Prandtl-Meyer Expansion

This module provides functions for analyzing Prandtl-Meyer expansion waves, which occur when supersonic flow turns around a convex corner or expands through a diverging channel.

Overview

Prandtl-Meyer expansion waves are isentropic compression waves that occur when supersonic flow encounters a convex corner or expansion. Unlike shock waves, expansion waves:

• Are isentropic (reversible process)
• Increase Mach number and velocity
• Decrease pressure, density, and temperature
• Maintain total temperature and pressure
• Can turn flow through large angles without losses
• Consist of infinitesimally weak waves (Mach waves)

Theory

Prandtl-Meyer expansion is governed by the Prandtl-Meyer function ν(M), which relates the Mach number to the maximum turning angle:

Prandtl-Meyer function: $\nu(M) = \sqrt{\frac{\gamma+1}{\gamma-1}} \arctan\sqrt{\frac{\gamma-1}{\gamma+1}(M^2-1)} - \arctan\sqrt{M^2-1}$

Turning angle relationship: $\theta_{12} = \nu(M_2) - \nu(M_1)$

Property ratios (from isentropic relations):

• Pressure: $\frac{p_2}{p_1} = \left(\frac{1 + \frac{\gamma-1}{2}M_1^2}{1 + \frac{\gamma-1}{2}M_2^2}\right)^{\frac{\gamma}{\gamma-1}}$
• Temperature: $\frac{T_2}{T_1} = \frac{1 + \frac{\gamma-1}{2}M_1^2}{1 + \frac{\gamma-1}{2}M_2^2}$
• Density: $\frac{\rho_2}{\rho_1} = \left(\frac{1 + \frac{\gamma-1}{2}M_1^2}{1 + \frac{\gamma-1}{2}M_2^2}\right)^{\frac{1}{\gamma-1}}$

Where:

• $M_1$, $M_2$ = upstream and downstream Mach numbers
• $\theta_{12}$ = turning angle from state 1 to state 2
• $\gamma$ = specific heat ratio

Functions

prandtl_meyer(M, gamma=1.4)

expand_mach2(M1, theta, gamma=1.4)

expand_p2(M1, theta, gamma=1.4)

theta_p(pratio, M1, gamma=1.4)

Function Details

prandtl_meyer

prandtl_meyer(M, gamma=1.4)

Calculate the Prandtl-Meyer function ν(M) which represents the maximum turning angle from sonic conditions.

Arguments:

• M::Real: Mach number (must be ≥ 1)
• gamma::Real=1.4: Specific heat ratio

Returns:

• Float64: Prandtl-Meyer angle ν in degrees

Formula: $\nu(M) = \sqrt{\frac{\gamma+1}{\gamma-1}} \arctan\sqrt{\frac{\gamma-1}{\gamma+1}(M^2-1)} - \arctan\sqrt{M^2-1}$

Example:

julia> prandtl_meyer(2.0)
26.379760813416906

julia> prandtl_meyer(3.0)
49.75681638417519

julia> prandtl_meyer(1.0)
0.0

expand_mach2

expand_mach2(M1, theta, gamma=1.4)

Calculate the downstream Mach number after a Prandtl-Meyer expansion through angle θ.

Arguments:

• M1::Real: Upstream Mach number
• theta::Real: Turning angle in degrees
• gamma::Real=1.4: Specific heat ratio

Returns:

• Float64: Downstream Mach number

Example:

julia> expand_mach2(2.0, 20.0)
2.3848314132746953

julia> expand_mach2(1.5, 10.0)
1.7985676229179285

expand_p2

expand_p2(M1, theta, gamma=1.4)

Calculate the pressure ratio (p₁/p₂) across a Prandtl-Meyer expansion.

Arguments:

• M1::Real: Upstream Mach number
• theta::Real: Turning angle in degrees
• gamma::Real=1.4: Specific heat ratio

Returns:

• Float64: Pressure ratio p₁/p₂

Example:

julia> expand_p2(2.0, 20.0)
1.687094471207049

julia> expand_p2(3.0, 15.0)
1.4720270270270274

theta_p

theta_p(pratio, M1, gamma=1.4)

Calculate the turning angle required to achieve a specified pressure ratio across an expansion.

Arguments:

• pratio::Real: Desired pressure ratio p₁/p₂
• M1::Real: Upstream Mach number
• gamma::Real=1.4: Specific heat ratio

Returns:

• Float64: Required turning angle in degrees

Example:

julia> theta_p(1.5, 2.0)
14.396394107665233

julia> theta_p(2.0, 2.5)
22.69419642857143

Applications

Supersonic Nozzle Design

Design the diverging section of a supersonic nozzle:

# Design requirements
M_exit = 3.5      # Exit Mach number
M_throat = 1.0    # Throat conditions
theta_max = 15.0  # Maximum wall angle

println("Supersonic Nozzle Design:")
println("Exit Mach: $M_exit")
println("Maximum wall angle: $theta_max°")

# Calculate required turning angle
nu_exit = prandtl_meyer(M_exit)
nu_throat = prandtl_meyer(M_throat)  # = 0 for M = 1
theta_required = nu_exit - nu_throat

println("Required turning angle: $(round(theta_required, digits=1))°")

# Check if design is feasible
if theta_required/2 <= theta_max
    println("Design feasible - wall angle: $(round(theta_required/2, digits=1))°")
    
    # Calculate area ratio
    area_ratio = area_ratio_at(M_exit)
    println("Area ratio A_exit/A*: $(round(area_ratio, digits=2))")
else
    println("Design not feasible - requires curved walls or multiple sections")
    
    # Calculate required sections
    n_sections = ceil(theta_required / (2 * theta_max))
    println("Minimum sections required: $n_sections")
end

Centered Expansion Fan

Analyze a centered expansion fan:

M1 = 1.5
theta_total = 30.0  # Total turning angle
n_waves = 6         # Number of Mach waves

println("Centered Expansion Fan Analysis:")
println("Initial Mach: $M1")
println("Total turning: $theta_total°")
println("Number of waves: $n_waves")

theta_step = theta_total / n_waves
println("\nWave\tθ (°)\tM\tμ (°)\tp/p₁")
println("----\t-----\t-----\t-----\t-----")

for i in 0:n_waves
    theta_current = i * theta_step
    
    if i == 0
        M_current = M1
        p_ratio = 1.0
    else
        M_current = expand_mach2(M1, theta_current)
        p_ratio = 1.0 / expand_p2(M1, theta_current)
    end
    
    mu = asind(1/M_current)  # Mach angle
    
    println("$i\t$(round(theta_current, digits=1))\t$(round(M_current, digits=3))\t$(round(mu, digits=1))\t$(round(p_ratio, digits=3))")
end

Flow Over Expansion Corner

Analyze flow over a sharp expansion corner:

# Supersonic flow over a backward-facing step
M1 = 2.2
step_angle = 12.0  # degrees

println("Flow Over Expansion Corner:")
println("Upstream Mach: $M1")
println("Corner angle: $step_angle°")

# Expansion analysis
M2 = expand_mach2(M1, step_angle)
p_ratio = expand_p2(M1, step_angle)

# Calculate other properties
T_ratio = (1 + 0.2*M1^2) / (1 + 0.2*M2^2)
rho_ratio = (T_ratio)^(1/0.4)

println("\nDownstream conditions:")
println("Mach number: $(round(M2, digits=3))")
println("Pressure ratio p₁/p₂: $(round(p_ratio, digits=3))")
println("Temperature ratio T₁/T₂: $(round(T_ratio, digits=3))")
println("Density ratio ρ₁/ρ₂: $(round(rho_ratio, digits=3))")

# Wave angles
mu1 = asind(1/M1)  # Upstream Mach angle
mu2 = asind(1/M2)  # Downstream Mach angle

println("\nWave characteristics:")
println("Upstream Mach angle: $(round(mu1, digits=1))°")
println("Downstream Mach angle: $(round(mu2, digits=1))°")

Method of Characteristics Application

Simple application showing characteristic line method concepts:

# Prandtl-Meyer expansion around a corner
M1 = 2.0
theta_total = 30.0  # Total turning angle
n_steps = 6         # Number of characteristic steps

println("Method of Characteristics Example:")
println("Initial Mach: $M1")
println("Total turning: $theta_total°")
println("Steps: $n_steps")

theta_step = theta_total / n_steps
M_current = M1

println("\nStep\tθ(°)\tM\tν(°)\tμ(°)")
println("----\t----\t-----\t-----\t-----")

for i in 0:n_steps
    theta_current = i * theta_step
    
    if i == 0
        nu_current = prandtl_meyer(M_current)
        mu_current = asind(1/M_current)  # Mach angle
        println("$i\t$(round(theta_current, digits=1))\t$(round(M_current, digits=3))\t$(round(nu_current, digits=2))\t$(round(mu_current, digits=1))")
    else
        M_current = expand_mach2(M1, theta_current)
        nu_current = prandtl_meyer(M_current)
        mu_current = asind(1/M_current)
        println("$i\t$(round(theta_current, digits=1))\t$(round(M_current, digits=3))\t$(round(nu_current, digits=2))\t$(round(mu_current, digits=1))")
    end
end

# Compare with exact solution
M_exact = expand_mach2(M1, theta_total)
println("\nExact solution M_final: $(round(M_exact, digits=3))")
println("Final step M: $(round(M_current, digits=3))")
println("Error: $(round(abs(M_exact - M_current)/M_exact * 100, digits=2))%")

Different Gases

Analyze Prandtl-Meyer expansion for different gases:

M1 = 2.0
theta = 20.0

gases = [
    ("Air", 1.4),
    ("Helium", 1.67),
    ("Argon", 1.67),
    ("CO₂", 1.3)
]

println("Gas Effects on Prandtl-Meyer Expansion:")
println("M₁ = $M1, θ = $theta°")
println("Gas\tγ\tM₂\tp₁/p₂\tν₁ (°)\tν₂ (°)")
println("---\t----\t----\t-----\t------\t------")

for (gas, gamma) in gases
    M2 = expand_mach2(M1, theta, gamma)
    p_ratio = expand_p2(M1, theta, gamma)
    nu1 = prandtl_meyer(M1, gamma)
    nu2 = prandtl_meyer(M2, gamma)
    
    println("$gas\t$gamma\t$(round(M2, digits=3))\t$(round(p_ratio, digits=2))\t$(round(nu1, digits=1))\t$(round(nu2, digits=1))")
end

Maximum Turning Angle

Find the maximum theoretical turning angle:

# For very high Mach numbers, ν approaches a maximum value
M_high = 100.0
nu_max = prandtl_meyer(M_high)

println("Maximum Prandtl-Meyer angle:")
println("ν(M→∞) ≈ $(round(nu_max, digits=1))°")

# Theoretical maximum for γ = 1.4
nu_theoretical = (sqrt((1.4+1)/(1.4-1)) - 1) * 90
println("Theoretical maximum: $(round(nu_theoretical, digits=1))°")

Validation Examples

Prandtl-Meyer Function Values

Calculate the Prandtl-Meyer function for different Mach numbers:

println("M\tν(M) (°)")
println("---\t-------")

for M in 1.0:0.2:4.0
    if M > 1.0  # Function only valid for M > 1
        nu = prandtl_meyer(M)
        println("$(round(M, digits=1))\t$(round(nu, digits=1))")
    else
        println("$(round(M, digits=1))\t0.0")
    end
end

Inverse Problem - Find Mach from Turning Angle

Given a turning angle, find the required initial Mach number:

M2_target = 3.0
theta_given = 25.0  # degrees

# Calculate required upstream Mach number
nu2 = prandtl_meyer(M2_target)
nu1 = nu2 - theta_given

# Find M1 by inverse of Prandtl-Meyer function (iterative)
M1_estimate = 1.5  # Initial guess
tolerance = 1e-6

for i in 1:100
    nu_calc = prandtl_meyer(M1_estimate)
    error = nu_calc - nu1
    
    if abs(error) < tolerance
        break
    end
    
    # Simple Newton-like iteration
    M1_estimate -= error / 10.0
    M1_estimate = max(M1_estimate, 1.001)  # Keep supersonic
end

println("Inverse Prandtl-Meyer Problem:")
println("Target M₂: $M2_target")
println("Given turning angle: $theta_given°")
println("Required M₁: $(round(M1_estimate, digits=3))")

# Verify
M2_check = expand_mach2(M1_estimate, theta_given)
println("Verification M₂: $(round(M2_check, digits=3))")

Limitations and Considerations

1. Isentropic assumption: Valid only for gradual expansions without shock waves
2. Perfect gas: Constant specific heats and ideal gas behavior
3. Steady flow: Time-invariant conditions
4. Two-dimensional flow: Assumes planar or axisymmetric geometry
5. Gradual expansion: Sharp corners may cause finite-strength waves
6. Supersonic flow: Functions only valid for M > 1

For viscous effects, unsteady phenomena, or real gas behavior, additional analysis methods are required.

See Also

• Oblique Shock Waves: For compression processes
• Isentropic Relations: For property ratios
• Normal Shock Waves: For normal compression
• Nozzle Analysis: For nozzle design applications