Validation and verification of Courant number insensitive CE/SE method for transient viscous flow simulations

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Abstract

In this paper, we report an extension of the space–time conservation element–solution element (CE/SE) framework-based viscous flow solver. With the accuracy of solution obtained through the use of a CE/SE-based solver closely related to the CFL number disparity across the mesh, a new formulation to make the solution insensitive to CFL number disparity is herein presented. The capability of the developed solver is then validated through simulation of 2D problems such as driven cavity, external flow over a flat plate, laminar flow over a square cylinder, etc. Investigations are also conducted to verify the sensitivity of results to grid spacing and mesh structure.

Introduction

In recent years computational fluid dynamics (CFD) has seen the attention increasingly shift away from steady-state Euler flows to the more complicated regime of unsteady viscous flows which may also involve shock. In such regimes, the traditional finite-difference methods such as the Lax–Wendorff scheme or the two-step MacCormack's scheme generally suffer from numerical wiggles near a discontinuity [12]. Motivated by the idea of von Neumann and Richtmyer [22] that this difficulty may be overcome by adding some form of artificial dissipation, many popular wiggle-suppressing shock-capturing schemes such as Harten's high-resolution scheme [9], Yee's total variation diminishing (TVD) scheme [25] and the essentially non-oscillatory method [10] came into existence. However, these schemes are burdened with artificial conditions such as monotonicity, entropy condition and TVD property which may not be consistent with flow physics [2]. Moreover, as rightly pointed out by Shu et al. [19], any excessive numerical damping can render these methods incapable of capturing small-scale flow features.

In contrast, there are other popular schemes such as spectral methods [8] and the compact finite-difference scheme with spectral like resolution [14] that have high-order accuracy and low numerical dissipation and thus can resolve small-scale features. However, these schemes are handicapped by their computational complexity and their limited capability to handle practical problems, e.g., those involving complex geometries and/or shock waves.

The conflict between stability and numerical accuracy continues to haunt most of the established methods. One example of this conflict is the degraded accuracy in time-accurate computation caused by too much numerical dissipation vs. the instability caused by too little numerical dissipation. This conflict calls for the development of an elusive stable numerical scheme with higher accuracy, wherein computations must be performed away from the edge of instability without going too far away from it. Only such a method can extend across flow regimes and disciplines with ease.

In this light, the space–time conservation element and solution element (CE/SE) [1], [2], [3], [4] framework, with its unique approach of enforcing flux conservation in both space and time, sidesteps most of the aforementioned issues. Dr. S.C. Chang and co-workers at the NASA Glenn Research Center are developing the method of CE/SE, a novel numerical framework for solving conservation laws with several attractive features. It is substantially different in both concept and approach from well-established methods, such as finite-difference, finite volume, and spectral methods. This high-resolution, multidimensional, numerical framework has been built from scratch with extensive consideration on physics and rigorous mathematical proof, thereby doing away with some of the limitations of traditional numerical simulation methods. In addition to being mathematically simple, it has very little or almost no numerical dissipation, making it an ideal candidate among the many available numerical schemes for problems involving flow instability. Its unique features include the following: (i) a unified treatment of both space and time; (ii) enforcement of local and global space–time flux conservation; (iii) use of a space–time staggered mesh that allows for evaluation of fluxes at the cell interfaces without solving the Riemann problem; (iv) schemes built from a non-dissipative core scheme, allowing for control of numerical dissipation (if needed) effectively and with mathematical justification (not ad hoc); (v) treatment of both the flow variable and its spatial derivatives as independent unknowns; (vi) the lack of directional splitting for flows in multiple spatial dimensions, leading to a truly multidimensional scheme; (vii) the avoidance of ad hoc numerical damping whenever possible. For inviscid flow problems (without flow discontinuity), the non-dissipative core scheme is a natural fit, because the flow is reversible in time. However, the characteristics of viscous flow and inviscid flow problems with shocks are irreversible in time, and thus the addition of numerical dissipation can be justified if its magnitude can be rigorously proved in mathematics. Since its inception, the CE/SE method has been successfully adapted to model several different applications in unsteady Euler flows, acoustic waves, traveling and interacting shocks, detonation waves, cavitation, etc. [1], [2], [3], [4], [6], [11], [13], [15], [16], [18], [23], [26], [27], [28], [29].

With the research interests of the authors lying in the area of unsteady flow problems, such as combustion instability and aero-acoustics, we recently extended the CE/SE method to model transient viscous flow problems through the development of a Navier–Stokes solver [20]. In spite of having achieved reasonable success with the developed flow solver, there was scope for further improvement. The accuracy of the solution obtained from a CE/SE-based framework, using a fixed time marching step, generally depends on the local CFL number (ν). This is a key issue with regards to the transient viscous solver, as it is built from a dissipative extension of the non-dissipative core scheme (for stability reasons). Although great care has been taken to develop the dissipative extension of the core scheme, so as to avoid the case of numerical dissipation overwhelming physical dissipation, the numerical accuracy of the solution can get influenced by the grid distribution. In viscous flow simulations, where meshes with heavy packing of the grid points near the wall are used, there can be a huge disparity in ν across the mesh making the solutions highly dissipative in regions where ν  1. Hence, by making the numerical solution insensitive to variation in ν, the solution obtained will have higher numerical accuracy. In this regard, Chang and Wang [5] recently proposed a multidimensional CFL number insensitive scheme for Euler solvers that demonstrates these characteristics. We extend this scheme to the already developed Navier–Stokes solver [20].

A rigorous validation and verification of the flow solver is necessary before the flow solver is utilized to do unsteady flow simulations with complex geometries like that of aero-acoustics, combustion instability, and direct numerical simulation (DNS) or large eddy simulation (LES) for turbulent flow. Also, to date, all the validation for the developed flow solver was carried out using quadrilateral meshes, while the CE/SE framework in two dimensions originally calls for the use of triangular meshes. Its extension to use polygon shaped meshes is only a recent development [29]. Hence it is proposed to verify and compare the effect of using triangular and quadrilateral meshes on the numerical accuracy. This effort will have dual benefits. First being, we will obtain a complete understanding on the effect of use of quadrilateral meshes on the Courant number insensitive scheme (CNIS), which has not been well-documented and tested. Secondly, use of triangular meshes for viscous simulations with most available CFD methods, results in a solution that has a lot of numerical dissipation, especially in the boundary layer region. With the CE/SE operating on a different framework, we will be able to see, if the method is able to overcome the disadvantages of the other methods have with the use of triangular meshes for viscous simulations.

Section snippets

Governing equations

Consider the two-dimensional unsteady Navier–Stokes equations in its non-dimensional conservation form:umt+fmx+gmyfvmxgvmy=0Here, m = 1, 2, 3, 4 represent the continuity, x-momentum, y-momentum and the energy equations. μm represents the flow variables, while fm and gm represent the inviscid part of the fluxes in x- and y-directions, respectively. fvm and gvm respectively represent the viscous part of the fluxes in x- and y-directions. These are defined asu1u2u3u4=ρρuρvρEf1f2f3f4=ρuρu2

The CE/SE framework preliminaries

In this section, details of the CE/SE framework, used in the development of the Navier–Stokes solver are described, along with the grid system employed. A set of physical conservation laws, in its original form, is nothing but a collection of statements of flux conservation in space–time. Hence, for a numerical framework to model the conservation laws accurately, it should also be constructed in such a way, as to guarantee conservation of flux not only in space alone, but in both space and

CFL insensitive scheme

As mentioned in the earlier sections, the behavior of the ‘a’ and ‘c’ schemes are closely hinged to the local CFL number |ν|. Although, both the ‘a’ and ‘c’ schemes tend to be accurate in the limit of |ν| approaching 1, there is a chance for numerical wiggles to appear in the ‘a’ scheme. In the limit of |ν| approaching 0, the numerical solution obtained through the ‘c’ scheme tends to become highly dissipative, while that obtained through the ‘a’ scheme is highly accurate and free of any

Numerical results

To validate the CNIS viscous flow solver, several benchmark test cases were simulated. The numerical results and comparisons are presented in this section. As mentioned earlier, when triangular meshes are used for viscous simulations, with available CFD methods, there is a problem of numerical dissipation overwhelming the accuracy of the solution, especially in the boundary layer region. In the CE/SE framework, as the spatial projection of the control volume is always a hexagon or a polygon of

Conclusion

In this paper, modification of a CE/SE-based transient viscous flow solver to make it Courant number insensitive was discussed. The accuracy of the scheme was validated through the simulation of both compressible and incompressible viscous flows. A grid convergence study was also performed to analyze the sensitivity of the results. Through the study it was inferred that CNIS enables us to conduct numerical simulations with a highly non-uniform mesh (packed meshes) and still maintain high

Acknowledgments

This work was performed under NNC04AA56A for the Glenn Research Center of the National Aeronautics and Space Administration. The authors are extremely grateful to Dr. Chau-Lyan Chang, NASA Langley Research Center and Dr. Xiao-Yen Wang, NASA Glenn Research Center, who have contributed to the success of this work through their valuable suggestions and indispensable experience. The authors would also like to thank the authors of the mesh partitioning software METIS that helped us reduce the

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