A high-order adaptive Cartesian cut-cell method for simulation of compressible viscous flow over immersed bodies

Abstract

A new adaptive finite volume conservative cut-cell method that is third-order accurate for simulation of compressible viscous flows is presented. A high-order reconstruction approach using cell centered piecewise polynomial approximation of flow quantities, developed in the past for body-fitted grids, is now extended to the Cartesian based cut-cell method. It is shown that the presence of cut-cells of very low volume results in numerical oscillations in the flow solution near the embedded boundaries when standard small cell treatment techniques are employed. A novel cell clustering approach for polynomial reconstruction in the vicinity of the small cells is proposed and is shown to achieve smooth representation of flow field quantities and their derivatives on immersed interfaces. It is further shown through numerical examples that the proposed clustering method achieves the design order of accuracy and is fairly insensitive to the cluster size. Results are presented for canonical flow past a single cylinder and a sphere at different flow Reynolds numbers to verify the accuracy of the scheme. Investigations are then performed for flow over two staggered cylinders and the results are compared with prior data for the same configuration. All the simulations are carried out with both quadratic and cubic reconstruction, and the results indicate a clear improvement with the cubic reconstruction. The new cut-cell approach with cell clustering is able to predict accurate results even at relatively low resolutions. The ability of the high-order cut-cell method in handling sharp geometrical corners and narrow gaps is also demonstrated using various examples. Finally, three-dimensional flow interactions between a pair of spheres in cross flow is investigated using the proposed cut-cell scheme. The results are shown to be in excellent agreement with past studies, which employed body-fitted grids for studying this complex case.

Introduction

Cartesian grid based Immersed Boundary (IB) methods for solving fluid flow over complex bodies are gaining popularity in recent years in view of their associated simplicity in grid generation and their ability to handle complex surface topologies [1], [2], [3], [4], [5]. Another major advantage of using IB based methods is that moving boundaries can be easily incorporated [6], [7], [8], [9]. Also using IB methods, complex boundaries and narrow gaps can be resolved without a major loss in grid quality. In contrast, traditional body-fitted approaches suffer from bad grid quality in the presence of complex surface features and gaps, adversely affecting the stability and convergence of the numerical solver [10]. Many engineering applications exist where it is of interest to understand the flow patterns over a group of objects that are separated by narrow gaps, such as flow past heat exchangers [11], struts, Internal Combustion (IC) engines, etc. Three-dimensional flow interactions between group of droplets or solid particles are also relevant in many applications such as air pollution control, combustion systems, and chemical processes [12]. IB methods are most suited for investigation of the flow physics for such configurations. However, to resolve the flow dynamics in such complex systems, a high-order of accuracy is often needed without which many critical flow phenomena cannot be captured unless a very fine numerical grid is employed. This requirement has been particularly challenging for IB methods as many of these schemes suffer from a lower order of accuracy, specifically near the embedded boundaries.

Many of the reported schemes for IB methods in the past, especially for viscous flows, are utmost second order accurate [13], [14], [15], [16], [5]. Besides, these methods also suffer from issues such as mass loss and noisy reconstruction of flow solution quantities such as wall shear stress, heat flux, etc. [17]. There have been very limited studies reported on higher order schemes for problems involving IB methods for viscous flows. Duan et al. [18] presented a finite-difference based cut-cell approach that was reported to be fourth-order accurate for simulation hypersonic flow over arbitrary surface roughness elements. The accuracy of the near wall solution such as the wall shear stress, was however not discussed in the study. In a recent work [19], a finite difference based extension method that can achieve a smooth reconstruction of pressure and wall shear stress at the immersed boundaries was presented. Although the scheme [19] was set to achieve up to sixth-order accuracy away from the immersed boundaries, the solution dropped to second order at the boundaries.

Cut-cell based finite volume IB methods [20], [21], [13] in comparison to finite difference ghost cell methods [43], [22], are attractive as they enforce strict conservation of mass, momentum and energy and thereby, avoids generation of spurious pressure fluctuations that are observed typically with ghost cell methods [16], [23], [24]. Nonetheless, there are two main problems that are often associated with cut-cell methods. The primary issue is the presence of very small cells, which results in an excessively small time step in case of an explicit scheme or a badly conditioned matrix in case of an implicit scheme. Several approaches have been suggested in the literature for the small cell problem such as (a) the cell-merging approach [14], [25], wherein the small cells are physically merged with the neighbor cell to create a net cell composed of big and small cells, (b) the cell linking approach [15], where the small cells are linked with a master neighbor cell to form a master/slave pair and, (c) cell mixing/redistribution [26], [27], approach where the numerical fluxes from the small cells are mixed with the surrounding cells in a conservative manner.

Among the different approaches for handling the small cell problem, the cell mixing is the easiest to implement as it does not require any changes to underlying cut-cell data structure [7]. The cell merging approach, on the other hand, require changes in the way cut-cells are indexed and stored. Furthermore, the cell merging process introduces new cell topologies, which can severely complicate the finite volume discretization process [27]. The additional complexity with cell merging approach is finding the appropriate neighbors for merging, which is non-trivial in three dimensions [27], [7]. In the cell linking approach, rather than actually merging the cells to form a single cell, the two cells are linked as a master/slave pair. The volumetric and surface information of the slave and master cells remain distinct. But the applicability of the method for linking more than two cells as encountered in three-dimensional cut-cell method has not been established. Additionally, the cell linking/merging procedure reduces the order of the numerical scheme locally [28]. To the best of authors knowledge, none of the above small cell treatments have been shown in the past to be more than second-order accurate. The other issue that often affects cut-cell based IB schemes is the oscillation of pressure and especially wall shear stress field values at the boundaries [29], [17]. The numerical oscillations are caused by the presence of irregularities in the stencil spacing of the numerical grid adjacent to the boundaries. All these limitations have to be addressed to develop a high-order accurate IB method.

The objective of this work is, therefore, to present a high-order accurate conservative adaptive Cartesian based cut-cell method for solving compressible viscous flow problems involving embedded boundaries. We demonstrate that high order of accuracy along with smooth reconstruction of flow field quantities such as heat flux, wall shear stress and pressure can be achieved with the proposed cut-cell method. Recently, Cecere et al. [16] showed that the Central Essentially Non-Oscillatory (CENO) scheme, a higher order approach originally developed for body fitted mesh [30], can be used for cut-cell based IB method. Although third order accurate reconstruction was demonstrated in [16], the overall scheme was reported to be only second order accurate. Our approach shares some features with that of Cecere et al. particularly in the use of k-exact high order polynomial approximation for flux reconstruction [31] (as employed in the CENO method). However, a number of key advances are made in the current effort. These are: (a) an extension of the CENO scheme for cut-cells up to fourth-order of accuracy for inviscid flux evaluation and third order accuracy for viscous flux evaluation, resulting in a formally third order accurate scheme for compressible viscous flows, (b) the use of local mesh refinement to resolve fine-scale surface features, (c) a novel (and easy to implement) small cell clustering algorithm to address the stability problem due to presence of small cells, and (d) the ability to achieve smooth reconstruction of wall shear stress and pressure on immersed interfaces. A nice feature of the new cut-cell approach, is that the method can be extended to arbitrary orders of accuracy both locally near embedded boundaries and globally.

The paper is organized as follows. The governing equations, the formulation of k-exact CENO reconstruction for cut-cells and the new cell clustering algorithm are described in Section 2. In Section 3, the method is validated and used for investigation of canonical laminar flow problems in two and three dimensions. Summary and future work are finally discussed in Section 4.