flowrayleigh

Rayleigh line flow relations

Syntax

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

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

Description

[mach,T,P,rho,velocity,T0,P0] = flowrayleigh(gamma,rayleigh_flow) returns an array for each Rayleigh line flow relation. This function calculates these arrays for a given set of specific heat ratios (gamma) for the Mach input mode.

[mach,T,P,rho,velocity,T0,P0] = flowrayleigh(gamma,rayleigh_flow,mtype) uses any one of the Rayleigh flow types mtype. Specify mtype types after all other input arguments.

Examples

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 this 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 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

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

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.

Input Arguments

gamma — Specific heat ratios

array | scalar | real numbers greater than 1

Specific heat ratios, specified an array or scalar of N real numbers greater than 1.

Dependencies

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

• Low-speed temperature ratio

• High-speed temperature ratio

• Subsonic total temperature

• Supersonic total temperature

• Subsonic total pressure

• Supersonic total pressure

Data Types: double

rayleigh_flow — One Rayleigh line flow

array | real numerical

One Rayleigh line flow, specified as an array of real numerical values. This argument can be one of these types.

Normal Shock Relation Types	Description
Mach numbers	Mach numbers, specified as a scalar or array of N real numbers greater than or equal to 1. If rayleigh_flow and gamma are arrays, they must be the same size. Use rayleigh_flow with the mtype value 'mach'. Because 'mach' is the default of mtype, mtype is optional when this array is the input mode.
Temperature ratio	Temperature ratios, specified as a scalar of real numbers: 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'.
Pressure ratios	Pressure ratios, specified as an array or scalar. normal_shock_relations must be a scalar or array of real numbers greater 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. . If rayleigh_flow and gamma are arrays, they must be the same size. Use rayleigh_flow with mtype value 'pres'.
Density ratios	Density ratios, specified as an array or scalar of real numbers that are 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'.
Velocity ratios	Velocity ratios, specified as an array or scalar of N real numbers: Greater than or equal to 0 Less than or equal to sqrt((gamma+1)/(gamma-1)) (as the Mach number approaches infinity) If flow_fanno and gamma are both arrays, they must be the same size. Use flow_fanno with mtype value 'velo'.
Total temperature ratio	Total temperature ratios, specified as a real scalar: 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'.
Total pressure ratio	Total pressure ratios, specified as a scalar: 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'.

Data Types: double

mtype — Input mode for Rayleigh

'mach' (default) | 'templo' | 'temphi' | 'pres' | 'dens' | 'velo' | 'totaltsub' | 'totaltsup' | 'totalpsub' | 'totalpsup'

Input mode for the Rayleigh flow in rayleigh_flow, specified as one of these types.

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.

Data Types: string

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 — Mach numbers

array

Mach numbers, returned as an array.

T — Temperature ratios

array

Temperature ratios, returned as an array.

P — Pressure ratios

array

Pressure ratios, returned as an array.

rho — Density ratios

array

Density ratios, returned as an array.

velocity — Velocity ratios

array

Velocity ratios, returned as an array.

T0 — Total temperature ratios

array

Total temperature ratios, returned as an array.

P0 — Total pressure ratios

array

Total pressure ratios, returned as an array.

Limitations

• This function assumes that:

  • The medium is a calorically perfect gas in a constant area duct.

  • The flow is steady, frictionless, and one dimensional.

  • 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.

  • 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. For thermally perfect gas correction factors, see [2]. 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.

More About

Temperature Ratio

Calculated as the local static temperature over the reference static temperature for sonic flow.

Pressure Ratio

Calculated as the static pressure downstream of the shock over the static pressure upstream of the shock.

Density Ratio

Calculated as the fluid density downstream of the shock over the density upstream of the shock.

Velocity Ratio

Calculated as local velocity over the reference velocity for sonic flow.

Total Temperature Ratio

Calculated as the static temperature downstream of the shock over the static temperature upstream of the shock.

Total Pressure Ratio

Calculated as static pressure downstream of the shock over the static pressure upstream of the shock..

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.