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The critical velocity of a moving fluid can be defined as the speed and direction at which a fluid can flow through a conduit without turbulence.
If you think about it, turbulence is said to be an unstable flow of fluid, one that fluctuates in magnitude and direction constantly. There are no layer disturbances in laminar flow or smooth flow, defined as fluid movement in parallel layers with no layer disturbance.
Critical Velocity: Equation and Dimensional Formula
The mathematical representation of Critical velocity and its dimensional formula is as given below:
Critical Velocity, Vc = kη/2rρ
Where,
K = Reynold’s number,
η = coefficient of viscosity of a liquid
r = radius of capillary tube and
ρ = density of the liquid.
Dimensional formula of Critical Velocity:
Reynolds number (k) = M0L0T0
Coefficient of viscosity (𝜂) = M1L-1T-1
Radius (r) = M0L1T0
The density of the fluid (⍴) = M1L-3T0
The dimensional formula of Critical Velocity:
Vc=M0L1T-1
Where,
Vc = Critical Velocity
M = Mass
L = Length
T= Time
What is Reynolds number?
The ratio of inertial forces to viscous forces is represented by the Reynolds number. It is a dimensionless number used to classify fluid systems in which the impact of viscosity is crucial in influencing velocities or flow patterns.
Mathematically, the Reynolds number is defined as:
Rc=uL=uLv
Where,
⍴: density of the fluid
𝜇: dynamic viscosity of the fluid
u: Velocity of the fluid
L: characteristic linear dimension
𝜈: kinematic viscosity of the fluid
Depending upon the value of the Reynolds number, flow type can be determined as follows:
The flow is streamlined or laminar if Re is between 0 and 2000.
The flow is unstable or turbulent if Re is between 2000 and 3000.
When Re exceeds 3000, the flow becomes extremely turbulent.
Critical Velocity of a Fluid
The Critical Velocity of a fluid is the velocity at which the liquid flow changes from streamlined to turbulent. When the fluid in the pipe has a low velocity, the streamlines are straight parallel lines. As the fluid’s velocity increases, the streamline remains straight and parallel to the pipe wall. When the velocity hits its limit, it begins to create patterns.
For example, the sewer pipes are progressively inclined to allow gravity to operate on the fluid flow, keeping the flow non-turbulent. Since solid particles are present in the flow, the excessive velocity of flow can induce pipe erosion, resulting in pipe damage.
Types of Critical Velocity
The critical Velocity is divided into two major types. They are:
Lower Critical Velocity
Higher Critical Velocity
Lower Critical Velocity
Lower Critical Velocity is the speed at which laminar flow terminates or transitions from laminar to transition phase. Prof. Reynolds Osborne first theorised it in the year 1883. Experiments have shown that when a laminar flow transitions to turbulence, it does not alter abruptly. However, there is a transition phase between the two types of flows.
Higher Critical Velocity
Higher Critical Velocity refers to the velocity at which turbulent flow begins or the velocity at which the flow changes from a transition phase to a turbulent flow.
Characteristics of Turbulent Flow
Irregularity: The flow is characterised by the erratic motion of the fluid particles. The movement of fluid particles is random. For this reason, turbulent flow is typically addressed statistically rather than deterministically.
Rotationality: Turbulent flow is characterised by a powerful three-dimensional vortex formation process. This technique is known as vortex stretching.
Dissipation: In a dissipative method, viscous shear stress transforms flow’s kinetic energy into internal energy.
Diffusivity: Inflow with a relatively constant velocity is referred to as diffusivity. Across a segment of the pipe, dispersal occurs, resulting in the entire fluid flowing at a single value and rapidly lowering extremely near to the pipe walls. Diffusivity is the feature that accounts for a flow’s superior mixing and exaggerated mass, momentum, and energy transfer rates.
Conclusion
We just learned the definition, formula, and examples of critical velocity for the flow of fluids. It is an important concept in fluid mechanics that helps us understand how a fluid flow changes from smooth or laminar to turbulent. It helps us in many real-world applications where we will be able to predict when a fluid flow turns turbulent.
