Aerodynamic and physical foundations of Supersonic de Laval Nozzle.
The converging-diverging nozzle invented by Gustaf de Laval represents the cornerstone of supersonic aerodynamics and modern rocket propulsion. In subsonic compressible flow, fluid accelerates as channel cross-sectional area contracts (dA < 0 implies dV > 0). At the minimum area throat (A = A*), the acoustic speed of sound is attained (Mach M = 1), choking the nozzle and establishing the maximum possible mass flow rate for given stagnation conditions. To achieve further acceleration into the supersonic regime (M > 1), the nozzle duct must diverge (dA > 0 implies dV > 0), converting thermal enthalpy into kinetic energy according to the area-velocity relation dA/A = (M^2 - 1) * dV/V. The exact isentropic area-Mach relation establishes that (A/A*)^2 = (1/M^2) * [(2/(gamma+1)) * (1 + (gamma-1)/2 * M^2)]^((gamma+1)/(gamma-1)). In this laboratory experiment, a transparent supersonic de Laval nozzle with an optical quartz sidewall is connected to a high-pressure air reservoir. Multiple static pressure taps along the divergent nozzle duct measure local static pressures P, from which local isentropic Mach numbers M are deduced. Stepping across divergent area ratios A/A* validates the supersonic expansion gradient dM / d(A/A*) and illustrates the three operating regimes: over-expanded with oblique/normal shock waves, perfectly expanded design point, and under-expanded expansion fans.
Viva voce questions & analytical theory
Why must a nozzle diverge to accelerate flow to supersonic speeds?
In compressible flow, the continuity equation yields dA/A = (M^2 - 1) * (dV/V). For subsonic flow (M < 1), dA and dV have opposite signs, so convergence accelerates flow. For supersonic flow (M > 1), (M^2 - 1) is positive, so dA and dV have the same sign—divergence is required to accelerate.
What is the physical meaning of sonic choking at the nozzle throat?
At the throat, acoustic speed of sound is reached (M = 1), and the mass flow rate reaches its absolute maximum for the given stagnation conditions: m_dot_max = (P_0 * A* / sqrt(T_0)) * sqrt(gamma/R) * (2/(gamma+1))^((gamma+1)/(2*(gamma-1))). Further lowering back-pressure cannot increase mass flow.
What is the critical pressure ratio P*/P_0 for air at sonic choking?
For a diatomic gas like air with gamma = 1.4, the isentropic critical pressure ratio is P*/P_0 = (2 / (gamma + 1))^(gamma / (gamma - 1)) = (2 / 2.4)^3.5 = 0.52828.
What characterizes an over-expanded nozzle condition?
An over-expanded nozzle occurs when the exit static pressure P_exit is lower than the ambient back-pressure P_amb. To compress the fluid up to ambient pressure, oblique shock waves or a normal shock train form at or inside the nozzle exit.
What characterizes an under-expanded nozzle condition?
An under-expanded nozzle occurs when exit static pressure exceeds ambient back-pressure (P_exit > P_amb). The flow expands further upon leaving the nozzle lip through a centered Prandtl-Meyer expansion fan, creating supersonic expansion plumes.
How does boundary layer growth on nozzle walls affect exit Mach number?
Viscous boundary layers grow along the divergent walls, creating a displacement thickness delta* that reduces the effective aerodynamic core cross-sectional area. Consequently, the effective area ratio (A/A*)_eff is smaller than geometric area ratio, yielding a slightly lower exit Mach number.
Experimental precautions and wind tunnel safety
- Ensure the compressed air supply is thoroughly desiccated to prevent water vapor condensation shocks in the supersonic test section.
- Wear mandatory hearing protection during high-pressure blowdown runs due to high-decibel acoustic exhaust shear noise.
- Monitor reservoir pressure decay to terminate runs before stagnation pressure drops below the minimum nozzle starting pressure.
- Inspect optical quartz viewing ports for structural stress cracks before pressurizing the nozzle duct.
Theoretical assumptions & flow limits
- Boundary layer displacement thickness on divergent walls constricts aerodynamic area, slightly reducing Mach number below geometric A/A*.
- Assumes one-dimensional, isentropic, calorically perfect gas expansion along the central streamline.
- If back-pressure is too high, normal or oblique shock waves will establish inside the divergent duct, violating isentropic flow.
- Real gas vibrational excitation effects are negligible at low stagnation temperatures (T_0 < 600 K).