Critical Velocity: Formula, Types & Reynolds Number

Critical velocity is the velocity when the fluid flows up to a break-even point after which it becomes turbulent. In terms of air, critical velocity can be defined as the speed of the object when the air resistance and gravitational pull of a free-falling object are both the same.

When a fluid passes through a tube, it can either flow in a streamlined or turbulent manner. Streamline flow is when the flow of water is parallel to the pipe without any zig-zag patterns or disruptions. But, as the velocity increases, the flow of water gets irregular, which is called a turbulent flow.

Key Takeaways: Critical Velocity, Critical Velocity Formula, Reynolds Number, Lower Critical Velocity, Upper critical Velocity

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What is Critical Velocity?

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Critical Velocity is the velocity at which a liquid can pass through the tube or passage without being turbulent. It is the breakeven velocity below which the flow is considered streamlined or laminar and above it is termed turbulent.

It can be understood in a way that when in instances of normal velocity the flow of water in parallel. But when external factors like velocity and viscosity are altered, change in flow is witnessed. For example, a less viscous liquid is more likely to get turbulent even with low velocity and similarly a more viscous liquid is more likely to be streamlined even at significantly higher velocity. 

So, the critical velocity of each liquid is different as they have different viscosities. 

Read More: Relative Velocity

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Critical Velocity Formula

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The formula for finding the Critical Velocity of a liquid is:

\(\color{red}{V_{\mathrm{critical}} = \frac{R \eta}{2 \rho r}}\)

Here,

R = Reynolds Number

V_critical = Critical Velocity

η = Coefficient of Viscosity

r = Radius of the tube or passage through which the liquid is flowing

ρ = Density of the fluid

The dimensions of the above variables present in the formula are:

Reynolds number (R) = M⁰L⁰T⁰

Coefficient of Viscosity (η) = M¹L^-1T^-1

Radius (r) = M⁰L¹T⁰

Density of the fluid (⍴) = M¹L^-3T⁰

The dimensional formula of Critical Velocity will look as

V_critical = M⁰L¹T^-1

The SI unit of Critical Velocity is called meter/sec. 

Read More: Escape Velocity

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Reynolds Number

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In a fluid, mostly two forces work together, the inertial force and the viscous force. The ratio of inertial forces to the viscous force gives us Reynolds Number. As it is a ratio, it has no unit and also no dimensions.

The Reynolds Number of fluid is given by the formula:

\(\color{red}{Re = \frac{\rho u L}{\mu}}\)

Re = Reynolds Number
ρ = Density of the fluid
u = flow speed
L = characteristic linear dimension
μ = dynamic viscosity of the fliud

Factors Affecting Reynolds Number

The Reynolds Number obtained from the formula is then used to determine whether the flow of the liquid is laminar or turbulent in nature.

There are three cases of Reynolds Number or Re:

1. Re greater than 0 and less than 2000 is said to be streamlined or laminar flow.
2. Re greater between 2000 and 3000 is said to be unstable or turbulent flow.
3. Re above 3000 is said to be a highly turbulent and dangerous flow.

Explanation of Reynolds Number

When Reynolds Number is low it means the viscous forces overpower the inertial forces resulting in a smooth flow.

When Reynolds Number is high it indicates that the inertial forces overpower the viscous forces and the liquid flows in high turbulence.

Representation of Laminar and Turbulent Flow

Read More: Mach Number

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Types of Critical Velocity

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The flow from laminar to turbulent does not occur abruptly. There is a small transition period between this change in the flow pattern. This transition can be classified as:

Lower Critical Velocity

Lower Critical Velocity is the speed at which the flow of liquid shifts from streamlined/laminar to a transition phase. The flow of liquid does not change directly from streamlined to turbulent, instead, it first enters a transition phase. This was proved by Prof. Reynolds Osborne in 1883. 

Upper Critical Velocity

Upper Critical Velocity is the speed at which the flow of liquid shifts from the transition period to turbulence. It is the speed at which turbulence of a flow begins or ends as the liquid enters into the transition from turbulent or from turbulent to the transition phase.

Read More: Force and Motion

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Things to Remember

1. Critical velocity is the velocity at which a fluid passes through a tube or a flow passage without being turbulent. 
2. Escape velocity is the speed with which an object must travel to break the gravitational pull of the earth and enter space. 
3. Critical velocity is dependent on Reynolds Number, the coefficient of viscosity of the fluid, the density of the fluid, and the radius of the pipe through which the fluid passes.
4. Reynolds Number is a dimensionless quantity that is evaluated by using the density of the fluid, dynamic viscosity of the fluid, length of the pipe through which the fluid flows, and the speed of flow.
5. Estimation of the Reynolds number facilitates in differentiating the nature of the flow as laminar or turbulent.
6. Reynolds number less than 2000 is termed as laminar flow.
7. Reynolds number between 2000-3000 is termed as turbulent.
8. Reynolds number greater than 3000 is termed a highly turbulent flow.
9. Lower critical and upper critical velocities help quantify the transition from laminar to turbulent flow.

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