{"id":180,"date":"2014-09-24T15:48:36","date_gmt":"2014-09-24T15:48:36","guid":{"rendered":"http:\/\/openwam.webs.upv.es\/docs\/?p=180"},"modified":"2014-09-24T15:48:36","modified_gmt":"2014-09-24T15:48:36","slug":"numerical-methods","status":"publish","type":"post","link":"https:\/\/openwam.webs.upv.es\/docs\/?p=180","title":{"rendered":"Numerical methods"},"content":{"rendered":"

\u00a0<\/p>\n

Conservation equations<\/h1>\n

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 <\/span>.<\/p>\n

  <\/span>   <\/span>\"\[\frac{{\partial {\bf{W}}}}{{\partial t}} + \frac{{\partial {\bf{F}}}}{{\partial x}} + {\bf{C}}_1 + {\bf{C}}_2 = 0\]\"<\/p>\n<\/p>\n

This conservation law system, composed by the continuity, momentum and energy equations, is complemented by the equation of state or the real gas properties <\/span>. 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:<\/p>\n

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

Lax-Wendroff scheme<\/h1>\n

The Lax-Wendroff scheme <\/span>\u00a0is 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 <\/span>:<\/p>\n

  <\/span>   <\/span>\"\[\begin{array}{l} {\bf{W}}_{j + {\textstyle{1 \over 2}}}^{n + {\textstyle{1 \over 2}}} = \frac{{{\bf{W}}_j^n + {\bf{W}}_{j + 1}^n }}{2} - \frac{{\Delta t}}{{2\Delta x}}\left( {{\bf{F}}_{j + 1}^n - {\bf{F}}_j^n } \right) - \frac{{\Delta t}}{4}\left( {{\bf{C}}_j^n + {\bf{C}}_{j + 1}^n } \right) \\ {\bf{W}}_j^{n + 1} = {\bf{W}}_j^n - \frac{{\Delta t}}{{\Delta x}}\left( {{\bf{F}}_{j + {\textstyle{1 \over 2}}}^{n + {\textstyle{1 \over 2}}} - {\bf{F}}_{j - {\textstyle{1 \over 2}}}^{n + {\textstyle{1 \over 2}}} } \right) - \frac{{\Delta t}}{2}\left( {{\bf{C}}_{j + {\textstyle{1 \over 2}}}^{n + {\textstyle{1 \over 2}}} + {\bf{C}}_{j - {\textstyle{1 \over 2}}}^{n + {\textstyle{1 \over 2}}} } \right) \\ \end{array}\]\"<\/p>\n

TVD scheme<\/h1>\n

The TVD flux limiter schemes were presented by Sweby <\/span> and consist of a first order flux combined with a limited second order flux. Davis <\/span> and Yee <\/span> 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.
\nOpenWAM uses an adaptation of the Sweby TVD flux limiter scheme <\/span> adapted by Gasc\u00f3n <\/span>. The way to adapt the schemes was to include the source term as part of the flux. The scheme can be written as:<\/p>\n

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

Where the second order accuracy flux can be calculated by means of the expression<\/p>\n

  <\/span>   <\/span>\"\[{\bf{\hat G}}_{j + {\textstyle{1 \over 2}}}^{SW} = \frac{1}{2}\left\{ {{\bf{F}}_j + {\bf{F}}_{j + 1} - {\bf{B}}_{j,j + {\textstyle{1 \over 2}}} + {\bf{B}}_{j + {\textstyle{1 \over 2}},j + 1} - {\bf{P}}_{j + {\textstyle{1 \over 2}}} h\left( {{\bf{\bar D}}_{j + {\textstyle{1 \over 2}}}^\psi } \right){\bf{Q}}_{j + {\textstyle{1 \over 2}}} \left( {{\bf{F}}_{j + 1} - {\bf{F}}_j + {\bf{B}}_{j,j + 1} } \right)_j } \right\}\]\"<\/p>\n

References<\/h1>\n\n
\r\n\t\t{2085454:U4ER2QKI};{2085454:BAU8M8X4};{2085454:UJIMZIQX};{2085454:UT8KICJA};{2085454:ZN4TZND7};{2085454:STKR62MP};{2085454:M65WR4X6};{2085454:ZN4TZND7};{2085454:JUTSZJXD}<\/span>\r\n\t\tIEEE<\/span>\r\n\t\tdefault<\/span>\r\n\t\tasc<\/span>\r\n\t\t<\/span>\r\n\t\t<\/span>\r\n\t\t<\/span>\r\n\t\t<\/span>\r\n\t\t<\/span>\r\n\t\t<\/span>\r\n\t\t<\/span>\r\n\t\t<\/span>\r\n\t\t<\/span>\r\n\t\t0<\/span>\r\n\t\t<\/span>\r\n\t\t180<\/span>
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