DNS approach to Turbulence and Vortex Dynamics in compressible and incompressible Fluids

As mentioned in the previous chapter, turbulence is characterized by the seemingly chaotic, yet correlated, motion of fluid elements (in both the Eulerian and Lagrangian sense) over a range of spatial and temporal scales. The distinguishing features of turbulence, however, are the lack of clear separation between the scales of motion (unlike in the kinetic theory of gases), a large number of degrees of freedom, and the large range of length and time scales (Kim et al. [2023]). Although it is accepted that the dynamics of turbulence can be described by the Navier–Stokes equations (NSE), it is prohibitively expensive to simulate all the length and time scales of motion. For many cases of interest, it is practically impossible to simulate turbulence without appropriate models such as LES and RANS (Kim et al. [2023]). The availability of coding languages offers a route for direct numerical simulation (DNS) of the Navier–Stokes equations for canonical flows on simple geometries and increasing Reynolds numbers. In this Thesis, a DNS code was developed in Python, and any part in C++ MPI (Message Passing Interface). This list represents a multi-processing method. The main problem of running code is the time processing, which is influenced by the use of the inverse method for resolving the linear system. The discretization in space of the Poisson equation, which appears in the Splitting-Method, produces a system in n equation, where n is the total number of the nodes. For this reason, the improvement of code in parallel computing (MPI) can be helpful. Another way to resolve the linear system can be the use of an iterative method. In the literature, overall approaches exist and can be synthesized through figure 4.1.
In this thesis, for resolving the Poisson equation, the Successive Over-relaxation (SOR) Method was used. It is a generalization of an improvement on the Jacobi Method (James and Riha [1975]).
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