We can build the threedimensional structure model based on the Gambit software and achieve the distribution of flow field in the pipe and reflux flow condition at the position of transducer in regard to the real position of transducer according to the Fluent software. Under the framework, define the reflux length based on the distance of reflux along the channel and evaluate the effect of reflux on flow field. Then we can correct the power factor with the transmission speed difference method in the ideal condition and obtain the matching expression of power correction factor according to the practice model. In the end, analyze the simulation experience and produce the sample table based on the proposed model. The comparative analysis of test results and simulation results demonstrates the validity and feasibility of the proposed simulation method. The research in this paper will lay a foundation for further study on the optimization of ultrasonic flowmeter, enhance the measurement precision, and extend the application of engineering.
Compared with the conventional flowmeter, the ultrasonic flowmeter has a better performance since it has no moving parts, no pressure loss, wide measuring range, excellent repeatability, and high precision [
With analysis of the related references, we found that
In order to estimate the measuring errors caused by disturbance, this paper proposed a novelty model that builds the practice structure of a singlepath ultrasonic transducer with a 50 mm pipe diameter and calculates the disturbance of transducer to the flow field approximately using the Fluent software for flow field analysis combined with test data; based on the above model we can analyze the measurement effects on the accuracy by quantitative methods.
This paper is structured in the following way. In Section
We can see the measurement principle of transmission speed difference method in the singlepath ultrasonic flowmeter [
The diagram of the principle of ultrasonic flowmeter.
It is assumed that the fluid will flow with velocity
Using (
Because of the presence of the actual fluid velocity distribution in the pipe crosssection, linear mean velocity
Then we can get that the flow of the pipe is
Considering the influences of pressure and temperature, the flow can be converted under the standard working conditions:
Based on the hydrodynamic theory, the fluid has viscosity so that the fluid shows different velocities at the points of different diameter in the crosssection. And the Reynolds number can be the only parameter that distinguishes moving patterns of viscous fluid. Whether the fluid moving as laminar or turbulent flow can be decided by the value of Reynolds number, there is a lower bound around 2000 for the critical Reynolds number, which transits laminar flow to turbulence. In the moving of laminar flows, the tiny disturbance in the flow field such as the roughness of pipe wall and free changes of surface will attenuate gradually so that the fluid flows as laminar flow. However, the tiny disturbance can be increased and flow becomes unstable if Reynolds number is bigger, so it is difficult to make sure the final status after disturbance increased as the equations are of nonlinearity, we can only conclude that the final stage is connected with structure of flow field and Reynolds number.
With regard to the ideal laminar flow shown in Figure
Based on the equation above, each point velocity distributed parabolically with radius
Through the simulation, we can get the flow results with parabolic distribution in Figure
The velocity profile in the laminar flow.
According to the distribution of flow velocity, the crosssectional area of the flow can be calculated as
We can achieve the relationship between the crosssection mean velocity, linear mean velocity, and the maximum flow rate based on the above theory; meanwhile the relationship between the crosssection mean velocity and linear mean velocity is obtained. However, the magnitude and position of maximum velocity cannot be measured directly in practice and engineering application.
The laminar flow velocity distribution and the value of power correction factor have been derived under the ideal circumstances. However, the pipe is not smooth in practice, and the pipe will be installed with temperature and pressure sensors inside it, which may disturb the flow field making the velocity of flow field dissatisfy the standard parabolic distribution. Therefore, the power correction factor
In this paper, we design the actual structure of ultrasonic flowmeter with small diameter and small flow as shown in Figure
The crosssection of ultrasonic flowmeter.
In the first step, we can use the Gambit software to build the geometric model of the flowmeter. The pipe is cylindrical with a 50 mmdiameter with holes at the 45degree angle along with pipe axis, where the transducer is installed; the pressure and temperature sensors are built separately inside the two holes on the left side.
Secondly, mesh the model. Since the pipe has a throughhole structure that the transducer and sensors are installed in, the shape of flow field is not cylindrical any more. Thence, the surface and volume of flow field can ensure the grid near the transducer and sensors is dense enough and can control the number of grids by choosing tetrahedral mesh.
Next, put the grid file into the Fluent software in order to do the fluid calculation. As the pipe is of small diameter, small flow, and small Reynolds number, we should employ the laminar flow model to make the fluid calculation.
At last, set the parameters for calculations. Using the Fluent software to deal with the laminar flow model when the minimum flow is 0.6 m^{3}/h and the corresponding Reynolds number
In Figures
The diagram of sound channel (
The diagram of sound channel (
In addition, the fluid velocity has parabolic distribution in the ideal laminar flow model and is parallel to the axis, but in the actual structure we can get the curve of fluid velocity along AB in Figures
The velocity curve in the
The velocity curve in the
The curve of crosssectional velocity distribution at the midpoint of output pipe (the midpoint of AB) can be seen in Figure
The curve of crosssectional velocity distribution at the midpoint of output pipe AB.
To the fluid, the Reynolds number can be estimated by
In the model shown in Figure
The length of reflux at point A.
The length of reflux at point B.
Simulating under different Reynolds numbers, we can get the reflux of fluid at the transducer and the length of refluxes
The length of refluxes
Number 



Percentage (%) 

1  145  0.01191  0.00798  20.3 
2  226  0.01198  0.00815  20.5 
3  459  0.01212  0.00956  22.1 
4  1168  0.01339  0.01110  25.0 
5  1853  0.01451  0.01361  28.7 
On the basis of Table
The relationship diagram of
From (
First, assume that the fluid is flowing parallel to the pipe axis in Figure
Second, suppose that the pipes are all smooth tubes; we can ignore the influences on the fluid of exact pipe structure. However, because of the actual structure of the transducer by intrusive installation, especially for the pipes with small diameters, the fluid flow will be affected.
In engineering, we can get the power correction factor generally from the test when correcting the flowmeter against the fluid with low Reynolds number, if, considering the actual shape, structure of pipe, and the influences on the measurement of the nonaxisparallel flowing fluid, the relationship between flow field that affects power correction factor and measurement error of pipe flow can be analyzed.
Reflux makes the linear average velocity less so that the measurement is lower and the error is negative. Now considering the influences of reflux, we can rewrite (
Considering that the type and size of the transducer in part A are generally the same as part B, so the hardware delay can be regarded as the same:
According to the simulation output data we can get
To test the effectiveness of simulation analysis, make a trial version of ultrasonic flowmeter shown in Figure
The analysis from the last section leads to the conclusion that the actual structure of the pipe generates reflux at points A and B; the reflux raises the downstream ultrasonic propagation time and lowers the upstream time so that the time difference is less, the flow measurement is lower, the errors will be negative, with the same diameters, and the measuring errors will increase gradually along with the increasing entrance velocity.
To estimate the exact influences on measurement of reflux, this paper is based on the output of Fluent and counts the propagation time and time difference of ultrasonic wave between two transducers, as Table
Time difference of ultrasonic wave at points A and B.
Number 




Percentage (%) 

1  145  1.874  1.608  97.5  3.6 
2  226  3.283  1.883  147.2  3.5 
3  459  5.60  3.810  277.1  3.4 
4  1168  17.723  2.366  683.4  2.94 
5  1853  23.10  1.59  1037.8  2.38 
The power correction factor
The result of mean linear velocity.
Number 

Mean linear velocity  Power correction factor 

1  145  0.0816  1.067 
2  226  0.123  1.105 
3  459  0.231  1.196 
4  1168  0.569  1.235 
5  1853  0.863  1.290 
Data from Table
The curve that indicates the relationship between the power factor and Reynolds number is drawn in Figure
The relationship of power factor
During the test,
The relationship of time difference and
Number 



1  145  90.5 
2  226  202.2 
3  459  346.8 
4  1168  746.4 
5  1853  1119.2 
Based on Table
The relationship curve of time difference and
The simulation and test outputs have the same trend with Reynolds number, but there are some offsets in Figure
Firstly, when installing two transducers along AB, some installation errors always exist.
Secondly, when building the finite element model of flow field, the meshing type and the size of grids will affect the accuracy and then generate the errors.
Thirdly, while using Fluent to simulate and calculate, the setting of related parameters in the laminar flow model will influence the accuracy of outputs.
It is effective to converge the tested and simulated results by improving the accuracy of meshing, setting the reasonable parameters and installation accuracy.
In this paper, we analyze the flow field of ultrasonic mono flowmeter with small diameter and low flow and discuss the influences on the flow field and power factor of exact pipe structure and the variation using different Reynolds number. The main conclusions are as follows.
The installation point of ultrasonic transducer and temperature/pressure sensor will disturb the laminar flow field, the velocity will not be standard parabolic distribution any longer, and the reflux is generated at the transducer; the length of reflux has the same trend with Reynolds number.
Near the transducer, the reflux decreases the linear average velocity and makes the measurement of flow lower; the errors will be negative.
The expression of power correction factor by simulated data is fit.
Through the test, the effectiveness of simulation is tested. Numerical simulation method can be a good reaction to flow state of flow field; it may be an important way to design and develop ultrasonic flowmeter.
In further work, we will take into account the main reason which causes the error between the test data and the simulation data and fit the power correction factor more accurately so that the proposed method is a more effective tool.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work was supported by the Fundamental Research Funds for the Central Universities (no. CDJZR10170007).