39]给出en

with the split pressures given by [39]

p^{+}_{L} = \begin{cases} p_L & \text{if } M_L \ge +1\\ \frac{p_L}{4}(M_L + 1)^2(2 - M_L) & \text{if } |M_L| < 1\\ 0 & \text{if } M_L \le -1\,, \end{cases} \tag{4.73}

而en

and

p^{-}_{R} = \begin{cases} 0 & \text{if } M_R \ge +1\\ \frac{p_R}{4}(M_R - 1)^2(2 + M_R) & \text{if } |M_R| < 1\\ p_R & \text{if } M_R \le -1\,. \end{cases} \tag{4.74}

当 $|M_{L/R}| < 1$ 时ï¼也可以采用如下低阶展开en

It is also possible to use the following lower-order expansion for $|M_{L/R}| < 1$

p^{\pm}_{L/R} = \frac{p_{L/R}}{2}\left(1 \pm M_{L/R}\right). \tag{4.75}

应当指出,AUSM也可以写成如下形式en

It should be noted that we can write AUSM also in the form

\begin{aligned} (\vec{F}_c)_{I+1/2} = {}& \frac{1}{2}(M_n)_{I+1/2}\left\{\begin{bmatrix} \rho c\\ \rho c u\\ \rho c v\\ \rho c w\\ \rho c H \end{bmatrix}_{L} + \begin{bmatrix} \rho c\\ \rho c u\\ \rho c v\\ \rho c w\\ \rho c H \end{bmatrix}_{R}\right\}\\ &- \frac{1}{2}\left|(M_n)_{I+1/2}\right|\left\{\begin{bmatrix} \rho c\\ \rho c u\\ \rho c v\\ \rho c w\\ \rho c H \end{bmatrix}_{R} - \begin{bmatrix} \rho c\\ \rho c u\\ \rho c v\\ \rho c w\\ \rho c H \end{bmatrix}_{L}\right\}\\ &+ \begin{bmatrix} 0\\ n_x(p^{+}_{L} + p^{-}_{R})\\ n_y(p^{+}_{L} + p^{-}_{R})\\ n_z(p^{+}_{L} + p^{-}_{R})\\ 0 \end{bmatrix}. \tag{4.76} \end{aligned}

方程(4.76)右端第一项代表左、右状态的马赫数加权平均——分别类似于通量平均方程(4.15)、(4.20)或(4.35)、(4.40)。第二项具有耗散性质ï¼由标量值 $|(M_n)_{I+1/2}|$ 缩放。en

The first term on the right-hand side of the above Eq. (4.76) represents a Mach number-weighted average of the left and right state - similar to the average of fluxes Eq. (4.15), (4.20) or Eq. (4.35), (4.40), respectively. The second term has a dissipative character. It is scaled by the scalar value $|(M_n)_{I+1/2}|$ .

实践证明,AUSM能清晰分辨强激波ï¼并给出精确的边界层结果。然而ï¼人们发现原始AUSM[40]、[54]在激波处以及流动与网格对齐的情形会产生局部压力振荡[57]。因此文献[57]、[58]建议在激波处切换到Van Leer格式。当对流马赫数 $(M_n)_{I+1/2}$ 趋于零时ï¼方程(4.76)中的耗散项也趋于零ï¼因而任何扰动都无法被格式阻尼。为解决流动对齐问题ï¼文献[57]建议如下修改耗散项的缩放en

AUSM proved to deliver a crisp resolution of strong shocks and accurate results for boundary layers. However, the original AUSM [40], [54] was found to generate local pressure oscillations at shocks and in cases where the flow is aligned with the grid [57]. In [57], [58] it was therefore suggested to switch at shocks to Van Leer's scheme. When the advection Mach number $(M_n)_{I+1/2}$ tends to zero, the dissipation term in Eq. (4.76) will approach zero as well. Thus, any disturbances cannot be damped by the scheme. In order to solve the flow alignment problem, it was proposed in [57] to modify the scaling of the dissipation term as follows

\left|(M_n)_{I+1/2}\right| = \begin{cases} \left|(M_n)_{I+1/2}\right| & \text{if } \left|(M_n)_{I+1/2}\right| > \delta\\ \dfrac{(M_n)^2_{I+1/2} + \delta^2}{2\delta} & \text{if } \left|(M_n)_{I+1/2}\right| \le \delta\,, \end{cases} \tag{4.77}

其中 $\delta$ 是一个小值 $(0 < \delta \le 0.5)$ 。这样ï¼数值耗散总是足够的。为了保持AUSM对边界层的精度ï¼可以按照与中心格式方程(4.57)相同的思路ï¼在壁面法向减小参数 $\delta$ 。文献[41]、[56]针对激波附近更好的表现提出了基本AUSM的进一步改进ï¼称为AUSM+。这些修改包括新的马赫数分裂与压力分裂ï¼分别取代关系式(4.63)、(4.64)与(4.73)、(4.74)。en

where $\delta$ is a small value $(0 < \delta \le 0.5)$ . Hence, there will always be a sufficient amount of numerical dissipation. In order to retain the accuracy of AUSM for boundary layers, the parameter $\delta$ could be reduced in the wall normal direction using the same idea as given for the central scheme by Eq. (4.57). Further improvements of the basic AUSM, with respect to better behaviour in the vicinity of shocks, was presented in [41], [56] as AUSM+. The modifications consist of new Mach and pressure splittings, which replace the relations (4.63), (4.64) and (4.73), (4.74), respectively.

CUSP Scheme CUSP格式

对流上游分裂压力(CUSP)格式的概念与AUSM十分相似。不过,CUSP方法的优点是可以表述为通量平均(但不像AUSM那样加权)减去一个耗散项。这一特点对于在显式混合多步格式中的实现至关重要。此外ï¼由于缩放因子与AUSM不同,CUSP格式在流动对齐情形下表现更为有利。CUSP格式由Jameson[42]、[59]、[60]提出ï¼随后由Tatsumi等人[43]、[61]修改。它既可在单元中心型空间离散中实现ï¼也可在(单元顶点)对偶控制体型空间离散中实现。en

The concept of the Convective Upwind Split Pressure (CUSP) scheme is quite similar to that of AUSM. However, the CUSP approach has the advantage to be formulated as an average of fluxes (but without weighting like within AUSM) minus a dissipation term. This feature is crucial for the implementation in an explicit, hybrid multistage scheme. Furthermore, because of the different scaling factors as compared to AUSM, the CUSP scheme behaves more favourably in the case of flow alignment. The CUSP scheme was introduced by Jameson [42], [59], [60], and subsequently modified by Tatsumi et al. [43], [61]. It can be implemented either within the cell-centred or the (cell-vertex) dual control-volume type of spatial discretisation.

通过控制体面的对流通量(方程(2.21))用通量的算术平均近似ï¼分别依照方程(4.15)、(4.20)、(4.35)或(4.40)。然后ï¼为使格式稳定ï¼从中心通量中减去耗散项。于是ï¼面 $$(I+1/2)$$ 处的总对流通量为en

The convective fluxes (Eq. (2.21)) through a face of the control volume are approximated using the arithmetic average of fluxes according to the Equations (4.15), (4.20), (4.35), or (4.40), respectively. The dissipation term is then subtracted from the central fluxes for stabilisation. Thus, the total convective fluxes at the face $$(I+1/2)$$ read

(\vec{F}_c)_{I+1/2} = \frac{1}{2}\left[\vec{F}_c(\vec{W}_R) + \vec{F}_c(\vec{W}_L)\right] - \vec{D}_{I+1/2}. \tag{4.78}

在对偶控制体离散的情形ï¼则相应使用 $$(i+1/2)$$ 。耗散项由状态向量之差与通量向量之差的线性组合构成ï¼可表示为en

In the case of the dual control-volume discretisation, $$(i+1/2)$$ would apply instead. The dissipation term, which is composed of a linear combination of the differences of the state and the flux vector, can be expressed as

\begin{aligned} \vec{D}_{I+1/2} = {}& \frac{1}{2}(\alpha^{*}c)_{I+1/2}\left\{\begin{bmatrix} \rho\\ \rho u\\ \rho v\\ \rho w\\ \rho\phi \end{bmatrix}_{R} - \begin{bmatrix} \rho\\ \rho u\\ \rho v\\ \rho w\\ \rho\phi \end{bmatrix}_{L}\right\}\\ &+ \frac{1}{2}\beta_{I+1/2}\left\{\begin{bmatrix} \rho V\\ \rho uV + n_x p\\ \rho vV + n_y p\\ \rho wV + n_z p\\ \rho HV \end{bmatrix}_{R} - \begin{bmatrix} \rho V\\ \rho uV + n_x p\\ \rho vV + n_y p\\ \rho wV + n_z p\\ \rho HV \end{bmatrix}_{L}\right\}. \tag{4.79} \end{aligned}

方程(4.79)中的 $\phi$ 项有两种选择:或取总能量 $$E$$ ï¼或取总焓 $$H$$ 。第一种情形称为E-CUSP格式[62]ï¼第二种相应称为H-CUSP格式。 $\phi = H$ 的表述保持总焓不变[43]ï¼因而适用于无黏流动。左( $$L$$ )、右( $$R$$ )状态的计算与MUSCL方法[29]类似ï¼采用限制插值(4.118)-(4.121)。方程(4.79)中的两个因子 $\alpha^{*}c$ 与 $\beta$ 定义为en

There are two choices for the term $\phi$ in Eq. (4.79). Either, $\phi$ is set equal to the total energy $$E$$ , or it is set to the total enthalpy $$H$$ . In the first case we speak of the E-CUSP scheme [62], the second choice is consequently called the H-CUSP scheme. The formulation with $\phi = H$ preserves the total enthalpy [43] and is therefore suitable for inviscid flows. The left ( $$L$$ ) and right ( $$R$$ ) state is evaluated similarly to the MUSCL approach [29], using the limited interpolation (4.118)-(4.121). The two factors $\alpha^{*}c$ and $\beta$ in Eq. (4.79) are defined as

\alpha^{*}c = \begin{cases} |V| & \text{if } \beta = 0\\ -(1 + \beta)\Lambda^{-} & \text{if } \beta > 0 \text{ and } 0 < M_n < 1\\ +(1 - \beta)\Lambda^{+} & \text{if } \beta < 0 \text{ and } -1 < M_n < 0\\ 0 & \text{if } |M_n| \ge 1 \end{cases} \tag{4.80}

而en

and

\beta = \begin{cases} +\max\left(0,\,\dfrac{V + \Lambda^{-}}{V - \Lambda^{-}}\right) & \text{if } 0 \le M_n < 1\\ -\max\left(0,\,\dfrac{V + \Lambda^{+}}{V - \Lambda^{+}}\right) & \text{if } -1 < M_n < 0\\ \mathrm{sign}(M_n) & \text{if } |M_n| \ge 1 \end{cases} \tag{4.81}

其中 $$M_n = V/c$$ 。上述公式(4.80)与(4.81)中的逆变量速度 $$V$$ 由速度分量的算术平均计算ï¼即en

with $$M_n = V/c$$ . The contravariant velocity $$V$$ in the above formulae (4.80) and (4.81) is computed from the arithmetic mean of the velocity components, i.e.,

V_{I+1/2} = \frac{1}{2}\left[(u_I + u_{I+1})n_x + (v_I + v_{I+1})n_y + (w_I + w_{I+1})n_z\right]. \tag{4.82}

计算 $$M_n$$ 所用的声速 $$c$$ 同样由算术平均量得到。方程(4.80)、(4.81)中的正、负特征值 $\Lambda^{+}$ 与 $\Lambda^{-}$ 是所谓Roe矩阵(Roe matrix)[63]的特征值,Roe矩阵将在4.3.3小节讨论。特征值为[60]en

The speed of sound $$c$$ in the evaluation of $$M_n$$ is also obtained from arithmetically averaged quantities. The positive and negative eigenvalues $\Lambda^{+}$ and $\Lambda^{-}$ in Eqs. (4.80), (4.81) are those of the so-called Roe matrix [63], which will be discussed in the Subsection 4.3.3. The eigenvalues are given by [60]

\Lambda^{\pm} = \frac{\gamma + 1}{2\gamma}\tilde{V} \pm \sqrt{\left(\frac{\gamma - 1}{2\gamma}\tilde{V}\right)^2 + \frac{\tilde{c}^2}{\gamma}}\,, \tag{4.83}

其中 $\tilde{V}$ 表示逆变量速度, $\gamma$ 为比热比, $\tilde{c}$ 表示声速。方程(4.83)中的流动变量须在控制体面上计算ï¼采用所谓的Roe平均(Roe averages)[63]得到en

where $\tilde{V}$ denotes the contravariant velocity, $\gamma$ is the ratio of specific heat coefficients and $\tilde{c}$ stands for the speed of sound. The flow variables in Eq. (4.83), which have to be evaluated at the faces of the control volumes, are obtained using the so-called Roe averages [63]

\begin{aligned} \tilde{u}_{I+1/2} &= \frac{u_L\sqrt{\rho_L} + u_R\sqrt{\rho_R}}{\sqrt{\rho_L} + \sqrt{\rho_R}}\\ \tilde{v}_{I+1/2} &= \frac{v_L\sqrt{\rho_L} + v_R\sqrt{\rho_R}}{\sqrt{\rho_L} + \sqrt{\rho_R}}\\ \tilde{w}_{I+1/2} &= \frac{w_L\sqrt{\rho_L} + w_R\sqrt{\rho_R}}{\sqrt{\rho_L} + \sqrt{\rho_R}}\\ \tilde{H}_{I+1/2} &= \frac{H_L\sqrt{\rho_L} + H_R\sqrt{\rho_R}}{\sqrt{\rho_L} + \sqrt{\rho_R}}\\ \tilde{c}_{I+1/2} &= \sqrt{(\gamma - 1)\left(\tilde{H} - \frac{\tilde{u}^2 + \tilde{v}^2 + \tilde{w}^2}{2}\right)_{I+1/2}}\\ \tilde{V}_{I+1/2} &= \tilde{u}_{I+1/2}n_x + \tilde{v}_{I+1/2}n_y + \tilde{w}_{I+1/2}n_z\,. \end{aligned} \tag{4.84}

因子 $\alpha^{*}c$ 与 $\beta$ 的定义使得:对超声速流动ï¼对流通量完全上风ï¼即 $\alpha^{*}c = 0$ 且 $\beta = \mathrm{sign}(M_n)$ ;而在亚声速流动( $\beta = 0$ )中ï¼耗散由 $$|V|$$ 缩放。这对黏性层的计算是一个理想的性质。在大长宽比单元的情形ï¼为保持稳健性ï¼显式时间推进格式通常需要沿单元较长一侧方向增大数值耗散。这可以通过采用类似方程(4.57)的谱半径之比来实现。更多细节见文献[37]ï¼该文献还包含CUSP格式与标量及矩阵人工耗散格式(4.3.1小节)的比较。en

The factors $\alpha^{*}c$ and $\beta$ are defined such that full upwinding of the convective fluxes results for supersonic flow, i.e., $\alpha^{*}c = 0$ and $\beta = \mathrm{sign}(M_n)$ . On the other hand, in subsonic flow (when $\beta = 0$ ) the dissipation is scaled by $$|V|$$ . This is a desirable property for the computation of viscous layers. In cases of large aspect ratio cells, explicit time-stepping schemes usually require increased numerical dissipation in the direction of the longer cell side in order to stay robust. This can be accomplished by employing ratios of the spectral radii similar to Eq. (4.57). More details can be found in Ref. [37], which also contains comparisons between the CUSP scheme and the scalar as well as the matrix artificial dissipation (Subsection 4.3.1) scheme.

把方程(4.84)的Roe平均换成算术平均、方程(4.83)的特征值换成 $\Lambda^{\pm} = V \pm c$ ï¼可以大大简化CUSP格式的实现。引入定义en

The implementation of the CUSP scheme can be considerably simplified by replacing the Roe averages from Eq. (4.84) by arithmetic averages and the eigenvalues from Eq. (4.83) by $\Lambda^{\pm} = V \pm c$ . Introducing the definition

\alpha^{*}c = \alpha c - \beta V\,, \tag{4.85}

修改后格式的因子为[60]、[43]en

the factors of the modified scheme read [60], [43]

\alpha = \begin{cases} |M_n| & \text{if } |M_n| \ge \delta\\ \dfrac{M_n^2 + \delta^2}{2\delta} & \text{if } |M_n| < \delta\,, \end{cases} \tag{4.86}

而en

and

\beta = \begin{cases} \max(0,\, 2M_n - 1) & \text{if } 0 \le M_n < 1\\ \min(0,\, 2M_n + 1) & \text{if } -1 < M_n < 0\\ \mathrm{sign}(M_n) & \text{if } |M_n| \ge 1\,. \end{cases} \tag{4.87}

$$M_n = V/c$$ 中的逆变量速度仍按方程(4.82)计算。方程(4.86)中的参数 $\delta$ 意在防止耗散在驻点处消失ï¼但这似乎并不总是必要。上述简化得到计算上非常高效的格式ï¼不过与采用Roe平均的原始表述相比ï¼激波分辨率略有下降。en

The contravariant velocity in $$M_n = V/c$$ is still computed as indicated in Eq. (4.82). The parameter $\delta$ in Eq. (4.86) is intended to prevent the dissipation from disappearing at stagnation points, but this does not seem to be always necessary. The above simplifications lead to a computationally very efficient scheme, however the shock resolution is slightly reduced as compared to the original formulation with Roe averages.