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12.6: d'Alembert's Paradox

Last updated

Mar 5, 2021
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- 12.5: The Working Equations for the Prandtl-Meyer Function
- 12.7: Flat Body with an Angle of Attack

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In ideal inviscid incompressible flows, the movement of body does not encounter any resistance. This result is known as d'Alembert's Paradox, and this paradox is examined here. Supposed that a two–dimensional diamond–shape body is stationed in a supersonic flow as shown in Figure 12.27.

Fig. 12.27 A simplified diamond shape to illustrate the supersonic d'Alembert's Paradox.

Again, it is assumed that the fluid is inviscid. The net force in flow direction, the drag, is

$D = 2 \left( \dfrac{w }{ 2} \, (P_2 - P_4)\right) = w \, (P_2 - P_4) \label{pm:eq:dragG}$

It can be observed that only the area that "seems'' to be by the flow was used in expressing equation (38). The relation between $P_2$ and $P_4$ is such that the flow depends on the upstream Mach number, $M_1$ , and the specific heat, $k$ . Regardless in the equation of the state of the gas, the pressure at zone 2, $P_2$ , is larger than the pressure at zone 4, $P_4$ . Thus, there is always drag when the flow is supersonic which depends on the upstream Mach number, $M_1$ , specific heat, $k$ , and the "visible'' area of the object. This drag is known in the literature as (shock) wave drag.

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.

Contributors and Attributions

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