Mathematical model of the boundary layer of airflow over a flat surface

Abstract

In fluid and gas mechanics, a subject that has been the focus of considerable scholarly attention is the modeling of flow phenomena on streamlined surfaces. These surfaces, which are typically installed in a parallel orientation to the direction of the free stream, play a pivotal role in the study of fluid dynamics. The boundary layer theory represents a pivotal branch of fluid dynamics, given the airflow plate is characterized by a high Reynolds number of airflow velocity entering the plate’s plane surface contexts. The complexity of the problem is due to the nonlinearity and multidimensional nature of the governing equations. This study proposes a straightforward numerical methodology that can address a range of nonlinear problems in surface flow mechanics, particularly in the context of near-wall boundary layers of a planar nature. The paper considers the process of air motion as an isotropic Newtonian medium on the surface as an isotropic Newtonian medium layer. The resulting differential equation is expressed in dimensionless quantities and then solved numerically using the Runge – Kutta method. The velocity distribution in the boundary layer on a flat airflow plate is obtained. As the airflow velocity entering the plate surface increases, so too do the tangential stresses. The nature of the change in tangential stresses is linear in the initial coordinate, corresponding to the onset of airflow entering the plate surface, with two transition points identified at Mach number M = 1 and M = 3. It is evident that along the entire length of the surface of the flat plate, the nature of the change in tangential stresses is not a linear dependence. Consequently, with an increase in the distance from the end face of the plate from 1 mm to 100 mm (a 100-fold increase), the tangential stresses decrease by 10 times within a given length interval. Furthermore, within the length interval ranging from 0.1 m to 1.0 m, the tangential stresses undergo a reduction by a factor of 3. The presented method of modeling the distribution of velocity and tangential stresses in the boundary layer on a flat airflow surface makes it possible to calculate the force loads on the surface in the entire range of flow velocities for an incompressible medium.