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OCR for page 23
Boundary-Layer Stability and Transition
W. S. Saric
Arizona State University
Tempe, USA
Abstract
Within the last five years, increased emphasis on
secondary instability analysis along with the
experimental observations of subharmonic instabilities
have changed the picture of the transition process for
boundary layers in low-disturbance environments.
Additional efforts with Navier-Stokes computations
have formed an impressive triad of tools that are
beginning to unravel the details of the early stages of
transition. This paper reviews these recent efforts.
Symbols
a chordwise complex wavenumber normalized by
disturbance amplitude
amplitude at R=Ro, usually Branch I
pressure coefficient
F
co/R= 6.28fv/Uo2: dimensionless frequency
f dimensional frequency thz]
L
~vx*/UO: boundary-Layer reference length.
N ln(A/AO): amplification factor
R
SIR,, = UoL/v boundary-layer Reynolds number
Rat initial boundary-layer Reynolds number, usually
Branch I
Rx UOx*/v: x-Reynolds number or chord Reynolds
number
U basic-state chordwise velocity normalized by UO
Uo
freestream velocity, [m/s]
v kinematic viscosity Em2/s]
cl)
2~fL/Uo: dimensionless circular frequency
x* dimensional chordwise coordinate [m]
x chordwise coordinate normalized with L
23
y normal-to-the-wall coordinate
z spanwise coordinate
1. Introduction
The problems of understanding the origins of
turbulent flow and transition to turbulent flow are the
most important unsolved problems of fluid mechanics
and aerodynamics. There is no dearth of applications
for information regarding transition location and the
details of the subsequent turbulent flow. A few
examples can be given here. (1) Nose cone and heat
shield requirements on reentry vehicles and the
"aerospace airplane" are critical functions of transition
altitude. (2) Vehicle dynamics and "observables" are
modulated by the occurrence of laminar-turbulent
transition. (3) Should transition be delayed with
Laminar Flow Control on the wings of large transport
aircraft, a 25% savings in fuel will result. (4) Lack of a
reliable transition prediction scheme hampers efforts to
accurately predict airfoil surface heat transfer and to
cool the blades and vanes in gas turbine engines. (5)
The performance and detection of submarines and
torpedoes are significantly influenced by turbulent
boundary-layer flows and efforts directed toward drag
reduction require the details of the turbulent processes.
(6) Separation and stall on low-Reynolds-number
airfoils and turbine blades strongly depend on whether
the boundary layer is laminar, transitional, or turbulent.
The common thread connecting each of these
applications is the fact that they all deal with bounded
shear flows (boundary layers) in open systems (with
different upstream or initial amplitude conditions). It is
well known that the stability, transition, and turbulent
characteristics of bounded shear layers are
fundamentally different from those of free shear layers
(Morkovin, 1969; Tani, 1969; Reshotko, 1976~.
Likewise, the stability, transition, and turbulent
characteristics of open systems are fundamentally
different from those of closed systems (Tatsumi, 1984~.
The distinctions are vital. Because of the influence of
indigenous disturbances, surface geometry and
roughness, sound, heat transfer, and ablation, it is not
possible to develop general prediction schemes for
transition location and the nature of turbulent structures
in bQundary-layer flows.
OCR for page 24
There have been a number of recent advances in
the mathematical theory of chaos that have been
applied to closed systems. Sreenivasan and Strykowski
(1984), among others, discuss the extension of these
ideas to open systems and conclude that the relationship
is still uncertain. It appears from a recent workshop
and panel discussion (Liepmann et al. 1986) that the
direct application of chaos theory to open systems is
still some distance away. However, the prospect of
incorporating some of the mathematical ideas of chaos
Into open system problems and of encouraging the
transfer of data to the mathematicians is good. Since
there is still some uncertainty in the direct application
of chaos theory to transition no further mention of this
will be given here.
The purpose of this report is to bring into
perspective certain advances to our understanding of
laminar-turbulent transition that have occurred within
the last five years. In particular, these advances have
been made by simultaneous experimental, theoretical,
and computational efforts.
1.1 Basic Ideas of Transition
With the increased interest in turbulent drag
reduction and in large scale structures within the
turbulent boundary layer, researchers in turbulence
have been required to pay attention to the nature of
lam~nar-turbulent transition processes. It is generally
accepted that the transition from laminar to turbulent
flow occurs because of an incipient instability of the
basic flow field. This instability intimately depends on
subtle, and sometimes obscure, details of the flow. The
process of transition for boundary layers in external
flows can be qualitatively described using the following
(albeit, oversimplified) scenario.
Disturbances in the freestream, such as sound or
vorticity, enter the boundary layer as steady and/or
unsteady fluctuations of the basic state. This part of the
process is called receptivity (Morkovin, 1969) and,
although it is still not well understood, it provides the
vital initial conditions of amplitude, frequency and
phase for the breakdown of laminar flow. Initially
these disturbances may be too small to measure and
they are observed only after the onset of an instability.
The type of instability that occurs depends on Reynolds
number, wall curvature, sweep, roughness, and initial
conditions. The initial growth of these disturbances is
described by linear stability theory. This growth is
weak, occurs over a viscous length scale, and can be
modulated by pressure gradients, mass flow,
temperature gradients, etc. As the amplitude grows
three-dimensional and nonlinear interactions occur in
the form of secondary instabilities. Disturbance growth
is very rapid in this case (now over a convective length
scale) and breakdown to turbulence occurs.
For many years, linear stability theory, with the
Orr-Sommerfeld equation as its keystone, served as the
basic tool for predictors and designers. Since the initial
growth is linear and its behavior can be easily
calculated, transition prediction schemes are usually
based on linear theory. However, since the initial
conditions (receptivity) are not generally known, only
correlations are possible and, most importantly, these
correlations must be between two systems with similar
environmental conditions. The impossibility of
matching or fully understanding these environmental
conditions has led to the failure of any absolute
transition prediction scheme for even the simple Blasius
flat-plate boundary layer.
The preceding does not always follow the observed
behavior. At times, the initial instability can be so
strong that the growth of linear disturbances is by-
passed (Morkovin, 1969) in such a way that turbulent
spots appear or secondary instabilities occur and the
flow quickly becomes turbulent. This phenomenon is
not well understood but has been documented in cases
of roughness and high freestream turbulence (Reshotko,
1986~. In this case, transition prediction schemes based
on linear theory fail completely.
1.2 Review of the Literature
The literature review follows the outline of the
process described above and begins with Reshotko
(1984a, 1986) on receptivity (i.e. the means by which
freestream disturbances enter the boundary layer). In
these papers, Reshotko summarizes the recent work in
this area and points out the difficulties in understanding
the problem. Indeed, the receptivity question and the
knowledge of the initial conditions are the key issues
regarding a transition prediction scheme. Of particular
concern to the transition problem are the quantitative
details of the roles of freestream sound and turbulence.
Aside from some general correlations, this is still an
opaque area. However. in section 3.2 below, a
demonstration of the role of initial conditions on the
observed transition phenomenon is discussed.
The details of linear stability theory are given in
Mack (1984b). This is actually a monograph on
boundary-layer stability theory and should be
considered required reading for those interested in all
aspects of the subject. It covers 58 pages of text with
170 references. In particular, his report updates the
three-dimensional (3-D) material in Mack (1969),
covering In large part Mack's own contributions to the
area.
The foundation paper with regard to nonlinear
instabilities is Klebanoff et al. (1962~. This seminal
work spawned numerous experimental and theoretical
works (not all successful) for the period of 20 years
after its publication. It was not until the experimental
observations of subharmonic instabilities by Kachanov
et al. (1977), Kachanov and Levchenko (1984), and
Saric and Thomas (1984), along with the work on
secondary instabilities, that additional progress was
made in this area. Recent papers of Herbert (1984a,b,c
1985; 1986a,b) cover the problems of secondary
instabilities and nonlinearities, i.e. those aspects of the
breakdown process that succeed the growth of linear
disturbances. It should be emphasized that two-
dimensional waves do not completely represent the
breakdown process since the transition process is
always three-dimensional in bounded shear flows.
Herbert describes the recent efforts in extending the
stability analysis into regions of wave interactions that
produce higher harmonics, three-dimensionality,
subharmonics, and large growth rates--all harbingers of
transition to turbulence. Recent 3-D Navier-Stokes
computations by Fasel (1980,1986), Spalart (1984),
24
OCR for page 25
Spalart and Yang (1986), Kleiser and Laurien (1985,
1986) Reed and co-workers (Singer et al. 1986, 1987;
Yang et al. 1987) have added additional understanding
to the phenomena. More is said about this in section
3.2.
The paper by Arnal (1984) is an extensive
description and review of transition prediction and
correlation schemes for two-dimensional flows that
covers 34 pages of text and over 100 citations. An
analysis of the different mechanisms that cause
transition such as Tollmien-Schlichting (T-S) waves,
Gortler vortices, and turbulent spots is given. The
effects that modulate the transition behavior are
presented. These include the influence of freestream
turbulence, sound, roughness, pressure gradient,
suction, and unsteadiness. A good deal of the data
comes from the work of the group at ONERA/CERT
part of which has only been available in report form.
The different transition criteria that have been
developed over the years are also described which gives
an overall historical perspective of transition prediction
methods.
In a companion paper, Poll (1984b) extends the
description of transition to 3-D flows. When the basic
state is three-dimensional, not only are 3-D
disturbances important, but completely different types
of instabilities can occur. Poll concentrates on the
problems of leading-edge contamination and crossflow
vortices, both of which are characteristic of swept-wing
flows. The history of these problems as well as the
recent work on transition prediction and control
schemes for 3-D flows are discussed by Reed and Saric
(1989).
Reshotko (1984b, 1985, 1986) and Saric (1985b)
review the application of stability and transition
information to problems of drag reduction and in
particular, laminar flow control. They discuss a variety
of the laminar flow control and transition control issues
which will not be covered here.
2. Review of T-S Waves
The disturbance state is restricted to two
dimensions with a one-dimensional basic state. The 2-
D instability to be considered is a viscous instability in
that the boundary-layer velocity profile is stable in the
inviscid limit and thus, an increase in viscosity (a
decrease in Reynolds number) causes the instability to
occur in the form of 2-D traveling waves called T-S
waves. All of this is contained within the framework of
the Orr-Sommerfeld equation, OSE. The historical
development of this work is given in Mack (1984b) and
a tutorial is given by Saric (1985a).
The OSE is linear and homogeneous and forms an
eigenvalue problem which consists of determining the
wavenumber, a, as a function of frequency, ce,
Reynolds number, R. and the basic state, U(y). The
Reynolds number is usually defined as R = UoL/v =
SIR,` and is used to represent *distance along the
surface. In general, L = Vex /UO is the most
straightforward reference length to use because of the
simple form of R and because the Blasius variable is
the same as y in the OSE. When comparing the
solutions of the OSE with experiments, the
dimensionless frequency, F. is introduced as F = m/R =
2~fv/Uo: where f is the frequency in Hertz.
Usually, an experiment designed to observe T-S
waves and to verify the 2-D theory is conducted in a
low-turbulence wind tunnel (u'/UO from 0.02% to
0.06%) on a flat plate with zero pressure gradient
(determined from the shape factor = 2.59 and not from
pressure measurements!) where the virtual-leading-
edge effect is taken into account by carefully controlled
boundary-layer measurements. Disturbances are
introduced by means of a 2-D vibrating ribbon using
single-frequency, multiple-frequency, step-function, or
random inputs (Pupator and Saric, 1989) taking into
account finite-span effects (Mack, 1984a). Hot wires
measure the U + u' component of velocity in the
boundary layer and d-c coupling separates the mean
from the fluctuating part. The frequency, F. for single-
frequency waves remains a constant.
When the measurements of are repeated along a
series of chordwise stations, the maximum amplitude of
the waves varies. At constant frequency, the
disturbance amplitude initially decays lintil the
Reynolds number at which the flow first becomes
unstable is reached. This point is called the Branch I
neutral stability point and is given by R~. The
amplitude grows exponentially until the Branch II
neutral stability point is reached which is given by Ru.
The locus of R~ and Rn points as a function of
frequency gives the neutral stability curve. If the
growth rate of the disturbances is defined as c' =
o(R,F), Fig. 1 is the locus of o(R,F) =0. For R > 600
the theory and experiment agree very well for Blasius
flow. For R < 600 the agreement is not as good
because the theory is influenced by nonparallel effects
and the experiment is influenced by low growth rates
and nearness to the disturbance source. Virtually all
problems of practical interest have R > 1000 in which
case the parallel theory seems quite adequate (Gaster,
1974; Saric and Nayfeh, 1977~.
By assuming that the growth rate, c, = o(R,F), to
hold locally (within the quasi-parallel flow
approximation), the disturbance equations are
integrated along the surface with R = it(x) to give:
A/Ao = exp(N)
where dN/dR = cs, A and Ao are the disturbance
amplitudes at R and R~, respectively, and R~ is the
Reynolds number at which the constant-frequency
disturbance f~rst becomes unstable (Branch I of the
neutral stability curve).
The basic design tool is the correlation of N with
transition Reynolds number, RT, for a variety of
observations. The correlation will produce a number
for N (say 9) which is now used to predict RT for cases
in which experimental data are not available. This is
the celebrated eN method of Smith and von Ingen (e.g.
Arnal, 1984; Mack, 1984b). The basic LFC technique
changes the physical parameters and keeps N within
reasonable limits in order to prevent transition. As long
as laminar flow is maintained and the disturbances
remain linear, this method contains all of the necessary
physics to accurately predict disturbance behavior. As
a transition prediction device, the eN method is certainly
the most popular technique used today. It works within
25
OCR for page 26
some error limits only if comparisons are made with
experiments with identical disturbance environments.
Since no account can be made of the initial disturbance
amplitude this method will always be suspect to large
errors and should be used with extreme care. When
bypasses occur, this method does not work at all. Mack
(1984b) and Arnal (1984) give examples of growth-rate
and eN calculations showing the effects of pressure
gradients, Mach number, wall temperature, and three
dimensionality for a wide variety of flows. These
reports contain the most up-to-date stability
. ~ .
ntormatlon.
3. Secondary Instabilities and Transition
There are different possible scenarios for the
transition process, but it is generally accepted that
transition is the result of the uncontrolled growth of
unstable three-dimensional waves. Secondary
instabilities with T-S waves are reviewed in some detail
by Herbert (1984b, 1985, 1986), Saric and Thomas
(1984) and Saric et al.(l984~. Therefore, only a brief
outline is given in section 3.1 in order to give the reader
some perspective of the different types of breakdown.
Section 3.2 discusses the very recent results.
3.1 Secondary Instabilities
The occurrence of three-dimensional phenomena in
an otherwise two-dimensional flow is a necessary
prerequisite for transition (Tan), 1981~. Such
phenomena were observed in detail by Klebanoff et al.
(1962) and were attributed to a spanwise differential
amplification of T-S waves through corrugations of the
boundary layer. The process leads rapidly to spanwise
alternating "peaks" and "valleys", i.e., regions of
enhanced and reduced wave amplitude, and an
associated system of streamwise vortices. The peak-
valley structure evolves at a rate much faster than the
(viscous) amplification rates of T-S waves. The
schematic of a smoke-strealdine photograph (Saric et
al. 1981) in Fig. 1 shows the sequence of events after
the onset of "peak-valley splitting". This represents the
path to transition under conditions similar to Klebanoff
et al. (1962) and is called a K-type breakdown. The
lambda-shaped (Hama and Nutant, 1963) spanwise
corrugations of streaklines, which correspond to the
peak-valley structure of amplitude variation, are a result
of weak 3-D displacements of fluid particles across the
critical layer and precede the appearance of Klebanoff's
"hair-pin" vortices. This has been supported by hot-
wire measurements and Lagrangian-type streakline
prediction codes (Saric et al.,l981; Herbert and
Bertolotti, 1985~. Note that the lambda vortices are
ordered in that peaks follow peaks and valleys follow
valleys.
Since the pioneering work of Nishioka et al.(l975,
1980), it is accepted that the basic transition
phenomena observed in plane channel flow are the
same as those observed in boundary layers. Therefore,
little distinction will be given here as to whether work
was done in a channel or a boundary layer. From the
theoretical and computational viewpoint, the plane
channel is particularly convenient since the Reynolds
number is constant, the mean flow is strictly parallel,
certain symmetry conditions apply, and one is able to
26
do temporal theory. Thus progress has been first made
with the channel flow problem.
Different types of three-dimensional transition
phenomena recently observed (e.g. Kachanov et al.
1977; Kachanov and Levchenko, 1984; Saric and
Thomas, 1984; Saric et al. 1984, Kozlov and
Ramanosov, 1984) are characterized by staggered
patterns of peaks and valleys (see Fig. 2) and by their
occurrence at very low amplitudes of the fundamental
T-S wave. This pattern also evolves rapidly into
transition. These experiments showed that the
subharmonic of the fundamental wave (a necessary
feature of the staggered pattern) was excited in the
boundary layer and produced either the resonant wave
interaction predicted by Craik (1971) (called the C-
type) or the secondary instability of Herbert (1983)
(called the H-type). Spectral broadening to turbulence
with self-excited subharmonics has been observed in
acoustics, convection, and free shear layers and was not
identified in boundary layers until the results of
Kachanov et al. (1977~. This paper reinitiated the
interest in subharmonics and prompted the
simultaneous verification of C-type resonance (Thomas
and Saric, 1981; Kachanov and Levchenko, 1984~.
Subharmonics have also been confirmed for channel
flows (Kozlov and Ramazanov, 1984) and by direct
integration of the Navier-Stokes equations (Spalart
19841. There is visual evidence of subharmonic
breakdown before Kachanov et al. (1977) in the work
of Hama (1959) and Knapp and Roache (1968) which
was not recognized as such at the time of their
publication. The recent work on subharmonics is found
in Herbert (1985, 1986a,b), Saric, Kozlov and
Levchenko (1984), and Thomas (1986~.
The important issues that have come out of the
subharmonic research is that the secondary instability
depends not only on disturbance amplitude, but on
phase and fetch as well. Fetch means here the distance
over which the T-S wave grows in the presence of the
3-D background disturbances. If T-S waves are
permitted to grow for long distances at low amplitudes
subharmonic secondary instabilities are initiated at
disturbance amplitudes of less than 0.3%Uo. Whereas,
~f larger amplitudes are introduced, the breakdown
occurs as K-type at amplitudes of 1% UO. Thus, there
no longer ex~sts a "magic" amplitude criterion for
breakdown.
A consequence of this requirement of a long
enough fetch for the subharmonic to be entrained from
the background disturbances is that the subharmonic
interaction will occur at or to the right of the Branch II
neutral stability point. Since this is in the stable region
of the fundamental wave, it was not likely to be
observed because the experimenters quite naturally
concentrated their attention of measurements between
Branch I and Branch II.
3.2 Recent Results
The surprise that results from the analytical model
of Herbert (1986a,b) and the Navier-Stokes
computations of Singer, Reed, and Ferziger (1986), is
that under conditions of the experimentally observed K-
Type breakdown, the subharmonic H-Type is still the
dominant breakdown mechanism instead of the
OCR for page 27
fundamental mode. This is in contrast to Klebanoff's
experiment, confirmed by Nishioka et al. (1975,
1980),Kachanov et al. (1977), Saric and Thomas
(1984), Saric et al. (1984), and Kozlov and Ramazanov
(1984) where only the breakdown of the fundamental
into higher harmonics was observed. Only Kozlov and
Ramazanov (1984) observed the H-type in their channel
experiments and only when they artificially introduced
the subharmonic.
This apparent contradiction was resolved by
Singer, Reed, and Ferziger (1987~. Here the full three-
dimensional, time-dependent incompressible Navier-
Stokes equations are solved with no-slip and
impermeability conditions at the walls. Periodicity was
assumed in both the streamwise and spanwise
directions. The implementation of the method and its
validation are described by Singer, Reed and Ferziger
(19861. Initial conditions include a two-dimensional T-
S wave, random noise, and streamwise vortices. No
shape assumptions are necessary, the spectrum is larger,
and random disturbances whether freestream or already
in the boundary layer can be introduced and monitored
for growth and interactions (Singer, Reed, and Ferziger,
1986~. Other advantages realized by computations are
1) the inclusion of boundary-layer growth, neglected in
linear theory but important to the growth of secondary
instabilities, 2) the generation of ensemble averages, 3)
the visualization of flow phenomena for comparison
with experiments (advanced graphics capability), and 4)
the calculation of vorticity and energy spectra, often
unavailable from experiments.
The streamwise vortices can alter the relative
importance of the subharmonic and fundamental
modes. Streamwise vortices of approximately the
strength of those that might be found in transition
experiments can explain the difficulty in experimentally
identifying the subharmonic route to turbulence
(Herbert 1983~.
The corresponding computational visualizations of
Singer et al. (1987) are shown in Figs. 3 and 4; flow is
from lower right to upper left. Figure 4 shows the
vortex structures, commonly seen in the transition
process, under the conditions of a forced 2-D T-S wave
and random noise as initial conditions. The
subharmonic mode is present as predicted by theory but
not seen experimentally. Other views of the vertical
structure are given by Herbert (1986a). However, when
streamwise vorticity (as is present in the flow from the
turbulence screens upstream of the nozzle) is also
included, the subharmonic mode is overshadowed by
the fundamental mode (as in the experiments!. The
resulting pattern, ordered peak-valley structure, is seen
in Fig. 4. Here is a case in which the computations
have explained discrepancies between theory and
experiments.
In the presence of streamwise vorticity, the
fundamental mode is preferred over the subharmonic;
this agrees with experimental observations, but not with
theory (which does not account for this presence).
Without streamwise vorticity, the subharmonic modes
dominate, as predicted by theory and confirmed by
computational simulations. In the presence of
streamwise vorticity characteristic of wind-tunnel
experiments, the K-type instability dominates and the
numerical simulations predict the experimental results.
Direct numerical simulations are playing an
increasingly important role in the investigation of
transition; the literature is growing, especially recently.
This trend is likely to continue as considerable progress
is expected towards the development of new, extremely
powerful supercomputers. In such simulations, the full
Navier-Stokes equations are solved directly by
employing numerical methods, such as finite-difference
or spectral methods. The direct simulation approach is
widely applicable since it avoids many of the
restrictions that usually have to be imposed in
theoretical models.
The Navier-Stokes solutions are taken hand-in-
hand with the wind tunnel experiments in a
complementary manner. The example of Singer, Reed,
and Ferziger (1987) illustrates that these two techniques
cannot be separated. The next step in the simulations
will be to predict the growing body of detailed data
being developed by Nishioka et al. (1980, 1981, 1984,
1985) on the latter stages of the breakdown process.
4 Transition Prediction and Control
.
When the recent work on subharmonics is added to
the discussion at the end of section 3 on the limitations
of the eN method, one indeed has an uncertainty
principle for transition (Morkovin, 1978~. Transition
prediction methods will remain conditional until the
receptivity problem is adequately solved and the bypass
mechanisms are well understood. In the mean time,
extreme care must be exercised when using correlation
methods to predict transition. Additional problems of
transition prediction and laminar flow control are
discussed by Reshotko (1985, 1986~. The main
principle of laminar flow control is to keep the
disturbance levels low enough so that secondary
instabilities and transition do not occur. Under these
conditions, linear theory is quite adequate and eN
methods can be used to calculate the effectiveness of a
particular LFC device.
The idea of transition control through active
feedback systems is an area Hat has received
considerable recent attention (Liepmann and
Nosenchuck, 1982; Thomas, 1983; Kleiser and Laurien,
1984, 1985; Metcalfe et al., 1985~. The technique
consists of first sensing the amplitude and phase of an
unstable disturbance and then introducing an
appropriate out-of-phase disturbance that cancels the
original disturbance. In spite of some early success,
this method is no panacea for the transition problem.
Besides the technical problems of the implementation
of such a system on an aircraft, the issue of three-
dimensional wave cancellation must be addressed. As
Thomas (1983) showed, when the 2-D wave is
canceled, all of the features of the 3-D disturbances
remain to cause transition at yet another location.
Some clear advantages over passive systems have yet to
be demonstrated for this technique.
Acknowledgements
This work is supported by the Air Force Office
of Scientific Research Contract AFOSR-85-NA-077.
27
OCR for page 28
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-
/
Off
~ f
Figure 1. Staggered peak-valley structure 01-type mode).
/
f
f
f
f
>
f
Figure 2. Peak-valley splitting structure (K-type mode).
31
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amp
Figs ~ Vortex lines with ~ = ~0~, f = 177. The Dow goes ham lower tight to upper
lea.
Figure 4. Vo~x lees with ~ = ~0~, ~ = 132. Tbo Cow goes Mom lower tight to upper
Ad. ElUpOcal s~c~wisc voices with maximum ~ pc~urbadon of 1 8~ arc
included.
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DISCUSSION
by F. Stern
Please comment on the influence of the
type of breakdown on the state of the
resulting turbulent boundary layer, which is
quite important both in experiments and
calculations. Also, are there any
similarities between the processes you have
classified for transition and those associated
with relaminarization?
Author's Reply
The first part of this question hits to
the heart of the motivation for doing work on
the latter stages of transition.
The type of transition is important
because of its influence on the large-scale
structure in "low" Reynolds number turbulent
boundary layers. The control of turbulent
boundary layers then rests on the type of
large scale structure that may be present.
For "high" Reynolds number turbulent boundary
layers, the situation is not so clear i.e. it
is hard to imagine that the details of the
transition process influence the structure of
fully developed turbulence.
It is unlikely that any of the structure
of the transition process is recovered during
relaminarization of a turbulent flow. I
believe relaminarization to be highly
dissipative due to large changes in the basic
state which cause a loss in the turbulence
production mechanisms. Unfortunately, there
is a dearth of detailed experiments in this
area.
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Representative terms from entire chapter:
boundary layers
