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12.6: d'Alembert's Paradox

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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

    \[ D = 2 \left( \dfrac{w }{ 2} \, (P_2 - P_4)\right) = w \, (P_2 - P_4) \label{pm:eq:dragG} \tag{38} \]

    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.

