High Order Finite Difference Schemes on Non-uniform Meshes with Good Conservation Properties

doi:10.1006/jcph.1999.6398

Journal of Computational Physics

Volume 157, Issue 2, 20 January 2000, Pages 746-761

https://doi.org/10.1006/jcph.1999.6398 Get rights and content

Abstract

Numerical simulation of turbulent flows (DNS or LES) requires numerical methods that are both stable and free of numerical dissipation. One way to achieve this is to enforce additional constraints, such as discrete conservation of mass, momentum, and kinetic energy. The objective of this work is to generalize the high order schemes of Morinishi et al. to non-uniform meshes while maintaining conservation properties of the schemes as much as possible. This generalization is achieved by preserving symmetries of the uniform mesh case. The proposed schemes do not simultaneously conserve mass, momentum, and kinetic energy. However, depending on the form of the convective term, conservation of either momentum or energy in addition to mass can be achieved. It is shown that the conservation properties of the generalized schemes are as good as those of the standard second order finite difference scheme on non-uniform meshes, while the accuracy of the new schemes is definitely superior. The predicted conservation properties are demonstrated numerically in inviscid flow simulations.

References (9)

S. Ghosal
An analysis of numerical errors in large-eddy simulations of turbulence
J. Comput. Phys.
(1996)
P. Spalart et al.
Spectral methods for the Navier–Stokes equations with one infinite and two periodic direction
J. Comput. Phys.
(1991)
J.K. Dukowicz et al.
Approximation as a higher order splitting for the implicit incompressible flow equations
J. Comput. Phys.
(1992)
F.H. Harlow et al.
Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface
Phys. Fluids
(1965)

There are more references available in the full text version of this article.

Cited by (129)

A new compact scheme-based Lax–Wendroff method for high fidelity simulations
2024, Computers and Fluids
The classical Lax–Wendroff (LW) method employs explicit second order central difference schemes to solve partial differential equations. Recently, the authors in Sengupta et al. (2023) have identified numerical parameter ranges in performing large eddy simulation (LES) using this classical LW method. In a bid to improve these numerical parameter ranges for high fidelity computations, a new non-uniform grid based compact scheme for LW method (NUCLW) is developed here. This new method uses sixth order non-uniform compact scheme (NUC6) for the first and higher order derivatives. Reported global spectral analysis confirms substantial improvement in the resolution and accuracy of NUCLW scheme over the classical LW scheme. A significant advantage of the present scheme is its ability to compute solutions for non-uniform, structured grids, and that does not suffer degradation in accuracy. The potential of the NUCLW scheme for high accuracy computations is demonstrated by solving 2D incompressible Navier–Stokes equations for the benchmark unsteady flow inside a square lid driven cavity problem for a subcritical Reynolds number of 3200 and a post-critical Reynolds number of 10,000 and simulating the instability of the Taylor–Green vortex problem. These demonstrate the ability of the NUCLW scheme to solve the Navier–Stokes equations for high fidelity simulations such as LES/DNS with enhanced ranges of numerical parameters with respect to enhanced accuracy and faster computations.
A family of spatio-temporal optimized finite difference schemes with adaptive dispersion and critical-adaptive dissipation for compressible flows
2023, Journal of Computational Physics
Citation Excerpt :
There are two formal methods can be used to generalize the application of the scale sensor to non-uniform meshes. In a first approach, a Jacobian transformation (JT) can be applied, by defining transformed co-ordinates [31] [32]. The coordinate transformation approach can be employed to transform the non-homogeneous meshes in physical space into homogeneous meshes in computational space.
A numerical scheme with good dissipation and dispersion properties is essential for simulations of complex compressible flows in resolving small-scale structures and capturing discontinuities. Although a sufficient numerical dissipation benefits the shock-wave capturing, the fine scales in smooth regions are dissipated either. In this work, a framework to construct arbitrarily high-order spatio-temporal optimized finite difference schemes with adaptive dispersion and critical-adaptive dissipation is proposed, where the dispersion and dissipation properties are characterized by two free parameters respectively. The proposed scheme can automatically adjust these two free parameters to control the dispersion and dissipation according to the local flow-field properties quantified by the scale sensor. As the first step to optimize the spectral properties of the fully discrete scheme, the total dispersion and dissipation errors induced by spatio-temporal discretization are studied. Then, the dispersion property is optimized by a new integrated error function devised for fully discrete scheme. To improve the precision of prediction, the scale sensor employed to quantify the local scaled wavenumber of flow fields is optimized in wavenumber space. Furthermore, a modified dispersion-dissipation condition is developed to characterize the relationship between the total dispersion and dissipation errors. Building upon the optimized scale sensor and modified dispersion-dissipation condition, the critical-adaptive dissipation parameter is obtained, which keeps the numerical dissipation as low as possible to resolve more small-scale structures in the low-wavenumber region and introduces sufficient dissipation to capture strong discontinuities successfully. Moreover, the adaptive dispersion parameter related to the local wavenumber and Courant number is obtained to reduce the total dispersion error significantly. Meanwhile, the critical-adaptive dissipation surface and adaptive dispersion surface are constructed for different Courant number utilized in actual flow simulations, which improves the spectral properties of the proposed scheme. Finally, some benchmark cases involving strong discontinuities and multi-scale structures are employed to verify the attractive performance of the proposed scheme.
Numerical treatment of incompressible turbulent flow
2022, Numerical Methods in Turbulence Simulation
Turbulence features a subtle balance between the energy input at the large scales of motion, the transfer of energy to smaller and smaller scales and the energy dissipation at the smallest scales. When numerically modeling this energy exchange, it should not be disturbed by nonphysical, numerical effects. In this vein, energy-conserving discretizations have been developed, which do not dissipate energy artificially and are inherently stable because they cannot spuriously generate kinetic energy. In this chapter, we consider these discretizations, with focus on finite-volume methods. Mathematically, the conservation of energy is a consequence of the symmetries of the differential operators in the incompressible Navier–Stokes equations. When the spatial discretization method preserves these symmetries, then the energy of the discrete system is conserved upon exact time integration. The latter is complicated by the algebraic incompressibility constraint. Together with the pressure gradient, it describes acoustical pressure waves with an infinite propagation speed, hence this acoustical part of the flow equations requires implicit time integration, whereas the remaining parts may be integrated explicitly.
Often it is too expensive to calculate all scales of motion, and coarse-grained models are used. To describe the dynamics of the larger eddies, the Navier–Stokes equations are filtered in space and the effect of the turbulence that is filtered out is modeled. Discretizing the resulting equations effectively introduces a second filter, by providing a good approximation for the low wavenumbers and suppressing the high wavenumbers. Consequently, modeling and discretization errors are entangled. In this chapter, this entanglement is analyzed for a finite-volume discretization. Because of the entanglement of discretization and modeling errors, it makes sense to use a unified idea about the nature of both the discretization (of the resolved part) and the subgrid model (for the unresolved part). Here, the discrete approximation is done so that fundamental properties (such as symmetries) are preserved. The subgrid modeling should then also be like that. We focus on asking how much dissipation the subgrid model should provide to counterbalance the nonlinear production of the small, unresolved scales of motion. This yields a minimum-dissipation condition.
High fidelity simulations in support to assess and improve RANS for modeling turbulent heat transfer in liquid metals: The case of forced convection
2021, Nuclear Engineering and Design
This paper attempts to give a brief overview of the work conducted through some recent EU funded projects under the Euratom research for innovative nuclear systems (THINS, SESAME and MYRTE) concerning the use of high fidelity (HiFi) simulations, namely Direct Numerical Simulations (DNS) and Large Eddy Simulations (LES), to adapt Reynolds Averaged Navier-Stokes (RANS) models for the computation of turbulent heat transfer in liquid metal flows.Here the focus is on forced convection only, that prevails in normal reactor operation, through some selected cases performed at UCLouvain, e.g. the channel flow and the impinging jet. The considered RANS approaches are those based on the simple gradient diffusion hypothesis or SGDH, and those based on the algebraic heat flux formulation (AHFM). Among the AHFM models the two possible formulations are assessed, i.e. the explicit form ( $k - ∊ - k_{θ} - ∊_{θ}$ of Manservisi and Menghini (2014) still based on the eddy diffusivity concept (gradient diffusion assumption), and the implicit form of the AHFM-NRG which is essentially a recalibration of the reference model of Kenjeres et al. (2005). The Results show the overall superiority of AHFM models, although SGDH-models using the Kay correlation for the turbulent Prandtl number and the dedicated thermal wall-function developed by Duponcheel et al. (2014) provide reasonable results at a much lower effort making them interesting for industrial applications. However further research is undoubtedly required to come closer to more universal models working for a wide range of Prandtl numbers and flow conditions.
Direct Numerical Simulation of Turbulent Heat Transfer at Low Prandtl Numbers in Planar Impinging Jets
2021, International Journal of Heat and Mass Transfer
DNS simulations of planar impinging jets are performed to provide reference data to validate RANS models for low-Prandtl turbulent heat transfer. In that configuration, a hot plane jet is blown from a slot at the top wall of a channel and impinges on the cooler bottom wall. The present DNS simulations are carried out at $R e = 4000$ and 5700, based on the jet width and velocity, and for $P r$ down to 0.01. Two cases are investigated: the case of a laminar uniform jet profile and the case of a fully turbulent inlet profile coming from an auxiliary channel flow simulation running in parallel. The DNS results show how the heat transfer in this configuration evolves from the turbulence-dominated case at $P r = 1$ to the molecular diffusion-dominate case at $P r = 0.01$ . The temperature field at $P r = 0.01$ is much smoother than at higher Prandtl number because the much larger heat diffusivity quickly diffuses the temperature fluctuations. The comparison between the laminar and the fully-developed turbulent inflows also show different heat transfer characteristics in the vicinity of the stagnation point, depending on the presence of velocity fluctuations in the bottom-wall boundary layers.
An efficient high-order least square-based finite difference-finite volume method for solution of compressible Navier-Stokes equations on unstructured grids
2021, Computers and Fluids
This paper presents an efficient high-order finite volume method for solution of compressible Navier-Stokes equations on unstructured grids. In this method, a high-order polynomial which is based on Taylor series expansion is applied to approximate the solution function within each control cell. The derivatives in the Taylor series expansion are approximated by the functional values at the cell centers of the considered cell and its neighboring cells using the mesh-free least square-based finite difference (LSFD) scheme. Naturally, this least square-based finite difference-finite volume (LSFD-FV) method inherits appealing characteristics of the LSFD scheme, i.e., the simple algorithm and easy implementation. In addition, since the LSFD scheme is mesh-free in nature, the developed high-order LSFD-FV method is endowed with the flexibility to handle the multi-dimensional problems with complex geometries on arbitrary grids. Different from other high-order methods, this LSFD-FV method applies a novel and effective strategy, i.e., the discrete gas-kinetic flux solver (DGKFS), to compute numerical fluxes at the cell interface. In this way, the inviscid and viscous fluxes are simultaneously and effectively evaluated by local reconstruction of solutions for the Boltzmann equation. The efficient time marching strategy coupled with the high-order LSFD-FV method is adopted to solve the resultant ordinary differential equations with the matching accuracy. A series of numerical examples are presented to validate the accuracy, robustness and flexibility of the developed method on unstructured grids. Numerical results demonstrate the superior performance of the developed high-order method on simulating compressible viscous flows, in comparison with the low-order counterpart and the k-exact method of the same order of accuracy.

View all citing articles on Scopus

^f1: [email protected]

¹: Permanent affiliation: Department of Mechanical and Aerospace Engineering, University of Missouri, Columbia, MO 65211.

View full text

Regular ArticleHigh Order Finite Difference Schemes on Non-uniform Meshes with Good Conservation Properties

Abstract

J. Comput. Phys.

J. Comput. Phys.

J. Comput. Phys.

Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface

Phys. Fluids

Regular Article
High Order Finite Difference Schemes on Non-uniform Meshes with Good Conservation Properties