CIRCULATION (CONT...)

CDEEP IIT Bombay

di' , v+ —y ay

CE 223 L f l /Slide

1) C

av , , + —ax

ax + —ay ay

an + —

Oy dy

11

dv

Ott U — dx + — dy ax ay

On ,

U —dx

ax

v + —0v

dx

V

Circulation around an elemental area

A B

CDEEP IIT Bombay IRROTATIONAL MOTION

CE 223 Lth/Slide.2_

Ferris Wheel

Fluid particles not rotating

--- --

.4P• •-•

h-rotational outer flow region

Velocity profile •

Rotational boundary layer region

7A-C■

"wit iStM-14(4k4.

Fluid particles rotating Wan

ROTATIONAL AND IRROTATIONAL FLOW

NEAR A WALL

CDEEP IIT Bombay

CE223 L I I 'Slid° •

Fluid elements in a rotational region of the flow rotate, but

those in an irrotational region of flow do not (souRea: car-tGaL cimsALA,2006)

VORTEX MOTION CDEEP

IIT Bombay

CE 223 L_UlSlide.

(a )

(b)

Streamlines and velocity profiles for flow A akin to solid body rotation, and flow B is irrotational everywhere except at the origin

z

4- Sz 1k

&Sy w

pzi8 yt5z pm+ —5x)5y5z

manly

CONTINUITY EQUATION

WEEP IIT Bombay

CE 223 L I I iSlide_5-

CONTINUITY EQUATION (CONTD)

CDEEP IIT Bombay

• The rate at which mass flows into the control CF 223 I IL/Slidell

volume, minus the rate at which the flow leaves the control volume must equal the net rate of change of mass within the control volume

a(pu) a(pv) „ a(pw) a. Sy

a(p Sr gy (5z) &Sy bz' by hi az z & =

at ay aZ at

CONTINUITY EQUATION (CONTD)

Divide right through by the volume (bx by bz), to arrive at the continuity equation in Cartesian

Op i_3(pu) a(pv) a(pw) =0 coordinates ox ay

CDEEP IIT Bombay

CE 223 1_ I I /Slide?

For incompressible flows, p is a constant and above can be

simplified as Cu ay au,

0 —+—+ — = ax "ay az

In case of steady compressible flows, the continuity equation

in the differential form is

a(pm) + a( pv) ± a(pw) =0

av tly az

STREAM FUNCTION

CDEEP IIT Bombay

• Lagrange introduced the stream function qi and the CI 223 I II /Slide S

corresponding velocity potential function

• Two dependent variables u and v are expressed in terms of one dependent variable

• At a given instant of time the stream function qi = tp(x,y)

STREAM FUNCTION (CONTD) CDEEP

IIT Bombay

CE 223

ovi ow _=u and = v • Property of a Lp function: cx

= const gives the equation of a streamline

LP exists for rotational and irrotational flows

V 2 v =0 For irrotational flow

The discharge between two streamlines w2 and 4J1

dip

VELOCITY POTENTIAL CDEEP

IIT Bombay

CI 223 L I I /Slide 1 0

• A velocity potential is introduced for irrotational flows

• The gradient of in any direction gives the velocity in that

direction: 30 30 ax ay

• 0 = constant represents an equipotential line

VELOCITY POTENTIAL (CONTD) CDEEP

IIT Bombay

• For an equipotential line dcp = 0

CE 223 L I I /Slide 1 1

a 0 a 0 dv u 610 = — dx + — dv=- u dx - v dy > =-

ax ay -

dx

• Similarly for a to = constant line dyi = 0 and dy = v

dx u

• Product of slopes of cl) and qi lines is (-1)

• cl) and iv lines are orthogonal

FLOWNET CDEEP

IIT Bombay

CE 223 L I I !Slide 12,

• A set of 1:1 and qi lines with spacing maintained equal

between cip, lines and qi lines is known as a flow net

• Flow net represents the solution of Laplace equation

\-7/*

= o

FLOWNET (CONTD)

• Methods of Solution :

r Graphical - free hand drawing

r Plotting equations for qi and cp lines

r Numerical solution of Laplace equation

r Conformal mapping

r Electrical and viscous flow analogy

CDEEP IIT Bombay

GE 223 IlL/Slide_13

• Velocity field is determined from continuity equation

and condition of irrotationality

• The equation of motion is not used

FLOWNET (CONTD) CDEEP

IIT Bombay

CE 223 L t i ;Slide 1 4

• Knowing the velocity field the pressure field

can be found based on Bernoulli's equation

• Superposition: Functions that individually satisfy

Laplace equation may be added together to describe a complex potential flow

• Inverse Method for getting ip and cD functions: Equations

are not directly solved

\ tra4 cs-tConcd-

CDEEP IIT Bombay

7 flif , 0

V zet?6 :a o

\71421 = 0 vrQeyia „ V"Er?,4-12)

a Le '

Sot eypos i 9/ c3 rt.

CE 223 Lit / Slide IS