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HELIUMAIR
simulation of hot jets.
The
noise emitted by fullyturbulent, perfectlyexpanded free jets depends on
three main factors:
 Jet
velocity (U) : the overall acoustic power goes as U^{n},
where n ranges from 68 for subsonic jets and approaches 3 for
veryhighspeed jets. The
value of U also affects the nature of jet noise emitted in the
downstream direction. At
low speeds, this noise comes from moving quadrupoles; at high speeds,
Mach wave emission becomes dominant.
 Jet Mach
number (M):
at fixed U, increasing M decreases the jet spreading rate, thus
elongates the noisesource region. This is primarily a compressibility
effect, not a density effect. A jet at U=700 m/s and M=2 is noisier
than a jet at U=700 m/s and M=1.5.
 Jet
density (r) : the
jettoambient density ratio affects modestly the jet growth rate.
In large facilities with
big budgets (typically industry or goverment) hot jets are used to replicate
the exhaust conditions of jet engines.
In a smallscale university setup, this is a very expensive
proposition fraught with safety concerns. An inexpensiveand much saferalternative is the use of
heliumair mixtures to match exactly the velocity and Mach number, and match
approximately the density, of hot jets.
Here is how this is done:
Matching the Mach number (M)
This is very easy. One simply
uses a nozzle with the same design Mach number as the hot jet and subjects
the nozzle to a pressure ratio that gives the desired exit Mach number.
Matching the velocity (U)
The velocity is
U= M a=M (gRT)^{1/2
}where a is the speed of sound, g is the ratio of specific
heats, R is the gas constant, and T is the static temperature
of the jet. In the hot air jet, the velocity is high because the temperature
is high:
U= M a=M (g_{air}R_{air}T_{air})^{1/2
}In the cold heliumair mixture (total temperature has room value), we
achieve exactly the same speed of sound, and hence jet velocity, by
increasing gR:
U= M a=M (g_{mix}R_{mix}T_{mix})^{1/2}
Matching (approximately) the density (r)
The density is
r = p/(RT)=gp/a^{2}
where p is the pressure at the jet exit (equal to ambient pressure for
pressurematched jets). Recall
that the speed of sound a is matched exactly, therefore
r_{mix} / r_{ai}_{r}= g_{mix} / g_{air}
so there is a
slight mismatch in density due to the higher g of the heliumair
mixture. Typical value of this mismatch is 10%.
Example: Simulation of a jet with M=1.5 and
U=700 m/s
HOT AIR JET

Gas
constant: R = 287 J/(Kg ^{o}K)
Ratio of specific heats: g = 1.4
Total temperature: T_{0}=786 ^{o}K (=1414 ^{o}R=954
^{o}F)
Mach number: M=1.5
Static temperature: T=542 ^{o}K
(=976 ^{o}R=515 ^{o}F)
Speed of sound: a=(gRT)^{1/2}=467 m/s (=1531 ft/s)
Jet velocity: U=Ma=700 m/s (=2296 ft/s)
Jet density*: r=p/(RT)=101325/(287*542)=0.651 kg/m^{3}
^{ }

^{COLD HELIUMAIR JET
Mixture of 26.4% helium and 73.6% air (by mass).}

Gas
constant: R = 760 J/(Kg ^{o}K)
Ratio of specific heats: g = 1.563
Total temperature: T_{0}=300 ^{o}K (room temperature)
Mach number: M=1.5
Static temperature: T=184 ^{o}K
Speed of sound: a=(gRT)^{1/2}=467 m/s (=1531 ft/s)
Jet velocity: U=Ma=700 m/s (=2296 ft/s)
Jet density*: r=p/(RT)=101325/(760*184)=0.724 kg/m^{3}
(11% higher than density of hot jet)

^{}

* Jet exit
pressure is ambient at sea level

The proof is the pudding... See comparison of noise spectra.
