💧Fluid Mechanics Unit 13 Review

Sound travels through fluids at different speeds depending on the medium's properties. The speed of sound in air is about 343 m/s at room temperature, while in water it's much faster at roughly 1,480 m/s. Temperature, compressibility, and molecular weight all play a role.

Mach number is the ratio of a flow's velocity to the local speed of sound. It's the single most important parameter for classifying compressible flow regimes: subsonic, transonic, supersonic, or hypersonic. Each regime brings different physics, different governing assumptions, and different engineering challenges.

Speed of Sound in Fluids

The speed of sound is the speed at which small pressure disturbances (sound waves) propagate through a fluid. You can think of it as how quickly the fluid "communicates" a pressure change from one location to another.

For an ideal gas, the speed of sound is:

$c = \sqrt{\gamma R T}$

where:

$\gamma$ = ratio of specific heats ( $c_p / c_v$ ). For air at standard conditions, $\gamma = 1.4$ . This parameter captures how the gas stores energy internally.
$R$ = specific gas constant (per unit mass, not the universal gas constant). For air, $R = 287 \text{ J/(kg·K)}$ .
$T$ = absolute temperature in Kelvin. Room temperature is about 293 K.

A few key trends to remember:

Temperature increases sound speed. Higher temperature means faster molecular motion, so disturbances propagate more quickly. Air at 20°C gives $c \approx 343$ m/s; air at 100°C gives $c \approx 386$ m/s.
Lower compressibility increases sound speed. Liquids are much less compressible than gases, which is why sound travels at ~1,480 m/s in water but only ~343 m/s in air.
Heavier molecules slow sound down. Molecular weight appears implicitly through $R$ (since $R = R_u / M_w$ , where $R_u$ is the universal gas constant and $M_w$ is molar mass). Helium (light) has $c \approx 1{,}007$ m/s, while carbon dioxide (heavy) has $c \approx 259$ m/s.

Speed of sound in fluids, 8.2: Relating Pressure, Volume, Amount, and Temperature: The Ideal Gas Law | General College ...

Calculation of Mach Number

The Mach number is a dimensionless ratio that quantifies how fast a flow moves relative to the local speed of sound:

$M = \frac{V}{c}$

$V$ = flow velocity (e.g., aircraft speed, gas velocity in a nozzle)
$c$ = local speed of sound, which depends on the fluid's properties and temperature at that point

The word "local" matters here. Because temperature can vary throughout a flow field, $c$ is not necessarily constant. A flow can be subsonic in one region and supersonic in another.

Steps to calculate Mach number:

Find the flow velocity $V$ from the problem statement, a pitot tube measurement, or a flow equation.
Calculate the local speed of sound $c$ using $c = \sqrt{\gamma R T}$ with the temperature at the point of interest.
Divide: $M = V / c$ .

Quick example: An aircraft flies at 680 m/s through air at 220 K (typical at cruise altitude). The local speed of sound is $c = \sqrt{1.4 \times 287 \times 220} \approx 297$ m/s. So $M = 680 / 297 \approx 2.29$ , which is supersonic.

Speed of sound in fluids, waves - Speed of sound at temperatures below 0 °C - Physics Stack Exchange

Flow Regimes vs. Mach Number

Regime	Mach Range	Key Characteristics	Examples
Subsonic	$M < 0.8$	Compressibility effects are minimal. Density changes are small, so the flow is often treated as incompressible.	Propeller aircraft, low-speed wind tunnels
Transonic	$0.8 \leq M \leq 1.2$	Mixed subsonic and supersonic regions coexist. Local shock waves can form on surfaces even if the freestream is below Mach 1. This is the trickiest regime to analyze.	Commercial jet aircraft at cruise, high-speed wind tunnels
Supersonic	$1.2 < M < 5$	Compressibility effects dominate. Oblique and normal shock waves create abrupt jumps in pressure, temperature, and density.	Fighter jets, rocket nozzle exhaust
Hypersonic	$M \geq 5$	Extreme temperatures behind shock waves can cause chemical dissociation and ionization of the gas. The thin shock layer and intense heating require specialized analysis.	Reentry vehicles, scramjets

The boundaries (0.8, 1.2, 5) are approximate conventions, not sharp physical thresholds. The transonic range is particularly important because small changes in Mach number can cause large changes in aerodynamic forces.

Significance of Mach Number

Mach number determines which physical effects you can ignore and which you can't:

Below about $M = 0.3$ , density changes are less than ~5%, so incompressible flow equations (like Bernoulli's) work well.
Above $M = 0.3$ , compressibility corrections become necessary. The full compressible flow equations (isentropic relations, normal/oblique shock relations) come into play.
Shock wave behavior depends directly on Mach number. Shock strength, angle, and the resulting pressure/temperature jumps are all functions of $M$ .

From a design standpoint, Mach number drives fundamentally different engineering choices:

Subsonic aircraft prioritize high-aspect-ratio wings for efficient lift and streamlined shapes for low drag.
Supersonic aircraft use swept or delta wings to manage shock waves, and they need thermal protection for aerodynamic heating.
Rocket nozzles use converging-diverging geometry specifically to accelerate flow from subsonic to supersonic, with the throat at $M = 1$ and the exit Mach number set by the area ratio.

💧Fluid Mechanics Unit 13 Review