Mathematical modeling of two-dimensional and three-dimensional turbulent wakes under pressure gradients

Research on turbulent wakes under pressure gradients has gradually attracted more attention due to its fundamental and applied significance. This study aims to propose new analytical solutions [including a new full solution (FS) and a new asymptotic solution (AS)] for two-dimensional and three-dimensional wakes. The model is developed based on the principle of the conservation of mass and momentum and the assumption that the velocity deficit in wakes follows the cosine distribution. The new asymptotic form is obtained by adding a high-order term to the streamwise momentum deficit equation. It is found that the new FS and the new AS are in excellent agreement with the experimental and numerical results. The new AS exhibits excellent convergence toward the FS, whereas significant discrepancies are observed between the original AS and the FS, particularly under adverse-pressure-gradient conditions. A comprehensive comparison between the FS and the ASs, as well as an in-depth analysis of the behavior of the FS, is conducted by varying the base flow inputs with different strengths and widths. The results further confirm a much better convergence performance of the new AS than that of the original one. It is also found that the FS exhibits instability characteristics when the maximum velocity deficit exceeds a critical threshold value. Compared to the relatively complicated FS and the original AS with lower accuracy, the new AS has the advantage of being a simple analytical form while retaining good predictive performance, which allows easy and accurate inspection of the wake development under pressure gradients. The mathematical model of wake flows presented in this paper is beneficial for scientific and engineering communities. © 2025 Author(s). Published under an exclusive license by AIP Publishing.