Fluid Mechanics

1. External versus Internal Flow
• Internal Flow - Flow is completely
confined/bounded on all sides. Eg: Pipe Flows,
blood flow inside arteries, etc.. The growth of
boundary layer is confined!
•
External Flow - Flow over bodies immersed in an
unbounded fluid Eg: flow over an airfoil or turbine
blades, channel flows, jets, and so on.

2. Friendship (Druzhba) oil pipeline runs from east Russia to Ukraine, Belarus, Poland, and so
on
4000 KM (2500 miles)

3. Example of external flow - Flow over an airfoil
Boundary layer
Entrance length L
Uo
Fully developed profile
u
Example of internal flow - Flow inside a pipe

4. Consider steady, incompressible flow through a piping system
1 2
lT
h
gz
V
P
gz
V
P





















 2
2
2
2
1
2
1
1
2
2




Mechanical energy per
unit mass at cross section 1
Mechanical energy per
unit mass at cross section 2
The difference between the mechanical energy at two locations, i.e, the total
head loss, is a result of the conversion of mechanical energy to thermal energy
due to frictional/viscous effects. Irreversible pressure loss due to
viscous effects. Pressure drop decides the power requirements and pump size
α is called
Kinetic energy
Factor, laminar flow
α=2, and turbulent
flow α=1
V is average
velocity
at a cross
section
Total head loss between
Cross sections 1 and 2
Calculation of Head Loss

5. Head loss in pipe flows- Major and
Minor losses
Total Head Loss( hLT) = Major Loss (hL)+ Minor Loss (hLM)
g
V
K
hlm
2
2

Due to sudden expansion,
contraction, fittings etc
K is loss coefficient must be
determined for each situation
g
V
D
L
f
h
Equation
s
Darcy l
2
'
2


Due to wall friction
In this experiment you will find
friction factor for various pipes
For Short pipes with multiple fittings, the minor losses are no longer
“minor”!!

6. Laminar vs Turbulent Flows
Laminar flow – smooth/undisturbed flow, occurs when a fluid
flows in parallel layers, with no disruption between the layers.
Occurs at typically low velocities. No eddies, swirls, or lateral
velocity. Highly ordered motion. Effect of viscosity – Significant.
Analytical solutions!
Transition to turbulence – The stage where flow ceases to be
laminar and becomes fluctuating. This is the regime before the
flow becomes fully turbulent. Depends on factors like geometry,
surface roughness, surface temperature, flow veloity, type of
fluid, fluctuations in inner flow, etc. Note: Transition to
turbulence doesn’t happen all of a sudden. The flow transitions
to turbulence over a region.

7. Laminar vs Turbulent Flows
Turbulent flow – The order of the flow is disrupted. Flow is
characterized by eddies or small packets of fluid particles which result
in lateral mixing, highly disordered motion. Rapid mixing due to
enhanced momentum transfer between particles. Flow is energetic
Measure of flow regime – Reynolds number (Re = ρVD/µ = VD/ν)
• Ratio of inertial to viscous forces
Re = ρVavgD/µ = VavgD/ν
Effect of viscosity becomes less significant with increasing Re number
μ = kg/s.m (dynamic or absolute viscosity)
ν = μ/ρ = m2/s (kinematic viscosity)
High Re = low viscous forces, less orderly flow. Viscous forces cannot prevent
fluctuations anymore

8. Pipe Flow-
Critical Reynolds number
Reynold’s number at which the flow starts to transition to
turbulence
• Re < 2000 (laminar)
• 2000< Re <4000 (transitional)
• Re >4000 (turbulent)
The transition from laminar to turbulent doesn’t take place
instantaneously, rather it is a gradual process.
Factors affecting transition
• Surface roughness
• Surface temperature
• Velocity
• Type of fluid
• Geometry
• Vibrations
• Pre-existing fluctuations in the upstream flow, etc.
“Most of the fluid flow fields in real life are
transitional or fully turbulent”

9. Entrance Region
• When fluid enters a pipe with a certain velocity, the no slip condition due to
viscosity causes the fluid particles close to the wall to be zero (Vwall = 0 )
• Consequently, it progressively slows down the fluid layers above the wall
• To compensate for the lower velocity close to the walls, the fluid particles in the
center of the pipe accelerate to keep the mass flow rate constant.
• A velocity gradient develops in the pipe where the fluid velocity changes from 0
at the wall to Vmax at the center of the pipe
This region near the wall where the shear forces due to viscosity are felt are called
viscous boundary layer or simply Boundary Layer
Boundary layer
Entrance length L
Uo
Fully developed profile
u

10. Entrance Region
In general, there are two regions of flow inside the pipe
1) Boundary layer region: Effects of viscosity are high, significant
velocity gradient
2) Free stream/core : Flow can be approximated as irrotational,
inviscid, viscous (frictional) effects are negligible (Fluid is always
viscous – Flow can be approximated to be inviscid- IMPORTANT)
The boundary layer grows in the downstream direction until it
reaches the center of the pipe.
The region from the pipe inlet to the point where the boundary
layers meet is called the ‘hydrodynamic entrance region’ and
its length is called ‘hydrodynamic entrance length’.
Boundary layer
Entrance length L
Uo
Fully developed profile
u
Remember
potential core in
a free jet

11. Entrance Region
Parabolic profile

12. Entrance Region
Note: Wall shear stress is related to the slope of the velocity profile at the
wall.
• At the pipe entrance, wall shear stress is maximum
• value decreases gradually to the fully developed value. In the fully
developed region, wall shear stress is constant. Velocity profile doesn’t
change.
• This means the pressure drop is the highest at the entrance of the tubes
• Different relations are used to estimate entrance lengths
Entrance effects may be negligible for long pipes. However, for shorter pipes,
entrance losses might constitute a major portion of the total loss

13. Fully developed pipe flow
Vavg
• No slip condition – viscosity
• Vwall = 0
• Vcenter = Vmax
• In experiments we usually use Vavg

14. Fully Developed Pipe Flow
Assumptions
1. Incompressible, fully developed, steady
2. The length of the pipe is much longer than the entrance region
(neglect entrance effects)
Vmax
Vavg
Pressure drop for a fully
developed pipe flow
is dependent
On wall shear stress.
P drops linearly with x
P2
P1
Force
balance

15. So far we have,
Average velocity in a pipe
Pressure gradient in a pipe is dependent on shear
stress and is constant. Pressure drops linearly with x
Also, solving force balance equation on a fluid element inside the pipe, we get,
Parabolic velocity profile for fully developed region
Plugging in u(r) in the equation for Vavg, we get,
Also, knowing and using equations for Vmax, we get,
This pressure drop is for laminar flow in a pipe which is a function of viscosity

16. What do you do for turbulent flows where analytical solutions are not available?
This is when use of empirical relations become necessary.
Also, a general formulation in terms of non-dimensional quantities will be
helpful even for laminar flows

17. Major loss (viscous
loss)
Physical problem is to relate pressure drop to fluid parameters and pipe
geometry
Differential Pressure Gauge-
measure ΔP
Pipe
V
L
D
ε
Using dimensional analysis we can show that
ρ μ

18. 
VD

Re
Experiments show that ΔP is directly proportional to L/D,
Defining friction factor ,
Notes:
• Vavg can simply be written
as V
• Both sides are divided by ’g’
to give represent hL as
column of fluid. In piping, it
is common to express head
loss in length scale
signifying additional height
fluid needs to be raised by
the pump
This is the generic form of major head loss in pipes due to friction or viscous losses.
The loss, as you would expect, is irreversible. The equation above works for both
laminar and turbulent flows.
NEXT STEP: FIND f for LAMINAR AND TURBULENT FLOW
f is called
Darcy’s friction
factor

19. Friction factor–
Laminar Flow
Generic equation for major loss
From analytical solution for laminar flow,
Using above equations, we can show,
Where, , friction factor is only a function of Reynold’s
number and NOT surface roughness, ε(Only for Laminar flow (Re
< 2300).
The friction factor can also be written in terms of wall shear stress
for laminar pipe flow as,
(Darcy’s friction factor)
(Fanning friction factor)

20. Watch your
units!!!
Loss in Mechanical energy
per unit weight of flowing fluid.
Unit of hL here is meters
Measuring ΔP, L, D, V f can be calculated
Loss in Mechanical energy
per unit mass of flowing fluid.
Unit of hL here is NOT meters
1 2
lT
h
gz
V
P
gz
V
P





















 2
2
2
2
1
2
1
1
2
2




Mechanical energy per
unit mass at cross section 1
Mechanical energy per
unit mass at cross section 2
Total head loss between
Cross sections 1 and 2
hLT = HL+ Hm

21. Friction Factor –
Turbulent Flows
• For Turbulent flow ( Re>4000) it is not possible to derive analytical
expressions.
• Empirical expressions relating friction factor, Reynolds number and
relative roughness are available in literature
In general, the friction factor f is determined experimentally and is usually
published in graphical form as a function of Reynolds number and non
dimensional surface Roughness ε/D.
L F Moody published this data first, usually referred to as Moody’s chart

22. Moody’s chart for friction factor
f
ReD
Laminar
f=64/Re
Transition
D
/
 Increases
Fully rough flow – Complete turbulence
f = function (Re, ɛ)

23. Laminar Flow: f = 64/Re (Re increases = viscous effect
decreases = smaller head loss coefficient
Transitional Regime : Critical, need experimental results
Fully turbulent regime
1) Smooth pipes – viscous effects decrease with increasing Re, decreasing f with
increasing Re
2) Very very rough pipes – f remains constant with increasing Re
3) Rough pipes – f decreases with increasing Re until the flow reaches some critical Re.
After that, f remains constant / For higher Re, f is a function of ɛ only.
f = function (Re, ɛ)

24. Friction factor correlations
f is not related explicitly Re and relative
roughness in this equation.
The following equation can be used instead























f
D
f
Equation
Colebrook
Re
51
.
2
7
.
3
/
log
0
.
2
1 
 
8
2
6
2
9
.
0
10
Re
5000
10
10
Re
74
.
5
7
.
3
ln
325
.
1























 

and
D
for
D
f



25. Minor Losses
These components interrupt the smooth motion of the flow and cause local
separation and recirculation of flow
Flow separation (locally) and associated viscous effects will tend to decrease
the flow energy. This results in losses.
The phenomenon is fairly complicated. Empirical loss coefficient ‘K’ will take
care of these complexities
Even though the losses are called “minor losses”, at times they make up a
large portion of the total losses (For instance, a short pipe system with a lot of
bends and valves, partially close control valves (decreased mass flow), etc.)
Valves Bends T joints Expansions Contractions
g
V
K
hlm
2
2

Entrance Exits

27. Minor Losses

28. Final Equation