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Tuoc-Laminar-Turbulent-Transition.pdf
On The Critical Reynolds Number For Transition From
Laminar To Turbulent Flow
Trinh, Khanh Tuoc
Institute of Food Nutrition and Human Health
Massey University, New Zealand
[email protected]
Abstract
In this visualisation, the transition from laminar to turbulent flow is characterised by
the intermittent ejection of wall fluid into the outer stream. The normalised thickness
of the viscous flow layer reaches an asymptotic value but the physical thickness drops
exponentially after transition. The critical transition pipe Reynolds number can be
obtained simply by equating it with the asymptotic value of the normalised thickness
of viscous flow layer.
Key words: Transition, critical stability Reynolds number, critical transition Reynolds
number, non-Newtonian pipe flow
1
1
Introduction
Reynolds (1883) was the first to propose a criterion for differentiation between
laminar and turbulent flows in his classic dye visualisation
Re
DVU
(1)
P
and suggested a critical value of Re
2100 for the upper limit of laminar flow. In a
second paper (Reynolds, 1895) he showed by time-averaging the Navier-Stokes
equations that new extra convection terms appeared in turbulence which have the
units of stress and are therefore called Reynolds stresses. Considerable effort has been
expanded in the last hundred and thirty years to understand the process of transition.
“The problem is simple in concept and yet the origins of the observed turbulent
motion remain largely mysterious despite more than a century of research” (Mullin
and Peixinho, 2006). The literature on the transition is vast, prompting Herbert
(1988), a well-known author, to note that "different reviews on shear-flow instability
may have little in common and a zero-overlap of cited literature. This curious fact
illustrates the many facets of the overall problem, the multitude of views, concepts,
and methods, and the need to remain open minded. It also grants me the right to
present my own view supported by a selection of references that I know is far from
complete". We will therefore not even attempt a survey of the present literature but
will concentrate on some developments directly relevant to the topic of this paper.
Kerswell (2005) noted that all experimental and theoretical evidence points to the fact
that the laminar flow state (which exists potentially for all flow rates) is linearly stable
to any infinitesimal disturbance. He also stated that the clear implication is that the
observed transition process can then only be initiated by finite amplitude disturbances.
This view is widely shared by other authors.
Reynolds himself observed that turbulence was triggered by inlet disturbances to the
pipe and the laminar state could be maintained to Re ป12,000 if he took great care in
minimizing external disturbances to the flow. By careful design of pipe entrances
Ekman (1910) has maintained laminar pipe flow up to a Reynolds number of 40,000
and Pfenniger (1961) up to 100,000 by minimising ambient disturbances. All
2
sufficiently small perturbations will decay e.g. Salwen et al. (1980) and Meseguer &
Trefethen (2003), who analyzed the problem up to Re
10 7 . Turbulence is only
triggered when the disturbances exceeded a finite critical amplitude of perturbation
Mullin and Peixinho (2006). Thus, to trigger transition, two thresholds have to be
crossed: The flow has to be sufficiently fast and a perturbation has to be strong
enough. Eckhardt et al. (2007) further noted that experiments by Hof et al. (2003)
further show that as the Reynolds number increases the critical threshold decreases so
that at sufficiently high Reynolds numbers the unavoidable residual fluctuations
always suffice to trigger turbulent flow.
Bayly et al.(1988) have reviewed very well the development of theoretical thinking on
the stability of shear flows. Its basis can be traced to Reynolds’concept that the
laminar pattern always represents a possible type of flow since it is a solution of the
Navier-Stokes equation but can be broken into a turbulent pattern by growing
perturbations.
The fundamental differential equation for the disturbance or stability equation was
extracted from the Navier-Stokes equations by Orr (1907) and Sommerfield (1908)
U c I cc D 2I U ccI
i
I cccc 2D 2I cc D 4I
D Re
(2)
where the disturbance has the form
\ ( x, y, t ) I ( y )e i Dx Et D is a real number and O
(3)
2S D is the wavelength of the disturbance. The quantity
E is complex
E
E r iE i
(4)
where E r is the circular frequency of the partial disturbance and E i is the
amplification factor. The results are often expressed in terms of the ratio of D and E
c
E
D
c r ici
(5)
Over a century ago, Lord Rayleigh (1880) analysed the problem for inviscid flow
where the right hand side of equation (2) can be neglected and formulated two
important theorems
1. Velocity profiles which posses an inflexion point are unstable
3
2. The velocity of propagation of neutral disturbances ci
0 in a boundary
layer is smaller than the maximum velocity of the main flow U m
In the last hundred years much effort has been directed to solving the Orr-Sommerfeld
equation for various flows but this equation is exceedingly difficult to analyze for the
large Reynolds numbers at which transition is observed to occur. Thus not much
progress were made until Tollmien and Schlichting obtained predictions for a curve of
neutral stability (Schlichting, 1979) that was verified experimentally a decade later by
Schubauer and Skramstad (1943) as shown in Figure 1.
Figure 1 Curve of neutral stability for neutral disturbances on a flat plate at zero
incidence from Schubauer and Skramstad (1943).
The minimum Reynolds number on the stability curve on flat plate is
ReG 1 , s
§ U fG 1 ·
¸
¨
© Q ¹ cs
(6)
520
where G 1 is the displacement thickness. At lower Reynolds numbers all disturbances
are dissipated but turbulence does not set in at that point. From this critical stability
Reynolds number onwards the disturbance can grow but transition to turbulence
occurs at U f x Q 3.5 u 10 5 10 6 corresponding to
ReG G
1
,T
(7)
950
For plane Poiseuille flow Lin (1955) predicted a critical stability Reynolds number of
5314. Thus the linear instability theory gives a satisfactory description of the initial
growth of very small disturbances but does not predict well the Reynolds numbers at
4
which transition is observed. In the linear instability theory the most dangerous
instabilities have fairly long wavelengths and low growth rates, in contrast to the
universally observed three-dimensional short-wavelength structure of turbulence. The
discrepancy may be caused by the neglect of the nonlinear self-interaction of the
disturbance, which always exists for finite perturbations no matter how small. The
next advance in the theory of stability was the modelling of non-linear two
dimensional waves but from the time of Reynolds' earliest experiments, it was
obvious that three dimensional disturbances are important at some stage in transition.
Three dimensional disturbances have been studied extensively since the 1960’s as
they provide a crucial link between classical instability theory and the observations of
transition in real flows. These developments have been well reviewed by Bayley et al.
(1988).
Eckhardt et al. (2007) have reviewed another approach based on the theory of strange
attractors. The background for these studies is the abstraction to consider the system
in its state space. In state space, there is one region dominated by the laminar flow.
The time-independent parabolic profile is a fixed point in this space. The parabolic
profile is linearly stable and, hence, all points in its neighbourhood evolve toward this
fixed point; these states form the basin of attraction of the laminar flow. The turbulent
dynamics take place in other parts of the state space where a turbulence basin of
attraction is situated. The spatially and temporally fluctuating dynamics of the
turbulent regions suggests that there are chaotic elements, such as horseshoe vortices,
just as in a regular attractor. However, the possibility of decay indicates that the basin
is not compact nor space filling; there must be connections to the laminar profile. In
dynamical systems such structures are known as chaotic saddles or strange saddles
(Eckhart et al., op. cit.).
Kerswell (2005) has reviewed another approach based on the developments and
promise of the recently discovered travelling wave solutions. Mullin and Peixinho
(2006) have reviewed the results of recent experimental investigations into transition
to turbulence in fluid flow through a circular straight pipe, at room temperature.
The failure of the formal approach to predict accurately the well known critical
Reynolds number for transition from laminar to real turbulent flows in different
5
geometries have led some authors to take a more heuristic approach that allows an
approximate estimate of the critical transition Reynolds number in situations of
practical interest. Of particular interest is the transition in non-Newtonian fluids for
which experimental data is still scarce and inaccurate e.g. Ryan and Johnson (1963a),
Hanks (1963a), Mishra and Tripathi (1971, 1973), Rudman et al.(2002) for power law
fluids, Hanks (1963a) for Bingham plastic fluids, Zamora et al (2005), Peixinho et al
(2005) for Herschel-Bulkley fluids and other geometries such as square ducts (Etemad
and Sadeghi, 2001). Wilson and Thomas (2006) tried to define an equivalent viscosity
for Bingham plastic fluids to better identify the transition Reynolds number, Güzel et
al. (2009)defined a new Reynolds number based on averaging local Reynolds
numbers across the pipe. Ryan and Johnson proposed a stability criterion
Z
RUU wU
W w wy
(8)
which is essentially an empirical estimate of the ratio of rate of energy supply to the
disturbance to rate of dissipation. Z must clearly be zero at the wall and the
centreline and passes through a maximum at an intermediate position. For a flow
where the laminar velocity profile is known Z max can be calculated. For Newtonian
pipe flow this coefficient can be shown to be a special case of the Reynolds number
4 DV
at
27 Q
Z max
r
R
1
(9)
3
The critical stability parameter for pipe flow is given at Re c
Zc
2100 as
(10)
808
Ryan and Johnson assumed that this value of Z c also applied to non-Newtonian
fluids. Hanks and his co-workers have applied the analysis of Ryan and Johnson to
power law fluids in concentric annuli and parallel plates (Hanks, 1963b, Hanks and
Valia, 1971, Hanks and Ricks, 1975) and Bingham plastic fluids (Hanks, 1963a,
Hanks and Dadia, 1971). For power law fluids Hanks predicted a critical Reynolds
number of
n2 n 2 n 1 n Re g ,c
6464
(11)
1 3n 2
where Re g is the generalised Metzner-Reed Reynolds number (Metzner and Reed,
1955). Mishra and Tripathi (1971, 1973) observed that the predictions of Ryan and
6
Johnson and Hanks were acceptable for moderate values of the behaviour index n of
power law fluids but did not agree with experimental evidence for highly nonNewtonian fluids (low n values) as shown in Figure 15. They proposed a different
empirical criterion based on the ratio of average kinetic energy of the fluid and
friction drag at the wall
Dc
Re g ,c
fc
UV 2
(12)
Average K .E. perunitvolume
2100D c
16
Re g ,c
(13)
0.0762
(14)
Dc
For power law fluids
Re g ,c
2
2100
4 n 2 5n 3
2
33n 1
(15)
Theoretical considerations
Following the advice of Herbert, and with the indulgence of the reader, I will present
the following views from a personal perspective and not use the traditional third
person narrative because my views have clear points of difference with the traditional
approach.
2.1
The four component decomposition of the instantaneous velocity in
turbulent flows and its implications
In previous publications (Trinh, 1992, 2009c, 2009a) I have proposed that much more
information can be extracted from measurements of turbulent flows by decomposing
the instantaneous velocity into four components instead of Reynolds’ two
components. The wall layer process has been well documented by many authors using
hydrogen bubble tracers (Kline et al., 1967, Kim et al., 1971, Corino and Brodkey,
1969) as shown schematically in Figure 2. It starts with an inrush of fast fluid towards
the wall that is deflected into a longitudinal vortex. The fluid beneath the vortex is
represented by a low-speed streak that develops during the so-called sweep phase.
The streaks tend to lift, oscillate and eventually burst in other violent ejections from
the wall towards the outer region. The sweep phase is much longer lasting than the
burst phase.
7
Outer
region
Inrush (time-lines
contorted by a
transverse vortex)
Ejection (burst)
Main flow
Log
law
Lift-up of wall dye:
developing subboundary layer
Start of viscous
sub-boundary
layer
Inner
region
Sweep
Ge
Wall layer
Wall
Figure 2. Visualisation of a cycle of the wall layer process drawn after the
observations of Kline et al.(1967) and regions in the flow field.
Figure 3 shows a trace of the instantaneous velocity u i corresponding to a cycle in the
wall layer. The local instantaneous velocity u i can be decomposed into a smoothed
velocity u~i in the sweep phase of the wall layer and a fast fluctuating velocity u ci
resulting from the passage of the longitudinal vortex above the wall.
ui = u~i + u ci
(16)
where u ci is periodic and does not contribute to the long time average velocity U i
f
³ uicdt = 0
(17)
0
Ui =
1 tQ ~
³ u i dt
tQ 0
(18)
ui = U i +U ci
(19)
8
Figure 3 Trace of instantaneous streamwise velocity in the wall layer after
measurements by Antonia et al. (1990).
~
U ci = U ci +u ci
(20)
~
Uci = ~
u i -Ui
(21)
~
u i = Ui + Uci + u ci
(22)
where
then
We may average the Navier-Stokes equations over the period t f of the fast
fluctuations. Bird, Stewart, & Lightfoot (1960), p. 158 give the results as
2
w( U ~
wp
ui ~
u j w u ci u c j
w ~
u i)
ui w ~
=+P
2
wt
w xi
w xj w xj
w xj
(23)
Equation (23) defines a second set of Reynolds stresses u ciu c j which we will call
"fast" Reynolds stresses to differentiate them from the standard Reynolds
stresses U ciU c j .
We may write the fast fluctuations in the form
u ic = u 0 ,i eiȦt + e-iȦt (24)
The fast Reynolds stresses uicu cj become
u ciu c j = u0,i u0, j ( e
2iZt
+ e-2iZt ) + 2 u0,i u0, j
9
(25)
Equation (25) shows that the fluctuating periodic motion u ci generates two
components of the "fast" Reynolds stresses: one is oscillating and cancels out upon
long-time-averaging, the other, u 0,i u 0, j is persistent in the sense that it does not depend
on the period t f . The term u 0,i u 0, j indicates the startling possibility that a purely
oscillating motion can generate a steady motion which is not aligned in the direction
of the oscillations. The qualification steady must be understood as independent of the
frequency Ȧ of the fast fluctuations. If the flow is averaged over a longer time than the
period tQ of the bursting process, the term u 0,i u 0, j must be understood as transient but
non-oscillating. This term indicates the presence of transient shear layers embedded in
turbulent flow fields and not aligned in the stream wise direction similar to those
associated with the streaming flow in oscillating laminar boundary layers (Schneck
and Walburn, 1976, Tetlionis, 1981). Thus a fourth component must be included in
the decomposition of the instantaneous velocity in turbulent flows
~
u i U i U ic u c ii (Zt ) u i , st
(26)
The streaming flow term u i , st dominates the bursting phase.
2.2
Evolution from two to four component velocities
To understand the transition process we must, in my view, show how the solution of
the Navier-Stokes transforms from the laminar state characterised by a twocomponent instantaneous velocity (equation 16) to the fully turbulent state
characterised by a four-component instantaneous velocity (equation 26). This
evolution also occurs in the process of turbulent wall shear flow which we consider
first. The analytical solution is extremely difficult and much greater progress has
been made by direct numerical simulations DNS. Unfortunately the database
generated by the DNS is usually massive and we cannot easily trace the evolution of
the terms in equation (26) let alone their interactions. Robinson (1991) for example
spent an entire PhD simply to describe and catalogue the coherent structures that
identified from the DNS of Moin and Kim (1982).
The sweep phase can be modelled as a Kolmogorov flow (Obukhov, 1983) a simple
two dimensional sinusoidal flow, or better still analysed with techniques borrowed
from laminar oscillating flow (Trinh, 2009c, Trinh, 1992). The traditional approach to
10
analyse these unsteady flows is by a method of successive approximations
(Schlichting, 1979, Tetlionis, 1981). The dimensionless parameter defining these
successive approximations is
H = Ue
LZ
(27)
where U e is the local mainstream velocity and L is a characteristic dimension of the
body. The smoothed velocity u~i is given by the solution of order H 0 which applies
when H 1 . The governing equation (Einstein and Li, 1956, Hanratty, 1956, Meek
and Baer, 1970, Trinh, 2009c) is a subset of the NS equations
w~
u
wt
w 2~
u
Q 2
wy
(28)
where u~ refers here to the smoothed velocity u~i in the x direction. It does not
require that there are no velocity fluctuations, only that they are small enough for their
effect on the smoothed phase velocity u~ to be negligible. Stokes (1851) has given the
solution to equation (28) as
~
u
erf (K s )
UQ
where
Ks =
GQ
y
4Qt
(29)
. The thickness of this sub-boundary layer is
4.16 UQ
(30)
where the velocity and normal distance have been normalised with the wall
parameters Q the kinematic viscosity and u*
W w U the friction velocity, W
w
the
time averaged wall shear stress and U the density.
To understand the effect of the fluctuations beyond simple growth, an analysis of
oscillating flow with a zero-mean velocity is particularly interesting since the basic
velocity fluctuations imposed by external means do not grow with time because there
is no mean motion along the wall. We may thus investigate the effect of the amplitude
and frequency of the fluctuations separately. The following treatment of the problem
is taken from the excellent book of (Tetlionis, 1981).
We define a stream function ȥsuch that
11
u=
w\
wy
v=
w\
wx
(31)
The basic variables are made non-dimensional
*
x =
x
L
*
y =
y
*
t = tZ
2Q/Z
(32)
§
2Q ·
¸
\ = \ ¨¨U f
¸
Z
©
¹
Ue
(x,t)
U (x,t) =
Uf
-1
*
*
e
(33)
where U f is the approach velocity for x o f . The system of coordinates x, y is
attached to the body. The Navier-Stokes equation may be transformed as:
*
*
*
2 *
*·
w \ * w 2\ *
1 w 3\ w U *e U e §¨ w \ w 2\
w\
* wU e ¸
+
=
+
U
e
w x* ¸
w y* w t* 2 w y* 3 w t* LZ ¨© w y* w y* w x* w x* w y* 2
¹
(34)
with boundary conditions
c
\* =
w\ *
w y*
=0
*
y =0
(35)
For large values of Z , the RHS of equation (34) can be neglected since
H = U e 1
LZ
(36)
In this case, Tetlionis reports the solution of equation (34) as:
ªU *0 ( x* )
\* = «
¬«
2
*
(1 - i)[1 - e(1+i) y ] +
*
U *0 y º it*
» e +C
2 ¼»
(37)
Tetlionis (op. cit. p. 157) points out that equation (37) may be regarded as a
generalisation of Stokes' solution (1851) for an oscillating flat plate often called
Stokes second problem. This latter solution describes an oscillating flow called the
Stokes layer which is often found embedded in other flow fields and has properties
almost independent of the host field.
In the case of a non-zero mean, the smoothed velocity can be described
by a
generalised form of Stokes solution for a flat plate started impulsively, often referred
to as Stokes’ first problem (equation 28). Then the stream function must incorporate
both of Stokes’ solutions. This composite solution, that we called of solution order
H 0 , is accurate only to an error of order İ. Tetlionis reports a more accurate solution
for the case when İ cannot be neglected (i.e. for larger H hence lower frequencies):
12
\* =
*
*
*
*
U 0 ( x ) * * it* * * -it*
[\ 0 ( y ) e \ 0( y ) e ] + H [\ *1 ( x* , y* ) e2it + \ *0 e- 2it ] + 0 ( H 2 )
2
(38)
where \ 0 and \ 1 are the components of the stream function of order İ 0 and H .
Substituting this more accurate solution into equation (34), we find that the
multiplication of coefficients of e it* and e it* forms terms that are independent of the
oscillating frequency, Ȧ, imposed on the flow field and were not anticipated in
equation (38). Thus the full solution of equation (34) is normally written (Tetlionis,
1981) as
\* =
*
U *0 ( x* ) * * it*
[\ 0 ( y ) e +\ *0( y* ) e-it ]
2
(39)
*
*
+ H [\ *st + [\ *1 ( x* , y* ) e2it +\ *1( x* , y* ) e-2it ] + O ( H 2 )
where the overbar denotes the complex conjugate and \ st* results from cancelling of
e it* and e it* terms.
The quantity \ st* shows that the interaction of convected inertial effects of forced
oscillations with viscous effects near a wall results in a non-oscillating motion that is
referred to in the literature as "Streaming". The problem has been known for over a
century (Faraday, 1831, Dvorak, 1874, Rayleigh, 1884, Carriere, 1929, Andrade,
1931, Schlichting, 1932) and studied theoretically (Riley, 1967, Schlichting, 1960,
Stuart, 1966, Tetlionis, 1981). The existence of this streaming flow, even in this
absence of any mainstream flow, is clearly demonstrated in Figure 4.
Figure 4 Streaming flow near a vibrating cylinder. After Schlichting (1960).
At the end of the sweep phase, the fluctuations have grown large enough for the
streaming flow to contain substantial amount of kinetic energy sufficient to eject wall
fluid from the wall layer.
13
The analytical solution of the stream function \ st* e.g. by Stuart (1966) presents
great difficulties and invariably involve simplifications that suppress critical
information. For example Stuart (1966) ignored the interaction terms between the
main flow and the streaming flow. The main crossflow deflects the wall ejection in a
streamwise direction. This is the first interaction. Immediately outside the wall layer,
the jet follows a quasi linear path before it becomes deflected into a curvilinear
pattern. As the main flow hits the jet, it is broken into small scales typical of
incoherent turbulence. By contrast the jet itself moves as a coherent structure.
2.3
Physical visualisation of transition
The ejections start to disturb the outer quasi-inviscid region beyond the wall layer and
dramatically increase the boundary layer thickness. At Reynolds numbers just above
the critical value, e.g. Re =2100 for pipe flow, only the far field section of the
intermittent jets penetrates the outer region. The disturbance to the previously “quasipotential” flow may be compared with that of a wall-parallel jet since the ejections are
here aligned in the direction of main flow. This region has been described by Cole's
law of the wake (Coles, 1956). As the Reynolds number increases further, so do the
fluctuations: the streaming flow strengthens and emerges at a cross flow angle with
the main flow. Millikan (1938) showed that an outer region that scale with the outer
parameters (Coles law of the wake in the present visualisation) and a wall region
which scales with the wall parameters (the solution of order H 0 ) must be linked by a
semi-logarithmic velocity profile.
U
A ln y B
1
(40)
ln y B
N
where N is called Karman’s universal constant with a canonical value of 0.40 . In the
present visualisation, upon transition, the first layer to be added to the wall layer is the
law-of-the-wake region then full turbulence is established when the log-law grows.
14
Figure 5. Penetration of the log-law into the outer region during a cycle oscillating
pipe flow. From Trinh (1992). Data of Akhavan et al. (1991) for Re Ȧ = 1080.
The transition from laminar to turbulent flow is clearly seen in the oscillating flow
experiments of Akhavan, Kamm, & Saphiro Akhavan (1991) . The flow is driven by a
reciprocating piston pump. The acceleration phase, where the pressure gradient is
favourable, is laminar. The velocity profile here exhibits only two regions: (a) a wall
layer which coincides very well with the profiles for laminar boundary layer flow and
those for the wall layer of steady turbulent pipe flow, and (b) a fluctuating potential
flow in the outer region.
Schneck and Walburn (1976) have argued in their study of pulsatile blood flow that
the secondary streaming flow results from a tendency of viscous forces to resist the
reversal of flow imposed by the oscillating motion of the main stream. This is
15
demonstrated more clearly in the experiments of Gad-el-Hak, Davis, McMurray, &
Orszag (1983) who generated an artificial bursting process in a laminar boundary
layer on a flat plate by decelerating it.
The magnitude of the deceleration oscillating flow and the corresponding adverse
pressure gradient must be sufficient to induce separation and ejection of low-speed
fluid from the wall. The ejections begin to penetrate the previously quasi-inviscid
region outside but because their energy content is still weak they are immediately
deflected in the direction of the main flow: The law of the wake layer makes it
appearance. The transition region ends when the ejections have gathered enough
strength for the rectilinear region of the jet to protrude from the wall layer: the loglaw layer appears. The growth of the log-law region in between the wall layer and the
law of the wake during the decelerating phase of oscillating flow is seen clearly in
Figure 5 where the original data of Akhavan et al. has been rearranged.
The evolution of the three layers (wall buffer layer, log-law layers and Cole’s law of
the wake layer) on a flat plate with increasing Reynolds number can be calculated
from published velocity profiles using one of several methods that are mutually
compatible (Trinh, 2010a) as shown in Figure 7.
2.4
Characteristics of the present view on transition
Since the seminal paper of Reynolds (1883), many researchers have approached
transition as “the issue of how and why the fluid flow along a circular pipe changes
from being laminar (highly ordered in space and time) to turbulent (highly disordered
in both space and time) as the flow rate increases” (Kerswell 2005). This view that
turbulence breaks down the previously laminar flow field is widely held. For example
Bayley et al. (1988) state that “The process by which turbulent flow develops and
replaces laminar flow is known as (turbulent) transition”.
The analysis in section 2.2 suggests however that the solution of order H 0 does not
disappear from the flow field even as it evolves from a two component to a four
component local instantaneous velocity. If we define laminar flow as one that only
involve exchange of viscous momentum between the wall and the fluid, which would
16
include unsteady oscillating laminar flow observed for example by Akhavan et al.
(1991), rather than focus narrowly on steady state laminar flows described by the
Hagen Poiseuille solution for pipes or Blasius solution for flat plates, then the
persistence of the solution order H 0 is fully compatible with the well known
observation that the laminar flow state (which exists for all flow rates) is linearly
stable to any infinitesimal disturbance. The inevitable conclusion in my mind was that
the development of turbulence does not break down or replace the viscous (laminar)
layer represented by the solution of order H 0 but actually transforms the previously
quasi inviscid flow outside that layer. This is a startling departure from the approach
pioneered by Reynolds. His experiment and others appear to indicate that indeed the
laminar flow field is completely disrupted and replaced at higher Reynolds numbers
by a turbulent apparently chaotic flow field. For example, Mullin and Peixinho
(2006) give a clear illustration of the effect of a turbulent puff on the flow field
(Figure 6).
Figure 6. Development of a perturbation where a small amount of fluid was
impulsively injected into the bulk flow through a 3mm hole. Re = 2000 and in the
snapshot sequence (a)–(f) the perturbation decays. Images taken at 1, 2, 3, 6, 14 and
30 diameters downstream from the injection point using a camera travelling at the
mean speed of the flow. In the sequence (g)–(l) a slightly larger amount of fluid was
injected and a puff is created.
17
How does one reconcile these two views? The first clues to this perplexing issue came
when I started to estimate the thicknesses of the different layers in various turbulent
flows in 1986-7 and eventually compiled Figure 7 reproduced from (Trinh, 2010a).
The reader should note that the normalised thickness of the buffer layer (the timeaveraged value of the wall layer thickness) grows up to the critical transition
Reynolds number then remained constant (Figure 7c). Many authors have shown that
the velocity profile in this layer is well described by the Stokes solution in equation
(29) (Einstein and Li, 1956, Meek and Baer, 1970, Trinh, 2005, Trinh, 2009a).
However since the normalised thickness of the boundary layer grows significantly
upon transition the solution of order H 0 becomes a smaller proportion of the entire
flow field. Since the wall shear stress W w and the shear velocity u*
W w U increase
rapidly after transition, a constant normalised wall or buffer layer results in a fast
decreasing physical thickness for this solution of order H 0 . In pipe flow for example,
this solution occupies the entire pipe up to Re
2100 then diminishes in physical
thickness dramatically with Reynolds number as shown for pipe flow in Figure 8.
Thus an observer or probe situated at the location Ob in Figure 8 may feel as though
the flow at that location has broken down to smaller turbulence scales but in fact the
solution of order H 0 has not been altered; it has simply moved out of the field of
vision of the observer.
The shape of the interface between the wall layer and the log-law layer in the vicinity
of the critical transition Reynolds number is quite similar to the shape factor for a flat
plate boundary layer as shown in Figure 8.
18
Figure 7 Representations of flow regimes (a) Reynolds flow regimes, (b)
Representative velocity traces (after Schubauer and Skramstadt (1947), (c) Additive
layers in normalised dimensions, (d) Physical thickness of layers (not to scale) and
path of ejections in transition and fully turbulent flow.
19
There is however a fundamental difference between the Hagen Poiseuille and Blasius
laminar flows which are steady state solutions of order H 0 for tQ o f and the wall
layer flow which is an unsteady state solution of order H 0 with a much shorter time
scale. The streaming flow expels fluid under the low-speed streaks when their kinetic
energy has reached a critical level.
Laminar
Turbulent
(a)
Axis
Ob position of probe or
observer
Wall
Re = 2100
(b)
Figure 8 (a) Variation of physical value of wall layer thickness in pipe flow with
Reynolds number (not to scale) and (b) boundary layer shape factor measured by
Schubauer and Klebanoff as quoted by Schlichting (1960).
20
This disrupts the solution of order H 0 and triggers an inrush of fluid from the
mainstream to start a new sweep phase. In that sense, the flow field is periodically
disrupted and more agitated even though the solution of order H 0 still applies for
most of the wall layer cycle since the sweep phase is the longest.
The time scale of the wall layer is given by the Stokes solution as
GQ
3.78tQ
tQ
u*
(41)
Where
tQ
(42)
Q
Since GQ is essentially constant in the turbulent regime at high Reynolds numbers, so
is tQ as shown by the measurements of Meek and Baer (1971) then tQ decreases as
Re increases because u* increases.
In fully turbulent flow the instantaneous velocity profile has a clear inflexion point
during the burst phase as shown for example by the measurements of Kim et al.
(1971) in Figure 9. Following the work of Lord Rayleigh (1880) this results in a roll
up of the fluid that rushes in to occupy the space vacated by the ejections. The
resulting travelling vortex immediately impresses a finite fluctuation on the low speed
streak in the sweep phase.
Figure 9 Smoothed phase velocity in a bursting cycle according to Kim et al. (1971)
21
By contrast the disturbances that come in from the pipe entrance or the leading edge
of a flat plate can be weak two dimensional waves typified by Tollmien-Schlichting
waves and can only grow above the minimum critical stability Reynolds number
predicted by Tollmien and Schlichting and verified by Schubauer and Skramstadt as
discussed in section 1. There is another way to visualise this critical Reynolds
number.
The log-law may be viewed as a direct consequence of the ejections from the wall. In
particular the Karman universal constant N in equation (40) can be interpreted as the
slope of the shear layers associated with the ejections (Trinh, 2009c, Trinh, 2010c).
As noted in section 2.2 these ejections, which are a defining element of turbulent
flow, arise from the streaming flow \ st part of the solution of order H . The
intersection between the solutions of order H 0 and H represents therefore the critical
point where the periodic fluctuations grow to a level where the solution of order H
can no longer be neglected.
In boundary layer flow past flat plates the turbulent friction factor given by the loglaw has been approximated by Prandtl as (Schlichting, 1960)
0.029
Re 0x.2
f
(43)
Its intersection point with the Blasius solution for laminar boundary layer flow
0.0664
Re 0x.5
f
(44)
occurs at Reynolds number Re x
34,000 or Re G 1 , s
G 1U f Q
330 which is of the
same order of magnitude as the result in equation (7).
For pipe flow, the intersection between the Hagen Poiseuille and the PrandtlNikuradse correlations
16
Re
f
1
f
4.0 log Re
(45)
f 0.4
(46)
22
occurs at Re s 1040 which is again quite close to “the lowest critical Reynolds
number… Re = 1250…(where there) appears… a saddle node bifurcation” (Eckhart
et al., 2007).
Above this critical stability Reynolds number, the magnitude of the wave disturbances
grow and therefore the term H increases. This means that the streaming function \ st
also increases. The interaction of the main flow with the streaming flow creates a
three dimensional flow field, even if the original boundary layer flow is two
dimensional. This interaction can modify substantially the velocity profile of the
Stokes solution but the effect only becomes noticeable when a wake begins to appear
behind the streaming flow. While it is not possible to predict the point where this
effect occurs in wall shear flow, a similar situation can be analysed quite readily. It is
well-known that surface roughness elements of scale e can disturb the wall layer
profile of turbulent flows ((Nikuradse, 1933) but only when the normalised roughness
scale exceeds a critical value e 5 . Nikuradse explains this limit by arguing that
only roughness elements that protrude from the so-called Prandtl laminar sub-layer
are effective. The modern view of the wall layer process (Figure 2) does not support
Prandtl’s assumption that a steady-state laminar sub-layer exists but it is easy to show
that the time-averaged Stokes solution does give the relation U y for
0 y 4.5 as observed by Prandtl and Nikuradse. An alternative explanation is
based on the development of a wake behind a sphere (Trinh 2009) that occurs at a
particle Reynolds number of Re p
a critical roughness e eu Q
20 (Garner and Keey, 1958) which gives
4.5 .
When the energy contained in the streaming flow is sufficiently high it induces
complete ejection of the wall fluid under the low-speed streaks into the outer region.
This defines, in my view, the critical transition Reynolds number which will depend
on the value of the factor
H = Ue
LZ
§ U e L ·§ Q ·
¸¨ 2 ¸
¨
© Q ¹© L Z ¹
(47)
This parameter is, in my view, a more appropriate criterion than the Reynolds
number. It still contains the Reynolds number when one equates the typical length L
with the diameter D but states that transition is also dependent on the frequency of the
original periodic disturbances.
23
Thus the most appropriate critical transition parameter should be H c . The fact that
there is reasonable agreement between authors on the critical transition Reynolds
number Re c,T is because the pipe entrance in most experimental works is the same
design. As noted before, Ekman (1910) and Pfenniger (1961) delayed turbulence
significantly by minimising entrance disturbances. On the other hand Tam and Ghajar
(1994) hastened the transition by using squared edged entrances (Figure 10). Kucur
and Uzal (2007) showed that Re c,T is dependent on the normalised height of trip rings
included in the pipe entrance.
0.007
0.006
StPr(-2/3)
0.005
0.004
Re-entrant
Square edged
Bell-mouth
0.003
0.002
0.001
0
100
1000
10000
100000
Re
Figure 10 Effect of pipe entrance design on the critical transition Reynolds number.
Data of Tam and Ghajar (1994).
The difficulty in the prediction of the critical transition Reynolds lies in estimating
the position and time where the kinetic energy contained in the terms u ciu c j have
overcome the viscous terms to eject the wall fluid. One way to address this problem is
to observe that the normalised distance y yu* Q is a kind of local instantaneous
Reynolds since it gives a ratio of kinetic to viscous forces. The end of the laminar
regime occurs, according to the representation in Figure 7, when the thickness of the
24
wall layer GQ reaches an asymptotic value. For laminar boundary layer flow on a flat
plate
GQ
GQ u*
Q
GQ U f
Q
f
2
5 Re
1/ 2
x
§ 0.664
¨
¨ 2 Re 1 / 2
x
©
·
¸
¸
¹
1/ 2
At the critical transition Reynolds number of Re x ,c
2.88 Re 1x / 4
(48)
2.5 3 u 10 5 , this value is
GQ 64.5 67.5 and represents the maximum penetration of viscous momentum
from the wall. For pipe flow, the thickness layer of order H 0 is
Ru* Re f
(49)
R
2 Re
2 2
Q
For Re c ,T
order
2100 2300 , Rc
65 67 . In fact this critical thickness of the layer
H 0 is the same for flow between parallel plates and between concentric
cylinders. It appears that it is independent of the flow geometry.
In my view, the distinctive feature of turbulent flow results from the growth of the
solution of order H in the region y ! 65 67 which is only possible when the fluid
in the layer of order H 0 is ejected into the region outside the wall layer. This event
necessarily interrupts the further growth of the solution of order H 0 which starts
again with an inrush of fluid from the main flow to replenish the wall layer. As far as
I know all the evidence shows that the streaming flow is not evenly distributed
throughout the surface but highly localised in space and time as shown in Figure 4.
Thus the inrushes that set up the low speed streaks are also localised and do not affect
significantly the flow pattern in the adjacent regions. This is compatible with the
formation of alternate high- and low-speed streaks first identified by Runstadler et.al.
(1967). Since the normalised time scale tQ of the solution of order H 0 reaches an
asymptotic value (Meek and Baer, 1970);(Trinh, 2009c), the physical time scale
tQ
§ t ·
Q ¨¨ Q ¸¸
© u* ¹
2
(50)
decreases as the Reynolds number increases. Thus the frequency of the streaming
flow increases with the Reynolds number. The phenomena described here have been
can be visualised in the following schematic diagram (Figure 11). In section A, the
Reynolds number, the time tQ required for disturbances of any strength to grow to the
point of ejection is infinite.
25
Re x , s
Section A
Solution
H0
only.
No distur-bance
growth
G
3.5 5.5 u 10 4
Re x ,T
2 3.5 u 10 5
Section B
Solutions of order H and
both apply
Disturbances grow
G
H
0
GQ
Section C
Fluid
ejected
into
y ! GQ
Section D
Log law
has
appeared
G !! GQ
region
GQ
G ! GQ
Law of
the wake
layer
Layer of order H
0
Log-law
layer
Figure 11 Transition of flow regimes along a flat plate
Therefore H is negligible and only the solution of order H 0 applies: the momentum
boundary layer G is equal to the thickness GQ of the layer of order H 0 . In section B,
Re x ! Re x , s and the disturbances begin to grow the streaming flow is still weak and
short lived except in the vicinity of Re x,T where they can disturb the solution of
order H 0 . The data of Rothfus and Senecal (Senecal and Rothfus, 1953) in Figure 12
illustrates that the velocity distribution in pipe flow can depart from the parabolic
profile predicted by the Hagen Poiseuille analysis at Re as low as 1500 but the
friction factor, which is a less sensitive measure, hardly departs from equation (45).
26
Thus we can say that in section B, G | GQ even though both the solutions of order H 0
and H apply. The evidence comes from the proof that the time averaged Stokes
solution adequately correlates all known measurements of velocity profile in the
sweep phase of the wall layer (Trinh, 2010d, Einstein and Li, 1956, Meek and Baer,
1970). In addition we can transform the Stokes solution into the Blasius equation
through an application of Taylor’s hypothesis (Trinh and Keey, 1992b, Trinh and
Keey, 1992a, Trinh, 2010b). The Blasius solution is well known to apply well in the
range Re x , s Re x Re x ,T .
Figure 12 Velocity profile in pipe flow. Re=1500. Data of Senecal and Rothfus
(1953).
In section C , the streaming flow has gathered just enough energy to eject wall fluid
into the region y ! GQ . The ejected fluid is immediately deflected in the streamwise
direction and forms in this visualisation a puff or slug. Immediately after Re x,T the
time scale tQ is still relatively long but the event of ejection triggers an inrush and a
vortex travelling along the wall. The wall process in Figure 2 quickens rapidly with
the Reynolds number because the interaction of the streamwise flow with the
streaming flow requires significant energy. The time scale tQ decreases exponentially
with increasing Reynolds number. A parallel observation is the exponential increase
27
of the time required for the disturbances to decay with increasing Reynolds number
(Peixinho and Mullin, 2006).
In section D, the emergence of the logarithmic law of the wall, linked with the
emergence of the rectilinear portion of the jet path signals the creation of wake
regions where the main stream is broken up and the areas of disturbance merge into a
seemingly chaotic mass.
Figure 13 Flow along a flat plate; turbulence originating from a disturbance of long
wavelength taken by Prandtl (1933) reproduced from (Schlichting, 1960)
The strengthening and increased frequency of the streaming flow with Reynolds
number is shown clearly in the series of photos that Prandtl has taken by following an
original disturbance with a camera travelling along the plate (Figure 13).
3
The critical transition Reynolds number in non-Newtonian
fluids
It is useful to examine the transition process in non-Newtonian and drag-reducing
fluids as the extra dimension allows to further probe different aspects of the problem.
28
3.1
Flat plate
The ratio of kinetic energy contained in the streaming flow to viscous resistance must
be estimated at the local point of ejection. Unfortunately traditional Reynolds
numbers in non-Newtonian fluid flow are based on the averaged wall shear stress.
When the wall layer thickness, the friction factor and the Reynolds number are
expressed in terms of the local instantaneous shear stress at the end of the sweep
phase, the data for non-Newtonian fluids collapse completely onto the Newtonian
curves (Trinh, 2009b). Thus the critical friction factor and Reynolds number are the
same for Newtonian and non-Newtonian fluids when expressed in terms of the
instantaneous shear stress. It was shown that for a flat plate the instantaneous friction
factor was related to its instantaneous counterpart by
f
n 1 f e
(51)
Then
n1
2
f c , power law
f c , Newtonian
3.2
(52)
Pipe flow
For pipe flow is the radius of curvature cannot be neglected and a more relevant
relation is
f c ,non Newtonian
f c , Newtonian
3 nc 1
4 nc
(53)
The term 3nc 1 4 nc appears in the Mooney-Rabinowitsch solution for laminar pipe
flow (Skelland, 1967)as
3nc 1
4 nc
Jw
8V D
Non Newtonian wall shear rate
Newtonian wall shear rate
(54)
Its use in equation (53) is suggested by the fact that the laminar velocity profile
appears as the limiting condition for t o f in studies of unsteady state laminar pipe
flow such as the Szymanski (1932) solution adapted to turbulent flow(Trinh, 2009c).
29
Since the generalised Hagen Poiseuille solution is
f
16
Re g
(55)
Equation (53) results in a transition Reynolds number for non-Newtonian fluids as
§ 3nc 1 ·
Re g ,c 2100¨
(56)
¸
© 4 nc ¹
There are very few experimental studies dedicated to the transition process in nonNewtonian fluids although interest is growing e.g. (Draad et al., 1998, Rudman et al.,
2002, Malasova et al., 2006, Mullin and Peixinho, 2006, Peixinho et al., 2005). Most
authors determine experimental values of Re c from published measurements of
friction loses. We can use either a plot of wall shear stress W w against the flow
function 8V D (Ryan and Johnson, 1959) or the friction factor vs. Reynolds
number as shown in Figure 14. The critical Reynolds number is often taken as the
intersection of line (1) representing laminar flow and line (2) representing fully
turbulent flow but this underestimates Re c . A better estimate is given by the
intersection of line (3) representing transition flow and line (2).
100
(2)
Predicted Rec
fReg
(3)
(1)
10
100
1000
Reg
10000
Figure 14. Estimate of Re g ,c . Data of Dodge(Dodge, 1959). 3% Carbopol, n
100000
.617
(1) laminar regime (2) turbulent (3) transition.
Unfortunately transition flow varies with the entrance conditions as discussed in
section 2. Many published works do not even include data points in that region.
30
There are also problems arising from difficulties in making accurate and relevant
value of the behaviour index. When the log-log plot of W w vs. 8V D is not straight
line we can follow the generalisation introduced by Dodge and Metzner (Dodge and
Metzner, 1959) and use the local gradient n c at the point of transition. This should be
based on the local instantaneous shear stress at the point of ejection as discussed in
section 2.3 but only the time averaged shear stress is available in published
experimental works. The difficulties in obtaining accurate measurements in nonNewtonian are evidenced in Figure 14 where the laminar flow data should follow the
line f Re g
16 but is actually a sloping line some 10% above the exact theory. It is
worth pointing out that the paper of Dodge and Metzner has been widely quoted in all
subsequent papers dealing with turbulent non-Newtonian flows and
most books
dealing with non-Newtonian flow e.g. (Skelland, 1967, Steffe, 1996; Chhabra and
Richardson, 1999). Given these uncertainties proofs of theoretical predictions of Re c ,
including this one, should be viewed with some reservation pending the availability of
more definitive and dedicated measurements. Nonetheless I have re-evaluated Re c
from the data of (Dodge, 1959); (Thomas, 1960); (Caldwell and Babitt, 1941);
(Carthew et al., 1983); (Güzel et al., 2009); (Malin, 1997) .
Only the data of Dodge uses power law fluids and gives values of n c experimentally.
The data from most other researchers are based on muds and suspensions better
described by the Bingham Plastic or Herschel-Bulkley models. Their laminar flow
data must be reverse engineered to give an estimate of n c at the point of transition. In
comparing the transition in fluids obeying different rheological models, it is more
useful to plot the critical frictional f c against n c than Re c against n c . The predicted
curves of Hanks, Mishra and Tripathi and equation (53) in Figure15 agree
substantially in the range 0.8 n c 1 but then Hanks’ curve shows an exponential
increase with decreasing n c , which is not observed in any real experimental
measurement. Equation (52) does not agree with any of these curves in the range
0.3 nc 1 but is closest to the curve of Mishra and Tripathi.
31
0.016
0.014
Dodge
Thomas
Caldwell
Guzel
Malin
Hanks
Mishra
Equation 53
Equation 52
0.012
0.01
fc 0.008
0.006
0.004
0.002
0
0
0.2
0.4
0.6
0.8
1
1.2
n'
Figure 15. Critical transition friction factor vs. behaviour index n c . Data from Dodge
(1959), Thomas (1960), Caldwell and Babbitt (1941), Güzel et al (2009), Carthew et
a. (1983), Malin (1997).
Predictions from equation (53) agree with those of Mishra and Tripathi down to
nc
0.5 but the predictions of Mishra and Tripathi (Mishra and Tripathi, 1971) tend
towards a limiting critical friction factor
f c | 0.4 while equation (53) tends
asymptotically to f c o 0.4 as n c o 0 indicating that the flow tends to remain laminar
with increasing levels of non-Newtonian behaviour. The Mooney-Rabinowitsch
equation gives (Skelland, 1967)
Ww
§ 8V ·
K c¨
¸
© D ¹
J w
8V
D
nc
(57)
§ 3n c 1 ·
¸
¨
© 4 nc ¹
(58)
The apparent viscosity at the wall is then
nc1
Pw
§ 8V ·
¨
¸
D ¹
©
Kc
§ 3nc 1 ·
¨
¸
© 4 nc ¹
(59)
and
32
P w o f as nc o 0
(60)
Therefore we should see f o 0 .
A different situation is encountered in drag-reducing flow. The viscoelasticity of
polymer solutions slows down the growth of the perturbations and results in thicker
wall layer at the point of transition (Trinh, 2010e, Trinh, 2009c). Similarly, wall
riblets constrain the lateral oscillations of the low-speed streaks and increase the time
scale tQ . As a result, the laminar flow regime is extended. Consequently, the thickness
of the wall layer is increased but the wall layer velocity profile still obeys the Stokes
solution (Trinh, 2009c, 2010f, 2010e). It is easy to show the relationship between the
between the increased thickness of the wall layer and the drag reduction in
viscoelastic fluids. For example using the velocity data of Pinho and Whitelaw (1990)
to determine GQ and U Q and forcing equation (40) together with their U m / V data,
we can predict the friction factor which is compared with their measurements in
Figure 16.
Figure 16. Comparison of calculated and experimental viscoelastic friction factors.
Data of Pinho and Whitelaw (1990)
The agreement is quite reasonable considering the considerable difficulties in the
measurement of accurate velocity profiles in viscoelastic flows. Some time ago
(Trinh, 1969) I proposed that the effect of viscoelasticity could be approached through
33
an analogy of the damping of waves in elastic media. Let ǻy be the shift increase in
penetration of viscous diffusion. From dimensional consideration, the only parameter
pertinent to the wall layer that can combined with this distance shift ǻy to give a time
scale is the friction velocity u* . Equation this time scale to the elastic time scale of the
fluid gives
ª u* º
« ǻy »
¬ ¼
ª ȝa º
«G»
¬ ¼
ª ǻyu* ȡ º
«
»
¬ ȝa ¼
(61)
ª IJw º
«G »
¬ ¼
(62)
Where the ratio W w G is called the Weissenberg number and represent a shift in
dimensionless distance of GQ above the non-elastic estimate. To estimate the value of
Re c,T we need to know how W w G varies with the shear rate J w . This information is
unfortunately rarely available in experimental studies of transition in viscoelastic
fluids.
4
Conclusion
In this visualisation, the laminar boundary layer and the wall layer in turbulent flow
are both associated with the solution of order H 0 for flow with periodic disturbances
of characteristic parameter H . The solution of order H induces streaming flows that
eventually lead to ejections from the wall. Transition begins when the ejections
penetrate the region outside the layer of order H 0 . At the point of transition the
thickness of the layer of order H 0 . GQ , is equal to the normalised pipe radius R .
This approach allows good predictions of the critical transition friction factor in nonNewtonian pipe flow.
5
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