Prandtl’s 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 the 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 lumps of the 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. It is assumed that the velocity fluctuation in the x-direction  u^| is related to the mixing length L as

u^| = L * du/ dy

And,

v^| is the fluctuation components of velocity in y-direction is of the same order of magnitude as u^| and hence,

v^| = L du/dy

Now,

u^| * v^| = u^| * v^|

or, u^| * v^|  = (L * du/ dy) *( L * du/ dy)

or, u^| * v^|   = L² (du/dy )²

Substituting the value of u^| * v^|  in equation of turbulent shear stress, we get the expression for shear stress in turbulent flow due to Prandtl’s  as

Thus,

The total shear stress at any point in turbulent flow is the sum of shear stress due to viscous shear and can be written as

Equation (2) is the general expression for shear stress distribution for turbulent flow in circular pipe.

Velocity Distribution

Prandtl assumed that the mixing length L is a linear function of the distance y from the pipe wall i.e.

L=Ky

Where, K is a constant known as Karman constant = 0.4

Substituting the value of L in equation (1) , we get

Using Boundary conditions

Y=R

u= U_max

[∴ K = 0.4 = Karman constant]

Or, u = U_max +2.5 u^* ln (y/R) ————–(4)

Above expression is called Prandtl universal velocity distribution for turbulent flow through pipe.

Dividing equation (4) by u^*, we get

In equation (5) ( U_max – u ) is known as velocity defect.

Darcy-Weisbach equation

Consider a uniform horizontal pipe having steady flow as shown above figure. Also, consider 1-1 and 2-2 are two sections of pipe.

Let,

P₁ = Pressure intensity at sections 1-1

P₂ = Pressure intensity at sections 2-2

v₁ = Velocity of flow at section 1-1

v₂ = Velocity of flow at section 2-2

L =  Length of the pipe between section 1-1 and 2-2

d = Diameter of pipe

f^| = Frictional resistance per unit wetted area per unit velocity

hf =  Loss of head due to friction

Applying Bernoulli’s equation between sections 1-1 and 2-2

Total head at 1-1 = Total head at 2-2 + loss of head due to friction between 1-1 and 2-2

But,

z₁ = z₂ as pipe is horizontal

v₁ = v₂ as diameter of pipe is same at 1-1 and 2-2

But, hf is the head loss due to friction and hence intensity of pressure will be reduced in the direction of flow by frictional resistance.

Now,

Frictional resistance = Frictional resistance per unit wetted area per unit velocity * wetted area * velocity²

Or, F₁ = f^| * π dL * v²

[∴ Wetted area = π dL, Velocity = v = v₁=v₂]

Or, F₁ = f^| *P*L*v² ————–(2)

[∴π d = Perimeter = P ]

The forces acting on the fluid between sections 1-1 and 2-2 are:

• Pressure force at section 1-1 = P₁ * A

where, A= Area of pipe

• Pressure force at sections 2-2 = P₂ *A
• Frictional force F₁ as shown in figure.

Resolving all forces in the horizontal direction, we have

P₁A- P₂A – F = 0 ————-(3)

Or, (P₁ – P₂ )A = F₁ = f^| * P *L*v²

[∴ From eqaution (2), F₁ = f^| PLv² ]

Or, P₁ – P₂ = (f^| PLv²)/A

But, from equation (1), P₁ – P₂ = ρgh_f

Equating the value of ( P₁ – P₂), we get

ρgh_f = (f^| PLv²)/A

or, h_f = (f^| /ρg) *(P/A)*L* v² ————-(4)

In equation (4)

P/A = Wetted perimeter/ Area

∴ h_f = (f^| /ρg) *(4/d)*L* v²

Or, h_f = (f^| /ρg) *(4Lv² /d)—————-(5)

Putting f^| /ρ = f/2

Where, f is known as coefficient of friction.

Equation becomes as

h_f = (4f/2g) * (Lv²/d)

or, h_f = (4f Lv²)/ (d*2g) ————–(6)

Equation (6) is known as Darcy Weisbach eqaution.

Also, equation(6) is written as

h_f = (fLv²) / d* 2g

Then, f is known as friction factor.

Nikuradse’s experiment:

He coated several different size of pipe with sand grains that had been sorted by sieving so as to obtain different sizes of grains of reasonably conform diameter. The diameter of grain sand grains may be represented by e, which is known as absolute roughness, e/d is relative roughness, where d is diameter of pipe.

The test results are plotted graphically as shown in figure:

The grain shows following character:

• Laminar and turbulent flow regime is indicated by change in relation between f and Re.
• Laminar regime is characterized by a single curve f= 64/Re for all roughness.
• In turbulent flow, curve of f versue Re exists for every relative roughness, e/d.
• At high Reynolds number friction factor becomes constant.

Friction coefficient for commercial pipe:

According to Blasius, F= 0.3164/Re1/4

According to Pradtl, 1/F = 2 log₁₀ Re F^1/2 – 0.8 for smooth pipes

According to Karman, 1/F^1/2 =  1.14 + 2 log₁₀ D/e for turbulent flow

Colebook-white equation

1/F^1/2 = 0.86 ln Re F^1/2 – 0.8 , for smooth pipe flow

1/F^1/2 = -0.86 ln(e/3.70 + 2.5/ReF^1/2), for transion zone

1/F^1/2 = -0.86 ln(e/3.7D), for complete turbulent zone.

Moody Chart:

Moody chart may be divided into four zones:

• The laminar zone where friction factor is function of Reynolds number.
• Critical zone where values are uncertain.
• Transition zone, f is friction of Re and e/d ratio.
• Turbulent zone f is independent of Re and depends on only e/d.

Hydraulic Gradient and Total Energy Line:

1. Hydraullic Gradient Line:

It is defined as the line which gives the sum of pressure headd P/W and datum head(Z) of flowing fluid in a pipe with respect to some reference line or it is the line which is obtained by joining the top of all vertical ordinates showing the pressure head P/W of a flowing fluid in a pipe from the centre of the pipe.

2. Total Energy Line:

It is defined as the line which gives the sum of pressure head, datum head and kinetic head of a flowing fluid in a pipe with respect to some reference line. It is also defined as the line which is obtained by joining the tops of all vertical ordinates showing the sum of pressure head and kinetic head from the centre of the pipe.

Minor Head ( energy) Losses:

The minor loss of energy (or head) includes the following cases:

1. Loss of head due to sudden enlargement
2. Loss of head due to sudden contraction
3. Loss of head at the entrance of a pipe
4. Loss of head at the exit of a pipe
5. Loss of head due to an obraction in a pipe
6. Loss of head due to bend in the pipe
7. Loss of head in various pipe fittings

Major Head Loss for Laminar flow through pipe ( Hagen Poisscuille Equation)

Consider a horizontal pipe of radius R. The viscous fluid is flowing from left to right in the pipe as shown in figure a. Consider a fluid element of radius r, sliding in a cylindrical fluid element of radius ( r+ dr). Let, the lenght of fluid element be △x.

If P is the intensity of pressure on the face AB, then the intensity of pressure on face CD will be (P + ∂P/∂x △x ).

Then,

The forces acting on the fluid element are:

• The pressure force, P*πr² on face AB.
• The pressure force (P + ∂P/∂x △x  ) πr² on face CD.
• The shear force, Ʈ * 2 πr△x on the surface of fluid element.

a. Shear Stress Distribution:

As there is no acceleration, hence the sum of all forces in the direction of flow must be zero i.e.

The shear stress Ʈ across a section varies with r as ∂P/∂x across a section is constant. Hence, shear stress distribution across a section is linear.

b. Velocity Distribution:

Taking relation,

Ʈ = µ(du)/(dy)

Here, y is measured from the pipe wall.

Hence,

y = R-r and

dy = -dr

∴ Ʈ = -µ(du)/(dy)

Substituting the value in equation (i)

Where,

c= Constant of integration and

 From boundary condition;

r=R, u=0

In equation (iii), values of µ, ∂P/∂x and R are constant, which means the velocity, u varies with the square of r. Thus, equation (iii) is a equation of parabola.

c. Ratio of maximum velocity to averange velocity:

The velocity is maximum, when r=0 in equation (iii)

Where, U_max = maximum velocity

Consider the flow through a circular ring element of radius r and thickness dr as shown in figure (b).

The fluid flowing per second through the elementary ring is

dQ = velocity at a radius r * area of ring element

      = u * 2 πrdr  

∴ Ratio of maximum velocity to average velocity = 2.

d. Drop of pressure for a given length (L) of a pipe:

From equation (v) we have,

Integrating the above equation w.r.t  x, we get

Where,

P₁-P₂ = Drop of pressure

∴ Loss of pressure head = (P₁-P₂)/ ρg

Equation (vi) is called Hagen Poisecuille Formula.

Difference between open channel and pipe flows:

1. Open channel flow has a free water surface.
2. Open channel flow is subjected to atmospheric pressure while pipe is not subjected to atmospheric pressure.
3. Open channel flow is not completely enclosed by boundaries , unlike pipe flow.
4. Open channel is always under the action of gravity , while pipe can be under gravity or may flow due to some external pressure.

On the basis of shape :

• Rectangular channel:

A channel which is rectangular in shape is called rectangular channel.

• Triangular channel:

A channel which is triangular shape is called triangular channel.

• Trapezoidal channel:

A channel which is trapezoidal in shape is called trapezoidal channel.

• Circular channel:

A channel which is circular in shape is called circular channel.

On the basis of occurrence:

• Natural channel:

Nature channel are generally irregular in shape, alignment and roughness of the surface.

Example: Streams, rivers, valley etc.

• Artificial channel :

Artificial channel are built for some specific purpose, such as irrigation, water supply, waste water, water power development and rain collection channels. These are regular in shape and alignment with uniform roughness of the boundary surface.

On the basis of boundary:

• Prismatic channel:

In this type of channel slope and cross section both don’t change.

• Non-Prismatic channel:

In this type of channel both slope and cross section or only slope or cross section change.

• Rigid channel:

In this type of channel boundary of channel do not change.

Example: Lined channel.

• Mobile channel:

In this type of channel boundary of channel will change.

Example: River, unlined channel.

On the basis of time:

• Steady flow:

A flow in which the velocity, depth of flow and discharge of the fluid at a particular fixed point does not change with time.

Mathematically,

∂v/∂t = 0 ,

∂Q / ∂t = 0 and

∂y / ∂t = 0

Where,

 v= Velocity

Q= Discharge

y =Depth of flow

• Unsteady flow:

A flow in which the velocity , discharge (i.e. rate of flow) and depth of flow of the fluid change with time.

Mathematically,

∂v/∂t  ≠ 0,

∂Q / ∂t ≠ 0 and

∂y / ∂t ≠ 0

Where,

 v= Velocity

Q= Discharge

y =Depth of flow

On the basis of space\length:

1. Uniform flow:

If for a given length of the channel, the velocity of flow, depth of flow, slope of the channel and cross- section remain constant, the flow is called uniform flow.

Mathematically,

∂v/∂s = 0 ,

∂y / ∂s = 0

2. Non-Uniform flow:

If for a given length of the channel, the velocity of flow, depth of flow, slope of the channel and cross-section donot remain constant the flow is called non-uniform flow.

Mathematically,

∂v/∂s ≠ 0 ,

∂y / ∂s ≠ 0

Also, non-uniform flow in open channels is also called varied flow.

Types of Non-Uniform flow:

• Rapidly varied flow(R.V.F)

It is defined as the flow in which depth of flow changes abruptly over a small length of the channel.

• Gradually varied flow(G.V.F)

It is defined as the flow in which depth of flow in a channel changes gradually over a long lengh of the channel,

• Spatially varied flow(S.V.F)

It is defined as the flow in which depth of flow changes gradually due to change in discharge.

On the basis of Reynolds number:

• Laminar flow:

It is defined as the type of flow in which the fluid particles move along well-defined paths or stream lines and all the stream lines and all the stream lines are straight and parallel.

Reynolds number(Re) is less than 500 or 600.

• Turbulent flow:

It is defined as the type of flow in which the fluid undergoes irregular fluctuations, or mixing, in contrast to laminar flow, in which the fluid moves in smooth paths or layers.

Reynolds number (Re) is more than 2000.

• Transitional flow:

The flow occurs between laminar and turbulent flow. Reynolds number ( Re) lies between 500 to 2000.

Mathematically,

Re= (ρvR)/ μ

On the basis of Froude number(F_r)

• Critical flow:

The flow in open channel is said to be critical flow if the Froude number(F_r) is equal to 1.

• Subcritical flow:

The flow in open channel is said to be critical flow if the Froude number(F_r) is less than 1.

• Supercritical flow:

The flow in open channel is said to be supercritical flow if the Froude number(F_r) is greater than 1.

Trapezoid:

• Depth of flow = y
  • Depth of flow section(y^’)

y^’ = ycosθ

• Top width(T)

T= B+2zy

• Area(A)

A = ( B+zy)y

• Wetted perimeter(P)

                  P= B+2y (Z² + 1)^1/2

• Hydraulic radius(R)

R= A/P

2. Rectangle

• Depth of flow= y

• Top width (T)

T= B

• Area(A)

A=By

• Wetted perimeter(P)

P= B+2y

• Hydraullic radius(R)

R= A/P

• Hydraullic depth(D)

D=A/T

Or D=y

3.Triangle:

• Depth of flow=y
• Top width(T)

T= 2zy

• Area(A)

A=zy²

• Wetted perimeter(P)

P=2y(z²+1)^1/2

• Hydraullic radius(R)

R=A/P

• Hydraulic depth (D)

D= A/T

Or, D= (½ )y