Reconstruction of Velocity Distribution in Partially-Filled Pipe Based on Non-Uniform Under-Sampling

In the process of research on the ow velocity distribution in a partially lled pipe, the under-sampling of measurement data oen occurs. For the rst time, this problem is solved by the improved non-uniform B-spline curve tting approximation (NBSC) method. e main innovation of this method is to reconstruct the ow velocity distribution tting curve with a small amount of non-uniform feature points containing ow velocity information. First, the curvature of a whole discrete sampled data is analyzed, then the weighted threshold is set, and the sampled points that satisfy the threshold are extracted as the initial velocity distribution feature points. Next the node vectors were constructed according to the initial feature points, and the initial interpolation tting curves are generated. Secondly, by using the relative deviation between the initial approximation curve and each sampled point, new feature points were added where the curve allowable deviation exceeded the specied tolerance, and then a new interpolation tting curve was obtained. e above procedure was repeated until the tting curve reached expected accuracy, thus the appropriate feature points were determined. Experimental results showed that, in the case of the same approximation deviation, the proposed NBSC method can solve the problem of under-sampling of measurement data better.


Introduction
In many engineering occasions, the velocity distribution of the uid is required to be obtained, for the velocity of full lled pipes, through the e orts of scienti c researchers, a relatively complete theory has been established. But in many cases, the uid in the pipeline is in a state of partially lled, for example, the transport of wastewater in sewer ows, the petro chemical industries etc. [1]. e velocity distribution in a partially lled pipe is di erent from that of full lled pipes, the factors a ecting velocity, such as wall friction and shear force, all vary with the change of the pipeline depth ratio. When the uid ow is a partially-lled pipe laminar, the velocity distribution is relatively simple. Ng and Lawrence [2] studied the velocity of the laminar ow in partially-lled pipes with numerical simulations. e analytical expressions of laminar ow velocity in partially lled pipes are given by Guo and Meroney [3], and Fullard and Wake [4] respectively, which are veri ed by digital simulation. It is di erent from the laminar ow velocity in partially lled pipes, where the uid is in a turbulent state, due to the in uence of free surface and secondary ow in the upper part of the uid, the velocity of the uid is much more complex than that of the laminar ow, so the research on turbulent ow velocity distribution in partially lled pipes is a focus and it is a di culty in the eld of hydraulics.
A lot of research on the velocity of partially lled pipes have been carried out. Knight and Sterling [5], and Sterling and Knight [6] have studied the velocity in the smooth section of partially lled pipes using a Pitot-static tube. Clark and Kehler [7] researched about the mean velocity distribution of corrugated culvert using acoustic Doppler velocimetry. Using a stereoscopic particle distribution velocimetry (PIV) system, Yoon et al. [8] studied the two-dimensional velocity distribution in partially-lled circular pipes. Henry et al. [1] investigated the stream wise velocity distribution using stereoscopic particle imaging velocimetry (S-PIV) in the partially lled pipes cross-stream plane. Chiu et al. [9][10][11][12][13], and Marini et al. [14] analyzed the two-dimensional velocity distribution formula with Shannon entropy, which represented the observed data reasonably well in a rectangular open channel. Luo and Singh [15], and Cui and Singh [16] derived the partially lled pipes velocity using the Tsallis entropy, which can reasonably describe the velocity near the boundary. ese studies greatly improved the research of partially-lled pipe turbulent velocity. But all of these methods need su cient sampled data. However, in many engineering applications, only a few measuring points can be obtained due to the measuring equipment, construction environment etc., and the known sampled data is non-uniformly distributed. erefore, it is necessary to reasonably reconstruct the uid velocity distribution in the case of non-uniform undersampled.
In the aspect of reconstructing the original eld in the case of undersampled, a series of studies have been carried out and many theoretical methods have been also put forward, such as ordered-subsets expectation-maximization method [17], Least squares tting algorithm [18,19], NURBS curve tting method [20,21], B-spline curve tting method [22,23], etc. Because the B-spline curve method has excellent properties such as geometric invariance, convex hull, and local support, this method is o en used in scienti c research and engineering applications, such as data analysis and distribution reconstruction [24][25][26][27]. However, the traditional B-spline tting curve do not directly pass through the sampled points, which leads to lower tting accuracy and cannot truly re ect the ow velocity. Especially, when the measurement data is small, the tting accuracy of this method is lower. How to get better ow velocity distribution reconstruction under the condition of under-sampling of measurement data, this would be the main problem to be solved in this paper.
In order to solve the previous problems, an improved non-uniform B-spline curve interpolation method called NBSC was proposed in this paper. According to the characteristics of velocity distribution, a small amount of sampling points containing velocity information are selected as featured points, and the velocity distribution curve was reconstructed according to the velocity information contained in the featured points, so as to solve the problem of under-sampling of measured data in velocity data acquisition. e main content of this method includes ve aspects: (a) introduction of traditional B-spline curve method, (b) the curvature of discrete sampled points and the extraction of distribution featured points, (c) reconstruction of the node vectors, (d) adjustment of feature points, and (e) reconstruction of the ow velocity distribution. where is the control points, + 1 is the number of control points, is the order of B-spline curve, = 0 , 1 , 2 , . . . , is the knot vector, and , ( ) is the th-order B-spline basis function, which is de ned as follows:

Traditional B-Spline Curve Method
e calculation process of the B-spline curve method can be summarized as follows: First, the sampled data was parameterized, then the node vector was de ned according to the parameterized data. Second, the basis function can be obtained according to Equations (2) and (3). ird, the control points of the tting curve can be calculated by using the sampling data and the basis function. Finally, with the obtaining of basis function and control points, the B-spline curve can be drawn according to Equation (1).

Uniform Node Vector with Parameter.
Node vector is an important parameter in the B-spline curve tting method. An appropriate node vector can not only ensure the curve re ecting the distribution of measurement points correctly, but also avoids matrix singularity when reverse calculating control vertexes. In any case, one needs to allocate a location parameter to each of the sampling data, then de ne a node vector, and nally compute the basis function and control points [20]. In this paper, the Cumulative Chord Length method [20] was used to parameterize the sampled data, then the node vectors was constructed with the parameterized sampled data.

Construction of Control Points.
e node vector of the sampling data can be calculated by Equation (5), and then the value of the basis function can be obtained by Equations (2) and (3). According to the basis function and the sampled data sequence , the linear equation of the control points is given as follows (4) 0 = 0, = 1, 3 Advances in Mathematical Physics e equation can be rewritten as where and −1 represent the matrix transpose. e equation can be solved with Gauss Elimination method [28], then the control points ( = 1, 2, . . . , − 1) are obtained.
With the obtaining of control points and basis function , the B-spline curve can be drawn according to Equation (1). Obviously, in the calculation process, the control points and basis function are the two main features of B-spline method. From the above analysis, the two parameters are mainly determined by node vector. erefore, the merits of the node vector directly determine the tting accuracy of the B-spline curve.

Initial B-Spline Fitting Curve.
A schematic diagram of partially lled circular pipe section is shown in Figure 1, where is the diameter of the measuring pipe, and ℎ (0 ≤ ℎ ≤ ) is the uid depth from the bottom of the pipe to the water surface, and (0 < ≤ ℎ) is the vertical height from the bottom of the pipe to the position of the measuring point. A Cartesian coordinate system is set in this way such that the bottom of the pipe has the coordinate origin, and presents the transverse distance from the centerline, and presents the vertical depth from axis upward positive. According to the de nition of traditional B-spline curve tting method, the ow velocity distribution curve of the partially lled pipe ow eld is shown in Figure 2.
In Figure 2, max is the maximum value of the ow velocity, where the curve obtained by the traditional B-spline curve tting method is generally close to the velocity sampled data, but there is obvious deviation between the tting curve and the sampled data. Moreover, all the sampled data were processed equally when the control points are inversely calculated, which makes the number of iterations increase, and greatly reduces the computational e ciency. But reducing iteration time it will cause further increase in the local deviation.
In many engineering applications, only insu cient sampled data can be obtained due to the in uence of construction environment, equipment, and other factors. e B-spline curve is tted with undersampled data, and the results are shown in Figure 3. Figure 3, it is clear that the conventional B-spline curve tting method based on a few sampled data reduces the computational complexity, however, there is a large deviation between the tted data and the sampled data. erefore, how to reconstruct the velocity image in the case of under-sampling is the main problem to be solved.

Improved Non-uniform B-Spline Curve Method
In many cases, only insu cient sample points can be measured. If the ow velocity information contained in these sample points can fully express the trend of the ow velocity distribution, then the B-spline curve method can be used to max =1, 2, ..., , = g ≤ , reconstruct the entire ow velocity distribution curve. Based on this idea, an improved non-uniform B-spline curve (NBSC) method was proposed. Compared with the traditional B-spline method, there are two obvious changes in this method. First, some sampled data were extracted as feature points, which have important in uence on the shape of approximation curve. Second, the non-uniform node vectors were reconstructed with these extracted feature points. e basis function and control points were updated with the new node vector.
With the obtaining of new control points the new tting curve were drawn. Obviously, the key to this method is to reasonably select feature points containing ow velocity information. In this paper, the curvature distribution method is used to determine the feature points of the tting curve.

e Curvature of Discrete Points.
In comparison with several methods of algorithm for curvature [29][30][31], the curvature distribution was solved with Local Estimation method in this paper. In the Local Estimation method, the parametric polynomials ( ) are established by using the front and back three points of the measured points, then the curvature of the sampled point in the data sequence can be expressed as where ὔ represent the rst derivative of the curve at parameter , ὔὔ represent the second derivative, and the magnitude of the curvature re ects the variation degree of ow velocity.

Selection of Unbalanced Feature
Points. According to Formula (8), the curvature of ow velocity at each sampled point on the central line of the pipeline can be obtained, the result is shown in Figure 4, where the abscissa is the height of the measuring point from the bottom of the tube, and the longitudinal coordinate is the curvature of the ow velocity at each measuring position.
Obviously, the curvature value of the curve is small where the velocity varies smooth, while the curvature value of the curve is relatively large the velocity varies sharply. erefore, in the sampled data sequence, the principle of selecting feature points are as follows: (1) e sampled point with maximum curvature max is the featured point. (2) e rst sampled point 0 and the end sampled point are the featured points. (3) e sampled points higher than the preset curvature threshold pre are the featured points. It should be noted that the curvature threshold is too high to re ect the characteristics of the curve. On the contrary, reducing the threshold will result in a large amount of calculation and lower computational eciency. erefore, the curvature weighted average w v is used as the preset threshold in this paper, where v is the average of curvature and w is the weight. uncertainty of measurement, resulting in greater uncertainty closer to the bottom of the pipeline, and vice versa. e measurement results are shown in Figure 6. Where ℎ is the ow depth, is the distance from the measuring position to the pipe bottom, and max is the maximum velocity on the central section of the pipeline.
From Figure 6, the position of maximum velocity max in the pipeline is below the free surface, and the position varies with di erent water depth, which is consistent with the measurement result of literature [1].

Comparisons and Discussions.
According to the proposed NBSC method, the average velocity along the pipeline center line is processed, and the approximation curve of velocity pro le at di erent ow depths are obtained. Under the condition of relative deviation less than 1%, the NBSC method was compared with the conventional B-spline curve method, and the results are shown in Figure 7. ese data are presented using both and max as the normalized velocity scale, the height of measuring point and the ow depth ℎ are used as normalized length scales. Figure 7 shows the comparison pro les of partially lled pipe uid for ow depths of 42%, 50%, 62%, 70%, and 79% respectively. As can be seen (1) e number of sampling points required for the improved B-spline curve tting method is far less than that required by the traditional method. When the ow depth of the pipeline are 50% and 79% respectively, the appropriate tting curve can be obtained only with four measuring points, and when the ow depth are 42%, 62%, and 70% respectively, only one feature point needs to be added. e workload of the measurement is greatly reduced.
(2) Except for the two feature points at the head end and the end point, the other feature points are located at a place where the ow velocity changes relatively, and the distribution of feature points are non-uniform.
where ε is the maximum relative tolerance of deviation, ( ), is the distance from measured data point to the nearest point of the tting curve, i.e., If g > , the corresponding sampled data point is the new featured point, and the node vector was reconstructed again with the non-uniform node vector construction method, then the approximation curve of quadratic optimization is obtained.

Experimental Measurement.
e schematic test equipment is shown in Figure 5. e Particle Image Velocimetry (PIV) system was adopted to measure the uid velocity of a single point, and the measurement accuracy was less than 0.5% in theory. e diameter of the transparent acrylic test pipe is = 0.04 m. In order to ensure the fully developed ow pro les in the measuring section, the distance between the pipe inlet and the observation point are 20 , and the distance between the pipe outlet and the measuring point are 15 . e hydraulic slope of the test pipe is = 0.0033, the pipe wall was hypothesized to be hydro-dynamically smooth, and the roughness coe cient is = 0.0085. A laser light emitted from a 30 mJ Nd:YAG laser to illuminate the test point, and the interval of laser pulse is 20 ns. Some glass spheres with diameter of 40 m were injected in order to speed the ow. e ow of the uid depends on the drive of the pump power, and there is no device for stabilizing the uid ow in the measurement system. e ow depth varies from 42% to 79% of .

Experimental
Results. e streamwise mean velocity pro les at pipe vertical bisector was measured by the PIV on the cross-sectional plane for 42%, 50%, 62%, 70%, and 79% ow depth. In the measurement sequence, the uncertainty of the i-th measurement point can be expressed as.
where is the number of repeated measurements of a single measurement point, and is the average value of repeated measurements. According to Equation (12), the uncertainty of measurement of all measurement points were calculated, the uncertainty ranges from 0.00012 to 0.00043. In the vicinity of the bottom of the pipeline, the tracer particles collide with the pipe wall under the action of gravity, resulting in disordered motion.
is disordered motion will a ect the (1) When the number of measurement points are the same, when compared with the traditional B-spline curve method, the proposed method has higher tting precision. e deviation of B-spline approximation curve determined by feature points is obviously smaller than that of the traditional method. (2) When the tting accuracy is the same, the workload of the proposed method is much lesser than that of the traditional B-spline curve method. e number of feature points obtained by extraction algorithm is obviously less than that of original measurement points, which reduces the computational complexity and improves the e ciency. (3) When the ow velocity changes drastically, more feature points are required, and when the ow velocity changes slowly, fewer feature points are required. is distribution law allows the measurements to be targeted more. (4) e method presented in this paper is easy to understand and apply to engineering applications. e analysis shows that this method has important practical signi cance for engineering applications. It can be used in steel smelting, aerospace, and other harsh environments, and can also be used in mold design, garment customization, big data analysis, etc., which can reduce the workload of design.
Data Availability e data used to support the ndings of this study are available from the corresponding author upon request. e measured object can be measured more specifically by using the proposed method in this paper.
(3) In the case of few feature points, there are a large deviation between the traditional B-spline tting curve and the measured data. e approximation curve obtained by the rst-order optimized B-spline curve tting method can be better described as the distribution of ow velocity, but there is an obvious deviation in the local region. is deviation can be e ectively reduced by using the quadratic optimization B-spline curve tting method. is means that in a ow eld where the ow velocity changes more drastically, the tting accuracy can be improved by appropriately increasing the feature points. (4) Using the method proposed in this paper, the ow velocity distribution can be reconstructed with only a small amount of feature points and water depth ratio. Under the condition of relative deviation less than 1%, using the improved B-spline curve tting method, the one-dimensional velocity distribution pro le can be reconstructed with at most ve feature points.

Conclusion
e velocity distribution in partially lled pipe is researched with non-uniform B-spline curve tting method. A B-spline curve approximation algorithm based on few feature points is proposed, which mainly includes feature point extraction algorithm based on discrete curvature analysis, and feature point adjustment algorithm based on deviation analysis. e conclusion are as follows: