The measure of the effect of compressibility in fluid flow is the magnitude of a dimensionless parameter known as
Mach number
When we study fluid flow, especially at high speeds, the fluid's density might change significantly. This change in density due to pressure variations is known as compressibility. To understand and quantify the effect of compressibility in fluid flow, engineers and scientists use dimensionless parameters.
Understanding Dimensionless Parameters in Fluid Flow
Dimensionless parameters are ratios of different physical forces or effects present in the fluid flow. They help us compare the relative importance of these forces and scale experiments. Several important dimensionless numbers exist, each associated with a different aspect of fluid behavior.
Identifying the Dimensionless Parameter for Compressibility
Let's look at the options provided and what physical effect each dimensionless number represents:
- Mach number: This is defined as the ratio of the flow velocity ($V$) to the speed of sound in the medium ($c$). Mathematically, it is represented as $Ma = \frac{V}{c}$. The speed of sound in a fluid is related to its compressibility. When the flow velocity approaches or exceeds the speed of sound, compressibility effects become very important. Therefore, the Mach number is a direct measure of the effect of compressibility in fluid flow.
- Newton's number: This number is related to the pressure coefficient and represents the ratio of pressure forces to inertial forces. It is sometimes used in specific contexts related to pressure drop. It does not directly measure compressibility.
- Weber number: This is defined as the ratio of inertial forces to surface tension forces. It is important in flows where interfaces exist, such as the formation of droplets or bubbles. It does not relate to compressibility.
- Euler number: Similar to Newton's number, the Euler number is related to the ratio of pressure forces to inertial forces. It is often used in problems involving pressure differences, like flow through pipes or around objects. It does not quantify compressibility effects.
| Dimensionless Number | Ratio of Forces/Velocities | Physical Effect Represented |
|---|---|---|
| Mach number ($Ma$) | Inertial forces / Elastic forces (or Flow velocity / Speed of sound) | Compressibility effects |
| Newton's number | Pressure forces / Inertial forces | Pressure effects |
| Weber number ($We$) | Inertial forces / Surface tension forces | Surface tension effects |
| Euler number ($Eu$) | Pressure forces / Inertial forces | Pressure effects |
From the definitions, it is clear that the Mach number is the specific dimensionless parameter used to measure the significance of compressibility effects in fluid flow. When the Mach number is low (typically less than 0.3), the fluid can often be treated as incompressible. As the Mach number increases, compressibility effects become more prominent.
Important Questions from Dimensionless Number
Match the following and select the correct answer from the codes given below the lists
List I
List II
A.
Steam Nozzle
1.
Mach number
B.
Compressible flow
2.
Reaction turbine
C.
Surface Tension
3.
Biot number
D.
Heat conduction
4.
Nusselt number
5.
Supersaturation
6.
Weber number
The Reynold’s number, used for critical velocity for turbulent flow of fluids, is given by the relation
Reynolds number for non - circular cross-section is:
[V = mean velocity, ν = kinematic viscosity, P = ratio of cross-sectional area to the wetted perimeter]
A) \(V.\frac{{4P}}{v}\)
B) \(\frac{{V.P}}{v}\)
C) \(\frac{{V.2P}}{{4v}}\)
D) \(\frac{{V.P}}{{4v}}\)
When the Mach number is less than unity, the flow is
The only possible dimensionless group that combines velocity ‘V’, body size ‘L’, fluid density ‘ρ’ & surface tension ‘σ’
