EVALUATING THE IMPACT OF SURFACE TRANSVERSE CURVATURE ON BOUNDARY LAYER FLOWS: AN EXTENDED THWAITES INTEGRAL APPROACH
Authors: Awais Muhammad , Mehmood Ahmer 
Published: December 2024

The existing Thwaites integral method does not account for the surface transverse curvature (TVC) effects on laminar boundary layer flows over axisymmetric bodies of revolution. Moreover, the method simplifies to the two-dimensional planer boundary layer case when it is applied to analyze the boundary layer flow over a circular cylinder. This study aims to extend the existing method of Thwaites to investigate the impacts of TVC on boundary layer flows and to extend its applicability to boundary layer flows over a circular cylinder. The classical Thwaites integral method is readily available for studying boundary layer flows where the boundary layer thickness is much smaller compared to the body’s radius, i.e., &delta; &#60;&#60; <i>r</i><sub>0</sub>. However, there are such flow situations where the boundary layer thickness is comparable to the radius of the body, i.e., &delta; &#8771; <i>r</i><sub>0</sub>. In such cases, the TVC assumes a prominent role, influencing the boundary layer flow. The fundamental premise of the existing method of Thwaites based on the assumption that a flow parameter m &#61; (&delta;<sup>2</sup><sub>m</sub>/&nu;)(dUe/dx) characterizes the growth rate of the boundary layer, without considering the impacts of surface TVC. In order to extend the applicability of the existing method of Thwaites to the axisymmetric boundary layer flows taking into account the TVC effects, the parameter m has been modified. Consequently, other Thwaites functions have also been modified for the boundary layer flows with TVC effects. The extended Thwaites integral method has the capability to account for TVC effects on axisymmetric boundary layer flows and has been applied to a retarded axisymmetric boundary layer flow over a circular cylinder, serving as a practical case to check its validation and accuracy. The newly proposed Thwaites parameters have been employed to calculate key boundary layer parameters, including momentum thickness, displacement thickness, skin-friction coefficient, and boundary layer separation against the various values of curvature parameter κ. It has been observed that integral values of the aforementioned quantities are in fairly good agreement with the numerical solution for 0 &le; &kappa; &le; 5. For instance, the maximum percentage error between the present and numerical values of the momentum thickness for &kappa; &#61; 0.50 is only &sim; 3&#37;. Similarly, for the calculation of the displacement thickness against &kappa; &#61; 0.75, the percentage error is noted to be 3&#37;. The maximum percentage error in the calculation of the skin-friction coefficient for &kappa; &#61; 0.75 is reported to be only 5.80&#37;. Additionally, the separation points against different values of κ have been calculated, with the maximum percentage error of 14&#37; for &kappa; &#61; 1.5. The findings have been presented through tabular and graphical representations, and comparisons have been given with the exact results. The study demonstrates a reasonable level of precision between the approximate and exact solutions, affirming the effectiveness of the extended Thwaites method in providing accurate results of the axisymmetric boundary layer flows under TVC effects.