5.4 The Shock Polar

The shock polar is a graphical representation of all possible flow deflection angles for a given Mach number. The shock polar is generated by plotting the normalized axial velocity component downstream of an oblique shock versus the normalized vertical velocity component. In essence the ratio of vertical and axial velocity components gives the deflection angle. Figure Figure 5.10 shows a set of shock polars generated for different upstream Mach numbers (indicated in the figure). The vertical and axial velocity components are normalized by the characteristic speed of sound (the speed of sound at sonic conditions). Each of the shock polars have two solutions where the vertical velocity component is zero, i.e. zero-deflection solutions. The zero-deflection solution furthest to the right represents the Mach wave solution and the solution to the left represents a normal shock. This is easy to realize as these are the only possible solutions that would result in zero flow deflection and the Mach wave solution is located where $V_x/a^\ast$ is greater than one, which means that the flow is supersonic on the downstream side, whereas the normal shock results in a subsonic flow on the downstream side ($V_x/a^\ast<1.0)$. Figures Figure 5.11-Figure 5.15 shows only the shock polar corresponding to an upstream mach number of 2.5. In Figure Figure 5.11, a unit half-circle is added to indicate which parts of the shock polar that represents supersonic solutions and which parts that represent supersonic solutions. Everything that falls inside of the unit circle represents subsonic solutions. i.e. the downstream Mach number is subsonic and all solutions that are outside of the circle represents solutions for which the flow is supersonic. Figure Figure 5.12 shows how the maximum possible flow deflection relates to the shock polar for a specific upstream Mach number. The line in the figure represents the flow deflection and when the line is tangent to the shock polar, the maximum flow deflection is reached. Further increase of the flow deflection angle leads to that the flow deflection line is outside of the shock polar and thus there are no possible solutions for angles greater than $\theta_{max}$. For the maximum deflection angle there is only one possible solution since the flow deflection line is a tangent to the shock polar. For all angles smaller than $\theta_{max}$, there are, however, two possible solutions (see Figure Figure 5.13. In most cases there is one supersonic solution (the weak solution) and one subsonic solution (the strong solution) as indicated in Figure Figure 5.13. However, for angles close to $\theta_{max}$ both solutions may fall inside of the unit circle and thus both solutions will be subsonic. This is in line with what we saw for the $\theta$-$\beta$-Mach relation earlier (see section The theta-beta-Mach Relation). Finally, we will have a look at how the the shock angle ($\beta$) is related to the shock polar. Figure Figure 5.14 shows how the shock angle ($\beta$) is found for the weak solution of a given upstream mach number and a given flow deflection ($\theta$). Draw a line starting at the Mach wave solution going through the the downstream solution (in this case the weak solution). Now, make another line starting at the origin that is perpendicular to the first line. The angle of the second line is the shock angle ($\beta$). In analogy, figure Figure 5.15 shows how to obtain the shock angle for the strong solution.