89
2011,23(1):89-94
DOI: 10.1016/S1001-6058(10)60092-3
THE CALCULATION OF THE PROFILE-LINEAR AVERAGE VELOCITY
IN THE TRANSITION REGION FOR ULTRASONIC HEAT METER BASED
ON THE METHOD OF LES*
LIU Yong-hui, DU Guang-sheng, TAO Li-li, SHEN Fang
School of Energy and Power Engineering, Shandong University, Jinan 250061, China,
E-mail: [email protected]
(Received October 31, 2010, Revised December 21, 2010)
Abstract: The measurement accuracy of an ultrasonic heat meter depends on the relationship of the profile-linear average velocity.
There are various methods for the calculation of the laminar and turbulence flow regions, but few methods for the transition region.
At present, the traditional method to deal with the transition region is to adopt the relationship for the turbulent flow region. In this
article, a simplified model of the pipe is used to study the characteristics of the transition flow with specific Reynolds number. The
k − ε model and the Large Eddy Simulation (LES) model are, respectively, used to calculate the flow field of the transition region,
and a comparison with the experiment results shows that the LES model is more effective than the k − ε model, it is also shown
that there will be a large error if the relationship based on the turbulence flow is used to calculate the profile-linear average velocity
relationship of the transition flow. The profile-linear average velocity for the Reynolds number ranging from 5 300 to 10 000 are
calculated, and the relationship curve is obtained. The results of this article can be used to improve the measurement accuracy of
ultrasonic heat meter and provide a theoretical basis for the research of the whole transition flow.
Key words: ultrasonic heat meter, Large Eddy Simulation (LES) model, profile average velocity, linear average velocity, pipe
Introduction
At present, the technology level of the
homemade ultrasonic heat meter is not high and the
flux measurement accuracy can not meet various
requirements. The measurement accuracy is mainly
depend on two factors : (1) the signal collection by the
electronic component, (2) the relationship of
profile-linear velocity of flow. Its flow measurement
principle is shown in Fig.1.
The ultrasonic Transducer A (TRA) and
Transducer B (TRB) not only can emit the ultrasonic
impulse (with an incident angle θ ), but also can
receive the ultrasonic signal. Because the ultrasonic
* Project supported by the National Natural Science
Foundation of China (Grant Nos. 10972123, 10802042), the
Natural Science Foundation of Shandong Province (Grant No.
Y2007A04).
Biography: LIU Yong-hui (1981-), Male, Ph. D. Candidate
Corresponding author: DU Guang-sheng,
E-mail: [email protected]
velocity in the downstream and upstream is different,
the time for the ultrasonic signal to reach the two
ultrasonic transducers is different, with a time
difference Δt . The linear average velocity vt is
obtained by capturing this time difference
vl =
Δtc 2
2 D ( tan θ )
−1
(1)
where D is the diameter of the pipe.
Fig.1 Flow measurement principle of the ultrasonic heat meter
90
The volume flow Q is
Q=
πD 2 kvl
4
(2)
where k is defined as the flow coefficient, which is
the ratio of the profile average velocity vs to the
linear average velocity vt .
k=
vs
vl
(3)
In order to insure the measurement accuracy, the
linear average velocity vl measured by the ultrasonic
flowmeter must be converted to the profile average
velocity vs .
The ultrasonic flowmeter is a much studied topic.
Iooss et al.[1], Raišutis[2], Willatzen and Kamath[3] and
Ashrafian et al.[4] studied the influences of structure,
roughness, velocity-profile, temperature on the
fully-developed turbulent flow, Inoue et al.[5],
Takeda[6] developed a system of ultrasonic flow
measurement by using the Ultrasonic Velocity Profile
(UVP) method and studied the influence of the
incidence angle on the measurement precision, Wu
and Zhang[7], Feng et al.[8] and Li et al.[9] also studied
the turbulent flow in pipes by using the DNS method,
and measured the flow field by the Digital Particle
Image Velocimetry (DPIV) method. The flow
coefficient k is documented in the Industrial
Automation Instrumentation Manual[10]:
When the flow is a laminar flow
k=
3
4
(4)
(
k = 1 + 0.01 6.25 + 431Re
)
−1
1. Physical model
In order to compare our results with the
experimental data of Westerweel[11], the pipe’s
diameter is 0.04 m and the length is 0.5 m in this
article.
The calculation’s stabilization is determined by
the size of the smallest mesh in the flow. According to
Ref.[12], the smallest mesh scale l needs to satisfy the
viscous length scale requirement
l = 5 y0+
(6)
where y0+ is the viscous length scale,
y0+ =
v
u∗
(7)
u ∗ is the friction velocity.
u∗ = u
f
2
(8)
and f is the fanning friction factor,
f = 0.079 Re −1/ 4
When the flow is a turbulent flow
−0.23
supplying system for new buildings, the flux for
100 m2 is less-than 0.25 m3/h, and the Re = 7 500 ,
which belongs to the transition region. In this article,
the characteristics of the transition flow with the
Re = 5 300 in pipes are studied, and the results are
compared with the result of experiments to validate
the calculation. The relationship between the
profile-linear average velocity and Reynolds number
in the range from 5 300 to 10 000 are also obtained
via the numerical simulation, which can provide a
theoretical basis for the research on the whole
transition flow.
(5)
These researches are mainly focused on the flow
measurement of the laminar and turbulent flows, and
not so much on the flow characteristics in the
transition region. The traditional method is to regard
the transition flow as the turbulent flow to obtain the
relationship of the profile-linear average velocity. This
simplified method must lead to measurement errors
when the water flow in the pipe is a transition flow.
There are many conditions that would lead to the
transition flows with supplying heat. For example, to
satisfy the energy efficiency requirement of heat
(9)
So, when Re = 5 300 , f = 0.00926 , u ∗ = 9.0 ×
10 −3 m / s
y0+ = 1.1 × 10−4 m
and
−4
l = 5 y = 5.5 × 10 m .
+
0
Fig.2 Cross section mesh
,
we
have
91
Figure 2 shows the cross section mesh. In this
section, the size function method is used to generate
the mesh, which satisfies the requirement of the
smallest structures mesh. Cooper method is used for
the flow direction. The number of mesh is about
2×106.
2. The calculation model and the validation of
results
2.1 Basic governing equations[13-19]
Continuity equation:
∂ρ ρ∂ (ui )
+
=0
∂t
∂xi
(10)
∂t
+
∂ ( ρ ui u j )
∂x j
=−
∂p
∂
+
∂xi ∂x j
⎛ ∂ui
⎞
− ρ ui′u j′ ⎟
⎜⎜ μ
⎟
⎝ ∂x j
⎠
(11)
(12)
where ρ is the density of water, u is the velocity
of flow, the superscript“ − ”expresses the quantity of
average over time, p is the pressure and μ is the
viscosity coefficient.
2.2 The k − ε model
k equation:
∂t
+
∂ ( ρ kui )
∂xi
=
∂
∂x j
⎡⎛
μt ⎞ ∂k ⎤
⎢⎜ μ +
⎥+
⎟
σ k ⎠ ∂x j ⎥⎦
⎢⎣⎝
Gk + Gb − ρε − YM + S
(13)
∂t
+
C1ε
μ
∂
∂ ⎡⎛
( ρε ui ) = ⎢⎜ μ + t
∂xi
∂x j ⎣⎢⎝
σε
ε
k
( Gk + C3ε Gb ) − C2ε ρ
and σ k , σ ε
are: C1ε = 1.44 , C2ε = 1.92 ,
The boundary conditions: The velocity-inlet is
used for the inlet condition. When Re = 5 300 ,
v = 0.133 m / s , the outlet condition is the free
outflow. The accuracy of calculation: The residual
error is less than 10−3.
2.3 The LES model
The equation of the LES model[20-22] is
∂τ ij
∂ui ∂ui ∂u j
∂ 2 ui
1 ∂p
+
+
=−
+ν
∂t
∂x j
∂x j ∂x j ∂x j
ρ ∂xi
where τ ij = ui u j − ui u j
(15)
is defined as the subgrid
stress, which concerns the momentum transport
between the filtrated small scale turbulence and the
part of the scale which can be calculated. The subgrid
stress is a term not closed in the equation. So, a certain
model is used to deal with the subgrid stress. In this
article, the Smagorinsky model is used
1
3
τ ij = 2μt S ij + τ kk δ ij
(16)
where μt is the subgrid turbulent viscous stress, S ij
is the transform ratio of the tensor
ε equation:
∂ ( ρε )
coefficients C3ε produced by the flotage are
considered for the compressible flow. When the fluid
is incompressible, the user-defined source terms are
not considered, and Gb , YM , Sk and Sε are equal
Cμ = 0.09 , σ k = 1.0 , σ ε = 1.3 .
ρ Cμ k 2
ε
∂ ( ρk )
C2ε and C3ε are constant coefficients, σ k and σ ε
are the turbulent Prandtl numbers related with k and
ε coefficients, Sk and Sε are the user-defined
source terms.
The turbulent kinetic energy Gb and related
Cμ
Viscosity coefficient of turbulence:
μt =
diffusivity of the compressible turbulent flow, C1ε ,
to zero. The values of constant coefficients C1ε , C2ε ,
Momentum equation:
∂ ( ρ ui )
turbulence kinetic energy produced by the flotage,
YM represents the panting action caused by the
ε2
k
⎞ ∂ε ⎤
⎥+
⎟
⎠ ∂x j ⎦⎥
+ Sε
1 ⎛ ∂u ∂u j ⎞
Sij = ⎜ i +
⎟
2 ⎜⎝ ∂x j ∂xi ⎟⎠
(14)
where Gk is the turbulent kinetic energy produced by
the gradient of the average velocity, Gb is the
(17)
This model is the basis of the subgrid model, which is
advanced by the Samagorin, and the model equation is
μt = ρ Ls 2 S
(18)
92
where Ls is the mixing length of the mesh, and
S = 2S ij S ij Cs , Cs = 0.1 .
The boundary conditions: The LES model is an
unsteady three-dimensional model, and it requires the
inlet instantaneous velocity profile. In order to obtain
a stabilized result as soon as possible, the outlet
profile velocity of the k − ε model is used for the
inlet velocity of the LES model[18], and the inlet
turbulence is provided by the method of Intensity and
Hydraulic Diameter. Time step: The time step is
determined via test calculations, which is 0.0005 s.
The accuracy of calculation is the same as the k − ε
model.
2.4 Comparison of the results between calculation and
experiment
The mean streamwise velocity profile obtained
by calculation and experiment with Re = 5 300 is
shown in Fig.3. Y-coordinate is the velocity,
X-coordinate is the position of the measurement points
in the cross section and D is the diameter.
3. The profile-linear average velocity of pipes
obtained with the LES model
3.1 The method of calculation of the linear average
velocity
The incidence angle is 20o[5], as shown in Fig.1.
The profile average velocity vs is constant because
of the constant flux. When the
Re = 5 300 ,
vs = 0.133 m / s . In order to have the statistical
significance of the results, the time-average method is
used for the linear average velocity.
Figure 4 shows the linear average velocity
against time. Y-coordinate is the linear average
velocity, and X-coordinate is the time.
Fig.4 The linear average velocity vs. time with Re = 5 300
Fig.3 Mean streamwise velocity profile obtained by calculation
and experiment with Re = 5 300
In Fig.3, the results of the k − ε model are
compared with the experimental data by
Westerweel[11] via DPIV, which shows that the
velocity profile obtained by the k − ε model
deviates from the experiment results. The k − ε
model is a fully-developed turbulent model, and is not
effective to be used to obtain the profile-linear
average velocity of the transition flow. It is shown that
the results obtained with the LES model are basically
the same as Westerweel’s, which verifies that the LES
model is more effective to be used to simulate the
characteristics of the transition flow with Re = 5 300 .
The mass, momentum and energy are mostly
transported by the moving of large eddies, and this
part can be determined by solving the N-S equation in
the LES model, so the calculation precision is
improved. When the Reynolds number is larger than
5 300, the LES model is also effective because of the
growing of the turbulent kinetic energy. So, in this
article the results of the LES model are used to study
the profile-linear average velocity of pipes.
As is shown, the linear average velocity
fluctuates with time, but the time-average value is
constant. The linear average velocity vs. time
fluctuates round a constant. This constant is the
time-average value of the linear average velocity, and
it is 0.1509 m/s. Therefore, the linear average velocity
of pipes vl
is this time-average value, vl =
0.1509 m / s .
3.2 A comparison with the results of the traditional
coefficient based on the turbulent flow
When Re = 5 300 , k can be calculated using
Eq.(5) provided by Ref.[10]
(
k = 1 + 0.01 6.25 + 431Re−0.23
)
−1
= 0.925
The flow coefficient k ′ obtained by the LES
model in this article is
k′ =
vs
= 0.884
vl
The difference Δ = ( k − k ′ ) / k ′ = 4.38% .
It shows that the profile-linear average velocity
of the transition flow is different from that of the
turbulent flow. According to the industry standard of
heat meter<CJ128-2007>, the flow measurement error
should not be more than 3%. The error of the method
adopting the relationship of the turbulent flow to
calculate the transition region exceeds this value.
93
Therefore, it is necessary to determine the profilelinear velocity in the transition flow region.
3.3 The relationship of profile-linear average velocity
in transition region
The flow fields with Re = 6 000, 7 000, 8 000,
9 000, 10 000 are calculated by the method of the LES
model, and the linear average velocities are shown in
Fig.5-Fig.9.
numbers, and the linear average velocity is denoted as
linear-v, and the profile average velocity as profile-v,
the relationship of profile-linear velocity calculated by
the tradition formula (Eq.(5)) is denoted as k, and
the relationship calculated in this paper is denoted as
kl . The results are shown in Table 1.
Fig.9 The linear average velocities vs. time with Re = 10 000
Fig.5 The linear average velocities vs. time with Re = 6 000
Table 1 compared the results of tradition method with
calculation
kl
Error
Re
linear-v profile-v
k
5 300
6 000
7 000
8 000
9 000
10 000
Fig.6 The linear average velocities vs. time with Re = 7 000
0.1509
0.1708
0.1994
0.2275
0.2566
0.2835
0.133
0.151
0.176
0.201
0.227
0.252
0.925
0.926
0.927
0.928
0.928
0.929
0.884
0.885
0.884
0.885
0.883
0.888
4.38%
4.45%
4.64%
4.57%
4.90%
4.43%
The curve of profile-linear average velocity vs.
Reynolds number ranging from 5 300 to 10 000 is
shown in Fig.10. The curve marked as “Eq.(5)” is the
result obtained by the traditional method of dealing
with the transition region, and the curve marked as
“Calculation” is the result of this article.
Fig.7 The linear average velocities vs. time with Re = 8 000
Fig.10 The curve of profile-linear average velocity vs.
Reynolds number ranging from 5 300 to 10 000
Fig.8 The linear average velocities vs. time with Re = 9 000
The time-average method is used to calculate the
linear average velocity with different Reynolds
It is shown that the difference between the results
obtained in this article and by the traditional formula
is 4.38% - 4.9%, which is more than what is required
by the standard of heat meter. Therefore, the
relationship of the turbulent flow region can not be
used to calculate the profile-linear average velocity in
the transition region, and it is necessary to obtain the
relationship in the transition region. But the
calculation of this article is not enough to obtain the
94
relations in the whole transition region, further work is
necessary.
4. Conclusion
It can be concluded that the LES model is more
effective than the k − ε model to calculate the flow
characteristics of pipes to have a good agreement with
the experimental data of Westerweel when Reynolds
number is 5 300, and the profile-linear average
velocity can be obtained by the LES model. When the
Reynolds number is larger than 5 300, the LES model
is also effective because of the growing of the
turbulent kinetic energy. The relationship of
profile-linear average velocity in the transition region
is studied by the LES model, and compared with the
results by the traditional method of dealing with the
transition region, with 4.38% - 4.9% of difference,
which is more than what is required by the standard of
heat meter <CJ128-2007>; the curve of relationship of
profile-linear average velocity with Reynolds number
in the range of 5 300 - 10 000 is obtained. For the
next step, one has to confirm the effectiveness of the
LES model for the flow with Re < 5 300 , so the
DNS method or the experimental method will be used
for this flow region to obtain the relationship of the
profile-linear average velocity.
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
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