The study of turbulent jets is of considerable practical and
theoretical interest [1,2]. When developing new methods for measuring
turbulence characteristics in liquid and gas flows, these objects were widely
used for testing equipment, representing a variety of flow modes. A large
volume of these data obtained by laser Doppler anemometry (LDA) is presented in
the traditional format in [3]. Some interesting results were published in
[4,5].
The model of a turbulent axisymmetric submerged jet with a profiled
nozzle was used also for testing equipment and for developing a method of
fiber-optic anemometry (FOA) [6-9].
The LDA method is widely represented in the scientific literature,
which cannot be said about the FOA method. Therefore, it would be useful to
briefly outline the principles and features of the work of the FOA.
The scheme of the projection-type FOA is shown in Fig. 1. The
measurement principle is based on photographing with a long exposure time the
image of the luminous end of the fiber-optic sensor introduced into the
investigated region of the turbulent flow.
Fig. 1. The scheme of a projection-type OVLA: 1 – a semiconductor
laser; 2 – a focusing optical system; 3 – an optical fiber; 4 – a frame of a
sensitive element; 5 – a sensor (a sensitive element of an optical fiber); 6 –
the flow under study; 7 – a projection lens; 8 – a screen; 9 – a digital camera
with an interference filter.
Photographing is carried out with a large averaging time compared to
the time scale of turbulence. The resulting picture is a blurred image of the
radiating end in the form of a spot with a brightness distribution
corresponding to the relative residence time of the end in this area. After
excluding distortions associated with the final dimensions of the end face,
this allows us to estimate the two-dimensional probability density of its
displacement.
Using Mathcad
programming tools, the photographs were represented as a matrix of values of
the relative probability of displacements of the sensor end. The following
statistical parameters were calculated using standard algorithms.
The moments of
the 1st and 2nd orders and the correlation coefficient
Rxy:
,
,
,
,
,
.
Moments of the 3th
order and related dimensionless shape coefficients (asymmetry coefficients):
,
,
,
,
,
,
,
.
Moments of the
4th order and related dimensionless coefficients of the form (kurtosis
coefficients):
,
,
,
,
,
,
,
,
,
.
All coefficients
of the form are defined in such a way that they are reset to zero for a normal
distribution.
The estimation biases due to
the finite dimensions of the sensor for a symmetric
instrument
function were taken into account according to the formulas:
,
,
,
,
,
,
,
,
,
,
.
In these formulas, the upper index M corresponds to the measurement
results of the primary biased distributions, A – to the parameters of the
instrument
function, T – to the restored turbulence
parameters.
The final dimensions of the luminous end of the fiber-optic sensor
lead to smoothing and blurring of the true picture of the probability
distribution of its displacements. The reconstruction of the true picture in
the general case requires the solution of the Fredholm integral equation, which
in a simplified symbolic form can be represented as
,
where
wM(x,y)
is the measured
distribution; the asterisk on the left side of the equality denotes the
integral convolution of the required distribution
wT(x,y)
and the
instrument function
wA(x,y).
FFT algorithms are used to restore the true
distribution. An example of numerical modeling of the problem of excluding the
instrument function is illustrated in Fig. 2.
Fig. 2. Rainbow cartograms of simulated FOA signals of the
projection type: a) ) – the signal
wM(x,y)
before the exclusion of the
hardware function; b) – the instrument function
wA(x,y)
(the signal from the stationary sensor) c)
– the reconstructed signal
wT(x,y).
The higher moments of the distribution
wT(x,y)
of interest
can be determined directly from the reconstructed signal and compared with the
results of calculations using the formulas of the previous section. This
allowed us to test the developed algorithms and programs. The use of relations
between moments is more preferable, since the reconstruction procedure is
unstable if there are even insignificant noises in the measured distributions.
The hydrodynamic part of the experimental device is schematically
shown in Fig. 3. It allowed creating a jet axisymmetric flow in a water-filled
cell with transparent walls. The initial velocity at the outlet of the nozzle
was
U0=5
m/s and was controlled by a
pressure indicator (Pitot tube). The nozzle diameter was
d
=3 mm, which
corresponded to the turbulent flow regime with the Reynolds number for the
nozzle diameter Re=15000. A sensor with a diameter of 120 microns and a length
of 7 mm was used. The Reynolds number for the sensor diameter at the flow rate
U0=5
m/s was Re=600.
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Fig.
3. Hydrodynamic circuit: 1 – a pipe with a profiled nozzle for forming a
fluid flow jet; 2 – a Pitot tube; 3 – a cuvette filled with water, 4 – the
pump.
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An example of the results of FOA measurements in the transverse
axial section of a turbulent jet is shown in Fig. 4.
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7
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5
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4
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3
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1
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3
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4
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7
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Fig.
4. An example of the FOA of measurements at the points of the cross-section of
a turbulent jet
(a
rainbow color scale is used).
The images shown
in Fig. 4 can be interpreted as a convolution of the two-dimensional
probability density of the flow velocity at a given point with the instrument
function of the fiber sensor. According to the formulas presented in the above,
it is possible to calculate the two-dimensional moments of turbulent
pulsations.
The convolution
of a circular function and a two-dimensional Gaussian function was used as a
three-dimensional model of the average velocity field. The parameters were
selected in such a way that in the initial region of the flow, the profile in
the axial section was close to rectangular, and then smoothly transformed into
Gaussian.
,
,
.
The
z
axis is directed along the
flow axis,
r0
is the initial radius of the jet.
As a three-dimensional model of the mean square pulsation axial
velocity
the Gaussian model was used:
.
To model the field of turbulent shear stresses
, the derivative of the
mean flow field along the radius was used.
The type and parameters of the approximating functions
were selected from the
conditions of simplicity and minimum deviations from the experimental data used
[10].
For a visual quantitative representation of the
fields of turbulence parameters, we then use the "Fire" color scale,
shown in Fig. 5.
Fig. 5. The "Fire" color scale.
The results of visualization of the average
velocity field in the axial section of the jet are shown in Fig.6. Figure 7
gives a visual representation of the transformation of the two-dimensional
profile of the average velocity in the cross section along the flow axis.
The field patterns and transverse profiles of the RMS pulsation velocity
are shown in Fig. 8 and Fig. 9.
Fig. 10 shows the field of shear turbulent stresses.
Fig. 6. a) Map of the average velocity field; b) Lines of
the same average velocity; c) Average velocity profiles in transverse axial sections.
Fig. 7. 2D profiles of the average velocity in various
cross-sections of the jet: à)
z/d
=0; b)
z/d
=15;
c)
z/d
=30.
Fig. 8. a) A map of the field of root-mean-square
pulsations; b) Lines of equal root-mean-square pulsations; c) Profiles of
root-mean-square pulsations in transverse axial sections.
Fig. 9. 2D profiles of root-mean-square velocity pulsations
in various cross-sections of the jet:
à)
z/d
=2;
b)
z/d
=8; c)
z/d
=15.
Fig.10. a) A map of the shear stress field; b)
Lines of equal shear stresses; c) Profiles of shear stresses in transverse
axial sections.
It should be noted that the presented results
for the root-mean-square pulsations of velocity and shear stresses in the
initial region of the jet
(z/d < 3) are very conditional, since the developed
turbulence has not yet been formed here.
The animations below show pictures of turbulence parameter fields in
various color maps (Fire, Rainbow, Neon, Royal, Topography).
Animation
1.
The average velocity field in
different color maps.
Animation
2.
The field of root-mean-square
pulsations in different color maps.
Animation
3.
The modulus of the shear stress
field in different color maps.
An example of the
measurement results of the shape coefficients is shown in Fig.11.
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Fig.
11. Measurement results in the axial cross-section
x/d
=5: profiles of
the kurtosis coefficients
gXXXX,
gYYYY,
gXXYY.
|
Deviations of the pulsation statistics from the normal law for the
near-axial regions may be due to the presence of coherent vortex structures in
the flow. Intermittency and return flows caused by the peculiarities of the
cuvette geometry and its finite dimensions can affect the peripheral regions.
The influence of methodological errors caused by the
optical-mechanical design of the sensor cannot be excluded.
A method of computer visualization of the fields of turbulence parameters
based on local measurements at individual points of the flow is developed.
Simple parametric models are proposed to describe the fields of turbulence
parameters in an axisymmetric jet stream. The results of quantitative
visualization are presented, which make it possible to calculate the energy
dissipation and other integral characteristics of the flow.
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jets. M.: Fizmatiz. 1978.
2. Belotserkovsky S. M., Ginevsky A. S.,
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vortices, Dokl.
RAS, 345:4(1995),479-482.
3. Smirnov V. I. Laser diagnostics of
turbulence: dissertation ... doctors of Physical and Mathematical Sciences:
01.02.05. - Moscow, 1997. - 258 p.: ill.
RSE OD, 71 99-1/7-8
4. Rinkevicius B. S. Laser Doppler
anemometry. - M.: Knorus. 2017.
5. Loginov A. A. Smirnov V. I. Three-beam
two-channel computerized laser Doppler measurements of turbulence
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Measurement
Techniques. 1996. No. 8. pp. 35-40.
6. Biryukova O. V., Smirnov V. I.
Measurement of statistical characteristics of turbulence by an optical system
with a hybrid fiber-optic sensor. //
Measurement
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and Technical Conference. conf. Optical methods of flow research.
Moscow: MEI. 2007. pp. 510-513.
8. Kondakov I. A., Panyukov R. A., Smirnov
V. I. Investigation of the operation of a laser fiber-optic turbulence sensor
in a liquid jet. Proceedings of the XI International Scientific and Technical
Conference. conf. "Optical methods of flow research" Moscow, June
27-30, 2011. Electronic version of the proceedings of the conference. 6 p.
9. Kerv Yu. V., I. A. Kondakov I. A.,
Smirnov V.I. Laser fiber-optic measurements of two-dimensional statistical
characteristics of turbulence of the 3rd and 4th order \ \ Optical methods of
flow research: XIII International Scientific and Technical Conference
[Electronic resource]: proceedings of the conference. - Electron. dan. -
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