# 11.5.2: Prandtl's Condition

- Page ID
- 1317

\[

M^{*} = \dfrac{ U }{ c^{*} } = \dfrac{c }{ c^{*} } \dfrac{U }{ c} =

\dfrac{c }{ c^{*} }\, M

\label{shock:eq:starMtoM} \tag{28}

\]

The jump condition across the shock must satisfy the constant energy.

\[

\dfrac{c^2 }{ k-1} + \dfrac{U^2 }{ 2 } =

\dfrac

```
Callstack:
at (Under_Construction/Purgatory/Book:_Fluid_Mechanics_(Bar-Meir)__-_old_copy/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} \tag{29}

\]

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} \tag{30} \]

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} \tag{31}

\]

After rearranging and dividing equation (31) the following can be obtained:

\[

U_1\,U_2 = {c^{*}}^2

\label{shock:eq:unPr} \tag{32}

\]

or in a dimensionless form

\[

{M^{*}}_1\, {M^{*}}_2 = {c^{*}}^2

\label{shock:eq:PrDless} \tag{33}

\]

### Contributors

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.