1

Basics of fluid flow

Types of flow

Fluid

Ideal/Real

Compressible/Incompressible

Flow

Steady/Unsteady

Uniform/Non-uniform

Laminar/Turbulent

Pressure/Gravity (free surface)

2

Basics of fluid flow (Chapter 4)

Basics of fluid flow, kinematics

Mechanics

Statics

Dynamics

Kinematics

Kinetics

Kinematics: deals with motion apart from considerations

of mass, force or energy

3

Basics of fluid flow

Path lines, streamlines and streak lines

Path line: the trajectory that a fluid particle would make as it moves

around with the flow

Streamline: line that shows the flow direction, local velocity vector is

tangent to the streamline at every point along the line at that instant

4

Partial derivative

Differentiating a function of more than one variable with respect to a particular

variable, with the other variables kept constant:

the notation ∂f/∂t means the partial derivative of the function f with respect to t

∂f/ ∂t : partial derivative

df/dt : total derivative

For more info:

http://apollo.lsc.vsc.edu/classes/met380/Fingerhuts_notes/driv.pdf

5

Basics of fluid flow

Types of flow

Steady flow: all fluid/flow properties at any point in the flow do not

change with time; however, conditions may be different at different

points.

Uniform flow: at every point in the flow, the velocity (in both magnitude

and direction) is identical at any given instant.

For steady flows:

6

Basics of fluid flow

One -, two -, and three- dimensional flows

This is the most general 3-D flow:

The flow is classified as 2-D if: The flow can be viewed as 1-D if:

7

Basics of fluid flow

Velocity and Acceleration

Convective (spatial)

acceleration

at

Local (temporal)

acceleration

an

8

Basics of fluid flow

Flow rate and Mean velocity

Flow rate: the rate at which fluid

crosses a known surface

volume flow rate mass flow rate

The volume flow rate passing through

the element of area dA (in yz plane) is

dQ = u(cosθ)dA=udA’

volume flow rate is equal to the magnitude of the mean

velocity multiplied by the flow area at right angles to the

direction of the mean velocity

9

Basics of fluid flow

Flow rate and Mean velocity

The volume flow rate passing through

the element of area dA is

dQ = u·dA =udA´

the local time mean velocity, u, will vary across the

section for real fluid

A

AVudAQ

QAVudAmA

10

Basics of fluid flow

Reynolds Transport Theorem & Continuity

QVAVA 2211

Copyright © The McGraw-Hill Companies, Inc.

FIGURE 5-24

Bernoulli’s Equation

(Energy per unit weight)

12

Derivation of the Bernoulli Equation

The forces acting on a fluid

particle along a streamline.

Steady, incompressible flow:

The sum of the kinetic, potential, and

flow energies of a fluid particle is

constant along a streamline during

steady flow when compressibility and

frictional effects are negligible.

Bernoulli

equation

The Bernoulli equation between any

two points on the same streamline:

Steady flow:

13

Energy in Steady Flow (Chapter 5)

Energies of a Flowing Fluid (Euler’s Equation)

Kinetic Energy

Potential Energy

1/2mV2 V2/2g

Wz z

Pressure Head

p = γh p/γ

Unit: L

(Energy per unit

weight)

14

Energy in Steady Flow (Chapter 5)

Bernoulli’s Equation

Unit: L

(Energy per unit weight)

Basic assumptions:

•Inviscid & incompressible fluid

•Steady flow

•Applies along a streamline

• No energy added or removed from the

fluid along the streamline

Piezometric pressure

Copyright © The McGraw-Hill Companies, Inc.

FIGURE 5-22

16

Energy in Steady Flow, Pipe Flow

V

p/γ p/γ

V2/2g

Bring moving water to a halt, and it'll

drive a column of water up to exactly

the height from which water would flow

to gain that velocity.

Pitot Tube

(Measures stagnation

pressure)

Free stream dynamic

pressure

Free stream static

pressure

17

Energy in Steady Flow, Free surface flow

V

V2/2g

Bring moving water to a halt, and it'll

drive a column of water up to exactly

the height from which water would flow

to gain that velocity.

Pitot Tube

(Measures stagnation

pressure)

18

Example: Bernoulli’s principle, Pitot Tube

http://www.youtube.com/watch?v=dk39ffdWq_E

19

Example:

Spraying Water

into the Air

Example: Water Discharge

from a Large Tank

20

The hydraulic

grade line (HGL)

and the energy

grade line (EGL)

for free discharge

from a reservoir

through a

horizontal pipe

with a diffuser.

Hydraulic grade line (HGL), P/g + z The line that represents the sum of

the static pressure and the elevation heads.

Energy grade line (EGL), P/g + V2/2g + z The line that represents the

total head of the fluid.

Dynamic head, V2/2g The difference between the heights of EGL and HGL.

21

Energy in Steady Flow

Stagnation pressure, ideal fluid (5.4)

1 2

V

V1 = V, p2 is the

stagnation

pressure

22

Energy in Steady Flow

General Energy Equation, steady flow, incompressible fluid

For an incompressible fluid with γ = const. and α =1:

23

Energy in Steady Flow

General Energy Equation, steady flow, incompressible fluid

For an incompressible fluid with γ = const. and α =1:

If there is no machine between points 1 and 2:

If head loss is neglected:

Real fluid

Ideal fluid

24

Energy in Steady Flow

Power considerations in fluid flow, Derivation of Power Equation

Power: P = (Force) x (Velocity) Power: P = Energy / Time

P = FV (F =ΔpA)

P = (ΔpA)V (Δp = γh)

P = (γhA)V (Q = AV)

P = γhQ

P = (Energy/Weight) x (Weight/Time)

P = ΔpQ

head (h) γQ

P = h γQ

25

Energy in Steady Flow

Power considerations in fluid flow, Units of Power

P = γhQ

Horsepower = P = γhQ/550

( [Q] = cfs, [h] = ft, [γ] = pcf )

Power in BG units

Kilowatts = P = γhQ/1000

( [Q] = m3/s, [h] = m, [γ] = N/m3 )

Power in SI units

P = γhQ P: power put into flow by a pump,

then h = hpump

P: power lost because of friction,

then h = hL

Pump efficiency, η = (power output) / (power input)

http://www.waterencyclopedia.com/Po-Re/Pumps-Traditional.html