# flowprandtlmeyer

Calculate Prandtl-Meyer functions for expansion waves

## Syntax

``````[mach,nu,mu] = flowprandtlmeyer(gamma,prandtlmeyer_array)``````
``````[mach,nu,mu] = flowprandtlmeyer(___,mtype)``````

## Description

### Default Input Mode

example

``````[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

example

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

## Examples

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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 = 3.6600 nu = 61 mu = 15.8564```

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 = 1.5557 1.3054 1.1547 1.0642 nu = 13.5505 6.3185 2.4868 0.7025 mu = 40 50 60 70```

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.

#### Dependencies

`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.

TypeDescription
`'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.

## Limitations

• 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.

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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.

## References

[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.

## Version History

Introduced in R2010a