3.5 Prandtl-Meyer expansion waves

Prandtl-Meyer expansion waves are a key concept in supersonic aerodynamics. They describe how supersonic flow behaves when expanding around corners or through nozzles, relating changes in flow direction to Mach number changes.

Understanding these waves is crucial for designing supersonic aircraft, engines, and wind tunnels. The theory assumes ideal conditions but provides valuable insights into high-speed flow behavior and forms the basis for more complex analyses.

Prandtl-Meyer expansion theory

• Describes the behavior of supersonic flow as it expands around a convex corner or through a diverging nozzle
• Based on the assumption of isentropic, irrotational, and inviscid flow
• Relates the change in flow direction to the change in Mach number through the Prandtl-Meyer function

Prandtl-Meyer function

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• Represents the angle through which a supersonic flow must turn to accelerate from Mach 1 to a given Mach number
• Denoted by the symbol $\nu(M)$ and expressed as a function of the specific heat ratio $\gamma$ and Mach number $M$
• Formula: $\nu(M) = \sqrt{\frac{\gamma+1}{\gamma-1}} \tan^{-1} \sqrt{\frac{\gamma-1}{\gamma+1}(M^2-1)} - \tan^{-1} \sqrt{M^2-1}$
• Tabulated values of the Prandtl-Meyer function are available for quick reference in aerodynamic calculations

Mach number vs turning angle

• As a supersonic flow expands, its Mach number increases while the flow direction changes
• The relationship between the turning angle $\theta$ and the change in Prandtl-Meyer function $\Delta \nu$ is given by: $\theta = \nu(M_2) - \nu(M_1)$
  • $M_1$ is the initial Mach number before expansion
  • $M_2$ is the final Mach number after expansion
• The maximum turning angle occurs when the flow expands to an infinite Mach number, known as the "vacuum" condition

Centered expansion waves

• Form when supersonic flow encounters a sharp convex corner, causing the flow to expand and accelerate
• Consist of an infinite number of Mach waves emanating from the corner, each representing a small increment in flow turning
• The expansion fan is bounded by the initial and final Mach waves, which define the region of flow affected by the expansion
• The flow properties (pressure, density, temperature) decrease smoothly across the expansion fan

Characteristics of expansion fan

• The expansion fan is isentropic, meaning that the entropy remains constant across the fan
• The flow is irrotational, implying that the velocity field has zero curl and can be represented by a potential function
• The expansion process is reversible, as the flow can be compressed back to its original state by passing through a convergent section
• The flow downstream of the expansion fan is uniform and parallel to the surface, with a higher Mach number than the upstream flow

Supersonic flow over convex corners

• When supersonic flow encounters a convex corner, an expansion wave forms to turn the flow and match the downstream conditions
• The flow properties change smoothly across the expansion fan, unlike the abrupt changes across oblique shock waves
• The expansion fan is centered at the corner and spreads out at the Mach angle $\mu$

Expansion wave geometry

• The Mach angle $\mu$ is related to the local Mach number $M$ by: $\sin \mu = \frac{1}{M}$
• The initial and final Mach waves of the expansion fan form angles $\mu_1$ and $\mu_2$ with the flow direction, respectively
• The total turning angle $\theta$ of the flow is equal to the difference in Prandtl-Meyer angles: $\theta = \nu(M_2) - \nu(M_1)$
• The geometry of the expansion fan can be constructed using the method of characteristics, which traces the paths of Mach waves through the flow field

Mach wave angle

• The Mach wave angle $\mu$ decreases as the Mach number increases, meaning that the expansion fan becomes narrower at higher Mach numbers
• At the vacuum condition (infinite Mach number), the Mach wave angle approaches zero, and the expansion fan collapses to a single line
• The Mach wave angle is a key parameter in determining the shape and extent of the expansion fan

Weak vs strong expansion waves

• Weak expansion waves occur when the turning angle is small, resulting in a relatively small change in Mach number across the fan
• Strong expansion waves involve large turning angles and significant increases in Mach number
• The strength of the expansion wave depends on the geometry of the convex corner and the initial Mach number of the flow
• In general, stronger expansion waves lead to more pronounced changes in flow properties and greater flow acceleration

Isentropic expansion process

• The expansion of supersonic flow through a Prandtl-Meyer fan is an isentropic process, meaning that the entropy remains constant
• Isentropic flow is a reversible process, implying that the flow can be compressed back to its original state without any losses
• The isentropic flow equations relate the changes in flow properties (pressure, density, temperature) to the change in Mach number

Compressible flow equations

• The isentropic pressure ratio: $\frac{p}{p_0} = \left(1 + \frac{\gamma-1}{2}M^2\right)^{-\frac{\gamma}{\gamma-1}}$
• The isentropic density ratio: $\frac{\rho}{\rho_0} = \left(1 + \frac{\gamma-1}{2}M^2\right)^{-\frac{1}{\gamma-1}}$
• The isentropic temperature ratio: $\frac{T}{T_0} = \left(1 + \frac{\gamma-1}{2}M^2\right)^{-1}$
  • $p_0$, $\rho_0$, and $T_0$ are the stagnation (total) pressure, density, and temperature, respectively
• These equations describe how the flow properties change as a function of Mach number in an isentropic expansion process

Stagnation properties in expansion

• Stagnation properties represent the conditions that would be achieved if the flow were brought to rest isentropically
• In an isentropic expansion, the stagnation pressure, density, and temperature remain constant throughout the process
• The stagnation properties are related to the local flow properties by the isentropic flow equations
• Knowing the stagnation properties and the local Mach number allows for the calculation of the local flow properties at any point in the expansion fan

Expansion ratio

• The expansion ratio is the ratio of the cross-sectional area of the flow after expansion to the area before expansion
• For a given initial Mach number and turning angle, the expansion ratio determines the final Mach number and flow properties
• The expansion ratio can be calculated using the area-Mach number relation for isentropic flow: $\frac{A}{A^*} = \frac{1}{M}\left[\frac{2}{\gamma+1}\left(1+\frac{\gamma-1}{2}M^2\right)\right]^{\frac{\gamma+1}{2(\gamma-1)}}$
  • $A^*$ is the critical area, corresponding to the sonic condition (Mach 1)
• Higher expansion ratios lead to greater flow acceleration and higher final Mach numbers

Applications of Prandtl-Meyer flow

• Prandtl-Meyer expansion theory has numerous applications in the design and analysis of supersonic flow devices and systems
• Understanding the behavior of expansion waves is crucial for optimizing the performance of supersonic nozzles, diffusers, and other components
• Prandtl-Meyer flow concepts are also used in the study of supersonic flow over airfoils, wings, and other aerodynamic shapes

Supersonic nozzle design

• Prandtl-Meyer expansion theory is used to design supersonic nozzles that efficiently accelerate the flow to high Mach numbers
• The contour of the nozzle is shaped to create a series of expansion waves that gradually turn and accelerate the flow
• The nozzle geometry is optimized to minimize losses and achieve the desired exit Mach number and flow properties
• Supersonic nozzles are used in rocket engines, wind tunnels, and other high-speed flow applications

Minimum length nozzles

• Minimum length nozzles are designed to achieve a specified exit Mach number in the shortest possible distance
• These nozzles use a sharp throat followed by a rapid expansion section, creating a strong expansion fan that quickly accelerates the flow
• Minimum length nozzles are often used in compact supersonic wind tunnels and rocket engines where space is limited
• Prandtl-Meyer flow theory is essential for determining the optimal contour and expansion ratio for minimum length nozzles

Thrust vectoring

• Thrust vectoring is a technique used to control the direction of the thrust generated by a rocket engine or jet engine
• One method of thrust vectoring involves the use of secondary injection ports that create local expansion waves to deflect the main flow
• By controlling the location and strength of these expansion waves, the thrust vector can be adjusted to steer the vehicle
• Prandtl-Meyer expansion theory is used to predict the effect of secondary injection on the main flow and optimize the thrust vectoring system

Numerical methods for expansion waves

• While Prandtl-Meyer theory provides analytical solutions for simple expansion wave geometries, numerical methods are often required for more complex flow fields
• Numerical methods discretize the flow domain into a grid of points and solve the governing equations (continuity, momentum, energy) at each point
• These methods can handle arbitrary geometries, non-ideal gas effects, and other complexities that are difficult to treat analytically

Method of characteristics

• The method of characteristics is a numerical technique that solves the supersonic flow equations along characteristic lines (Mach waves)
• The flow domain is divided into a network of characteristic lines, and the flow properties are calculated at the intersection points
• The method of characteristics is particularly well-suited for handling expansion waves, as the characteristic lines naturally follow the Mach wave angles
• This method can be used to construct the complete flow field, including the expansion fan and the uniform flow regions upstream and downstream

Finite difference schemes

• Finite difference schemes are numerical methods that approximate the partial derivatives in the flow equations using finite differences
• The flow domain is discretized into a structured or unstructured grid, and the flow properties are calculated at each grid point
• Finite difference schemes can be explicit (calculating the flow properties at the next time step directly) or implicit (solving a system of equations for the flow properties at the next time step)
• These schemes are more versatile than the method of characteristics and can handle a wider range of flow conditions and geometries

Boundary conditions for expansion

• Proper treatment of boundary conditions is essential for accurately simulating expansion waves using numerical methods
• At the inflow boundary, the flow properties (Mach number, pressure, density, etc.) must be specified based on the upstream conditions
• At the outflow boundary, the flow is typically assumed to be supersonic, and no information can propagate upstream (characteristic boundary condition)
• Along solid walls, the flow must satisfy the no-penetration condition (zero normal velocity) and the tangency condition (flow parallel to the wall)
• For inviscid flow, the wall pressure is determined by the Prandtl-Meyer expansion theory, while for viscous flow, the no-slip condition (zero velocity at the wall) must be applied

Limitations of Prandtl-Meyer theory

• While Prandtl-Meyer expansion theory is a powerful tool for analyzing supersonic flow, it is based on several simplifying assumptions that limit its applicability in certain situations
• Understanding these limitations is important for interpreting the results of Prandtl-Meyer calculations and knowing when more advanced methods may be necessary

Assumptions and simplifications

• Prandtl-Meyer theory assumes that the flow is steady, inviscid, and adiabatic (no heat transfer)
• The gas is assumed to be ideal, with constant specific heats and a constant specific heat ratio $\gamma$
• The flow is assumed to be irrotational and isentropic, meaning that there are no shocks or other sources of entropy production
• These assumptions are reasonable for many high-speed flow applications but may break down in the presence of strong viscous effects, heat transfer, or non-ideal gas behavior

Real gas effects

• At very high Mach numbers or in flows with significant temperature variations, real gas effects become important
• Real gases may exhibit variable specific heats, chemical reactions (dissociation, ionization), and other non-ideal behavior
• These effects can alter the expansion process and the resulting flow properties, leading to deviations from the ideal Prandtl-Meyer solutions
• Real gas effects are particularly important in hypersonic flows (Mach > 5) and in flows with strong shocks or expansions

Viscous and heat transfer effects

• Prandtl-Meyer theory neglects the effects of viscosity and heat transfer, which can be significant in certain flow situations
• Viscous effects, such as boundary layer growth and flow separation, can modify the effective geometry of the expansion and alter the flow properties
• Heat transfer between the flow and the surrounding surfaces can change the temperature distribution and affect the expansion process
• In flows with strong viscous or heat transfer effects, more advanced computational fluid dynamics (CFD) methods may be required to accurately predict the flow behavior