forced vortex
TRANSCRIPT
Vortex Motion• Consider flow along a curved path
– Mass in dA: ρdAdr– Radial acceleration: v2/r– Centrifugal force:– Pressure= P at r ; P+dP at r+dr
• So, the force balance (along the curved path) is:– (dP) (dA) = (ρdAdr) (v2/r)
Or,
• So the pressure gradient is:
dr dA
P
P+dPV (tangential velocity)
streamlines
drrvdP
2
ρ=
rv
drdP 2
ρ=
[(P+dP)-P]dA= resultant force = mass x accel
Forced and Free Vortex•Pressure gradient:
is useful for investigating conditions in direction normal to streamlines.• e.g.: straight streamlines: r ∞ and dP 0• curved streamlines: at r r+dr, then P P+dP • Exactly how they change depends on how v varies with r• Consider motion of a rotating cylinder:
– Angular velocity is ω– Linear velocity v is given by : v = ω r– Linear velocity decreases towards the center– i.e. no relative motion between wall of cylinder and fluid– The rotation is a solid body rotation and is called Forced Vortex
rv
drdP 2
ρ=
ω
Forced and Free Vortex• Flow in forced vortex is rotational
• In a forced vortex, we apply a torque:
Free Vortex (Potential Vortex):• No torque is applied.
• So, vr = constant = K
( )
( )mvrdtdFvT
mvdtdF
==
= Linear momentum
Angular momentumTorque
( ) ( ) 0or 0 == vrdtdmvr
dtd
rv constant=
A
A
AA
B
B
B
B
r
Forced and Free Vortex• Free Vortex
– The flow is irrotational,–– In cylindrical coordinates:
• x = r cos θ, y = r sin θ, z = z
( ) 0=mvrdtd
A
A
B
B
A
B
A
B
0=×∇ v
( ) ( ) 0ˆˆˆ1
0
=
∂∂
−∂∂
+
∂∂
−∂∂
+
∂∂
−∂∂
=
=∂∂
∂∂
∂∂
=×∇
zvrvrz
vzvrrv
zv
r
vrvvzr
zrv
rzrz
zr
θθ
θ
θ
θ
θθ
θ
( )
( ) 0
0
0
=∂∂
−∂∂
=∂∂
−∂∂
=
∂∂
+∂∂
−∂∂
=∂∂
−∂∂
θ
θθ
θ
θθθ
r
zr
zz
vrvr
zv
zv
zvr
zrvvrv
zv
dzdr
rdθ
∂r
∂r
Forced and Free Vortex• Circular symmetry
– ONLY vθ ≠ 0, and vr = 0 and vz = 0 (no flow).
– So, i.e. r vθ = constant, =>– Note: vθ ∞ when r 0.– This cannot be true for real fluid because of fluid viscosity.– Fluid viscosity slows down the motion.– Viscous friction causes fluid outside the core to rotate like solid
body and flow to be rotational.
( ) 0
0
=∂∂
−∂∂
=∂∂
−∂∂
θθr
zr
vrvr
zv
zv
0 0
0 0
0
( ) 0=∂∂
θrvr r
v constant=θ
( ) =∂∂
−∂∂θ θ
z rvz
v0
Forced and Free VortexRankine Vortex• ωr r < a (forced vortex rotational flow)
constant/r = ωa2/r r > a (free vortex irrotational flow)
• vr = 0 and vz = 0
vθ =
vθ =ωa when r =a
vθ
r r
v=rω v=ωa2/r
a aForcedVortex(rotational)
FreeVortex(irrotational)
W
Vorticity
Rotational flow
z
rzrr
zrvW ˆor
00
1 ωω
ωθ
θ=
∂∂
∂∂
∂∂
=×∇=
Zero vorticity (irrot)0=×∇= vW
Forced and Free Vortex• Pressure Distribution
( )
( )
( ) ( ) ( ) ( )22
212
2
222
21
211
22
2
21
2
12
2211
22
21
2
12
2
12
22
13
212
2
13
22
1
22
1
2
222
Kor rK :Notes
112
1211
vvrrv
rrv
rK
rKPP
rvrvv
rrKPP
rKdr
rKPPdr
rKdr
rvdP
rv
drdP
−=
−=
−=−⇒
===
−=−
−==−⇒==
=
∫∫∫∫
ρρρ
ρ
ρρρρ
ρ
222
211 22
vPvP ρρ+=+
Consider r ∞, v=K/r 0, and P2 P∞
121
211 2
and 2
PvPPvP =−=+ ∞∞ρρ
21
2
1 2
rKPP ρ
−= ∞
As r1 small, P1 small
∴Low pressure near center when r smalle.g. whirlpool and tornado
=>
Examples of Vortex Motion
von Karman Vortices near Guadalupe Island, 260 km west of Baja California
Multiple-exposure photograph of the tip vortex on a rectangular
wing of aspect ratio 1
Wake Vortex Study at Wallops Island
http://eosweb.larc.nasa.gov/HPDOCS/misr/misr_html/von_karman_vortex.html http://www.asc.nasa.gov/media_room/photo1.htmlhttp://www.nd.edu/~mav/research.htm
Examples of Vortex Motion
Re=9.6
Re=13.1
Re=26
Re=2000
Re=10,000
Flow past a cylinder Vortex formation in microfluidic channel
M.Van Dyke, An Album of Fluid Motion, Parabolic Press (Standford, 1982) Lim DSW, Shelby JP, Kuo JS, Chiu DT , Applied Physics Letters 83 (6): 1145-1147 Aug 11 2003
Examples of Vortex Motion
http://www.pma.caltech.edu/Courses/ph136/yr2002/chap17/0217.1.pdf
Rayleigh-Bernard convection. A fluid is confined between two horizontal surfaces separated by a vertical distance d. When the temperature difference between the two plates ∆T is increased sufficiently, the fluid will start to convect heat vertically.
Silicone oil on a uniformly heated copper plate.M.Van Dyke, An Album of Fluid Motion, Parabolic Press (Standford, 1982)
http://www.ldeo.columbia.edu/users/jcm/Topic3/Topic3.html
InstabilityScope
1. Couette Flow: The flow of fluid in an annulus between two concentric spinning cylinders.
2. Linear stability analysis
1. Couette Flow• For steady circular flow• Consider simplified Navier-Stokes in cylindrical coordinates:
• Solution:
r2
r1 r1≤r≤ r2
02
22 =−+ θ
θθ udr
durdr
udr
rBAru +=θ
Instability• If r1 rotates at Ω1 and r2 rotates at Ω2, then:
• Consider the fluid element going around the ring:Velocity: uθ(r)=rωAngular momentum: ρr uθ(r)= ρr2 ω
2. Linear stability analysisConsider conservation of angular momentum:
( )2
12
2
22
2121
21
22
211
222 ,
rrrrB
rrrrA
−Ω−Ω
=−Ω−Ω
=
r2
r1
( ) ( ) 0or 0 22 == ωωρ rDtDr
DtD
Instability• A small change in angular momentum:
• Now, consider stability of a small perturbation:– i.e. if we displace a fluid from r r+δr such that the angular
momentum is conserved.– Can pressure in surrounding fluid able to maintain equilibrium?– Compare pressure in perturbed and equilibrium state.
• Consider centrifugal pressure:
drr
d
rddrdrdrrdrrdrrd
ωω
ωωωω
ωωω
2
2 2or 0)2()(
2
22
−=⇒
−=⇒−=
=+=
2222
222
21
2
or
rrP
drrdPrr
udrdP
ρωρω
ωρωρρ θ
==⇒
=== ∫∫
Instability
• Stability criteron: Pdist – Pundist < 0
( ) rrrdrdωP
rP
undist
undist
δδωδρ
ωρ
+
++=
=
at (r)rr21
21
22
22
( ) [ ]
( )2
2
22
2)(21
)(21
−+=
++=
rr
rrrP
drrrP
dist
dist
δωωδρ
ωωδρ
The next ring at r + δr
Same ring is being displace over, so we need to look at dω based on the conservation of angular momentum
( ) ( )
( ) ( ) ( )
( ) 0222
24
0424)()(21
(r)rr212)(
21
2
2
2
22
22
22222
22
22
<−=
+−=
−−≈
>
−+−−−+=
++−
−+=−
rdrrd
rr
drd
r
rdrdr
r
rdrdωdr
rr
drdr
rrrrr
rdrdωr
rrrrPP undistdist
δωωδωωω
δωωδω
δωδωωδωωωδρ
δωδρδωωδρ
Instability( )
( ) 0
0
2
2
<
>
drrddrrd
ω
ω For stability• This implies that:
• Recall:
• Since,
For instability
rBArru +== ωθ
BArrur +==⇒ 22θω
( ) 022 >=+⇒ ArBArdrd
022 21
22
211
222
21
22
211
222
>
−Ω−Ω
=
−Ω−Ω
=
rrr
rrAr
rrrrA
211
222 rr Ω>Ω For stability
Rayleigh stability critereon
Taylor-Couette Cell• Taylor-Couttte flow is the flow of an incompressible, viscous fluid contained in the gap between two concentric, rotating cylinders. • When angular velocity of the inner cylinder exceeds a critical values, flow patterns developed that consists ofaxisymmetric vortices stacked on top of one another in the axial direction, with radial inflows and outflows.•Some flow patterns observed are:
http://www.students.ncl.ac.uk/a.j.youd/tcf/tcf.html
Kelvin-Helmholtz Instability
• Kelvin-Helmholtz instability occurs when two fluids of different densities flowing at various velocities.
Formation of clouds generated by Kelvin-Helmholtz instability.http://www.colorado-research.com/~werne/eos/text/turbulence.html
Two stably stratified fluids are flowing from left to right with the uppermost low density fluid traveling 3.5 times faster than the lower heavy fluid
http://www.itsc.com/movvkv.htm
Waves that grow on jets of high or low density fluid such as the hot bouyant jet of gas.
von Karman Vortex
A heated plate with a thick initial thermal boundary layer is suddenly accelerated to 5 m/s in air
• Vortices that shed alternatively from two sides of a body (e.g. flat plate or cylinder) in highly structured and unsteady pattern.
Smoke at various levels in vortex sheetM.Van Dyke, An Album of Fluid Motion, Parabolic Press (Standford, 1982) Periodic vortex street created by a flat plate
http://www.itsc.com/movvkv.htm
Viscous Fingering• Frontal instabilities resulted
when a low viscosity fluid displaces a high viscosity fluid in a porous medium
Nitrogen injection into mineral oil in a Hele-Shaw-type cell
Nitrogen injection into mineral oil in a Hele-Shaw-type cell but with half of the cell etched with rectangular lattice
Viscous finger of associating polymer: from normal fingering to fractals.
http://www.phyast.pitt.edu/groups/cond_mat/research/pattern_formation/pattform.html