Isentropic Relations

This module provides functions for calculating isentropic flow relations, which describe the relationships between flow properties in compressible flow without heat transfer or friction.

Overview

Isentropic flow relations are fundamental to compressible flow analysis. They relate static and stagnation (total) properties through the Mach number for flows where entropy remains constant. These relationships are essential for:

• Nozzle design and analysis
• Pitot tube measurements
• Flow property calculations
• Wind tunnel calibration

Theory

For an ideal gas undergoing isentropic flow, the relationships between stagnation and static properties are:

• Temperature: $\frac{T_0}{T} = 1 + \frac{\gamma-1}{2}M^2$
• Pressure: $\frac{p_0}{p} = \left(1 + \frac{\gamma-1}{2}M^2\right)^{\frac{\gamma}{\gamma-1}}$
• Density: $\frac{\rho_0}{\rho} = \left(1 + \frac{\gamma-1}{2}M^2\right)^{\frac{1}{\gamma-1}}$

Where:

• $M$ is mach number
• $\gamma$ is specific heat ratio (1.4 for air at standard conditions)
• Subscript 0 denotes stagnation conditions

Functions

t0_over_t(M::Real, gamma::Real=1.4)

p0_over_p(M::Real, gamma::Real=1.4)

rho0_over_rho(M::Real, gamma::Real=1.4)

Function Details

t0_over_t

t0_over_t(M, gamma=1.4)

Calculate the stagnation to static temperature ratio for isentropic flow.

Arguments:

• M::Real: Mach number
• gamma::Real=1.4: Specific heat ratio

Returns:

• Float64: Stagnation to static temperature ratio (T₀/T)

Formula: $\frac{T_0}{T} = 1 + \frac{\gamma-1}{2}M^2$

Example:

julia> t0_over_t(2.0)
1.8

julia> t0_over_t(1.5, 1.67)  # Helium
2.005

p0_over_p

p0_over_p(M, gamma=1.4)

Calculate the stagnation to static pressure ratio for isentropic flow.

Arguments:

• M::Real: Mach number
• gamma::Real=1.4: Specific heat ratio

Returns:

• Float64: Stagnation to static pressure ratio (p₀/p)

Formula: $\frac{p_0}{p} = \left(1 + \frac{\gamma-1}{2}M^2\right)^{\frac{\gamma}{\gamma-1}}$

Example:

julia> p0_over_p(2.0)
7.824481684411352

julia> p0_over_p(1.0)
1.8929441933097926

rho0_over_rho

rho0_over_rho(M, gamma=1.4)

Calculate the stagnation to static density ratio for isentropic flow.

Arguments:

• M::Real: Mach number
• gamma::Real=1.4: Specific heat ratio

Returns:

• Float64: Stagnation to static density ratio (ρ₀/ρ)

Formula: $\frac{\rho_0}{\rho} = \left(1 + \frac{\gamma-1}{2}M^2\right)^{\frac{1}{\gamma-1}}$

Example:

julia> rho0_over_rho(2.0)
4.346837646954653

julia> rho0_over_rho(3.0)
14.621473116765412

Applications

Wind Tunnel Calibration

Calculate test section conditions from stagnation measurements:

# Known stagnation conditions
p0_measured = 150000  # Pa
T0_measured = 300     # K
M_test = 1.8

# Calculate static conditions
p_static = p0_measured / p0_over_p(M_test)
T_static = T0_measured / t0_over_t(M_test)

println("Test section conditions:")
println("Static pressure: $(round(p_static/1000, digits=1)) kPa")
println("Static temperature: $(round(T_static, digits=1)) K")

Pitot Tube Analysis

Calculate dynamic pressure coefficient:

M = 0.8  # Subsonic flow

# Pressure coefficient from pitot measurements
p0_p_ratio = p0_over_p(M)
q_coefficient = p0_p_ratio - 1

println("Dynamic pressure coefficient: $(round(q_coefficient, digits=3))")

# Compare with incompressible approximation
q_incompressible = 0.5 * 1.4 * M^2
error = abs(q_coefficient - q_incompressible) / q_coefficient * 100

println("Incompressible error: $(round(error, digits=1))%")

Nozzle Exit Conditions

Calculate exit conditions for known stagnation state:

# Rocket nozzle conditions
p0_chamber = 2e6  # Pa (20 bar)
T0_chamber = 3000 # K
M_exit = 3.5

# Exit conditions
p_exit = p0_chamber / p0_over_p(M_exit)
T_exit = T0_chamber / t0_over_t(M_exit)
rho_ratio = rho0_over_rho(M_exit)

println("Rocket nozzle exit conditions:")
println("Exit pressure: $(round(p_exit/1000, digits=1)) kPa")
println("Exit temperature: $(round(T_exit, digits=1)) K")
println("Density ratio ρ₀/ρ: $(round(rho_ratio, digits=2))")

Different Gases

Compare properties for different working fluids:

M = 1.5
gases = [
    ("Air", 1.4),
    ("Helium", 1.67),
    ("CO₂", 1.3),
    ("Steam", 1.33)
]

println("M\tGas\tγ\tT₀/T\tp₀/p\tρ₀/ρ")
println("---\t---\t----\t----\t----\t----")

for (gas, gamma) in gases
    T_ratio = t0_over_t(M, gamma)
    p_ratio = p0_over_p(M, gamma)
    rho_ratio = rho0_over_rho(M, gamma)
    
    println("$M\t$gas\t$gamma\t$(round(T_ratio, digits=2))\t$(round(p_ratio, digits=2))\t$(round(rho_ratio, digits=2))")
end

Validation Examples

Consistency Check

Verify thermodynamic consistency:

M = 2.5
gamma = 1.4

T_ratio = t0_over_t(M, gamma)
p_ratio = p0_over_p(M, gamma)
rho_ratio = rho0_over_rho(M, gamma)

# Check ideal gas law: p₀/p = (ρ₀/ρ)(T₀/T)
consistency = p_ratio - (rho_ratio * T_ratio)

println("Consistency check (should be ≈ 0): $(round(consistency, digits=10))")

Mach Number Range

Analyze property variations:

println("M\tT₀/T\tp₀/p\tρ₀/ρ")
println("---\t----\t----\t----")

for M in 0.1:0.2:3.0
    T_ratio = t0_over_t(M)
    p_ratio = p0_over_p(M)
    rho_ratio = rho0_over_rho(M)
    
    println("$(M)\t$(round(T_ratio, digits=2))\t$(round(p_ratio, digits=2))\t$(round(rho_ratio, digits=2))")
end

Limitations

1. Ideal Gas Assumption: Functions assume ideal gas behavior with constant specific heats
2. Isentropic Process: No heat transfer, friction, or shock waves
3. Steady Flow: Time-invariant conditions
4. Calorically Perfect Gas: Constant γ throughout the process

For real gas effects or non-isentropic processes, additional corrections may be necessary.

See Also

• Normal Shock Waves: For flows with entropy changes
• Nozzle Analysis: Applications in nozzle design
• Atmospheric Model: Real gas properties at different conditions