Fluid Mechanics :Fluid Dynamics and Flow Measurements Notes

A liquid is flowing from left to right. By the continuity, ρ₁A₁u₁ =  ρ₂A₂u₂

 As we are considering a liquid, 

Velocities in pipes coming from a junction 

mass flow into the junction = mass flow out

ρ₁Q₁= ρ₂Q₂ + ρ₃Q₃

When incompressible,

Q₁ = Q₂ + Q₃

A₁u₁=A₂u₂ + A₃u₃

Vortex flow

• This is the flow of rotating mass of fluid or flow of fluid along curved path.

Free vortex flow

• No external torque or energy required. The fluid rotating under certain energy previously given to them. In a free vortex mechanics, overall energy flow remains constant. There is no energy interaction between an external source and a flow or any dissipation of mechanical energy in the flow.
• Fluid mass rotates due to the conservation of angular momentum.
• Velocity inversely proportional to the radius.
• For a free vortex flow

vr= constant 

v= c/r

• At the center (r = 0) of rotation, velocity approaches to infinite, that point is called singular point.
  
• The free vortex flow is irrotational, and therefore, also known as the irrotational vortex.
• In free vortex flow, Bernoulli’s equation can be applied.

Examples include a whirlpool in a river, water flows out of a bathtub or a sink, flow in centrifugal pump casing and flow around the circular bend in a pipe.

Forced vortex flow

• To maintain a forced vortex flow, it required a continuous supply of energy or external torque.
• All fluid particles rotate at the constant angular velocity ω as a solid body. Therefore, a flow of forced vortex is called as a solid body rotation.
• Tangential velocity is directly proportional to the radius.
  • v = r ω           
  • ω = Angular velocity. 
  • r = Radius of fluid particle from the axis of rotation.
• The surface profile of vortex flow is parabolic.

• In forced vortex total energy per unit weight increases with an increase in radius.
• Forced vortex is not irrotational; rather it is a rotational flow with constant vorticity 2ω.

Examples of forced vortex flow is rotating a vessel containing a liquid with constant angular velocity, flow inside the centrifugal pump.

Energy Equations

• This is the equation of motion in which the forces due to gravity and pressure are taken into consideration. The common fluid mechanics equations used in fluid dynamics are given below
• Let, Gravity force F_g, Pressure force F_p, Viscous force F_v , Compressibility force F_c , and Turbulent force F_t.

F_net = F_g + F_p + F_v + F_c + F_t

• If fluid is incompressible, then F_c = 0

∴ F_net = F_g + F_p + F_v + F_t

This is known as Reynolds equation of motion.

• If fluid is incompressible and turbulence is negligible, then, F_c = 0, F_t = 0

∴ F_net = F_g + F_p + F_v

This equation is called as Navier-Stokes equation.

• If fluid flow is considered ideal then, a viscous effect will also be negligible. Then

F_net = F_g + F_p

This equation is known as Euler’s equation.

• Euler’s equation can be written as:

Bernoulli’s Equation

It is based on law of conservation of energy. This equation is applicable when it is assumed that

• Flow is steady and irrotational
• Fluid is ideal (non-viscous)
• Fluid is incompressible

It states in a steady, ideal flow of an incompressible fluid, the total energy at any point of the fluid is constant.

The total energy consists of pressure energy, kinetic energy and potential energy or datum energy. These energies per unit weight of the fluid are:

• Pressure energy

• Kinetic energy

• Datum energy = z

Bernoulli’s theorem is written as:

• Bernoulli’s equation can be obtained by Euler’s equation

As fluid is incompressible, ρ = constant

where, Head = Energy / Weight

• Restrictions inthe application of Bernoulli’s equation
  • Flow is steady
  • Density is constant (incompressible)
  • Friction losses are negligible
  • It relates the states at two points along a single streamline, (not conditions on two different streamlines)

The Bernoulli equation is applied along streamlines like that joining points 1 and 2 

Total head at 1 = Total head at 2

This equation assumes no energy losses (e.g. from friction) or energy gains (e.g. from a pump) along the streamline. It can be expanded to include these simply, by adding the appropriate energy terms

Note Point: 

The Bernoulli equation is often combined with the continuity equation to find velocities and pressures at points in the flow connected by a streamline.

Kinetic Energy Correction Factor (α)

In a real fluid flowing through a pipe or over a solid surface, the velocity will be zero at the solid boundary and will increase as the distance from the boundary increases. The kinetic energy per unit weight of the fluid will increase in a similar manner.

The kinetic energy in terms of average velocity V at the section and a kinetic energy correction factor α can be determined as:

In which m = ρAVdt is the total mass of the fluid flowing across the cross-section during dt. By comparing the two expressions for kinetic energy, it is obvious that,

The numerical value of α will always be greater than 1

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