Control Volume V
3.5 Numerical Methods for Fluid Dynamics
with νsgs being the kinematic eddy viscosity. The most known formulation for the eddy viscosity was obtained by Smagorinsky in 1963 [100] and is represented by the following equation:
νsgs = (Cs∆)2| ¯S| (3.63) with | ¯S| = √
2 ¯S ¯S, ∆ being the filter width (volume of the cell) and Cs a constant that usually takes a value between 0.1 and 0.2. This model assumes that an equilibrium exists between the energy production in the large scales and dissipation in the small scales, which is not always a correct assumption.
3.4.3 DNS Models
The main concept behind DNS modelling is to solve the Navier-Stokes equation on the fluid domain without the use of any turbulence model. This means that the mesh is fine enough in all the regions of the domain to be able to solve the smallest dissipative scales (Kolgomorov scales). The increasing availability of computational resources has increased the use of DNS simulations in the liter-ature, but these are usually restricted to simple geometries and low Reynolds numbers, therefore not yet universally used. The number of mesh points (N ) has to respect the following [92]:
N3 ≥ Re2.25 (3.64)
therefore we can understand the enormous amount of resource needed in order to simulate even a flow with low Reynolds number. DNS simulations are however used in many cases to validate turbulence models because they represent techni-cally the real flow and give much more information respect to experiments, where physical quantities are usually measured in a point.
3.5 Numerical Methods for Fluid Dynamics
As stated in the previous section, there is currently no universal analytical solu-tion for the conservasolu-tion equasolu-tions. Therefore one approach used to solve fluid dynamics problems is by Computational Fluid Dynamics (CFD), by solving the equations numerically using modern computers. There are other approaches for the solution of fluid dynamics problems, such as the use of experimental
facili-3.5. NUMERICAL METHODS FOR FLUID DYNAMICS
ties, which have the advantage of representing the real physics, in the limitations given by the apparatus. Experimental analysis provides excellent results in the measurements of flow quantities such as the lift of an object, or the velocity field in a domain. The main problems are the intrusion caused by the measurement tools, and the limitation on the number of quantities and points that can be measured, not to mention the difficulty in reproducing real flows by matching the non-dimensional quantities.
On the other hand, CFD has theoretically no limitations in the representa-tion of a problem, and the quantities given are available in all the mesh cells of the domain, providing a more detailed view of the whole flow field. Still, there are issues related to CFD solutions, such as the mesh sensitivity, stability and convergence of the solutions. In the past years, due to the exponential increase of computational resources, CFD solutions have become much more accurate, and the number of users taking advantage of them has grown massively.
Any method used for a numerical solution needs to go through some base com-ponents such as the mathematical model, the discretisation method, the choice of a computational grid, finite approximations of physical quantities, a solution method and eventually a convergence criterion. All these steps will be discussed briefly.
3.5.1 Mathematical Model
The first step for the solution of a CFD problem is the choice of a specific set of equations. The general conservation equations are too complicated, and one needs to make some assumptions in order to simplify them (2D flows, incom-pressible, isothermal, etc.). This is done because it is unachievable to obtain a set of equations valid for every type of problem, and some simplifications are only valid for a particular kind of flows (for example assuming that the flow around a supersonic aircraft is incompressible would lead to entirely wrong results).
3.5.2 Discretisation Method
The next step to be taken is to choose which type of discretisation to apply to the specific set of equations. This usually means to build a specific set of algebraic equations that represent the original partial differential equations in particular points of the domain. The solution of algebraic equations is easier and obtained by the use of iterative methods. There is a wide range of approaches that lead
3.5. NUMERICAL METHODS FOR FLUID DYNAMICS
to different formulations, but the most used ones are the Finite Difference (FD), Finite Volume (FV) and Finite Element (FE) methods. All three methods lead to the same solution if the mesh is fine enough in all the approaches. The choice of one or the other depends on the type of problem and the developer.
Most of the commercial and open-source software is currently using the Finite Volume Method, because it is highly conservative, and this will be the only one described in details.
3.5.3 Computational Grid
As the CFD process leads to a set of discretised equations, the solution is available in determined points or cells, depending on how the grid was obtained. A wide range of types of grid exists, but the primary classification is: structured or unstructured.
• A structured grid is usually defined by regular connectivity, and each point or cell of the grid can be numbered consecutively, allowing a high space efficiency, better convergence and higher resolution. An example of a 2D structured grid is shown in Figure 3.5.
Figure 3.5: Structured Grid around a semi-cylinder
Even though they are advantageous from a solution point of view, structured grids are limited to simple geometries, and due to their nature, a fine spacing in one specific area of interest leads to a waste of resources in other zones where fine spacing is not required, increasing the data size and computational time.
• Unstructured grids are the most flexible because they can fit any arbitrary solution domain boundary. Connectivity is irregular in such grids. Therefore it is not simple to express them as an array of points in a simple way, such as
3.5. NUMERICAL METHODS FOR FLUID DYNAMICS
structured grids, and for this reason, the solution is usually slower if compared to a structured mesh. The most used types of unstructured grids are triangular in 2D domains and tetrahedral in 3D ones, but there is no restriction in the shape of the elements nor the number of neighbours of each cell. An example of an unstructured grid is shown in Figure 3.6.
Figure 3.6: Unstructured Grid around a cylinder
• Structured and unstructured grids can be used in the same domain, defining a so called ”hybrid” grid. These type of grids are somehow advantageous.
Figure 3.7: Hybrid Grid used to model the boundary layer
To conclude this section on computational grids, it is worth mentioning the meaning and use of staggered grids. While on collocated (also known as non-staggered) grids the variables are all stored in the same location, in staggered grids the scalar variables are stored in the cell centres of the control volume, while vectorial quantities such as velocity are located at the cell faces. This allows to face coupling problems between pressure and velocity. On the other hand, the