Viscous Flow in Pipes
Introduction
•
Flow of viscous, incompressible fluid in pipes and ducts will be considered.
•
Pipe is of round cross section, duct is not round.
•
Basic components of a typical pipe system include pipes, fittings, valves, pumps or
turbines.
•
Even simple pipe systems are quite complex in terms of analytical considerations.
•
“Exact” analysis of the simplest pipe flow topics (such as laminar flow in long,
straight, constant diameter pipes) and dimensional analysis considerations
combined with experimental results for the other pipe flow topics will be used.
General Characteristics of Pipe Flow
•
Unless otherwise specified, we will assume that the conduit is round (pipe).
•
Pipe is assumed to be completely full of the flowing fluid.
•
For open-channel flow (b), gravity alone is the driving force – water flows down a
hill.
•
For pipe flow (a), gravity may be important, but the main driving force is a pressure
gradient along the pipe.
•
If pipe is not full, it is not possible to maintain this pressure difference.
Laminar of Turbulent Flow
•
Flow may be laminar, transitional, or turbulent (figure).
•
For laminar flow in a pipe there is only one component of velocity, V = ui
•
For turbulent flow predominant component of velocity is also along the pipe, but it
is unsteady (random) and accompanied by random components normal to the pipe
axis, V = ui + vj + wk (figure).
•
Pipe flow characteristics are dependent on the value of the Reynolds number
Re 
VD

•
Reynolds number range for which laminar, transitional, or turbulent pipe flows are
obtained cannot be precisely given.
•
Flow in round pipe is laminar if Re is less than approximately 2100
•
Flow is round pipe is turbulent if Re is greater than approximately 4000
Entrance Region and Fully Developed Flow
•
Region of flow near where fluid enters the pipe is termed the entrance region
•
Fluid typically enters the pipe with a nearly uniform velocity profile at section (1).
Entrance Region and Fully Developed Flow
•
As the fluid moves through the pipe, viscous effects cause it to stick to the pipe
wall (no-sip condition). Boundary layer in which viscous effects are important is
produced along the wall. Initial velocity profile changes with distance along pipe,
x.
Entrance Region and Fully Developed Flow
•
Beyond section (2) velocity profile does not vary with x. Boundary layer has grown
in thickness to completely fill the pipe.
Entrance Region and Fully Developed Flow
•
Viscous effects are of considerable importance within the boundary layer.
•
For fluid outside the boundary layer (within inviscid core) viscous effects are
negligible
Entrance Region and Fully Developed Flow
•
Dimensionless entrance length:
le
 0.06 Re for laminar flow
D
le
16
 4.4  Re 
for turbulent flow
D
Entrance Region and Fully Developed Flow
•
Flow between (2) and (3) is termed fully developed.
•
Fully developed flow is interrupted by bend, valves etc. Beyond the interruption
flow gradually begins its return to its fully developed character.
Pressure and Shear Stress
•
Flow in horizontal pipe is driven by pressure difference.
•
Nonzero pressure gradient is a result of viscous effects (if viscosity were zero,
pressure would not vary with x). Work done by pressure forces is needed to
overcome the viscous dissipation of energy through the fluid.
•
In fully developed flow region viscous force exactly balances pressure force (fluid
flows with no acceleration).
•
In non-fully developed flow regions fluid accelerates or decelerates (velocity
profile changes). Thus, in the entrance region there is a balance between pressure,
viscous, and inertia (acceleration) forces.
•
Magnitude of the pressure gradient, p/x, is larger in the entrance region than in
the fully developed region, where it is constant, p/x = –Δp/l < 0 (figure)
•
Laminar flow differs from turbulent due to different nature of shear stress in
laminar and turbulent flows.
Fully Developed Laminar Flow
•
Velocity profile in fully developed flow is the same at any cross section of the pipe.
•
If velocity profile is known, other flow information such as pressure drop, head
loss, flowrate and the like can be obtained.
•
Details of the velocity profiles are different for laminar and turbulent flows.
•
Equation for velocity profile in fully developed laminar flow (and other important
results) can be derived:
– from F = ma applied directly to fluid element,
– from Navier-Stokes equations of motion,
– from dimensional analysis.
•
Regardless of method used results are the same
Fully Developed Laminar Flow
Consider steady, fully developed, laminar flow of an incompressible, viscous fluid in a
horizontal pipe.
Velocity variation in radial direction, combined with fluid viscosity, produces shear
stress
If gravity force is neglected, the pressure varies only in x direction. If pressure
decreases in x direction, then
p2  p1  p
 p  0 
Fully Developed Laminar Flow
Apply Newton’s second law of motion to cylindrical fluid element.
p 2

l
r
Basic pipe flow is governed by a balance between viscous and pressure forces.
(a)
Fully Developed Laminar Flow
Shear stress varies from zero (at r = 0) to the wall shear stress (at r = D/2)
2 r
 w
D
Pressure drop and wall shear stress are related by
4l w
p 
D
(b)
(c)
For laminar flow of Newtonian fluid
  
du
dr
(d)
Fully Developed Laminar Flow
Combining eqs. (a) and (d) and integrating obtain velocity profile (details)
 pD 2 
 2r 
u r  
  1   
16

l
 D 

 
2
2

 2r
 V c 1  
  D

(e)
where Vc is the centerline velocity (V at r=0) . In terms of wall shear stress:
2
 wD 
 r 
u r 
 1   
4  
 R 
(f)
 R 2Vc
Q   udA 
2
(g)
Volume flowrate:
Average velocity
Q Vc pD 2
V  
A 2 32 l
back
(h)
Fully Developed Laminar Flow
Volume flow rate:
 D 4 p
Q
128 l
(i)
Thus, for a horizontal pipe the flowrate is
(a) directly proportional to the pressure drop,
(b) inversely proportional to the viscosity,
(c) inversely proportional to the pipe length,
(d) proportional to the pipe diameter to the fourth power
The flow is termed Hagen-Pouseuille flow and equation (e) is commonly referred to as
Poiseuille’s law
Fully Developed Laminar Flow
For nonhorizontal pipes
Fully Developed Laminar Flow
For nonhorizontal pipes
  p   l sin   D 4
Q
128 l
Fully Developed Laminar Flow
Darcy friction factor
Fully Developed Laminar Flow
p 
From equation (h)
32  lV
D2
Dividing by V 2/2 we obtain
p
or
1
V 2
2
32  l D 



 64
2
1
V 2
2
l V 2
p  f
D 2
f  p  D l   V 2 2 
where Darcy friction factor
For laminar flow
   l
64
 l


 
 
Re
 D
 VD  D
f 
64
Re
or
f 
8 w
V 2
Fully Developed Laminar Flow
Consider energy equation for the flow considered
p1
V12
p2
V22
 1
 z1 
 2
 z2  hL

2g

2g
where kinetic energy coefficient α accounts for nonuniform velocity profile. For fully
developed flow α1 = α2 and
With
p   l sin  2

l
r
 p1
 
p2 

z

 z2  hL
1 




 

hL 
2 l 4l w

r
D
Head loss in a pipe is a result of the viscous shear stress on the wall
Fully Developed Laminar Flow. Summary
Velocity profile
 pD 2 
 2r 
u r  
  1   
 D 
 16  l  
2
2
 wD 
 r 
u r 
 1   
4  
 R 
Centerline velocity and average velocity
pD 2
Vc 
16  l
Vc pD 2
V 
2 32  l
Poiseuille’s law
 D 4 p
Q
128 l
Head loss
hL 
2 l 4l w

r
D
  p   l sin   D 4
Q
128 l
Fully Developed Turbulent Flow
Transition from Laminar to Turbulent Flow
Transition from laminar to turbulent flow in a pipe occurs at 2100 < Re < 4000
Turbulent Shear Stress
Turbulent flow parameters can be described in terms of mean and fluctuating parameters.
If u = u(x,y,z,t) is x component of instantaneous velocity, then its time average value is
u
1 t0 T
u  x, y, z, t  dt

t
0
T
The fluctuating part of the velocity, u’ is that time-varying portion that differs from the
average value
u  u  u
or
u  u  u
more
details
Turbulent Shear Stress
du
 
?
dy
du
 
dy
du
 
?
dy
Turbulent Shear Stress
du
 
dy
(a) Laminar flow shear stress caused by random motion of molecules.
du
 
?
dy
Turbulent Shear Stress
du
 
dy
(a) Laminar flow shear stress caused by random motion of molecules.
(b) Turbulent flow as a series of random, three-dimensional eddies.
Turbulent Shear Stress
•
Laminar flow involves randomness on the molecular scale.
•
Turbulent flows involve random motions of finite-sized particles. It consists of
series of random, three-dimensional eddy type motions.
•
These eddies range in size from very small to fairly large diameter. They move
about randomly, conveying mass with an average velocity u  u  y 
•
Eddy structure greatly promotes mixing within the fluid and greatly increases
transport of x momentum across the plane A-A.
•
Thus, finite parcels of fluid (not individual molecules as in laminar flow) are
randomly transported across this plane, resulting an a relatively large shear stress.
•
Random velocity components that account for this momentum transfer (hence, the
sear force) are u’ (for the x component of velocity) and v’ (for the rate of mass
transfer crossing the plane).
Turbulent Shear Stress
Shear stress on plane A  A is given by
 
du
  u v   lam   turb
dy
If flow is laminar, than u  v  0, and above equation reduces to laminar shear stress
For turbulent flow it is found that turbulent shear stress,  turb    u v, is positive. Hence,
shear stress is greater in turbulent flow than in laminar flow.


Term of the form   u v or   vw, etc. are called Reynolds stresses.
Note, turbulent shear stress depends on the fluid density
Turbulent Shear Stress
Structure of turbulent flow in a pipe. (a) Shear stress. (b) Average velocity
more...
Dimensional Analysis of Turbulent Flow
Most turbulent pipe flow information is based on experimental data and semiempirical
formulas, even if flow is fully developed.
Fundamental difference between laminar and turbulent flow is that the shear stress for
turbulent flow is a function of the density of the fluid,  (Reynolds stresses).
For turbulent flow there is a relatively thin viscous sublayer formed in the fluid near
the pipe wall. If a wall roughness element protrudes sufficiently far into (or even
through) this layer, the structure and properties of the viscous sublayer (along with Δp
and τw) will be different than if the wall were smooth. Thus, for turbulent flow pressure
drop is the function of the wall roughness.
Pressure drop for steady, incompressible turbulent flow in a horizontal round pipe of
diameter D can be written in functional form as
p  F  V , D , l ,  ,  ,  
where V is the average velocity, l is the pipe length, and ε is a measure of the
roughness of the pipe wall.
Dimensional Analysis of Turbulent Flow
In dimensionless form:
 VD l  
p


, , 

1
2
V
  D D
2
where ε/D is a relative roughness. Assuming that pressure drop is proportional to the
pipe length we have
p
l 



Re,


1
2
D
D

V


2
With
where

f  p D l V 2 2
 pressure drop can be written as
l V 2
p  f
D 2


f    Re, 
D

Dimensional Analysis of Turbulent Flow
For laminar fully developed flow, the value of f is f = 64/Re, independent of ε/D.
For turbulent flow, functional dependence f = φ(Re, ε/D) is a complex one that cannot,
as yet, be obtained from theoretical analysis. Results are obtained from experiment
and presented in terms of curve-fitting formula or graphical form.
From energy equation for steady incompressible fully developed flow in constant
diameter (D1 = D2 so that V1 = V2) horizontal pipe follows that
l V2
hL  f
D 2g
This equation, called the Darcy-Weisbach equation, is valid for any fully developed,
steady, incompressible pipe flow – whether pipe is horizontal or on a hill.
Friction Factor
Functional dependence of friction factor, f, on Reynolds number and relative
roughness is obtained from experiments conducted by J. Nikuradse in 1933.
Original data of Nikuradse were correlated by L.F. Moody and C.F. Colebrook and
presented in Moody chart and Colebrook formula
  D
1
2.51 
 2.0 log 



3.7
f
Re f 


Typical roughness values for various pipe surfaces are given in table
Example 8.5 Air under standard conditions flow through a 4.0-mm-diameter drawn
tubing with an average velocity of V = 50 m/s. For such conditions the flow would
normally be turbulent. However, if precautions are taken to eliminate disturbances to
the flow (the entrance to the tube is very smooth, the air is dust free, the tube does not
vibrate, etc.), it may be possible to maintain laminar flow. (a) Determine the pressure
drop in a 0.1-m section of the tube if the flow is laminar. (b) Repeat the calculations if
the low is turbulent.
Example 8.5 Air under standard conditions flow through a 4.0-mm-diameter drawn
tubing with an average velocity of V = 50 m/s. For such conditions the flow would
normally be turbulent. However, if precautions are taken to eliminate disturbances to
the flow (the entrance to the tube is very smooth, the air is dust free, the tube does not
vibrate, etc.), it may be possible to maintain laminar flow. (a) Determine the pressure
drop in a 0.1-m section of the tube if the flow is laminar. (b) Repeat the calculations if
the low is turbulent.
Solution With density and viscosity known, Reynolds number:
Re  VD   13700
(a) If the flow were laminar, then f = 64/Re = 0.00467, and the pressure drop:
l V 2
p  f
 0.179 kPa
D 2
Example 8.5 Air under standard conditions flow through a 4.0-mm-diameter drawn
tubing with an average velocity of V = 50 m/s. For such conditions the flow would
normally be turbulent. However, if precautions are taken to eliminate disturbances to
the flow (the entrance to the tube is very smooth, the air is dust free, the tube does not
vibrate, etc.), it may be possible to maintain laminar flow. (a) Determine the pressure
drop in a 0.1-m section of the tube if the flow is laminar. (b) Repeat the calculations if
the low is turbulent.
Solution With density and viscosity known, Reynolds number:
Re  VD   13700
(a) If the flow were laminar, then f = 64/Re = 0.00467, and the pressure drop:
l V 2
p  f
 0.179 kPa
D 2
(b) If the flow were turbulent, then from table ε = 0.0015 mm so that ε/D = 0.000375.
From Moody chart, with Re = 1,37x104 and ε/D = 0.000375, f = 0.028. Pressure drop:
l V 2
p  f
 1.076 kPa
D 2
p  43.7 kN/m 2
(1)
l
h1
p  43.7 kN/m 2
h2
(2)
(1)
Answer
p  43.7 kN/m 2
h  18.5 m
(2)
Minor Losses
Minor Losses
Losses occur in straight pipes (major losses) and pipe system components (minor
losses)
Major losses can be calculated by use of friction factor obtained from Moody chart
Minor losses are given in terms of loss coefficient, which is defined as
KL 

hL
p

1
2

V
V 2 2g
2

Then pressure drop
V 2
p  K L
2
and head loss
V2
hL  K L
2g
Minor Losses
In most cases of practical interest the loss coefficients for components are function of
geometry
K L    geometry 
Minor losses are sometimes given in terms of an equivalent length, leq
In this terminology, head loss through component is given in terms of the equivalent
length of pipe that would produce the same head loss as the component. That is
leq V 2
V2
hL  K L
 f
2g
D 2g
or
leq 
KL D
f
where D is based on the pipe containing the component
Minor Losses
Entrance flow conditions and loss coefficient. (a) Reentrant, KL = 0.8, (b) sharp-edged, KL =
0.5, (c) slightly rounded, KL = 0.2, well-rounded, KL = 0.04
Minor Losses
Flow pattern and pressure distribution for a sharp-edged entrance
Minor Losses
Entrance loss coefficient as a function of rounding of the inlet edge
Minor Losses
Exit flow conditions and loss coefficient. (a) Reentrant, KL = 1.0, (b) sharp-edged, KL = 1.0,
(c) slightly rounded, KL = 1.0, well-rounded, KL = 1.0
Minor Losses
Loss coefficient for a sudden contraction
Minor Losses
Loss coefficient for a sudden expansion
Noncircular Conduits
Noncircular duct calculations are based on hydraulic diameter
Noncircular duct
Dh 
4A
P
Noncircular Conduits
Fully Developed Laminar Flow
Friction factor
f 
C
Re h
where constant C depends on the shape of the duct, and Reh is based on hydraulic
diameter
Re h 
VDh

Hydraulic diameter is also used in definition of relative roughness, ε/Dh , and head loss
l V2
hL  f
Dh 2 g
Noncircular Conduits
Fully Developed Turbulent Flow
Calculations for fully developed turbulent flow in ducts of noncircular cross section
are usually carried out by using the Moody chart data for round pipes with the
diameter replaced by the hydraulic diameter and the Reynolds number based on the
hydraulic diameter.
Such calculations are usually accurate to within about 15%.
If grater accuracy is needed, a more detailed analysis based on the specific geometry
on interest is needed
Single Pipes Problem Solutions
END
Supplementary slides
Typical Pipe System
back
(a) Experiment to illustrate type of flow. (b) Typical dye streaks
back
Time dependence of fluid velocity at a point
back
Pressure distribution along a horizontal pipe
back
Turbulent Flow
Time average of the fluctuations is zero
u 

Average of the square of fluctuation is positive
 u  
2

t0 T
1 t0 T
1 t0 T
1
u

u
dt

udt

udt



 Tu  Tu   0



t
t
t
0
0
0
T
T
T
1 t0 T
2

u
dt  0



t
T 0
Average of products of the
fluctuations, such as u v ,
may be zero or nonzero (either
positive of negative)
back
Turbulent Velocity profile
In viscous sublayer (law of the wall)
u
yu*

u*
v
In overlap region
 yu *
u
 2.5ln 
  5.0
*
u
 v
where friction velocity:
u* 
Typical structure of the
turbulent velocity profile
in a pipe
w

back
Turbulent Velocity profile
In outer turbulent layer (power-law velocity profile)
u 
r
  1 
Vc 
R
1
n
Exponent, n, for power-law velocity profiles
back
Turbulent Velocity profile
Typical laminar flow and
turbulent flow velocity profile
back
Flow in the viscous sublayer
near rough and smooth walls
back
Moody Chart
back
back
back
Turbulent Flow
Time-averaged, u , and fluctuating, u, description of a parameter for turbulent flow