11.5.2: Prandtl's Condition
- Page ID
- 808
\[
M^{*} = \dfrac{ U }{ c^{*} } = \dfrac{c }{ c^{*} } \dfrac{U }{ c} =
\dfrac{c }{ c^{*} }\, M
\label{shock:eq:starMtoM}
\]
The jump condition across the shock must satisfy the constant energy.
\[
\dfrac{c^2 }{ k-1} + \dfrac{U^2 }{ 2 } =
\dfrac
Callstack:
at (Bookshelves/Civil_Engineering/Book:_Fluid_Mechanics_(Bar-Meir)/11:_Compressible_Flow_One_Dimensional/11.5_Normal_Shock/11.5.2:_Prandtl's_Condition), /content/body/p[2]/span, line 1, column 2
\label{shock:eq:momentumC}
\]
Dividing the mass equation by the momentum equation and combining it with the perfect gas model yields
\[
{{c_1}^2 \over k\, U_1 } + U_1 = {{c_2}^2 \over k\, U_2 } + U_2 \label{shock:eq:massMomOFS} \]
Combining equation (29) and (30) results in
\[ \dfrac{1 }{ k\,U_1} \left[ \dfrac{k+1 }{ 2 }\, {c^{*}}^2
- {k-1 \over 2 } U_1 \right] + U_1 =
\dfrac{1 }{ k\,U_2} \left[ {k+1 \over 2 } {c^{*}}^2
- \dfrac{k-1 }{ 2 }\, U_2 \right] + U_2
\label{shock:eq:combAllR}
\]
\[
U_1\,U_2 = {c^{*}}^2
\label{shock:eq:unPr}
\]
or in a dimensionless form
\[
{M^{*}}_1\, {M^{*}}_2 = {c^{*}}^2
\label{shock:eq:PrDless}
\]
Contributors and Attributions
Dr. Genick Bar-Meir. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or later or Potto license.