Fluid mechanics: Laminar & Turbulent flow, & Losses in Pipe Flow

Laminar flow & Turbulent Flow

• Laminar flow is the flow which occurs in the form of lamina or layers with no intermixing between the layers.
• Laminar flow is also referred to as streamline or viscous flow.
• In case of turbulent flow there is continuous inter mixing of fluid particles.

Reynold’s Number:

• The dimensionless Reynolds number plays a prominent role in foreseeing the patterns in a fluid’s behaviour. It is referred to as Re, is used to determine whether the fluid flow is laminar or turbulent.

Reynold’s no, R_e

V=mean velocity of flow in pipe

D = Characteristic length of the geometry

μ = dynamic viscosity of the liquid (N – s /m²)

ν = Kinematic viscosity of the liquid (m²/s)

Where,

A_c = Cross – section area of the pipe

P = Perimeter of the pipe

Case–I

Laminar flow in a pipe

Shear stress and velocity distribution in laminar flow through a pipe

Shear Stress distribution:

Fluid Velocity Variation:

Discharge, Q = A × u_avg

Ratio of maximum velocity to average velocity

Thus, Average velocity for Laminar flow through a pipe is half of the maximum velocity of the fluid which occurs at the centre of the pipe.

Pressure variation in Laminar flow through a pipe over length L

On integrating the above equation on both sides:

Head loss in Laminar flow through a pipe over length L

Above equation is hagen poiseuille equation.

As we know,

 Thus,

   LAMINAR FLOW BETWEEN TWO FIXED PARALLEL PLATES

Velocity Distribution (u):

Thus, velocity varies parabolically as we move in y-direction as shown in Figure.

Velocity and shear stress profile for turbulent flow

Discharge (Q) between two parallel fixed plates:

The average velocity is obtained by dividing the discharge (Q) across the section by the area of the section t ×1.

Ratio of Maximum velocity to average velocity:

Thus,

Pressure difference between two parallel fixed plates

Turbulent flow

Turbulent flow is a flow regime characterized by the following points as given below

Shear stress in turbulent flow

In case of turbulent flow there is huge order intermission fluid particles and due to this, various properties of the fluid are going to change with space and time.

Average velocity and fluctuating velocity in turbulent flow

 Boussinesq Hypothesis:

Similar to the expression for viscous shear, J. Boussinesq expressed the turbulent shear mathematical form as

where   τ_t = shear stress due to turbulence

η = eddy viscosity

u bar = average velocity at a distance y from boundary. The ratio of η (eddy viscosity) and (mass density) is known as kinematic eddy viscosity and is denoted by ϵ (epsilon). Mathematically it is written as

If the shear stress due to viscous flow is also considered, then …. shear stress becomes as

The value of η = 0 for laminar flow.

Reynolds Expression for Turbulent Shear Stress.

Reynolds developed an expression for turbulent shear stress between two layers of a fluid at a small distance apart, which is given as:

where u’, v’ = fluctuating component of velocity in the direction of x and y due to turbulence.   

As u’ and v’ are varying and hence τ will also vary.

Hence to find the shear stress, the time average on both the sides of the equation

The turbulent shear stress given by above equation is known as Reynold stress.

Prandtl Mixing length theory:

According to Prandtl, the mixing length l, is that distance between two layers in the transverse direction such that the lumps of fluid particles from one layer could reach the other layer and the particles are mixed in the other layer in such a way that the momentum of the particles in the direction of x is same.

VELOCITY DISTRIBUTION IN TURBULENT FLOW

In above equation, the difference between the maximum velocity u_max, and local velocity u at any point i.e. (u_max - u) is known as ‘velocity defect’.

Velocity distribution in turbulent flow through a pipe

Laminar sublayer thickness

Energy losses in pipes

• sudden expansion of pipe
• sudden contraction of pipe
• bend in pipe
• any obstruction in pipe

Major Loss: It is calculated by Darcy Weisbach formula

Loss of head due to friction: 

where:

L = Length of pipe

V = Mean velocity of flow

d = Diameter of pipe

f = friction factor

friction factor (f) = 4 ×coefficient of friction (f')

Chezy’s Formula: In fluid dynamics, Chezy’s formula describes the mean flow velocity of steady, turbulent open channel flow.

Average velocity V is given by: 

Where: 

i = Loss of head per unit length of pipe  (hydraulic slope tan θ)

Minor Loss: The another type of head loss in minor loss is induced due to following reasons

Loss due to Sudden Enlargement:

Head loss:

Loss due to Sudden Contraction:

Head loss: 

Remember v₁ is velocity at point which lies in contracted section.

Loss of Head at Entrance to Pipe:

Head loss: 

Loss at Exit from Pipe

Head loss: 

Combination of Pipes: Pipes may be connected in series, parallel or in both. Let see their combinations.

Pipe in Series: As pipes are in series, the discharge through each pipe will be same.

In series pipes:

(i). Q = A₁v₁ = A₂v₂ = A₃v₃

(ii). The total head loss will be the sum of the head losses of each individual pipe.

Pipes in Parallel: In this discharge in main pipe is equal to sum of discharge in each of parallel pipes.

For Parallel pipes:

(i). Total discharge: Q = Q₁ + Q₂

(ii). Loss of head in each parallel pipe is same.

i.e. Loss of head for branch pipe 1 = Loss of head for branch pipe 2

Hydraulic Gradient Line (HGL) and Total Energy Line (TEL):

HGL → It joins piezometric head  at various points.

TEL → It joins total energy head at various points

Note:

1. HGL is always parallel but lower than TEL by velocity head.
2. For stationary bodies such as reservoirs or lakes, the EGL and HGL coincide with the free surface of the liquid.
3. A steep jump or droop occurs in EGL and HGL whenever mechanical energy is added to the fluid (by a pump or mechanical energy is removed from the fluid (by a turbine) respectively.
4. 

Water Hammer: When a liquid is flowing through a long pipe fitted with a vale at the end of the pipe and the valve is closed suddenly a pressure wave of high intensity is produced behind the valve. This pressure wave of high intensity is having the effect of hammering action on the walls of the pipe. This phenomenon is known as water hammer.

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