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A Critical Assessment of Simluations for Transitional
and Turbulent Flows
Tapan K Sengupta
High Performance Computing Laboratory, Dept. of Aerospace Engg.,
Indian Institute of Technology Kanpur,
Kanpur 208 016, India.
E-mail: [email protected], [email protected]
In the symposium we have seen a very wide range of simulations for transitional and turbulent flows. In writing this epilogue, the author takes the responsibility of identifying some common elements which add to success of desired DNS and LES. Of course, the single common element of any such activities is to be aware of numerical dispersion relation and how it relates to physical dispersion relation. We also investigate suitability of a specific combined implicit-explicit (IMEX) time integration method in performing DNS, which has been used to explore the so-called bypass transition. While IMEX methods have often been used, yet its properties and potentials are not assessed. Specifically, error analysis by solving a model equation that mimics some of the physical processes in DNS is not performed. A model equation with exact solution allows tracking of error with time for IMEX time integration methods, with either explicit or implicit spatial discretization. This analysis is motivated by recent DNS of flow transition over a flat plate reported in Bhaumik and Sengupta (Phys. Rev. E, 89, 043018, 2014) and Sayadi et al. (J. Fluid Mech., 724, 480-509, 2013), both trying to mimic the classical experimental results on flow transition in Schubauer and Skramstad (J. Aero. Sci., 14, 69, 1947). These two simulations show distinctly different computed solutions, yet the average flow quantities show remarkable similarity. In Bhaumik and Sengupta (2014), explicit dispersion relation preserving time integration method has been used everywhere in the domain. In contrast, in Sayadi et al. (2013), implicit time integration method is used very close to the plate, while explicit time integration method is used in rest of the domain. Such splitting of domain into explicit and implicit regions causes spurious internal reflection of the signal at the interface between these regions, creating unphysical, spurious and upstream-propagating q-waves which considerably contaminates solution. This is investigated in detail along with other issues of high resolution computing here to show suitable methods for DNS.
Keywords: DNS; LES; Transitional Flow; Turbulent flow; Error analysis; q-waves; Compact schemes; IMEX method; Anisotropic wave propagation; Round-off error
1. Introduction
Direct numerical simulation (DNS) is attempted by many investigators for canonical problems. Despite its widespread use as a tool of scientific computing, yet there are no known accepted definition of what constitutes DNS. It is not adequate to state that not using any model(s) of transition and turbulence in simulating high Reynolds number flows qualifies as DNS. In that case, all laminar flow computations should have been considered as DNS. It is understood in fluid mechanics context, that simulating flow fields with wide-band energy and enstrophy spectrum without models constitute DNS. In a generic simulation of dynamical system governing space-time dynamics, this definition can be broadened to include simulations of wide-band spectral events without any numerical artifacts. Such numerical artifacts could be wide ranging from altering the governing dynamical equation; trying to study so called free vibration problems by studying forced vibration; adding spurious dispersion and dissipation of numerical methods, so much so that the physical dispersion relation (if it can be identified in the first place) is poorly followed by adopted numerical dispersion relation and adding other sources of numerical errors. In this context, the main issue is one of evaluating the numerical dispersion relation. It is somewhat ironic that the celebrated error analysis of von Neumann used all over the world today, and which remained classified during the second world war years due to its supposed importance, has been the main reason, which has spawned misunderstanding of numerical dispersion relation. This has been corrected by the author’s group and reported in Refs. 33 and 35, following an early lead in Ref. 14. This will be amply demonstrated by a simple test case and rudimentary metrics identified for correct error analysis. This may provide another qualitative description of what constitute DNS. In short, DNS is one, in which dispersion relation preserving (DRP) method is used and which minimizes all known sources of error. This has been supported by Prof. Sheu and Prof. Deville in the panel discussion that DRP schemes have to be established as the principal tool for DNS and LES.
Thus it appears that not obtaining the correct numerical dispersion relation is responsible for the slow growth of scientific computing, apart from nonlinearity of the governing differential equation. However, computers are used to solve nonlinear problems to understand the physics of the problem. So the inability to solve nonlinear problems in closed form provide opportunity for scientific computing to bring the second wave of development in mathematical physics. Establishment of calculus brought the first wave of scientific knowledge creation. So it is going to be very interesting to be traveling with the second wave! Computing and continuum mechanics is again the crucible where ideas and applications are fine-grained and analyzed for further development. However, we will keep our attention focused mainly on fluid dynamical problems in this section.
The second-most important issue of scientific computing related to inability of handling non-periodic problems. Lack of analysis tools for numerical methods has somewhat delayed progress so far. The community (specially who are more mathematically inclined) has spent a lot of time to study so-called periodic problems (note the warning of Prof. Deville in the panel discussion in this context). A global spectral analysis (GSA) has appeared in literature over more than a decade from the author’s group (see Refs. 28 and 36 for complete details) for full domain analysis of non-periodic problems. It is equally relevant to note that GSA can accommodate all types of discretization schemes and nothing escapes its scrutiny, be it a finite difference, finite volume, finite element or Fourier spectral method – all such schemes have been analyzed29,32. The existence of such a tool should be viewed as a great advantage for all sentient researchers in the field.
Of course GSA requires adopting a model equation for calibration and the one adopted here is the linear wave equation or one-dimensional (1D) convection equation. The reason for the choice of this model equation needs some elaboration. In the context of the symposium theme, we gathered to discuss about transitional and turbulent flow. The governing Navier-Stokes equation (NSE) is a statement of dynamical equilibrium between mainly two processes: convection and diffusion; apart from imposed pressure gradient and unsteadiness. For high Reynolds number transitional and turbulent flows, convection mainly dictates the dynamics. So from that perspective alone, adopting linear convection equation appears as the lowest common denominator. It is not relevant to distinguish between 1D and multi-dimensional convection, as it has been amply demonstrated in Ref. 39, as to how GSA can be extended to multi-dimensions and related errors due to anisotropy of numerical methods can also be studied. The second reason for adopting linear wave equation is due to the perspective that...
