Anisotropic turbulence in a flow of incompressible fluid between rotating coaxial cylinders

Abstract

The problem of developed turbulent flow of a viscous incompressible fluid between circular coaxial cylinders (Couette flow) is solved. The inner cylinder rotates with constant angular velocity, and the outer cylinder is fixed. The region of flow is divided into two boundary layers adjacent to the cylinders, and the kernel of the flow. Experimental and theoretical studies have shown /1–4/ that the turbulent flow of a Newtonian fluid near a rigid wall has a structure. The structure consists of an ordered system of boundary vortices which determine, at every point of the stream, a characteristic direction, and at the rigid wall itself the vortices are directed along the streamline. Therefore, the turbulent fluid in the boundary region must be treated as anisotropic /5, 6/ and in /6/ it is assumed that the viscous anisotropy of a turbulent fluid is analogous to the anisotropy of liquid crystals. It was shown in /7/ that the turbulent flow of a Newtonian fluid near a plane wall can, in fact, be described within the framework of the Ericksen-Leslie model /8, 9/ of an oriented fluid, provided that certain additional conditions can be imposed on the defining constants of the model. In the present paper the turbulent fluid in the boundary regions is regarded as an oriented fluid /7/ and as a viscous fluid with a turbulent viscosity that is constant over the transverse cross-section at the kernel. Unlike existing solutions the model remains unchanged when changing from the flow between two parallel walls to the flow between rotating cylinders. The solution obtained shows good agreement with experimental data /10/.