Enter the Mach number, area ratio (A/A*), and specific heat ratio (γ) into the calculator to determine the missing variable.

Mach Number From Area Ratio Calculator

Enter any 2 values to calculate the missing variable. Note: for A/A* > 1 there are generally two Mach solutions (subsonic and supersonic), so choose a branch when solving for Mach.

Isentropic Area–Mach Relation (Area Ratio and Mach Number)

The isentropic (quasi-1D) area–Mach relation for an ideal gas relates Mach number and the area ratio. To find Mach number from a given area ratio, you generally solve this relation numerically. For A/A* > 1 there are typically two solutions: one subsonic (M < 1) and one supersonic (M > 1).

\frac{A}{A^*}=\frac{1}{M}\left[\frac{2}{\gamma+1}\left(1+\frac{\gamma-1}{2}M^2\right)\right]^{\frac{\gamma+1}{2(\gamma-1)}}

Variables:

  • M is the Mach number (dimensionless)
  • A/A* is the area ratio (local area A divided by the critical area A* where M = 1)
  • γ is the ratio of specific heats (γ = cp/cv), dimensionless

For isentropic nozzle/duct flow of an ideal gas, the minimum possible area ratio is A/A* = 1 at M = 1. For area ratios greater than 1, a subsonic and a supersonic Mach number may both satisfy the relation.

What is Mach Number?

The Mach number is a dimensionless quantity in fluid dynamics representing the ratio of flow velocity to the local speed of sound. It is named after Austrian physicist and philosopher Ernst Mach. When the Mach number is less than 1, the flow is subsonic; when it is equal to 1, the flow is sonic; and when it is greater than 1, the flow is supersonic. Flow in the vicinity of Mach 1 is often referred to as transonic. In high-speed flows where the Mach number is greater than about 5, the flow is often referred to as hypersonic.

How to Calculate Mach Number From Area Ratio?

The following steps outline how to calculate the Mach number from the area ratio and specific heat ratio.

  1. First, determine the area ratio (A/A*).
  2. Next, determine the specific heat ratio (γ).
  3. Next, choose the Mach branch (subsonic M < 1 or supersonic M > 1) if A/A* > 1.
  4. Next, use the isentropic area–Mach relation and solve it numerically for M.
  5. After inserting the values and calculating the result, check your answer with the calculator above.

Example Problem : 

Use the following variables as an example problem to test your knowledge.

Area ratio (A/A*) = 1.5

Specific heat ratio (γ) = 1.4

Solving the area–Mach relation gives two possible Mach numbers: M ≈ 0.430 (subsonic branch) or M ≈ 1.855 (supersonic branch).