Compressible Flow and Shock Waves

Oblique shock relations

An oblique shock is a shock wave inclined at an angle to the upstream velocity, and it forms when a supersonic stream is turned by a wedge or a compression corner. The inclination of the wave is the shock angle \beta, measured from the upstream flow direction, and the flow deflection \theta is related to \beta and the upstream Mach number M_1 by the \theta-\beta-M relation. Only the velocity component normal to the shock, V_{n1}=V_1\sin\beta, is changed by the wave, so the normal-shock relations apply to the normal Mach number M_{n1}=M_1\sin\beta, while the tangential velocity V_t passes through unchanged. For a given M_1 and \theta, the solution with the smaller \beta is the weak shock, which is the usual attached case.

The wave is adiabatic, so the total temperature T_0 is conserved across it, while the process is irreversible, so the total pressure falls and the entropy increases. Because the tangential velocity is preserved but the static temperature rises behind the shock, the speed of sound a=\sqrt{\gamma RT} increases, and the tangential Mach number M_t=V_t/a therefore decreases. The speed of sound changes across the wave for this reason, so it is not a conserved quantity across the shock.

An attached shock must satisfy \sin\beta\ge 1/M_1. Since the lower bound \sin^{-1}(1/M_1) decreases as M_1 grows, a ramp setting that produces a given shock angle can always be found at a higher Mach number. For a fixed wedge the shock angle decreases as the Mach number increases, so the shock lies closer to the surface at higher Mach number, and at low Mach number the wave may detach into a curved bow shock.

Across the wave the static pressure ratio is p_2/p_1=1+\frac{2\gamma}{\gamma+1}(M_{n1}^2-1), and the surface pressure coefficient is C_p=\frac{2}{\gamma M_1^2}\left(\frac{p_2}{p_1}-1\right). For M_1=3 and \beta=40^\circ with \gamma=1.4, M_{n1}\approx 1.93, p_2/p_1\approx 4.17, and C_p\approx 0.50.

Nozzle over-expansion and under-expansion

A nozzle is perfectly expanded when its exit static pressure equals the ambient pressure, and then no waves appear at the exit. Off-design operation is classified by comparing these two pressures.

• Over-expanded — exit pressure below ambient. The jet is compressed at the lips by oblique shocks that turn the stream inward, so the jet boundary converges toward the axis.
• Under-expanded — exit pressure above ambient. Prandtl-Meyer expansion fans form at the lips and turn the jet outward as the pressure drops toward ambient.

For a convergent-divergent nozzle designed for exit Mach number M_e, the reservoir and exit pressures are linked by p_0/p_e=\left(1+\frac{\gamma-1}{2}M_e^2\right)^{\gamma/(\gamma-1)}. If the exit pressure obtained this way exceeds the ambient pressure, the nozzle is under-expanded, and the expansion fans at the lip disappear only when the reservoir pressure is lowered until the exit pressure equals ambient. For example, a nozzle designed for M_e=2 with a reservoir pressure of about 7.8 bar has an exit pressure of about 1 bar.

Fanno flow relations

Fanno flow is steady, adiabatic, one-dimensional flow in a constant-area duct in which wall friction is the only irreversible effect. Because no heat or work crosses the boundary, the total temperature T_0 stays constant along the duct, and the static temperature follows T_0/T=1+\frac{\gamma-1}{2}M^2. Friction makes the entropy rise along the duct, and the entropy is largest at the sonic point M=1. The maximum-entropy point of a Fanno curve on the T-s diagram therefore marks the sonic state.

Friction accelerates subsonic flow and decelerates supersonic flow, so both branches approach M=1 downstream. The sonic static temperature is T^*=\frac{2T_0}{\gamma+1}, which for air with \gamma=1.4 equals T_0/1.2. For example, air entering at M=0.5 with a static temperature of 300 K has T_0=315 K, and its sonic temperature is about 262.5 K.

The friction length between two sections is obtained from the tabulated Fanno parameter. Between two Mach numbers, \frac{4fL}{D}=\left(\frac{4fL^*}{D}\right)_{M_1}-\left(\frac{4fL^*}{D}\right)_{M_2}, where f is the average Fanning friction factor. The friction factor is therefore found by dividing the difference of the table values by 4L/D.

Rayleigh flow relations

Rayleigh flow is steady, one-dimensional, inviscid flow in a constant-area duct with heat addition or removal and no friction. The momentum balance gives p+\rho V^2=\text{constant}, and continuity gives \rho V=\text{constant}. The energy equation gives T_0=T+\frac{V^2}{2c_p}, so heat addition always raises the stagnation temperature, and the stagnation temperature downstream of the heating is higher than upstream. The stagnation pressure falls under heating, because the process is irreversible and the duct area does not change the stagnation pressure of an inviscid flow.

With constant area, the downstream density follows from continuity as \rho_2=\rho_1V_1/V_2, and the momentum balance then gives p_2=p_1+\rho_1V_1^2-\rho_2V_2^2. For example, with p_1=120 kPa, \rho_1=1.5 kg/m^3, V_1=80 m/s and V_2=160 m/s, the density becomes 0.75 kg/m^3 and the static pressure becomes 110.4 kPa.

In a diverging duct, the area increase and the heat addition act on the Mach number and the static pressure in competing directions. Neither the sign of the Mach number change nor that of the pressure change is therefore fixed without knowing which effect dominates. Among the idealized one-dimensional models, Rayleigh flow is consistent with steady inviscid one-dimensional flow, whereas Fanno flow requires wall friction and oblique shocks require a change of flow direction that one-dimensional flow cannot provide.

Choked convergent nozzle flow

A convergent nozzle chokes when the back pressure falls below the critical pressure p^*=p_0\left(\frac{2}{\gamma+1}\right)^{\gamma/(\gamma-1)}, which is about 0.53\,p_0 for air. When the nozzle is choked, the exit Mach number is exactly unity, and the mass flow rate is fixed by the reservoir state and the exit area through \dot m=\rho^*a^*A_e. For example, with a reservoir pressure of 6 bar the critical pressure is about 3.17 bar, so any back pressure below that value chokes the exit.

The choked mass flow rate depends on the exit area and the reservoir state. Enlarging the exit area of a choked convergent nozzle, with reservoir and ambient conditions unchanged, keeps the nozzle choked and increases the mass flow. The same larger mass flow then passes through an upstream section of unchanged area, so the local velocity and Mach number there increase.

In a choked nozzle the throat static temperature T^*=\frac{2T_0}{\gamma+1} depends only on the reservoir temperature. Raising the reservoir pressure scales the static pressure and density at the exit with p_0, while the Mach number, the static temperature and the jet velocity remain unchanged.

Convergent-divergent nozzle throat condition

In a convergent-divergent duct the throat is the section of minimum area. The isentropic area-Mach relation \frac{A}{A^*}=\frac{1}{M}\left[\frac{2}{\gamma+1}\left(1+\frac{\gamma-1}{2}M^2\right)\right]^{\frac{\gamma+1}{2(\gamma-1)}} gives the area ratio for any Mach number, and A^* is the sonic area. For steady isentropic flow the throat Mach number can be either sonic or supersonic. When the inflow is supersonic, the Mach number falls as the area narrows, so the smallest value reachable at the throat is unity. A subsonic throat would require a deceleration through a normal shock, which an isentropic process cannot contain.