Engineering Fluid Mechanics

Chapter 4
Fluids in Motion
Dr. Ahmad Sakhrieh
American University of Ras Al-Khaimah
Department of Mechanical Engineering
Introduction
This chapter focuses on the Kinematics part of fluid in
motion, i.e. Velocity, Acceleration and Flow Visualization
and next chapter will discuss the Dynamics of fluids, i.e.
Force and Momentum.
Objectives:
Understand velocity, acceleration, and flow visualization
Distinguish between convective and local acceleration.
Introduce new flow terminologies such as streamlines,
steady and uniform flow
Distinguish between Laminar and Turbulent flow
Pressure varaition in flowing fluid
Eulers Equation
Bernoullis Eqaution
Pressure distribution about bodies
Separation
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 Velocity; Lagrangian and Eulerian Viewpoints
Lagrangian viewpoint
Lagrangian viewpoint in which the motion of a specific fluid particle is
recorded for all time ( Watch individual, single particle all the time, at all
locations x,y, and z). This is the familiar approach in dynamics.

Using the Cartesian coordinate system,

the position vector is expressed as
r(t)=x i +y j+z k
where i, j, and k are the unit vectors in the
x, y, and z directions
the particle velocity is obtained
by differentiating the particles position
vector with respect to time.

v(t)=u i +v j+w k
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Eulerian viewpoint
The other way to express fluid velocity is to focus on a certain point in
space and describe the motion of fluid particles that pass this point as
time goes on. This is known as the Eulerian approach. In this case, the
fluid particle velocity depends on the point in space and time.
u = u(x, y, z, t) ; v = v(x, y, z, t) ; w = w(x, y, z, t)
In this method we observe the motion of particles passing a specific point
in space, as opposed to the Lagrangian method in which we track the
position of a specific particle as time passes. In order to describe the entire
flow field, we must know the fluid motion at all points in the field.

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Streamlines and Flow Patterns
To visualize the flow field it is desirable to
construct lines that show the flow direction.
Such a construction is called a flow pattern,
and the lines are called streamlines.
The streamline is defined as a line drawn
through the flow field in such a manner that
the local velocity vector is tangent to the
streamline at every point along the line at
that instant.

Thus, the tangent of the streamline at

a given time gives the direction of the
velocity vector.

A streamline, however, does not

indicate the magnitude of the velocity.

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Dividing Streamline:
Whenever flow occurs around a body, part of it will go to one side and
part to the other as shown in the figure for flow over an airfoil section.
The streamline that follows the flow division (that divides on the
upstream side and joins again on the downstream side) is called the
dividing streamline. At the location where the dividing streamline
intersects the body, the velocity will be zero with respect to the body.
This is the stagnation point.

The fluid velocity can be expressed as:

V=V(S,t)
where s is the distance traveled by a fluid
particle along a path, and t is the time

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Uniform flow:
Flow in which the velocity is the same in both magnitude and direction at a given
instant at every point in a fluid (Streamlines are straight and parallel)

Non-Uniform flow:
The velocity changes along a fluid path (Streamlines are not straight and parallel)

V
0
S

(a) Converging flow. (b) Vortex flow. 7

Steady Flow:
Velocity does not change ( magnitude and direction) with time. Flow in a
pipe is an example of steady flow.
V
0
Un-Steady Flow: t
Velocity does change ( magnitude and direction) with time. for example
tank discharge liquid. V
0
Ideal Flow versus Real Flow t
Ideal fluid flow: hypothetical fluid. It assumes no friction (viscosity = 0,
called inviscid flow). In real fluid flow implies friction effect (viscosity exist,
called viscous fluid flow).

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Laminar and Turbulent Flow
Laminar flow is a well-ordered state of flow in which adjacent fluid layers move
smoothly with respect to each other. A typical laminar flow would be the flow of
honey or thick syrup from a jug. Laminar flow in a pipe has a smooth, parabolic
velocity distribution as shown in the figure (a)
Turbulent flow is an unsteady flow characterized by intense cross-stream
mixing. For example, the flow in the wake of a ship is turbulent. The eddies
observed in the wake cause intense mixing. The transport of smoke from a smoke
stack on a windy day also exemplifies a turbulent flow. The mixing is apparent as
the plume widens and disperses. An instantaneous velocity profile for turbulent
flow in a pipe is shown in Figure (b). See Movie

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 Laminar flow:
Motion of the fluid particles is very orderly
all particles moving in straight lines
parallel to the pipe walls.

Turbulent flow:
Motion is, locally, completely random but the
overall direction of flow is one way.

Laminar flow: Re < 2000

Transitional flow: 2000 < Re < 10000
Turbulent flow: Re > 10000

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One-Dimensional and Multi-Dimensional Flows
The dimensionality of a flow field is characterized by the number of spatial
dimensions needed to describe the velocity field.

Figure (a) shows the velocity distribution for an axisymmetric flow in a circular
duct. The flow is uniform, or fully developed, so the velocity does not change inthe
flow direction (z). The velocity depends on only one dimension, namely the radius
r, so the flow is one-dimensional.

Fig. (b) shows the velocity

distribution for uniform flow in a
square duct. In this case the
velocity depends on 2D, namely
x and y, so the flow is 2D.

Figure (c) shows the velocity

distribution for the flow in a square
duct but the duct cross-sectional
area is expanding in the flow
direction so the velocity will be
dependent on z as well as x and y.
This flow is 3D.
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Acceleration
Eulerian formulation
the velocity at a point in the flow field is a function of both space and time
Thus V = u(x,y,z,t) i + v(x,y,z,t)j + w(x,y,z,t)k
The acceleration of a fluid particle in the x direction is given by
ax = du(x,y,z,t)/dt
where u is the x component of velocity.
Coordinates Approach:
By using the chain rule for differentiation of a multivariable function, we can
express this as:

But,

Local Acceleration

Similarly,
Convective Acceleration

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Streamline Approach:
Using normal and tangential components, the velocity of a fluid particle on
a streamline may be written as:

V V S ,t e t
where V(s, t) is the speed of the
particle, which can vary with
distance along the pathline, s, and time, t.
The direction of the velocity

vector is
given by a unit vector e t


dV d et dV (I)
a V e t
dt dt dt


In Eq. (I), the derivative of the unit vector d et / dt is nonzero because

the direction of the unit vector changes with time as the particle moves
alongthe pathline. The derivative is:
d et V
en (II)
dt r 13
To evaluate the derivative of speed in Eq. (I), the chain rule for a function
of two variables can be used.
dV S ,t V S V

dt S t t
dV S ,t V V
V (III)
dt S t
Substituting (II) and (III) into (I) yield,

V V V 2
a V e t e n
S t
Centripetal Acceleration
r
Convective Acceleration
Local Acceleration
The acceleration component along the streamline depends on two terms:
The first one, the variation of velocity with time at a point on the pathline, namely
V / t, is called the local acceleration. In steady flow the local acceleration is zero.
The other term, VV / s, depends on the variation of velocity along the pathline
and is called the convective acceleration.
In a uniform flow, the convective acceleration is zero.
The acceleration with magnitude V2 / r, which is normal to the pathline and
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directed toward the center of rotation, is the centripetal acceleration.
 Pressure Variation in flowing Fluid: Eulers Equation
Force Balance
F ma
Fpressure Fgravity ma
Fpressure pA p p A
pA
z
Fgravity W

z
pA A Aa Important Notes:
1- It shows that the acceleration is equal to the
Divide by A change in piezometric pressure with distance. The
minus sign means the acceleration is in the direction
p z
a of decreasing piezometric pressure.
2- In a static body of fluid, Euler's equation reduces to
the hydrostatic differential equation
dp dz 3- In the direction normal to the wall, the n
a direction, the acceleration is zero.
d d
Applying Euler's equation in the n
d
p z a direction gives n(p + z) /= 0, so
the piezometric pressure is constant in
d the normal direction. 15
Euler Equation
 Pressure Distribution in Rotating Flows
Apply Eulers Equation in r-direction
d
p z ar
dr
V 2
ar
r
V2
d
p z
dr r
V r
d
p z r 2 Important Notes:
dr 1- If the angular speed is zero, the
Integrating with respect to r equations reduces to the hydrostatic
equation
r 2 2
p z cons tan t 2- The equation for pressure variation in a
2 rotating flow is used to predict the surface
profile of a liquid in a rotating tank.
2r 2
p z C 3- The maximum pressure in the tank
occurs at the outer radius and the deepest
2 point
These equations describe the
pressure variation in rotating
flow. 16
Example:
A U-tube is rotated about one leg, as shown. Before being rotated the in the tube fills
0.25 m of each leg. The length of the base of the U-tube is 0.5 m, and each leg is 0.5 .
What would be the maximum rotation rate in (rad/s) to ensure that no liquid is expelled
from the other leg.

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 The Bernoulli Equation Along a Streamline
Start from Eulers Eqn. along a streamline
Important Notes:
d
p z at 1- Steady 2- Irrotational 3- Non-viscous fluid (= 0)
dS 4- Incompressible fluid 5- Apply along the streamline
6- If V=0, the equations reduces to hydrostatic Eq.
V V
at V 7-Even though the real fluid is viscous, the effects of
S t viscosity are small for short distances. Also, the effects
Assume steady state of viscosity on pressure change are negligible compared
to the pressure change due to velocity variation.
dV
at V
dS
V
dV 1 d 2
V
dS 2 ds
d V 2
d
p z
ds ds 2
d V2
p z 0
ds 2
p1 V12 p2 V22 p3 V32
z1 z2 z3
V2
p z
2g 2g 2g
C
2 Bernoullis Equation 18
Kerosene at 20 C flows through a
contraction section as shown. A pressure
gauge connected between the upstream
pipe and throat section shows a pressure
difference of 20 kPa. The gasoline velocity in
the throat is 8 m/s. What is the velocity in the
upstream (m/s) in the upstream pipe?

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 Application of the Bernoulli Equation to Velocity
Measurement Devices
Stagnation tube:
A Stagnation tube (sometimes called a total head tube) is an open-ended
tube directed upstream in a flow and connected to a pressure sensor.
Because the velocity is zero at the tube opening, the pressure measured
corresponds to stagnation conditions.
V12 V22
p1 z1 p 2 z 2
2 2
Note V2=0
V12 p 2 p1
2

By the equations of hydrostatics
(there is no acceleration normal to
the streamlines where the
streamlines are straight and parallel),
p1 = d and p2 = (l + d).
Therefore, Eq. (4.20) can be written as
V1 2 gl
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The Pressure Coefficient
Describing the pressure distribution is important because pressure gradients
influence flow patterns and pressure distributions acting on bodies which create
resultant forces. A common dimensionless group for describing the
pressure distribution is called the pressure coefficient:
p z p zo h ho
Cp 2
Vo / 2 Vo /( 2 g )
2

Pressure Distribution around a Circular CylinderIdeal Fluid

where p is local pressure; p0 and V0 are the free-stream pressure and velocity.
Note that the minimum pressure (Cp = - 3.0) occurs at the midsection, point C,
where the velocity is highest.
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Concept of favorable and adverse pressure gradient:
From Euler's equation for pressure gradient
and acceleration along a streamline,
(neglecting gravitational effects) one has: dp
at
ds
One notes that at > 0 if p/s < 0; that is,
the fluid particle accelerates if the pressure decreases
with distance along a pathline. This is a favorable pressure gradient.

On the other hand, at < 0 if p/s > 0, so the fluid particle

decelerates if the pressure increases along a pathline.
This is an adverse pressure gradient.

First, (from A-B) it decelerates from the free-stream velocity

to zero velocity at the forward stagnation point as it travels in
an adverse pressure gradient.
Then, as it passes from B to C, it is in a favorable pressure gradient, and it
accelerated to its highest speed.
From C to D the pressure increases again toward the rearward stagnation point,
and the particle decelerates.
Finally, the pressure decreases from D to E, and this favorable pressure
gradient accelerates the particle back to the free-stream velocity.
Understanding this qualitative description of how the fluid particle travels from
one point to another will be helpful when the phenomenon of separation. 22
Separation
Flow separation occurs when the fluid
pathlines adjacent to body deviate from
the contour of the body and produce
a wake.
This flow condition is very common.
It tends to increase drag, reduce lift, and
produce unsteady forces that can lead to
structural failure.

Consider the flow of a real (viscous) fluid past a cylinder as shown in the figure.
The flow pattern upstream of the midsection is very similar to the pattern for an
ideal fluid.

However, in a viscous fluid the velocity at the surface is zero (no-slip), whereas
with the flow of an inviscid fluid the surface velocity need not be zero.

Because of viscous effects, a thin layer, called a boundary layer, forms next to
the surface. The velocity changes from zero at the surface to the free-stream
velocity across the boundary layer. Over the forward section of the cylinder,
where the pressure gradient is favorable, the boundary layer is quite thin.
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Separation
In real flow, downstream of the midsection, the
pressure gradient is adverse and the fluid particles
in the boundary layer, slowed by viscous effects,
can only go so far and then are forced to detour
away from the surface.

This is called the separation point. A recirculatory flow called

a wake develops behind the cylinder.
The flow in the wake region is called separated
flow.

The pressure distribution on the cylinder surface in

the wake region is nearly constant, as shown in
the figure.
The reduced pressure in the wake leads to increased
drag.

A photograph of an airfoil section with flow

separation near the leading edge is shown
in the figure. Separation on an airfoil surface leads
to stall and loss of lift. 24
Separation around blunt bodies
Separation and the development of a wake region also occurs on blunt objects
and cross sections with sharp edges, as shown in the figure.
In these situations, the flow cannot negotiate the turn at the sharp edges and
separates from the body, generating eddies, a separated region, and wake flow.
The vortices shed from the body can produce lateral oscillatory forces than can
induce vibrations and ultimately lead to structural failure, as evidenced by the
collapse of the Tacoma Narrows Bridge in 1940.
The prediction and control of separation is a continuing challenge for engineers
involved with the design of fluid systems.

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