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Applied Mathematical Sciences, Vol. 9, 2015, no. 100, 4957 - 4970
HIKARI Ltd, www.m-hikari.com
http://dx.doi.org/10.12988/ams.2015.56451
Mechanical Model of the Turbulence
Generation in the Boundary Layer
Arkadiy Zaryankin
National Research University “Moscow Power Engineering Institute”
Krasnokazarmennaya str. 14, Moscow, Russian Federation
Andrey Rogalev
National Research University “Moscow Power Engineering Institute”
Krasnokazarmennaya str. 14, Moscow, Russian Federation
Copyright © 2015 Arkadiy Zaryankin and Andrey Rogalev. This article is distributed under
the Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
Abstract
Contemporary commercial software suites for computation of fluid dynamics are
based on numerical solutions of unclosed motion equations proposed by Osborne
Reynolds. Closure of these equations is achieved by the application of various
semi-empirical or phenomenological turbulence theories. However, the
mechanism responsible for the destruction of a laminar (layered) flow has no
physical description so far.
This submission reviews the mechanical model of transition to a turbulent flow
based on Helmholz’s theorem for a generalized motion of a fluid element.
Keywords: turbulence, turbulence model, boundary layer
1 Introduction
The proliferation of commercial software suites for computation of fluid
dynamics (CFD) software suites intended for simulating flows of various media in
various channels has brought about the illusion that almost all important problems
involving flows of liquid and gaseous media can now be solved.
This stance had removed much of the motivation for continued development
of theoretical fluid dynamics, precipitating a wind-down of experimental research
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Arkadiy Zaryankin and Andrey Rogalev
and sometimes prompting outright neglect of any experimental testing for
computational results.
Whenever these results contradicted any sporadic bits of experimental data,
the discrepancy was explained either by suboptimal choice of computation grids
or by constraints of the particular turbulence “theory” in use. The latter
circumstance has led to the emergency of poorly substantiated or completely
unsubstantiated turbulence “theories” over the recent years, yet no one dares to
question whether the initial unclosed set of differential equations does correctly
describe the flow of the working medium at hand.
However it is the validity of the initial assumptions that is ultimately
responsible for the reliability of results of computations.
The issue is further complicated by the fact that flow patterns of working
media flowing through various channels are highly diverse, and hardly any single
set of differential equations could describe it adequately.
Therefore, for example, the flow of liquid or gaseous media through a Laval’s
nozzle features the formation of a laminar or laminar-turbulent boundary layer at
its inlet confusor side. Outside of this layer, the flow, unperturbed by viscosity
forces, appears to be close to the potential flow.
Around the narrow passage section of the nozzle, where the transition from
confusor to the diffusor flow takes place (in case of subsonic velocities), there are
very strong perturbations prompting either a detachment of the flow from walls or
its localized (wall-side) additional turbulization with subsequent drop of pressure
pulsations in the direction of the main flow. If the transition to diffusor flow
pattern is not accompanied by the detachment of the flow from the streamlined
surface, the flow beyond the boundary layer remains virtually identical to
potential flow that, for an uncompressible fluid, is described by Euler’s motion
equations. For laminar boundary layers, high-precision computations can be
performed using Navier-Stokes’ equations or the simpler Prandtl’s equations for
boundary layer. Finally, Reynolds’ equations (or averaged Navier-Stokes’
equations) will hold for the turbulent boundary layer. In other words, the three
various flow patterns necessitate the use of three different sets of equations.
A further complication for the real situation in the “classical” channel at hand
is that, as the flow approaches the narrow section of Laval’s nozzle, it
significantly accelerates locally (close to the wall) in the confusor part, matched
by a similarly intense slowdown of flow as it passes into the diffusor part of the
channel.
As shown in [3], the local acceleration of the flow in the confusor part in case
of a turbulent boundary layer leads to its laminarization (a drop in the
completeness of velocity profile) and subsequent detachment of the boundary
layer from the streamlined surface upon transition into the diffusor flow area.
It is very impossible to describe the boundary layer laminarization process
with the equations mentioned above.
This structural non-uniformity of the flow occurs virtually in any channel, and
no single set of differential equations (such as unclosed Reynolds’ equations)
would suffice to describe it reliably.
Mechanical model of the turbulence generation in the boundary layer
4959
In essence, Reynolds’ equations will not lead to an unambiguous solution even
with a completely turbulent flow. The following is noted in [13] with regard to
that: “Turbulent motion equations always appear as unclosed, therefore problems
of the turbulence theory cannot generally be reduced directly to determining the
single solution of some differential equation (or equations) specified by known
initial and boundary values.”
Nevertheless, almost all commercial software is based on unclosed Reynolds’
equations relying on “novel” turbulence theories for closure.
Their justification in physics is no better than that of classical semi-empirical
turbulence theories put forward by Prandtl, Karman, Taylor, Kolmogorov and
Obukhov.
Therefore, it would be reasonable to recapitulate the physical nature of the
causes of turbulent flows and make an endeavor at representing at least qualitative
pattern of these flows’ origination.
2 The mechanical model of turbulence origination
Numerous experimental studies have identified the major factors influencing
the transition from laminar to turbulent flow in boundary layer [5, 7, 8, and 12].
These include the ruggedness of streamlined surface, flow turbulence at the
exterior of surface layer, and effective longitudinal pressure gradient.
However, as Reynolds numbers increase, even the maximum negative
influence of said factors fails to prevent the development of a turbulent boundary
layer [9, 14, 16, 18, and 19].
This statement is confirmed theoretically by a number of mathematical studies
of Navier-Stokes equations in terms of their resilience to various perturbing
factors [11, 15].
The fundamental problem of fluid dynamics related to the emergency of
turbulent flows is establishing the relationship between additional stresses arising
in the turbulent flow and averaged velocity components. Both classical and
numerous emergent turbulence theories are aimed at identifying this relationship.
At the same time, the physical mechanism responsible for the appearance of
turbulence remains largely hidden. This situation is most accurately described by
G. Schubauer and M. Tchen [17] whose statements are worth citing in full.
“Turbulence arises irrespective of the shape of boundary surfaces. The root
cause is the mechanism that excites the motion of particles in the directions other
than that in which they are sheared. This very mechanism should be sought by us
inside the flow itself.”
“According to contemporary data, the initial spike of turbulence appears
spontaneously as perturbations lead to disintegration of the laminar flow pattern.”
“We will only concern ourselves with sheared flows. This is because only this
kind of flows is capable of bringing forth and sustaining turbulent pressures?”
“If rather the turbulence is discovered in a flow devoid of any notable tange-
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Arkadiy Zaryankin and Andrey Rogalev
ntial pressures, the respective turbulent motions would comprise disintegrating
remnants of sheared flows originating at some point upstream.”
These considerations lead to the following conclusions:
1. The mechanism of transition from laminar to turbulent flow is external rather
than internal to the flow.
2. Turbulence arises spontaneously.
3. Sheared flows are a necessary condition for turbulence.
Based on the foregoing conclusions, it can be postulated that the existence of
two flow patterns in fluids owes itself to the capacity of these fluids to allow a
slight deformation of their liquid elements.
This capacity is reflected in the well-known Helmholz’s theorem that claims fluid
motion (in contrast to solids motion) to be comprised by three rather than two
motion components: translational, rotational and deformational (the latter missing
in the case of motion of solids).
For laminar flow, the translational motion is determined by the velocity vector
c  iu  jv , the rotational motion is determined by the rotational speed
1 v u
xy  (  ) , and the deformational motion of a laminar fluid element is
2 x y
1 v u
determined by the angular deformation velocity  xy  (  ) .
2 x y
Within the boundary layer
v
x
u
, therefore it can be assumed with a
y
relatively small margin of error that  and  are equal to z 
z 
1 u
,
2 y
1 u
, accordingly.
2 y
The change in the transversal velocity gradient in any boundary layer section
causes a change in the angular deformation velocity  for an elementary fluid
element. At the same time, the center of rotation of this fluid element tries to
maintain its original position while the deformational motion displaces the center
of elemental mass from the center of rotation by е . Figure 1 shows a graphical
representation of this process for a flat “liquid” element.
Mechanical model of the turbulence generation in the boundary layer
4961
у
dR
dy
a,b
e
bl
ωz
х
dx
Fig. 1: Liquid element deformation diagram
If, at the initial point in time, the liquid element is shaped as an elementary
rectangle with sides measuring dx and dy , and its center of rotation (the point a )
matches its center of mass (the point b ), then, at any angular velocity (   0 ), the
rotation and deformation will cause the rectangular shape of the element to change
its configuration after the passage of time dt (the new configuration is shown in
Figure 1 with dashed lines). As a result, the center of rotation (the point a )
maintains its position while the center of mass (the point b ) becomes displaced
into the position bl by a distance of e, prompting the emergency of a centrifugal
force dR equal to
dR  dm   2 e
(1)
At low angular deformation velocities  z (smaller lateral velocity gradient
du
) viscosity forces will act to offset the arising eccentricity e , naturally
dy
“balancing” the rotating elementary particles of liquid. In this sense the laminar
flow mode comprises a “balanced” flow condition. When angular deformations of
elementary “liquid” volumes accelerate, molecular friction forces become
insufficient to compensate for the additive force dR , causing the “liquid” element
at hand to depart from its flow line, dragging with it a portion of fluid within the
circle circumscribed by the rotation of the center of mass (point b in Figure 1)
around the center of rotation (point a in Figure 1). The liquid within the circle
with the radius e forms the original vortex core that already rotates in accordance
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Arkadiy Zaryankin and Andrey Rogalev
with the laws for solid bodies while the flow remains laminar. (In a real
three-dimensional flow this is already the original 3D vortex core).
For a laminar liquid flow the elementary mass of liquid dm shown in Figure
1 would be equal to
dm    dxdy  l
(2)
Then, the force dR    dxdy  l   2 e , and sites lying parallel to the x-z plane
will be exposed to an additive tension τ equal to

dR
   e  dyz2
dx  l
(3)
1 u
and assuming the linear value of
2 y
dy  const  e  l , we end up formally with the L . Prandtl’s formula for turbulent
motions in a sheared flow
Considering
that
z 
 du 
  l  
 dy 
2
2
(4)
In this case, the linear dimension l  const  e depends directly on the
angular deformation velocity  z and, as a consequence and in contrast to the
“displacement length” l in L . Prandtl’s formula, it peaks out relative to the
boundary layer near the wall and tends toward zero at the outer boundary of said
layer.
In line with the foregoing and in accordance with M.E. Zhukovsky’s theorem,
the emergent primary discrete vortex cores in the flow are affected by a force
perpendicular to the translational velocity of the liquid. This force promotes
intense ejection of particles from boundary layer areas adjacent to walls, resulting
in a similarly intense traversal motion of fluid particles toward streamlined
surfaces.
When there is a transversal liquid transfer at the wall, the transversal velocity
gradient du
rises sharply, causing the laminar flow mode to be destroyed in a
dy
burst. That is, the turbulent flow is an illustrative example of an unbalanced flow
where a relatively narrow zone close to the walls is the origin of a rather intense
turbulence. The role of this turbulence generation zone has so far been largely
unexplored in scientific literature although it is responsible for and explanative of
a number of known empirical facts.
In particular, studies concerning the effect of external turbulence on the
friction factor have failed to identify any clear regularity as long as the degree of
Mechanical model of the turbulence generation in the boundary layer
4963
external turbulence stood within E0 = 5-6%. Only at E0 > 6% did the factor
increase [2].
3 Shielding feature of turbulence origination area in turbulent
boundary layer
High turbulence level in near wall area of flow damps outside disturbance
considerably. Therefore, it is an original screen, which is defending flowed
surfaces from mentioned disturbance.
Such a pattern is quite natural considering the structure of the boundary layer
described above. The turbulence generation area is a reliable “baffle” shielding
the zone adjacent to walls from external perturbations. In other words, as long as
external turbulence remains smaller than the turbulence generated by the
boundary layer itself, liquid flow conditions immediately at the wall remain
unchanged. These conditions only begin to change when external perturbations
become commensurate with pulsations in the layer adjacent to the wall. This
situation can be rarely seen in practice as any artificially created turbulence will
decay rapidly, declining to within 4-5% of its former magnitude at relatively small
distances from the origin of the turbulence.
Thus, the turbulence generation area screens the wall from external
perturbations, reducing the potential level of dynamic loads.
Protective properties of the boundary layer should be visible not only in the
delay of external perturbations but also in the protection of the external flow
against perturbations progressing from the wall.
The proof of this proposition calls for a review of findings from studies of
boundary layer on vibrating surfaces [10, 20]. In these experiments the surface
where boundary-layer measurements were made has been made to vibrate at a
fixed frequency using an actuator adjustable between 0 and 500 Hz. Findings
from these tests are summarized in Figure 2 showing the velocity profile and the
distribution of relative turbulence degrees in the cross-section of the boundary
layer.
The maximum turbulence in case of vibrations’ absence ( f  0) was used as a
scale value for the degree of turbulence. Quite visibly, perturbations progressing
from the wall only modify the velocity profile in a narrow zone at the wall. The
outer part of the boundary layer remains unchanged at all frequencies. Nor does
the distribution of turbulence degrees across the boundary layer show any
variation.
These findings justify a completely new assessment of the role played by the
boundary layer. Until now, this layer was only viewed as the source of energy
losses and as an immediate cause of detachment of flow from streamlined surfaces.
However, findings reported by us render this description inexhaustive.
The model relied upon to explain the destruction of laminar boundary layer is
in full agreement with experimental data at hand.
For example, Figure 3 (curve #1) plots data [15] illustrative of the nature of
change in the degree of turbulence across the boundary layer.
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Arkadiy Zaryankin and Andrey Rogalev
In the area adjacent to the wall, where the lateral velocity gradient reaches its
maximum, the degree of turbulence similarly peaks out at 8%, declining to zero
over a distance 30% greater than the thickness of the boundary layer (the 30%
decay zone for turbulence generated in the boundary layer).
Therefore, the model at hand has the presence of transversal velocity gradients
in the stream as the necessary condition of turbulence generation. Unless this
condition is met, any turbulence will decline somewhat rapidly under the
influence of molecular viscosity forces. It follows that Reynolds’ equations are
only applicable within the turbulent part of the boundary layer.
Fig. 2: Distribution of relative speed u  u
EE
umax and relative turbulence degree
Emax in the cross-section of the boundary layer
4 Laminar sublayer in a turbulent boundary layer
Now it would be reasonable to address another widespread hypothesis of a
certain laminar sublayer between the streamlined wall and the turbulent boundary
layer.
This hypothesis has been initially put forward to confer physical sense to the
logarithmic velocity profile in the turbulent boundary layer that results from integ-
Mechanical model of the turbulence generation in the boundary layer
4965
rating the equation (4) provided that the turbulent friction tension  T remains
unchanged across the boundary layer while the linear dimension l (the
displacement distance in L . Prandtl’s interpretation) is linearly related to the
transversal coordinate y (l  ky) .
The introduction of the laminar sublayer hypothesis has made it possible to
reject the single boundary condition of zero flow velocity at streamlined surface
(the sticking hypothesis) in favor of a condition calling for the equality of
velocities at the outer boundary of the laminar sublayer and inner boundary of the
turbulent sublayer.
This shifting of the lower boundary of the turbulent boundary layer, besides
appearing rather logical, eliminates a logarithmic peculiarity at a streamlined wall
(at y  0, u   ).
Many papers [1, 4, and 6] on problems of computations in the turbulent
boundary layer justify the linearity of change in displacement path length with
increasing transversal coordinate y by invoking the physical impossibility of
velocity pulsations at the wall, this circumstance being an indirect confirmation
that the laminar sublayer is real.
So, for instance, [15] states that tensions of apparent turbulent friction is the
immediate vicinity of a wall are low compared to viscous tensions of laminar flow.
It follows from here that any turbulent flow behaves mostly like a laminar flow in
the thinnest layer lying immediately close to the wall. Velocities in this narrow
layer – called the laminar sublayer – are so small that velocity forces overwhelm
inertial forces here. This means that no turbulence can arise.
However, a question remains to be answered then: what was measured by
low-inertia pressure sensors mounted on streamlined walls? Absent velocity
pulsations, there could be no pressure pulsations either. However, pressure
oscillograms plotted by low-inertia sensors mounted on walls are consistent with
pressure pulsations of a sufficiently high amplitude. The only attempt to address
this question is made in [10] by invoking turbulent motion of liquid in the
laminary layer (therefore the generally accepted term, “laminary sublayer”,
becomes an inopportune choice). The only similarity with laminar motion is that
the average velocity in this case obeys the same distribution as the true velocity of
a laminar flow would, ceteris paribus. Of course, there is no clear-cut boundary
separating the viscous sublayer from the rest of the flow; the notion of a viscous
sublayer is qualitative in this sense.
These considerations reinforce the physical justification of the turbulence
generation model described above to the detriment of the validity of the laminar
sublayer hypothesis. Any experimental attempts at proving the existence of a
laminary sublayer by measuring velocity pulsations in transversal cross-sections
of the boundary layer directly may better serve as proofs of absence thereof, as
velocity pulsations remain sufficiently high at distances as short as 30  100 m
4966
Arkadiy Zaryankin and Andrey Rogalev
from the wall. In this sense, findings from measurements of velocity pulsations in
the boundary turbulent layer reported in [15] and plotted in Figure 3 are the most
instructive.
Fig. 3: Distribution of turbulent velocity pulsations in a boundary layer on a
lengthwise streamlined plate
All RMS pulsations of velocity components u, v, w within the boundary layer
in this case are related to the average velocity u at its outer boundary (curves #1,
#2, #3). In addition, the curve #4 illustrates the varying correlation of pulsational
velocity components u ' v ' across the boundary layer at in the case at hand and,
consequently, the change of frictional tension in this layer.
Relationships illustrated above testify to a pronounced increase in relative
pulsations of all velocity components toward the streamlined surface up to the
point of physically touching its wall. Notably, longitudinal pulsations defined by
curve #1 in Figure 3 experience the greatest upward gradient near in the area
adjacent to the wall. For laminar flows, this relationship comprises a variation in
'2
turbulence degree Ei in the cross-section of the boundary layer Ei  u
. At
u
a distance y from the wall equal to y  0.0052 (  being the boundary layer
thickness) the measured turbulence peaks out at more than 10%.
Mechanical model of the turbulence generation in the boundary layer
4967
At y  0.0015 i.e. virtually at the wall (at   20mm, y  30m ,
comparable to the ruggedness of a thoroughly machined wall), turbulence stood at
6% (see the callout in Figure 3). The wall is also where the maximum relative
correlation occurs between the pulsational velocity components u and v (curve
#4) determining the turbulent friction tension.
All these findings are in a full agreement with the turbulence model described
earlier whereby the linear dimension l enters the equation (4) determining
tangential tension in a turbulent flow as a function of angular deformation  of
u
liquid particles (  ) . As this velocity reaches its maximum at the wall,
y
correlations of velocities u ', v ' and pressure pulsations similarly reach their
maxima.
In fact, the foregoing data, together with Laufer’s experiments [15],
significantly undermines the practical legitimacy of the laminar sublayer
hypothesis underlying all previous attempts at using L . Prandtl’s formula as a
proxy for the real pattern of variations of several unknown quantities entering this
formula.
5 Conclusions
1. The model of laminar flow destruction in the boundary layer has been
justified physically based on generalized Helmholz’s liquid flow theorem
emphasizing the significant role that the deformational motion of fixed liquid
elements plays in contrast to solid bodies.
2. If an angular velocity vector is present in the liquid, angular deformations
will cause the center of rotation of “liquid” elements to try to maintain its initial
position even as the center of mass would be displaced relative to the center of
rotation, bringing about an elemental centrifugal force working to drag this
element away from its stationary flow line. At small Reynolds’ numbers this force
would be dampened by molecular viscosity forces, while at greater Reynolds’
numbers the “liquid” elements of working fluids no longer hold to their stationary
flow lines, and the flow becomes random (turbulent) in character.
3. It has been shown that the proposed model of transition from a laminar
boundary-layer flow to a turbulent mode makes it easy to arrive at L . Prandtl’s
well-known formula linking turbulent frictional tension with average velocity. In
this case, the linear dimension l going into this formula would be interpreted not
as an agitation path but rather as a value determined by the angular deformation
velocity of “liquid” elements.
Acknowledgements. This study conducted by National Research University
“Moscow Power Engineering Institute” has been sponsored financially by the
4968
Arkadiy Zaryankin and Andrey Rogalev
Russian Science Foundation under Agreement for Research in Pure Sciences and
Prediscovery Scientific Studies No. 14-19-00944 dated July 16, 2014.
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Arkadiy Zaryankin and Andrey Rogalev
Received: July 6, 2015; Published: July 27, 2015