Gandhinagar Institute of TechnologySubject :- Fluid Mechanics

Prepared By – Pavan Narkhede [130120119111] Darshit Panchal [130120119114]

Topic :- Turbulent Flow

Guided by :Prof.Jyotin kateshiya

MECHANICAL ENGINEERING 4th - B : 2

INTRODUCTION:

Laminar Flow: In this type of flow, fluid particles moves along smooth straight parallel

paths in layers or laminas, with one layer gliding smoothly over an adjacent layer, the paths

of individual fluid particles do not cross those of neighbouring particles.

Turbulent Flow: In turbulent flow, there is an irregular random movement of fluid in

transverse direction to the main flow. This irregular, fluctuating motion can be regarded as

superimposed on the mean motion of the fluid.

Laminar

Transitional

Turbulent

Types of flow depend on the Reynold number , ρVd Re = -------- µ Re < 2000 – flow is laminar

Re > 2000 – flow is turbulent

2000 < Re < 4000 – flow changes from laminar to turbulent.

Magnitude of Turbulence : - It is the degree of turbulence, and measures how strong, violent or intence the turbulence.

- Magnitude of Turbulence = Arithmetic mean of root mean square of turbulent fluctuations

= Or =

t

dtut 0

21

t

dtut 0

21 222

31 wvu

Intensity of turbulence :

- It is the ratio of the magnitude of turbulence to the average flow velocity at a

point in the flow field

- So, Intensity of Turbulence =

222

222

31

wvu

wvu

Expression for co-efficient of friction :Darcy – Weisbach equation

From the experimental measurement on turbulent flow through pipes, it has observed

That the viscous friction associated with fluid are proportional to

(1) Length of pipe (l)

(2) Wetted perimeter (P)

(3) Vn , where V is average velocity and n is index depending on the material

(normally, commertial pipe turbulent flow n=2

f – friction factor

L – length of pipe

D – diameter of pipe

v – velocity of flow

OR

gpp

h f 21

gD

lVCh ff 2

2

Moody Diagram :

Developed to provide the friction factor for turbulent flow for various values of Relative roughness and

Reynold’s number!

From experimentation, in turbulent flow, the friction factor (or head loss) depends upon velocity of fluid

V, dia. of pipe D, density of fluid ρ, viscosity of fluid µ, wall roughness height ε.

So, f = f1 (V,D, ρ, µ, ε)

By the dimensional analysis,

, Where called relative roughness.

0

,1 DVDff

D

Key points about the Moody Diagram –

1. In the laminar zone – f decreases as Nr increases!

2. 2. f = 64/Nr.

3. 3. transition zone – uncertainty – not possible to predict -

4. Beyond 4000, for a given Nr, as the relative roughness term D/ε increases (less rough), friction

factor decreases

5. For given relative roughness, friction factor decreases with increasing Reynolds number till the

zone of complete turbulence

6. Within the zone of complete turbulence – Reynolds number has no affect.

7. As relative roughness increases (less rough) – the boundary of the zone of complete turbulence

shifts (increases)

Co-efficient of friction in terms of shear stress :

We know, the propelling force = (p1 - p2) Ac ---- (1)

Frictional resistance in terms of shear stress = As Where = shear stress ----(2)

By comparing both equation,

(P1 – P2) = OR

( co-efficient of frictionin terms of shear stress)

0 0

Vf 2

02

vu

dAuvdA

dAdF

t

Shear stress in turbulent flow

In turbulent flow, fluid particles moves randomly, therefore it is impossible to trace the

Paths of the moving particles and represents it mathematically

u b

u

= mean velocity of particles moving along layer A

= mean velocity of particles moving along layer B The relative velocity of particle along layer B and with respect to layer A

= - since , >

This relative velocity is the cause of shear stress between the two layers

u au b

u a

u au b

Prandtl’s mixing length theory :

Prandtl’s assumed that distance between two layers in the transverse direction

(called mixing length l) such that the lumps of fluid particles from one layer could reach the other

Layer and the particles are mixed with the other layer in such a way that the momentum of the

Particles in the direction of x is same, as shown in below figure :

Total shear

where , (Viscosity)

n = 0 for laminar flow. For highly turbulent flow, .

tv

222

dyduyk

dydu

dydu

dydu

dydu

dydu

22 yk

Hydrodynamically Smooth andRough Pipe Boundaries

Hydronamically smooth pipe : The hight of roughness of pipe is less than thickness of

laminar sublayer of flowing fluid.

ɛ < δ′

Hydronamically rough pipe : The hight of roughness of pipe is greater than the

thickness of laminar sublayer of flowing fluid.

ɛ > δ′

From Nikuradse’s experiment

Criteria for roughness: Hydrodynamically

smooth pipe Hydrodynamically

rough pipe Transiton region

region in a pipe In terms of Reynold number1. If Re → Smooth boundary

2. If Re ≥100→Rough boundary

3. If 4<Re <100 →boundary is in transition stage.

625.0

25.0

6

4

The Universal Law of The Wall

dydvyk

2

22

0

CRy

Kv LogV

e

*

Ryv LogVv e*max

5.2

yRv LogVV

ve*

*

max 5.2

Velocity Distribution for turbulentflow

Velocity Distribution

in a hydrodynamically

smooth pipe

Velocity Distribution

in a hydrodynamically

Rough Pipes

y

Vv

elog5.25.8*

R

Vv

elog5.275.4*

Velocity Distribution for turbulentflow in terms of average Velocity (V)

Velocity Distribution

in a hydrodynamically

smooth pipe

Velocity Distribution

in a hydrodynamically

Rough Pipes

RV

VV

e

*5.275.1* log

R

VV

elog5.275.4*

Resistance to flow of fluid in smooth and rough pipes

The frictional head loss

- Where f = frictional co-efficient or friction factor - Pressure loss in pipe is given by

gDflVh f 2

4 2

Dl

DVP ,Re,2

2

- From experimental result, the pressure loss is the function of l/D to the first power

- But friction factor

- From equation ,the friction factor f is a function of Re and ratio of ε/D.

DDlVP Re,

2

2

f

lVDP

2

2

Df Re,

For laminar flow

- We know, in laminar flow the f is function of only re and it is independent of ε/D ratio.

For terbulent flow- In terbulent flow, f is a function of Re and type of pipe. So f is also depend on

boudary.

a) Smooth pipe

b) Rough pipe

Re16

f

(a) Smooth pipe

For smooth pipe ,f is only a function of Re. For 4000<Re<

laminar sublayer (δ′>>ε).- The blasius equation for f as

,For 4000<Re< laminar sublayer in smooth pipe.

From Nikuradse’s experimental result for smooth pipe

510

41

Re

079.0f 510

8.0)4(Re0.241 log

10 f

f

(b) Rough pipe

- In rough pipe δ′<<ε, the f is only function of ratio ε/D and it is independent of Re.

- From Nikuradse’s experimental result for rough pipe

74.1241 log10

R

f

THANK YOU….