Course Details

Purpose of this course is the development of students’ ability to describe mathematically  fluid- mechanics problems. This includes formulation of the defining equations of the fluids motion and proper setting of the boundary conditions. Furthermore, through the presentation of analytical solutions to different flows and the observation of various flow fields, the course attempts to acquaint the students with the topology and development of the fluid flows and to introduce them into the crucial problem of Fluid Mechanics, that of the understanding and prediction of turbulence.

The first part of the course includes description of fluids kinematics in the context of continuum mechanics. The Lagrange and Euler representations of the flow and the motion of the fluids (normal and shear deformation, rotation),considered as continua, are discussed. The fluid’s state of stresses is also described (normal and shear stresses) and the concept of viscosity is introduced.

The second part includes the mathematical formulation of the main fluid mechanics conservation laws. Mass conservation, Newton’s second law and momentum theorem, the energy conservation equation (first and second thermodynamic laws), vorticity equations are some of the topics dealt with in this part. The basic stress-strain  laws are discussed and the Navier-Stokes equations of a viscous flow in different coordinate systems are developed (Cartesian, orthogonal, curvilinear).

The third part includes introduction to turbulence and its mathematical models, with emphasis to flows of technological interest. Description of the problem and possible solutions. The Reynolds stresses and their order of magnitude. The two types of flow, laminar and turbulent. The linear stability theory. The definition of turbulence. The balance of the average kinetic energy  of the fluid in turbulent flows, the turbulent boundary layer close to solid boundaries, the differential equation for the Reynolds stresses, the conservation of turbulence kinetic energy, the turbulence kinetic energy balance in a boundary layer, the turbulent flow in the vicinity of a solid boundary and special forms of the ?wall law?, the Boussineq hypothesis (1877) are some of the topics dealt with in this part. Furthermore, Prandtl?s zero order (algebraic) model – one equation model, and differential one-equation models are presented. Modeling the turbulence kinetic energy equation, Bradshaw model, two equation models of turbulent flows (k – ε) Launder, turbulence kinetic energy distribution and length scales in recirculating flows, the Reynolds-stresses model, and the effect of external forces in turbulence production. Correlation of two velocities, the inertial subdomain, measurements of the one-dimensional energy spectrum in fully developed pipe flow, the length scale of small eddies, the probability density and the intermittency factor, Fourier transformations and characteristic functions. The large- eddy-simulation (LES) model, and finally the most recent developments in the mechanism of turbulence production.