Disturbance region update method for steady compressible flows

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Abstract

A new methodology capable of achieving considerable acceleration in iteration together with lower memory requirements, named as the disturbance region update method (DRUM), is presented for steady compressible flow simulation. The methodology is constructed based on the characteristics of the time-marching solution process, utilizing a time-dependent dynamic computational domain. The update of the solution needs only be executed within the dynamic domain containing solely the disturbed cells with non-convergent solutions, while the update of the dynamic domain can be realized by measuring the propagation of disturbances. Eight test cases, including a subsonic, three mixed subsonic–supersonic, and four supersonic flow problems, are chosen to validate the developed methodology. Numerical results of these test cases demonstrate that, firstly, DRUM is equally robust for different reconstruction, spatial discretization and time-marching schemes, initializations as well as for discontinuous walls; secondly, it accomplishes remarkable convergence speed for solving all compressible flow problems, benefiting from the reduction in the computational effort per iteration and the reduction in the total number of iterations; thirdly, it may also save notably in terms of the maximum memory requirements for supersonic problems.

Introduction

Continued efforts have been devoted to improvements of computational efficiency in computational fluid dynamics (CFD). And there are constant demands for sustained increases in the execution speed of scientific and engineering codes [1], e.g., in the studies on complex flows [[2], [3]] and in the aircraft design [4]. For steady compressible flows, the prevailing number of numerical methods in CFD applies the time-marching technique [5]. The technique utilizes time-dependent governing equations, starting from arbitrarily assumed initial conditions, and integrates with respect to time to attain an asymptotic limit at large time as the steady-state solution. In order to develop efficient time-marching procedures, large efforts have been paid on three classes of techniques, which are focused on the schemes coupled with the time-marching procedure, on the implementation and on the grid, respectively.

The first class attempts to enhance the convergence per iteration of the time-marching technique, such that the total number of iterations required for reaching the same convergence threshold could be lessened. One theoretical foundation of this class is to increase the maximum allowable time step of the time integration. Successful examples include the local time-stepping strategy [6], the multigrid methodology [7], the large time step schemes [8], and the implicit time-marching schemes (e.g., the lower-upper symmetric Gauss–Seidel (LU-SGS) method [9]). A time step is proportional to the size of the corresponding control volume and the stability-correlated Courant–Friedrichs–Lewy (CFL) number [5]. The local time-stepping strategy [6] allows each control volume to employ its own maximum allowable time step; i.e., with the same CFL number, a larger control volume is advanced in time with larger time steps and hence converges faster. Similarly, the basic idea of the multigrid methodology [7] is to utilize the coarse grids with larger time steps to drive the solution on the finest grid faster to the steady state. As opposed to the former two, the latter two make use of increasing the CFL number. The large time step Godunov scheme proposed by Qian and Lee [8] enables the Godunov scheme to elevate its upper limit of the admissible CFL number via a multi-wave approximation to the expansion fan. Moreover, the implicit time-marching schemes are unconditionally stable so that they are well capable of enhancing convergent performance by significantly larger CFL numbers.

The other theoretical foundation of the first class is based on the idea of damping the error in residuals, such as methods like the residual smoothing technique [10], the convergence error estimation method based on eigenvalue analysis [11], and the implicit time-marching schemes based on Newton–Krylov method (e.g., the generalized minimal residual (GMRES) technique [12]). For these schemes, residuals associated with fluxes are replaced by improved corrections before updating the solution. By substituting the average weighted with the residuals of adjacent control volumes for the residual of a specific control volume, the residual smoothing technique [10] would provide a better damping on the high-frequency error. The convergence error estimation method proposed by Eyi [11] replaces the residual with the convergence error approximated by the average weighted in terms of the residuals at several time levels. The GMRES technique [12] seeks a correction in the Krylov subspace, which is capable of minimizing the global residual to achieve the quadratic convergence. In addition, the multigrid methodology [7] also serves the purpose of damping the low-frequency error components on the finest grid via iterations over coarser grids.

The second class, and quite an attractive one at present, relies on parallelization. Methods of this class could accelerate the convergence per second by means of distributing the work to multiple processors; that is, the method itself solely shares the computational effort owned originally by a single processor rather than reducing the total amount of computations. Parallel computing on the central processing unit (CPU) and the graphics processing unit has shown significant speedup in comparison with the sequential execution time on single CPU calculations [13]. Numerous efforts were accomplished to improve computational efficiency, e.g., parallel implementation for various schemes [[13], [14]], for different grids [[15], [16]], and for diverse flow problems [[17], [18]], as well as strategies to optimizing parallelization [19].

The third class, which is rather intuitive, is drawn based on reducing the number of grid cells so as to reduce the computational effort per iteration. Nowadays this class can only be implemented via the grid generation, e.g., via rational grid topologies and distributions, as well as the adaptive mesh refinement technique [[16], [20]]. The grid generation attempts to concentrate the computational effort in regions near shock discontinuities and sharp profiles, while degrading grid resolution of the regions with small gradients. The technique is capable of controlling the grid refinement, coarsening and deformation, but unable to vary the computational domain flexibly throughout computation execution. It is worthwhile to note that the low flexibility inherent in the computational domain may be one of the dominant obstacles to the reduction of the computational effort. The reason is twofold: Firstly, the selection of the computational domain that plays a crucial role in the numerical result is highly dependent on experience, in which an inadequate domain may lead to a failure of computations, while a redundant one would waste computational resources. Secondly, a static computational domain may not fully take the characteristics of the time-marching solution process into account, which would also result in wasting the computational effort.

To circumvent this problem, an attempt is made in the present work to develop a convergence acceleration methodology based on the idea of varying the computational domain synchronously with the propagation of disturbances in the flowfield. The remainder of this paper is organized as follows. In Section 2, the motivations and aims of the present work are first specified following a numerical experiment on a supersonic, inviscid flow over a wedge. The currently implemented mathematical and numerical models are introduced briefly in Section 3. The principle and the implementation of the developed methodology are described in Section 4, and some numerical test cases for validating the methodology are reported in Section 5. Section 6 summarizes the concluding remarks.

Section snippets

Motivations and aims

The time-marching solution process of a steady flow can be viewed as roughly reproducing the time history of an unsteady flow starting from the assumed initial conditions. Take an inviscid supersonic flow over a two-dimensional (2D) wedge with a wedge angle of 5°at Mach 2 as an example. The solution process initialized from the freestream conditions and driven by a finite-volume Euler solver on a uniform structured grid is illustrated in Fig. 1, in which Nmax denotes the total number of

Governing equations

For clarity, the proposed methodology, DRUM, would be introduced by utilizing inviscid, two-dimensional (2D) or axisymmetric compressible flow problems. The Euler equations governing the flows can be written in integral form of the conservation laws as ddtΩωaWdΩ+ΩωaF,GndS=ΩωQdΩwith ωa=1ω+ωy.

Here, a Cartesian frame of reference (x, y) is chosen and the time variable is denoted by t. Ω is a 2D control volume, Ω the boundary of Ω, dS the elemental surface area of Ω, and n

Disturbance region update method

The idea of DRUM is inspired in accordance with the observation on some convergence histories of the time-march-ing method. In the time-marching solution process, disturbances characterizing the change in flow properties would start from the place where the steady-state governing equations cannot be fulfilled. On the one hand, the disturbances would propagate to the surrounding flowfield with finite wave speeds. The upstream solution, on the other hand, would converge faster than that of the

Validation and discussion

In this section, eight relevant problems are chosen to demonstrate the capacity of the developed DRUM, as listed in Table 2. DRUM is implemented in Fortran 95 on Windows, running on a Intel dual-core CPU with 2.9 GHz and 3.33 GB of RAM. All norms are taken to be the root mean square or L2 norm. Except the transonic test case C, the CCFL in all other tests are set to be 0.9 and 10 for the explicit and the implicit schemes, respectively. Other than the test case F focused on testing the

Conclusion and discussions

The present work introduces a new methodology, the disturbance region update method (DRUM), for solving steady compressible flow problems. The methodology is constructed based on the finite volume technique and characterized by advancing in time with a high-flexible, dynamic computational domain (DCD), which retains solely the disturbed cells with non-convergent solutions. A circular doubly linked list is utilized to store the information of the DCD, and enables the developed DRUM procedure to

Acknowledgments

This work was supported by grants from the National Natural Science Foundation of China (Nos. 11372028, 11721202), and the Fundamental Research Funds for the Central Universities .

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