If you are thinking of finding the supersonic flow Mach number after an expansion wave, this Prandtl-Meyer expansion calculator is the perfect match for you. Our calculator will help you find the downstream Mach number along with the downstream flow properties.
Please read the following article to learn about:
- What an expansion wave is, and how it is generated;
- What the Prandtl-Meyer theory is, and how to evaluate the Prandtl-Meyer function; and
- How to use the Prandtl-Meyer expansion calculator.
What is an expansion wave?
When a supersonic flow encounters a surface that folds outward, the flow has more space to move, leading to an expansion of the flow. This expansion is assisted by a fan of waves called an expansion wave, which acts as a continuous boundary where the expansion happens.
Next time, take a closer look at a rocket launch! Shock waves, in particular, are essential characteristics of rocket engines. The aft part of a rocket engine is called a nozzle. During the ascending phase of a rocket, the flow coming out of the nozzle goes through different conditions:
- Optimal expansion: the flow coming out of the nozzle has the same area as the nozzle cross-sectional area. In this case, we do not see any shock waves.
- Under-expansion: the flow coming out of the nozzle has a smaller cross-sectional area than the nozzle. In this condition, we are able to see shock waves.
- Over-expansion: the flow coming out of the nozzle has a larger cross-sectional area than the nozzle. In this condition, we see an expansion wave helping the supersonic flow expand after leaving the nozzle.
So, keep an eye on the launcher's nozzle and look for shock and expansion waves!
What is the Prandtl-Meyer expansion theory?
In 1907, German fluid dynamicist Ludwig Prandtl studied expansion waves, followed by his student Theodor Meyer in 1908. They developed a theory called the Prandtl-Meyer expansion to explain the behavior of expansion waves.
A flow with Mach number M₁ (M₁ > 1) flows along a parallel surface. Because of an outward-folding surface, an expansion wave is generated as a fan of waves. The flow passing through this fan changes direction. After the expansion wave (region 2), the flow has a Mach number M₂ with its direction changed by the deflection angle θ. The properties after the expansion wave are denoted by T₂, P₂, and ρ₂.
According to the Prandtl-Meyer theory, you can find the downstream flow properties using the following steps:
1. Find the Prandtl-Meyer function ν(M₁) for the upstream Mach number M₁ with known specific heat ratio γ of the fluid medium:
ν(M) = √((γ+1)/(γ−1)) · arctan√((γ−1)(M²−1)/(γ+1)) − arctan√(M²−1)
2. Use the deflection angle θ of the surface to find ν(M₂):
ν(M₂) = θ + ν(M₁)
3. Use ν(M₂) to find the Mach number in the downstream region with the known γ by applying the same Prandtl-Meyer function. The equation is transcendental, so M₂ is solved either by an iterative method or by using Prandtl-Meyer function tables. This calculator solves it iteratively for you.
Because the expansion wave is isentropic, the flow properties follow the isentropic relations. Once you know the upstream properties, you can find the pressure, temperature, and density downstream:
T₂/T₁ = (1 + [(γ−1)/2]M₁²) / (1 + [(γ−1)/2]M₂²)
P₂/P₁ = (T₂/T₁)γ/(γ−1)
ρ₂/ρ₁ = (T₂/T₁)1/(γ−1)
Good job! You now understand how to find the downstream properties of an expansion wave. Use our calculator to skip all these tricky calculation steps and get answers at supersonic speed ;).
How to use the Prandtl-Meyer expansion calculator?
To use our Prandtl-Meyer expansion calculator, follow these instructions:
- First, based on your requirement, choose YES/NO for the "Do you want flow properties?" option. If you want only the Mach number and Mach angles, set it to NO: you need only the upstream Mach number M₁ and the deflection angle θ. If you want the downstream properties (pressure, temperature, and density), choose YES: along with the upstream Mach number M₁ and deflection angle θ, you must also provide the upstream temperature, pressure, and density.
- Pick your unit system — metric (K, kPa, kg/m³) or American (°R, psi, lb/ft³). The Mach number and angles are dimensionless and do not depend on the unit system.
- Insert the upstream Mach number, deflection angle, specific heat ratio, and (optionally) the upstream flow properties.
Our calculator will provide you with the following results:
- The downstream Mach number after the expansion wave, M₂;
- The Prandtl-Meyer function values ν(M₁) and ν(M₂);
- The Mach angles μ₁ and μ₂; and
- (If requested) the downstream temperature T₂, pressure P₂, and density ρ₂.
A numerical example
Let's expand air (γ = 1.4) with an upstream Mach number M₁ = 2.0 through a deflection angle θ = 10°.
- ν(M₁) for M₁ = 2.0 ≈ 26.380°
- ν(M₂) = θ + ν(M₁) = 10° + 26.380° = 36.380°
- Solving the Prandtl-Meyer relation gives M₂ ≈ 2.385
With T₁ = 288.15 K, P₁ = 101.325 kPa, and ρ₁ = 1.225 kg/m³, the isentropic relations give T₂ ≈ 254.3 K, P₂ ≈ 65.6 kPa, and ρ₂ ≈ 0.899 kg/m³ — the flow accelerates and cools as it expands.
FAQs
What is an expansion wave?
An expansion wave is a fan of weak waves that forms when a supersonic flow turns around a convex (outward-folding) corner. Through the fan, the flow gradually accelerates, while its pressure, temperature, and density decrease. The process is isentropic (no entropy increase), unlike a shock wave.
How do you find the downstream Mach number?
Compute the Prandtl-Meyer function ν(M₁) of the upstream flow, add the deflection angle θ to get ν(M₂) = θ + ν(M₁), then invert the Prandtl-Meyer function to solve for M₂. Because the relation is transcendental, M₂ is found iteratively (or from tables).
Is the Prandtl-Meyer expansion isentropic?
Yes. An expansion fan is made of an infinite number of infinitesimally weak Mach waves, so the flow remains isentropic. This is why the simple isentropic relations can be used to find the downstream temperature, pressure, and density.
What is the maximum turning angle?
As the Mach number approaches infinity, the Prandtl-Meyer function reaches a maximum value νmax = (π/2)(√((γ+1)/(γ−1)) − 1), which is about 130.45° for air (γ = 1.4). A flow cannot turn by more than νmax − ν(M₁); beyond that it would expand to a vacuum.
What is the Mach angle?
The Mach angle μ = arcsin(1/M) is the angle between the flow direction and a Mach wave. As the flow accelerates through the expansion fan, the Mach number increases and the Mach angle decreases.