How the speed of sound relates to fluid dynamics

The speed of sound is a measure of fluid compressibility. Soundwaves compress fluids in an adiabatic isentropic manner, adding internal energy to them.

Fundamental point

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Fundamental point

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Sound is a mechanical wave produced by the back and forth vibration of the particles in the medium through which the sound wave travels. If a sound wave moves from left to right through air, air particles will be displaced both rightward and leftward as the sound wave's energy passes through it. The particle motion is parallel (and anti-parallel) to the direction of energy transport. This is what distinguishes sound waves in the air as longitudinal waves. Because of the longitudinal motion of the air particles, there are areas in the air where the air particles are compressed together and areas where the air particles are spread apart. These areas are referred to as compressions and rarefactions, respectively. The compressions are high-pressure zones, whereas the rarefactions are low-pressure zones. A sound wave is called a longitudinal wave as said above because it is produced by compressions and rarefactions in the air. The air particles vibrate parallel to the direction of propagation.

Background reading

When fluid moves through the air molecules near the fluid are disturbed and move around it as the fluid moves through the air. When the fluid travels at a low speed, typically less than 250 mph, the density of the air remains constant. At higher speeds, some of the fluid's energy is expended compressing the air and changing the density of the air locally. Because this force is proportional to air density, the compressibility effect modifies the amount of resulting force on the fluid. As the speed of fluid increases, the effect becomes more pronounced.

The speed of sound is the transfer of vibrational energy

The speed of sound in a medium depends on how quickly vibrational energy can be transferred through the medium. For this reason, the derivation of the speed of sound in a medium depends on the medium and on the state of the medium. In general, the equation for the speed of a mechanical wave in a medium depends on the square root of the restoring force, or the elastic property, divided by the inertial property,

v = √ [Elastic property/Inertial property]

What is a wave?

A wave is a disturbance in a medium that carries energy without a net movement of particles.

Any energy carrier is a vibration, be it electromagnetic radiation emitted by the sun or a little applied push to a glass of water.

The next obvious question arises why an energy carrier generates vibrations? The answer is that any energy added to a system generates an energy imbalance in the system. The system immediately distributes the added energy over a larger volume of the substance by generating entropy.

Why waves are sinusoidal?

A vibration generates compression on the medium. The medium releases the compression by expansion. This goes on as the fluid propagates taking the shape of a sine curve until it hits an obstruction.

Mach number

The Mach number is the ratio of the speed of the fluid, to the speed of sound in the gas. The speed of sound is equal to the speed of transmission of small, isentropic disturbances in the flow.

In a fluid, the speed of sound depends on the bulk modulus and the density, v=√Bρ.

Mach = u/c

where:

M is the local Mach number,

u is the local flow velocity with respect to the boundaries, and

c is the speed of sound in the medium, which in air varies with the square root of the thermodynamic temperature.

By definition, at Mach 1, the local flow velocity u is equal to the speed of sound. At Mach 0.65, u is 65% of the speed of sound (subsonic), and, at Mach 1.35, u is 35% faster than the speed of sound (supersonic).

As the Mach number is defined as the ratio of two speeds, it is a dimensionless number. If M < 0.2–0.3 and the flow is quasi-steady and isothermal, compressibility effects will be small and simplified incompressible flow equations can be used

For subsonic flow [ M<1] the density is relatively constant

For transonic flow [ M=1] the density change is nearly equal to the velocity change

For supersonic flow [M>1] the density changes faster than velocity by M^2

How speed of sound links fluid dynamics?

Explanation

Please refer to the image above. You would notice that there are two cases. Case 1 relates to Mach number < 0.2-0.3, here the fluid jetting through the constriction is acting as an incompressible fluid. The pressure drop at the constriction is a linear function of the fluid flow. There is no change in the density of the fluid at the nozzle. The sound waves are not able to compress the air.?Case 2 relates to Mach 1, here the sound and the fluid have the same speed. Sound waves have completely compressed the fluid jetting through the constriction.?As a result of the compression and expansion of the fluid at the constriction, there is a reduction of the density of the fluid. The relationship flow between the flow and pressure drop at the constriction is no longer linear. This erratic fluid flow behavior at the constriction is called choked flow. At Mach1, the relative speed between the sound waves and fluid reaches zero.

What does sound do to fluid? Adiabatic isentropic compression

Sound waves are approximately adiabatic, and the sound speed is determined by adiabatic compressibility. The classical sound wave causes adiabatic compression, in which the heat of the compression does not have enough time to escape the pressure pulse and thus contributes to the pressure caused by the compression. The compression of fluid is approximated as isentropic

During the process of compression and expansion of the air, no heat is added or removed from the system. A process where heat is not added or removed from the system is known as an adiabatic system. For an adiabatic process, pV^γ= constant, where p is the pressure, V is the volume, and gamma (γ= Cp/Cv ) is a constant that depends on the gas. For air, γ = 1.40. The density equals the number of moles times the molar mass divided by the volume, so the volume is equal to V = nxMxρ. The number of moles and the molar mass are constant and can be absorbed into the constant p (1ρ) γ = constant

Credit: Google