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Metallurgical Engineering

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Shock Polar

Let V_x be the component of velocity in x-direction while V_y be the component of velocity in y direction. Hence V_x1 is the x-direction velocity component for upstream velocity and V_x2 is the x-direction velocity component for post shock velocity. Similarly V_y1 and V_y2 are the is the y-direction velocity component for upstream and post shock velocities respectively. If the upstream velocity is parallel to x axis then V_y1 is equal to zero since V_x1 is equal to V₁. Now lets plot velocities along their respective directions as shown in Fig. 2. Here the resultant velocity V₁ is parallel to x axis while V₂ makes an angle θ with x-axis, which is the flow deflection angle.

Now if we increase the deflection angle then Vx₂ will decrease and Vy₂ will increase since in such cases resultant velocity V₂ will have to make an increased angle θ with the x-axis. Figure 3 represents the same process.

In the same way if we change the wedge angle for all possible attached shock solutions and join them together then such a plot is called as hodograph.

In the same way we can plot for various free streams Mach number for all the possible deflection angles. However, if we increase the free stream Mach number by increasing free stream velocity then it becomes impossible to plot such a graph when Vx₁ becomes ∞ for M₁ equal to ∞. To make such a plot possible, lets divide x axis and y axis by a* which is the reference or stared quantity. This non-dimensionalsiation makes it possible to represent Vx₁ since it will be represented by M₁*. We can divide by a* to post shock velocities as well, since a* is constant in the flow field for an adiabatic flow. Hence V₂ gets transformed to M₂* which will make an angle equal to the flow deflection angle with the x-axis. This plot is called as shock polar. Such a shock polar can be plotted for very high Mach numbers M₁ also due to the fact that, if M₁ is equal to ∞, M₁* is equal to 2.54. Therefore this plot is equally helpful for representing the pre and post shock velocities, flow deflection angle and the shock angle. Typical shock polar is shown in Fig. 4.