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.