Fluid Dynamics
ESci 146 | AY 2018 – 2019
Flow Concepts
Flow can be classified in many ways:
1. Real or Ideal
2. Turbulent or Laminar
3. Steady or Unsteady
4. Uniform or Non-uniform
5. Rotational or Irrotational
6. Reversible or Irreversible
Flow Concepts
Additional terms:
Conduit – channel though which a fluid is
conveyed
Streamline – a continuous line drawn through
the fluid so that it has the direction of the
velocity at every point
Streamtube – the tube made by all the
streamlines passing through a small, closed
curve.
Basic Equations
All flow situations, regardless of their nature
are subjected to the following relations:
1. Newton’s Laws of Motion
2. Law of Conservation of Mass
3. First & Second Law of Thermodynamics
4. Boundary Conditions
5. Newton’s Law of Viscosity
Basic Equations
1. Continuity Equation
2. Euler’s Equation
3. Bernoulli’s Equation
4. Steady-State Energy Equation
5. Momentum Equations
a) Linear-Momentum Equation
b) Moment-of-Momentum Equation
Continuity Equation
Fluid Dynamics
Basic Equations
Continuity Equation:
𝑨𝟏 𝒗𝟏 = 𝑨𝟐 𝒗𝟐
𝑄 = discharge (m3/s or cfs)
𝐴1 & 𝐴2 = cross-sectional area (m2 of ft2)
𝑉1 & 𝑉2 = velocity (m/s or ft/s)
Basic Equations
Continuity Equation:
𝑨𝟏 𝒗𝟏 = 𝑨𝟐 𝒗𝟐
This continuity equation is for
incompressible & steady flow only.
Example No. 1
Continuity Equation
Water at 20°C flows in a pipe system at Section 1 (D1 = 2 m)
with a velocity of 3.0 m/s. It leaves the system at Section 2
(D2 = 3 m). Compute for the (a) the velocity of flow, (b) the
volume flux in m3/s, (b) the mass flux in kg/s, and (c) the
weight flux in N/s at Section 2.
Seatwork
Whole sheet paper | 20 April 2019
Air at 30°C and 100 kPa flows through a 15 × 30 cm
rectangular duct at 15 N/s. Compute (a) the mass
flux in kg/s, (b) the volume flux in m3/s, and
(c) the average velocity in m/s.
Example No. 2
Continuity Equation
Water flows steadily through a box at three sections as
shown on the board. Section 1 has a diameter of 75 mm
and the flow in is 28 lps. Section 2 has a diameter 50 mm
and the flow out is 9 m/s average velocity. Compute the
average velocity and volume flux at section 3 if D3 = 25
mm. Is the flow at 3 in or out?
Seatwork
Whole sheet of paper | 30 April 2019
Two fluids of different specific gravity enter at
sections 1 and 2 as shown. If the flow is steady and
mixing is complete before exit, calculate the average
velocity, mass flux and specific gravity of the mixture
leaving at section 3.
Quiz
07 May 2019
You will only need a 1/2 sheet of paper.
Make your solutions neat & systematic.
Item No. 1
Q6 15 mins
Three pipes steadily deliver water
at 20°C to a large exit pipe in
Figure 2. The velocity 𝑉2 = 5 m/s,
and the exit flow rate 𝑄4 = 120
m3/h. If it is known that increasing
𝑄3 by 20 percent would increase 𝑄4
by 10 percent, find (a) 𝑉1 , (b) 𝑉3 ,
and (c) 𝑉4 . Assume incompressible
flow.
Bernoulli’s Equation
Fluid Dynamics
Bernoulli’s
Energy Theorem
Neglecting friction, the total head, of
the total amount of energy per unit
weight, is the same at every point in
the path of flow.
Daniel Bernoulli, 1738
𝑬𝒏𝒆𝒓𝒈𝒚
𝑯𝒆𝒂𝒅 =
𝑾𝒆𝒊𝒈𝒉𝒕
Elevation Pressure Velocity
Head Head* Head
Potential energy Kinetic energy
per unit weight per unit weight
𝒅𝒑
+ 𝒈 𝒅𝒛 + 𝒗 𝒅𝒗 = 𝟎
𝝆
Euler’s Equation
𝟐 𝟐
𝒑𝟏 𝒗𝟏 𝒑𝟐 𝒗𝟐
𝒛𝟏 + + = 𝒛𝟐 + +
𝜸 𝟐𝒈 𝜸 𝟐𝒈
Bernoulli’s Equation
Bernoulli’s Equation
Assumptions when applying Bernoulli’s
Equation:
1. Constant of integration varies from one point to
another but remains constant along a streamline.
2. Steady flow
3. Frictionless flow
4. Incompressible flow
Example No. 1
Bernoulli’s Equation
Water is flowing in an open channel at a depth of 2 m and a
velocity of 3 m/s. It then flows down a chute into another
channel where the depth is 1 m and the velocity is 10 m/s.
Assuming frictionless flow, determine the difference in
elevation of the channel floors.
Bernoulli’s Equation
Modifications of Assumptions Underlying
Bernoulli’s Equation:
1. When all streamline originates from a reservoir, where
the energy content is everywhere the same, the
constant of integration does not change from one point
to another.
2. In the flow of a gas, as in a ventilation system, where
the change in pressure is only a small fraction of the
absolute pressure, the gas may be considered
incompressible.
Example No. 2
Bernoulli’s Equation
Air at standard atmosphere flows through a 100 mm
constriction from a 200 mm pipeline. When flowrate is 11.1
N/s, the pressure in the constriction is 100 mm Hg. Calculate
the pressure in the 200 mm section neglecting air’s
compressibility.
Bernoulli’s Equation
Modifications of Assumptions Underlying
Bernoulli’s Equation:
3. For unsteady flow with gradually changing
conditions (e.g. emptying a reservoir), Bernoulli’s
equation may be applied without applicable error.
Example No. 3
Bernoulli’s Equation
A nozzle attached to the wall of a reservoir has a
diameter of 100 mm. Determine the velocity of efflux
from the nozzle if fluid surface in the reservoir is 4 m
above the center line of the nozzle. Find the
discharge through the nozzle.
Bernoulli’s Equation
Modifications of Assumptions Underlying
Bernoulli’s Equation:
3. For unsteady flow with gradually changing
conditions (e.g. emptying a reservoir), Bernoulli’s
equation may be applied without applicable error.
4. Bernoulli’s equation is of use is analyzing real-fluid
cases by first neglecting viscous shear to obtain
theoretical results. The equation may then be modified
so that it conforms to the actual physical case.
Seatwork
Bernoulli’s Equation
A venturi meter, consisting of a converging portion followed
by a throat portion of constant diameter and then a gradually
diverging portion, is used to determine the rate of flow in a
pipe (drawn on the board). The diameter at section 1 is 15 cm,
and at section 2 it is 10 cm. Find the discharge through the
pipe when 𝑝1 − 𝑝2 = 20 𝑘𝑃𝑎 and oil,𝑆 = 0.90, is flowing.
Bernoulli’s Equation
Bernoulli’s Equation in
real fluid flow:
1. Flow in Orifices, Tubes & Weirs
2. Pipe Flow
3. Open Channel Flow
Quiz
10 May 2019
You will only need a ½ sheet of paper.
Make your solutions neat & systematic.
Item No. 1
Q7 15 mins
The jet from a fountain rises to a height, ℎ.
The fountain discharges water at 2 m3/s
through a nozzle of 220 mm. (a) Find ℎ. (b)
What is the diameter of the stream at a
height of ℎ/2?
Item No. 2
Q7 15 mins
Water at 20°C, in the
pressurized tank as shown,
flows out and creates a
vertical jet. Assuming steady
frictionless flow, determine
the height 𝐻 to which the jet
rises.
