Symmetry-preserving discretization of Navier–Stokes equations on collocated unstructured grids
Introduction
We consider the simulation of turbulent, incompressible flows of Newtonian fluids. Under these assumptions, the dimensionless governing equations in primitive variables are where Re is the dimensionless Reynolds number. The basic physical properties of the Navier–Stokes (NS) equations can be deduced from the symmetries of the differential operators (see [1], for instance). In a discrete sense, it suffices to retain such operator symmetries to preserve the analogous (invariant) properties of the continuous equations. It may be argued, especially if the method is going to be used on unstructured meshes, that accuracy may need to take precedence over the properties of the operators. However, in this work, we have adopted the same philosophy followed by Verstappen and Veldman [2], [3]: symmetries of the convective and diffusive operators are critical to the dynamics of turbulence and must be preserved.
Reconciling accuracy and stability has always been a great challenge for numerical simulation of turbulence. Upwind-like schemes have been very popular because they are very stable: the convective term introduces artificial numerical dissipation that systematically damps kinetic energy. Central difference schemes do not add non-physical dissipation, however, they do not guarantee stability. In the latter case, kinetic energy may need to be damped explicitly. Nevertheless, in both cases, the artificial false dissipation interferes with the subtle balance between convection transport and diffusive dissipation. This usually affects the smallest scales of motion, the essence of turbulence [3]. One solution to guarantee stability is to construct numerical approximations of the convective term that preserve total kinetic energy.
The importance of conservative discretization methods was realized in the pioneering works by Arakawa [4] and Arakawa and Lamb [5]. They showed that, for hydrostatic systems, the conservation of both kinetic energy and enstrophy by convective schemes prevent systematic and unrealistic energy cascade towards high wavenumbers, a cause of non-linear instability. Morinishi et al. [6] reviewed existing conservative, second-order finite-difference schemes for structured meshes, and introduced a fourth-order conservative scheme. However, for non-uniform meshes the truncation error was not fourth-order, so they chose to sacrifice conservation for accuracy and developed a “nearly conservative” fourth-order scheme. Later, Vasilyev [7] generalized the numerical schemes of Morinishi et al. to non-uniform meshes using a mapping technique. However, their schemes do not simultaneously preserve momentum and kinetic energy: it depends on the form chosen for the convective term [7].
Verstappen and Veldman [2], [3] proposed to exactly preserve the symmetry properties of the underlying differential operators. The convective operator is represented by a skew-symmetric matrix and the diffusive operator by a symmetric, positive-definite matrix. The authors showed that such conditions are enough to ensure stability. With regard to the accuracy, they recalled the paper by Manteuffel and White [8]. In this work, the authors emphasized that local truncation error is not decisive: given stability, a second-order local truncation error is a sufficient, but not a necessary, condition for a second-order global truncation error. Moreover, they proved that the second-order symmetry-preserving discretization proposed in [2], [3] yields a second-order accurate solution although its local truncation error is indeed first-order on non-uniform meshes. Finally, a fourth-order symmetry-preserving scheme is obtained removing the leading terms of the discretization error by means of a Richardson extrapolation. On uniform grids, conservative schemes by Verstappen and Veldman [2], [3] become identical to those proposed by Morinishi et al. [6].
The way for accurate DNS and LES simulations on more complicated domains was opened by Perot [9] and Zhang et al. [10]. They derived a conservative staggered mesh scheme for unstructured grids. Later, Mahesh et al. [11] developed both staggered and collocated conservative schemes for LES in complex domains. More recently, Hicken et al. [12], presented a fully-conservative method for staggered unstructured grids. They use ‘shift’ transformations to obtain staggered operators from easy-to-define collocated operators. Thus, their formulation intends to lead to a generalization of the works by Perot [9] and Verstappen and Veldman [3]. However, it has only been put in practice on orthogonal unstructured grids. The extension of the ideas of Hicken et al. [12] to general unstructured meshes seems unlikely due to the impossibility to construct proper ‘shift’ transformations on such grids [13]. For further information about conservative schemes on unstructured grids, the reader is referred to the review article by Perot [14]. To complete this section, it is worth mentioning other related methods: namely, the Keller box schemes [15], [16] and the mimetic schemes or support operators methods by Shashkov and Steinberg [17] and Hyman and Shashkov [18], [19]. Finally, discrete calculus methods proposed by Perot and Subramanian [20] can be viewed as a general methodology for developing numerical methods that capture physics well. In this sense, most of the above-described conservative methods can be derived as discrete calculus methods. We can conclude, that nowadays it is generally accepted that the quality of the results is not automatically improved by simply increasing the order of accuracy of the numerical scheme [21]. Instead, the numerical schemes should retain the symmetry properties of the continuous equations. Very recent works in this vein can be found in [22], [23], [24], [25], [26], [27], [28], [29], [30], for instance.
The rest of the paper is arranged as follows. In the next section, it is shown that retaining the symmetry properties of the underlying continuous operators leads to spatial discretizations that exactly conserve the total kinetic energy for inviscid flows. Here, a collocated-mesh formulation is preferred over a staggered one due to its simpler form. This is also presented in Section 2. Special emphasis is given to the well-known checkerboard problem (see also Appendix A Computing and projecting the cell-centered predictor velocity, Appendix B Origin of the checkerboard spurious modes) and the properties of the ‘shift’ linear operator that relates the cell-centered and the staggered auxiliary velocity fields. A novel approach, based on a fully-conservative regularization of the convective term, is proposed to mitigate the checkerboard spurious modes. Then, in Section 3 the discrete operators are constructed taking into account all the constraints imposed by the global properties derived in the previous section. Finally, the new discretization method is numerically tested: (i) a simple experiment is performed to show the supraconvergence for a Poisson problem in Section 4 and (ii) DNS results for a turbulent differentially heated cavity are presented to show the accuracy and robustness of the method in Section 5. Finally, relevant results are summarized and conclusions are given.
Section snippets
Symmetry-preserving discretization on collocated grids
In this section, a conservative finite-volume discretization of the incompressible NS equations (1a), (1b) on general unstructured grids is presented. Despite the intrinsic errors due to the improper pressure gradient formulation [6], [31], [32], here a collocated-mesh scheme is preferred over a staggered one due to its simpler form for unstructured grids. In this case, both the pressure and the velocities are stored at the center of the control volume whereas a secondary velocity field is
Constructing the discrete operators
The discretization of the operators preserving the global properties is addressed in this section. In general, the constraints imposed by the operator (skew-)symmetries strongly restrict the form of the local approximations limiting, in some cases, the local truncation error.
Supraconvergence
The accuracy of a numerical method has been traditionally analyzed in terms of its local truncation error. However, Manteuffel and White [8] showed that the local truncation error only provides a lower bound for the global truncation error. In practice, it is possible that low-order terms cancel providing a better real accuracy. This is the case for symmetry-preserving discretizations on structured Cartesian grids [3], [12]. In the context of finite-volume (finite-difference also) schemes this
Test-case: turbulent differentially heated cavity
In this section the performance of the symmetry-preserving discretization presented in this work is assessed through application to a 2D buoyancy-driven turbulent flow in an air-filled () differentially heated cavity (see Fig. 6, left). This configuration has been chosen because it can be discretized with both structured Cartesian and unstructured meshes. The cavity is subjected to a temperature difference, , across the vertical isothermal walls while the top and bottom
Concluding remarks
The essence of turbulence are the smallest scales of motion. They result from a subtle balance between convective transport and diffusive dissipation. Mathematically, these terms are governed by two differential operators differing in symmetry: the convective operator is skew-symmetric, whereas the diffusive is symmetric and positive-definite. On the other hand, accuracy and stability need to be reconciled for numerical simulations of turbulent flows around complex configurations. With this in
Acknowledgements
This work has been financially supported by the Ministerio de Ciencia e Innovación, Spain (Ref. ENE2010-17801). Also by a Generalitat de Catalunya Beatriu de Pinós postdoctoral fellowship (2006 BP-A 10075) and a Juan de la Cierva postdoctoral contract (JCI-2009-04910) by the Ministerio de Ciencia e Innovación. Calculations have been performed on the IBM MareNostrum supercomputer at the Barcelona Supercomputing Centre. The authors thankfully acknowledge these institutions. We also thank the
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