Efficient high order accurate staggered semi-implicit discontinuous Galerkin methods for natural convection problems
Introduction
Natural convection problems play an important role in computational fluid dynamics. They appear in numerous engineering applications and natural phenomena ranging from the design of cooling devices in industrial processes, electronics, building isolation or solar energy collectors, to the simulation of atmospheric flows. In the last decades, the scientific community has put a lot of efforts into the study of these phenomena, see e.g. [5], [6], [7], [8], [9], [10] for a non-exhaustive overview. Nowadays, the main challenge is to develop efficient high order numerical methods which are able to capture even small scale structures of the flow, avoiding the use of RANS turbulence models (see [11], [12]). In this paper, we propose a novel family of high order accurate staggered semi-implicit discontinuous Galerkin (DG) methods, which extends the works presented in [1], [3], [4] appropriately to deal also with gravity driven flows.
Depending on the magnitude of the temperature perturbation and on the importance of density changes, natural convection problems are usually divided into two main groups. If the Mach number and the temperature fluctuations are small, the incompressible Navier-Stokes equations under the usual Boussinesq assumption can be applied. Otherwise, the full compressible Navier-Stokes equations have to be employed. In the following, we will mainly focus on the first case. However, in this paper also the compressible model will be considered, thus allowing for a direct comparison of the results obtained using the two different systems of governing partial differential equations. Therefore, we will be able to further validate the applicability of the Boussinesq approach for the flow regimes we are interested in.
In the literature there are numerous approaches that have been proposed for the solution of the Navier-Stokes equations, such as finite difference methods [13], [14], [15], [16] or continuous finite element schemes [17], [18], [19], [20], [21], [22], [23]. Nevertheless, the construction of high order numerical methods, and especially of high order discontinuous Galerkin (DG) finite element schemes, is still a very active research field, which has started with the pioneering works of Bassi and Rebay [24], and Baumann and Oden [25], [26]. Later, several high order DG methods for the incompressible and compressible Navier-Stokes equations have been proposed, see for example [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39]. We also would like to mention recent works on semi-implicit DG schemes that can be found in [40], [41], [42], [43], [44], to which our approach is indirectly related.
The algorithm proposed in this article makes use of the novel family of staggered semi-implicit DG schemes that has been introduced in [1], [2], [3] for the incompressible Navier-Stokes equations in two and three space dimensions and which was later also extended to the full compressible regime in [4], following the work outlined in [45], [46], [47]. These arbitrary high order accurate DG schemes are constructed on staggered unstructured meshes. The pressure, the density and the energy are defined on the triangular or tetrahedral primal grid, whereas the velocity is computed on a face-based staggered dual mesh. While the use of staggered grids is a very common practice in the finite difference and finite volume framework (see e.g. [13], [14], [48]), its use is not so widespread in the context of high order DG schemes. The first staggered DG methods, which adopted a vertex-based dual grid, have been proposed in [49], [50]. Other recent high order staggered DG algorithms that rely on an edge-based dual grid have been advanced in [51], [52]. For high order staggered semi-implicit discontinuous Galerkin schemes on uniform and adaptive Cartesian meshes, see [53], [54].
Focusing on the incompressible model, we propose two different approaches for the computation of the nonlinear convective terms. On the one hand, we consider the methodology already introduced in [3]. There, the convective subsystem for the velocity is solved considering the Rusanov flux function for an explicit upwind-type discretization of the nonlinear convective terms. Instead, the viscous terms are discretized implicitly, making again use of the dual mesh in order to obtain the discrete gradients, without needing any additional numerical flux function for the viscous terms. One of the major drawbacks of this approach is its high computational cost coming from the small time step dictated by the CFL stability condition due to the explicit discretization of the convective terms. Moreover, to avoid spurious oscillations, a limiter should be used (see [4]). As an alternative option, which is at the same time able to deal with large gradients and substantially reduces the computational cost, we propose the use of an Eulerian-Lagrangian approach, recently forwarded in [55] also in the context of high order in space staggered DG schemes. The trajectory of the flow particles is followed backward in time by integrating the associated trajectory equations at the aid of a high order Taylor series expansion, where time derivatives are replaced by spatial derivatives using the Cauchy-Kovalevskaya procedure, similar to the ADER approach of Toro and Titarev [56], [57], [58]. The high order spatial discretization of the DG scheme is then employed to obtain a high order time integration for each point needed to solve numerically the advection part of the governing equations. For further information on efficient semi-Lagrangian and Eulerian-Lagrangian schemes we refer the reader to [59], [60], [61], [62], [63], [64], [65], [66], [67], [68], [69].
The use of the Boussinesq assumption yields the coupling of the incompressible Navier-Stokes equations with an additional conservation equation for the temperature. The computation of the related advection and diffusion terms is performed similarly to what is done for the velocity in the momentum equation. Nevertheless, let us remark that the temperature is defined on the primal mesh so that interpolation from one mesh to the other is avoided in the fully Eulerian approach. Once the new temperature is known, the gravity source term in the momentum equation can be evaluated. Finally, the pressure Poisson equation is solved and the velocity at the new time step is computed.
Regarding the compressible Navier-Stokes equations, we extend the numerical scheme introduced in [4] to consider the additional gravity terms. To this end, two new terms are included in the pressure system which has been obtained by formal substitution of the discrete momentum equation into the discrete energy equation. The first gravity term, coming from the momentum equation, is computed jointly with the convective and viscous terms of the momentum equation at the beginning of each time step. Meanwhile, the gravity term embedded in the energy equation is computed at each Picard iteration using the updated values of the linear momentum density.
The rest of the paper is organized as follows. In Section 2 we recall the incompressible and compressible Navier-Stokes equations. For the incompressible model, the Boussinesq assumption is made to account for fluid flow with small temperature variations under gravity effects. Concerning the compressible model, we consider the full Navier-Stokes equations including the conservation law for the total energy density and assuming here the equation of state for an ideal gas. Section 3 is devoted to the description of the semi-implicit staggered DG method used to solve the incompressible model in two and three space dimensions. We start by recalling some basic definitions about the usage of staggered meshes and the polynomial spaces which are employed. Next, we derive the numerical method considering two different frameworks for the discretization of convective and diffusive terms, namely an Eulerian and an Eulerian-Lagrangian approach. The extension of the algorithm to the compressible case is described in Section 4. Several benchmarks are presented in Section 6, aiming at assessing the validity, efficiency and the robustness of our novel numerical schemes. The main pros and cons of the Eulerian and the Eulerian-Lagrangian approaches are analyzed as well. Finally, we compare the results obtained with the incompressible solver against those computed with the compressible solver in the low Mach number regime.
Section snippets
Governing equations
As already mentioned, natural convection problems may be studied using two different models: the incompressible and the compressible Navier-Stokes equations, both including proper gravitational terms. The choice of the model usually depends on specific features of the flow under consideration, like the magnitude of the temperature fluctuations or the importance of capturing density variations. Here, we are mainly interested in small temperature changes, so that we will firt focus on the
Numerical method for the incompressible model
System (2)-(4) will be solved starting by the staggered semi-implicit discontinuous Galerkin scheme detailed in [1], [2], [3]. Here, we recall the main ingredients of the algorithm, while for an exhaustive description the reader is referred to the aforementioned references.
Numerical method for the compressible model
The staggered semi-implicit discontinuous Galerkin scheme described in the previous section for the incompressible model has been extended in [4] to solve the compressible Navier–Stokes equations at all Mach numbers. Concerning semi-implicit finite volume schemes for all and low Mach number flows, we refer the reader to [84], [85], [86], [87], [88].
To simulate natural convection problems, some modifications are needed in order to incorporate the gravitational terms. In what follows, we will
Time step restriction
The maximum time step is restricted by a CFL-type condition based on the local flow velocity:with CFL < 1/d, d the space dimension, hmin the smallest insphere diameter (in 3D) or incircle radius (in 2D) and vmax is the maximum convective speed. If viscous terms are present, the eigenvalues of the viscous operator have to be considered as well (see [4], [33]).
As commented in Section 3.6, if we employ the Eulerian–Lagrangian approach, the scheme becomes unconditionally
Numerical test problems
In this section, classical benchmarks for natural convection problems are used in order to verify the validity and the efficiency of the novel algorithms presented in this work. Moreover, these tests allow us to analyze the strengths and drawbacks of our numerical schemes.
Conclusions
In this work, we have presented a new high order accurate staggered semi-implicit discontinuous Galerkin finite element scheme for the solution of natural convection problems. The algorithm is based on the work proposed in [1], [3], [4]. A unified framework for the discretization of incompressible and compressible Navier–Stokes equations with gravity terms has been introduced. The computational cost of the global algorithm has been reduced thanks to the development of a novel
Acknowledgments
This work was financially supported by INdAM (Istituto Nazionale di Alta Matematica, Italy) under a Post-doctoral grant of the research project Progetto premiale FOE 2014-SIES; M.T. and M.D. acknowledge partial support of the European Union’s Horizon 2020 Research and Innovation Programme under the project ExaHyPE, grant no. 671698 (call FETHPC-1-2014); W.B. was partially financed by the GNCS group of INdAM and the program Young Researchers Funding 2018. The simulations were performed on the
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