Non-equilibrium Flow

From CFD-Wiki

On the level of kinetic theory, even for an ideal gas with atoms having spherically symmetric potentials as well as for a molecular gas with the ability to store energy in internal degrees of freedom, there have been few attempts have been made to develop a model suited for wide ranges of cell Knudsen numbers Knc (Knc = lambda/Delta x, where lambda = mean free path, Delta x = typical cell dimension). In the collision dominated limit Knc --> 0 usually this new situation is faced by modifying the transport coefficients (typical example: Eucken correction ); the extension of the Enskog-Chapman perturbation method to molecular gases, however, already shows that, beyond these modifications, new phenomena (bulk viscosity ) appear. When the different degrees of freedom have largely different relaxation times, then relaxation equations may be added to the conservation (Navier{Stokes) equations. In the other extreme case of Knc > 1, as a pragmatic engineering approach, usually a Monte Carlo method is extended by some statistical model to describe the energy exchange between translational and internal degrees of freedom.

The availability of high speed numerical calculation allows to treat many problems on the level of kinetic theory. It is now possible to study individual particle interactions of arbitrary complexity and to introduce these results in form of transition probabilities into the kinetic equations. Since, with increasing complexity of molecules, also the dimensionality increases drastically, it is one of the major tasks in the model development to simplify in such a way that the problem still remains tractable while preserving the pertinent features of the physical phenomena. The interlaced system of a path-integral form of the kinetic equation and conservation equation has been used as unified solution for monatomic gases where the state of a gas flow changes drastically. Some examples of such flows are: free jet flows with very high stagnation to ambient pressure ratio, a gas in the gravitational field of a planet where density varies exponentially with altitude, gas rotating inside a high speed spinning cylinder with enormous density change over a small radius, or shock waves, boundary and shear layers. Approximation of the path-integral equation and different levels of approximation in the collision term of the kinetic equation allow an adequate and efficient description of flows at any degree of non-equilibrium. The basic idea behind interlacing conservation equations with the path integral (form of the kinetic equation) is to extend the capability of the kinetic equation to describe flow fields for very small cell Knudsen number Knc << 1, where the information cannot be passed along the grid using kinetic equation alone. By interlacing, while conservation equations, at one level, moves information along the grid, the kinetic equation, at molecular level, describes the degree of non-equilibrium correctly.