Content area
Abstract
The combined problem posed by turbulence and viscoelasticity has been considered as the most difficult problem in physics even in 1990s. However, Direct Numerical Simulations (DNS) of Newtonian and non-Newtonian turbulent flows have become more popular with the increasing computer capabilities (high performance clusters of parallel supercomputers), the development of advanced viscoelastic models tailored for dilute polymer solutions, based on the kinetic theory, and accurate spectral methods with high efficiency. Based on an initial breakthrough from our research group in 1997, significant work has been conducted here and elsewhere in developing a theoretical understanding of the nature of polymer-induced drag reduction phenomena through DNS of viscoelastic turbulent channel flow in a straight channel flow geometry. However, no work has addressed so far the combined effect of turbulence, viscoelasticity and a moderately complex, non-planar flow geometry.
Turbulent flows over wall boundaries more complex than planar are of interest in many industrial and engineering problems, for example, heat exchanger designs, ocean flow simulations, pipeline transport, blood flow through arteries, etc. In particular, when the fluid is viscoelastic it is of interest to examine the effects of wall roughness on polymer drag reduction as we attempt to transfer laboratory test results to industrial flow applications. However, no numerical calculations have been performed so far to study the viscoelastic turbulent flows over rough surfaces, even idealized as wavy boundaries.
The first goal of the present work that we set and achieved was the development and validation of all necessary tools to perform high accuracy direct numerical simulations of viscoelastic turbulent flows within moderately complex flow geometries, such as a channel with a wavy wall. The second goal was to carefully validate the developed code against especially constructed limiting case solutions. The third was to test the new code extensively for Newtonian flows in a case of relatively high Reynolds number where experimental data and previous simulation results are available. The fourth was to develop and test the code performance in turbulent viscoelastic flows in a wavy channel in order to carefully understand the critical parameters for a successful simulation. In proceeding towards to completion of the last goal we have found necessary the development of a totally novel numerical representation that allows the preservation of a key characteristic in the viscoelastic flow simulations, that of the positive definiteness of an internal structural parameter of the viscoelastic simulations, the polymer conformation tensor.
More specifically, we first developed a new code based on an efficient implementation of spectral methods in non-orthogonal stretched coordinates in order to study the effects of complex flow boundaries acting on the flow behavior for both Newtonian and viscoelastic fluids. Second, the code has been validated against Newtonian laminar flow results. Detailed quantitative comparisons of the numerical and perturbation results have confirmed that the new code provides a very accurate spectral approximation for zero as well as non-zero Reynolds numbers. Third, large scale parallel computations involving DNS of Newtonian turbulent flows were performed at three different Reynolds numbers and for three different mesh resolutions. To achieve those results, a specially constructed initial guess velocity was used that was obtained through the intermediate use of a pseudoconformal transformation. DNS results obtained at the lowest friction Reynolds number (Reτ =160) for two different mesh sizes allowed us to show the robustness of the code and convergence with mesh refinement.
Very large scale parallel computations, involving more than 200,000 CPU hours at the finest mesh resolution, executed at a high enough Reynolds number (Reτ =310) in order to compare the simulation results against previously available experimental data, allowed us to improve the best previously available numerical results for Newtonian turbulent flow in an undulating channel in all aspects: better spatial approximation (fully spectral as opposed to seventh order spectral element used in the literature before), better time approximation (second order as opposed to mixed first-second order before), better pressure boundary conditions, finer mesh resolution (192 x 513 x 480 along the spanwise, shearwise and streamwise direction respectively), larger computational domain and larger time of integration for the statistics. Those results showed good agreement with available experimental data where the latter were accurate (far from wall) while simultaneously demonstrated their limitation in the region close to the wall where experimental measurements are particularly difficult. The results demonstrated a significant change to the near wall boundary layer structure due to the surface waveness. This change affected all quantities: averaged values, statistics and especially coherent structures.
Direct numerical simulations for viscoelastic turbulent flows in a wavy channel were also performed by implementing a second order semi-implicit time integration scheme on both the momentum and constitutive model equations, the same technique that has been used before in straight channel simulations. However, preliminary simulation results showed the necessity of a significant improvement in the way the viscoeleastic constitutive equation was solved in order to ensure stability to the calculations. This led to the implementation of the exponential mapping for the conformation tensor using a multigrid diffusion correction based on second order finite difference. This substantial new development was successfully implemented and tested in straight channel viscoelastic simulations where it showed enhanced stability in comparison to the previous simulations and much more physically meaningful results. However, the turbulent viscoelastic flow in a wavy channel, when the undulations are high enough to allow for flow reversal and recirculation (as was the case tried here), presents a much bigger computational challenge. We believe that a fully implicit time integration scheme for the viscoelastic model equation is needed in connection with a more effective fine tuning of the diffusional stabilization of the constitutive equation using a suitably adjusted variable diffusivity. All the tools needed are available and/or have been developed in this work.
In summary, in this work we have developed and successfully tested all the necessary ingredients for a successful simulation of viscoelastic turbulent flow in a wavy channel. It is only a matter of time in future work to address the fine tuning of parameters such as the diffusivity, time integration scheme, etc. in order to successfully perform the very large scale simulations needed to reproduce viscoelastic turbulent flows in a wavy channel flow geometry.
