The Reynold’s number, used for critical velocity for turbulent flow of fluids, is given by the relation
Reynolds Number: Understanding Fluid Flow
The Reynolds number (Re) is a dimensionless quantity that helps predict flow patterns in different fluid flow situations. It is a crucial parameter in fluid dynamics, especially for determining when flow transitions from laminar to turbulent, or for characterizing fully turbulent flow. The question specifically mentions its use for critical velocity for turbulent flow of fluids.
Defining Reynolds Number
The Reynolds number is fundamentally a ratio that compares the inertial forces to the viscous forces within a fluid. This ratio helps to characterize the relative importance of these two types of forces for given flow conditions.
It is expressed mathematically as:
\(\text{Re} = \dfrac{\rho u L}{\mu}\)
Where:
- \(\rho\) is the fluid density (mass density)
- \(u\) is the fluid flow speed (or characteristic velocity, often related to critical velocity)
- \(L\) is a characteristic linear dimension (e.g., pipe diameter)
- \(\mu\) is the dynamic viscosity of the fluid
Alternatively, it can also be expressed using kinematic viscosity (\(\nu = \mu/\rho\)):
\(\text{Re} = \dfrac{u L}{\nu}\)
Forces Governing Fluid Flow
In fluid dynamics, two primary forces are often considered when analyzing the nature of flow, especially concerning the Reynolds number:
- Inertia Force: This force is related to the fluid's tendency to resist changes in its state of motion. It is proportional to the fluid's density, its velocity, and a characteristic length. It represents the kinetic energy of the fluid.
- Viscous Force: This force arises from the fluid's internal friction (viscosity), which opposes relative motion between adjacent layers of fluid. It tends to damp out disturbances in the flow.
Relationship of Forces in Reynolds Number
The Reynolds number is defined as the ratio of these two fundamental forces:
\(\text{Reynolds Number} = \dfrac{\text{Inertia Force}}{\text{Viscous Force}}\)
Let's consider why this ratio is so important for turbulent flow and critical velocity:
- When inertia forces are dominant (high Reynolds number), the flow tends to be chaotic and unpredictable, leading to turbulent flow. This means the fluid's momentum causes it to continue moving in complex, swirling patterns even after disturbances.
- When viscous forces are dominant (low Reynolds number), the flow tends to be smooth and orderly, known as laminar flow. The fluid's internal friction effectively dampens out any disturbances.
The transition from laminar to turbulent flow occurs at a specific range of Reynolds numbers, known as the critical Reynolds number, which corresponds to the critical velocity for a given system.
Analyzing the Options
Let's evaluate the given options based on our understanding of the Reynolds number:
| Option | Analysis |
|---|---|
| \(\rm \dfrac{Pressure\;head}{Viscous \;force}\) | This ratio is not the definition of the Reynolds number. While pressure head is important in fluid flow, it is not part of the fundamental definition comparing inertia and viscous forces. |
| \(\rm \dfrac{Viscous \;force}{Mass\;density}\) | This ratio does not represent a dimensionless number like the Reynolds number, nor does it correctly relate the key forces involved in flow regimes. |
| \(\rm \dfrac{Viscous \;force}{Pressure\;head}\) | Similar to option 1, this ratio does not define the Reynolds number. It involves forces and pressure, but not in the standard dimensionless form for flow characterization. |
| \(\rm \dfrac{Inertia\;force}{Viscous \;force}\) | This option correctly represents the definition of the Reynolds number. It is the ratio of inertia force to viscous force, which determines the flow regime (laminar, transitional, or turbulent flow) and is directly linked to the concept of critical velocity. |
Therefore, the relation defining the Reynolds number for characterizing turbulent flow and critical velocity is the ratio of Inertia force to Viscous force.
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
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’
- Which of the following is not a non-dimensional parameter?
