# Research on Hyperbola Fitting Algorithm for Turbulence Level Measurement Test Data

Authors: Yufeng Du; Long Wu; Xunnian Wang; Jun Lin; Neng Xiong
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

Hyperbola fitting of test data is an extremely important process in turbulence level measurement test in wind tunnels. The solution of the overdetermined equations (SOE) method is often used to solve hyperbola fitting parameters to obtain turbulence level. However, due to unsteady flow characteristics, the SOE method often results in overfitting phenomena, which makes it impossible to solve turbulence level accurately. This paper proposes using the constrained least-squares (CLS) method to convert the problem of hyperbola fitting of test data into the inequality constrained optimization problem and then using the Lagrange programming neural network (LPNN) method to solve turbulence level iteratively. The stability of the LPNN method is analysed, and three sets of typical turbulence level measurement test data are processed using the LPNN method. The results verify the feasibility of applying the LPNN method to iteratively solve the turbulence level of wind tunnels.


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## Body

## 1. Introduction

Wind tunnel test is the most effective method for aerodynamic research. Even if the computer-based numerical simulation technology and the model flight test technology are rapidly improving, wind tunnel test is still an indispensable method for the research of complex aerodynamic characteristics during the research and development of aircraft. The precise design of advanced aircraft requires high accuracy of wind tunnel test results. However, there exist many factors and phenomena which finally result in the inaccuracy of measurements in the wind tunnel, such as the influence of flow qualities, Reynolds and Mach numbers, standard model test results, wind tunnel walls, and supporting system interference in 2D and 3D tests [1–7].As an important flow quality in wind tunnels, turbulence level will affect the accuracy of wind tunnel test results, such as calculation of aircraft force and moment coefficients [1], measurement of test model attitude angle [8], and measurement of transition characteristics of the boundary layer on test model surface [9]. For aircraft design, the errors in wind tunnel tests mean that there will be design errors in aerodynamic parameters such as lift and drag coefficients. The design errors will lead to estimation errors in aircraft weight, which severely restricts the economy and safety of aircraft [10]. Therefore, it is very important to evaluate the turbulence level in wind tunnels accurately and quantitatively.Hot-wire anemometry (HWA) is currently the most widely used method for turbulence level measurement due to its advantages such as high-frequency response, high sensitivity, and cost effectiveness [11–14]. In the compressible flow, the hot-wire response function derived by the changing overheat ratio method conforms to a hyperbolic relationship. Therefore, the problem of solving turbulence level can be converted into the problem of solving hyperbola fitting parameters of a set of two-dimensional scattered points [15]. However, in the actual measurement and data acquisition process, the two-dimensional scattered points may deviate from hyperbolic distribution due to unsteady flow characteristics. In some cases, there are even straight lines and other conic distributions, which make it difficult to solve turbulence level precisely.Many researchers have studied hyperbola fitting, and the methods utilized differ due to different application scenarios. Hough transform (HF) is a method of graph detection and fitting based on pattern recognition. The HF method converts scattered points in the image space to those in the parameter space and converts the problem of scattered points fitting to the problem of solving extreme value in the parameter space, thereby realizing the detection and fitting of the hyperbola. The HF method has the advantages of strong robustness, but its shortcomings of low computational efficiency limit its real-time application [16–20]. The least-squares (LS) method is the most commonly used method for hyperbola fitting. The key idea of the LS method is to establish a suberror function for each scattered point and use the sum of the squares of all suberror functions as the error function of hyperbola fitting. The suberror function can be expressed by residuals or orthogonal distances between scattered points and fitted hyperbolic curves. Hyperbolic curves are obtained by calculating the extreme value of the error function. The LS method has been widely used due to its ease of application and high computational efficiency [21–25]. Both Lebiga VA [12, 26, 27] and Radespiel R [28–31] have done a lot of research in the field of turbulence level measurement and obtaining the fitting solution in compressible flow. They conducted a comprehensive and thorough analysis of fluctuation measurement results, which provided data support for the uncertainty analysis of subsequent wind tunnel tests. However, they have not conducted too much research on the hyperbola fitting method. Instead, the solution of the overdetermined equations (SOE) method is used directly to calculate the hyperbolic parameters to complete the hyperbola fitting. The SOE method is based on the LS method, which means it has high computational efficiency, while the SOE method cannot obtain precise turbulence level results due to overfitting if the scattered points deviate from hyperbolic distribution.In order to solve the problem mentioned above, this paper proposes using the constrained least-squares (CLS) method to convert the problem of hyperbola fitting to the inequality constrained optimization problem and then using the Lagrange programming neural network (LPNN) method to solve turbulence level iteratively. The results show that the LPNN method is superior to the traditional SOE method, which verify the feasibility of the LPNN method for solving turbulence level in wind tunnels.


## 2. The SOE Method

According to the literature [26–32], the response function of the constant temperature hot-wire anemometer (CTA) in compressible flow is (1)ΔEE¯=FCTAΔmm¯−GCTAΔT0T0¯,where E is the output voltage of CTA, m and T0 are the gas mass flow rate and total temperature of the hot-wire probe measuring point, and FCTA and GCTA are the mass flow rate and total temperature sensitivity coefficients of CTA, respectively. FCTA and GCTA are only related to the overheating ratio, namely, the working temperature, of the hot-wire probe under a fixed flow condition. Dividing equation (1) by GCTA and defining θ=1/GCTAΔE/E¯ as the response function and r=FCTA/GCTA as the independent variable, we can get(2)θ=Δmm¯r−ΔT0T0¯.Take the mean square value of equation (2):(3)θ2¯2=Δmm¯2¯r2−2Δmm¯ΔT0T0¯¯r+ΔT0T0¯2¯.From equation (3), we can know that the response function of CTA conforms to the hyperbolic relationship with r as the independent variable and θ2¯ as the dependent variable. The two-dimensional scattered points R,Θ to be fitted can be obtained by continuously changing the overheat ratio n times and recording the output voltage of CTA at the same time:(4)R=r1r2…rnT,Θ=θ12¯θ22¯…θn2¯T.Construct the coefficient matrixA, the variable vector X to be sought, and the nonhomogeneous term vector Π of overdetermined equations:(5)A=r12r11r22r21⋮⋮⋮rn2rn1,X=Δmm¯2¯−2Δmm¯ΔT0T0¯¯ΔT0T0¯2¯T,Π=θ12¯2θ22¯2…θn2¯2T.Therefore, the flow fluctuations can be obtained by solving the following overdetermined equations:(6)A⋅X=Π.SinceA is not a square matrix, it is impossible to solve the overdetermined equations by solving the inverse matrix of A. Instead, the generalized inverse matrix of A can be solved by the singular value decomposition (SVD) method to solve the overdetermined equations. Any real matrix A (n × q) can be decomposed:(7)A=U⋅Wz000⋅VT,where U(n×n) and V(q×q) are orthogonal matrices, Wz=diagw1,w2,…,wz, w1,w2,…,wz are the nonzero singular values of matrix A, and w1≥w2≥⋯≥wz>0, z is the rank of A. When the rank of A is full, that is, z = q, the SVD form of A is(8)A=U⋅Wz0⋅VT.According to equation (8), the generalized left inverse matrix of A is (9)A+=V⋅Wz−10⋅UT,which satisfies A+⋅A=I. Therefore, the solution of the overdetermined equations (6) using the SVD method is(10)X=A+⋅Π.After obtaining the flow fluctuations, the following equations can be used to solve the turbulence levelTu of the flow: (11)Δmm¯=Δρuρu¯=Δρρ¯+Δuu¯,ΔT0T0¯=αΔTT¯+βΔuu¯,α=11+γ−1/2M2β=γ−1M21+γ−1/2M2,Δpp¯=Δρρ¯+ΔTT¯,Δρρ¯=1γΔpp¯,where ρ, u, T, and p are the gas density, velocity, static temperature, and static pressure of the hot-wire probe measuring point, respectively, and M is the Mach number. Turbulence level Tu can be solved by the following equations: (12)Tu=Δuu¯2¯1/2=H2HGG2⋅X1/2,(13)H=11−M2,G=M21−M2⋅1β.In the actual measurement process, conventional wind tunnels cannot guarantee the strict steadiness of flow due to the long duration of the turbulence level measurement test by using the changing overheat ratio method. Therefore, there will be cases where the two-dimensional scattered points do not strictly obey hyperbolic distribution, and even the distribution is similar to straight lines and other conic curves. The schematic diagram is shown in Figure1.Figure 1
Schematic diagram of scatters and fitted curves of ideal and disturbed conditions.The ○ points and the solid blue line in Figure1 are the scattered points under ideal conditions and the hyperbola obtained by the SOE method, and the + points and the dotted red line are the scattered points with flow disturbances and the curve obtained by the SOE method. As a result of overfitting, the fitting curve is approximately a straight line, which has deviated from the ideal hyperbola. In this situation, the SOE method will not accurately solve the turbulence level.


## 3. Proposed Algorithm for Solving Turbulence Level

In order to solve the problem mentioned above, the CLS method is used to convert the problem of hyperbola fitting to the inequality constrained optimization problem, and then the LPNN method is used to obtain turbulence level iteratively.


### 3.1. The CLS Method

Compared with the traditional LS method, the CLS method can limit the parameters to be fitted within a certain range to prevent overfitting. Suppose that the fitted hyperbolic equation of the two-dimensional scattered pointsR,Θ is (14)f2r=c1r2+c2r+c3,where r is the independent variable, fr is the dependent variable, and c1, c2, and c3 are the hyperbola fitting coefficients. According to equation (14), the dependent variable fitting values FR and residuals E are (15)FR=fr1,fr2,…,frnT,E=Θ−FR=e1,e2,…,enT,ei=θi2¯−fri,i=1,2,…,n.Select the error functionS of the CLS method as the sum of squares of the residuals of each scatter: (16)S=∑i=1nei2.Comparing equations (3) and (14), it can be seen that the hyperbola fitting coefficients c1, c2, and c3 in the CLS method are the flow fluctuations to be sought; namely, (17)C=c1,c2,c3T=X^.According to the physical meanings of the variables in equations (5) and (17), the constraints are established as follows: (1)
Since the mean square values ofm and T0 fluctuations are always positive, we can say c1>0 and c3>0.(2)
The variances ofm and T0, their covariance, and correlation coefficient are defined, respectively, as follows:(18)σm2=Δm2¯,σT02=ΔT02¯,σmT0=ΔmΔT0¯.,(19)ρmT0=σmT0σmσT0,−1≤ρmT0≤1.From equations (18) and (19), the correlation coefficient ρmT0 can be rewritten as (20)ρmT0=ΔmΔT0¯Δm2¯ΔT02¯=−12−2Δm/m¯ΔT0/T0¯¯Δm/m¯2¯ΔT0/T0¯2¯.Constraints can be established based on the range of the correlation coefficientρmT0: (21)c22−4c1c3≤0.In summary, the problem of hyperbola fitting of two-dimensional scattered points can be converted to the following optimization problem:(22)argminCSC,s.t.c1>0&c3>0&c22−4c1c3≤0.


### 3.2. The LPNN Method

To solve the nonlinear inequality constrained optimization problem in equation (22), the LPNN method is applied [33]. The inequality constraints can be rewritten as follows: (23)g1C=−c1+τ≤0,g2C=−c3+τ≤0,g3C=c22−4c1c3≤0,where τ is the correction value of coefficients c1 and c3 and τ=10−10. Its value is determined according to the flow quality values of the conventional wind tunnels. The Lagrange function L with inequality constraints is defined as(24)LC,ν,λ=SC+∑i=13λigiC+νi2,where λ=λ1,λ2,λ3T are the constraint coefficients and λi≥0 for i=1,2,3.ν=ν1,ν2,ν3T are the relaxation variables. There are three types of neurons in the LPNN method: variable neurons control C, Lagrange neurons control λ, and relaxation neurons control ν. According to the KKT condition, the state equations of neurons can be obtained as follows: (25)dCdt=−∂LC,ν,λ∂C,dνdt=−∂LC,ν,λ∂ν,dλdt=∂LC,ν,λ∂λ,where t is characteristic time. The Euler iteration equation is (26)Ck+1=Ck−∂LCk,νk,λk∂Ck⋅Δt,νk+1=νk−∂LCk,νk,λk∂νk⋅Δt,λk+1=λk+∂LCk,νk,λk∂λk⋅Δt,where k is the number of iterations and Δt is the iteration time step. The component form of equation (26) is (27)cik+1=cik−∂SCk∂cik+∑j=13λjk∂gjCk∂cik⋅Δt,i=1,2,3,νlk+1=νlk−2λlkνlk⋅Δt,l=1,2,3,λjk+1=λjk+gjCk+νjk2⋅Δt,i=1,2,3,The neural network architecture diagram of the entire iteration process is shown in Figure2.Figure 2
Architecture diagram of LPNN for inequality constrained optimization problem.


### 3.3. Stability Analysis of the LPNN Method

According to the literature [33], when performing stability analysis, it is necessary to verify that the iterative equilibrium point obtained by the LPNN method is the asymptotically stable point of the neural network. Let the iterative equilibrium point be C∗,ν∗,λ∗, and this point can be obtained by the following equation: (28)dC¯dtdλdt=−∇C¯C¯2LC∗,ν∗,λ∗∇C¯gC∗,ν∗−∇C¯gC∗,ν∗0⋅C¯−C¯∗λ−λ∗=−G⋅C¯−C¯∗λ−λ∗,where C¯=CT,νTT and giC,ν=giC+νi2 for i=1,2,3. Additionally, we have (29)∇C¯gC∗,ν∗=∇CgC∗,ν∗∇νgC∗,ν∗=−10−4c3∗002c2∗0−1−4c1∗2ν1∗0002ν2∗0002ν3∗.∇C¯gC∗,ν∗ can be easily verified to be a full column rank. To be more precise, the gradients above at the equilibrium point are linearly independent, which means that C¯∗ is a regular point and C∗,ν∗,λ∗ is a Kuhn–Tucker point of the Lagrange function L [33]; namely, the following equation holds: (30)∇CC2LC∗,ν∗,λ∗>0.The specific form of∇C¯C¯2LC∗,ν∗,λ∗ in matrix G in equation (28) is (31)∇C¯C¯2LC∗,ν∗,λ∗=∇CC2LC∗,ν∗,λ∗00002λ1∗00002λ2∗00002λ3∗.According to the results of equation (30) and λi≥0 for i=1,2,3, we can say that ∇C¯C¯2LC∗,ν∗,λ∗ is a strict positive definite matrix and the coefficient matrix G in equation (28) is a negative semidefinite matrix, which means that the equilibrium point C∗,ν∗,λ∗ is the asymptotically stable point of the network.


## 3.1. The CLS Method

Compared with the traditional LS method, the CLS method can limit the parameters to be fitted within a certain range to prevent overfitting. Suppose that the fitted hyperbolic equation of the two-dimensional scattered pointsR,Θ is (14)f2r=c1r2+c2r+c3,where r is the independent variable, fr is the dependent variable, and c1, c2, and c3 are the hyperbola fitting coefficients. According to equation (14), the dependent variable fitting values FR and residuals E are (15)FR=fr1,fr2,…,frnT,E=Θ−FR=e1,e2,…,enT,ei=θi2¯−fri,i=1,2,…,n.Select the error functionS of the CLS method as the sum of squares of the residuals of each scatter: (16)S=∑i=1nei2.Comparing equations (3) and (14), it can be seen that the hyperbola fitting coefficients c1, c2, and c3 in the CLS method are the flow fluctuations to be sought; namely, (17)C=c1,c2,c3T=X^.According to the physical meanings of the variables in equations (5) and (17), the constraints are established as follows: (1)
Since the mean square values ofm and T0 fluctuations are always positive, we can say c1>0 and c3>0.(2)
The variances ofm and T0, their covariance, and correlation coefficient are defined, respectively, as follows:(18)σm2=Δm2¯,σT02=ΔT02¯,σmT0=ΔmΔT0¯.,(19)ρmT0=σmT0σmσT0,−1≤ρmT0≤1.From equations (18) and (19), the correlation coefficient ρmT0 can be rewritten as (20)ρmT0=ΔmΔT0¯Δm2¯ΔT02¯=−12−2Δm/m¯ΔT0/T0¯¯Δm/m¯2¯ΔT0/T0¯2¯.Constraints can be established based on the range of the correlation coefficientρmT0: (21)c22−4c1c3≤0.In summary, the problem of hyperbola fitting of two-dimensional scattered points can be converted to the following optimization problem:(22)argminCSC,s.t.c1>0&c3>0&c22−4c1c3≤0.


## 3.2. The LPNN Method

To solve the nonlinear inequality constrained optimization problem in equation (22), the LPNN method is applied [33]. The inequality constraints can be rewritten as follows: (23)g1C=−c1+τ≤0,g2C=−c3+τ≤0,g3C=c22−4c1c3≤0,where τ is the correction value of coefficients c1 and c3 and τ=10−10. Its value is determined according to the flow quality values of the conventional wind tunnels. The Lagrange function L with inequality constraints is defined as(24)LC,ν,λ=SC+∑i=13λigiC+νi2,where λ=λ1,λ2,λ3T are the constraint coefficients and λi≥0 for i=1,2,3.ν=ν1,ν2,ν3T are the relaxation variables. There are three types of neurons in the LPNN method: variable neurons control C, Lagrange neurons control λ, and relaxation neurons control ν. According to the KKT condition, the state equations of neurons can be obtained as follows: (25)dCdt=−∂LC,ν,λ∂C,dνdt=−∂LC,ν,λ∂ν,dλdt=∂LC,ν,λ∂λ,where t is characteristic time. The Euler iteration equation is (26)Ck+1=Ck−∂LCk,νk,λk∂Ck⋅Δt,νk+1=νk−∂LCk,νk,λk∂νk⋅Δt,λk+1=λk+∂LCk,νk,λk∂λk⋅Δt,where k is the number of iterations and Δt is the iteration time step. The component form of equation (26) is (27)cik+1=cik−∂SCk∂cik+∑j=13λjk∂gjCk∂cik⋅Δt,i=1,2,3,νlk+1=νlk−2λlkνlk⋅Δt,l=1,2,3,λjk+1=λjk+gjCk+νjk2⋅Δt,i=1,2,3,The neural network architecture diagram of the entire iteration process is shown in Figure2.Figure 2
Architecture diagram of LPNN for inequality constrained optimization problem.


## 3.3. Stability Analysis of the LPNN Method

According to the literature [33], when performing stability analysis, it is necessary to verify that the iterative equilibrium point obtained by the LPNN method is the asymptotically stable point of the neural network. Let the iterative equilibrium point be C∗,ν∗,λ∗, and this point can be obtained by the following equation: (28)dC¯dtdλdt=−∇C¯C¯2LC∗,ν∗,λ∗∇C¯gC∗,ν∗−∇C¯gC∗,ν∗0⋅C¯−C¯∗λ−λ∗=−G⋅C¯−C¯∗λ−λ∗,where C¯=CT,νTT and giC,ν=giC+νi2 for i=1,2,3. Additionally, we have (29)∇C¯gC∗,ν∗=∇CgC∗,ν∗∇νgC∗,ν∗=−10−4c3∗002c2∗0−1−4c1∗2ν1∗0002ν2∗0002ν3∗.∇C¯gC∗,ν∗ can be easily verified to be a full column rank. To be more precise, the gradients above at the equilibrium point are linearly independent, which means that C¯∗ is a regular point and C∗,ν∗,λ∗ is a Kuhn–Tucker point of the Lagrange function L [33]; namely, the following equation holds: (30)∇CC2LC∗,ν∗,λ∗>0.The specific form of∇C¯C¯2LC∗,ν∗,λ∗ in matrix G in equation (28) is (31)∇C¯C¯2LC∗,ν∗,λ∗=∇CC2LC∗,ν∗,λ∗00002λ1∗00002λ2∗00002λ3∗.According to the results of equation (30) and λi≥0 for i=1,2,3, we can say that ∇C¯C¯2LC∗,ν∗,λ∗ is a strict positive definite matrix and the coefficient matrix G in equation (28) is a negative semidefinite matrix, which means that the equilibrium point C∗,ν∗,λ∗ is the asymptotically stable point of the network.


## 4. Results and Analysis

### 4.1. Results of the Proposed Algorithm

Three sets of typical turbulence level measurement test results are selected to be processed. The independent variable and dependent variable data in equation (3) are shown in Table 1.Table 1
Data on turbulence level measurement test.
M = 0.449M = 0.516M = 0.599Rθ2¯/%rθ2¯/%rθ2¯/%0.1460.0930.1470.0580.1460.0740.1770.1100.1790.0680.1780.0730.2140.1120.2170.0800.2150.0790.2570.1110.2610.0920.2590.1000.4130.1810.4260.1590.4210.1800.7420.3140.7880.2730.7730.279The LPNN method is used to iteratively solve the turbulence level corresponding to the three sets of test data above in Table1. Considering convergence speed, the initial value of iteration is selected as 10−4, 10−5, 10−6, and 10−7, respectively, and the iteration time step is Δt=10−8. The iteration results of variable C are shown in Figure 3. The iteration results of variables ν and λ have no concern with solving the turbulence level, so they are not listed here.Figure 3
Results of variables from the iteration process. (a)C1 for M = 0.449. (b) C2 for M = 0.449. (c) C3 for M = 0.449. (d) C1 for M = 0.516. (e) C2 for M = 0.516. (f) C3 for M = 0.516. (g) C1 for M = 0.599. (h) C2 for M = 0.599. (i) C3 for M = 0.599.
(a)(b)(c)(d)(e)(f)(g)(h)(i)From Figure3, we can easily find that variable C converges to a fixed value within approximate 4×106 steps, and consistent results can be obtained with different initial values. The results indicate good and stable convergence of the LPNN method.


### 4.2. Comparison with the SOE Method

The comparison results of the fitted hyperbola obtained by the LPNN method and the traditional SOE method are shown in Figure4. The turbulence level and other flow qualities by the two methods are shown in Table 2.Figure 4
Comparison of hyperbola fitting from LPNN and SOE results. (a)M = 0.449. (b) M = 0.516. (c) M = 0.599.
(a)(b)(c)Table 2
Comparison of flow quality from LPNN and SOE results.
MethodΔm/m¯2¯ (C1)−2Δm/m¯ΔT0/T0¯¯ (C2)ΔT0/T0¯2¯ (C3)Tu (%)Goodness of fittingM = 0.449LPNN2.03×10−5−2.97×10−69.27×10−70.540.9981SOE2.02×10−5−2.90×10−69.11×10−70.540.9981M = 0.516LPNN1.26×10−5−4.51×10−71.54×10−70.440.9974SOE9.40×10−62.41×10−6−2.93×10−70.460.9992M = 0.599LPNN1.33×10−59.36×10−81.61×10−70.480.9864SOE6.28×10−66.29×10−6−7.92×10−70.520.9936For the caseM = 0.449, Figure 4(a) shows that the scattered points of turbulence level measurement test results basically conform to the hyperbolic distribution, and the fitted hyperbolae obtained by the LPNN method and the traditional SOE method basically coincide. The results in Table 2 show that the turbulence level and other flow qualities are basically equal for M = 0.449. Therefore, in the case when the scattered points conform to the hyperbolic distribution, both the LPNN method and the SOE method can be used to solve the turbulence level accurately.For the casesM = 0.516 and M = 0.599, Figures 4(b) and 4(c) show that the scattered points deviate from the hyperbolic distribution due to the unsteadiness of the flow, and there are even straight lines and other conic curves for the scattered points distribution. The fitting curve obtained by the traditional SOE method is an ellipse, which is inconsistent with the theoretical results from equation (3). Table 2 shows the high goodness of fitting values and negative C3 values for the SOE results of M = 0.516 and M = 0.599. C3 represents the mean square value of T0; thus, C3 should be positive for all situations according to its physical meaning. However, the results of C3 obtained by the SOE method are negative, which indicate that overfitting has occurred for M = 0.516 and M = 0.599. The scattered points are overfitted into ellipses by the SOE method, which results in negative C3 values. In this situation, although equations (12) and (13) can still be used to solve turbulence levels, the results are not true values.The fitting curve for the casesM = 0.516 and M = 0.599 obtained by the LPNN method is still hyperbola with the influence of the unsteadiness of the flow, and the goodness of fitting values are high, which is consistent with the theoretical results. From the results, we can easily find that the LPNN method is better than the SOE method, and the results verify the feasibility of applying the proposed algorithm to solve the turbulence level in wind tunnels.


## 4.1. Results of the Proposed Algorithm

Three sets of typical turbulence level measurement test results are selected to be processed. The independent variable and dependent variable data in equation (3) are shown in Table 1.Table 1
Data on turbulence level measurement test.
M = 0.449M = 0.516M = 0.599Rθ2¯/%rθ2¯/%rθ2¯/%0.1460.0930.1470.0580.1460.0740.1770.1100.1790.0680.1780.0730.2140.1120.2170.0800.2150.0790.2570.1110.2610.0920.2590.1000.4130.1810.4260.1590.4210.1800.7420.3140.7880.2730.7730.279The LPNN method is used to iteratively solve the turbulence level corresponding to the three sets of test data above in Table1. Considering convergence speed, the initial value of iteration is selected as 10−4, 10−5, 10−6, and 10−7, respectively, and the iteration time step is Δt=10−8. The iteration results of variable C are shown in Figure 3. The iteration results of variables ν and λ have no concern with solving the turbulence level, so they are not listed here.Figure 3
Results of variables from the iteration process. (a)C1 for M = 0.449. (b) C2 for M = 0.449. (c) C3 for M = 0.449. (d) C1 for M = 0.516. (e) C2 for M = 0.516. (f) C3 for M = 0.516. (g) C1 for M = 0.599. (h) C2 for M = 0.599. (i) C3 for M = 0.599.
(a)(b)(c)(d)(e)(f)(g)(h)(i)From Figure3, we can easily find that variable C converges to a fixed value within approximate 4×106 steps, and consistent results can be obtained with different initial values. The results indicate good and stable convergence of the LPNN method.


## 4.2. Comparison with the SOE Method

The comparison results of the fitted hyperbola obtained by the LPNN method and the traditional SOE method are shown in Figure4. The turbulence level and other flow qualities by the two methods are shown in Table 2.Figure 4
Comparison of hyperbola fitting from LPNN and SOE results. (a)M = 0.449. (b) M = 0.516. (c) M = 0.599.
(a)(b)(c)Table 2
Comparison of flow quality from LPNN and SOE results.
MethodΔm/m¯2¯ (C1)−2Δm/m¯ΔT0/T0¯¯ (C2)ΔT0/T0¯2¯ (C3)Tu (%)Goodness of fittingM = 0.449LPNN2.03×10−5−2.97×10−69.27×10−70.540.9981SOE2.02×10−5−2.90×10−69.11×10−70.540.9981M = 0.516LPNN1.26×10−5−4.51×10−71.54×10−70.440.9974SOE9.40×10−62.41×10−6−2.93×10−70.460.9992M = 0.599LPNN1.33×10−59.36×10−81.61×10−70.480.9864SOE6.28×10−66.29×10−6−7.92×10−70.520.9936For the caseM = 0.449, Figure 4(a) shows that the scattered points of turbulence level measurement test results basically conform to the hyperbolic distribution, and the fitted hyperbolae obtained by the LPNN method and the traditional SOE method basically coincide. The results in Table 2 show that the turbulence level and other flow qualities are basically equal for M = 0.449. Therefore, in the case when the scattered points conform to the hyperbolic distribution, both the LPNN method and the SOE method can be used to solve the turbulence level accurately.For the casesM = 0.516 and M = 0.599, Figures 4(b) and 4(c) show that the scattered points deviate from the hyperbolic distribution due to the unsteadiness of the flow, and there are even straight lines and other conic curves for the scattered points distribution. The fitting curve obtained by the traditional SOE method is an ellipse, which is inconsistent with the theoretical results from equation (3). Table 2 shows the high goodness of fitting values and negative C3 values for the SOE results of M = 0.516 and M = 0.599. C3 represents the mean square value of T0; thus, C3 should be positive for all situations according to its physical meaning. However, the results of C3 obtained by the SOE method are negative, which indicate that overfitting has occurred for M = 0.516 and M = 0.599. The scattered points are overfitted into ellipses by the SOE method, which results in negative C3 values. In this situation, although equations (12) and (13) can still be used to solve turbulence levels, the results are not true values.The fitting curve for the casesM = 0.516 and M = 0.599 obtained by the LPNN method is still hyperbola with the influence of the unsteadiness of the flow, and the goodness of fitting values are high, which is consistent with the theoretical results. From the results, we can easily find that the LPNN method is better than the SOE method, and the results verify the feasibility of applying the proposed algorithm to solve the turbulence level in wind tunnels.


## 5. Conclusions

In this paper, we propose a new hyperbola fitting algorithm for turbulence level measurement test data, based on the Lagrange programming neural network method, to solve the turbulence level in wind tunnels iteratively. The LPNN method is stable and the equilibrium point is proved to be asymptotically stable. The results show that, better than the SOE method, the LPNN method can be utilized to solve turbulence level, and the results will not be affected by the unsteadiness of the flow. The results of this paper will be mainly used to solve the turbulence level more precisely to ensure the high accuracy of wind tunnel test results.Future research will continue to further develop the hyperbola fitting algorithm on the basis of the LPNN method to improve the accuracy of turbulence level results. In addition, the LPNN method will also be improved to be applied in data processing for wind tunnel test results.



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*Source: 5620195-2020-10-23.xml*