Prandtl Meyer Expansion Calculator

Created by Vishnuvardhan Shakthibala
Reviewed by Steven Wooding
Last updated: Apr 19, 2022

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 Mach number along with downstream flow properties.

Please read the following article to learn about:

  • What is an expansion wave, and how are they generated?
  • What is the Prandtl Meyer theory, and how to evaluate Prandtl Meyer function?
  • 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 will have 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 the rocket launch!
Shock waves, in particular, are essential characteristics of rocket engines. The rocket engine’s aft part is called a nozzle. During the ascending phase of a rocket, the flow coming out of the nozzle will go through different conditions. They are:

  • Optimal expansion: The flow coming out from the nozzle will have the same area as the nozzle cross-sectional area. In this case, we will not see any shock waves.
  • Under expansion: The flow coming out from the nozzle will have a reduced cross-sectional area than the nozzle cross-sectional area. In this condition, we will be able to see shock waves.
  • Over expansion: The flow coming out from the nozzle will have an increased cross-sectional area compared to the nozzle cross-sectional area. In this condition, we will see an expansion wave helping the supersonic waves expand after coming out of 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 the expansion waves, followed by his student Theodor Meyer in 1908. They developed a theory called Prandtl Meyer expansion to explain the behavior of expansion waves. We will briefly look at the theory and the equations behind our Prandtl Meyer expansion calculator in the following paragraphs.

Below, in the figure, you can observe a surface folding in the outward direction. A flow with Mach number M1(M1>1)M_1 ( M_1>1)flows along the parallel surface (a-b). Because of the outward folded surface (after surface a-b), we notice the generation of an expansion wave. This expansion wave lends itself as a fan of waves. The flow passing through this fan of waves changes the direction of the flow. After the expansion wave (region 2), the flow will have a Mach number M2M_2 with its direction changed by an angle μθ\mu-\theta. The properties in the region after expansion wave are denoted by T2T_2, P2P_2, and ρ2\rho_2.

The figure shows supersonic flow over a surface which turning outward.
Schematic demonstration of an expansion wave generation via a surface turning outward.

According to Prandtl Meyer theory, you can find the downstream flow properties by using the following steps:

  1. Find the Prandtl Meyer function ν(M1)\nu(M1) for the upstream Mach number M1M_1 with known specific heat ratio γ\gamma of the fluid medium . We can use the following equation to do that:
ν(M2)=γ+1γ1arctan(γ1)(M121)(γ+1)arctanM121\scriptsize \begin{align*} \nu(M_2)&=\sqrt \frac{\gamma+1}{\gamma-1} \arctan\sqrt\frac{{(\gamma-1)} (M_1^2-1)}{(\gamma+1)} \\ & -\arctan\sqrt{M_1^2-1} \end{align*}
  1. Now use the deflection of the surface θ\theta and find the ν(M2)\nu(M2) using the following equation:
ν(M2)=θ+ν(M1)\footnotesize \qquad \nu(M_2)=\theta+\nu(M_1)
  1. Use the ν(M2)\nu(M2) to find the Mach number in the downstream region with known γ\gamma of the fluid medium. You can again make use of the Prandtl Meyer function:
ν(M2)=γ+1γ1arctan(γ1)(M221)(γ+1)arctanM221\scriptsize \begin{align*} \nu(M_2)&=\sqrt \frac{\gamma+1}{\gamma-1} \arctan\sqrt \frac{{(\gamma-1)} (M_2^2-1)}{(\gamma+1)} \\ & -\arctan\sqrt{M_2^2-1} \end{align*}
  1. The above equation is transcendental, and you can solve for M2M_2 either by iterative method or by using the Prandtl Meyer function – Mach number tables.

  2. As expansion wave is isentropic, we can use the isentropic flow equations to find the flow properties. Once you know the upstream properties, you can use the following equations to find pressure, temperature, and density, respectively:

T2T1=(1+[(γ1)/2]M121+[(γ1)/2]M22)\scriptsize \qquad \frac{T2}{T1}=\left(\frac{1+[(\gamma-1)/2]M_1^2}{1+[(\gamma-1)/2]M_2^2}\right)

P2P1=(1+[(γ1)/2]M121+[(γ1)/2]M22)γ/(γ1)\scriptsize \qquad \frac{P2}{P1}=\left(\frac{1+[(\gamma-1)/2]M_1^2}{1+[(\gamma-1)/2]M_2^2}\right)^{\gamma/(\gamma-1)}
ρ2ρ1=(1+[(γ1)/2]M121+[(γ1)/2]M22)1/(γ1)\scriptsize \qquad \frac{\rho_2}{\rho_1}=\left(\frac{1+[(\gamma-1)/2]M_1^2}{1+[(\gamma-1)/2]M_2^2}\right)^{1/(\gamma-1)}

Good job! You have successfully understood how to find the downstream properties of an expansion wave. Use our calculator to avoid 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, use the following instructions:

  1. First, based on your requirement, you can choose YES/NO for the "Do you want flow properties ?" option. If you want only the Mach number and Mach angles, you can set it to "NO": you need only the upstream Mach number M1M_1 and deflection angle θ\theta. If you want downstream properties: pressure, temperature, and density, you should choose "YES": along with upstream Mach number M1M_1 and deflection angle θ\theta, you must provide the upstream temperature, pressure, and density.
  2. Let us assume that we choose YES for the "Do you want flow properties ?" option as this is more general.
  3. Now, you can insert upstream Mach number, deflection angle, and upstream flow properties: temperature T1T_1 , pressure P1P_1, and density ρ1\rho_1.
  4. Our calculator will provide you with the following results:
    • Downstream Mach number or Mach number after the expansion wave M2M_2;
    • Downstream flow properties: temperature T2T_2, pressure P2P_2, and density ρ2\rho_2; and
    • Mach angles and Prandtl meyer function evaluations for M1M_1 and M2M_2.

Hooray! You learned to use our calculator to find flow properties after the expansion waves. If you are curious about oblique shock waves, please check our oblique shock wave calculator. We are sure you will like it ;).

A numerical example

Now, it's time to look at an example.

  1. Let us consider an upstream flow with the following properties:

    • Mach number M1M_1 = 1.5;
    • Pressure P1P_1 = 1 atm;
    • Temperature T1T_1 = 288 K;
    • Density ρ1\rho_1 = 1.22586 kg/m3; and
    • Deflection angle θ\theta = 15 degrees.
  2. For our example, let us assume we need all the downstream properties. Therefore, first, choose "YES" for the "Do you want flow properties ?" option. Now you can insert the inputs, and our calculator will provide you with the results right away. Click the advanced mode button to see forward and rearward Mach angles and Prandtl Meyer function evaluations. Following are the results from the calculator:

    • M2M_2 = 2.0;
    • ν2\nu_2 = 26.91 degrees (advanced mode);
    • ν1\nu_1 = 11.91 degrees (advanced mode);
    • Pressure P2P_2 = 0.469 atm;
    • Temperature T2T_2 = 230.0336 K;
    • Density ρ2\rho_2 = 0.69896 kg}/m3;
    • Forward Mach angle μ1\mu_1 = 41.81 degrees (advanced mode);
    • Rearward Mach angle μ2\mu_2 = 29.6874 degrees (advanced mode); and
    • Flow deflection with respect to horizontal plane = ν2θ\nu_2-\theta = 14.68737 degrees.

Cool! We hope now you are confident to tackle any challenges that arise when you want to deal with Prandtl Meyer expansion waves!


How do I calculate the pressure downstream of an expansion fan?

To find the pressure downstream of the expansion wave, follow these steps:

  1. Find the downstream Prandtl Meyer function using the upstream Mach number and the deflection angle.
  2. Using the result from step 1, use the Prandtl Meyer function – Mach number table or solve the Prandtl Meyer equation for downstream Mach number to obtain the downstream Mach number.
  3. Calculate the total pressure using the upstream pressure and isentropic pressure ratio – Mach number table.
  4. Obtain the downstream pressure by using the total pressure and the isentropic pressure ratio, and Mach number table.

What is the Mach angle for Mach number of 1.5?

Mach angle is 41.81 degrees for Mach number 1.5. You can calculate the Mach angle by taking arcsin of one divided by the Mach number.

What happens to total pressure through an expansion wave?

The total pressure remains constant through an expansion wave. This constancy is due to the fact that there is no discontinuity in the expansion wave, unlike shock waves where discontinuity changes the total pressure.

Are expansion waves isentropic?

Yes, expansion waves are isentropic. Due to the continuous nature of the expansion wave, there is no loss due to dissipation; as a result, entropy and stagnation properties remain constant. This condition makes expansion waves isentropic.

Vishnuvardhan Shakthibala
Image of a circle with radius, diameter, circumference and area marked.
Upstream: region 1
Do you want flow properties?
Gamma (γ)
Theta (θ)
Pressure (P₁)
Temperature (T₁)
Density (ρ₁)
lb/cu ft
Downstream: region 2
Pressure (P₂)
Temperature (T₂)
Density (ρ₂)
lb/cu ft
Flow deflection
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