Multimedia files -4/13
Contents:
1. Boundary layers
2. Normal mode spectrum
3. Continuous spectrum
4. Effect of the boundary layer growth
5. F.X. Wortmann and M. Strunz experiment
6. Further reading
1

The solutions depend on both x and y, but by a coordinate transformation the dependence on x can be eliminated leading to the so-called self-similar forms of the solutions.
wedge flows
Falkner-Skan (Hartree) profiles
Blasius profile
U(x,y)
Falkner-Skan
(Stewartson) profiles
2

In contrast to the flow along the flat plate at zero angle of attack, pressure gradieent dp(x)/dx in this case is a constant, rather than zero. Then the governing equation is: nonlinear f ' ' '
 ff ' '
 
H

1
 f '
2


0 ,
Hartree parameter equation with boundary conditions f ( 0 )
 f ' ( 0 )

0 , f ' (

)

1 ,
While the free stream velocity depends on x as:
U fs
( x )

U
0
 x
 x
0

2


H

H
, so that
U ( x ,

)

U
0 f ' (

)
 x
 x
0
 
2

H

H
Because of the exponent, the constant m is used sometimes instead of m

2


H

H

H
:
• only one solution exists at 
H
≥ 0 (accelerated boundary layer)
• two solutions exist at -0.199< 
H
< 0 (pre- or post-separation profiles)
• zero, one, two or infinite number of solutions exist at 0.199<

H
(near-wall jets)
Hartree solution
  m

1
2
U
 x y

2
2
 
U
 x y two solutions at

H
=-0.13
Stewartson solution
3

temporal w r
/a
=1 spatial downstream propagating badly resolved modes w/a r
=0 w/a r
=1 w/a r
=0 upstream propagating
4

Least-stable discrete mode discrete modes well resolved
Continuous spectrum standing waves
(only for instability in space propagates slowly propagate with free stream velocity
5

d
The boundary layer approximation is not valid at flat plate leading edge
6

U(x,y)
   a r
=const
a i
=const
Re d
*
= w
=2 p f
U
0 d
* /
 d
* /U
0 changes downstream!
Dimensional frequency, however, does not change downstream.
neutral curve
F =const
F= w
/Re=2 p f

/U
0
2 is independent of dimensionless and does not change downstream
7

TS-wave generated by means of oscillating strip
Injection of tracers
8

Photograph (see below) of wave-like streak lines in a water channel obtained with the aid of the flow tinting method by
F.X. Wortmann; disturbance was created artificially by an oscillating strip (3 × 800 × 0.03 mm ).
Two-dimensional waves are observed at right part of the pattern; the rolling up of streak lines downstream is a consequence of the instability of the perturbation waves.
9

Linear evolution of the Tollmien-Schlichting
(TS) wave in a laminar boundary layer
10

2D TS-wave visualization in water tunnel by flow tinting method
Click to play
11

Distortion of the instantaneous velocity profiles caused by the 2D TS-wave downstream evolution at x = 0 mm
Click to play
12

Distortion of the instantaneous velocity profiles caused by the 2D TS-wave downstream evolution at x = 35 mm
Click to play
13

Distortion of the mean velocity profile caused by the TS-wave downstream evolution
This figure is taken from Schlichting, H. and Gersten, K. (2000). Boundary layer theory
(Sth ed.). Berlin Heidelberg New York: Springer.
Patterns of streamlines and velocity distribution for a neutral disturbance in the boundary layer on a flat plate at zero incidence.
U(y)
– mean flow; U(y)+u’(x,y,t) – disturbed velocity distribution; U
Reynolds number; l
= 40 d
1
- disturbance wave length; c r
= 0.35 U
∞
 d
1
/

= 893
– waves velocity propagation;
∫  u’ 2 dy = 0.172 U
∞ d
- disturbances intensity (calculations by Schlichting).
14

• Schlichting H.
(2000) The boundary layer theory,
Springer.
•
Drazin P. G. and Reid W. H.
(1981) Hydrodynamic
Stability, Cambridge University Press, p. 1‒14.
•
Schmid P.J. and Henningson D.S
. (2000) Stability and transition in shear flows, Springer, pp. 1‒60.
15