Introduction to Hydrodynamic Instability
黃美嬌
台灣大學機械系
計算熱流研究室
Computational Thermo-fluid Research Lab
(CTRL)
presented at 東海應數 on Nov. 18, 2004
OUTLINE
Overview of hydrodynamic instability
Examples:
Rayleigh-Benard Instability
Taylor (Dean) Instability
Hydrodynamic Instability
• Which type, laminar or turbulent, is more likely to occur?
laminar when the Reynolds number is very low
turbulent at larger Reynolds number
• Reynolds number = UL/n = (L2/n)(L/U)-1
• The equations of hydrodynamics allow some flow patterns.
Given a flow pattern , is it stable?
If the flow is disturbed, will the disturbance gradually
die down, or will the disturbance grow such that the
flow departs from its initial state and never recovers?
Free-shear flows:
mixing layers, wakes, jets, etc
smaller critical Reynolds number
less sensitive to the form of the basic flow
inviscid instability leads to coherent structures
not affected by viscosity if it is small enough
Wall-bounded flows:
boundary layers, pipe flows, etc
basic flows without inflexion point
viscosity plays a role
sensitive to the form of the basic flow
Hydrodyanmics
Du u
1
=
 ( u  ) u = - P  g  n 2u
Dt t

DT T
=
 ( u  ) T =  2T
Dt
t
 u = 0
~ n/ = momentum/thermal diffusivities (m2/sec)
~ incompressible viscous Newtonian flows
~ mass,momentum,energy conservation
~ negligible viscous dissipation heat
Kelvin-Helmholtz (inviscid) instablility:
z
U 2 , 2
U1 , 1
x
linear stability analysis+normal mode disturbance:
k   (U1 - U 2 )  ( 12 - 22 ) g k x2  k y2
2
x 1 2
2
Kelvin-Helmholtz instablility
A long rectangular tube, initially horizontal, is filled with water above
colored brine. The fluids are allowed to diffuse for about an hour, and the
tube then quickly tilted six degrees, setting the fluids into motion. The brine
accelerates uniformly down the slope, while the water above similarly
accelerates up the slope. Sinusoidal instability of the interface occurs after a
few seconds, and has here grown nonlinearly into regular spiral rolls.
U 2 , 2
Kelvin-Helmholtz (inviscid) instablility:
k   (U1 - U 2 )  ( 12 - 22 ) g k x2  k y2
2
x 1 2
2
U1 , 1
~ instability due to heavy fluid on the upside
~ instability due to shear
~ instability due to an rapid downward vertical acceleration
and heavy fluid rests below
~ instability for all cases
Wall Shear Flows
~ inviscidly unconditionally stable (Rayleigh analysis)
~ viscously unstable (Orr-Sommerfeld analysis)
Re =
UD
 5772
n
~ unstable in labs as Re > 2000
Vortex Shedding
Hope Bifurcation
velocity signals
vortex shedding behind a vertical plate
Rayleigh-Benard Instability:
Low DT: motionless, pure thermal conduction
Higher DT: steady convection roll
T
Even Higher DT: unsteady, turbulent
fluid
TDT
 Driving force: buoyancy
 Damping force: viscous dissipation
Centrifugal Instability:
Low W: laminar, concentric streamlines
Higher W: steady convection roll
Even Higher W: unsteady, turbulent
 Driving force: centrifugal force
 Damping force: viscous dissipation
W
Görtler Instability
U
d
~ instability due to an imbalance between the centrifugal
force and the restoring normal pressure gradient
~ concave walls, e.g. lower side of airfoils; turbine blades
Görtler Vortex (streamwise vorticity)
U
U d d

Görtler number G =
n
R
R = radius of curvature
Blasius boundary layer
~ uniform flow over an semi-infinite flat plate
~ Tollmien-Schlichting waves
~ temporary/spatial growth
Curves of marginal stability
based on the parallel and nonparallel stability theory and
experimental data.
(temporary instability)
Surface tension instability
Examples:
• Rayleigh-Benard instability
• Taylor instability
§ Rayleigh-Benard Convection
TL
z
H
x
TH
Du u
1
=
 ( u  ) u = - P  g  n 2u
Dt t

DT T
=
 ( u  ) T =  2T
Dt
t
 u = 0
 (T )  -
steady stationary solution
uh = 0
Ph = h g
 2Th = 0
1 
 T
under Boussinesq approximation
h  0 = constant
Linear stability analysis:
T = Th  
u = uh  u  = u 
P = Ph  P
 = h  
u 
= -P  g  n 2u 
t

 ( u  ) Th =  2
t
 u = 0
 = h 
2
 2


2
2 2
2
 2 - Pr   2 -    w = Pr Ra  H w
 t
 t

2
2
 = 2 2
x y
characteristic length = H
2
2
2
 = 2 2 2
x y z
characteristic velocity = /H
2
H
2
Pr =
characteristic time = H2/
n
= Prandtl number

gH 4   h 
Ra =
  = Rayleigh number
n z  0 
Rayleigh number:
g
gH 4   h 
N2
Ra =
 =
n z  0  ( n H 2 )(  H 2 )
  h 
2
-2
=
N
(sec
)
 
z  0 
~ characteristic frequency of gravity wave
Ra = the relative importance of buoyancy effects
compared to momentum and thermal diffusive effects.
Normal mode approach: w( x, y, z, t ) = W ( z ) exp i ( k x x  k y y - t )
2
2
 d2


d
d
i 
2
2
2
2
k
k

i

k

W
=
Ra

k
W
 2
 2
 2

Pr 
 dz
 dz
 dz
k 2 = kx2  k y2
Given Pr and Ra, if there exists any mode (kx,ky) such that its
(eigenvalue)  has positive imaginary part, then the system is
linearly unstable.
• free and constant-temperature surfaces:
u  v
w =
=
=  H  = 0
z
z
• rigid surfaces:
u = v = w =  H  = 0
• free and constant-temperature surfaces:
W ( z ) = sin j z , 0  z  1 and j integer
2 2
2 3

  -1 (1  Pr ) ( j   k )  - Pr ( j   k ) - k 2 Ra  = 0


2
2 2
2
Im ( )  0 (unstable) if Ra  ( j   k
2
(
Racr = Ra j = 1, k = 
274
2 =
 657.5
4
)
2
)
2 3
k2
unstable if Ra  ( j   k
2
2
j=3
Ra
j=2
k
)
2 3
j=1
k2
(
Racr = Ra j = 1, k = 
274
2 =
 657.5
4
)
W ( z ) = sin j z , 0  z  1 and j integer
j = 1, k x = 
H
2 2H
2 , ky = 0
• rigid surfaces:
~ no analytical solution yet
~ numerical solutions available and show
Racr  1708
As Ra increases, more and more unstable modes are
inspired and the flow transition to turbulence via
successive bifurcations.
Rayleigh-Benard Instability:
Low DT: motionless, pure thermal conduction
Higher DT: steady convection roll
T
Even Higher DT: unsteady, turbulent
fluid
TDT
 Driving force: buoyancy
 Damping force: viscous dissipation
Lorenz: modes with j = 1
k
 ( x, z , t ) = X sin ( kx ) sin ( z )
2
2
2 (   k )
Ra
 = Y cos ( kx ) sin ( z ) - Z sin ( 2z )
2 Racr DT
X and Y : rising warm fluid and descending cold fluid
Z : distortion of the vertical temperature profile
from linearity
X (t ) = Y (t ) = Z (t ) = 0  stationary pure conduction state
nonzero constant X (t ), Y (t ), Z (t )  steady convection roll
Lorenz equations:
dX
= Pr(Y - X )
dt
dY
Ra
= - XZ 
X -Y
dt
Racr
dZ
= XY - Z
dt
4 2
= 2 2
 k
Fixed points: (0,0,0) = pure (stationary) conduction state
~ the only fixed point if Ra<Racr
~ always exists
~ stable if Ra<Racr
~ unstable if Ra>Racr
Fixed points:
(
)
 ( Ra Racr - 1) ,   ( Ra Racr - 1) , Ra Racr - 1
~ exists only if Ra>Racr
~ steady convection roll
~ stable if
Ra Pr ( Pr    3)

Racr
( Pr -  - 1)
~ pitchfork bifurcation at Ra = Racr
Ra
Racr
Pr = 10, = 8 3 (corresponding to k = 
Ra Ra cr = 1.25 
Pr ( Pr    3)
( Pr -  - 1)
= 24.7
2)
Ra Ra cr = 4 
Pr ( Pr    3)
( Pr -  - 1)
= 24.7
Ra Ra cr = 25 
Pr ( Pr    3)
( Pr -  - 1)
= 24.7
Ra Ra cr = 25 
Pr ( Pr    3)
( Pr -  - 1)
= 24.7
Taylor Instability: inviscid fluid
ur ur 1 u uz
 

=0
r
r r 
z
Dur u2
1 P
- =Dt
r
 r
Du ur u
1 1 P

=Dt
r
 r 
Duz
1 P
=Dt
 z
D 
 u 

=  ur

 uz
Dt t
r r 
z
W
base solution
ur = u z = 0
u (r ) = r W(r )
P(r ) = Ph (r ) =  r W 2 dr
axis-symmetric disturbance
u
u
u u
u
uu
1 1 P
 ur      u z   r  = t
r
r 
z
r
 r 
ru
ru u ru
ru Dru DH
 ur

 uz
=
=
=0
t
r
r 
z
Dt
Dt
~ Angular momentum per unit mass of a fluid element
about the axis (z-axis) remains constant.
motion in the r-z plane：
r-direction：pressure force + centrifugal force
z-direction：pressure force
z
r
d 1 2
2
Centrifugal force = u / r = - ( u )
dr 2
Dru
=0
Dt
~ potential-energy-like
Consider two fluid particles originally located at r1 and r2
respectively and later interchange their locations at later time.
H1 = r1u1 ( r1 , t ) = r2u1 ( r2 , t  Dt )
H 2 = r2u2 ( r2 , t ) = r1u2 ( r1 , t  Dt )
1
1
1

1

E (t ) =   u21   u22  dV =   H12 / r12   H 22 / r22  dV
2
2
2

2

1
1

2
2
E (t  Dt ) =   H1 / r2   H 22 / r12  dV
2
2

The change in the kinetic energy is
1
1 1
E (t  Dt ) - E (t ) = ( H - H )  2 - 2    dV
r2  2
 r1
2
2
2
1
0
if r2  r1 and H 2  H1
0
dH dru
if
=
0
dr
dr
azimuthal kinetic energy is released
instability possible in the r-z motion
Linear stability analysis:
u = (ur' , rW  u' , uz' )
P = Ph  P
Normal mode approach + axis-symmetric disturbance:
(ur' , u' , u z' , P ' /  )  (u, v, w, p)exp ( st  ikz )
 d  d 1
k2
2
    - k  u (r ) = 2   u (r )
s
 dr  dr r 

1 d 2 2
 = 3 (r W)
r dr
 d  d 1
k2
2
    - k  u (r ) = 2   u (r )
s
 dr  dr r 

~ classical Sturm-Liouville eigenvalue problem
Rayleigh quotient:
s 2  0 if (r )  0 everywhere
(stable to axis-symmetric disturbance)
s 2  0 for some k if (r ) changes signs.
(unstable to axis-symmetric disturbance)
Couette Flow
W1
W2
2 2
W2 R22 - W1 R12 R1 R2 ( W1 - W2 ) 1
W(r ) =
+
2
2
R2 - R1
R22 - R12
r2
W2 R22 - W1 R12
 =4
W(r )
2
2
R2 - R1
• Cylinders rotate in the same direction.
W2 R12
>0 (stable) if
> 2
W1 R2
• Cylinders rotate in different directions.
  0 (unstable) close to the inner cylinder
W2 W1 = 0.5
2
u
0.25
W 2 W1 = 0.1
r
W 2 W1  0
2
u
r
Viscous damping
Taylor number
T=
4W R
2 4
1 1
2
(1 -  ) (1 -   2 )
(1 -  )
n
2 2
 = W 2 W1
 = R1 R2
Given  and  , there exist a Tcr .
unstable if T  Tcr
narrow gap approximation: Tcr 
3416
for 0    1
1 
kcr  3.12 R2
W1 n
W2 n
Hydrodynamic instability
~ free shear
~ wall effect
~ buoyancy-induced
~ stratification effect
~ centrifugal-force induced
~ surface tension
~ others
Figure 1: A stable fluid
chain.
Figure 2: The instability
generated by increasing
flow rate, as seen with
the naked eye.
Figure 3: The same
instability visualized
with a strobe lamp.
Increasing the flow rate serves to broaden the fishbones.
In the wake of the fluid fish, a regular array of drops obtains,
the number and spacing of which is determined by the pinchoff of the fishbones.
polygonal fluid sheet
Thank you for
your attention!!
We examine the form of the free surface flows resulting from the collision of equal jets at an
oblique angle. Glycerol-water solutions with viscosities of 15-50 cS were pumped at flow rates
of 10-40 cc/s through circular outlets with diameter 2 mm. Characteristic flow speeds are 1-3 m/s.
Figures 3-9 were obtained through strobe illumination at frequencies in the range 2.5-10 kHz.
Figure 1: At low flow rates, the resulting stream takes the form of a steady fluid chain, a
succession of mutually orthogonal fluid links, each comprised of a thin oval sheet bound by
relatively thick fluid rims. The influence of viscosity serves to decrease the size of successive
links, and the chain ultimately coalesces into a cylindrical stream. As the flow rate is increased,
waves are excited on the sheet, and the fluid rims become unstable. The rim appears blurred to
the naked eye (Figure 2); however, strobe illumination reveals a remarkably regular and striking
flow instability ( Figures 3-6). Droplets form from the sheet rims but remain attached to the fluid
sheet by tendrils of fluid that thin and eventually break. The resulting flow takes the form of fluid
fishbones, with the fluid sheet being the fish head and the tendrils its bones. Increasing the flow
rate serves to broaden the fishbones. Figures 7-9: In the wake of the fluid fish, a regular array of
drops obtains, the number and spacing of which is determined by the pinch-off of the fishbones.
Some of these photos have appeared in Hasha & Bush (2002, Phys. Fluids, Gallery of Fluid
Motion).
A combined theoretical and investigation of fluid chains and fishbones is under review.