A shape optimization method for resistance reduction of local piping components with multiscale validation

Abstract

Reducing the flow resistance of local components in building transmission and distribution systems is a key pathway to building energy conservation. Here we propose a novel low-resistance optimization method applied to U-bend shape design, with the minor axis a, major axis b, and offset c defined as shape features. The sample size is 150, and the parameter ranges are as follows: [50 mm, 150 mm] for a, [50 mm, 150 mm] for b, and [−30 mm, 30 mm] for c. On this basis, five representative machine learning regression modeling paradigms, including ridge regression, support vector regression, random forest, multilayer perceptron, and Gaussian process regression, are systematically compared, and the model with the best predictive performance is selected as the surrogate model for U-bend optimization. The proposed method is validated through full-scale experiments, numerical simulations, and turbulent energy dissipation. The results show that within a Reynolds number range of 1.0 × 10⁵ to 2.4 × 10⁵, the optimized U-bend achieves a resistance reduction rate of 13–24% relative to the traditional circular U-bend. This study provides a reference for the low-resistance design and energy-saving optimization of building transmission and distribution systems.

Ao Tian and colleagues propose a low-resistance optimization method for design of building fluid distribution systems, demonstrated in U-bends, which reduces flow resistance by up to 24%. This method supports energy savings and carbon emission reduction

Introduction

The building sector accounts for significant shares of global energy consumption and carbon emissions, namely, approximately 40% of global energy consumption and 28% of carbon emissions¹. With the increasing scale and complexity of buildings, the number of transmission and distribution systems that are used to transport hot and cold water, air, and other media within buildings is increasing, and the energy consumption of these systems now accounts for more than 30% of the total energy consumption of buildings². Fluid machinery requires enormous amounts of energy to overcome flow resistance within pipes to maintain the continuous operation of these transmission and distribution systems^3,4. Local components, such as U-bends, tees, and reducers, are major sources of resistance in fluid transmission and distribution systems⁵. Therefore, in-depth research and optimization of the flow profiles of local components to effectively reduce their resistance losses are urgent technical requirements for achieving the carbon neutrality goal of the building industry.

U-bends are common components in HVAC systems. Their bending geometry can lead to significant flow separation and energy loss, making research on U-bends a hot topic. Zhu et al.⁶ studied the erosion of a U-bend by solid‒liquid two-phase flow on the basis of the Euler‒Lagrange method and reported that there were two severe erosion zones at the outer bend of the U-bend at 40° ≤ θ ≤ 80° and 110° ≤ θ ≤ 150°. Wang et al.⁷ studied the flow and heat transfer characteristics of supercritical fluid inside a U-bend by combining numerical simulations and experiments and reported that the influence of the U-bend on the entire flow field was reflected mainly in the downstream region, which reached 175.5 times the pipe diameter. Han et al.⁸ used numerical simulations to study the effects of secondary flow in a U-bend on flow and heat transfer and reported that the Damköhler number of the inner wall of the U-bend was 102.4, which was significantly greater than that of a straight pipe, for which the corresponding value was 24.4. Through numerical simulation, Zhang et al.⁹ studied the flow field characteristics of gas‒liquid two-phase flow in a U-bend and optimized the shape of the U-bend to suppress small bubble aggregation. Liu et al.¹⁰ studied the flow interference between adjacent pipes and U-bends. Ligus et al.¹¹ combined particle image velocimetry and numerical simulation to study high Reynolds number flow in a U-bend and reported that as the bending radius decreased, the turbulent kinetic energy increased by up to 45%. Ishaque et al.¹² studied the icing phenomenon on the outer surface of a U-bend under low-temperature conditions. Rezaei et al.¹³ studied the flow of an ice slurry inside a U-bend. Liang et al.¹⁴ reported that the presence of droplets increased the total wall Nusselt number of low-water-content jet fuel inside a U-bend by 4–60%. You et al.¹⁵ reported that the average heat transfer coefficient inside a U-bend was 6–53% greater than that inside a straight pipe. Existing research has focused mainly on the flow field structure, heat transfer characteristics, and wear characteristics of U-bends and provided observations and empirical summaries of macroscopic flow phenomena; however, relatively insufficient attention has been given to the quantitative correlation between the geometry of U-bends and local resistance loss, and systematic shape optimization research with the explicit goal of reducing local resistance loss is lacking.

In contrast, research on resistance reduction optimization for other local components of pipelines has yielded relatively notable results. Idelchik’s handbook revealed that modifying the wall structure of local components can effectively reduce flow resistance¹⁶. Logachev et al.¹⁷ proposed two methods for measuring the resistance of exhaust hoods and optimized the shape of a circular exhaust hood, with a maximum resistance reduction rate of 90%. Liu et al.¹⁸ designed a low-resistance duct elbow on the basis of the principle of field synergy, which reduced the resistance by up to 31%. Yin et al.¹⁹ studied resistance reduction via the addition of guide vanes inside a water pipe tee and achieved a resistance reduction of more than 8%. Wang et al.²⁰ optimized the structural shape of a butterfly valve using numerical simulation, reducing the flow resistance by 21%. Najafi et al.²¹ studied the influence of guide vanes on the flow field of duct elbows through numerical simulation and reported that when the curvature ratio of the elbow is in the range of 3.3–5.1, installing guide vanes can reduce flow resistance. Yagmur et al.²² proposed the insertion of nonplanar guide vanes inside the bend to reduce local resistance, achieving a resistance reduction rate of up to 38%. However, on the basis of the published literature, existing research in the field of resistance reduction optimization for local components has focused mainly on components, such as tees, elbows, and valves, rather than U-bends. Therefore, it is necessary and practically important in the field of engineering to systematically develop data-driven shape optimization methods for U-bends to reduce their resistance.

In recent years, the introduction of machine learning in engineering shape optimization has proven to be an efficient strategy. The core advantage of this approach lies in the ability of machine learning algorithms to quickly search for optimal solutions in high-dimensional design spaces using surrogate models, thereby reducing time costs. Kim et al.²³ proposed a shape optimization method for adhesive pillars by combining machine learning with Bézier curves. An optimized adhesive pillar can improve the uniformity of the stress distribution. Fan et al.²⁴ used machine learning to optimize the structure of a cylinder for flow around the cylinder and reduced the flow resistance by 30%. Li et al.²⁵ reviewed the application of machine learning in aerodynamic shape optimization and reported that machine learning is an efficient shape optimization method. Sun et al.²⁶ used a neural network model to optimize the shape of a two-dimensional airfoil, and the optimal lift-to-resistance ratio reached 0.76. Tran et al.²⁷ used a dataset that covered a wide range of automotive geometries and aerodynamic performance in the industry and proposed a low-resistance method for optimizing automotive shape in which machine learning is used. Savage et al.²⁸ applied machine learning to assist in the design of new shapes of chemical reactors and improved experimental plug flow performance by 60%. Tian et al.²⁹ used a random forest model to optimize the resistance reduction of circular water pipe bends and achieved a resistance reduction of 19–26%. Therefore, the use of machine learning to optimize the flow shape of U-bends is an effective method for reducing their local resistance.

Compared with existing related work, this paper differs fundamentally in the following three aspects. First, in this paper, an optimization framework is constructed on the basis of a comparison of multiple modeling paradigms and the selection of the optimal surrogate model, rather than relying on a single preset machine learning model for optimization analysis, making the proposed method more robust and universal. Second, in the strategy for determining the optimal configuration, the approach shifts from experience-based trial-and-error optimization to surrogate model-driven global prediction of the parameter space, achieving direct determination of the optimal combination of geometric parameters. Third, the optimization results are verified using a U-bend as the optimization object and through dual-diameter experiments. In this study, a method for optimizing the shapes of local components is proposed, and its effectiveness is verified. First, the shape parameters of the local components are identified as features, and the local resistance coefficients that correspond to different shape parameters are calculated as labels. Then, a machine learning regression model is constructed, and five representative modeling paradigms are compared to determine the optimal surrogate model and predict the optimal solution. Next, the resistance reduction mechanism is revealed through turbulent energy dissipation analysis. Finally, a physical prototype is manufactured using 3D printing, and the effectiveness of the proposed method is verified through full-scale experiments and numerical simulations. A flowchart of this study is shown in Fig. 1. This study provides a new reference for the field of resistance reduction and energy dissipation in local components.

Fig. 1. Research framework.

Logical sequence of the methods used in this study.

Results

U-bend shape optimization

To address the local resistance characteristics of a U-bend, in this study, a computational model was constructed, as shown in Fig. 2a. Typically, sufficient turbulence development requires an upstream straight pipe length of at least 10 times the pipe diameter. Therefore, computational sections 1 and 4 were set at 40D upstream and downstream of the U-bend, respectively. By applying an area-weighted average to the pressure in each section, the corresponding total pressure values P₁ and P₄ were obtained, and their difference is denoted as ΔP_1–4. Furthermore, the friction loss ΔP_f,1–4 between the two computational sections was determined by removing the straight pipe from the U-bend. On this basis, the formula for calculating the local resistance coefficient ξ of the U-bend is as follows:

$$\xi =\frac{\mathrm{\Delta }{P}_{1-4}-\mathrm{\Delta }{P}_{f,1-4}}{P_v}=\frac{\left(P_1-P_4\right)-\left(P_1^{’}-P_2^{’}\right)-\left(P_3^{’}-P_4^{’}\right)}{\rho \left(v^2/2\right)}$$ 1

where P_v represents the dynamic pressure, Pa; ρ represents the density of the fluid, kg m⁻³; and v represents the average velocity at the U-bend inlet, m s⁻¹.

Fig. 2. Numerical model and geometric parameter definition for the U-bend.

a Numerical model for calculating U-bend resistance. b Schematic diagram of the U-bend shape parameters.

The reduction rate of the local resistance coefficient is represented by η, and its formula is as follows:

$$\begin{array}{c}\eta =\frac{\mid {\xi }_{tra}-{\xi }_{opt}\mid }{\mid {\xi }_{tra}\mid }=\frac{\mid \mathrm{\Delta }\xi \mid }{\mid {\xi }_{tra}\mid }\hfill \end{array}$$ 2

where ξ_tra represents the local resistance coefficient of the traditional U-bend, ξ_opt represents the local resistance coefficient of the optimized U-bend, and Δξ represents the reduction in the local resistance coefficient.

The resistance of the U-bend is quantitatively characterized by the local resistance coefficient ξ, defined as the ratio of the local total pressure drop to the dynamic pressure. The resistance reduction rate η is defined as the percentage reduction in the local resistance coefficient ξ of the optimized U-bend relative to that of the conventional U-bend. The shape parameters of the U-bend are shown in Fig. 2b. The shape of the U-bend can be determined on the basis of the shape and position of the central cross-section. The central cross section can be defined as an ellipse, with the minor axis denoted as a, the major axis denoted as b, and the offset of the center denoted as c (where the downward direction is positive and the upward direction is negative).

In this study, five representative machine learning regression algorithms were compared. As described in the “Machine learning” section, a nested cross-validation strategy was used to select the regression model best suited for U-bend shape optimization. The average R² and average RMSE of the five candidate regression models for the test set are shown in Table 1.

Table 1.

Average R² and RMSE of the five candidate regression models for the test set

Candidate regression model	Average R2	Average RMSE
Ridge regression	0.368	0.032
Support vector regression	−3.55	0.080
Random forest regression	0.880	0.014
Multilayer perceptron regression	0.748	0.020
Gaussian process regression	0.952	0.007

The results show that Gaussian process regression yields the highest average R² and the lowest average RMSE. Compared with the second-best-performing model, namely, the random forest, Gaussian process regression increases the average R² by 8% and decreases the average RMSE by 50%, demonstrating the best predictive performance. Therefore, Gaussian process regression is selected as the best surrogate model for the local resistance coefficient of the U-bend³⁰. The local resistance coefficient of the U-bend exhibits complex nonlinear coupling with the shape parameters, and the response trends vary across different parameter directions. As a result, the ability of support vector regression to characterize this problem is limited, leading to relatively weak predictive performance in the present study. This study involves a small-sample, low-dimension continuous variable regression problem. The training time for all five models is less than one minute, and the difference in computational cost is not significant; thus, it is not a major limiting factor in model selection. Therefore, the superior performance of Gaussian process regression in this study is methodologically justified. The final selected Gaussian process regression model included a Matérn-5/2 kernel with optimized hyperparameters of σ²_f = 64.64, ℓ = 6.34, and σ²_n = 0.0025.

Visualizations of the four performance evaluations of Gaussian process regression are shown in Fig. 3. A scatter plot that compares the predicted and actual values is shown in Fig. 3a, with all the data points clustered closely around the ideal line. The learning curve of the model is shown in Fig. 3b; as the number of training samples increases, the R² values of both the training and validation sets increase significantly and tend to stabilize, whereas the R² curve of the validation set converges rapidly and eventually stabilizes above 0.95. A scatter plot of the residuals is shown in Fig. 3c, with the vast majority of residual values below 0.02. Histograms and violin plots are presented in Fig. 3d to illustrate the distribution of the residuals; the residual values are concentrated near zero, and their frequency distribution approximates a normal distribution.

Fig. 3. Schematic diagram of the performance of the Gaussian process regression model.

a Comparison between the predicted and actual values obtained from the Gaussian process regression model. b R² values of the Gaussian process regression model for the training and validation datasets. c Scatter plot of the residuals of the Gaussian process regression model. d Histogram and violin plot of the residuals of the Gaussian process regression model.

In summary, the Gaussian process regression model demonstrates excellent performance in terms of fitting accuracy, generalization ability, and error distribution, which makes it a suitable surrogate model for this study. With the Gaussian process regression model, the local resistance coefficient ξ of each shape parameter is predicted for all combinations within its range of values. The parameter combination with the minimum local resistance coefficient ξ is then selected as the optimal solution. In other words, this model is used to perform a grid search over all integer combinations of the three shape features—minor axis a, major axis b, and offset c—within their range of values. Finally, the optimal shape of the DN80 U-bend was obtained with a = 130 mm, b = 65 mm, and c = −6 mm. According to the numerical simulation results, the resistance reduction rate η reached 19%.

Local sensitivity and stability analysis of optimal shape parameters

To analyze the reliability and stability of the optimal parameters, we applied perturbations within ±10% to three geometric parameters, centered on the optimal solution, and analyzed the changes in the local resistance coefficient. The local resistance coefficients corresponding to the perturbations of the optimal parameters were calculated through numerical simulation while consistent settings were maintained. First, we applied perturbations of ±10% and ±5% to one of the optimal parameters (minor axis a, major axis b, and offset c) sequentially while keeping the other two optimal parameters unchanged. We found that the local resistance coefficient of the U-bend increased to varying degrees, with the optimal parameter consistently corresponding to the lowest resistance level. Specifically, as shown in Fig. 4a, the change in the resistance coefficient is continuous and smooth; when the perturbation amplitude is ±10%, the relative increase in the local resistance coefficient is 2–5%. Next, we fixed one optimal parameter and applied perturbations of ±10% to the other two optimal parameters. Three two-dimensional parametric response surfaces near the optimal point were established. As shown in Fig. 4b, the local resistance coefficient in the a–b parameter plane ranges from 0.295 to 0.317. The optimal parameter corresponds to the smallest local resistance coefficient, with a maximum relative increase of 7% and no local abrupt changes. As shown in Fig. 4c, the maximum relative increase in the local resistance coefficient in the a–c parameter plane is 6%. As shown in Fig. 4d, the maximum relative increase in the local resistance coefficient in the b–c parameter plane is 4%. A combination of the one-dimensional and two-dimensional parameter analysis results reveals that when the geometric parameters deviate from the predicted optimal values, the local resistance coefficient increases to varying degrees, and the optimal parameter always corresponds to the lowest resistance level. As the parameter values change within the neighborhood of the optimal parameter, the change in the resistance coefficient is continuous and smooth and does not exhibit abrupt changes or high sensitivity to parameter perturbations.

Fig. 4. Local perturbation response analysis of optimal parameters.

a Changes in the local resistance coefficient under single-factor perturbations with respect to the optimal parameters. b Local resistance coefficient response surface in the a–b parameter plane. c Local resistance coefficient response surface in the a–c parameter plane. d Local resistance coefficient response surface in the b–c parameter plane.

In summary, the obtained optimal geometric parameters are not random results caused by local discrete points but rather reliable solutions with stable low-resistance characteristics within their neighborhood. Furthermore, in this study, the resistance reduction effectiveness corresponding to the optimal parameters was further verifiTurbulent energy dissipation and secoed through the energy dissipation principle and full-scale experiments.

Resistance reduction effect of the normalized U-bend

Quantifying the resistance reduction effect of the optimized U-bend under different working conditions through numerical simulation is a necessary task. In this study, the optimal shape of the U-bend for DN80 was determined, with the straight pipe diameter assumed to be DN80. However, in actual engineering, the U-bend is not limited to a single pipe diameter^31,32. Optimizing the resistance reduction of the U-bend for each pipe diameter separately is inefficient. Therefore, in this study, the optimal shape feature of the U-bend that corresponded to DN80 was normalized relative to the pipe diameter. The obtained dimensionless number can make the optimization results applicable to the U-bend for all pipe diameters. The normalization results of the three shape features were a/D = 1.625, b/D = 0.813, and c/D = −0.075. In this study, the effectiveness of the normalization operation was verified through numerical simulation. Except for the turbulence model validation, all the numerical simulations in this study used the k–ε Realizable model. For each operating condition, the numerical simulation yielded a unique result, and the difference between the numerical and experimental results was less than 2%. The resistance reduction effect of the optimized U-bend for six pipe diameters for four typical Reynolds numbers is shown in Fig. 5. In this study, the Reynolds number was used to verify the resistance-reduction stability of the optimized structure under different flow velocities, indicating that the studied operating conditions were within the fully turbulent range. In this study, local resistance was reduced by modifying the wall geometry of the U-bend. Geometric relationships reveal that the wall area of a circular pipe is inversely proportional to the internal fluid volume with respect to the pipe diameter. As the pipe diameter increases, the influence of the wall geometry on the internal fluid weakens, and the resistance reduction rate decreases to some extent². However, the optimized U-bend maintains a stable resistance reduction trend across different pipe diameters, with a resistance reduction rate ranging from 13 to 24%.

Fig. 5. Comparison of resistance under various working conditions.

a Local resistance coefficients of the traditional U-bend and the optimized U-bend for different pipe diameters and Reynolds numbers. b Schematic comparison of the geometric models of the traditional U-bend and the optimized U-bend.

Turbulent energy dissipation and secondary flow analysis

Energy dissipation is a key physical quantity for measuring the rate at which mechanical energy is converted into heat energy in a fluid system, and its magnitude can directly reflect the strength of flow resistance. Since high energy dissipation corresponds to high resistance, comparing the energy dissipation cloud maps of the optimized U-bend and the traditional U-bend can qualitatively demonstrate whether the resistance of the optimized U-bend has been reduced, and the specific proportion of resistance reduction is reflected by the resistance reduction rate η³³. The energy dissipation function ϕ per unit volume can be calculated with the following formula:

$$\varphi =\mu \left\{2\left[{\left(\frac{\partial u_x}{\partial x}\right)}^2+{\left(\frac{\partial u_{\mathrm{y}}}{\partial y}\right)}^2+{\left(\frac{\partial u_z}{\partial z}\right)}^2\right]+{\left(\frac{\partial u_x}{\partial y}+\frac{\partial u_y}{\partial x}\right)}^2+\right)\left({\left(\frac{\partial u_y}{\partial z}+\frac{\partial u_z}{\partial y}\right)}^2+{\left(\frac{\partial u_z}{\partial x}+\frac{\partial u_x}{\partial z}\right)}^2\right\}$$ 3

where μ represents the dynamic viscosity coefficient, N s m⁻², and $u_x$, $u_y$, and $u_z$ represent the partial velocities in the x, y, and z directions, respectively, m s⁻¹.

For the entire flow field V, the total energy dissipation can be expressed as Φ, and the formula for calculating Φ is as follows:

$$\mathit{\Phi }=\int \int {\int }_V\varphi dV$$ 4

The resistance effect of a local component is reflected not only in the pressure loss at the component but also, more importantly, in its continuous effect on the downstream flow field organization. After passing through a local component, the fluid often undergoes velocity distribution reconstruction and experiences increased turbulence intensity and secondary flow. These unsteady or nonuniform flow characteristics continue to develop along the flow direction over a certain distance. Therefore, the effects of local components on the overall flow performance of the system are usually concentrated in the downstream region. In other words, comparing the turbulent energy dissipation of the downstream flow field of a local component is key to evaluating whether the optimized component effectively reduces resistance. In this study, the turbulent dissipation rate was used as an indicator, and a visual analysis of the turbulent dissipation rate of the downstream flow field of the optimized U-bend and the traditional U-bend under the typical operating condition of Re = 1.2 × 10⁵ at the inlet was performed, as shown in Fig. 6b, c. The difference in the turbulent dissipation field of the downstream flow field between the optimized U-bend and the traditional U-bend was reflected mainly in the first 3D region, with the optimized U-bend reducing the area of the high-energy dissipation region. To quantitatively compare the differences in energy dissipation in the downstream flow field between the traditional U-bend and the optimized U-bend, a volume integration of the energy dissipation function of the flow field within a 5D range downstream of the U-bend is performed. The results show that the total energy dissipation of the traditional U-bend is 0.94 W, whereas that of the optimized U-bend is 0.78 W, representing a 17% reduction compared with that of the traditional structure.

Fig. 6. Comparison of flow field distributions.

a Schematic illustration of the selected cross-sections in the downstream flow field of the U-bend. b Turbulent energy dissipation field of the selected cross-sections for the traditional U-bend, where D denotes the pipe diameter. c Turbulent energy dissipation field of the selected cross-sections for the optimized U-bend. d Total pressure field comparison between the traditional and optimized U-bends. e Velocity field and streamline comparison between the traditional and optimized U-bends.

The magnitude of energy dissipation depends on the integral value of the velocity gradient within the flow field volume. Changing the shape of the U-bend alters the intensity of the velocity gradient and the integral boundary conditions. The optimal parameters are set in a way that balances the effects of the velocity gradient and the integral boundary conditions, minimizing both energy dissipation and resistance. In this study, the resistance reduction effect of the optimized U-bend is quantified based on the resistance reduction rate η; the optimized U-bend for DN80 achieves a resistance reduction rate of 19%.

Previous studies have shown that secondary flow and related complex flow structures are generated in curved channels. This issue has been extensively discussed from different perspectives, including experimental measurements, numerical simulations, and flow physics analysis, providing important references for understanding the complex flow characteristics in curved channels^34–36. Figure 6d, e shows a comparison between the total pressure field and the velocity field. Compared with the traditional U-bend, the optimized U-bend exhibits more uniform total pressure and velocity distributions at the outlet. Further combined with the streamlined distribution, it can be found that the streamlines corresponding to the optimized U-bend are generally straight and consistent with the main flow direction. In contrast, the streamlines corresponding to the traditional U-bend exhibit obvious lateral deviation and bending. This indicates that the optimized U-bend reduces flow distortion³⁷. To more accurately quantify the secondary flow intensity and the degree of flow distortion, in this study, the dimensionless number Se is used to characterize the magnitude of the secondary flow intensity, and the specific values of the secondary flow intensity in the downstream flow field are compared. The dimensionless number Se can be calculated using the following formula:

$$Se=\frac{\rho D_h^2}{\mu }\frac{{\iint }_A∣{\omega }^n∣dA}{A}$$ 5

where ρ represents the fluid density, kg m⁻³; D_h represents the equivalent pipe diameter, m; μ represents the dynamic viscosity coefficient, N s m⁻²; A represents the cross-sectional area perpendicular to the mainstream direction, m²; n represents the normal direction of the cross-section perpendicular to the mainstream direction; and ωⁿ represents the vorticity component in the normal direction of the cross-section.

Table 2 lists the specific values of the dimensionless number Se in the downstream flow field of the traditional U-bend and the optimized U-bend. The secondary flow intensity of both gradually decreases in the downstream direction. In contrast, the secondary flow intensity of the traditional U-bend is consistently greater than that of the optimized U-bend, with the difference exceeding 21%.

Table 2.

Downstream Se comparison of traditional and optimized U-bends

Downstream position	Traditional U-bend	Optimized U-bend
0D	178366.6	130255.6
1D	79403.0	62673.1
2D	48117.4	37616.6
3D	37055.6	24992.7
4D	31266.4	18107.0
5D	27326.3	14026.5

Experimental verification analysis

To evaluate the resistance reduction performance of the optimized U-bend under actual working conditions, full-scale experimental systems for DN80 and DN25 U-bend components were constructed in this study. The local resistance coefficients of the optimized and traditional structures were tested and compared with the numerical simulation results. The results for the DN80 pipe diameter are shown in Fig. 7. The optimized U-bend maintained a stable, low resistance performance at different Reynolds numbers, and the numerical simulation results were in good agreement with the full-scale experimental results. The conclusions obtained for the DN25 pipe diameter were consistent with those for the DN80 pipe diameter, and the detailed results are provided in Supplementary Fig. 1. The optimal U-bend that was obtained in this study significantly reduces the local resistance of fluid flowing through the component, reduces pressure loss during pipeline system operation, and provides certain technical support for energy conservation, emission reduction, and low-carbon operation in the building and environmental fields.

Fig. 7. Comparison of the traditional DN80 U-bend and optimized DN80 U-bend.

a Local resistance coefficients of the traditional U-bend and the optimized U-bend at different Reynolds numbers. The error bars represent ± one standard deviation calculated from eight independent experiments. b Photographs of the 3D-printed physical models of the traditional U-bend and the optimized U-bend.

To further assess the significance of the differences between the two structures from a statistical inference perspective, an independent-samples t-test of the differences in local resistance coefficients between the traditional U-bend and the optimized U-bend under various Reynolds number conditions with a DN80 pipe diameter was conducted on the basis of the results of eight repeated experiments. The statistical results are shown in Table 3. The results show that within the studied Reynolds number range, the differences between the two structures reached statistical significance (p < 0.001), indicating that the optimized U-bend exhibits significant resistance reduction effects under different operating conditions.

Table 3.

Independent-samples t-test results for local resistance coefficients

Re	Traditional U-bend mean	Optimized U-bend mean	t value	p value
1.0 × 105	0.367	0.298	13.311	<0.001
1.2 × 105	0.370	0.301	13.669	<0.001
1.4 × 105	0.375	0.304	14.532	<0.001
1.6 × 105	0.376	0.307	14.911	<0.001
1.8 × 105	0.379	0.308	14.182	<0.001
2.0 × 105	0.381	0.310	13.860	<0.001
2.2 × 105	0.384	0.314	14.557	<0.001
2.4 × 105	0.387	0.316	14.692	<0.001

A comparison of the U-bend geometries before and after optimization shows that the optimized U-bend involves a reshaping of the local contour in the curved region. This geometric reconstruction alters the local curvature distribution and transition characteristics within the bend, which helps moderate the fluid turning process in the U-bend and mitigate the local variations in transverse pressure gradient induced by the bend. As a result, the secondary flow in the optimized structure is alleviated to some extent, ultimately leading to a lower local resistance coefficient.

Discussion

U-bends are widely used local components in pipeline systems and represent important sources of resistance loss. To address the resistance loss induced by U-bends, in this study, a low-resistance local component optimization method was developed, and an optimized U-bend structure was obtained. For the multifactor regression problem associated with U-bend shape optimization, five representative machine learning modeling paradigms were evaluated and compared, among which Gaussian process regression exhibited the highest fitting accuracy, demonstrating its effectiveness as a surrogate model for predicting the local resistance coefficient of U-bends. On the basis of this model, a global exhaustive search was conducted to determine the optimal combination of shape features for the DN80 U-bend, yielding optimal parameters of a = 130 mm, b = 65 mm, and c = −6 mm. Numerical simulations and full-scale experimental results consistently revealed that the resistance reduction rate of the optimized structure reached 19%, confirming the effectiveness and reliability of the proposed optimization framework. Furthermore, normalization of the optimal geometric parameters with respect to the pipe diameter resulted in dimensionless shape characteristics of a/D = 1.625, b/D = 0.813, and c/D = −0.075, which were subsequently applied to U-bends with six different pipe diameters. At four representative Reynolds numbers, the resistance reduction rate ranged from 13 to 24%, indicating that the optimized structure maintains favorable low-resistance performance across different flow conditions and geometric scales. Turbulent energy dissipation analysis further revealed that the optimal U-bend structure effectively suppressed the formation and development of high-energy dissipation regions in the downstream flow field, providing a physical explanation for the observed resistance reduction. The strong agreement between the full-scale experimental measurements and numerical simulation results confirms that the optimized U-bend structure exhibits stable low-resistance performance under actual operating conditions. Overall, this study demonstrates that data-driven shape optimization combined with physical flow analysis provides a practical and scalable pathway for reducing local resistance losses in pipeline systems.

Here are some considerations regarding the results presented in this work. First, the parameter space considered in this work has covered a considerably wide region, and while we could explore an even larger region as future work, the optimal solution is not expected to vary significantly. Second, for pipe diameters other than DN80, the normalization process provides effective solutions that are close to the optimal ones. Furthermore, considering practical engineering applications, only liquid water was used as the working fluid, and the manufacturing variations introduced by 3D printing were not taken into account. Finally, because the local resistance coefficient varies little with the Reynolds number in the fully turbulent region, the optimal geometries corresponding to different Reynolds number datasets were not compared. These aspects will be further investigated in future work.

Methods

Machine learning

In this study, the shape of a U-bend was optimized. The resistance reduction problem for local components is essentially a multifactor regression problem²⁹. Machine learning is an effective method for solving various regression problems^38–41. In most studies, designers’ experience has been used to select a suitable regression model. However, for different problems, systematically comparing the prediction effects of various types of regression models to determine the most suitable regression model is more scientific. There are five representative modeling paradigms for different types of machine learning models: linear models, kernel method models, ensemble tree models, neural network models, and statistical probability models^42–44. In this study, a classic regression algorithm for each modeling paradigm was selected, and the most suitable regression model was determined via comparison. Compared with using only a single modeling paradigm, the five modeling paradigms can be used to systematically compare the applicability of different types of learning frameworks and reduce the randomness of model selection. In this study, ridge regression was chosen as the linear modeling approach. Support vector regression was chosen as the kernel modeling approach. Random forest regression was chosen as the ensemble tree modeling approach. A multilayer perceptron was chosen as the neural network model. Gaussian process regression was chosen as the statistical probability modeling approach.

Before training the model, the sample set needs to be determined. The shape of the U-bend can be determined by determining the values of three features: the minor axis a, the major axis b, and the offset c. The corresponding local resistance coefficient ξ was obtained by numerical simulation. The three features and one label form a sample set. Considering both accuracy and efficiency, a total of 150 sample sets were selected. This research focuses on the U-bend of a DN80 circular water pipe. Here, DN denotes the nominal pipe size. The DN80 U-bend used in this study had an outer diameter of 88.9 mm, a wall thickness of 4.5 mm, and an inner diameter of 79.9 mm. The value ranges of the three features were initially determined through preliminary experiments as follows: [50 mm, 150 mm] for a, [50 mm, 150 mm] for b, and [−30 mm, 30 mm] for c. Specifically, preliminary numerical simulations conducted within a wider parameter range revealed potential for a sharp increase in resistance or structural inefficiency. Numerical simulations were then used to select a relatively broad and reasonable parameter range. All simulations in the sample dataset were performed at a Reynolds number of Re = 1.2 × 10⁵, which is a typical Reynolds number in engineering applications. Each sample corresponds to a specific U-bend flow field. Because the dimensional tolerance for the local components was approximately 1 mm, the features were limited to integers. A total of 150 feature combinations were randomly sampled from the prescribed value ranges, and the corresponding labels were obtained through numerical simulations.

After the sample set was determined, to ensure the scientific rigor and objectivity of the model evaluation, a nested cross-validation strategy was used for model selection in this study⁴⁵. The nested cross-validation consisted of two layers: an outer fivefold cross-validation layer for estimating model performance and an inner threefold cross-validation layer for hyperparameter tuning. The evaluation metric was R². After the most suitable surrogate model was selected, a grid search over all integer parameter combinations within the defined ranges of the three features was performed to determine the optimal low-resistance U-bend shape, which is the optimal feature combination that minimizes the local resistance of the U-bend. The machine learning modeling in this study was implemented in Python 3.11.7 using the scikit-learn 1.2.2 library. The flowchart of the machine learning part of this study is shown in Supplementary Fig. 2. As reported in the subsequent sections, the resistance reduction effect of the optimal feature combination under different operating conditions was verified through numerical simulations and full-scale experiments.

Numerical simulations

Combining numerical simulations with full-scale experiments is an efficient method for studying low-resistance local components^46–48. The local resistance coefficient is defined and calculated on the basis of the time-averaged pressure drop; therefore, a steady-state numerical simulation method is used in this work. The geometric model was created in SpaceClaim software, and the numerical simulation was performed in Fluent software. Velocity boundary conditions were used at the inlet, and pressure boundary conditions were used at the outlet. No-slip boundary conditions were applied to the walls, and the surface roughness was set to 0.15 mm. The SIMPLE algorithm was used to determine the coupling between pressure and velocity, and the convection term was discretized using a second-order upwind scheme. During the calculation, the residual convergence criterion was set to 10⁻⁶, and the convergence criteria for key physical quantities (average velocity and pressure) were set to 10⁻³. The fluid was liquid water with a density ρ of 998.2 kg m⁻³, a dynamic viscosity μ of 1.002 × 10⁻³ N s m⁻², and a temperature of 20 °C.

Turbulence model validation

The appropriate selection of a turbulence model is crucial for accurately predicting the local resistance characteristics within a U-bend⁴⁹. In this study, the turbulence model was validated using results from full-scale experiments. Taking a traditional U-bend as the object and maintaining consistency in terms of the mesh, boundary conditions, and other solution settings, in this study, the calculation results of five typical turbulence models were compared. The wall function method is used to handle near-wall flow without fully analyzing the viscous sublayer. A scalable wall function is selected, with the wall y⁺ value ranging from 20 to 100 in this study. The inlet turbulence intensity is set to 5%. A comparison between the predicted values of each turbulence model and the results of the full-scale experiments is shown in Fig. 8a. The results of the k–ε Realizable model were closest to those of the full-scale experiments. With the inlet Reynolds number Re = 1.4 × 10⁵ as an example, the errors of the average results of the full-scale experiments and the five turbulence models were 7.7%, 5.1%, 3.5%, 1.9%, and 12.1%, respectively, with the k–ε realizable model having the smallest error. Therefore, the k–ε realizable model was selected on the basis of the experiments as the turbulence model⁵⁰. It is necessary to clarify the error of the k–ε realizable model across the entire Reynolds number range. Within the Reynolds number range of 1.0 × 10⁵–2.4 × 10⁵, the errors between the k–ε Realizable model result and the average of the full-scale experiments for the eight operating conditions were 1.7%, 1.4%, 1.9%, 1.3%, 1.9%, 1.9%, 1.8%, and 1.6%, respectively.

Fig. 8. Validation of the turbulence model, the velocity profile, and the mesh.

a Comparison of the results of different turbulence models and experimental results. The error bars represent ± one standard deviation calculated from eight independent experiments. b Validation of the velocity profile at calculation section 1. c Schematic diagram of mesh generation.

To demonstrate the rationality of the computational domain length, the velocity profile was verified using computational section 1 at 40D upstream of the U-bend as an example. As shown in Fig. 8b, the velocity distribution in this section exhibits typical characteristics of fully developed pipe flow, with an overall axisymmetric distribution. The velocity gradually increases near the pipe wall and tends to level off in the central region of the pipe, without significant distortion or offset, indicating that the selected upstream computational domain length is reasonable.

Mesh independence verification

The geometry of the U-bend exhibits obvious tortuous characteristics, and polyhedral meshes are advantageous because of their flexible element shapes, adaptability to complex surfaces, and high convergence efficiency^51,52. Therefore, in this study, a polyhedral mesh generation strategy was adopted, and local refinement was applied to the U-bend region and the wall boundary layer region. The specific details of the mesh generation are shown in Fig. 8c. The minimum skewness is 6.1 × 10⁻⁵, the maximum is 7.4 × 10⁻¹, and the average is 1.4 × 10⁻²; the minimum orthogonal quality is 0.3, and the maximum aspect ratio is 26.8.

In numerical simulation studies, the number of grid cells directly affects the reliability of the calculation results and the solution efficiency. Therefore, determining a reasonable grid density through grid independence verification is necessary^53–55. For the traditional U-bend, seven grid schemes with different fineness levels were established in this study, with 431k, 793k, 1441k, 1947k, 2989k, 3767k, and 4571k elements. Three calculation points were selected on the central axis of the cross-section 3D downstream of the U-bend. When the inlet Re = 1.2 ×  10⁵, the total pressure values and the local resistance coefficient values of the U-bend at the three calculation points under different numbers of grid elements were calculated, as shown in Fig. 9b, c. The results that corresponded to the grid with 2989k elements stabilized, and the difference from those of the finer grid with 4571k elements was less than 2%. Therefore, in this study, the grid division scheme that corresponded to the grid with 2989k elements was selected.

Fig. 9. Schematic diagram of mesh independence verification.

a Schematic illustration of the locations of the calculation points. b Total pressure values at the three calculation points in cases with different mesh sizes. c Local resistance coefficients of the U-bend for different mesh sizes.

Full-scale experiments

Full-scale experiments provide the most reliable basis for verifying the accuracy of numerical simulations and the effectiveness of machine learning predictions. In this study, the resistance reduction effect of optimizing the U-bend was evaluated through experiments. The structure of the DN80 experimental system is shown in Fig. 10. The DN80 U-bend was connected to the DN80 straight pipe by a flange for easy replacement. The U-bend was manufactured using 3D printing with white resin as the material. The layer thickness was 0.1 mm, and the surface finish (Ra) ranged from 11 to 25 μm. Postprocessing included cleaning and curing⁵⁶. The experimental system was driven by a frequency-adjustable pipeline circulation pump supplied by a vertical water tank, and the water level was kept constant during the experiment. The pipeline flow rate was measured by an electromagnetic flow meter, the pressure at the test point was collected by a pressure transmitter, and a distance of more than 15D was set in front of the test point. The upstream and downstream test points were both 40D away from the U-bend, and the pressure transmitters were mounted vertically upward⁵⁷. To avoid the influence of water temperature variations on experimental accuracy, an intermittent operation mode was adopted to control the water temperature. The water pump was shut down for half an hour to cool after running continuously for 2 h. The water temperature was measured every half hour during the experiment, and the temperature variation limits were controlled within 2 °C. The measurement results revealed that the temperature range during the experiment ranged from 19.7 to 21.1 °C, with a maximum fluctuation of 1.4 °C. Within this temperature range, the density and viscosity of water changed little, so their influence on the experimental results was ignored.

Fig. 10. Schematic diagram of the DN80 full-scale test bench system.

① butterfly valve. ② water tank. ③ Y-type filter. ④ rubber flexible joint. ⑤ pipeline circulation pump. ⑥ pressure regulating tank. ⑦ electromagnetic flowmeter. ⑧ pressure transmitter.

All the measurement instruments were newly purchased and factory-calibrated by the manufacturer. The pressure transmitters conformed to IEC 62828-2:2017⁵⁸, and the electromagnetic flowmeters conformed to ISO 20456:2017⁵⁹. During each experiment, after the system was adjusted to the target flow rate, it was considered to have reached a steady state when the fluctuations in the volumetric flow rate and differential pressure did not exceed ±1% of their average values over a continuous 60 s. After the steady state was reached, the data were continuously collected for 60 s, and the temporal average was determined based on the result. All the measurement data were saved by a paperless recorder. The basic parameters of the experimental equipment and measuring instruments are listed in Table 4.

Table 4.

Parameters of the experimental equipment and measuring instruments

Item	Instrument	Range	Resolution	Accuracy
Static pressure	Pressure transmitter	0–100 kPa	0.01 kPa	0.5%
Volumetric flow rate	Electromagnetic flowmeter	8–80 m3 h−1	0.01 m3 h−1	0.5%
Increases in the flow rate and head	Pump	10–60 m3 h−1	–	–

The calculation method for the local resistance of the U-bend is similar to that in the “Numerical simulations” section. For each operating condition, 10 measurements were performed when the local resistance coefficient was tested. After the maximum and minimum values were removed, the average value was calculated using the remaining 8 measurements. The experimental error is represented by the standard deviation s_x:

$$s_x=\sqrt{\frac{1}{n-1}{\sum }_{i=1}^n{\left(x_i-\overline{x}\right)}^2}$$ 6

where n represents the number of experiments, x_i represents the measured value, and represents the mean value.

In this study, the main sources of uncertainty were instrument accuracy uncertainty (u_ins), repeatability uncertainty (u_rep), and installation uncertainty (u_set). Instrument accuracy uncertainty is assumed to follow a uniform distribution, and repeatability uncertainty is calculated on the basis of the standard deviation (s_x) of eight repeated experiments⁶⁰. For the pipeline experiments, the uncertainty (u_set) associated with the installation conditions typically ranged from 0.2 to 1.2%; in this study, 0.6% was used as a representative value^61–63. The formula for calculating the relative combined standard uncertainty (u_c) for each operating condition is as follows:

$$u_c=\sqrt{u_{ins}^2+u_{rep}^2+u_{set}^2}=\sqrt{{\left(\frac{0.005}{\sqrt{3}}\right)}^2+{\left(\frac{s_x}{\sqrt{8}\xi }\right)}^2+{0.006}^2}$$ 7

To accurately quantify the uncertainty level of the experimental results, the experimental data for the DN80 pipe diameter were selected as an example, and the relative combined standard uncertainty u_c of the traditional U-bend and optimized U-bend under various operating conditions was calculated. The specific data are shown in Table 5.

Table 5.

Relative combined standard uncertainty of the DN80 experimental results

Re	Traditional U-bend	Optimized U-bend
Re = 1.0 × 105	1.1%	1.3%
Re = 1.2 × 105	1.2%	1.4%
Re = 1.4 × 105	1.2%	1.4%
Re = 1.6 × 105	1.1%	1.2%
Re = 1.8 × 105	1.1%	1.3%
Re = 2.0 × 105	1.3%	1.5%
Re = 2.2 × 105	1.0%	1.1%
Re = 2.4 × 105	1.1%	1.3%

The results show that the range of u_c under various operating conditions is 1.0–1.5%, which is significantly smaller than the resistance reduction rate. This finding indicates that the measurement uncertainty has a limited effect on the determination of the resistance reduction effect.

Furthermore, the same experimental scheme was used consistently, and a full-size experimental platform was also constructed for the DN25 U-bend, as detailed in Supplementary Fig. 3. The corresponding DN25 U-bend had an outer diameter of 33.7 mm, a wall thickness of 3.2 mm, and an inner diameter of 27.3 mm.

Author contributions

A.T., R.G. and A.G.L. conceived the idea and designed the experiments. A.T., J.K.R., Y.Y.W. and Y.B.G. performed the numerical simulations. A.T., R.Y.J. and Y.W. recorded and processed the data. Y.T. analyzed the data. A.T., J.K.R., R.G., Z.J.F. and S.H. contributed to writing and editing of the manuscript and to the interpretation of the data.

Peer review

Peer review information

Communications Engineering thanks Bin Xu, Luís F.N. Sá, and the other anonymous reviewer(s) for their contribution to the peer review of this work. Primary handling editors: [Philip Coatsworth].

Funding

R.G. discloses support for the research of this work from the National Natural Science Foundation of China General Program [grant number 52178090], the Outstanding Youth Science Foundation Program of Shaanxi Province [grant number 2024JC-JCQN-50], the Youth Innovation Team of Shaanxi Universities, the Principle of Low Resistance Turbulent Flow Topology Design for Building Fluid Distribution Network [grant number X20250071], and the Outstanding Scholar Collaboration Program. All other authors declare no relevant funding.

Data availability

The data that support the findings of this study are available from the corresponding authors upon reasonable request.

Competing interests

The authors declare no competing interests.

Supplementary information

The online version contains Supplementary material available at https://doi.org/10.1038/s44172-026-00705-5.