6.5 Prandtl-Meyer expansion waves

Prandtl-Meyer expansion waves are key to understanding supersonic flow behavior. They occur when supersonic flow encounters convex corners, causing the flow to expand and accelerate. This process is crucial for designing high-speed aerodynamic components.

These waves fan out from corners, decreasing pressure, density, and temperature while increasing velocity and Mach number. The Prandtl-Meyer function relates Mach number to flow deflection angle, helping engineers predict and control supersonic flow in various applications.

Prandtl-Meyer expansion waves

• Prandtl-Meyer expansion waves are a fundamental concept in compressible fluid dynamics that describe the behavior of supersonic flow as it expands around convex corners or surfaces
• Expansion waves occur when a supersonic flow encounters a sudden change in geometry, such as a sharp corner, causing the flow to expand and accelerate
• Understanding Prandtl-Meyer expansion waves is crucial for designing supersonic nozzles, airfoils, and other aerodynamic components in high-speed applications

Definition of expansion waves

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• Expansion waves are defined as a series of Mach waves that fan out from a convex corner or surface in a supersonic flow
• These waves cause a continuous decrease in pressure, density, and temperature, while increasing the flow velocity and Mach number
• The expansion process is isentropic, meaning that it occurs without heat transfer or entropy change

Assumptions in Prandtl-Meyer theory

• The Prandtl-Meyer theory assumes that the flow is steady, inviscid, and adiabatic
• It also assumes that the gas is ideal and calorically perfect, with constant specific heats
• The theory neglects the effects of boundary layers, shock waves, and flow separation

Prandtl-Meyer function

• The Prandtl-Meyer function, denoted as $\nu(M)$, relates the Mach number to the flow deflection angle in an expansion wave

• It is defined as: $\nu(M) = \sqrt{\frac{\gamma+1}{\gamma-1}} \tan^{-1} \sqrt{\frac{\gamma-1}{\gamma+1}(M^2-1)} - \tan^{-1} \sqrt{M^2-1}$

  where $\gamma$ is the specific heat ratio and $M$ is the Mach number

• The Prandtl-Meyer function is a key parameter in determining the flow properties across an expansion wave

Relationship between Mach number and flow deflection

• The Prandtl-Meyer function establishes a unique relationship between the Mach number and the flow deflection angle
• As the flow expands around a corner, the deflection angle increases, and the Mach number increases accordingly
• The maximum deflection angle that can be achieved through an expansion wave is called the Prandtl-Meyer angle, which corresponds to the maximum Mach number

Expansion wave geometry

• Expansion waves originate from the corner or surface where the flow is deflected
• The waves propagate at the local speed of sound relative to the flow, forming a fan-like structure
• The expansion fan consists of an infinite number of Mach waves, each representing a small increment in flow deflection and Mach number

Centered expansion waves

• Centered expansion waves occur when the flow is deflected by a sharp corner with a finite angle
• In this case, the expansion waves emanate from a single point (the corner) and spread out in a radial pattern
• The flow properties (pressure, density, temperature, and velocity) vary continuously across the expansion fan

Prandtl-Meyer expansion fan

• The Prandtl-Meyer expansion fan is the region bounded by the first and last Mach waves in an expansion wave
• Within the expansion fan, the flow properties change gradually and continuously
• The flow outside the expansion fan remains unaffected by the expansion process

Mach waves in expansion flow

• Mach waves are weak disturbances that propagate at the local speed of sound relative to the flow

• In an expansion wave, the Mach waves are inclined at the Mach angle, which is defined as: $\mu = \sin^{-1} \left(\frac{1}{M}\right)$

  where $M$ is the local Mach number

• The Mach waves carry information about the changes in flow properties across the expansion fan

Weak vs strong expansion waves

• Expansion waves can be classified as weak or strong, depending on the magnitude of the flow deflection
• Weak expansion waves occur when the flow deflection angle is small, resulting in a gradual change in flow properties
• Strong expansion waves, on the other hand, involve large deflection angles and rapid changes in flow properties
• The distinction between weak and strong expansion waves is important for accurately predicting the flow behavior and avoiding flow separation

Supersonic flow over convex corners

• When a supersonic flow encounters a convex corner, it undergoes a Prandtl-Meyer expansion
• The flow expands around the corner, resulting in an increase in Mach number and a decrease in pressure, density, and temperature
• The flow deflection angle is determined by the corner angle and the upstream Mach number
• Designing supersonic airfoils and nozzles often involves carefully shaping the convex surfaces to control the expansion process

Prandtl-Meyer expansion in nozzles

• Prandtl-Meyer expansion is a key phenomenon in the design of supersonic nozzles
• In a converging-diverging nozzle, the flow accelerates through the converging section and becomes supersonic in the diverging section
• The diverging section of the nozzle acts as a series of small convex corners, causing the flow to undergo a continuous Prandtl-Meyer expansion
• By carefully designing the nozzle contour, engineers can control the expansion process and achieve the desired exit Mach number and flow properties

Numerical methods for expansion waves

• Analyzing Prandtl-Meyer expansion waves often requires numerical methods, especially for complex geometries or non-ideal flow conditions
• Finite difference and finite volume methods are commonly used to solve the governing equations of compressible flow
• These methods discretize the flow domain into a grid and solve the conservation equations at each grid point
• Numerical simulations can provide detailed information about the flow field, including the distribution of Mach number, pressure, density, and temperature

Compressible flow analogy for expansion

• The Prandtl-Meyer expansion process can be understood through the compressible flow analogy
• In this analogy, the expansion wave is treated as a series of infinitesimal waves, each causing a small change in flow properties
• The analogy allows for the application of the method of characteristics, which is a powerful technique for solving hyperbolic partial differential equations
• By tracing the characteristic lines (Mach waves) through the flow field, the flow properties can be determined at any point in the expansion fan

Prandtl-Meyer function tables and charts

• Prandtl-Meyer function tables and charts are valuable tools for quickly determining the flow properties across an expansion wave
• These tables and charts provide the relationship between the Mach number, flow deflection angle, and Prandtl-Meyer function for a given specific heat ratio
• Engineers and researchers often use these resources to estimate the flow conditions without the need for complex calculations
• Prandtl-Meyer function tables and charts are particularly useful for preliminary design and analysis of supersonic flow systems

Applications of Prandtl-Meyer expansions

• Prandtl-Meyer expansion waves have numerous applications in aerospace engineering and high-speed fluid dynamics
• Supersonic aircraft design: Prandtl-Meyer expansions are used to design the contours of supersonic airfoils and wings to minimize drag and optimize performance
• Rocket nozzles: The diverging section of a rocket nozzle is designed using Prandtl-Meyer expansion theory to achieve the desired exit Mach number and thrust
• Wind tunnels: Supersonic wind tunnels often employ Prandtl-Meyer expansions to generate high-speed flow for testing aircraft and spacecraft models
• Gas dynamics: Prandtl-Meyer expansions are important in understanding the behavior of compressible gases in various industrial and scientific applications, such as gas pipelines, turbomachinery, and combustion systems