On the Localized Transition of Pipe Poiseuille Flow Part II: The Role of Tensile Energy Flux Vector

This is the second article on the mechanical mechanism of laminar turbulent transition in pipe Poiseuille flow, which is one of the most important topics in turbulence research [1,2], as a representative of a large category of wall-bounded flows [3]. Traditional fluid mechanics stability research focuses on the effects of different disturbances and pays less attention to the mechanical properties of flow structures [4]. In this paper, the tensile energy flux, which is renamed from the viscous energy flux vector [5], and its divergence are deduced and visualized in pipe Poiseuille flow. The tensile energy flux vector is both zero at wall and in the center, and at a critical position 0.707R, the divergence of tensile energy flux vector is zero. Once the tensile force flow reaches its critical value [4], the critical position 0.707R is just the local position where onset of turbulence occurs, consistent with some experimental results [6]. This predicted position has a zenith angle of 45° if membrane theory of spherical shell is applied on the fluid [4], and this angle may be analogous to the cracks angle in the uniaxial compressive strength experiment of rock specimen subjected to uniaxial compression [7]. This article also proves that the critical Reynolds number during laminar turbulent transition in a circular tube is not a constant, but the ratio of critical tensile energy flux to average kinetic energy flux inside the tube is inversely proportional to the Reynolds number, similar to the inverse relationship between laminar flow resistance coefficient and Reynolds number.