Calculate Mach number, area ratio, or gamma from the isentropic area-Mach relation for isentropic flow, with subsonic or supersonic branch options.

Ideal gas, constant ฮณ and quasi-one-dimensional isentropic flow. A* is critical sonic area; it equals the geometric throat area only when that throat is choked.

Isentropic Areaโ€“Mach Relation (Area Ratio and Mach Number)

The isentropic (quasi-1D) areaโ€“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 two positive mathematical solutions: one subsonic (M < 1) and one supersonic (M > 1).

(A) / (A^ ร—) = (1) / (M)[(2) / (ฮณ + 1)(1 + (ฮณ - 1) / (2)Mยฒ)](ฮณ + 1) / (2(ฮณ - 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)
  • ฮณ is the ratio of specific heats (ฮณ = 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, one subsonic and one supersonic Mach number satisfy the relation. A* is the critical sonic area; it equals the geometric throat area only for a choked throat.

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. Select Mach number as the solve mode, then determine the area ratio (A/A* โ‰ฅ 1).
  2. Next, determine the specific heat ratio (ฮณ).
  3. Next, choose the Mach branch (subsonic M < 1 or supersonic M > 1) if A/A* > 1.
  4. Next, use the isentropic areaโ€“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 (ฮณ) = 1.4

Solving the areaโ€“Mach relation gives two possible Mach numbers: M โ‰ˆ 0.430 (subsonic branch) or M โ‰ˆ 1.855 (supersonic branch).