Gas Dynamics &

PrandtlMeyer ExpansionIn this section the expansion of a flow is considered. A supersonic flow negotiating a convex corner undergoes expansion. As Expansion Shocks are not possible for steady flow, then expansions will be produced by a series on Mach waves. The expansion will be an isentropic process. Figure 33: Flow expansion through a shock? The waves may be centered at a convex corner or spread out as in the case of a convex surface. The Mach Waves are divergent in both the cases. A centered wave is called a PrandtlMeyer Expansion fan. Figure 34 : PrandtlMeyer Expansion A PrandtlMeyer fan is shown in Fig. 34, through which a flow expands from a Mach Number, $M_1$ to $M_2$. The leading wave is inclined to the flow at an angle $μ_1=\sin^{1}(1\/M_1)$ and the expansion terminates in a wave inclined at an angle, $μ_2=\sin^{1}(1\/M_2)$ . An expression connecting the flow turning angle $θ$ and the change in Mach Number can be found. Considering a differential element within the fan (Fig. 35) then the previous rules for 1D flow with weak waves shown in the previous sections canbe applied. The Mach Number changes from $M$ to $M + dM$ as the flow turns through an angle $dθ$. Figure 35 : PrandtlMeyer Expansion, continued For a weak disturbance (Mach Wave), where $w$ is the initial flow velocity before the wave. On integrating it gives, where a new function $ν$ has been introduced. This is called the PrandtlMeyer function. Starting from the results of the previous 1D flow section, then, after manipulation and substitution, The PrandtlMeyer function is a significant tool for calculating supersonic flows. Note that for $M=1, ν=0$. For every Mach Number greater than one there is a unique PrandtlMeyer function value. Tables for supersonic flow at the end of this section list $ν$ as a function of Mach Number. The practice, $ν$ is the angle through which a sonic flow should be turned in order to reach a Mach Number of $M$. In addition, a flow turning through an angle $θ$ will experience a change in PrandtlMeyer function value of, With the knowledge of $ν, M_2$ can be calculated for any value of $θ$. This function can also calculate the Mach Number following an isentropic compression, Figure 36 : Using PrandtlMeyer Function It should be noted that $ν$ decreases in compression and increases in expansion. See Fig. 36 