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Calculate Prandtl-Meyer functions for expansion waves


Default Input Mode


[mach,nu,mu] = flowprandtlmeyer(gamma,prandtlmeyer_array) returns an array containing Mach numbers mach, Prandtl-Meyer angles nu, and Mach angles mu. flowprandtlmeyer calculates these arrays for a given set of specific heat ratios, gamma, for the Mach input mode.

Specify Input Mode


[mach,nu,mu] = flowprandtlmeyer(___,mtype) uses any one of the isentropic flow types mtype. Specify mtype types after all other input arguments.


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Calculate the Prandtl-Meyer functions for gases with specific heat ratios. This example yields a 1 x 4 array for nu, but only a scalar for mach and mu.

gamma = [1.3,1.33,1.4,1.67];
[mach,nu,mu] = flowprandtlmeyer(gamma,1.5)
mach =
    1.5000    1.5000    1.5000    1.5000

nu =
   12.6928   12.4455   11.9052   10.2042

mu =
   41.8103   41.8103   41.8103   41.8103

Calculate the Prandtl-Meyer relations for air (gamma = 1.4) for Prandtl-Meyer angle 61 degrees. This example returns a scalar for mach, nu, and mu.

[mach,nu,mu] = flowprandtlmeyer(1.4,61,'nu')
mach =

nu =

mu =

Calculate the Prandtl-Meyer angles for a specific heat ratio of 1.4 and range of Mach angles from 40 degrees to 70 degrees. This example uses increments of 10 degrees and returns a 4 x 1 column array for mach, nu, and mu.

[mach,nu,mu] = flowprandtlmeyer(1.4,(40:10:70)','mu')
mach =

nu =

mu =

Calculate the Prandtl-Meyer relations for gases with specific heat ratio and Mach number combinations as shown. This example returns a 1 x 2 arrayeach for nu and mu, where the elements of each vector correspond to the inputs element-wise.

gamma = [1.3,1.4];
prandtlmeyer_array = [1.13,9];
[mach,nu,mu] = flowprandtlmeyer(gamma,prandtlmeyer_array)
mach =
    1.1300    9.0000

nu =
    2.0405   99.3181

mu =
   62.2461    6.3794

Input Arguments

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Specific heat ratios, specified as an array or scalar of N specific heat ratios.


gamma must be a real, finite scalar greater than 1 for these input modes:

  • Subsonic area ratio

  • Supersonic area ratio

Data Types: double

Prandtl-Meyer types, specified as an array of one of these types.

Prandtl-Meyer TypeDescription
Mach numbers

Mach numbers, specified as a scalar or array of N real numbers greater N real numbers greater than or equal to 0. If prandtlmeyer_array and gamma are arrays, they must be the same size.

Use prandtlmeyer_array with the mtype value 'mach'. Because 'mach' is the default of mtype, mtype is optional when this array is the input mode.

Prandtl-Meyer angle

Prandtl-Meyer angle, specified as a scalar or array of N real numbers greater than or equal to 0 in degrees. prandtlmeyer_array must be:

  • Real scalar greater than or equal to 0 (at Mach number equal 1)

  • Less than or equal to 90 * (sqrt((gamma+1)/(gamma-1)) - 1) (as the Mach number approaches infinity).

Use prandtlmeyer_array with mtype value 'nu'.

Mach angles

Mach angles, specified as a scalar or array of N in degrees. A Mach angle is a function of Mach number only.

Data Types: double

Input mode of Isentropic flow, specified as one of these types.

'mach'Mach number.
'nu'Prandtl-Meyer angle.
'mu' Mach angle.

Data Types: double

Output Arguments

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Mach numbers, returned as an array.

Prandtl-Meyer angles, returned as an array.

Mach angles, returned as an array.


  • The function assumes that the flow is two-dimensional. The function also assumes a smooth and gradual change in flow properties through the expansion fan.

  • This function assumes that the environment is a perfect gas. It cannot assume a perfect gas environment if:

    • There is a large change in either temperature or pressure without a proportionally large change in the other.

    • The stagnation temperature is above 1500 K. The function cannot assume constant specific heats. In this case, you must consider it a thermally perfect gas. For thermally perfect gas correction factors, see [2].

    • The local static temperature is so high that molecules might dissociate and ionize (static temperature 5000 K for air). In this case, you cannot assume a calorically or thermally perfect gas.

More About

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Prandtl-Meyer Angle

Angle change required for a Mach 1 flow to achieve a given Mach number after expansion.

Mach angle

Angle between the flow direction and the lines of pressure disturbance caused by supersonic motion in degrees.


[1] James, John E. A. Gas Dynamics. 2nd ed. Boston: Allyn and Bacon 1984.

[2] Ames Research Staff. NACA Technical Report 1135. Moffett Field, CA: National Advisory Committee on Aeronautics, 1953. 667–671.

Introduced in R2010a