Basic mathematical questions about an equation are: Is there a solution, is it unique and is it stable in some sense? In the case of the Navier-Stokes equations, there are no satisfactory answers to these questions. The existing results provide roughly speaking either long time results for very strict conditions on the initial data or short time results for weak conditions on the initial data. For a review we refer to Lions [125].
2.9.1 Analysis of the Euler equations
As for the Navier-Stokes equations, the analysis of the Euler equations is extremely diffi-cult and the existing results are lacking [36]. First of all, the Euler equations are purely hyperbolic, meaning that the eigenvalues of ∇fc·n are all real for any n. In particular, they are given by
λ1/2 =|vn| ±c,
λ3,4,5 =|vn|. (2.30)
Thus, the equations have all the properties of hyperbolic equations. In particular, the solu-tion is monotone, total variasolu-tion nonincreasing,l1-contractive and monotonocity preserving.
Furthermore, we know that when starting from nontrivial smooth data, the solution will be discontinuous after a finite time.
From (2.30), it can be seen that in multiple dimensions, one of the eigenvalues of the Euler equations is a multiple eigenvalue, which means that the Euler equations are not strictly hyperbolic. Furthermore, the number of positive and negative eigenvalues depends on the relation between normal velocity and the speed of sound. For vn > c, all eigenvalues are positive, for vn < c, one eigenvalue is negative and furthermore, there are
2.9. ANALYSIS OF VISCOUS FLOW EQUATIONS 33 zero eigenvalues in the cases vn = c and vn = 0. Physically, this means that for vn < c, information is transported in two directions, whereas forvn> c, information is transported in one direction only. Alternatively, this can be formulated in terms of the Mach number.
Of particular interest is this for the reference Mach numberM, since this tells us how the flow of information in most of the domain looks like. Thus, the regime M < 1 is called the subsonic regime, M > 1 the supersonic regime and additionally, in the regime M, we typically have the situation that we can have locally subsonic and supersonic flows and this regime is called transonic.
Regarding uniqueness of solutions, it is well known that these are nonunique. However, the solutions typically violate physical laws not explicitely modelled via the equations, in particular the second law of thermodynamics that entropy has to be nonincreasing. If this is used as an additional constraint, entropy solutions can be defined that are unique for the one dimensional case.
2.9.2 Analysis of the Navier-Stokes equations
For a number of special cases, exact solutions have been provided, in particular by Helmholtz, who managed to give results for the case of flow of zero vorticity and Prandtl, who derived equations for the boundary layer that allow to derive exact solutions.
Chapter 3
The Space discretization
As discussed in the introduction, we now seek approximate solutions to the continuous equations, where we will employ the methods of lines approach, meaning that we first dis-cretize in space to transform the equations into ordinary differential equations and then discretize in time. Regarding space discretizations, there is a plethora of methods avail-able. The oldest methods are finite difference schemes that approximate spatial derivatives by finite differences, but these become very complicated for complex geometries or high orders. Furthermore, the straightforward methods have problems to mimic core properties of the exact solution like conservation or nonoscillatory behavior. While in recent years, a number of interesting new methods have been suggested, it remains to be seen whether these methods are competitive, respectively where their niches lies. In the world of elliptic and parabolic problems, finite element discretizations are standard. These use a set of ba-sis functions to represent the approximate solution and then seek the best approximation defined by a Galerkin condition. For elliptic and parabolic problems, these methods are backed by extensive results from functional analysis, making them very powerful. However, they have problems with convection dominated problems in that there, additional stabiliza-tion terms are needed. This is an active field of research in particular for the incompressible Navier-Stokes equations, but the use of finite element methods for compressible flows is very limited.
The methods that are standard in industry and academia are finite volume schemes.
These use the integral form of a conservation law and consider the change of the mean of a conservative quantity in a cell via fluxes over the boundaries of the cell. Thus, the methods inherently conserve these quantities and furthermore can be made to satisfy additional properties of the exact solutions. Finite volume methods will be discussed in section 3.2.
A problem of finite volume schemes is their extension to orders above two. A way of achieving this are discontinous Galerkin methods that use ideas originally developed in the finite element world. These will be considered in section 3.8.
35
3.1 Structured and unstructured Grids
Before we describe the space discretization, we will discuss different types of grids, namely structured and unstructured grids. The former are grids that have a certain regular struc-ture, whereas unstructured grids do not. In particular, cartesian grids are structured and also so called O- and C-type grids, which are obtained from mapping a cartesian grid using a M¨obius transformation. The main advantage of structured grids is that the data structure is simpler, since for example the number of neighbors of a grid cell is a priori known, and thus the algorithm is easier to program. Furthermore, the simpler geometric structure also leads to easier analysis of numerical methods which often translates in more robustness and speed.
Figure 3.1: Example of unstructured (left) and structured (right) triangular grids
On the other hand, the main advantage of unstructured grids is that they are geomet-rically much more flexible, allowing for a better resolution of arbitrary geometries.
When generating a grid for the solution of the Navier-Stokes equations, an important feature to consider is the boundary layer. Since there are huge gradients in normal direction, but not in tangential direction, it is useful to use cells in the boundary layer that have a high aspect ratio, the higher the Reynolds number, the higher the aspect ratio. Furthermore, to avoid cells with extreme angles, the boundary layer is often discretized using a structured grid.
Regarding grid generation, this continues to be a bottle neck in CFD calculations in that automatic procedures for doing so are missing. Possible codes are for example the commercial software CENTAUR [35] or the open source Gmsh [67].