flowrayleigh - Rayleigh line flow relations

Syntax

[mach, T, P, rho, velocity, T0, P0] = flowrayleigh(gamma, rayleigh_flow, mtype)

Description

[mach, T, P, rho, velocity, T0, P0] = flowrayleigh(gamma, rayleigh_flow, mtype) returns an array for each Rayleigh line flow relation. This function calculates these arrays for a given set of specific heat ratios (gamma), and any one of the Rayleigh line flow types. You select the Rayleigh flow type with mtype.

This function assumes that the medium is a calorically perfect gas in a constant area duct. It assumes that the flow is steady, frictionless, and one dimensional. It also assumes that the main mechanism for the change of flow variables is heat transfer.

This function assumes that the environment is a perfect gas. In the following instances, 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, it cannot assume a perfect gas environment. If the stagnation temperature is above 1500 K, do not assume constant specific heats. In this case, the medium ceases to be a calorically perfect gas; you must then consider it a thermally perfect gas. See 2 for thermally perfect gas correction factors. The local static temperature might be so high that molecules dissociate and ionize (static temperature 5000 K for air). In this case, you cannot assume a calorically or thermally perfect gas.

Input Arguments

gamma

Array of N specific heat ratios. gamma must be either a scalar or an array of N real numbers greater than 1. gamma must be a real, finite scalar greater than 1 for the following input modes: low speed temperature ratio, high speed temperature ratio, subsonic total temperature, supersonic total temperature, subsonic total pressure, and supersonic total pressure.

rayleigh_flow

Array of real numerical values for one Rayleigh line flow. This argument can be one of the following:

Array of Mach numbers. This array must be a scalar or an array of N real numbers greater than or equal to 0. If rayleigh_flow and gamma are arrays, they must be the same size.
Use rayleigh_flow with mtype value 'mach'. Because 'mach' is the default of mtype, mtype is optional when this array is the input mode.
Scalar value of temperature ratio. The temperature ratio is the local static temperature over the reference static temperature for sonic flow. rayleigh_flow must be a real scalar:
- Greater than or equal to 0 (at the Mach number equal 0 for low speeds or as Mach number approaches infinity for high speeds)
- Less than or equal to 1/4*(gamma+1/gamma)+1/2 (at mach = 1/sqrt(gamma))
Use rayleigh_flow with mtype values 'templo' and 'temphi'.
Array of pressure ratios. The pressure ratio is the local static pressure over the reference static pressure for sonic flow. rayleigh_flow must be a scalar or array of real numbers less than or equal to gamma+1 (at the Mach number equal 0). If rayleigh_flow and gamma are arrays, they must be the same size.
Use rayleigh_flow with mtype value 'pres'.
Array of density ratios. The density ratio is the local density over the reference density for sonic flow. rayleigh_flow must be a scalar or array of real numbers. These numbers must be greater than or equal to:
gamma/(gamma+1) (as Mach number approaches infinity)
If rayleigh_flow and gamma are arrays, they must be the same size.
Use rayleigh_flow with mtype value 'dens'.
Array of velocity ratios. The velocity ratio is the local velocity over the reference velocity for sonic flow. rayleigh_flow must be a scalar or an array of N real numbers:
- Greater than or equal to 0
- Less than or equal to (gamma+1)/gamma (as Mach number approaches infinity)
If rayleigh_flow and gamma are both arrays, they must be the same size.
Use rayleigh_flow with mtype value 'velo'.
Scalar value of total temperature ratio. The total temperature ratio is the local stagnation temperature over the reference stagnation temperature for sonic flow. In subsonic mode, rayleigh_flow must be a real scalar:
- Greater than or equal to 0 (at the Mach number equal 0)
- Less than or equal to 1 (at the Mach number equal 1)
In supersonic mode, rayleigh_flow must be a real scalar:
- Greater than or equal to (gamma+1)^2*(gamma-1)/2/(gamma^2*(1+(gamma-1)/2))) (as Mach number approaches infinity)
- Less than or equal to 1 (at the Mach number equal 1)
Use rayleigh_flow with the mtype values 'totaltsub' and 'totaltsup'.
Scalar value of total pressure ratio. The total pressure ratio is the local stagnation pressure over the reference stagnation pressure for sonic flow. In subsonic mode, rayleigh_flow must be a real scalar.
- Greater than or equal to 1 (at the Mach number equal 1)
- Less than or equal to (1+gamma)*(1+(gamma-1)/2)^(-gamma/(gamma-1)) (at Mach number equal 0)
In supersonic mode, rayleigh_flow must be a real scalar greater than or equal to 1.
Use rayleigh_flow with mtype values 'totalpsub' and 'totalpsup'.

mtype

A string that defines the input mode for the Rayleigh flow in rayleigh_flow.

Type Description

'mach' Default. Mach number.

'templo' Low speed static temperature ratio. The low speed temperature ratio is the local static temperature over the reference sonic temperature. This ratio for when the Mach number of the upstream flow is less than the critical Mach number of 1/sqrt(gamma).

'temphi' High speed static temperature ratio. The high speed temperature ratio is the local static temperature over the reference sonic temperature. This ratio is for when the Mach number of the upstream flow is greater than the critical Mach number of 1/sqrt(gamma).

'pres' Pressure ratio.

'dens' Density ratio.

'velo' Velocity ratio.

'totaltsub' Subsonic total temperature ratio.

'totaltsup' Supersonic total temperature ratio.

'totalpsub' Subsonic total pressure ratio.

'totalpsup' Supersonic total pressure ratio.

Output Arguments

All output ratios are static conditions over the sonic conditions. All outputs are the same size as the array inputs. If there are no array inputs, all outputs are scalars.

`mach`	Array of Mach numbers.
`T`	Array of temperature ratios. The temperature ratio is the local static temperature over the reference static temperature for sonic flow.
`P`	Array of pressure ratios. The pressure ratio is the local static pressure over the reference static pressure for sonic flow.
`rho`	Array of density ratio. The density ratio is the local density over the reference density for sonic flow.
`velocity`	Array of velocity ratios. The velocity ratio is the local velocity over the reference velocity for sonic flow.
`T0`	Array of total temperature ratios. The temperature ratio is the local static temperature over the reference static temperature for sonic flow.
`P0`	Array of total pressure ratios. The total pressure ratio is the local stagnation pressure over the reference stagnation pressure for sonic flow.

Examples

Calculate Rayleigh Line Flow Relations Given Air

Calculate the Rayleigh line flow relations for air (gamma = 1.4) for supersonic total pressure ratio 1.2.

[mach,T,P,rho,velocity,T0,P0] = flowrayleigh(1.4,1.2,'totalpsup')

mach =

    1.6397

T =

    0.6823

P =

    0.5038

rho =

    0.7383

velocity =

    1.3545

T0 =

    0.8744

P0 =

    1.2000

This example returns scalar values for mach, T, P, rho, velocity, T0, and P0.

Calculate Rayleigh Line Flow Relations for Specific Heat Ratios in Array

Calculate the Rayleigh line flow relations for gases with specific heat ratios given in the following 1 x 4 row array for the Mach number 0.5.

gamma = [1.3,1.33,1.4,1.67];
[mach,T,P,rho,velocity,T0,P0] = flowrayleigh(gamma,0.5)

mach =

    0.5000    0.5000    0.5000    0.5000

T =

    0.7533    0.7644    0.7901    0.8870

P =

    1.7358    1.7486    1.7778    1.8836

rho =

    2.3043    2.2876    2.2500    2.1236

velocity =

    0.4340    0.4371    0.4444    0.4709

T0 =

    0.6796    0.6832    0.6914    0.7201

P0 =

    1.1111    1.1121    1.1141    1.1202

This example returns a 1 x 4 row array for mach, T, P, rho, velocity, T0, and P0.

Calculate Rayleigh Line Flow Relations for Specific Heat Ratios and High Speed Temperature

Calculate the Rayleigh line flow relations for a specific heat ratio of 1.4 and high speed temperature ratio 0.70.

[mach,T,P,rho,velocity,T0,P0] = flowrayleigh(1.4,0.70,'temphi')

mach =

    1.6035

T =

    0.7000

P =

    0.5218

rho =

    0.7454

velocity =

    1.3416

T0 =

    0.8833

P0 =

    1.1777

This example returns scalar values for mach, T, P, rho, velocity, T0, and P0.

Calculate Rayleigh Line Flow Relations for Gases with Specific Heat Ratio and Static Pressure

Calculate the Rayleigh line flow relations for gases with specific heat ratio and static pressure ratio combinations as shown.

gamma = [1.3,1.4];
P = [0.13,1.7778];
[mach,T,P,rho,velocity,T0,P0] = flowrayleigh(gamma,P,'pres')

mach =

    3.5833    0.5000

T =

    0.2170    0.7901

P =

    0.1300    1.7778

rho =

    0.5991    2.2501

velocity =

    1.6692    0.4444

T0 =

    0.5521    0.6913

P0 =

    7.4381    1.1141

This example returns a 1 x 2 array for mach, T, P, rho, velocity, T0, and P0 each. The elements of each array correspond to the inputs element-wise.

References

1. James, J. E. A., Gas Dynamics, Second Edition, Allyn and Bacon, Inc, Boston, 1984.

2. NACA Technical Report 1135, 1953, National Advisory Committee on Aeronautics, Ames Research Staff, Moffett Field, Calif. Pages 667–671.

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