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Conservation equations

The governing equations describing the 1-D non-homentropic gas flow, with the consideration of area change, friction and heat transfer in a pipe, form a non-homogeneous hyperbolic system .

$\frac{{\partial {\bf{W}}}}{{\partial t}} + \frac{{\partial {\bf{F}}}}{{\partial x}} + {\bf{C}}_1 + {\bf{C}}_2 = 0$

This conservation law system, composed by the continuity, momentum and energy equations, is complemented by the equation of state or the real gas properties . In previous equation, $\bf{W}$ is the desired state vector of the solution, $\bf{F}$ is the flux vector and $\bf{C}$ the source term separating the effect of the area changes from the effect of friction and heat transfer. The 1-D gas flow governing equations were traditionally arranged in the vector form:

$\begin{array}{c} {{\bf{W}}\left( {x,t} \right) = \left[ {\begin{array}{c} {\rho S} \\ {\rho uS} \\ {S\left( {\rho \frac{{u^2 }}{2} + \frac{p}{{\gamma - 1}}} \right)} \\ \end{array}} \right]} \hfill & {{\bf{F}}({\bf{W}}) = \left[ {\begin{array}{c} {\rho uS} \\ {\left( {\rho u^2 + p} \right)S} \\ {uS\left( {\rho \frac{{u^2 }}{2} + \frac{{\gamma p}}{{\gamma - 1}}} \right)} \\ \end{array}} \right]} \hfill \\ {{\bf{C}}_1 \left( {x,{\bf{W}}} \right) = \left[ {\begin{array}{c} 0 \\ { - p\frac{{dS}}{{dx}}} \\ 0 \\ \end{array}} \right]} \hfill & {{\bf{C}}_2 \left( {\bf{W}} \right) = \left[ {\begin{array}{c} 0 \\ {g\rho S} \\ { - q\rho S} \\ \end{array}} \right]} \hfill \\ \end{array}$

Lax-Wendroff scheme

The Lax-Wendroff scheme is a centred second-order accuracy numerical scheme in which the flow is approximated by Taylor series. OpenWAM applies the two-steps version proposed by Richtmyer and Morton :

$\begin{array}{l} {\bf{W}}_{j + {\textstyle{1 \over 2}}}^{n + {\textstyle{1 \over 2}}} = \frac{{{\bf{W}}_j^n + {\bf{W}}_{j + 1}^n }}{2} - \frac{{\Delta t}}{{2\Delta x}}\left( {{\bf{F}}_{j + 1}^n - {\bf{F}}_j^n } \right) - \frac{{\Delta t}}{4}\left( {{\bf{C}}_j^n + {\bf{C}}_{j + 1}^n } \right) \\ {\bf{W}}_j^{n + 1} = {\bf{W}}_j^n - \frac{{\Delta t}}{{\Delta x}}\left( {{\bf{F}}_{j + {\textstyle{1 \over 2}}}^{n + {\textstyle{1 \over 2}}} - {\bf{F}}_{j - {\textstyle{1 \over 2}}}^{n + {\textstyle{1 \over 2}}} } \right) - \frac{{\Delta t}}{2}\left( {{\bf{C}}_{j + {\textstyle{1 \over 2}}}^{n + {\textstyle{1 \over 2}}} + {\bf{C}}_{j - {\textstyle{1 \over 2}}}^{n + {\textstyle{1 \over 2}}} } \right) \\ \end{array}$

TVD scheme

The TVD flux limiter schemes were presented by Sweby and consist of a first order flux combined with a limited second order flux. Davis and Yee also worked on these techniques approaching homentropic flow. In these flow conditions, the Davis flux limiter technique can be obtained just by adding a viscous term to the second step of Ritchmyer and Morton scheme.
OpenWAM uses an adaptation of the Sweby TVD flux limiter scheme adapted by Gascón . The way to adapt the schemes was to include the source term as part of the flux. The scheme can be written as:

${\bf{W}}_j^{n + 1} = {\bf{W}}_j^n - \frac{{\Delta t}}{{\Delta x}}\left[ {{\bf{\hat G}}_{j + {\textstyle{1 \over 2}}}^{SW} - {\bf{\hat G}}_{j - {\textstyle{1 \over 2}}}^{SW} } \right] - \frac{{\Delta t}}{{\Delta x}}\left[ {{\bf{B}}_{j - {\textstyle{1 \over 2}},j} + {\bf{B}}_{j,j + {\textstyle{1 \over 2}}} } \right]$

Where the second order accuracy flux can be calculated by means of the expression

${\bf{\hat G}}_{j + {\textstyle{1 \over 2}}}^{SW} = \frac{1}{2}\left\{ {{\bf{F}}_j + {\bf{F}}_{j + 1} - {\bf{B}}_{j,j + {\textstyle{1 \over 2}}} + {\bf{B}}_{j + {\textstyle{1 \over 2}},j + 1} - {\bf{P}}_{j + {\textstyle{1 \over 2}}} h\left( {{\bf{\bar D}}_{j + {\textstyle{1 \over 2}}}^\psi } \right){\bf{Q}}_{j + {\textstyle{1 \over 2}}} \left( {{\bf{F}}_{j + 1} - {\bf{F}}_j + {\bf{B}}_{j,j + 1} } \right)_j } \right\}$

Numerical methods

Conservation equations

Lax-Wendroff scheme

TVD scheme

References

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