An adaptive multiscale finite volume solver for unsteady and steady state flow computations☆
Introduction
Due to the increasing computer power more and more realistic and consequently more complex models have become tractable by numerical simulation. Studying the interaction of aerodynamics and structural dynamics is a typical example. Here several severe obstructions, such as time-dependency of the involved processes, varying complex geometries and the coupling of physical regimes with different characteristic features come together. In order to resolve a typically singular behavior of the solution meshes with several millions of cells are required. However, improved hardware or purely data oriented strategies such as parallel computing are not sufficient to overcome the arising difficulties. As important and necessary these aspects may be they have to be complemented in the long run by mathematical concepts that aim at minimizing in the first place the size of arising discrete problems.
This paper summarizes some recent attempts in this direction. We present an integral concept for designing a finite volume solver for compressible flow computations. The three main components of this new solver QUADFLOW consist of (i) a standard finite volume discretization for arbitrary grid topologies, (ii) a block-structured grid generation using parametric mappings based on B-splines and (iii) a local grid adaptation based on a local multiscale analysis. These tools are appropriately adjusted to each others needs. Here the core ingredient is the adaptation strategy that is based on a mathematically well-founded concept. The ultimate goal of the present work is to verify that this concept is no longer confined to academic problems on Cartesian grids only but has become mature. In particular, it can be employed for the investigation of problems arising, for instance, in aerodynamics.
In the literature, several adaptive strategies have been discussed or are under current investigation. A standard strategy is to base local mesh refinements on local indicators which are typically related to gradients in the flow field, see [13], [14], or local residuals, see [48], [66], [67]. Although these concepts turn out to be very efficient in practice they offer no reliable error control. For this purpose, a posteriori estimates have been derived which aim at equilibrating local errors. So far, this type of error estimator is only available for scalar problems, see [50]. In the present work, however, we employ recent multiresolution techniques. The starting point is a proposal by Harten [45] to transform the arrays of cell averages associated with any given finite volume discretization into a different format that reveals insight into the local behavior of the solution. The cell averages on a given highest level of resolution (reference mesh) are represented as cell averages on some coarse level where the fine scale information is encoded in arrays of detail coefficients of ascending resolution. This information is essentially used in Harten’s original strategy to gain computational time by avoiding expensive flux evaluations in regions where the solution is smooth. Instead cheap finite differences are employed in major parts of the domain. The solution remains within the same accuracy as the reference scheme, i.e., the scheme on the finest computational mesh that uses the expensive flux evaluation throughout the entire domain. Successful implementations of this strategy have been carried out for two-dimensional Cartesian meshes [15], [16], [26], [27], [62], curvilinear meshes [33] and unstructured meshes [2], [17], [30]. However, since one works still on a uniform mesh the computational complexity stays proportional to the number of cells on the finest grid which in 3D computations with the above objectives is prohibited.
In contrast to this, the detail coefficients will be used here to create locally refined meshes on which the discretization is performed. Of course, the crux in this context is to arrange this procedure in such a way that at no stage of the computation there is ever made use of the fully refined uniform mesh. A central mathematical problem is then to show that the solution on the adapted mesh is of the same accuracy as the solution on the reference mesh. This genuine adaptive approach has been presented in [42] and has been investigated in [31]. A self-contained account of the adaptive concept for conservation laws can be found in [59]. By now the new adaptive multiresolution concept has been employed by several groups with great success to different applications, see [23], [32], [58], [63].
The adaptive concept is based on a hierarchy of meshes. This requires a new grid generation strategy. Accepting the Navier–Stokes equations as the model of choice we give preference to quadrilateral and hexahedral cells that still facilitate best boundary fitted anisotropic meshes. Local refinement gives rise to meshes of quadtree and octree type, respectively. A key idea is to represent such meshes as parametric mapping from the computational domain into the physical domain. In this way one overcomes the restriction to Cartesian meshes as employed in the literature mentioned above. Such mappings can be realized using well established concepts from computer aided geometric design (CAGD), see for instance [19], [39]. To retain sufficient geometric flexibility this is combined with block structuring.
Finally, one needs a discretization scheme that meets the requirements of the adaptation concept and fits well with the mesh generation. The adaptive method crucially depends on the assumption, that the fluxes of the underlying discretization are conservative. Furthermore, to avoid complicated mesh management within each block and to keep the discretizations of the individual blocks as independent as possible and, in particular, to avoid global geometrical constraints, we insist on meshes with hanging nodes. Both requirements are met by the development of a finite volume scheme that can cope, in particular, with fairly general cell partitions.
We organize the remainder of this paper as follows: Section 2 gives a short description of the governing equations as used in the present work. Section 3 is the essential part of the paper describing the multiscale analysis and the grid adaptation algorithm. Section 4 is concerned with the representation and generation of parametric meshes using B-spline methods. Section 5 offers a self-contained account of the discretization scheme including the realization of spatial second order, the choice of limiters, the treatment of convective and viscous fluxes and the time integration. In Section 6 we present several applications to well-known fluid dynamical test cases that highlight the features of the whole flow solver. We conclude this paper in Section 7 with some remarks on future developments.
Section snippets
Governing equations
In the present study, laminar viscous fluid flow is described by the Navier–Stokes equations for a compressible gas. In order to solve problems in time dependent domains, including moving boundaries, we consider the governing equations in its arbitrary Lagrangian Eulerian (ALE) formulation. Neglecting body forces and volume supply of energy, the conservation laws for any control volume V with boundary ∂V and outward unit normal vector n on the surface element dS⊂∂V can be written in integral
Grid adaptation concept
In this section we outline the grid adaptation concept. It is based on a multiscale analysis of an array of cell averages. For this purpose we first summarize the multiscale setting. Finally we explain how to perform an adaptive mesh refinement employing the multiscale analysis.
Parametric meshes
The multiscale setting outlined in Section 3 is based on a hierarchy of nested grids. From this point of view the most natural way to discretize the flow domain would be to employ adaptive Cartesian grids. On the other hand it is widely accepted that boundary conforming meshes are preferable for the discretization of viscous flows because they facilitate best the generation of anisotropic grid cells that are necessary for a stable and accurate resolution of boundary layers. In this approach one
Finite volume method
The occurrence of hanging nodes due to local mesh adaptation poses particular difficulties concerning the discretization of the governing equations. In the following section we present a finite volume method, which is capable to operate on meshes of any arbitrary topology. This approach offers a unified way to incorporate hanging nodes. Its main ingredients will be discussed in detail, including data structures, realization of spatial second order accuracy, treatment of convective and viscous
Numerical results
In the following, various test cases are considered to demonstrate the benefits of the adaptive concept for a wide range of applications. First, we investigate the inviscid, stationary flow about the NACA0012 airfoil in transonic Mach number regime, the inviscid hypersonic flow over a double ellipse and the transonic flow over a swept wing in a channel. To evaluate the method for viscous flows, the laminar flow about the NACA0012 airfoil and the laminar flow over a flat plate are investigated.
Conclusion and outlook
We have presented the still intermediate state of development of the new flow solver QUADFLOW that integrates dynamic adaptation, mesh generation and finite volume discretization. Its main features have been illustrated by numerous steady and unsteady computations governing a wide class of flow problems including transonic and hypersonic flows, laminar flows as well as moving boundaries. Without a-priori knowledge of the solution all physical relevant effects (shocks, boundary layers, etc.) are
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This work has been performed with funding by the Deutsche Forschungsgemeinschaft in the Collaborative Research Centre SFB 401 ‘Flow Modulation and Fluid–Structure Interaction at Airplane Wings’ of the RWTH Aachen, University of Technology, Aachen, Germany.