Critical Velocity

Introduction to Critical Velocity

The velocity with which the liquid flow changes from streamlined to turbulent known as the critical velocity of the fluid. The fluid's streamlines are straight parallel lines when the velocity is less for the fluid in the pipe. As the velocity of the fluid gradually increases, the streamline continues to be in straight and parallel to the pipe wall. Once the velocity reaches the breaking point, it forms patterns. Throughout the pipe, the critical velocity will disperse the streamlines. 

To keep the flow non-critical, the sewer pipes are gradually sloped so gravity works on the fluid flow. The excess velocity of flow can cause erosion of the pipe since solid particles are present in the flow, which will lead to damage to the pipe. By using the trenchless method like cured-in-place-pipe, pipe bursting and slip lining, pipe damaged by the action of high-velocity fluid can be rectified.

The fluid's Critical Velocity can be calculated using the Reynolds number, which characterizes the flow of streamlined or turbulent air. It is a dimensionless variable which can be calculated by using the formula. 

Critical velocity formula

The mathematical representation of critical velocity with the dimensional formula given below:

Critical velocity vc = (kη/rρ)

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:

  • Reynolds number (Re) =  M0L0T0

  • Coefficient of viscosity (𝜂) =  M1L-1T-1

  • Radius (r) =  M0L1T0

  • The density of fluid (⍴) = M1L-3T0

  • Critical velocity\[{V_c} = \frac{{\left| {{M^0}{L^0}{T^0}} \right|\left| {{M^1}{L^{ - 1}}{T^{ - 1}}} \right|}}{{\left| {{M^1}{L^{ - 3}}{T^0}} \right|\left| {{M^0}{L^1}{T^0}} \right|}}\]

\[\therefore {V_c} = {M^0}{L^1}{T^{ - 1}}\]

SI unit of critical velocity is meter/sec

Reynolds number

The ratio between inertial forces and viscous forces is known as the Reynolds number. Reynold’s number is a pure number that helps identify the nature of the flow and critical velocity of a liquid through a pipe. 

The number is mathematically represented as follows:

\[{R_c} = \frac{{\rho uL}}{\mu } = \frac{{uL}}{v}\]

Where,

⍴: density of the fluid in kg.m^-3

𝜇: dynamic viscosity of the fluid in m^2s

u: velocity of the fluid in ms^-1

L: characteristic linear dimension in m

𝜈: kinematic viscosity of the fluid in m2s-1

By determining the value of the Reynolds number, flow type can decide as follows:

  • If the value of Re is between 0 to 2000, the flow is streamlined or laminar

  • If the value of Re is between 2000 to 3000, the flow is unstable or turbulent

  • If the value of Re is above 3000, the flow is highly turbulent

Reynolds number concerning laminar and turbulent flow regimes are as follows:

  • When the value of Reynolds number is low then the viscous forces are dominant, laminar flow transpires and are categorized as a smooth, constant fluid motion

  • When the value of the Reynolds number is high, then the inertial forces are dominant, turbulent flow occurs and tends to produce vortices, flow uncertainties, and disordered eddies.

Following is the derivation of Reynolds number:


\[{R_c} = \frac{{ma}}{{\tau A}} = \frac{{\rho V.\frac{{du}}{{dt}}}}{{\mu \frac{{du}}{{dy}}.A}}\alpha \frac{{\rho {L^3}\frac{{du}}{{dt}}}}{{\mu \frac{{du}}{{dy}}{L^2}}} = \frac{{\rho L\frac{{du}}{{dt}}}}{\mu } = \frac{{\rho {u_0}L}}{\mu } = \frac{{{u_0}L}}{v}\]

Where,

t= time

y = cross-sectional position

u = \[\frac{{dx }}{dt}\] : flow speed

τ = shear stress in Pa

A = cross-sectional area of the flow

V = volume of the fluid element

\[{u_0}\] = a maximum speed of the particle relative to the fluid in ms^-1

L = a characteristic linear dimension

𝜇 = fluid of dynamic viscosity in Pa.s

𝜈 = kinematic viscosity in m^2s

⍴ = density of the fluid in kg.m^-3

Critical Velocity Ratio

The idea of critical velocity was established that will make a channel free from silting and scouring. From long observations, a relation between critical velocity and full supply depth was formulated as

The values of C and n were found out as 0.546 and 0.64 respectively, thus v0=0.546 D^0.64

However, in the above formula, the critical velocity was affected by the grade of silt. So, another factor (m) was introduced which was known as the critical velocity ratio (C.V.R).

V0=0.546mD^0.64

Critical Velocity Ratio (C.V.R) is otherwise known as the ratio of mean velocity 'V' to the critical velocity 'Vo' where Vo is known as the critical velocity ratio (CVR). 

It is denoted by m i.e. CVR (m) = V/Vo

When m = 1, there will be no silting or scouring

When m>1, scouring will occur,

When m<1, silting will occur.

So, by finding the value of m, the condition of the canal can be predicted whether it will have silting or scouring.

FAQ (Frequently Asked Questions)

1. What does one mean by the Critical velocity of a fluid?

The limiting value in which the flow streamlined and above which the velocity becomes turbulent is known as the critical velocity. The speed and direction in which the flow of a liquid changes from through tube smooth to turbulent is known as the critical velocity of the fluid. There are multiple variables on which the critical velocity depends, but whether the flow of the fluid is smooth or turbulent is determined by the Reynolds number.

2. What is the critical velocity of a non-viscous liquid?

This is actually a very difficult problem to solve, as the critical velocity of liquid flow is the point where the flow changes from laminar to turbulent. 

The critical velocity depends on Reynolds Number, 

R = ρVD/μ

A Reynolds Number of 2320 or less defines laminar flow, and higher than that 4000 defines turbulent flow. 

Here, the fluid is specified to be non-viscous. As such, μ = 0 and Reynolds Number is infinity.

The interpretation of this specific phenomenon is wide open to speculation.


3. How to explain the relationship between critical velocity and density of fluid?

The critical velocity is the velocity of the flow of liquid up to which the flow is streamlined and above which it becomes turbulent. This is denoted by Vc and depends on:

1.  Coefficient of viscosity of liquid (η)

2.  Density of liquid

3.  Radius of tube

This brings us to the relation

Vc = (K η)/ρr

For the flow to be streamlined, the value of Vc should be very much higher. For this, the coefficient of viscosity must be large and ρ, r must be small.

This means the Reynolds Number Nr should be within 0 to 2000.