Hierarchy of Vortexes in 3-D Confined Flow
A. Malahov, Y. Vorobyov, V. Demidenko, A. Kaluev
Odessa State Maritime University, Odessa State Naval Academy
[email protected]
ABSTRACT
In the work there was provided an analysis of the main features of the interaction between the flow, that
is confined by the walls of rigid cylindrical pipe and bodies with different shapes. The main results were
obtained during numerical experiments based on the finite-difference solution of the full 3-D non-stationary
Navier-Stokes system of equations. Is was shown, that in confined spaces in contrast to unrestricted flows
the vortex structures would be arising not only behind the bodies, but also before their frontal surfaces.
Introduction
Particularly almost unstudied and very interesting, for consideration is the problem of the confined
flow streamlining of those bodies, which are located in closed spaces. It's worth to be underlined, that
questions connected with the interaction of the flow with the body in non-restricted spaces in all details
were discussed in the tens of thousands of scientific publications. In full contradiction to this fact we can
establish, that completed theory about streamlining of the body by the flow, which is restricted by the
arbitrary walls, in fluid mechanics literature at the present time is absent. Thus, for example, it's unknown,
in which area the flow separates from the surface of the body or what kind of geometrical shape has the
near-wake behind it. Nevertheless we can say easily, that results, which obtained in solution of the very
class of problems are fundamental. Practically without changes in the mathematical formulation of the
problem in the first approximation it is possible in full volume to produce the modeling of the work of the
artificial valve of the human heart. In the same way it is possible to produce the modeling of the work of the
jet automatics of the supersonic flying or space aircrafts or to solve the problem about the motion of the
shock wave inside the explosive material with the foreign insertion located inside. So, as we can see,
despite of the polar areas of its application from the hydromechanical point of view these problems are
almost identical.
In the following, as an example, we'll consider the problem about the streamlining of the plain
bodies, which are located inside the tough and unmovable cylinder. The common scheme of this problem
presented in figure 1. It should be noted, that problem formulation and received results are universal and
correct in other, more common cases. From the begining the problem is considering as a three dimensional
and the non-stationary. Finally we are interested in the field of the flow, which will be received after the
long period of time. It's worth to point out, that in the very class of the problems because of the dissipation
of the energy in the described processes it is impossible to receive the stationary or quasi-stationary flows.
Wall
Calculation unit
common scheme
(a)
discretization of the flow
(b)
FIGURE 1. Internal fluid dynamics problem.
CP663, Rarefied Gas Dynamics: 23rd International Symposium, edited by A. D. Ketsdever and E. P. Muntz
© 2003 American Institute of Physics 0-7354-0124-1/03/$20.00
Problem Formulation And Main Results
The problem, has been solving with the usage of numeric method. The discretization of the
calculation volume has been provided according to the principle of equal laying out. In this case the shape
of the cross section of the pipe is not principal. All calculation volumes, as it is shown at the figure 1-(b)
were located on the concentric circles, that overspreading equivalent areas or volumes, if we are talking
about the length of the calculation space. In such a way during an approach to the rigid walls (to the zone of
the flow with a high gradients of velocity) we automatically receive thickening of the calculation mesh, that
is well observed in figure 2. Along the length of the modeling volume the calculation sections were located
with a constant step. As a starting equations we have used the non-stationary Navier-Stokes equations,
closed by a continuity equation [1]. But in this formulation with consideration of initiary and boundary
conditions it's quite difficult to receive the connection between the velocity and the pressure and also in this
case very often are arising great difficulties, connected with the stability of the calculation process. Because
of this reason in the quality of closing equation there was used the von Neumann condition. For getting rid
of the calculation instability we have used split method. The starting three dimensional system of the nonstationary Navier-Stokes equations in the aggregate with the von Neumann condition [2] were divided both
in dimension and in time to three systems. The solution of the problem was received with the second order
of accuracy. The calculation scheme was explicit. The time step was equal 0.001.
All results, which in the following will be considered are answering to numerical experiments that
were modeling streamlining of the disk and the rectangle, which were displaced at one percent in relation to
the axis of symmetry of the calculation area. In all pictures the flow moves from left to the right. The main
result, which has been received in numerical experiments shown at figures 3-4. As much as the time grows
before the frontal surface of the streamlining body there has been observing the beginning, development
and following stable existence of the very complex vortex area, whose character of the behavior could be
described as hierarchical. For this zone there has always been observed the time-dimension invariance in
relation to the small disturbances.
50
Y, mm
40
30
20
10
0
-10
-20
-30
-40
-50
-50
X, mm
-40
-30
-20
-10
0
10
20
30
40
50
FIGURE 2. Numeric mesh in the cross cut of the calculation volume.
.
∂Vi
1

+ ( V j ∇ )V i = −
gradP + ν ∇ 2 V i 
∂t
ρ

divV = 0
∂Vi
1
(1).






+ ( V j ∇ )V i = − gradP + ν ∇ 2 V i
∂t
ρ
∂V y ∂V
∂V y ∂V
 ∂V ∂V y ∂V y ∂V

x
x +
z −
z + ∂V z ∂V x − ∂V z ∂V x 
∇ 2 P = 2⋅ρ 
−
 ∂x ∂y
∂x ∂y
∂y ∂z
∂z ∂y
∂z ∂x
∂x ∂z 

FIGURE 3. Isolines of the vorticity field of the flow passing by a disk. Re=500, t=378,7 s.
(2).
t = 378,7 s. Re = 500
(a)
t = 17 s. Re = 104
(b)
FIGURE 4. Vortex zone before the disk.
At figure 3 presented the plain cross-cut of all calculation volume in the horizontal section of the
pipe. This field of vorticity answers to the streamlining of the disk. An analysis of received results points
out onto the fact, that before the disc there exists a stable system of spatial ring type vortexes, which are in
constant interaction between each other. From the concerned vortex area there takes place the emission of
the vortex formations, that are passing away downstream by the flow. At the figure 3 one of the very
passing away vortexes located between the cross-sections 4.8 and 5.2. In the figure 4 (a-b) as a single area
shown the received system of vortexes. Even by sight it could be seen, that despite of differences in
Reynols parameters, that are equal 500 and 10000 accordingly, the discovered phenomenon is universal.
The difference is usually enclosed in the quantity and the shape of the vortexes only, and also, in the time of
their development.
At the figure 5 presented the field of lines of the flow near the streamlining disk when the time
moment equals 7 seconds. Upper horizontal line answers to the wall of the pipe. Here, are well observed
two vortexes, which has already separated from the surface of the streamlining body and it possible to see
as on the smooth frontal surface of the disc there takes place the birth of the third vortex.
FIGURE 5. Internal fluid dynamic problem.
On the base of received results has been formulated the following scenario for the behavior of the
restricted flow: before the body there takes place the begining of the compound area of the spatial vortex
motion. In its development it always creates self subordinate and self similar structures, and it behaves in
hierarchic mode. In that part of the flow, which answers to the area of the middle cross-section of the
streamlining body with the start of the flow motion there takes place particularly instantaneous rising and
during a certain and quite little interval of time the development of three dimensional vortexes. When these
vortex structures achieve dynamically stable condition during the following time moments there will be
observing their rotation in contrary directions. At the same time the cross cut dimensions of the vortexes are
staying non variable. The proposed scenario for the change of the flow topology is universal, because as it
could be seen at the figure 6 the mechanism of the streamlining and the character of the vortexes behavior
in time are not changing with the co-axial addition of the second body, which is identical to the primary
one. The flow endures the main changes in the rear part of the streamlining bodies only, so there, where is
observing the decreasing of the stagnant area of the flow. One of the application results of the very
decreasing of the stagnant area in the rear part of the fairing is a decreasing of the energy loss while the
flow passing by a body, that is fixed inside a pipe. The rarefaction, which appears inside the internal disk
space results in redistribution of the vorticity in the compound vortex area before the first fairing and the
total vorticity of the considered vortex area in this case should be decreasing.
Quite untraditional result is a character of the restricted flow separation from the streamlining body.
In particular as it well observed at the figure 7 during the streamlining of the disc the separation starts not at
the first line of the break of the surface contour, but a bit lower, that is directly on the frontal surface. This
result sometimes takes place in a usual physical experiment.
At the figure 8 presented the dependence of the drag coefficient on a Reynolds number when the
time moment equals two hundred seconds. Received changing of the drag coefficient is a result of the
existence in the flow such a called "Lateral Bernoulli forces", which ones in some cases of the motion of the
body inside a pipe bring to appearing of the eccentricity, and sometimes even to the rotation of the body.
Exactly the influence of the lateral forces in a case of the laminar motion of the neutrally floating solid
particles inside a cylindrical pipe is the reason of their displacement from the center of the pipe to the
direction of the walls and contrary, that finally results in their concentration in a certain circular area located
approximately in the middle between the pipe axis and its walls. This fact is invariant in relation to the
initial picture of the flow and to the primary location of particles inside a pipe.
FIGURE 6. Isolines of the vorticity field of the flow passing by a single disk and
a tandem of two thin disks. Re=50000, t=33,8 sec
FIGURE 7. Separation of the flow.
FIGURE 8. Change of the drag coefficient of the disk (thickness 3 mm)
with increasing of Re number of a flow.
Inasmuch as in every numerical experiment there always arises the question about the correctness of
results received in the end, then there is a sense to consider in more details the testing criterions. In all
experiments the relation of the mesh and the full Reynolds parameters didn't exceed the value equals 0.15.
At figure 9 with enlargement in one million times we can see as in time is changing the maximal
value of the error for the solving equations. And finally, the most tough criterion is an integral of an error in
time in every calculation unit related to the flow velocity in this calculation unit. The modulo of it should
always be less then one.
t

0

(3)
∆ =  ∫ ε i ⋅ dt  / Vi << 1
From the point of view of the physics of the process this estimation answers to the account of the influence
of the artificial force, which appears because of the discretization of the continuous media, and also because
of the errors of the differential scheme and truncation errors of the computer. As it could be seen from the
figure 10 for all calculation area this parameter attains large values in the area of the vortex zone only, and
its maximum reaches the value, which equals 0.57. This chart answers to the data of the figure 3.
The first equation.
(a)
The second equation.
(b)
The third equation.
(c)
FIGURE 9. Maximal error in solution of the Navier-Stokes equations.
FIGURE 10. Distribution of the error in solution of the Navier-Stokes equations.
Conclusions
1.
2.
The discovered phenomena is universal in relation to such two dynamical parameters as time t and
Reynolds parameter Re.
The character of the behavior of the described vortex zones possess a certain analogue of the
"kinematic memory". The main characteristics of the received vortex area are hierarchic structure and
also the dimension-time scale invariance in relation to small disturbances.
REFERENCES
1. Malahov, A. V., The Hydromechanics Of Confined Flows, Astroprint, Odessa. UA, 1999.
2. Yanenko N. N. Method Of Fractional Steps For Solution Of Multi Dimensional Problems Of Mathematical Physics.
Nauka, Sybirian department., Novosibirsk, 1967.
3. J. Happel, H. Brenner., Low Reynolds Number Hydrodynamics With Special Applications To Particle Media,
Prentice-Hall, 1965.