Improved Momentum-Integral Framework for Modeling Viscous and Thermal Boundary Layers

Abstract

This work advances the classical Kármán–Pohlhausen (KP) momentum-integral method by applying an improved nearfield formulation, developed by Majdalani and Xuan, to several canonical problems. This formulation resolves a century-old inconsistency in boundary-layer predictions. The original KP formulation, while foundational, imposed a mathematically convenient but physically inaccurate condition on the second derivative of the velocity profile at the boundary-layer edge. Building on recent work by Majdalani and Xuan, this study adopts a corrected fourth-order polynomial representation (MX4) that eliminates this constraint, significantly improving the accuracy of both viscous and thermal boundary-layer predictions. The enhanced MX4 profile is first applied to external flows over circular cylinders under potential farfield conditions and then under real Hiemenz flow conditions. Compared to traditional KP profiles, the updated method yields more accurate estimates of separation angles, wall shear, and heat transfer metrics. Further improvements are demonstrated by incorporating the Hiemenz flow as a more realistic farfield approximation. This adjustment enhances the agreement with experimental data and computational benchmarks across a wide subcritical Reynolds number range. To generalize the method for more adaptable geometries, the analysis is extended to elliptic cylinders in crossflow with arbitrary aspect ratios. The study shows that increasing the aspect ratio delays separation and reduces both skin friction and drag, with the MX4 model performing especially well for slender configurations. Given the absence of consistent farfield models for non-circular geometries, new empirical velocity profiles are developed from RANS simulations, enabling practical implementation across a broader range of Reynolds numbers. The framework is further adapted to internal swirling flows within a cylindrical tube, a configuration motivated by propulsion applications such as the Vortex Combustion Cold-Wall (VCCW) chamber. By solving the axial, tangential, and thermal momentum-integral equations as a coupled system, the model captures the complex interplay between the swirl and Reynolds numbers. Results highlight the competing effects of swirl-induced thickening and Reynolds-driven thinning of the boundary layers, with corresponding impacts on wall shear and heat transfer. Overall, this work demonstrates the effectiveness and versatility of the momentum-integral approach in capturing key boundary-layer characteristics and behavior. The resulting methodology serves as a practical and efficient alternative to full numerical simulations for a variety of external and internal flow configurations relevant to engineering design.