Large Eddy Simulation Of Incompressible Turbulent Flow

Since the absence of valid methods to predict turbulent flow, present applications of Computational Fluid Dynamics (CFD) to relatively complex flows, which are commonly seen in engineering problems, lead to poor results. Turbulent flow is the major challenge for CFD to become an accurate quantitative prediction technique, and it is the bottleneck of CFD technology development and new aircraft design. The flow field are divided into Resolved-Filtered-Scale (RFS) motions and Sub-Filter-Scale (SFS) motions in the Large Eddy Simulation (LES) of turbulent flow by applying a low pass filter. The RFS motions are computed directly by the filtered Navier-Stokes equations, and the SFS motions are modeled by using SFS model. Because of the universal characteristics of the small-scale motions in high-Reynolds-number turbulent flows, it is expected that there may exist some universal SFS models, independent of the flow geometry. It is believed that LES is the most promising method to increase the accuracy of turbulent prediction and LES will replace Reynolds-Averaged Navier-Stokes (RANS) equations simulation, the most popular engineering strategy currently, as the most widely used engineering design tool for turbulent prediction in the foreseeable future.Since the lack of understanding of the nature of the small-scale motions of turbulent flows, the accuracy of available SFS models is not so satisfactory, which introduce considerable modeling errors in LES. On the other hand, since the multi-scale nonlinear systems dealt by LES are very different to those traditional CFD applications, the numerical strategies successfully used in traditional CFD applications, such as the simulations of steady flows, laminar flows and/or the Reynolds-Averaged turbulent flows, can introduce large numerical errors in LES. The numerical errors interact with modeling errors in the complex nonlinear dynamic procedure, which leads to considerable uncertainty in the results of LES simulation currently.A LES strategy with considerably reduced numerical errors has been established in this dissertation, based on the study of spatial discretization errors and the explicit filtering approach. The key idea is to reduce the differentiation errors induced by the finite-difference schemes as much as possible by scheme optimization, then carefully designed explicit filtering algorithm is applied to control the numerical errors to the magnitude below that of the SFS stress. First, an optimized fourth-order central tridiagonal compact scheme (denoted as optC4) with five-point stencil is developed, based on the minimization of the fourier-analysis-based dispersion errors. Then, an explicit filtering algorithm which combine the explicit filtering and numerical stability filtering into only one filtering procedure is designed. After that, a 3-D incompressible flow solver capable of doing large eddy simulations of turbulent flow with considerably reduced numerical errors is developed based on the foregoing works.The SFS modeling is studied based on the LES solver developed above. First, a new expression of the filter width of the explicit-filtering filters is developed based on the estimation of the rate of SFS kinetic energy dissipation. Then, the reversibility and irreversibility of the small-scale motions of turbulent flow are studied, and an explanation why some SFS models underestimates the kinetic energy dissipation is presented. Based on these analysis, a criterion of SFS modeling, say all appropriate SFS models should consist of a time-reversible part and time-irreversible part, is presented, and the average kinetic energy dissipation produced by the time-reversible part is proved to be zero, which means the average kinetic energy dissipation is totally on account of the time-irreversible part. This criterion gives a physical basis for the so-called mixed models as well as a vanishing dissipation restriction for the time-reversible part of SFS model. SFS modeling based on these works is studied in the last section, and several new dynamic mixed models are presented for some possible choices of the two SFS model parts. Two of the new models, of which the time-reversible part is modeled by the similarity model and the time-irreversible part is modeled by the Smagorinsky model, is used to LES simulation of the decaying isotropic turbulence, and the results show that the new models can give more accurate kinetic energy dissipation than the available ones.This dissertation is divided into seven chapters as follows:The first chapter is the introduction. A literature survey of the importance and difficulty of turbulent flow research, the evolution of the physical picture of turbulent flow, the research methods of turbulent flow, and the position of numerical simulation methods (especially the position of LES method) in turbulent flow research is presented. The history and current status of large eddy simulation method of turbulent incompressible flow are reviewed, and the works of this dissertation are briefly summarized.In the second chapter, the basic concepts of LES are concisely provided, together with the clarification of the concepts, methods and contents of this dissertation. First, the physical foundation and the governing equations of LES approach are briefly introduced. Then, the error analysis and reducing of LES, the importance of explicit filtering approach to error-reducing, and the error-reducing strategies used in this dissertation are discussed. After that, the test methods of SFS models and the two most widely used SFS models are introduced.The third chapter provides the numerical methods. First, the numerical solution method of the incompressible Navier-Stokes equations used in this dissertation, say Artificial Compressible Method (ACM), is presented, and the LES equations based on ACM is introduced, together with the formulations in generalized coordinates. Subsequently, the spatial discretization methods and the temporal integration methods are discussed. After that, the technique adopted to ensure freestream preservation on three-dimensional curvilinear meshes is presented.In the fourth chapter, the spatial discretization schemes suffice for the requirements of the error-reducing strategies is studied. First, the necessity of high order central scheme for the high level error-reducing strategies of LES is argue based on the review of literatures. Then, the fourier analysis of spatial discretization errors is introduced, and the differentiation error and aliasing error of finite-difference scheme are studied. Enlightened by the "3/2-rule" for de-aliasing in spectral method, we argue that the wave number range of scheme optimization should be restricted withinω= kΔ∈[0, 2/3π)]. With this understanding, an optimized fourth-order central tridiagonal compact scheme (optC4) with five-point stencil is developed. Subsequently, the excellent resolution characteristics of optC4 as well as its validity in the error-reducing strategies of LES are evaluated. In the last section, the impact of several spatial discretization errors are evaluated by numerical simulations with carefully designed explicit filtering approach.In the fifth chapter, an explicit filtering algorithm that combine the explicit filtering and numerical stability filtering in only one filtering procedure is developed. First, the commonly used explicit filtering algorithm, the filter of which is applied to the non-linear term of the filtered Navier-Stokes equations, is introduced, and the drawback of such algorithm that the result governing equations are generally not Galilean invariant is reviewed. Then, the cumulative effect of numerical stability filtering, the filter of which is applied to the whole conserved variable field, is discussed. Subsequently, a new explicit filtering algorithm, the filter of which is applied to the increment of the conserved variable field, is developed. The increment filtering algorithm can fulfill the purposes of explicit filtering and numerical stability filtering in one filtering procedure without breaking the Galilean invariant of the governing equations and it introduces no cumulative effect of filtering. In the following section, a general set of rules proposed in literature for constructing discrete filters that the commutation error has an accuracy of order O(Δ~n) are presented. We show that the constraints that make sure the commutation error has an accuracy of order O(Δ~n) are equivalent to the constraints that make sure the formal truncation error of the filter is order O(Δ~n) of accuracy, so filters constructed under these constraints can fulfill the requirements the explicit filtering and numerical stability filtering. After that, the filter constructing algorithms based on the Modified-Wave-Number of spatial discretization scheme are developed, and filters coupled with optC4 scheme are constructed following these algorithms. In the last section, the explicit filtering algorithm developed in the current chapter is verified by the large eddy simulations of the decaying isotropic turbulence as well as turbulent channel flow.The sixth chapter studies SFS modeling. First, SFS models commonly appear in literature are reviewed. Then, a new expression of the filter width of the explicit filter is developed based on the estimation of the rate of Sub-filter kinetic energy dissipation. The results of numerical simulation show that the new filter-width expression is much more accurate and robust compared to the available ones. Subsequently, the reversibility and irreversibility of the small-scale motions of turbulent flow are studied, and an explanation why some SFS models underestimate the kinetic energy dissipation is presented. With this background, a criterion of SFS modeling, say all appropriate SFS models should consist of a time-reversible part and time-irreversible part, is presented, and the average kinetic energy dissipation produced by the time-reversible part is proved to be zero, which means the average kinetic energy dissipation is totally on account of the time-irreversible part. This criterion gives a physical basis for the so-called mixed models as well as a vanishing dissipation restriction for the time-reversible part of SFS model. SFS modeling based on these works is studied in the last section, and several new dynamic mixed models are presented for some possible choices of the two SFS model parts. Two of the new models, of which the time-reversible part is modeled by the similarity model and the time-irreversible part is modeled by the Smagorinsky model, are used to LES simulation of the decaying isotropic turbulence, and the results show that the new models can give more accurate kinetic energy dissipation than the available ones.The conclusions are presented in the seventh chapter. The works of this dissertation are summarized, the possible future development of LES as well as some opportunities for further research are discussed.