Hegedus Aerodynamics

Note: the equations below require MathML to be displayed.

Starting Equations

The following was obtained from reference (1).

$${\widehat{F}}_v=\frac{1}{J}\left[\begin{array}{c}0\\ {\xi }_x{\tau }_{xx}+{\xi }_y{\tau }_{xy}+{\xi }_z{\tau }_{xz}\\ {\xi }_x{\tau }_{xy}+{\xi }_y{\tau }_{yy}+{\xi }_z{\tau }_{yz}\\ {\xi }_x{\tau }_{xz}+{\xi }_y{\tau }_{yz}+{\xi }_z{\tau }_{zz}\\ {\xi }_x{b}_x+{\xi }_y{b}_y+{\xi }_z{b}_z\end{array}\right]\text{,}{\widehat{G}}_v=\frac{1}{J}\left[\begin{array}{c}0\\ {\eta }_x{\tau }_{xx}+{\eta }_y{\tau }_{xy}+{\eta }_z{\tau }_{xz}\\ {\eta }_x{\tau }_{xy}+{\eta }_y{\tau }_{yy}+{\eta }_z{\tau }_{yz}\\ {\eta }_x{\tau }_{xz}+{\eta }_y{\tau }_{yz}+{\eta }_z{\tau }_{zz}\\ {\eta }_x{b}_x+{\eta }_y{b}_y+{\eta }_z{b}_z\end{array}\right]\text{,}{\widehat{H}}_v=\frac{1}{J}\left[\begin{array}{c}0\\ {\zeta }_x{\tau }_{xx}+{\zeta }_y{\tau }_{xy}+{\zeta }_z{\tau }_{xz}\\ {\zeta }_x{\tau }_{xy}+{\zeta }_y{\tau }_{yy}+{\zeta }_z{\tau }_{yz}\\ {\zeta }_x{\tau }_{xz}+{\zeta }_y{\tau }_{yz}+{\zeta }_z{\tau }_{zz}\\ {\zeta }_x{b}_x+{\zeta }_y{b}_y+{\zeta }_z{b}_z\end{array}\right]$$ $$b_x=u{\tau }_{xx}+v{\tau }_{xy}+w{\tau }_{xz}-q_x$$ $$b_y=u{\tau }_{xy}+v{\tau }_{yy}+w{\tau }_{yz}-q_y$$ $$b_z=u{\tau }_{xz}+v{\tau }_{yz}+w{\tau }_{zz}-q_z$$ $${\tau }_{xx}=\frac{2}{3}\mu \left(2\frac{\partial u}{\partial x}-\frac{\partial v}{\partial y}-\frac{\partial w}{\partial z}\right)$$ $${\tau }_{yy}=\frac{2}{3}\mu \left(2\frac{\partial v}{\partial y}-\frac{\partial u}{\partial x}-\frac{\partial w}{\partial z}\right)$$ $${\tau }_{zz}=\frac{2}{3}\mu \left(2\frac{\partial w}{\partial z}-\frac{\partial u}{\partial x}-\frac{\partial v}{\partial y}\right)$$ $${\tau }_{xy}=\mu \left(\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}\right)$$ $${\tau }_{xz}=\mu \left(\frac{\partial w}{\partial x}+\frac{\partial u}{\partial z}\right)$$ $${\tau }_{xz}=\mu \left(\frac{\partial w}{\partial x}+\frac{\partial u}{\partial z}\right)$$ $${\tau }_{yz}=\mu \left(\frac{\partial v}{\partial z}+\frac{\partial w}{\partial y}\right)$$ $$q_x=-\kappa \frac{\partial a^2}{\partial x}\text{,}{q}_y=-\kappa \frac{\partial a^2}{\partial y}\text{,}{q}_z=-\kappa \frac{\partial a^2}{\partial z}$$ $$\frac{\partial }{\partial x}={\xi }_x\frac{\partial }{\partial \xi }+{\eta }_x\frac{\partial }{\partial \eta }+{\varsigma }_x\frac{\partial }{\partial \zeta }$$ $$\frac{\partial }{\partial y}={\xi }_y\frac{\partial }{\partial \xi }+{\eta }_y\frac{\partial }{\partial \eta }+{\varsigma }_y\frac{\partial }{\partial \zeta }$$ $$\frac{\partial }{\partial z}={\xi }_z\frac{\partial }{\partial \xi }+{\eta }_z\frac{\partial }{\partial \eta }+{\varsigma }_z\frac{\partial }{\partial \zeta }$$ $$\kappa =\frac{\mu }{\mathrm{Pr}\left(\gamma -1\right)}$$

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3D Vector From

The above equations are placed in vector form to be used in Aero Troll CFD.  The derivation is shown here.

$${\widehat{F}}_v=\frac{1}{J}\left[\begin{array}{c}0\\ \begin{array}{cc}{\overrightarrow{\widehat{f}}}_{v\text{,}\mathrm{\rho U}}& \end{array}\\ {\widehat{f}}_{v\text{,}{\mathrm{\rho e}}_0}\end{array}\right]\text{,}{\widehat{G}}_v=\frac{1}{J}\left[\begin{array}{c}0\\ \begin{array}{cc}{\overrightarrow{\widehat{g}}}_{v\text{,}\mathrm{\rho U}}& \end{array}\\ {\widehat{g}}_{v\text{,}{\mathrm{\rho e}}_0}\end{array}\right]\text{,}{\widehat{H}}_v=\frac{1}{J}\left[\begin{array}{c}0\\ \begin{array}{cc}{\overrightarrow{\widehat{h}}}_{v\text{,}\mathrm{\rho U}}& \end{array}\\ {\widehat{h}}_{v\text{,}{\mathrm{\rho e}}_0}\end{array}\right]$$ $${\overrightarrow{\widehat{f}}}_{v\text{,}\mathrm{\rho U}}=\mu \left[\overrightarrow{\xi }\frac{1}{3}\left(\overrightarrow{\xi }·\Delta {\overrightarrow{U}}_{\xi }+\overrightarrow{\eta }·\Delta {\overrightarrow{U}}_{\eta }+\overrightarrow{\zeta }·\Delta {\overrightarrow{U}}_{\zeta }\right)+\left(\overrightarrow{\xi }·\overrightarrow{\xi }\right)\Delta {\overrightarrow{U}}_{\xi }+\left(\overrightarrow{\xi }·\overrightarrow{\eta }\right)\Delta {\overrightarrow{U}}_{\eta }+\left(\overrightarrow{\xi }·\overrightarrow{\zeta }\right)\Delta {\overrightarrow{U}}_{\zeta }+\Delta {\overrightarrow{U}}_{\eta }\times \left(\overrightarrow{\eta }\times \overrightarrow{\xi }\right)+\Delta {\overrightarrow{U}}_{\zeta }\times \left(\overrightarrow{\zeta }\times \overrightarrow{\xi }\right)\right]$$ $${\widehat{f}}_{v\text{,}{\mathrm{\rho e}}_0}={\overrightarrow{\widehat{f}}}_{v\text{,}\mathrm{\rho U}}·\overrightarrow{U}+\kappa \left(\left(\overrightarrow{\xi }·\overrightarrow{\xi }\right)\frac{\partial a^2}{\partial \xi }+\left(\overrightarrow{\xi }·\overrightarrow{\eta }\right)\frac{\partial a^2}{\partial \eta }+\left(\overrightarrow{\xi }·\overrightarrow{\zeta }\right)\frac{\partial a^2}{\partial \zeta }\right)$$ $${\overrightarrow{\widehat{g}}}_{v\text{,}\mathrm{\rho U}}=\mu \left[\overrightarrow{\eta }\frac{1}{3}\left(\overrightarrow{\xi }·\Delta {\overrightarrow{U}}_{\xi }+\overrightarrow{\eta }·\Delta {\overrightarrow{U}}_{\eta }+\overrightarrow{\zeta }·\Delta {\overrightarrow{U}}_{\zeta }\right)+\left(\overrightarrow{\eta }·\overrightarrow{\xi }\right)\Delta {\overrightarrow{U}}_{\xi }+\left(\overrightarrow{\eta }·\overrightarrow{\eta }\right)\Delta {\overrightarrow{U}}_{\eta }+\left(\overrightarrow{\eta }·\overrightarrow{\zeta }\right)\Delta {\overrightarrow{U}}_{\zeta }+\Delta {\overrightarrow{U}}_{\xi }\times \left(\overrightarrow{\xi }\times \overrightarrow{\eta }\right)+\Delta {\overrightarrow{U}}_{\zeta }\times \left(\overrightarrow{\zeta }\times \overrightarrow{\eta }\right)\right]$$ $${\widehat{g}}_{v\text{,}{\mathrm{\rho e}}_0}={\overrightarrow{\widehat{g}}}_{v\text{,}\mathrm{\rho U}}·\overrightarrow{U}+\kappa \left(\left(\overrightarrow{\eta }·\overrightarrow{\xi }\right)\frac{\partial a^2}{\partial \xi }+\left(\overrightarrow{\eta }·\overrightarrow{\eta }\right)\frac{\partial a^2}{\partial \eta }+\left(\overrightarrow{\eta }·\overrightarrow{\zeta }\right)\frac{\partial a^2}{\partial \zeta }\right)$$ $${\overrightarrow{\widehat{h}}}_{v\text{,}\mathrm{\rho U}}=\mu \left[\overrightarrow{\zeta }\frac{1}{3}\left(\overrightarrow{\xi }·\Delta {\overrightarrow{U}}_{\xi }+\overrightarrow{\eta }·\Delta {\overrightarrow{U}}_{\eta }+\overrightarrow{\zeta }·\Delta {\overrightarrow{U}}_{\zeta }\right)+\left(\overrightarrow{\zeta }·\overrightarrow{\xi }\right)\Delta {\overrightarrow{U}}_{\xi }+\left(\overrightarrow{\zeta }·\overrightarrow{\eta }\right)\Delta {\overrightarrow{U}}_{\eta }+\left(\overrightarrow{\zeta }·\overrightarrow{\zeta }\right)\Delta {\overrightarrow{U}}_{\zeta }+\Delta {\overrightarrow{U}}_{\xi }\times \left(\overrightarrow{\xi }\times \overrightarrow{\zeta }\right)+\Delta {\overrightarrow{U}}_{\eta }\times \left(\overrightarrow{\eta }\times \overrightarrow{\zeta }\right)\right]$$ $${\widehat{h}}_{v\text{,}{\mathrm{\rho e}}_0}={\overrightarrow{\widehat{h}}}_{v\text{,}\mathrm{\rho U}}·\overrightarrow{U}+\kappa \left(\left(\overrightarrow{\zeta }·\overrightarrow{\xi }\right)\frac{\partial a^2}{\partial \xi }+\left(\overrightarrow{\zeta }·\overrightarrow{\eta }\right)\frac{\partial a^2}{\partial \eta }+\left(\overrightarrow{\zeta }·\overrightarrow{\zeta }\right)\frac{\partial a^2}{\partial \zeta }\right)$$

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Axisymmetric

The axisymmetric equations in the ξ and ζ are used from the 3D set, except $\Delta {\overrightarrow{U}}_{\eta }$ is replaced with the following:

$$\Delta {\overrightarrow{U}}_{\eta }=\overrightarrow{U}\times \Delta \overrightarrow{\Theta }$$ where $\Delta \overrightarrow{\Theta }$ is along the axis.

The equations in the η direction are:

$${\Delta }_{\eta }{\overrightarrow{\widehat{g}}}_{v\text{,}\mathrm{\rho v}}=-{\mu }_{i\text{,}j\text{,}k}\left[\frac{1}{2}\frac{\left({\overrightarrow{S}}_{i+1\text{,}j\text{,}k}^{\xi }-{\overrightarrow{S}}_{i-1\text{,}j\text{,}k}^{\xi }\right)+\left({\overrightarrow{S}}_{i\text{,}jk+1}^{\zeta }-{\overrightarrow{S}}_{i\text{,}j\text{,}k-1}^{\zeta }\right)}{{\mathrm{vol}}_{i\text{,}j\text{,}k}}\frac{2}{3}{\left(-\overrightarrow{\xi }·\Delta {\overrightarrow{U}}_{\xi }+2\left(\Delta \overrightarrow{\Theta }\times \overrightarrow{\eta }\right)·\overrightarrow{U}-\overrightarrow{\zeta }·\Delta {\overrightarrow{U}}_{\zeta }\right)}_{i\text{,}j\text{,}k}\right]$$ $${\Delta }_{\eta }{\widehat{g}}_{v\text{,}{\mathrm{\rho e}}_0}=0$$ Where $\overrightarrow{S}$ is the area vector for a given cell side (η in this case) and vol is the volume of the cell.

The following were assumed to derive the η equation set.

$$\overrightarrow{\eta }·\overrightarrow{\xi }=0$$ $$\overrightarrow{\eta }·\overrightarrow{\zeta }=0$$ $$\overrightarrow{\eta }\times \Delta {\overrightarrow{U}}_{\eta }=0$$ $$\overrightarrow{\eta }·\Delta {\overrightarrow{U}}_{\xi }=0$$ $$\overrightarrow{\eta }·\Delta {\overrightarrow{U}}_{\zeta }=0$$ $$\overrightarrow{\xi }·\Delta {\overrightarrow{U}}_{\eta }=0$$ $$\overrightarrow{\zeta }·\Delta {\overrightarrow{U}}_{\eta }=0$$ The derivation is shown here.

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2D

The 2D set is as follows:

$${\overrightarrow{\widehat{f}}}_{v\text{,}\mathrm{\rho U}}=\mu \left[\overrightarrow{\xi }\frac{1}{3}\left(\overrightarrow{\xi }·\Delta {\overrightarrow{U}}_{\xi }+\overrightarrow{\zeta }·\Delta {\overrightarrow{U}}_{\zeta }\right)+\left(\overrightarrow{\xi }·\overrightarrow{\xi }\right)\Delta {\overrightarrow{U}}_{\xi }+\left(\overrightarrow{\xi }·\overrightarrow{\zeta }\right)\Delta {\overrightarrow{U}}_{\zeta }+\Delta {\overrightarrow{U}}_{\zeta }\times \left(\overrightarrow{\zeta }\times \overrightarrow{\xi }\right)\right]$$ $${\widehat{f}}_{v\text{,}{\mathrm{\rho e}}_0}={\overrightarrow{\widehat{f}}}_{v\text{,}\mathrm{\rho U}}·\overrightarrow{U}+\kappa \left(\left(\overrightarrow{\xi }·\overrightarrow{\xi }\right)\frac{\partial a^2}{\partial \xi }+\left(\overrightarrow{\xi }·\overrightarrow{\zeta }\right)\frac{\partial a^2}{\partial \zeta }\right)$$ $${\Delta }_{\eta }{\widehat{g}}_{v\text{,}\mathrm{\rho U}}=0$$ $${\Delta }_{\eta }{\widehat{g}}_{v\text{,}{\mathrm{\rho e}}_0}=0$$ $${\overrightarrow{\widehat{h}}}_{v\text{,}\mathrm{\rho U}}=\mu \left[\overrightarrow{\zeta }\frac{1}{3}\left(\overrightarrow{\xi }·\Delta {\overrightarrow{U}}_{\xi }+\overrightarrow{\zeta }·\Delta {\overrightarrow{U}}_{\zeta }\right)+\left(\overrightarrow{\zeta }·\overrightarrow{\xi }\right)\Delta {\overrightarrow{U}}_{\xi }+\left(\overrightarrow{\zeta }·\overrightarrow{\zeta }\right)\Delta {\overrightarrow{U}}_{\zeta }+\Delta {\overrightarrow{U}}_{\xi }\times \left(\overrightarrow{\xi }\times \overrightarrow{\zeta }\right)\right]$$ $${\widehat{h}}_{v\text{,}{\mathrm{\rho e}}_0}={\overrightarrow{\widehat{h}}}_{v\text{,}\mathrm{\rho U}}·\overrightarrow{U}+\kappa \left(\left(\overrightarrow{\zeta }·\overrightarrow{\xi }\right)\frac{\partial a^2}{\partial \xi }+\left(\overrightarrow{\zeta }·\overrightarrow{\zeta }\right)\frac{\partial a^2}{\partial \zeta }\right)$$

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3D Thin Layer Set

$${\overrightarrow{\widehat{f}}}_{v\text{,}\mathrm{\rho U}}=\mu \left[\overrightarrow{\xi }\frac{1}{3}\left(\overrightarrow{\xi }·\Delta {\overrightarrow{U}}_{\xi }\right)+\left(\overrightarrow{\xi }·\overrightarrow{\xi }\right)\Delta {\overrightarrow{U}}_{\xi }\right]$$ $${\widehat{f}}_{v\text{,}{\mathrm{\rho e}}_0}={\overrightarrow{\widehat{f}}}_{v\text{,}\mathrm{\rho U}}·\overrightarrow{U}+\kappa \left(\overrightarrow{\xi }·\overrightarrow{\xi }\right)\frac{\partial a^2}{\partial \xi }$$ $${\overrightarrow{\widehat{g}}}_{v\text{,}\mathrm{\rho U}}=\mu \left[\overrightarrow{\eta }\frac{1}{3}\left(\overrightarrow{\eta }·\Delta {\overrightarrow{U}}_{\eta }\right)+\left(\overrightarrow{\eta }·\overrightarrow{\eta }\right)\Delta {\overrightarrow{U}}_{\eta }\right]$$ $${\widehat{g}}_{v\text{,}{\mathrm{\rho e}}_0}={\overrightarrow{\widehat{g}}}_{v\text{,}\mathrm{\rho U}}·\overrightarrow{U}+\kappa \left(\overrightarrow{\eta }·\overrightarrow{\eta }\right)\frac{\partial a^2}{\partial \eta }$$ $${\overrightarrow{\widehat{h}}}_{v\text{,}\rho U}=\mu \left[\overrightarrow{\zeta }\frac{1}{3}\left(\overrightarrow{\zeta }·\Delta {\overrightarrow{U}}_{\zeta }\right)+\left(\overrightarrow{\zeta }·\overrightarrow{\zeta }\right)\Delta {\overrightarrow{U}}_{\zeta }\right]$$ $${\widehat{h}}_{v\text{,}{\mathrm{\rho e}}_0}={\overrightarrow{\widehat{h}}}_{v\text{,}\mathrm{\rho U}}·\overrightarrow{U}+\kappa \left(\overrightarrow{\zeta }·\overrightarrow{\zeta }\right)\frac{\partial a^2}{\partial \zeta }$$ The derivation is shown here.

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Viscous Jacobian

The viscous Jacobians are derived from the 3D thin layer set.

$$\frac{\partial {\widehat{F}}_v}{\partial Q}=\mu \left(\overrightarrow{k}·\overrightarrow{k}\right)\frac{1}{\rho }\left[\begin{array}{ccccc}0& 0& 0& 0& 0\\ -\left(\frac{1}{3}{\tilde{k}}_x\left(\overrightarrow{\tilde{k}}·\overrightarrow{U}\right)+u\right)& \frac{1}{3}{\tilde{k}}_x{\tilde{k}}_x+1& \frac{1}{3}{\tilde{k}}_x{\tilde{k}}_y& \frac{1}{3}{\tilde{k}}_x{\tilde{k}}_z& 0\\ -\left(\frac{1}{3}{\tilde{k}}_y\left(\overrightarrow{\tilde{k}}·\overrightarrow{U}\right)+v\right)& \frac{1}{3}{\tilde{k}}_y{\tilde{k}}_x& \frac{1}{3}{\tilde{k}}_y{\tilde{k}}_y+1& \frac{1}{3}{\tilde{k}}_y{\tilde{k}}_z& 0\\ -\left(\frac{1}{3}{\tilde{k}}_z\left(\overrightarrow{\tilde{k}}·\overrightarrow{U}\right)+w\right)& \frac{1}{3}{\tilde{k}}_z{\tilde{k}}_x& \frac{1}{3}{\tilde{k}}_z{\tilde{k}}_y& \frac{1}{3}{\tilde{k}}_z{\tilde{k}}_z+1& 0\\ -\left(\frac{1}{3}{\left(\overrightarrow{\tilde{k}}·\overrightarrow{U}\right)}^2+\overrightarrow{U}·\overrightarrow{U}\right)& \left(\frac{1}{3}{\tilde{k}}_x\left(\overrightarrow{\tilde{k}}·\overrightarrow{U}\right)+u\right)& \left(\frac{1}{3}{\tilde{k}}_y\left(\overrightarrow{\tilde{k}}·\overrightarrow{U}\right)+v\right)& \left(\frac{1}{3}{\tilde{k}}_z\left(\overrightarrow{\tilde{k}}·\overrightarrow{U}\right)+w\right)& 0\end{array}\right]+\frac{\mu }{\mathrm{Pr}}\left(\overrightarrow{k}·\overrightarrow{k}\right)\frac{\gamma }{\rho }\left[\begin{array}{ccccc}0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\\ -\left(\frac{\left(\rho e_0\right)}{\rho }-\overrightarrow{U}·\overrightarrow{U}\right)& -u& -v& -w& 1\end{array}\right]$$ The eigenvalues for this are:

$$\lambda =\frac{\mu \left(\overrightarrow{k}·\overrightarrow{k}\right)}{\rho }\left(0\text{,}1\text{,}1\text{,}\frac{4}{3}\text{,}\frac{\gamma }{\mathrm{Pr}}\right)$$ The derivation is shown here.

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Diagonal Viscous Jacobian

Two forms of the diagonal viscous Jacobian matrix are tested.  The first form is determined from the maximum viscous Jacobian eigenvalue (2).

$$\frac{\partial {\widehat{F}}_v}{\partial Q}=\frac{\mu \left(\overrightarrow{k}·\overrightarrow{k}\right)}{\rho }\max \left(\frac{4}{3}\text{,}\frac{\gamma }{\mathrm{Pr}}\right)\text{I}$$ The second form is derived from the eigenvalues of the transformed viscous Jacobian.

$$\frac{\partial {\widehat{F}}_v}{\partial Q}=\frac{\mu \left(\overrightarrow{k}·\overrightarrow{k}\right)}{\rho }\left(\left[\begin{array}{ccccc}1& 0& 0& 0& 0\\ 0& 1& 0& 0& 0\\ 0& 0& 1& 0& 0\\ 0& 0& 0& \frac{4}{3}& 0\\ 0& 0& 0& 0& \frac{4}{3}\end{array}\right]+\frac{2-\gamma }{\mathrm{Pr}}I\right)$$ The derivation for the eigenvalues of the transformed viscous Jacobian is shown here.

After some testing, the first form was chosen for Aero Troll CFD.  The first form did consistently better.  However, it should be noted that the diagonal viscous Jacobian drops cross coupling and therefore the DADI method using the diagonal viscous Jacobian does noticeably worse for some cases in regards to convergence than the ADI method using the viscous Jacobian.

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Spatial Discretization

Finite differences are used to discretize the viscous terms.

Some terms are required half way between nodes.  Examples of their representation are given below.

$$\frac{1}{J_{i+1/2}}{\overrightarrow{\xi }}_{i+1/2}·{\overrightarrow{\eta }}_{i+1/2}=\frac{1}{2}\left(\frac{{\overrightarrow{S}}_{i+1}^{\xi }·{\overrightarrow{S}}_{i+1}^{\eta }}{{\mathrm{vol}}_{i+1}}+\frac{{\overrightarrow{S}}_i^{\xi }·{\overrightarrow{S}}_i^{\eta }}{{\mathrm{vol}}_i}\right)$$ $$\frac{1}{J_{i+1/2}}{\overrightarrow{\xi }}_{i+1/2}\times {\overrightarrow{\eta }}_{i+1/2}=\frac{1}{2}\left(\frac{{\overrightarrow{S}}_{i+1}^{\xi }\times {\overrightarrow{S}}_{i+1}^{\eta }}{{\mathrm{vol}}_{i+1}}+\frac{{\overrightarrow{S}}_i^{\xi }\times {\overrightarrow{S}}_i^{\eta }}{{\mathrm{vol}}_i}\right)$$ $$\frac{1}{J_{i+1/2}}{\overrightarrow{\xi }}_{i+1/2}\left({\overrightarrow{\eta }}_{i+1/2}·\Delta {\overrightarrow{U}}_{i+1/2}^{\eta }\right)=\frac{1}{2}\left(\frac{{\overrightarrow{S}}_{i+1}^{\xi }\left({\overrightarrow{S}}_{i+1}^{\eta }·\Delta {\overrightarrow{U}}_{i+1/2}^{\eta }\right)}{{\mathrm{vol}}_{i+1}}+\frac{{\overrightarrow{S}}_i^{\xi }\left({\overrightarrow{S}}_i^{\eta }·\Delta {\overrightarrow{U}}_{i+1/2}^{\eta }\right)}{{\mathrm{vol}}_i}\right)$$ Where $\overrightarrow{S}$ is the area vector for a given cell side (ξ, η, and ζ) and vol is the volume of the cell.

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References

1) Krist, S. L., Biedron, R. T., and Rumsey, C. L., "CFL3D User's Manual (Version 5.0)," NASA TM 1998-208444, June 1998.
2) Coakley, T. J., "Implicit Upwind Methods for the Compressible Navier-Stokes Equations," AIAA-83-1958, 1992.
3) Vinokur, M., "An Analysis of Finite-Difference and Finite-Volume Formulations of Conservation Laws," NASA CR-177416, NASA, June 1986.