FLUID MECHANICS

COPYRIGHT BY ELETTRONICA VENETA SPA

FREE and FORCED

VORTEX

mod. HB14/EV

THEORY AND EXERCISES

HB14$$$101E0.DOC

CONTENTS

Page

1. SAFETY RULES 1

2. INTRODUCTION 2

3. THEORY 3 3.1 Free vortex 3 3.2 Spiral vortex 5 3.3 Forced vortex 7

4. TECHNICAL SPECIFICATIONS 8

5. INSTALLATION 9

6. OPERATING INSTRUCTIONS 10 6.1 Free vortex 10 6.2 Forced vortex 11

7. EXERCISES 12 7.1 Free vortex 12 7.2 Forced vortex 14

1

1. SAFETY RULES

Before installing and setting this equipment at work, read

and understand the contents of this handbook.

Any maintenance operation must be carried out by a

skilled technician.

Keep attention to the surface measuring probes and vortex

diameter gauge: they are very sharp

2

2. INTRODUCTION

The apparatus allows the study of the shape of "free and forced

vortices" and consists of a transparent cylindrical vessel (D1) having

two pairs of diametrically opposed inlet tubes of 3.0 mm and

12.5mm diameter.

The 12.5 diameter inlet tubes form an angle of 15 with the diameter,

so that a swirling motion is imparted to the liquid entering the vessel;

they are used as entry tubes for the free vortex experiment.

An outlet with interchangeable orifices (8, 12, 16, 24) is centrally

positioned in the base of the vessel.

The profile of the free vortex formed at the top of the vessel is

determined by a gauge mounted diametrically which measures the

diameter of the vortex at various depths. This gives the co-ordinate

points required to plot the vortex profile.

The velocity at 3 different radii may be measured using the pitot

tubes supplied.

Measure the total head at different radii with the Pitot tubes and

calculate the correspondent linear velocity:

The forced vortex is created in the vessel by using as the inlet the 3.0

mm bore tubes which are angled at 60 to the diameter. The inlet

water hits against a four blades paddle which acts as a stirrer.

The water leaves the vessel via the 12.5 mm diameter angled tubes

which are used as inlet tubes for the free vortex experiment.

The four blades paddle rotates on a vertical shaft supported by a plug

in the hole used as the outlet for the free vortex experiment.

A bridge piece with needles probes allows to determine the co-

ordinates of the vortex profile to be measured.

3

3. THEORY

rph Radius of pitot head arm mm

v Linear velocity mm/s

xd Pitot reading above datum

i.e. peripheral water height

mm

Angular velocity radians/s

3.1 Free vortex

When water flows out of a vessel through a central hole in the base,

a free vortex is formed; the sense of rotation being dependent on

initial disturbance.

The water moves spirally towards the centre with stream line motion,

so that, neglecting losses caused by viscosity, the energy per unit

mass remains constant.

If, while the mass of water is rotating, the central exit hole is

plugged, the flow of water in the vertical plane ceases and the

motion becomes one of simple rotation in the horizontal plane, and is

known as a Free Cylindrical Vortex.

Since stream line motion applies, Bernoullis theorem holds, and

cost.zg2

v

W

P 2

in any horizontal plane

cost.g2

v

W

P 2

4

Differentiating with respect to r.

0dr

dv

g

v

dr

dP

W

1 (Eq. 3.1.1)

Consider a pair of stream lines a distance dr apart, lying on the same

horizontal plane, and connected by a column of fluid or area da. The

centrifugal force on the column is balanced by the difference in

pressure between the two ends, i.e.

drdadr

dP

rg

vdrdaW

2

dr

dP

rg

vW

2

(Eq. 3.1.2)

Combining Eq. 3.1.1 and Eq. 3.1.2

0r

v

dr

dv

0dr

dv

g

v

rg

v2

and

v

dv

r

dr

Integrating,

ln r + ln r = const

that is

v r = const = k

r

kv

5

Then, in a free cylindrical vortex the velocity varies inversely as the

distance from the axis of rotation.

To determine the equation governing the surface profile, the

equation for the curve of equal pressure (atmosphere) is derived

from Bernoullis theorem

cconst.zg2

v2

r

kv

czrg2

k2

2

czrg2

k2

2

2

2

rg2

kzc

which is equation to a hyperbolic curve of nature yx2 = A which is

asymptotic to the axis of rotation and to the horizontal through z = C.

3.2 Free spiral vortex

The motion in a free spiral vortex differs from that in a free

cylindrical vortex in that in the former there is a radial flow towards

the centre. The equation governing radial flow towards the centre is

derived as follows.

Consider the flow of water across a segment of circle towards its

diameter; then the energy through any stream tube is constant so that

6

const.zg2

v

W

P 2

If A is area of channel at some point where the velocity is v,

11 vAconst.vA

where A1 and v1 are the area and velocity at some point distant r,

from the centre of the circular plane.

Putting A = k r

Then:

r

vrv 11

and if z is constant:

crg2

vr

W

P2

2

1

2

1

2

2

1

2

1

rg2

vrc

W

P

and

cg2

v

W

P2

11

2

2

1

2

1

2

2

1

2

1

2

11

r

r1

g2

v

rg2

vr

g2

v

W

P-P

A free spiral vortex may be considered as a case of cylindrical vortex

and radial motion combined. In each case the velocity is inversely

proportional to the radius. The angle between the stream lines and

the corresponding radius vector at any point will be constant, the

stream lines forming a series of spirals.

7

3.3 Forced vortex

Since angular velocity is constant:

v = r

Increase in radial pressure is given by:

2

1

2

2

2

12

r

r

2

P

P

22

rrg

WPP

drrg

WdP

rg

W

r

v

g

W

dr

dP

2

1

2

1

r = 0 when P = P

2

2

10 rg2

W

P-P

or since hW

P

2

2

10 r

g2

hh

2

2

10 r

g2

hh

which is the equation of a parabola.

8

4. TECHNICAL SPECIFICATIONS

The main technical characteristics of this unit are indicated here

below:

Supporting framework of AISI 304 stainless steel

Cylindrical vessel, code D1, made of transparent methacrylate

with 2 12.5 mm inlet tube at 15 to the diameter and with 2 3

mm inlet tube at 60 to the diameter

Set of orifice d = 8, 12, 16, 24 mm

Stirrer made of AISI 304 stainless steel

9

5. INSTALLATION

Place the apparatus on the hydraulics bench working surface

mod. HB/EV so that the central outlet in the base is located over

the channel

Level the apparatus using the adjustable feet

Using the pipe with quick connection provided, connect the inlet

of the equipment mod. HB14/EV with the bench mod. HB/EV

Position the outlet pipe of the equipment mod. HB14/EV over

the tank of the hydraulic bench

10

6. OPERATING INSTRUCTIONS

6.1 Free vortex

Select the orifice and place this into the central outlet located in

the base of the apparatus

Screw the quick connection onto the discharge available on the

bottom of the flow channel of the bench mod. HB/EV

Close the apparatus outlet valve V3

Close valve V1 and open partially valve V2 of mod. HB14/EV

so that water flows into the cylindrical vessel via the two inlet

ports set at 15 to the diameter

Close the pump outlet valve V1 of hydraulic bench and start the

bench pump G1

Slowly open valve V1 of hydraulic bench and adjust the valve

V2 of mod. HB14/EV until water just begins to flow out of the

cylindrical vessels overflow cut outs; maintain the water at this

level by regulating the water flow by means of the valve V2 of

mod. HB14/EV

When stable conditions are attained, the profile of the vortex is

obtained by measuring the vortex diameter at a number of planes,

the distance of the planes from the fixed datum being also

measured

Pitot tubes can be used to obtain measurements of the velocity of

the fluid at a number of difference radii: 15, 25, 30 mm. Replace

the profile measuring gauge with the 15mm radius arm pitot

tube. Immerse the tube until the "nose" is approximately 5mm

from the vortex core profile surface. Note these scale readings.

Repeat the test using the 25mm and 30mm pitot tube.

Switch off pump G1

11

6.2 Forced vortex

Position the blanking plug with shaft in the central hole located

in the base of the vessel

Open the apparatus outlet valve V3

Close valve V2 and open partially valve V1 of the apparatus so

that water flows into the cylindrical vessel via the two inlet ports

set at 60 to the diameter

Close the pump outlet valve V1 and start the bench pump G1

Slowly open valve V1 of hydraulic bench and adjust the valve

V1 and V3 of mod. HB14/EV until water just begins to flow out

of the cylindrical vessels overflow cut outs; maintain the water

at this level by regulating the water flow by means of the valve

V3 of mod. HB14/EV. Water will now flow through the ports at

60 and impinge on the paddle wheel before flowing out of the

apparatus via the two ports set at 15 and not in use

Ensure that the flexible outlet pipe is completely filled with

water for maintaining a syphonic action and hence increasing the

discharge capacity through the outlet valve

The speed of rotation of the paddle wheel is determined by the

rate of flow of water into the apparatus that is proportional to the

degree of opening of valve V1 of mod. HB14/EV. For each value

of flow rate the outlet valve of the apparatus should be adjusted

until water just flows out of the overflow cut outs

The profile of the water surface is determined by the surface

measuring probes (needles) which are adjusted until each probe

just breaks the water surface.

After speed of rotation of the paddle wheel has been measured by

timing a number of red paddle rotations, the measuring probe

bridge piece is removed from the apparatus and the length of

each probe is measured using the metallic ruler supplied

12

7. EXERCISES

7.1 Free vortex

The co-ordinate points for the vortex profile should be plotted

using the depth gauge

Measure the hydraulic head using the 3 Pitot tubes

Repeat the experiment changing the diameter of the orifice

Radius, r

(mm)

Measured Depth,

x (mm)

1/r2

(mm)

40.0 14 0,000625

35.0 19 0,000816

30.0 22 0,0011

25.0 31 0,0016

20.0 43 0,0025

15.0 75 0,0044

Table 1: Free Vortex with orifice diameter = 24 mm

Plotting x vs. 1/r2, we obtain a straight line with slope:

g2

km

2

13

17432g2

km

2

k = (2 98100 17432) = 18493

Measure the total head at different radii with the Pitot tubes and

calculate the correspondent linear velocity:

hg2vPitot

h140h98102hg2v

Compare this velocity with the velocity calculated with the formula

v = k/r.

Radius of Pitot

mm

Pitot head

mm

v = k/r

mm/s

h140vPitot

mm/s

15 76 1232 1220

25 42 740 907

30 33 616 804

14

7.2 Forced Vortex

Plot the co-ordinate points experimentally obtained for the

vortices at various speeds of rotation using the depth gauges

(needles)

Calculate the angular velocity using stopwatch and rotating

paddle

Plot and compare the theoretically obtained curves of vortex

surface profile with the experimentally determined forced

vortices

Number

revolution

Time

(sec.)

rps r (mm)

110 90 70 50 30 0

50 35 1.43 190 203 218 231 236 238 xmeas.

49.7 33.3 20.1 10.3 3.7 0 hcalc.

188.3 204.7 217.9 227.3 234.3 238 xcalc. = (238-hcalc)

2

2

10 r

g2

hh

22

0calc. rg2

hhh

Example of calculation

232

2

calc. r101.4r98102

1.432h

For r = 0, hcalc. = 0 and xcalc. = 238 mm

For r = 30, hcalc. = 3.7 and xcalc. = 234.3 mm

ELETTRONICA VENETA spa - 31045 Motta di Livenza (Treviso) ITALYVia Postumia. 16 Tel. +39 0422 7657 r.a. Fax +39 0422 861901www.elettronicaveneta.com

All rights reserved. No part of this publication may be reproduced, stored in any retrieval system, or transmitted in any form or by anymeans, electronic, mechanical, photocopying, recording, or otherwise without the prior writen permission of Elettronica Veneta S.p.a.