Oblique Shock

Figure 13.3: A typical oblique shock schematic

The shock occurs in reality in situations where the shock has three-dimensional effects. The three-dimensional effects of the shock make it appear as a curved plane. However, for a chosen arbitrary accuracy it requires a specific small area, a one-dimensional shock can be considered. In such a case, the change of the orientation makes the shock considerations two-dimensional. Alternately, using an infinite (or a two-dimensional) object produces a two-dimensional shock. The two-dimensional effects occur when the flow is affected from the ``side,'' i.e., a change in the flow direction^13.4.

To match the boundary conditions, the flow turns after the shock to be parallel to the inclination angle. Figure (13.3) exhibits the schematic of the oblique shock. The deflection angle, $\delta$ , is the direction of the flow after the shock (parallel to the wall). The normal shock analysis dictates that after the shock, the flow is always subsonic. The total flow after the oblique shock can also be supersonic, which depends on the boundary layer.

Only the oblique shock's normal component undergoes the ``shock.'' The tangent component does not change because it does not ``move'' across the shock line. Hence, the mass balance reads

$$\rho_1 {U_1}_n = \rho_2 {U_2}_n$$ (13.1)

The momentum equation reads

$$P_1 + \rho_1 {{U_1}_n } ^{2} = P_2 + \rho_2 {{U_2}_n } ^{2}$$ (13.2)

The momentum equation in the tangential direction yields

$${U_1}_t = {U_2}_t$$ (13.3)

The energy balance reads

$$C_p T_1 + {{{U_1}_n } ^{2} \over 2} = C_p T_2 + {{{U_2}_n } ^{2} \over 2}$$ (13.4)

Equations (13.1), (13.2), and (13.4) are the same as the equations for normal shock with the exception that the total velocity is replaced by the perpendicular components. Yet the new relationship between the upstream Mach number, the deflection angle, $\delta$ , and the Mach angle, $\theta$ has to be solved. From the geometry it can be observed that

$$\tan \theta = {{U_1}_n \over {U_1}_t}$$ (13.5)

and

$$\tan ( \theta - \delta ) = {{U_2}_n \over {U_2}_t}$$ (13.6)

Unlike in the normal shock, here there are three possible pairs^13.5 of solutions to these equations. The first is referred to as the weak shock; the second is the strong shock; and the third is an impossible solution (thermodynamically)^13.6. Experiments and experience have shown that the common solution is the weak shock, in which the shock turns to a lesser extent^13.7.

$${\tan \theta \over \tan ( \theta - \delta ) } = {{U_1}_n \over {U_2}_n }$$ (13.7)

The above velocity-geometry equations can also be expressed in term of Mach number, as

$$\sin \theta = {{M_1}_n \over {M_1}}$$ (13.8)

$$\sin (\theta - \delta ) = {{M_2}_n \over {M_2}}$$ (13.9)

$$\cos \theta = {{M_1}_t \over {M_1}}$$ (13.10)

$$\cos (\theta - \delta ) = {{M_2}_t \over {M_2}}$$ (13.11)

The total energy across an oblique shock wave is constant, and it follows that the total speed of sound is constant across the (oblique) shock. It should be noted that although, ${U_1}_t = {U_2}_t$ the Mach number is ${M_1}_t \neq {M_2}_t$ because the temperatures on both sides of the shock are different, $T_1 \neq T_2$ .

As opposed to the normal shock, here angles (the second dimension) have to be determined. The solution from this set of four equations, (13.8) through (13.11), is a function of four unknowns of $M_1$ , $M_2$ , $\theta$ , and $\delta$ . Rearranging this set utilizing geometrical identities such as $\sin\alpha = 2\sin\alpha\cos\alpha$ results in

$$\tan \delta = 2 \cot \theta \left[{M_1}^{2} \sin^2 \theta - 1 \over {M_1}^{2} \left(k + \cos 2 \theta \right) +2 \right]$$ (13.12)

The relationship between the properties can be determined by substituting $M_1 \sin \theta$ for of $M_1$ into the normal shock relationship, which results in

$${P_2 \over P_1} = {2k {M_1 }^{2} \sin^2 \theta - (k-1) \over k + 1}$$ (13.13)

The density and normal velocity ratio can be determined by the following equation

$${\rho_2 \over \rho_1} = {{U_1}_n \over {U_2}_n} = { (k+1) {M_1}^{2} \sin^2\theta \over (k-1) {M_1}^2 \sin^2\theta + 2}$$ (13.14)

The temperature ratio is expressed as

$${T_2 \over T_1} = {2k {M_1}^2 \sin^2\theta - (k-1) \left[(k-1) {M_1}^2 + 2 \right] \over (k+1)^2 {M_1}}$$ (13.15)

Prandtl's relation for oblique shock is

$$U_{n_1}U_{n_2} = c^{2} - {k -1 \over k+1} {U_t}^2$$ (13.16)

The Rankine-Hugoniot relations are the same as the relationship for the normal shock

$${P_2 - P_1 \over \rho_2 - \rho_1} = k { P_2 - P_1 \over \rho_2 - \rho_1}$$ (13.17)