Shock waves - Revision history

https://wiki.g3dflow.com/index.php?action=history&feed=atom&title=Shock_waves Shock waves - Revision history 2026-04-16T18:02:11Z Revision history for this page on the wiki MediaWiki 1.45.1 https://wiki.g3dflow.com/index.php?title=Shock_waves&diff=499&oldid=prev Nian at 07:57, 1 April 2026 2026-04-01T07:57:09Z

<p></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 07:57, 1 April 2026</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l14">Line 14:</td> <td colspan="2" class="diff-lineno">Line 14:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>--><noinclude><noinclude>{{#vardefine:secno|3}}{{#vardefine:secno|3}}{{#vardefine:eqno|<del style="font-weight: bold; text-decoration: none;">18</del>}}{{#vardefine:eqno|<ins style="font-weight: bold; text-decoration: none;">19</ins>}}</noinclude></noinclude><!--</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> </table>

Nian https://wiki.g3dflow.com/index.php?title=Shock_waves&diff=494&oldid=prev Nian at 07:49, 1 April 2026 2026-04-01T07:49:05Z

<p></p> <a href="https://wiki.g3dflow.com/index.php?title=Shock_waves&diff=494&oldid=245">Show changes</a>

Nian https://wiki.g3dflow.com/index.php?title=Shock_waves&diff=245&oldid=prev Nian at 19:40, 20 March 2026 2026-03-20T19:40:06Z

<p></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 19:40, 20 March 2026</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l1">Line 1:</td> <td colspan="2" class="diff-lineno">Line 1:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[Category:Compressible flow]]</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[Category:Compressible flow]]</div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">[[Category:Topic]]</del></div></td><td colspan="2" class="diff-side-added"></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[Category:One-dimensional flow]]</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[Category:One-dimensional flow]]</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[Category:Inviscid flow]]</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[Category:Inviscid flow]]</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[Category:Wave solution]]</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[Category:Wave solution]]</div></td></tr> <tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"><noinclude></ins></div></td></tr> <tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">[[Category:Compressible flow:Topic]]</ins></div></td></tr> <tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></noinclude></ins></div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>__TOC__</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>__TOC__</div></td></tr> </table>

Nian https://wiki.g3dflow.com/index.php?title=Shock_waves&diff=193&oldid=prev Nian at 17:13, 20 March 2026 2026-03-20T17:13:28Z

<p></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 17:13, 20 March 2026</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l1">Line 1:</td> <td colspan="2" class="diff-lineno">Line 1:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[Category:Compressible flow]]</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[Category:Compressible flow]]</div></td></tr> <tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">[[Category:Topic]]</ins></div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[Category:One-dimensional flow]]</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[Category:One-dimensional flow]]</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[Category:Inviscid flow]]</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[Category:Inviscid flow]]</div></td></tr> </table>

Nian https://wiki.g3dflow.com/index.php?title=Shock_waves&diff=147&oldid=prev Nian: /* Shock Waves */ 2026-03-17T13:36:40Z

<p><span class="autocomment">Shock Waves</span></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 13:36, 17 March 2026</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l6">Line 6:</td> <td colspan="2" class="diff-lineno">Line 6:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>__TOC__</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>__TOC__</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">==Shock Waves==</del></div></td><td colspan="2" class="diff-side-added"></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><!--</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><!--</div></td></tr> </table>

Nian https://wiki.g3dflow.com/index.php?title=Shock_waves&diff=35&oldid=prev Nian: Nian moved page Shock Waves to Shock waves 2026-03-16T10:07:52Z

<p>Nian moved page <a href="/index.php?title=Shock_Waves" class="mw-redirect" title="Shock Waves">Shock waves</a> to <a href="/index.php?title=Shock_waves" title="Shock waves">Shock waves</a></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <tr class="diff-title" lang="en"> <td colspan="1" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="1" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 10:07, 16 March 2026</td> </tr><tr><td colspan="2" class="diff-notice" lang="en"><div class="mw-diff-empty">(No difference)</div> </td></tr></table>

Nian https://wiki.g3dflow.com/index.php?title=Shock_waves&diff=34&oldid=prev Nian at 10:07, 16 March 2026 2026-03-16T10:07:15Z

<p></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 10:07, 16 March 2026</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l1">Line 1:</td> <td colspan="2" class="diff-lineno">Line 1:</td></tr> <tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">[[Category:Compressible flow]]</ins></div></td></tr> <tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">[[Category:One-dimensional flow]]</ins></div></td></tr> <tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">[[Category:Inviscid flow]]</ins></div></td></tr> <tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">[[Category:Wave solution]]</ins></div></td></tr> <tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr> <tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">__TOC__</ins></div></td></tr> <tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Shock Waves==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Shock Waves==</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> </table>

Nian https://wiki.g3dflow.com/index.php?title=Shock_waves&diff=33&oldid=prev Nian: Created page with "==Shock Waves==  The starting point is to set up the governing equations for one-dimensional steady compressible flow over a control volume enclosing the normal shock (Fig. \ref{fig:shock:cv}). <!-- \begin{figure}[ht!] \begin{center} \includegraphics[]{figures/standalone-fig..." 2026-03-16T10:06:37Z

<p>Created page with "==Shock Waves==  The starting point is to set up the governing equations for one-dimensional steady compressible flow over a control volume enclosing the normal shock (Fig. \ref{fig:shock:cv}). <br /> <br /> The starting point is to set up the governing equations for one-dimensional steady compressible flow over a control volume enclosing the normal shock (Fig. \ref{fig:shock:cv}).<br /> <br /> <br /> <br /> continuity:<br /> <br /> <math display="block"><br /> \rho_1 u_1=\rho_2 u_2<br /> </math><br /> <br /> momentum:<br /> <br /> <math display="block"><br /> \rho_1 u_1^2+p_1=\rho_2 u_2^2+p_2<br /> </math><br /> <br /> energy:<br /> <br /> <math display="block"><br /> h_1 + \frac{1}{2}u_1^2=h_2 + \frac{1}{2}u_2^2<br /> </math><br /> <br /> Divide the momentum equation by <math>\rho_1 u_1</math><br /> <br /> <br /> <math display="block"><br /> \frac{1}{\rho_1 u_1}\left(\rho_1 u_1^2+p_1\right)=\frac{1}{\rho_1 u_1}\left(\rho_2 u_2^2+p_2\right)=\left\{\rho_1 u_1=\rho_2 u_2\right\}=\frac{1}{\rho_2 u_2}\left(\rho_2 u_2^2+p_2\right) \Rightarrow<br /> </math><br /> <br /> <math display="block"><br /> \frac{p_1}{\rho_1 u_1}-\frac{p_2}{\rho_2 u_2}=u_2-u_1<br /> </math><br /> <br /> For a calorically perfect gas <math>a=\sqrt{\gamma p/\rho}</math>, which if implemented in Eqn. \ref{eq:governing:mom:b} gives<br /> <br /> <math display="block"><br /> \frac{a_1^2}{\gamma u_1}-\frac{a_2^2}{\gamma u_2}=u_2-u_1<br /> </math><br /> <br /> The energy equation (Eqn. \ref{eq:governing:energy}) with <math>h=C_p T</math><br /> <br /> <math display="block"><br /> C_p T_1 + \frac{1}{2}u_1^2=C_p T_2 + \frac{1}{2}u_2^2<br /> </math><br /> <br /> Replacing <math>C_p</math> with <math>\gamma R/(\gamma-1)</math> gives<br /> <br /> <math display="block"><br /> \frac{\gamma RT_1}{\gamma-1} + \frac{1}{2}u_1^2=\frac{\gamma RT_2}{\gamma-1} + \frac{1}{2}u_2^2<br /> </math><br /> <br /> With <math>a=\sqrt{\gamma RT}</math> this becomes<br /> <br /> <math display="block"><br /> \frac{a_1^2}{\gamma-1} + \frac{1}{2}u_1^2=\frac{a_2^2}{\gamma-1} + \frac{1}{2}u_2^2<br /> </math><br /> <br /> Eqn. \ref{eq:governing:energy:d} can be set up between any two points in the flow. Specifically, we can use the relation to relate the flow velocity, <math>u</math>, and speed of sound, <math>a</math>, in any point to the corresponding flow properties at sonic conditions (<math>u=a=a^*</math>).<br /> <br /> <math display="block"><br /> \frac{a^2}{\gamma-1} + \frac{1}{2}u^2=\frac{\gamma+1}{2(\gamma-1)}{a^*}^2<br /> </math><br /> <br /> If Eqn. \ref{eq:governing:energy:e} is evaluated in locations 1 and 2, we get<br /> <br /> <math display="block"><br /> \begin{aligned}<br /> &a_1^2 = \frac{\gamma+1}{2}{a^*}^2 - \frac{\gamma-1}{2}u_1^2\\<br /> &a_2^2 = \frac{\gamma+1}{2}{a^*}^2 - \frac{\gamma-1}{2}u_2^2<br /> \end{aligned}<br /> </math><br /> <br /> Since the change in flow conditions over the shock is adiabatic (no heat is added inside the shock), critical properties will be constant over the shock. Especially <math>a^*</math> will be constant.<br /> <br /> Eqn. \ref{eq:governing:energy:f} inserted in \ref{eq:governing:mom:c} gives\\<br /> <br /> <br /> <math display="block"><br /> \frac{1}{\gamma u_1}\left(\frac{\gamma+1}{2}{a^*}^2 - \frac{\gamma-1}{2}u_1^2\right)-\frac{1}{\gamma u_2}\left(\frac{\gamma+1}{2}{a^*}^2 - \frac{\gamma-1}{2}u_2^2\right)=u_2-u_1 \Rightarrow</math><br /> <br /> <math display="block"><br /> \left(\frac{\gamma+1}{2\gamma}\right){a^*}^2\left(\frac{1}{u_1}-\frac{1}{u_2}\right)=\left(\frac{\gamma+1}{2\gamma}\right)\left(u_2-u_1\right) \Rightarrow<br /> </math><br /> <br /> <math display="block"><br /> {a^*}^2\left(\frac{1}{u_1}-\frac{1}{u_2}\right)=\left(u_2-u_1\right) \Rightarrow<br /> </math><br /> <br /> <math display="block"><br /> {a^*}^2\left(\frac{u_2}{u_1 u_2}-\frac{u_1}{u_1 u_2}\right)=\left(u_2-u_1\right) \Rightarrow<br /> </math><br /> <br /> <math display="block"><br /> \frac{1}{u_1 u_2}{a^*}^2\left(u_2-u_1\right)=\left(u_2-u_1\right) \Rightarrow<br /> </math><br /> <br /> <math display="block"><br /> {a^*}^2=u_1 u_2<br /> </math><br /> <br /> Eqn. \ref{eq:prandtl} is sometimes referred to as the Prandtl relation. Divide the Prandtl relation by <math>{a^*}^2</math> on both sides gives<br /> <br /> <math display="block"><br /> 1=\frac{u_1}{a^*}\frac{u_2}{a^*}=M^*_1M^*_2<br /> </math><br /> <br /> or <br /> <br /> <math display="block"><br /> M^*_2=\frac{1}{M^*_1}<br /> </math><br /> <br /> The relation between <math>M^*</math> and <math>M</math> is given by<br /> <br /> <math display="block"><br /> {M^*}^2=\frac{(\gamma+1)M^2}{2+(\gamma-1)M^2}<br /> </math><br /> <br /> from which is can be seen that <math>M^*</math> will follow the Mach number <math>M</math> in the sense that<br /> <br /> * <math>M=1\Rightarrow M^*=1</math><br /> * <math>M<1\Rightarrow M^*<1</math><br /> * <math>M>1\Rightarrow M^*>1</math> <br /> <br /> Eqn. \ref{eq:MachStar} inserted in Eqn. \ref{eq:NormalMach} gives<br /> <br /> <br /> <math display="block"><br /> \frac{(\gamma+1)M_1^2}{2+(\gamma-1)M_1^2}=\frac{2+(\gamma-1)M_2^2}{(\gamma+1)M_2^2}<br /> </math><br /> <br /> <math display="block"><br /> M^2_2=\frac{1+\left[(\gamma-1)/2\right]M^2_1}{\gamma M^2_1-(\gamma-1)/2}<br /> </math><br /> <br /> The Mach number relations above effectively show that if the Mach number upstream of the shock is greater than one, the downstream Mach number must be less than one and vice versa. We can also see that a sonic upstream flow <math>M_1=1.0</math> gives sonic flow downstream of the shock. So, apparently the relation as such holds for both supersonic and subsonic upstream flow mathematically. The question is if it is also physically correct. For a supersonic upstream flow we will get a discontinuous compression and if the flow upstream of the control volume is subsonic we will instead get a discontinuous expansion inside the control volume but, again, is this physically correct? We will get the answer by analyzing the entropy change over the control volume.<br /> <br /> Analyzing the energy equation and the second law of thermodynamics shows that there is a direct relation between entropy increase and total pressure drop.<br /> <br /> <math display="block"><br /> s_2-s_1=C_p\ln\dfrac{T_2}{T_1}-R\ln\dfrac{p_2}{p_1}<br /> </math><br /> <br /> <math display="block"><br /> s_2-s_1=C_p\ln\dfrac{T_2}{T_{o_2}}\dfrac{T_{o_1}}{T_1}\dfrac{T_{o_2}}{T_{o_1}}-R\ln\dfrac{p_2}{p_{o_2}}\dfrac{p_{o_1}}{p_1}\dfrac{p_{o_2}}{p_{o_1}}<br /> </math><br /> <br /> using the isentropic relations we get<br /> <br /> <math display="block"><br /> s_2-s_1=C_p\ln\dfrac{T_{o_2}}{T_{o_1}}-R\ln\dfrac{p_{o_2}}{p_{o_1}}<br /> </math><br /> <br /> and since the process is adiabatic and thus <math>T_{o_2}=T_{o_1}</math> the change in entropy is directly related to the change in total pressure as<br /> <br /> <math display="block"><br /> s_2-s_1=-R\ln\dfrac{p_{o_2}}{p_{o_1}}<br /> </math><br /> <br /> or<br /> <br /> <math display="block"><br /> \dfrac{p_{o_2}}{p_{o_1}}=e^{-(s_2-s_1)/R}<br /> </math><br /> <br /> Figure~\ref{fig:shock:entropy} shows the entropy change over a normal shock. As can be seen in the figure, a subsonic upstream Mach number leads to a reduction of entropy, which once and for all rules out all such solutions as non-physical and thus the question about the upstream conditions can now be considered answered. This in turn implies that the Mach number downstream of a normal shock will always be subsonic, which can be seen in Fig~\ref{fig:shock:downstream:Mach} below.<br /> <br /> <br /> <br /> By rewriting the right-hand side of Eqn.\ref{eq:NormalMach:b}, it is easy to realize that the downstream Mach number <math>M_2</math> approaches a finite value for large values of the upstream Mach number, <math>M_1</math>.<br /> <br /> <math display="block"><br /> \left.M_2^2\right|_{M_1\rightarrow\infty}=\left.\dfrac{2/M_1^2+(\gamma-1)}{2\gamma-(\gamma-1)/M_1^2}\right|_{M_1\rightarrow\infty}=\dfrac{\gamma-1}{2\gamma}<br /> </math></div>

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