058:0160
Professor Fred Stern
Chapter 6- part2
1
Fall 2009
Chapter 6: Viscous Flow in Ducts
6.2 Stability and Transition
Stability: can a physical state withstand a disturbance and
still return to its original state.
In fluid mechanics, there are two problems of particular
interest: change in flow conditions resulting in (1)
transition from one to another laminar flow; and (2)
transition from laminar to turbulent flow.
(1) Transition from one to another laminar flow
(a) Thermal instability: Bernard Problem
A layer of fluid heated from below is top heavy,
but only undergoes convective “cellular” motion
for
bouyancy force
viscous force
1 
α = coefficient of thermal expansion =   
  T 
  T / d   dT
   1  T 
dz
Raleigh #:
Ra 
gd
gd 4

 Racr
w / d 2
k
0
d = depth of layer
k, ν =thermal, viscous diffusivities
P
058:0160
Professor Fred Stern
Fall 2009
Chapter 6- part2
2
w=velocity scale: convection (w) = diffusion (k/d)
from energy equation, i.e., w=k/d
Solution for two rigid plates:
Racr = 1708
for progressive wave disturbance
^ i ( x ct )
ct
αcr/d = 3.12
w  we
 e i [cos( x  ct )  sin( x  ct )]
^ i ( x  ct )
T T e
λcr = 2π/α = 2d
α = αr
c=cr + ici
For temporal stability
αr = 2π/λ=wavenumber
cr = wave speed
ci : > 0
=0
<0
unstable
neutral
stable
Ra > 5 x 104 transition
to turbulent flow
Thumb curve: stable for low Ra < 1708 and very long or
short λ.
058:0160
Professor Fred Stern
Chapter 6- part2
3
Fall 2009
(b) finger/oscillatory instability: hot/salty over
cold/fresh water and vise versa.
Rs  gd ds dz  /k
4
(Rs – Ra)cr = 657
S
   (1  T  S )
0
058:0160
Professor Fred Stern
Fall 2009
Chapter 6- part2
4
(c) Centrifugal instability: Taylor Problem
Bernard Instability: buoyant force > viscous force
Taylor Instability: Couette flow between two
rotating cylinders where centrifugal force (outward
058:0160
Professor Fred Stern
Chapter 6- part2
5
Fall 2009
from center opposed to centripetal force) > viscous
force.
Ta 
r c  
i
2
2
i
o


2
c  r  r  r
0
i
i
 centrifugal force/ viscous force
Tacr
Tatrans
= 1708
= 160,000
αcrc = 3.12  λcr = 2c
Square counter rotating vortex
pairs with helix streamlines
058:0160
Professor Fred Stern
Fall 2009
Chapter 6- part2
6
058:0160
Professor Fred Stern
Chapter 6- part2
7
Fall 2009
(d) Gortler Vortices
Longitudinal vortices in concave curved wall
boundary layer induced by centrifugal force and
related to swirling flow in curved pipe or channel
induced by radial pressure gradient and discussed
later with regard to minor losses.
For δ/R > .02~.1
and
Reδ = Uδ/υ > 5
058:0160
Professor Fred Stern
Chapter 6- part2
8
Fall 2009
(e) Kelvin-Helmholtz instability
Instability at interface between two horizontal
parallel streams of different density and velocity with
heavier fluid on bottom, or more generally
ρ=constant and U = continuous (i.e. shear layer
instability e.g. as per flow separation). Former case,
viscous force overcomes stabilizing density
stratification.


g 22  12  12 U1  U 2 
U U
1
2
2
 ci  0 (unstable)
large α i.e. short λ always unstable
Vortex Sheet
1
2
1   2  ci  (U1  U 2 )
>0
Therefore always unstable
058:0160
Professor Fred Stern
Fall 2009
Chapter 6- part2
9
058:0160
Professor Fred Stern
Chapter 6- part2
10
Fall 2009
(2) Transition from laminar to turbulent flow
Not all laminar flows have different equilibrium states,
but all laminar flows for sufficiently large Re become
unstable and undergo transition to turbulence.
Transition: change over space and time and Re range of
laminar flow into a turbulent flow.
Re 
cr
U

δ = transverse viscous thickness
~ 1000
Retrans > Recr
with
xtrans ~ 10-20 xcr
Small-disturbance (linear) stability theory can predict Recr
with some success for parallel viscous flow such as plane
Couette flow, plane or pipe Poiseuille flow, boundary
layers without or with pressure gradient, and free shear
flows (jets, wakes, and mixing layers).
Note: No theory for transition, but recent DNS helpful.
058:0160
Professor Fred Stern
Chapter 6- part2
11
Fall 2009
Outline linearized stability theory for parallel viscous
flows: select basic solution of interest; add disturbance;
derive disturbance equation; linearize and simplify; solve
for eigenvalues; interpret stability conditions and draw
thumb curves.
^
u  u u
^
v vv
^
p  p p
u, v  mean flow, which is solution steady NS
^ ^
u,v  small 2D oscillating in time disturbance is
solution unsteady NS
^
^ ^
^ ^
^
1^
u t  u u x  u u x  v u y  v u y   p x   2 u

^
^ ^
^ ^
^
1^
v t  u v x  u v x  v v y  v v y   p x   2 u

^ ^
ux  v x  0
^
Linear PDE for
^
^
u, v, p
for ( u , v ,
p ) known.
Assume disturbance is sinusoidal waves propagating in x
direction at speed c: Tollmien-Schlicting waves.
^
i ( xct )
 ( x, y , t )   ( y ) e
^
^ 
i ( xct )
u
  e
y
^
^

i ( xct )
v
 i e
x
Stream function
y =distance across
shear layer
058:0160
Professor Fred Stern
^
Fall 2009
^
u v 0
x
Chapter 6- part2
12
Identically!
y
    i = wave number = 2 
c  c  ic = wave speed = 

r
r
i
i
Where λ = wave length and ω =wave frequency
Temporal stability:
Disturbance (α = αr only and cr real)
ci
>0
=0
<0
unstable
neutral
stable
Spatial stability:
Disturbance (cα = real only)
αi
<0
=0
>0
unstable
neutral
stable
058:0160
Professor Fred Stern
^
Chapter 6- part2
13
Fall 2009
^
Inserting u , v into small disturbance equations and
^
eliminating p results in Orr-Sommerfeld equation:
inviscid Raleigh equation
(u  c)( ''  2 )  u ''  
u  u /U
i
( IV  2 2 ''  4 )
 Re
Re=UL/υ
y=y/L
4th order linear homogeneous equation with
homogenous boundary conditions (not discussed here) i.e.
eigen-value problem, which can be solved albeit not
easily for specified geometry and ( u , v , p ) solution to
steady NS.
058:0160
Professor Fred Stern
Fall 2009
Chapter 6- part2
14
Although difficult, methods are now available for the
solution of the O-S equation. Typical results as follows
(1) Flat Plate BL:
U
Re 
 520


crit
(2) αδ* = 0.35
 λmin = 18 δ* = 6 δ (smallest unstable λ)
 unstable T-S waves are quite large
(3) ci = constant represent constant rates of damping
(ci < 0) or amplification (ci > 0). ci max = .0196 is
small compared with inviscid rates indicating a
gradual evolution of transition.
058:0160
Professor Fred Stern
Fall 2009
Chapter 6- part2
15
(4) (cr/U0)max = 0.4  unstable wave travel at average
velocity.
(5) Reδ*crit = 520  Rex crit ~ 91,000
Exp: Rex crit ~ 2.8 x 106 (Reδ*crit = 2,400) if care taken,
i.e., low free stream turbulence
058:0160
Professor Fred Stern
Chapter 6- part2
16
Fall 2009
Falkner-Skan Profiles:
(1) strong influence of 
Recrit   > 0

fpg
Recrit   < 0

apg
Reδ*crit :
67 sep bl
520 fp bl
12,490 stag point bl
058:0160
Professor Fred Stern
Fall 2009
Chapter 6- part2
17
058:0160
Professor Fred Stern
Fall 2009
Chapter 6- part2
18
058:0160
Professor Fred Stern
Fall 2009
Chapter 6- part2
19
Extent and details of these processes depends on Re and
many other factors (geometry, pg, free-stream, turbulence,
roughness, etc).
058:0160
Professor Fred Stern
Fall 2009
Rapid development of spanwise flow, and initiation of
nonlinear processes
- stretched vortices disintigrate
- cascading breakdown into
families of smaller and smaller
vortices
- onset of turbulence
Note: apg may undergo
much more abrupt
transition. However, in
general, pg effects less
on transition than on
stability
Chapter 6- part2
20
058:0160
Professor Fred Stern
Fall 2009
Chapter 6- part2
21
058:0160
Professor Fred Stern
Fall 2009
Chapter 6- part2
22
Some recent work concerns recovery distance: