Fluid Mechanics dynamics of viscous fluid

The dynamics of real fluids is dominated by the forces of molecular viscosity, on the one hand, and by their importance in relation to the forces of inertia of the flow, on the other. When viscosity forces are large relative to inertial forces, the flow is regular, with the velocity field, or more generally fluid parameters, varying monotonically both in space and, possibly, in time. In this type of flow, known as laminar flow, any instability is dissipated by the fluid's viscosity. This is not the case, however, as soon as the forces of inertia become greater than the forces of viscosity. Instabilities, unavoidable in practice, develop in the form of vortices of various sizes: the flow becomes turbulent. In this flow mode, all transfers are improved, which is an advantage, but irreversibilities are greater, which is obviously a disadvantage.

Due to viscosity, any presence of a material wall implies a relatively strong change in the velocity field. If the flow is of the "external" type, i.e. when the walls occupy only a small part of the flow, variations are only felt in a zone close to the walls, called the boundary layer. Outside this boundary layer, the flow behaves like a perfect fluid. In "internal" flows, where the walls delimit a relatively small flow zone, the entire velocity field is subject to gradients.

Whatever the type of flow, the general balance equations (mass, momentum and energy) are applicable. In the vast majority of practical cases, however, it is impossible to solve these generally coupled partial differential equations analytically. In such cases, numerical solutions are required. This is true for laminar flows; it becomes a general rule for turbulent flows. Fluctuations in the fluid's thermokinematic parameters introduce additional unknowns that complicate the resolution considerably. The most common method currently used in industrial problems is the statistical method, in which only the mean values of the flow parameters are considered. Because of the non-linearity of the basic equations, this method requires modeling fluctuations and introducing a number of additional equations, known as closure equations, and coefficients that need to be calibrated against experience. Among the various models studied and proposed by specialists, the most common is that which uses the notion of turbulent viscosity and closure equations based on the transfer of turbulent kinetic energy k and its dissipation rate ϵ. To facilitate the resolution of industrial problems, various software packages are offered by companies specializing in this field.

Important: for notations and symbols, please refer to the end of this article.