Calculation of streaklines for time periodic flows

Abstract

The calculation of streaklines for complex time-dependent flows is confronted with severe difficulties due to the huge data sets involved. In the present work we propose a storage reduction method for tune-periodic flows. The method is based on the observation that many time-periodic flows can be accurately approximated using a small number of Fourier terms. By storing only the significant harmonics of the Fourier decomposition, the storage (disk space and core memory) required for calculating streaklines can be reduced by one order of magnitude or more. The reduced storage permits the calculation of the streaklines after completing the solution of the equations, rather than calculating a set of predefined streaklines simultaneously with the solution of the flow. This significantly enhances the capability of studying interactively complex flowfields. Test cases confirm the assumption that a small number of Fourier terms is adequate for calculating accurately the streaklines of complex flows. HE simulation of time-dependent flows is one of the major top- ics of interest in contemporary computational fluid dynamics (CFD) studies. With the increase in computing power availability, complex unsteady flows can be calculated, creating huge data sets. In typical three-dimensional cases, <9(106) mesh points and (9(104) time steps are required, generating data sets of (9(1010) words of storage. In two-dimensional cases less storage is required (<9(108) words), but still it might be excessive even when large supercom- puters are used, not to mention workstations that are mostly used for postprocessing the results. These large data sets confront difficult challenges in the postpro- cessing of flow simulations. One of the common flow visualization techniques is based on the calculation of the streaklines that mimic dye-injection experiments. This calculation is computationally in- tensive. Yet, the main obstacle in the case of time-dependent flows is the need to store (both on disk and in memory) large data sets (the whole evolution of the flow). One way to overcome this problem is to calculate the streaklines simultaneously with the solution of the flow equations. This necessarily means that the release points and release rate of the particles should be predetermined. The flow cannot be studied interactively, which is a severe shortcoming in the case of complex flowfields. Time-periodic flows are a special class of time-dependent flows. Time-periodic flows are obtained as self-excited flows (e.g., the von Karman vortex street) or by forced oscillating perturbations (e.g., the flow in blood vessels or certain hydraulic devices). Sim- ple time-periodic flows (with a narrow spectra) are mostly found in low or intermediate Reynolds (Re) number cases, but certain high Reynolds number flows can be also approximated as periodic flows. In the present article, we suggest how to overcome some of the problems associated with the calculation of streaklines in time- periodic flows by using the Fourier transform of the solution. The so- lution itself is obtained from any unsteady flow solver. This approach might be appealing if a small number of harmonics can accurately approximate the time-dependent solution. We will demonstrate that in certain cases the saving in storage space and core memory might be more than one order of magnitude. This saving can be utilized for calculating streaklines interactively even for solutions employ- ing a very large number of mesh points, a task that is in many cases impossible otherwise.