[Physics Class Notes] on Critical Velocity Pdf for Exam

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 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 (?) =  M1L-1T-1

• Radius (r) =  M0L1T0

• The density of fluid (⍴) = M1L-3T0

  [V_c=frac{mid M^0L^0T^0mid mid M^1L^{-1}T^{-1}mid }{mid M^1L^{-3}T^0mid mid M^0L^1T^0mid }]

∴Vc=M0L1T-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

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

u: Velocity of the fluid in ms^-1

L: characteristic linear dimension in m

?: 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}{TA}=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 Lfrac{dy}{dt}}{mu }=frac{rho u_0L}{mu }=frac{u_0}{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

U0 = a maximum speed of the particle relative to the fluid in ms^-1

L = a characteristic linear dimension

? = fluid of dynamic viscosity in Pa.s

? = 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.

Critical Velocity Definition

The speed at which gravity and air resistance on a falling object are equalised is defined as the speed at which the object reaches its destination Critical Velocity. The speed and direction at which a fluid will flow through a conduit without becoming turbulent is the alternative approach of elaborating Critical Velocity. Turbulent flow is defined as a fluid flow that is erratic and changes amplitude and direction continuously.

Characteristics of Turbulent Flow

At higher velocities, low viscosity, and larger associated linear dimensions, turbulent flow is more likely to occur. A turbulent flow is defined as one with a Reynolds number greater than Re > 3500.

Ir
regularity: The irregular motion of the fluid particles characterizes the flow. Fluid particles travel in a haphazard manner. Turbulent flow is frequently treated statistically rather than deterministically for this reason.

Diffusivity: Inflow with a relatively constant Velocity Dispersal occurs across a section of the pipe, resulting in the entire fluid flowing at a single value and rapidly dropping extremely close to the walls. Diffusivity is the property that accounts for the better mixture and exaggerated rates of mass, momentum, and energy transfers in a flow.

Rotationality:Turbulent flow is distinguished by a strong three-dimensional vortex production process. Vortex stretching is the name for this mechanism.

Dissipation: A dissipative approach is one in which viscous shear stress moulds the K.E. of flow into internal energy.

Critical Velocity-Formula, Units

With the dimensional formula, the following is a mathematical demonstration of Critical Velocity:

VC= Reηρr

Where,

Vc: Critical Velocity

Re: Regarding the Reynolds figure (ratio of mechanical phenomenon forces to viscous forces)

?: coefficient of viscosity

r: radius of the tube

⍴: density of the fluid

Critical Velocity Dimensional Formula:

                          Vc = M0L1T-1

Unit of Critical Velocity:
SI unit of Critical Velocity is meter/sec

Types of Critical Velocity

Lower Critical Velocity:

Lower Critical Velocity is the speed at which laminar flow ceases or shifts from laminar to transition period. There is a transition time between laminar and turbulent flow. It has been discovered experimentally when a laminar flow turns into turbulence, it does not change abruptly. But there’s a transition period between 2 forms of flows. This experiment was first performed by Prof. Reynolds Osborne in 1883.

Upper Critical Velocity:

The speed at which turbulence in a flow begins or ends. Greater or higher Critical Velocity refers to the Velocity at which a flow transitions from a transition period to turbulent flow.