CDEEP IIT Bombay

CI= 223 I_ 2S/Slide

HIGH REYNOLDS NUMBER FLOWS

• The other limiting approximation is for laminar

flows where the viscous forces are relatively very

small compared to the inertia forces- the laminar boundary

layer

• However, the entire viscous terms cannot be dropped in the

N-S equations

• If viscous terms are ignored, the order of the differential

equation changes from two to one

HIGH REYNOLDS NUMBER FLOWS (Cont...) CDEEP

• The solution can only satisfy one boundary IIT Bombay

CE 223 L251/Slicte_a,_

condition, the normal velocity component is zero

• The solution is not complete as the no-slip condition cannot

be satisfied

• On the other hand, the limiting condition that the viscous

effects are very small should be applied after integration

• This is the essence of Prandtl's boundary layer theory

CDEEP IIT Bombay

CE 223 L 1,7/SlicleAk 3

CREEPING MOTION

Stokes Solution-

• The solution deals with creeping motion past a spherical

body of diameter D

• The body force is ignored and the variation of the

hydrostatic pressure is negligible because of the exceedingly

small size of the object

• By dropping the inertia terms, the x-component of the N-S

equations gets simplified

• The three equations for creeping motion can be expressed

as Vp = tiV2 q where, q is the velocity vector

CREEPING MOTION (Cont...) CDEEP

• The density of the fluid does not appear in the IIT Bombay

CE 223 L2E/SI de,41-?4 simplified governing equations

• The above equation was solved analytically making use of

the boundary conditions that the normal and the tangential

velocities are zero on the surface of the sphere

• The governing equations show that the pressure forces in

the flow are large enough to balance the viscous forces on

the right side

CDEEP IIT Bombay

CE 223 L255iSiideS1

CREEPING MOTION (Cont...)

• From Stokes solution, the viscous drag and the

pressure drag were evaluated as

(FD ) VIS = 2n pVD

And, ( FD)pressure = n pVD

• The total drag force on the sphere,

FD = 2n pVD + n pVD = 3n pVD

• The coefficient of drag is defined as

CD 1

pV 2 A

ED

CREEPING MOTION (Cont...)

• Therefore, for the creeping flow CDEEP

IIT Bombay

CE 223 LIE/Slidetg

= 37r p

/R i

7 D _ 24/2 _ 24

MP G92 ) P VD Re

• The above expression agrees very well with the

experimental values at low Reynolds number, R e < 1.

• When a spherical particle has a density greater than the

column of fluid in which it falls, it settles with a certain

velocity V

• The forces that act are the weight of the particle, the

buoyancy force and the drag

CD

CDEEP IIT Bombay

CE 223 L!Sltde 4a7

STOKES LAW (Cont...)

• The weight of the particle acts downward while the

buoyancy and the drag forces act upward

FA (b )

Fld) F(d)

F(g)

Forces on the sphere

STOKES LAW (Cont...) CDEEP

IIT Bombay

CL 223 L 26 /side 54-se the terminal velocity, Vt

• The terminal velocity is attained when there is a balance

between the upward and the downward forces

• One can write

TrD 3 rrD 3

6

Ys = 6 yr +

• Therefore, the terminal velocity of fall

V D2(Ps Pf )9

t— 18#

• The particle settles at a constant velocity known as

STOKES LAW (Cont...) CDEEP

IIT Bombay

CE 223 L25/Slide_.1.2,

settling basins, silting of reservoirs, finding the time

of settlement of dust particles from volcanic eruptions, etc.

• The settling velocity is useful in the design of

STOKES LAW (Cont...)

• For dust particles settling in air, the unit weight of CDEEP

IIT Bombay

CE 223 L25:/Slide .01510 air being negligible compared to that of the solid

particle, any change in the air density with height would not

affect the terminal velocity

• Thus for a given size of the particle the settling velocity

primarily depends on the viscosity

• If the variation in the viscosity with height is not significant,

one may assume the settling velocity to be uniform over the

entire height of fall

HELE-SHAW MODEL COEEP

IIT Bombay

CF. 223 L251./Slitie_141)

• This is a case where analogy between potential flow and

viscous fluid motion exists

• The justification for such analogy can be worked out

based on the creeping motion solution

• Experiment enables to observe the streamline pattern

around any shape of the body which represent the

potential flow field

HELE-SHAW MODEL (Contd.) CDEEP

IIT Bombay

• The contour of interest of the body is

CF 223 LIS/Slide _Pia

sandwiched between two parallel glass plates having a

very small gap thickness

• Dye is injected at the upstream end of the tray though

discrete points and the colored lines trace out he stream

lines

• Flow visualization is helpful for complicated shapes of

obstructions for which analytical solutions cannot be

obtained easily

• The steak lines show the mean flow pattern in the plane

of the parallel plates

HELE-SHAW MODEL (Contd.) CDEEP

IIT Bombay

CE 223 L25./Slide Ate/3

• Let x and y represent the plane midway between

the two plates and h be the height of the gap

• The velocity component w in the vertical direction does not

exist

• Since it is creeping motion, the internal terms may be

dropped from the left side of the N-S equation

a2 v a2 v =

a2 v)

(

r - ax2 ay 2 az 2 And, ap ay

HELE-SHAW MODEL (Contd.)

• The simplified governing equation of the motion

will be:

CDEEP IIT Bombay

CE 22:3 l 261Shdea4

ap = ax

a 2 i, a 2 i, (

821

—+ + ax2 ay 2 az 2

• As the gap thickness is exceedingly small, the gradients in

the z-direction are far bigger than the velocity gradients in

the x and y direction

HELE-SHAW MODEL (Contd.)

• The above equation can be simplified as CDEEP

lIT Bombay

CE 223 L2E/Slide jg"-/‘

op = ( a2z 2u

; Boundary conditions, z= ± h/2, u = 0 ax

Op — = 021 ay az 2

; Boundary conditions, z= ± h/2, v = 0

• The continuity equation to be satisfied is

au+ = ax dy

HELE-SHAW MODEL (Contd.) CDEEP

IIT Bombay

CE 223 L 2 61side I6

...c..........„---_-_,„

----- "-------t---T eir

He le-Shaw Apparatus

HELE-SHAW MODEL (Contd.) CDEEP

IIT Bombay

CI- 223 t 2i Slide 2e017 • The above equation can be solved independently

for the two velocity components u and v

• The governing equations and the boundary conditions are

identical to the case of plane Poiseuille flow

• The distribution of u and v with respect to z will be parabolic

in nature and the flow is rotational

HELE-SHAW MODEL (Contd.) CDEEP

IIT Bombay

,.. CE 223 LZWSlicies2d7se

• The relationship between the average velocity and

the pressure gradient can be expressed as

2 __hltiny = 12pax h— (—) Ox 12p

1 ap 2 1-1av

= — —12p

—ay n = i ph 2

ay 12//

HELE-SHAW MODEL (Contd.) CDEEP

IIT Bombay

CE 223 leciSlidea, 7C1 • The rotation about the z-axis in terms of the

average velocities can be written as

1 (au m , (him) ) 1 ( 82 (ph2 32 ph2 _) ( = 0 = —2 ay ax 2 k ayax izie axay C up)}

• Since u and v satisfy the continuity equation at every point,

their averages too will satisfy the continuity equations

a u a v a va v — = 0 ax ay

HELE-SHAW MODEL (Contd.) CDEEP

IIT Bombay

CE 223 LI-5/Sltdea 20

• Thus, the average velocities satisfy the basic

requirements of a potential flow

• The fact remains that the distribution of the velocities u and

v within the gap is parabolic and the flow is rotational

• In the experiment the average motion is seen as the streak

lines, which also represent the streamlines of a potential

flow around the object of interest