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# VISCOUS INCOMPRESSIBLE FLOW Notes | EduRev

## : VISCOUS INCOMPRESSIBLE FLOW Notes | EduRev

``` Page 1

NPTEL – Mechanical – Principle of Fluid Dynamics

Joint initiative of IITs and IISc – Funded by MHRD                                                            Page 1 of 72
Module 5 : Lecture 1
VISCOUS INCOMPRESSIBLE FLOW
(Fundamental Aspects)

Overview
Being highly non-linear due to the convective acceleration terms, the Navier-Stokes
equations are difficult to handle in a physical situation. Moreover, there are no general
analytical schemes for solving the nonlinear partial differential equations.  However,
there are few applications where the convective acceleration vanishes due to the
nature of the geometry of the flow system. So, the exact solutions are often possible.
Since, the Navier-Stokes equations are applicable to laminar and turbulent flows, the
complication again arise due to fluctuations in velocity components for turbulent
flow. So, these exact solutions are referred to laminar flows for which the velocity is
independent of time (steady flow) or dependent on time (unsteady flow) in a well-
defined manner. The solutions to these categories of the flow field can be applied to
the internal and external flows. The flows that are bounded by walls are called as
internal flows while the external flows are unconfined and free to expand. The
classical example of internal flow is the pipe/duct flow while the flow over a flat plate
is considered as external flow. Few classical cases of flow fields will be discussed in
this module pertaining to internal and external flows.
Laminar and Turbulent Flows
The fluid flow in a duct may have three characteristics denoted as laminar, turbulent
and transitional. The curves shown in Fig. 5.1.1, represents the x-component of the
velocity as a function of time at a point ‘A’ in the flow. For laminar flow, there is one
component of velocity
ˆ
V ui =
?
and random component of velocity normal to the axis
becomes predominant for turbulent flows i.e.
ˆ ˆˆ
V u i v j wk = + +
?
. When the flow is
laminar, there are occasional disturbances that damps out quickly. The flow Reynolds
number plays a vital role in deciding this characteristic. Initially, the flow may start
with laminar at moderate Reynolds number. With subsequent increase in Reynolds
number, the orderly flow pattern is lost and fluctuations become more predominant.
When the Reynolds number crosses some limiting value, the flow is characterized as
turbulent. The changeover phase is called as transition to turbulence. Further, if the
Page 2

NPTEL – Mechanical – Principle of Fluid Dynamics

Joint initiative of IITs and IISc – Funded by MHRD                                                            Page 1 of 72
Module 5 : Lecture 1
VISCOUS INCOMPRESSIBLE FLOW
(Fundamental Aspects)

Overview
Being highly non-linear due to the convective acceleration terms, the Navier-Stokes
equations are difficult to handle in a physical situation. Moreover, there are no general
analytical schemes for solving the nonlinear partial differential equations.  However,
there are few applications where the convective acceleration vanishes due to the
nature of the geometry of the flow system. So, the exact solutions are often possible.
Since, the Navier-Stokes equations are applicable to laminar and turbulent flows, the
complication again arise due to fluctuations in velocity components for turbulent
flow. So, these exact solutions are referred to laminar flows for which the velocity is
independent of time (steady flow) or dependent on time (unsteady flow) in a well-
defined manner. The solutions to these categories of the flow field can be applied to
the internal and external flows. The flows that are bounded by walls are called as
internal flows while the external flows are unconfined and free to expand. The
classical example of internal flow is the pipe/duct flow while the flow over a flat plate
is considered as external flow. Few classical cases of flow fields will be discussed in
this module pertaining to internal and external flows.
Laminar and Turbulent Flows
The fluid flow in a duct may have three characteristics denoted as laminar, turbulent
and transitional. The curves shown in Fig. 5.1.1, represents the x-component of the
velocity as a function of time at a point ‘A’ in the flow. For laminar flow, there is one
component of velocity
ˆ
V ui =
?
and random component of velocity normal to the axis
becomes predominant for turbulent flows i.e.
ˆ ˆˆ
V u i v j wk = + +
?
. When the flow is
laminar, there are occasional disturbances that damps out quickly. The flow Reynolds
number plays a vital role in deciding this characteristic. Initially, the flow may start
with laminar at moderate Reynolds number. With subsequent increase in Reynolds
number, the orderly flow pattern is lost and fluctuations become more predominant.
When the Reynolds number crosses some limiting value, the flow is characterized as
turbulent. The changeover phase is called as transition to turbulence. Further, if the
NPTEL – Mechanical – Principle of Fluid Dynamics

Joint initiative of IITs and IISc – Funded by MHRD                                                            Page 2 of 72
Reynolds number is decreased from turbulent region, then flow may come back to the
laminar state. This phenomenon is known as relaminarization.

Fig. 5.1.1: Time dependent fluid velocity at a point.
The primary parameter affecting the transition is the Reynolds number defined as,

Re
UL ?
µ
= where, U

is the average stream velocity and L is the characteristics
length/width. The flow regimes may be characterized for the following approximate
ranges;
23
34
46
6
0 Re 1: Highly viscous laminar motion
1 Re 100 : Laminar and Reynolds number dependence
10 Re 10 : Laminar boundary layer
10 Re 10 : Transition to turbulence
10 Re 10 : Turbulent boundary layer
Re 10 : Turbulent and Reyn
< <
< <
< <
< <
< <
> olds number dependence

Fully Developed Flow
The fully developed steady flow in a pipe may be driven by gravity and /or pressure
forces. If the pipe is held horizontal, gravity has no effect except for variation in
hydrostatic pressure. The pressure difference between the two sections of the pipe,
essentially drives the flow while the viscous effects provides the restraining force that
exactly balances the pressure forces. This leads to the fluid moving with constant
velocity (no acceleration) through the pipe. If the viscous forces are absent, then
pressure will remain constant throughout except for hydrostatic variation.
Page 3

NPTEL – Mechanical – Principle of Fluid Dynamics

Joint initiative of IITs and IISc – Funded by MHRD                                                            Page 1 of 72
Module 5 : Lecture 1
VISCOUS INCOMPRESSIBLE FLOW
(Fundamental Aspects)

Overview
Being highly non-linear due to the convective acceleration terms, the Navier-Stokes
equations are difficult to handle in a physical situation. Moreover, there are no general
analytical schemes for solving the nonlinear partial differential equations.  However,
there are few applications where the convective acceleration vanishes due to the
nature of the geometry of the flow system. So, the exact solutions are often possible.
Since, the Navier-Stokes equations are applicable to laminar and turbulent flows, the
complication again arise due to fluctuations in velocity components for turbulent
flow. So, these exact solutions are referred to laminar flows for which the velocity is
independent of time (steady flow) or dependent on time (unsteady flow) in a well-
defined manner. The solutions to these categories of the flow field can be applied to
the internal and external flows. The flows that are bounded by walls are called as
internal flows while the external flows are unconfined and free to expand. The
classical example of internal flow is the pipe/duct flow while the flow over a flat plate
is considered as external flow. Few classical cases of flow fields will be discussed in
this module pertaining to internal and external flows.
Laminar and Turbulent Flows
The fluid flow in a duct may have three characteristics denoted as laminar, turbulent
and transitional. The curves shown in Fig. 5.1.1, represents the x-component of the
velocity as a function of time at a point ‘A’ in the flow. For laminar flow, there is one
component of velocity
ˆ
V ui =
?
and random component of velocity normal to the axis
becomes predominant for turbulent flows i.e.
ˆ ˆˆ
V u i v j wk = + +
?
. When the flow is
laminar, there are occasional disturbances that damps out quickly. The flow Reynolds
number plays a vital role in deciding this characteristic. Initially, the flow may start
with laminar at moderate Reynolds number. With subsequent increase in Reynolds
number, the orderly flow pattern is lost and fluctuations become more predominant.
When the Reynolds number crosses some limiting value, the flow is characterized as
turbulent. The changeover phase is called as transition to turbulence. Further, if the
NPTEL – Mechanical – Principle of Fluid Dynamics

Joint initiative of IITs and IISc – Funded by MHRD                                                            Page 2 of 72
Reynolds number is decreased from turbulent region, then flow may come back to the
laminar state. This phenomenon is known as relaminarization.

Fig. 5.1.1: Time dependent fluid velocity at a point.
The primary parameter affecting the transition is the Reynolds number defined as,

Re
UL ?
µ
= where, U

is the average stream velocity and L is the characteristics
length/width. The flow regimes may be characterized for the following approximate
ranges;
23
34
46
6
0 Re 1: Highly viscous laminar motion
1 Re 100 : Laminar and Reynolds number dependence
10 Re 10 : Laminar boundary layer
10 Re 10 : Transition to turbulence
10 Re 10 : Turbulent boundary layer
Re 10 : Turbulent and Reyn
< <
< <
< <
< <
< <
> olds number dependence

Fully Developed Flow
The fully developed steady flow in a pipe may be driven by gravity and /or pressure
forces. If the pipe is held horizontal, gravity has no effect except for variation in
hydrostatic pressure. The pressure difference between the two sections of the pipe,
essentially drives the flow while the viscous effects provides the restraining force that
exactly balances the pressure forces. This leads to the fluid moving with constant
velocity (no acceleration) through the pipe. If the viscous forces are absent, then
pressure will remain constant throughout except for hydrostatic variation.
NPTEL – Mechanical – Principle of Fluid Dynamics

Joint initiative of IITs and IISc – Funded by MHRD                                                            Page 3 of 72
In an internal flow through a long duct is shown in Fig. 5.1.2. There is an
entrance region where the inviscid upstream flow converges and enters the tube. The
viscous boundary layer grows downstream, retards the axial flow
( ) , u rx ??
??
at the wall
and accelerates the core flow in the center by maintaining the same flow rate.
constant Q u dA = =
?
(5.1.1)

Fig. 5.1.2: Velocity profile and pressure changes in a duct flow.

Page 4

NPTEL – Mechanical – Principle of Fluid Dynamics

Joint initiative of IITs and IISc – Funded by MHRD                                                            Page 1 of 72
Module 5 : Lecture 1
VISCOUS INCOMPRESSIBLE FLOW
(Fundamental Aspects)

Overview
Being highly non-linear due to the convective acceleration terms, the Navier-Stokes
equations are difficult to handle in a physical situation. Moreover, there are no general
analytical schemes for solving the nonlinear partial differential equations.  However,
there are few applications where the convective acceleration vanishes due to the
nature of the geometry of the flow system. So, the exact solutions are often possible.
Since, the Navier-Stokes equations are applicable to laminar and turbulent flows, the
complication again arise due to fluctuations in velocity components for turbulent
flow. So, these exact solutions are referred to laminar flows for which the velocity is
independent of time (steady flow) or dependent on time (unsteady flow) in a well-
defined manner. The solutions to these categories of the flow field can be applied to
the internal and external flows. The flows that are bounded by walls are called as
internal flows while the external flows are unconfined and free to expand. The
classical example of internal flow is the pipe/duct flow while the flow over a flat plate
is considered as external flow. Few classical cases of flow fields will be discussed in
this module pertaining to internal and external flows.
Laminar and Turbulent Flows
The fluid flow in a duct may have three characteristics denoted as laminar, turbulent
and transitional. The curves shown in Fig. 5.1.1, represents the x-component of the
velocity as a function of time at a point ‘A’ in the flow. For laminar flow, there is one
component of velocity
ˆ
V ui =
?
and random component of velocity normal to the axis
becomes predominant for turbulent flows i.e.
ˆ ˆˆ
V u i v j wk = + +
?
. When the flow is
laminar, there are occasional disturbances that damps out quickly. The flow Reynolds
number plays a vital role in deciding this characteristic. Initially, the flow may start
with laminar at moderate Reynolds number. With subsequent increase in Reynolds
number, the orderly flow pattern is lost and fluctuations become more predominant.
When the Reynolds number crosses some limiting value, the flow is characterized as
turbulent. The changeover phase is called as transition to turbulence. Further, if the
NPTEL – Mechanical – Principle of Fluid Dynamics

Joint initiative of IITs and IISc – Funded by MHRD                                                            Page 2 of 72
Reynolds number is decreased from turbulent region, then flow may come back to the
laminar state. This phenomenon is known as relaminarization.

Fig. 5.1.1: Time dependent fluid velocity at a point.
The primary parameter affecting the transition is the Reynolds number defined as,

Re
UL ?
µ
= where, U

is the average stream velocity and L is the characteristics
length/width. The flow regimes may be characterized for the following approximate
ranges;
23
34
46
6
0 Re 1: Highly viscous laminar motion
1 Re 100 : Laminar and Reynolds number dependence
10 Re 10 : Laminar boundary layer
10 Re 10 : Transition to turbulence
10 Re 10 : Turbulent boundary layer
Re 10 : Turbulent and Reyn
< <
< <
< <
< <
< <
> olds number dependence

Fully Developed Flow
The fully developed steady flow in a pipe may be driven by gravity and /or pressure
forces. If the pipe is held horizontal, gravity has no effect except for variation in
hydrostatic pressure. The pressure difference between the two sections of the pipe,
essentially drives the flow while the viscous effects provides the restraining force that
exactly balances the pressure forces. This leads to the fluid moving with constant
velocity (no acceleration) through the pipe. If the viscous forces are absent, then
pressure will remain constant throughout except for hydrostatic variation.
NPTEL – Mechanical – Principle of Fluid Dynamics

Joint initiative of IITs and IISc – Funded by MHRD                                                            Page 3 of 72
In an internal flow through a long duct is shown in Fig. 5.1.2. There is an
entrance region where the inviscid upstream flow converges and enters the tube. The
viscous boundary layer grows downstream, retards the axial flow
( ) , u rx ??
??
at the wall
and accelerates the core flow in the center by maintaining the same flow rate.
constant Q u dA = =
?
(5.1.1)

Fig. 5.1.2: Velocity profile and pressure changes in a duct flow.

NPTEL – Mechanical – Principle of Fluid Dynamics

Joint initiative of IITs and IISc – Funded by MHRD                                                            Page 4 of 72
At a finite distance from entrance, the boundary layers form top and bottom wall
merge as shown in Fig. 5.1.2 and the inviscid core disappears, thereby making the
flow entirely viscous. The axial velocity adjusts slightly till the entrance length is
reached ( )
e
xL = and the velocity profile no longer changes in x

and ( ) u ur ˜ only.
At this stage, the flow is said to be fully-developed for which the velocity profile and
wall shear remains constant. Irrespective of laminar or turbulent flow, the pressure
drops linearly with x . The typical velocity and temperature profile for laminar fully
developed flow in a pipe is shown in Fig. 5.1.2. The most accepted correlations for
entrance length in a flow through pipe of diameter ( ) d , are given below;
( )
( )
( )
1
6
,,, ;
so that Re
Laminar flow : 0.06 Re
Turbulent flow : 4.4 Re
e
e
e
e
L f dV V Q A
Vd
Lg
L
d
L
d
?µ
?
µ
= =
??
= =
??
??
˜
˜
(5.1.2)
Laminar and Turbulent Shear
In the absence of thermal interaction, one needs to solve continuity and momentum
equation to obtain pressure and velocity fields. If the density and viscosity of the
fluids is assumed to be constant, then the equations take the following form;

2
Continuity: 0
Momentum:
u vw
x y z
dV
pg V
dt
? ?µ
? ??
++ =
?? ?
= -? + + ?
?
?
(5.1.3)

Page 5

NPTEL – Mechanical – Principle of Fluid Dynamics

Joint initiative of IITs and IISc – Funded by MHRD                                                            Page 1 of 72
Module 5 : Lecture 1
VISCOUS INCOMPRESSIBLE FLOW
(Fundamental Aspects)

Overview
Being highly non-linear due to the convective acceleration terms, the Navier-Stokes
equations are difficult to handle in a physical situation. Moreover, there are no general
analytical schemes for solving the nonlinear partial differential equations.  However,
there are few applications where the convective acceleration vanishes due to the
nature of the geometry of the flow system. So, the exact solutions are often possible.
Since, the Navier-Stokes equations are applicable to laminar and turbulent flows, the
complication again arise due to fluctuations in velocity components for turbulent
flow. So, these exact solutions are referred to laminar flows for which the velocity is
independent of time (steady flow) or dependent on time (unsteady flow) in a well-
defined manner. The solutions to these categories of the flow field can be applied to
the internal and external flows. The flows that are bounded by walls are called as
internal flows while the external flows are unconfined and free to expand. The
classical example of internal flow is the pipe/duct flow while the flow over a flat plate
is considered as external flow. Few classical cases of flow fields will be discussed in
this module pertaining to internal and external flows.
Laminar and Turbulent Flows
The fluid flow in a duct may have three characteristics denoted as laminar, turbulent
and transitional. The curves shown in Fig. 5.1.1, represents the x-component of the
velocity as a function of time at a point ‘A’ in the flow. For laminar flow, there is one
component of velocity
ˆ
V ui =
?
and random component of velocity normal to the axis
becomes predominant for turbulent flows i.e.
ˆ ˆˆ
V u i v j wk = + +
?
. When the flow is
laminar, there are occasional disturbances that damps out quickly. The flow Reynolds
number plays a vital role in deciding this characteristic. Initially, the flow may start
with laminar at moderate Reynolds number. With subsequent increase in Reynolds
number, the orderly flow pattern is lost and fluctuations become more predominant.
When the Reynolds number crosses some limiting value, the flow is characterized as
turbulent. The changeover phase is called as transition to turbulence. Further, if the
NPTEL – Mechanical – Principle of Fluid Dynamics

Joint initiative of IITs and IISc – Funded by MHRD                                                            Page 2 of 72
Reynolds number is decreased from turbulent region, then flow may come back to the
laminar state. This phenomenon is known as relaminarization.

Fig. 5.1.1: Time dependent fluid velocity at a point.
The primary parameter affecting the transition is the Reynolds number defined as,

Re
UL ?
µ
= where, U

is the average stream velocity and L is the characteristics
length/width. The flow regimes may be characterized for the following approximate
ranges;
23
34
46
6
0 Re 1: Highly viscous laminar motion
1 Re 100 : Laminar and Reynolds number dependence
10 Re 10 : Laminar boundary layer
10 Re 10 : Transition to turbulence
10 Re 10 : Turbulent boundary layer
Re 10 : Turbulent and Reyn
< <
< <
< <
< <
< <
> olds number dependence

Fully Developed Flow
The fully developed steady flow in a pipe may be driven by gravity and /or pressure
forces. If the pipe is held horizontal, gravity has no effect except for variation in
hydrostatic pressure. The pressure difference between the two sections of the pipe,
essentially drives the flow while the viscous effects provides the restraining force that
exactly balances the pressure forces. This leads to the fluid moving with constant
velocity (no acceleration) through the pipe. If the viscous forces are absent, then
pressure will remain constant throughout except for hydrostatic variation.
NPTEL – Mechanical – Principle of Fluid Dynamics

Joint initiative of IITs and IISc – Funded by MHRD                                                            Page 3 of 72
In an internal flow through a long duct is shown in Fig. 5.1.2. There is an
entrance region where the inviscid upstream flow converges and enters the tube. The
viscous boundary layer grows downstream, retards the axial flow
( ) , u rx ??
??
at the wall
and accelerates the core flow in the center by maintaining the same flow rate.
constant Q u dA = =
?
(5.1.1)

Fig. 5.1.2: Velocity profile and pressure changes in a duct flow.

NPTEL – Mechanical – Principle of Fluid Dynamics

Joint initiative of IITs and IISc – Funded by MHRD                                                            Page 4 of 72
At a finite distance from entrance, the boundary layers form top and bottom wall
merge as shown in Fig. 5.1.2 and the inviscid core disappears, thereby making the
flow entirely viscous. The axial velocity adjusts slightly till the entrance length is
reached ( )
e
xL = and the velocity profile no longer changes in x

and ( ) u ur ˜ only.
At this stage, the flow is said to be fully-developed for which the velocity profile and
wall shear remains constant. Irrespective of laminar or turbulent flow, the pressure
drops linearly with x . The typical velocity and temperature profile for laminar fully
developed flow in a pipe is shown in Fig. 5.1.2. The most accepted correlations for
entrance length in a flow through pipe of diameter ( ) d , are given below;
( )
( )
( )
1
6
,,, ;
so that Re
Laminar flow : 0.06 Re
Turbulent flow : 4.4 Re
e
e
e
e
L f dV V Q A
Vd
Lg
L
d
L
d
?µ
?
µ
= =
??
= =
??
??
˜
˜
(5.1.2)
Laminar and Turbulent Shear
In the absence of thermal interaction, one needs to solve continuity and momentum
equation to obtain pressure and velocity fields. If the density and viscosity of the
fluids is assumed to be constant, then the equations take the following form;

2
Continuity: 0
Momentum:
u vw
x y z
dV
pg V
dt
? ?µ
? ??
++ =
?? ?
= -? + + ?
?
?
(5.1.3)

NPTEL – Mechanical – Principle of Fluid Dynamics

Joint initiative of IITs and IISc – Funded by MHRD                                                            Page 5 of 72
This equation is satisfied for laminar as well as turbulent flows and needs to be solved
subjected to no-slip condition at the wall with known inlet/exit conditions. In the case
of laminar flows, there are no random fluctuations and the shear stress terms are
associated with the velocity gradients terms such as, , and
u u u
xy z
µµ µ
?? ?
?? ?
in x-
direction.  For turbulent flows, velocity and pressure varies rapidly randomly as a
function of space and time as shown in Fig. 5.1.3.

Fig. 5.1.3: Mean and fluctuating turbulent velocity and pressure.

One way to approach such problems is to define the mean/time averaged turbulent
variables. The time mean of a turbulent velocity ( ) u is defined by,
0
1
T
u u dt
T
=
?
(5.1.4)
where, T

is the averaging period taken as sufficiently longer than the period of
fluctuations. If the fluctuation ( ) u uu ' = - is taken as the deviation from its average
value, then it leads to zero mean value. However, the mean squared of fluctuation
( )
2
u ' is not zero and thus is the measure of turbulent intensity.
( )
22
00 0
11 1
0; 0
TT T
u u dt u u dt u u dt
TT T
'' ' ' = = - = = ?
?? ?
(5.1.5)
In order to calculate the shear stresses in turbulent flow, it is necessary to
know the fluctuating components of velocity. So, the Reynolds time-averaging
concept is introduced where the velocity components and pressure are split into mean
and fluctuating components;
;; ; u u u v v v w w wp p p '' ' ' =+=+ =+ =+                               (5.1.6)
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