high reynolds number flows
TRANSCRIPT
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
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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
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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
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• 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
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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
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,.. 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