In urban water supply and drainage systems, rainwater channels or pipes are rectangular in design to help control the flow rate and adapt well to limited space. When the Bernoulli equation in fluid mechanics is used to solve the head loss of rectangular pipelines, the velocity parameter used in the kinetic energy term is usually the instantaneous or average velocity of the section at a certain point. Given that this velocity parameter is in exponential form, the smaller the error is, the greater the impact on the result will be. Thus, the kinetic energy term must be corrected. This study focuses on establishing a cross section velocity distribution model in a rectangular pipe and deriving the kinetic energy correction coefficient through the velocity distribution. Based on the Navier–Stokes equation, the partial differential equation describing the velocity distribution is further refined and simplified. Combined with the boundary conditions of the pipeline, the method of separating variables and Fourier transform are used to solve the equation. An example shows how to establish the velocity distribution model and find the analytical solution. Finally, the analytical formula of the kinetic energy correction factor of different cross section parameters and fluid properties is derived. To verify the accuracy of the analytical formula, the Fluent numerical simulation software is used for empirical verification, and then the Deming regression method is used to analyze the error between the theoretical and experimental values. The regression results of the kinetic energy correction coefficient prediction model established in this study are consistent with the actual values, and the confidence interval reaches 95%. This work provides strong guidance for the prediction of the kinetic energy correction coefficient in fluid mechanics and has an important theoretical and practical value.

Hydraulic sand-jetting perforation is widely employed in oil and gas reservoir stimulation and downhole operations, where the nozzle profile design is crucial for enhancing jet energy utilization and improving perforation quality. However, traditional nozzle design often relies on empirical methods and trial-and-error approaches, making it challenging to achieve an optimal contour that balances high efficiency and low energy loss. Against this background, this study focuses on the working conditions of hydraulic sand-jetting perforation and utilizes the Euler–Lagrange (EL) model in combination with the Shooting method to theoretically model and numerically solve for the optimal converging nozzle profile. By formulating a variational optimization problem and incorporating comparisons of turbulent kinetic energy, analyses of pressure and velocity distributions, as well as comparative simulations of different nozzle contours (including a straight conical nozzle, a hyperbolic nozzle, a Vitoshinsky-profiled nozzle, and the EL-based nozzle), the results show that the Vitoshinsky and hyperbolic nozzles achieve increases in outlet velocity of approximately 6.81% and 5.48%, respectively, compared to the straight conical nozzle. In contrast, due to its initial expansion followed by contraction, the EL nozzle experiences a slight reduction in outlet velocity (around 5.82%) while maintaining a more stable internal flow field and a shortened local acceleration zone. This configuration facilitates more rational energy distribution and reduces turbulence-induced losses. The findings provide feasible theoretical and engineering guidance for nozzle profile optimization, laying a foundation for improving the efficiency of hydraulic sand-jetting perforation and advancing nozzle design.

In the fields of oil and gas, aerospace, and marine engineering, elliptical pipes are utilized for optimizing space usage, reducing aerodynamic drag, and enhancing structural resilience against bending and torsional forces. Predicting and optimizing fluid dynamics within such configurations remains challenging due to complex velocity distributions impacting kinetic energy calculations. This study explores the interaction between the kinetic energy correction factor and velocity distribution in elliptical pipes by uniquely solving the Navier–Stokes equations in an elliptical coordinate system. We begin by deriving transformation relationships between elliptical and Cartesian coordinates and defining gradient and Laplacian operators to express fundamental fluid dynamics quantities such as velocity vectors and pressure distributions. After analyzing the steady-state Navier–Stokes equations, we establish a set of partial differential equations that describe fluid velocity and pressure. Employing the method of separation of variables and linearizing into an eigenvalue problem, we analytically solve these equations. To confirm the validity of our theoretical models, we utilize computational fluid dynamics with the Wray–Agarwal turbulence model, alongside physical experiments with Preston tubes, which corroborate the predicted velocity distributions in elliptical pipes. Comparative analysis with experimental data using the root mean square error (RMSE) and Bland–Altman methods demonstrates that RMSE values are less than 0.05 and all differences fall within the 95% confidence interval. This level of precision significantly improves the reliability of kinetic energy correction factor predictions and supports the design and study of fluid dynamics in elliptical pipes.