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.
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: 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: 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: In supersonic mode, rayleigh_flow must
be a real scalar: 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. 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
expand all
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
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
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
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 elementwise.
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.
See Also
flowfanno  flowisentropic  flownormalshock  flowprandtlmeyer