Modeling of the Flow due to Double Rotations Causing Phenomenon of
Negative Pressure
IVAN KAZACHKOV
Department of Energy Technology of the Royal Institute of Technology,
Brinellvägen 68, 10044, Stockholm,
SWEDEN
also with
Department of Information Technology, Physical, Mathematical and Economic Sciences,
Nizhyn Mykola Gogol State University,
Grafska Street 2, 16600, Nizhyn, Chernihivska Oblast,
UKRAINE
Abstract: - This paper is devoted to mathematical modeling and computational experiments of a flow with
negative pressure. A previously unknown class of fluid flow under the action of counter-current centrifugal
forces is in focus. Volumetric forces in a non-conducting fluid can arise from gravity, vibrations, or rotations.
In this paper, we consider controlled variable volumetric forces in a system with two rotations around the
vertical axis and the tangential axis of a horizontal disk rotating around the vertical axis. The study of the
coordinate system during double rotation showed that the double rotation about two perpendicular axes, one of
which moves along a tangential direction to the rotating horizontal disk, is equal to the rotation around the
oscillating axis inclined at some angle to the vertical axis.
Key-Words: - Centrifugal Forces; Double Rotations; Curvilinear Gap; Stretching; Negative Pressure;
Cavitation.
Received: January 14, 2023. Revised: November 18, 2023. Accepted: December 16, 2023. Published: December 31, 2023.
1 Introduction
A boundary value problem is formulated for a
system of partial differential equations (Navier-
Stokes), which describe the motion of fluid inside a
turbine with the subsequent transition of its internal
channel into the cavitation section, which has three
parallel narrow slit cylindrical channels with curved
walls and flow holes between channels (a patent of a
Swedish company United Science and Capital
Sweden AB, [1]). Then differential equation array is
integrated across the narrow channel to simplify the
equations. The resulting mathematical model
allowed analyzing and creating the numerical
method.
Mathematical and computer simulations revealed
an intense oscillating fluid flow in a curved gap with
periodic regions of negative and positive pressure
due to constant and variable centrifugal forces. In
the cross section of the channel, there is an area
close to the outside of the turbine (with cavitator),
where at high speeds of double rotation there is a
stretching of the fluid by significant opposite
centrifugal forces. The latter provides the conditions
for an intense cavitation phenomenon, but due to the
negative pressure, other as yet unknown phenomena
are also possible. Further development of the
mathematical model is planned, as well as
experimental study of these phenomena on the
prepared device.
2 Statement by Flow under Double
Rotations and Negative Pressure
2.1 Description of the System
The device, [1], is based on the principle of the
cavitation process inside the working chamber (with
the turbine at the entrance or without it in another
version) due to the two high-speed independent
rotations in two perpendicular directions.
The complex flow in the rotational channel
placed on the rotating horizontal disk is considered
according to the schematic in Figure 1, [1]. The first
rotation is going around the z-axis as shown in
Figure 1, with the rotation speed Ω. The other
rotation has the rotation speed ω regarding the
tangential axis to the main rotation circle, at the
WSEAS TRANSACTIONS on FLUID MECHANICS
DOI: 10.37394/232013.2023.18.25
Ivan Kazachkov
E-ISSN: 2224-347X
259
Volume 18, 2023
distance R0 from the central axis z (x=y=0) of the
device.
Fig. 1: Double rotating coordinate system for the
device below: around vertical axis z and in channel
around tangential axis to a circle of radius R0
3 cavitators are rotating with the rotation speed ω
installed around the axis z on the distance R0 from
the center, equally distributed by the circle of the
radius R0. Their radiuses are r0.
The centrifugal forces in the turbines (cavitators)
shown in Figure 2 are due to the main rotation (red
color), the forces direct in all points of the flow to
the left in the picture (edge of the main rotation
circle). The centrifugal forces due to the turbine
rotation (black) act along the radius of the turbine.
Therefore, in situation 1 the centrifugal forces act in
opposite directions causing a strong stretch of liquid
(condition for the cavitation). To the left in the
picture (situation 2), both forces act in the same
direction creating the highest pressure (condition for
a burst of cavitation bubbles). In all other places
(e.g. 3 and 4) liquid moves from point 1 to point 2
counter-currently from the top and the bottom of the
turbine.
2.2 Cylindrical Coordinates in the Channel
The rotating coordinate system has the vertical axis
z or shifted from the central axis on some distance
R0 as shown in Figure 1. The rotation is going
around the vertical axis z and also around the axis
tangential to the circle of the radius R0. Intensive
rotation and mixing flow are fascinating phenomena
and may be highly effective in several applications:
engineering, technological, natural processes, [2],
[3], [4], [5], [6].
Many theoretical aspects have been studied for
the diverse rotational flows. Nevertheless, it is still a
problem to learn more in deep for many theoretical,
as well as practical applications. The described
system with double rotations is considered at first in
the world. In the local cylindrical coordinate system
( , , )rz
connected to the channel, the coordinate
surfaces are cylinders
r const
, semi-planes
const
and planes
z const
. The focus of the paper
is on the flow regimes in the gap of two rotating
channels, therefore the coordinate z is now directed
along the axis of the cylinder.
Fig. 2: Schematic of the centrifugal forces in the
turbine
In Figure 1, we used z coordinate only for a
general illustration of the system under
consideration. But in the equations for the flow
inside the turbine, the axis z is directed along the
axis of the turbine, which is tangential to the
rotating disk in the Figure 1.
2.3 Studies of Negative Pressure in Liquid
From the above analysis, the liquid may get into
stretching conditions in some localities, where
unknown dramatic regimes with negative pressure
and cavitation are available. Normally cavitation is
WSEAS TRANSACTIONS on FLUID MECHANICS
DOI: 10.37394/232013.2023.18.25
Ivan Kazachkov
E-ISSN: 2224-347X
260
Volume 18, 2023
supposed to be under depressurization below the
saturation level, due to which liquid starts
vaporizing. Negative pressure corresponds to a local
stretching of liquid that can break the bonds of
molecules causing the cavitation process to be much
less known. A stretched liquid is under negative
pressure. This is the unstable, metastable state of a
liquid, possibly due to the Van der Waals forces of
attraction between the molecules of the liquid: both,
between themselves and between them and the walls
of the vessel.
The gaseous state of a really existing substance,
[7], is a gas that is not described exactly by the
Clapeyron - Mendeleev equation, in contrast to its
simplified model, hypothetical ideal gas. There is
also another classification, according to which a
highly superheated vapor is called a real gas, the
state of which slightly differs from the state of an
ideal gas. Superheated vapor, the state of which
differs significantly from an ideal gas, and saturated
vapor (two-phase equilibrium system liquid - vapor)
does not obey the laws of an ideal gas, [7]. This
phenomenon can be observed in the Torricelli
experiment. Similarly, mercury in the medical
thermometer, after the contact with the body has
ceased, is in a stretched state. Moreover, it is in the
maximum thermometer when the temperature
begins dropping after the maximum, [8].
It is available to stretch the thoroughly cleaned
and degassed water. In experiments, the short-term
tensile stresses of 23-28 MPa were achieved, [9].
Technically pure liquids containing suspended
solids and the smallest gas bubbles cannot withstand
even minor tensile stresses. Nevertheless, this is a
method of raising liquid working in trees, [10]. The
superheated (metastable) liquid heated above its
boiling point causes such specific dynamic
phenomena as explosive boiling due to a stored
heat, instability of liquid-vapor interface, and
formation of a phase transition front in several
regimes, [11].
Water is one of the substances that present
density anomalies, [12], which may cause different
unique phenomena, e.g. cavitation and abnormal
behaviors. The negative pressure despite a long
history of study is still a very little known
phenomena, [13], [14], [15], [16], [17], [18], [19],
[20], [21], [22], [23], [24], [25], [26], [27],
e.g. paper, [11], shows that a high average stress
difference on the interface of phase change is due to
the negative stresses in the interface because the
water belongs to a class of substances with density
anomalies.
The negative pressure region of the phase
diagram proves to be paramount in understanding
the unusual behavior of this class of substances.
Any condensed (solid or liquid) phase can exist in
absolute negative pressure regimes, while the same
is not true for gas phases. Theoretical arguments and
experimental evidence demonstrated this. While in a
gas phase pressure is proportional to density, this
does not necessarily occur in condensed phases. It is
convenient to extend the definition of pressure. In
liquids and solids, pressure ought to be treated as
3x3- tensor P, rather than scalar, [14]. The authors
[12] have shown how the negative pressure region
of the phase diagram proves to be paramount in
understanding the unusual behavior of this class of
substances and in liquids and solids.
Several experiments made during the Royal
Society Meeting, [15], did not provide any
explanation for the experiment because adhesion
and cohesion were not known for them yet. To
generate a very high negative pressure in a liquid
one ought to use extremely small amounts of
sample, [16], [17], [18], [19], [20], [21], [22], [23],
[24], [25]. The boiling of superheated and stretched
liquids has been studied in a series of papers, [28],
[29], [30], [31], [32]. The suppression effect for
cavitation centers of a heterogeneous nature with
low-boiling impurities was discovered
experimentally. The effect of a pulsed electric field
on the limiting overheating of liquid at negative
pressures was revealed. It was shown that for short-
term exposure to an electric field that does not lead
to the formation of a noticeable amount of
electrolysis products, the tension field of 107 V/cm
is not enough to change the temperature of the
limited liquid overheating.
2.4 Negative Pressure due Stretching Liquid
The physical situation described above by Figure 1
and Figure 2 revealed negative pressure oscillations
due to a strong variation of the volumetric forces in
a fluid flow by amplitude and direction. In the
numerical simulation below, for the flow under
double rotations, the amazing features (oscillations
of flow parameters and pressure) from the high
positive to the high negative values we revealed.
3 Mathematical Modelling of the Flow
3.1 Equations of Flow in Double Rotations
The differential equation array for the fluid flow in a
cylindrical coordinate system is as follows, [33]:
( ) ( ) ( ) =0,
u u v w
t r r r z
WSEAS TRANSACTIONS on FLUID MECHANICS
DOI: 10.37394/232013.2023.18.25
Ivan Kazachkov
E-ISSN: 2224-347X
261
Volume 18, 2023
2
2
0
2
00
( cos ) cos 2 cos
u u v u u v
u w r
t r r z r
r R v w
2
2
( ) ( )
()
( ) 2
p u u
r r r r
u u v u
z z r r r
,
2
00
( cos ) sin 2 sin
v v v v w v uv
u
t r r z r
R r w u
2
2
1( ) ( )
()
( ) (2 - ),
p v v
r r r r
v v u v
z z r r r
(1)
2 sin cos
w w v w w
u w v u
t r r z
2
()
( ) ( ) ( ) ,
p w w w w
z r r r z z r r
V
T T v T w T u u v w
c u p
t r r z r r r z
2
()
( ) ( ) ( )
T T T T Qc
r r r z z r r
.
where z is directed along the axis of the turbine.
Here are:
, , , , ,v u v w p T
- density, velocity
vector, pressure, and temperature, respectively, and,
µ,
- dynamic viscosity and heat conductivity
coefficients, cV- heat capacity,
c
Q
- internal heat
generation due to cavitation.
3.2 Simplifications of the Model
We consider flow in a thin gap surrounding the
wavy channel of the turbine rotating around its axis
with a frequency ω so that centrifugal force is acting
by the radius of the turbine; r0 is the radius of the
turbine. We neglect the width of the thin layer
around the turbine in a narrow channel with wavy
walls. The forces are projected on the coordinates r
and φ, with account of the distance from the center
of rotation.
4 Correlations for Curvilinear
Channel
In the gap channel, we can simplify the equation
array (1) due to small changes of the flow
parameters across the thin layer, transforming the
problem from 3D to 2D geometry. The integration is
performed by radial coordinate r across the
channel’s layer from one surface
10 sin r
ar k z
to another surface
21
b
. Here r0 is the radius of
the turbine without a curvilinear channel, a is the
amplitude of the surface wave, b is the distance
between the walls of the channel,
2rdr
is the
integrating element in the cylindrical coordinate
system.
4.1 Dimensionless Integral Correlations
For numerical modeling and simulation, it is better
to use the equations in dimensionless form
accepting the following scales for the velocity,
length and pressure, correspondingly:
0
r
, r0,
2
0
0.5 r
,
From (1), the dimensionless equations are got, [33]:
00
3 2 2 =0
z
vw
b
,
0 0 0 0 2
0
00
44
2
5
42 2 cos 3
v u v u
bv
zb
b w v b
b
2
62 1 sin ( cos ) cos 1a z b
b
,
2
00
0
2
2
2 1 3
5
3 2 1 sin (cos ) sin
vv
p
bw z
a z b
00
2
2
0
0
2
4 2 2 sin
2
13 6 sin
Re
w
b
u b b
v
b a v z
z
, (2)
2
00 0
00
42
55
13 1 sin 1.5
2
vw wb
b u w
z
p
a z b z
00
22
00
22
4 cos 3 sin
2
11
6 2 2
Re
b
u b v b
ww
b
bz
.
Here are:
2
0
Re / ,r
/
.
22
2
2
u v v v w w u
r r r z r r z
22
0
/ 0.5 ,p p r
0
/,z z r
0
/,b b r
0 0 0
/,v v r
0 0 0
/,u u r
0
/,a a r
0 0 0
/,w w r
0,
r
kr
00
/,Rr
WSEAS TRANSACTIONS on FLUID MECHANICS
DOI: 10.37394/232013.2023.18.25
Ivan Kazachkov
E-ISSN: 2224-347X
262
Volume 18, 2023
4.2 The Parameters and Initial Conditions
The Reynolds number for our conditions ( ,
, ) is , which
means the highly developed turbulent flow, so that
the turbulent viscosity must be accounted. If the
“new Prandtl formula” is used, the above expression
can be kept, with the turbulent viscosity coefficient.
The following initial data are stated:
Ω=6000 rpm, γ =1/3, a=1.5 mm, b=3 mm, =0.05,
β=3, α=60, kr=103, =162 kN/m2=162
kPa= 1.62 Bar. The water is supplied to the turbine
by flow rate 1.7 l/s or 1.7 kg/s, so that by 6000 rpm
at the R0= 18 cm it creates the force 30.6 N, which
is for the gap of the turbine by radius 6 cm and
width 3 mm approximately P0=(30.6/1.13)103
kN/m2=27.1 kPa. If the cross-section for the water
supply is open only to the 10% of the total turbine
cross-section, then P0=271 kPa, or 2.71 Bar.
5 Solution of the Equation Array
5.1 Statement of the Boundary Problem
The equation array (2) was solved numerically by
the following boundary conditions:
z
=0,
0
u
=0,
0
w
=-0.042,
0
v
=1,
p
=1, (3)
satisfying the above considered physical situation.
Now all terms in (2) are estimated and the small
ones are omitted compared to the bigger ones, to
simplify the equations. Zero indexes are removed
together with the tildes over the dimensionless
functions, just for simplicity. Then it yields:
2
4 4 4
2 2 2 cos 3
53
vu vu v w v
z
2
62 1 0.5 sin ( cos ) cos 1z
,
2
=- 2
3z
vw
, (4)
2
2
2
2 1 3 3 2 1 0.5 sin (cos ) sin
5
v v p
wz
z
2
2
2
1
4 2 2 sin 3 3 sin
2 Re
wv
u v z
z
,
2
41
2 3 1 0.5 sin 1.5
5 5 2
vw w p
uw z
zz
22
22
4 cos 3 sin
2
11
6 2 2
Re
uv
ww
z
,
where
b
,
0.5a
.
5.2 The Non-Viscous Approximation
Because by the estimation made, the
non-viscous case (due to big inertia forces compared
to the viscous ones) can be analyzed. For the stated
parameters, neglecting
0.05 in (4) compared to
the 1 and similar, simplifies the (4) as follows:
4
=- 3
vw
z
,
2
8 4 4
80 cos 3
5 3 3
vu vu v w v
z
1
240 (3 cos ) cos 1
9
, (5)
221
2 3 (cos 3) sin 4 sin
3 20 3
v v p u
ww
z
,
2
8 3 1
0.01 cos 3 sin
5 2 3 5
vw w p u
uw v
zz
.
But
2
880
5
vu vu v
z
because u<<v, therefore,
from the first equation (5), follows:
22
3 cos 1/3cos / 20 / 45cosv v w
.
Here
/ 20v
and
/ 45cosw
may be of the same order
if w grows dramatically inside the turbine compared
to v. But there are no reasons for this. At the
entrance to a turbine, they are respectively 0.05 and
-0.00093. Therefore, finally, it yields:
22
/ 20 3 1/ 3cos cos 0vv
.
The solution of this algebraic equation yields
2
2 2 2
1 1 1 1
3 cos cos
40 1600 3 40
1 1 1
3 cos cos 2 2sin cos
3 40 2 3
v
(6)
5.3 Phenomenon of Two Counter-Current
Rotational Flows under Double
Rotations
As the correlation (6) shows, there are two counter-
current rotational flows with approximately the
same velocities, which correspond to the analysis of
physical processes, namely the action of the
resulting centrifugal force from the line
=0 to the
=π from the top and from the bottom, both sides
(as shown in Figure 2):
2n
,
11.53
40
v
;
2n
,
11.73
40
v
;
06r sm
18000rpm
62
10 /ms
6
Re 1.08 10
b
22
0
0.5 r
6
Re 1.08 10
WSEAS TRANSACTIONS on FLUID MECHANICS
DOI: 10.37394/232013.2023.18.25
Ivan Kazachkov
E-ISSN: 2224-347X
263
Volume 18, 2023
21n
,
12.08
40
v
; (7)
where
0, 1, 2, 3,...n
The plus-minus in (6), (7),
and later on correspondingly to the counter-current
flows going in the opposite directions from the
plane
=0, symmetrically pushing the fluid from
=0 to
=π.
This is a very unusual complex flow due to the
double rotations, which cause centrifugal forces in
the flow inside the turbine, when all the fluid is
pushed in the cross-section from
=0 to
=π and
along the symmetry axis to the exit of the turbine.
All the time, due to the rotation of the turbine, the
new portion of water is coming from
=0 to
=π.
Then derivative
/v
from (6) and put into (5):
22
3sin sin 2
1
8 2 2sin cos
23
w
z
, (8)
1
22
3sin sin 2 0.042+A
1
8 2 2sin cos
23
wz
,
2n
,
0.042w
;
2n
,
0.042 0.22wz
;
21n
,
0.042w
;
where
1
A
- an arbitrary function of φ computed
from (3),
0, 1, 2, 3,...n
1
A
=0 if no initial
distribution by φ is stated for w. As seen from (8),
the velocity of the component flow along the axis of
the turbine is always in the same direction and the
same amplitude as from the inlet
0.042w
. But at
the upper and bottom points of the turbine (
/2
) it has two components:
0.042 0.216wz
,
0.042 0.216wz
, respectively, which give at z=-1,
correspondingly,
0.174w
and
0.258w
.
5.4 Calculation of the Flow Pressure
From the third equation of the system (5), where it
is possible to neglect the term
2/w v z
comparing
to the
2/v
because w<<1 and
/0vz
, yields
2
1 2 4
(cos 3)sin sin
3 9 3 20 3
wp v u
,
As far as
uw
, we can neglect the term
/ 20u
in
the above equation as the small one comparing to
the
/ 3sinw
, and then omit
4/ 9sinw
comparing to
2/ 3sin
(also because we have no part of the w in
(8), which is responsible for the dependence of w on
). Therefore, it results in
2
12
(cos 3)sin
39
pv
,
where from follows after partial integration by angle
coordinate
, with account of (6), (8):
2
22
2
1
1 1 1
2 2sin cos
3 40 2 3
1cos 0.67 cos
9
p
Cz
(9)
And the total pressure is got as total integral
from the total differential:
.
pp
dp d dz
z
. (10)
Therefore, let us compute also partial integral by
coordinate z using the last equation of the system
(5).
The second partial integral (now it is by
coordinate z) is got according to (10):
22
42
2 2 1 1
cos sin 2 2sin cos
45 3 40 2 3
cos cos 2
2.8 10 8 12
uz
pz
uz z
3
32
2
22
4.7 10 3sin sin 2 1.04 10 ( )
1
2 2sin cos
23
zz C r
+
2
4
22
2
22
3sin sin 2
4.2 10 1
2 2sin cos
23
3sin sin 2
16 0.042 .
15 1
2 2sin cos
23
z
z
(11)
2()Cr
is got from boundary conditions. Finally,
2
22
22
1 1 1
2 2sin cos
3 40 2 3
2 1 1
sin 2 2sin cos
3 40 2 3
p
z
(12)
2 4 2
2 1 cos cos 2
0.67cos cos cos 2.8 10
45 9 8 12
uz uz z
2
4
22
2
32
22
3sin sin 2
4.2 10 1
2 2sin cos
23
3sin sin 2
16 0.042 1.04 10
15 1
2 2sin cos
23
z
zz
WSEAS TRANSACTIONS on FLUID MECHANICS
DOI: 10.37394/232013.2023.18.25
Ivan Kazachkov
E-ISSN: 2224-347X
264
Volume 18, 2023
3
22
12
22
4.7 10 3sin sin 2
1
2 2sin cos
23
3sin sin 2 sin
0.055 ( )
1
2 2sin cos
23
z
z d C z C r
.
The (9), and (11) were inserted into (10) yielding
the total pressure distribution (12) by the
coordinates z and φ. By the coordinate z,
1
Cz
is
easily computed according to the boundary
condition (3), while the pressure distribution at the
inlet to the turbine
2()Cr
is not so simple question,
therefore, we put
2()Cr
=0. Then pressure
distribution in the turbine flow is:
2
22
22
1 1 1
1 2 2sin cos
3 40 2 3
2 1 1
sin 2 2sin cos
3 40 2 3
p
z
(13)
2 4 2
2 1 cos cos 2
0.67cos cos cos 2.8 10
45 9 8 12
uz uz z
2
4
22
2
32
22
3sin sin 2
4.2 10 1
2 2sin cos
23
3sin sin 2
16 0.042 1.04 10
15 1
2 2sin cos
23
z
zz
3
2 2 2 2
4.7 10 3sin sin 2 3sin sin 2 sin
0.055
11
2 2sin cos 2 2sin cos
2 3 2 3
zzd
.
6 Computer Simulation
6.1 Analysis of the Terms in Solution and
Simplification of the Model
In the equation (13), the following terms may be
omitted as the small ones comparing to the other
similar terms:
2
4
22
3sin sin 2
4.2 10 ,
1
2 2sin cos
23
z
4
2.8 10 ,uz
32
1.04 10 z
,
3
22
4.7 10 3sin sin 2
1
2 2sin cos
23
z
.
Then the equation (13) is simplified as follows:
2
22
2
1 1 1
1 2 2sin cos
3 40 2 3
12
0.67 cos cos cos
9 45
p
u
22
22
2 1 1
sin 2 2sin cos
3 40 2 3
3sin sin 2
0.022 1
2 2sin cos
23
(14)
22
2
2
22
3sin sin 2 sin
0.055 1
2 2sin cos
23
3sin sin 2
cos cos 2 1.07 1
8 12 2 2sin cos
23
dz
z
.
6.2 Preliminary Analysis of Flow
Peculiarities
The obtained correlation (14) describes the pressure
distribution in the two flows: up and down of the
turbine from its right side to its left side. At the
entrance there is about 25% loses comparing to the
pressure in front of the turbine. The turbine is
rotating but the liquid is flowing counter-currently
from φ=0 to
from the top and to
from
the bottom of the turbine due to the double
centrifugal forces, which move the liquid both sides
of the turbine to the region
, where from it is
pressurized and pushed along the axis to the exit of
turbine.
The upper sign in the “plus-minus” expressions
of (13) correspond respectively to the upper and
down flows in the turbine. Therefore, we assume
that at the plane φ=0, where the pressures are
subtracting each other and velocities are counter-
current, the liquid is stretched by plus and minus
forces, so that cavitation may happen. But at the
meeting point of the two counter-current flows at
the plane
, the pressure is doubled due to a
meeting of two opposite flows. Thus, according to
the above, from the (13) yields the following
pressure difference at φ=0:
1/ 30 7 / 3 0.05pp
Here, the pressure is 0.56 and the total pressure
(with an account of dynamic pressure) is 2.9. The
dynamic pressures are the same acting in counter-
current directions resulting in zero. Thus, the
opposite flows and negative pressure are good
WSEAS TRANSACTIONS on FLUID MECHANICS
DOI: 10.37394/232013.2023.18.25
Ivan Kazachkov
E-ISSN: 2224-347X
265
Volume 18, 2023
condition for the cavitation. At the upper and
downsides of the turbine
/2
, the pressure is 0
and the total pressure is 3.0. At the external part of
turbine,
, the pressure is -4.11 and the total
pressure, with account of the two meeting counter-
current flows with the dynamic pressures 4.32, is
4.53!
As seen from the above estimation, due to the
counter-current flow by the coordinate φ, the flow
pressure in the turbine is substantially decreased
(depressurized flow) to the side of turbine φ =0,
while it is increased (pressurized) to the opposite
side of the turbine (
). Then up and down of the
turbine (
/2
), it is symmetrically prone to a
slight pressurization.
The pressure difference between the sections
/2
, φ=0 is
0.56p
, and between the
sections
and
/2
it is
4.11p
, so
that flow is intensively accelerated in the last semi-
sphere. We estimate dependence of the pressure on
longitudinal coordinate z, which is negative due to a
liquid flow opposite to the turbine movement by
tangential to the main rotation circle. For the z=-1,
from (14) follows:
2
22
2
1 1 1
1 2 2sin cos
3 40 2 3
12
0.67 cos cos cos
9 45
p
u
22
22
2 1 1
sin 2 2sin cos
3 40 2 3
3sin sin 2
0.022 1
2 2sin cos
23
(15)
22
2
22
3sin sin 2 sin cos
0.055 8
1
2 2sin cos
23
3sin sin 2
cos 2 1.07 .
1
12 2 2sin cos
23
d
Neglecting the small terms in (15) yields
2 2 2
1 1 1 1 cos 2
2sin cos 0.79cos cos
3 3 2 3 9 12
p
2
22
22
3sin sin 2
21
sin 2 2sin cos 1.07 1
3 2 3 2 2sin cos
23
and calculation gives:
φ=0: p=0.82;
/2
: p=-1.97;
: p=-1.43.
The pressure difference between the sections
/2
and φ=0 is
2.79p
, between the
sections
and
/2
it is
0.54p
, so that
the water flow is shifted to a middle section of the
turbine
close to the axis at the exit.
6.3 Computer Simulation of the Flow
The approximate models (6), (8), and (13) were also
implemented for the computer simulation in a wide
range of parameters using the prepared FLEX PDE
computer program. This is compared to the full
model.
As numerical simulations showed, the above
approximate mathematical model is unique in the
point that it allowed revealing the counter-current
flow in a rotational turbine while direct numerical
simulation on the computer does not allow this
because the numerical solution cannot treat
simultaneous flow in two opposite directions (two
different solutions at the same time!). A
combination of the detailed analysis of the
approximate analytical solution and direct computer
simulation is important because it allows revealing
all possible regimes of the device functioning.
In a curvilinear channel, the radial velocity can
be estimated through the following correlation
taking into account the form of the channel:
sinr b a kz
. Thus,
/ cos /u r t ak kz z t
=
cosak kz w
, which yields for a=1.5 mm, k=628,
2
10 2k
, one wave on 1 cm length of
the channel, the following estimation:
cosu w kz
.
Now, accounting (5), and (6), with this
estimation for the radial oscillating velocity, let us
take approximate pressure from (14) neglecting the
estimated small terms and considering separately the
solutions for the upper and the down parts of the
turbine, correspondingly:
221
2 3 (cos 3)sin 4 sin
3 20 3 w
v v p u
wz
,
2 2 2
2
1 2 1 1
sin 2 2sin cos
3 3 2 60 2 3
22
0.67 cos cos cos cos
9 45
p
wkz
22
22
21
sin 2 2sin cos
3 2 3
3sin sin 2
0.022 1
2 2sin cos
23
z
(16)
WSEAS TRANSACTIONS on FLUID MECHANICS
DOI: 10.37394/232013.2023.18.25
Ivan Kazachkov
E-ISSN: 2224-347X
266
Volume 18, 2023
2
2
22
3sin sin 2
cos cos 2 1.07 1
8 12 2 2sin cos
23
z
,
22
11
2 2sin cos
40 2 3
v
;
221
2 3 (cos 3) sin 4 sin
3 20 3 w
v v p u
wz
,
2 2 2
2
1 2 1 1
sin 2 2sin cos
3 3 2 60 2 3
22
0.67 cos cos cos cos
9 45
p
wkz
22
22
21
sin 2 2sin cos
3 2 3
3sin sin 2
0.022 1
2 2sin cos
23
z
(17)
2
2
22
3sin sin 2
cos cos 2 1.07 1
8 12 2 2sin cos
23
z
,
22
11
2 2sin cos
40 2 3
v
.
The equation arrays (16) and (17) for the upper and
down parts of the turbine were solved numerically.
6.4 The Results of Computer Simulation
The results of calculations for regions 0 ≤
≤ π ,
−π ≤
≤ 0 and a few cross-sections of the
channel in the range −3 ≤ 𝑧 ≤ 0 are given below.
By the upper part of the channel according to (16)
are presented in Figure 3, Figure 4 and Figure 5 (x is
assigning the variable
in the graphs).
Thus, flow in the channel at section z=-0.5 is
along the axis of the channel only in a vicinity of
φ=0. In the main volume, it oscillates decreasing the
flow rate along the axis. What is more, with an
account of the sharp pulses close to the opposite
side of the channel (φ=π) it may be concluded that
at this cross-section, at the distance z=-0.5, the flow
rate is nearly zero (just shaking along the axis).
Fig. 3: Initial data w,v,u,p at turbine’s entrance (z=0)
WSEAS TRANSACTIONS on FLUID MECHANICS
DOI: 10.37394/232013.2023.18.25
Ivan Kazachkov
E-ISSN: 2224-347X
267
Volume 18, 2023
Fig. 4: Flow parameters w, v, u, p at z=-0.5
depending on 0 ≤
≤ π (in radians, from 0 to
3.14)
The rotational velocity v has every positive
direction being substantially oscillating. Flow
pressure oscillates a lot being negative in half of the
cross-section.
Figure 5 shows a similar tendency by z=-1 but
the axial velocity has a lower peak close to φ=0
(w=-3.6) but a big negative peak at φ =0.86π (w=-
6).
Fig. 5: Flow parameters w, v, u, p at the cross-
section z=-1 depending on angle 0 ≤
≤ π
Mostly it oscillates in range (-2, 3), and
integrally very little flow rate along the axis,
prevailing oscillation. The rotational velocity v has
everywhere positive direction being even more
oscillating than before, with increase of the
amplitude from φ =0 to φ =π.
WSEAS TRANSACTIONS on FLUID MECHANICS
DOI: 10.37394/232013.2023.18.25
Ivan Kazachkov
E-ISSN: 2224-347X
268
Volume 18, 2023
Fig. 6: Axial and rotational flow velocities in
channel by z,φ (y,x in figure)
Fig. 7: Cross-sectional velocity-pressure by z,φ (y,x)
The flow pressure is higher than at z=-0.5 and
the negative pressure region is bigger (more than
half of the whole section and up to p=-4 around φ
=0.68π). The general tendency of the longitudinal
and rotational velocities is given by both
coordinates in the region in Figure 6, where from is
seen that the most impressive anomalies are at the
end of the channel.
Similarly, the cross-sectional velocity and
pressure in the region are presented in Figure 7,
where it is obvious that the region of negative
pressure is mostly approximately after a distance of
one radius of the channel. It is quite big and the
highest values are at the end of the channel from
φ=π/2 to φ=3π/4. At the beginning of the channel,
the negative pressure is absent. The highest positive
pressure is in the range of approximately φ=0 to
φ=π/4 in the whole channel.
The flow parameters at the exit from the channel
are given for the presented simulation in Figure 8.
Fig. 8: Flow parameters w,v,u,p at exit (z=-3)
7 Conclusion
The developed mathematical model and the
computer simulation have shown the peculiarities of
the thermal hydraulic processes inside the device
under double rotations. The governing parameters
are all interconnected and are sensitive to variation.
After obtaining the first experimental data it is
WSEAS TRANSACTIONS on FLUID MECHANICS
DOI: 10.37394/232013.2023.18.25
Ivan Kazachkov
E-ISSN: 2224-347X
269
Volume 18, 2023
possible to solve the problem of optimization of the
main parameters of the processes and device. This is
the base for the calculation of the optimal
parameters in further industrial construction.
The control parameters are the speed of rotation
and flow rate. As it is seen from the data obtained,
in the turbine, there is an available pressure increase
of up to 5 times compared to the entering pressure to
the turbine. Using the developed models and
computer programs it is possible to perform detailed
calculations and optimization of the processes based
on the experimental data. Express analysis of the
requested parameters is available even in a
notebook.
The most remarkable is that the abrupt increase
of the pressure locally due to the cavitation process
does not influence the whole region but just locally
as a concern to pressure. The velocity components
are growing substantially and oscillating a lot. This
means that such intensive spatial variation of the
flow parameters creates the conditions for the
production of the cavitation bubbles in the internal
part of the turbine where a huge decrease of the
pressure happens, while in the opposite part of the
turbine, the dramatic increase of pressure leads to
explosion of the bubbles causing the intensive fluid
heating.
The mathematical model allows studying the
main regularities of a new class of problems related
to the flow in channels under the influence of
double rotations, in which volumetric variable
centrifugal forces arise. The centrifugal forces create
significant oscillations of the flow parameters,
including areas where the forces are opposite,
resulting in fluid stretching and, as a result, negative
pressure and strong cavitation with a break of the
molecules. This may produce hydrogen from water
or be used for water desalination or purification of
the sewage waters.
For the US&C facility, it was possible to find
simplification of the mathematical model, which
allowed obtaining the numerical-analytical solution
of the problem for further optimization of the
parameters. The future theoretical and experimental
research of the described new direction is focused
on the regimes with dynamic negative pressure and
their influence on the fluid as concerned to
cavitation and disintegration of its molecules in a
flow.
The considered flows may cause different unique
phenomena with applications in energy, chemical
technology, etc.
References:
[1] Ivan Kazachkov, Kujtim Hyseni,
Mohammad Al Fouzan, Yevgen Chesnokov,
Development and application of the device
with double rotation turbine, Proc. 3d Int
Conf on Innovative Studies of Contemporary
Sciences. Tokyo, Japan, 2021, February 19-
21.
[2] Pascal Belleville, Lhadi Nouri, and Jack
Legrand, Mixing Characteristics in the Torus
Reactor, Chem. Eng. Technol., No.15, 1992,
pp. 282–289.
[3] S.T. Wereley & R.M. Lueptow, Inertial
particle motion in a Taylor Couette rotating
filter, Phys. Fluids, 11, 1999, pp. 325.
[4] A. Smits, B. Auvity & M. Sinha, Taylor-
Couette flow with a shaped inner cylinder,
Bull. Am. Phys. Soc., 45(9), 2000, pp. 37.
[5] A. Andersen, T. Bohr, B. Stenum, J.J.
Rasmussen, B. Lautrup Anatomy of bathtub
vortex, Phys. Rev. Lett., Vol.91, 2003, pp.
104502-1−4.
[6] CNET. The gaping vortex in this Texas lake
is big enough to suck in a boat, [Online].
https://www.cnet.com/science/a-vortex-
draining-this-texas-lake-is-like-something-
out-of-a-cartoon/ (Accessed Date: Jannuary
11, 2024).
[7] J.S. Hsieh, Engineering Thermodynamics.
Prentice-Hall, 1993.
[8] Torricelli's experiment. Simple barometer,
PhysicMax, Retrieved 7 December 2016.
[9] Pascal// Encyclopedia of Physics/ Ed. А.М.
Prokhorov. М.: Great Russian Encyclopedia,
Vol. 3, 1992: Magnetoplasma - Poynting's
theorem, pp. 549-550.
[10] Veritasium. The Most Amazing Thing about
Trees, [Online].
https://www.tes.com/teaching-resource/the-
most-amazing-thing-about-trees-6328295
(Accessed Date: Jannuary 11, 2024).
[11] Simcha Srebnik and Abraham Marmur,
Negative Pressure within a Liquid−Fluid
Interface Determines Its Thickness,
Langmuir, vol. 36 (27), 2020, pp. 7943-7947.
[12] R. Attila, Katalin Martinás, L.P.N Rebelo,
Thermodynamics of Negative Pressures in
Liquids, J. Non-Equilibrium
Thermodynamics, January, vol. 23, No 4,
1998, pp. 351-375.
[13] W. Kauzmann, Kinetic theory of gases, W.A.
Benjamin Inc., NY, 1996.
[14] S. Hess, Rheology and shear-induced
structures in fluids. In: Lecture notes in
physics 381, Rheological modeling:
WSEAS TRANSACTIONS on FLUID MECHANICS
DOI: 10.37394/232013.2023.18.25
Ivan Kazachkov
E-ISSN: 2224-347X
270
Volume 18, 2023
Thermodynamical and statistical
approaches. Eds.: Casas-Vasquez J., Jou D.,
p. 51, Springer-Verlag, Berlin-Heidelberg,
1991.
[15] G.S. Kell, Early observations of negative
pressures in liquids. American J. Phys., 51,
1983, pp. 1038.
[16] Q. Zheng, D.J. Durben, G.H. Wolf, C.A.
Angel, Liquids at large negative pressures:
water at the homogeneous nucleation limit,
Science, 254, 1991, pp. 829.
[17] D.H. Trevena, Cavitation and tension in
liquids, Adam Hilger: Bristol, 1987.
[18] H.N.V. Temperley, L.L.G. Chambers, The
behavior of water under hydrostatic tension:
I, Proc. Roy. Soc., 58, 1946, pp. 420.
[19] H.N.V. Temperley, The behavior of water
under hydrostatic tension: II, Proc. Roy. Soc.,
58, 1946, pp. 436.
[20] H.N.V. Temperley, The behavior of water
under hydrostatic tension: III, Proc. Roy.
Soc., 59, 1947, pp. 199.
[21] L.J. Briggs, Limiting negative pressure of
water, J. Appl. Phys., 21, 721, 1950.
[22] A.T.J. Hayward, Measuring the extensibility
of liquids, Nature, 202, 1964, pp. 481.
[23] R.E. Apfel, The tensile strength of liquids,
Sci. Amer., 227, 1972, pp. 581.
[24] S.J. Henderson, R.J. Speedy, A Berthelot-
Bourdon tube method for studying water
under tension, J. Phys. E. Sci. Instrum., 13,
1980, pp. 778.
[25] S.J. Henderson, R.J. Speedy, Temperature of
maximum density in water at negative
pressure, J. Phys. Chem., 91, 1987, pp. 3062.
[26] S.J. Henderson, R.J. Speedy, Melting
temperature of ice at positive and negative
pressure, J. Phys. Chem., 91, 1987, pp. 3069.
[27] J.L. Green, D.J. Durben, G.H. Wolf, C.A.
Angell, Water and solutions at negative
pressure: Raman spectroscopy study to -80
megapascals, Science, 249, 1990, pp. 649.
[28] V.E. Vinogradov, E.N. Sinitsin, N.A.
Kotelnikov, Effect of Subcooling on the
Reaction Force in Discharge of Flashing
Water, Fluid Mechanics-Soviet Research,
vol. 16, No. 5, 1987, pp. 39-41.
[29] V.E. Vinogradov, S.V. Gissatulina, Ye.N.
Sinitsyn, Maximum Tensile Forces in n-
Pentane, Fluid Mechanics Research, vol. 21,
No.4, 1992, pp. 50-54.
[30] V.E. Vinogradov, P.A. Pavlov, Liquid
Boiling-up at negative pressures, Proc. of the
Int. Symposium on the Physics of Heat
Transfer in Boiling and Condensation, M.,
1997, pp. 57-60.
[31] V.E. Vinogradov, P.A. Pavlov, Limiting
superheat of aqueous solutions at negative
pressures, NATO science series: 2: 42 Math.,
Physics and Chemistry, Vol. 84: Liquids
under Negative Pressure, 2002. pp. 13-22.
[32] V.E. Vinogradov, P.A. Pavlov, Cavitation
strength of water solutions, J. of Engineering
Thermophysics, Vol. 11(4), 2002, pp. 353-
363.
[33] Ivan V. Kazachkov, Mathematical Modelling
and Computer Simulation of the Flow in
Thin Gap Channel due to Alternating
Volumetric Mass Forces, WSEAS
Transactions on Fluid Mechanics, Vol. 15,
2020, pp.202-212,
https://doi.org/10.37394/232013.2020.15.20.
Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
The author contributed in the present research, at all
stages from the formulation of the problem to the
final findings and solution.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
The article was funded by Dr. Mohammad Al
Fouzan and company US&C Sweden AB.
Conflict of Interest
The author has no conflicts of interest to declare.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en
_US
WSEAS TRANSACTIONS on FLUID MECHANICS
DOI: 10.37394/232013.2023.18.25
Ivan Kazachkov
E-ISSN: 2224-347X
271
Volume 18, 2023
