Elsevier

Computer Physics Communications

Volume 244, November 2019, Pages 97-116
Computer Physics Communications

Zonal disturbance region update method for steady compressible viscous flows

https://doi.org/10.1016/j.cpc.2019.06.015Get rights and content

Abstract

The zonal disturbance region update method (zDRUM) presented in this work is an extension of the disturbance region update method (DRUM) to steady compressible viscous flows, capable of achieving convergence acceleration together with lower memory requirements. In the frame of DRUM and the zonal method, the new methodology taking advantage of the characteristics of the time-marching solution process employs two time-dependent dynamic computational domains where solely disturbed cells with non-convergent solutions are updated while the inviscid and the viscous flows are treated separately. A new data structure inspired by the pin art is introduced to store the dynamic computational domains more efficiently. Numerical results of six test cases in a wide range of Mach and Reynolds numbers demonstrate that, firstly, zDRUM accomplishes remarkable convergence speed for solving all compressible viscous flow problems, benefiting from the reduction in the computational effort per iteration; secondly, it is equally robust and efficient for different dimensions and flow types, for various reconstruction, spatial discretization and time-marching schemes; thirdly, it may reduce the maximum memory requirements.

Introduction

Although technology improvements in computer speed have enabled the solution of the complete Navier–Stokes (N–S) equations, numerical simulation of compressible viscous flows over practical configurations on grids having a sufficient resolution still requires a large amount of computational effort. Therefore, there are constant demands for sustained speedup of scientific and engineering codes [1], e.g., in the studies on complex flows [2], [3] and in the aircraft design [4], [5].

When it comes to the acceleration of the numerical simulation for viscous flows, the most straightforward approaches are constructed by replacing the N–S equations with reduced forms that are cheaper to solve. It is well accepted that flows with negligible viscous effects can be determined by the Euler equations, which truncate the viscous and thermal conduction terms of the N–S equations [6]. If the inviscid flow is also irrotational and isentropic, the Euler equations can be further simplified as the full-potential (F-P) equations by representing the velocity components as derivatives of a scalar potential function [6]. For flows dominated by viscosity, Prandtl in 1904 propounded the famous boundary-layer (B-L) theory, in which the B-L equations [7] are a first-order simplification of the N–S equations at high Reynolds number. To increase the approximation order up to the second, Van Dyke [8] derived a second-order boundary-layer equation considering the pressure gradient in the normal direction, by performing an order-of-magnitude analysis on the terms in the N–S equations. On the basis of this second-order B-L equations, Davis et al. [9] developed the viscous shock-layer technique to deal with hypersonic flow problems. The parabolized N–S equations [10] belong to a more accurate approximation in which only the viscous terms involving derivatives in the streamwise direction are dropped. For turbulent flows, Reynolds in 1895 firstly presented the approximate treatment of the chaotic fluctuations of turbulent flow variables, decomposing flow variables into a mean and a fluctuating part. Nowadays, the density-weighted Reynolds-averaged Navier–Stokes equations (RANS) [11] are widely used to simulate compressible turbulent flows. In the frame of the RANS equations, turbulence models of varying complexity, e.g., first-order closures based on the eddy-viscosity hypothesis of Boussinesq [12], [13] and non-linear eddy-viscosity closures, are developed to close the equation system.

The B-L theory is constructed on the phenomenon that viscous friction plays virtually no role in the vast region of the flowfield away from the wall while becomes dominant in the thin layer of fluid immediately adjacent to the wall [7]. Therefore, another class of the acceleration approaches that subdivide the flow into inviscid and viscous regions have attracted wide attention, termed the zonal technique. Numerous efforts are paid on coupling the aforementioned equations in different zones to reduce computational requirements. One could utilize either the N–S equations or a reduced form, or both, for viscous flows while describe inviscid flows by either the Euler or F-P equations. With the assumption that the viscous region has no effect on the inviscid portion of the flowfield, the classic Euler/ B-L method [10] solved the Euler and the B-L equations separately, in which results of the Euler equations solved first are supplied as input to the integral of the B-L equations. In order to fit viscous separation areas and wakes, Saint-Martin et al. [14] added a N–S solver into the classic Euler/ B-L method. By comparison, Veldman et al. [15] took the interaction between inviscid and viscous flows into account, iteratively coupling the transonic F-P equations with the B-L equations in a quasi-simultaneous way, where the information exchange of regions took place at the wall and viscous results attained by the B-L equations were fed back to the inviscid-flow solver as wall conditions. Payne et al. [16] associated a parabolized N–S and a complete N–S solver together, in which regions where parabolizing assumptions are valid were solved by a parabolized N–S solver with downstream-marching schemes while the N–S solver was responsible for the remaining regions. Summa et al. [17] solved the inviscid portion of the flow over a subsonic airfoil or wing with a F-P code while the viscous portion with a N–S method. Su [18] combined the F-P equations with the RANS equations to solve high Reynolds number turbulent flows.

Although the zonal technique aimed at taking full advantage of the flow nature of each zone is a promising approach and has achieved considerable speedup, it also suffers from the following disadvantages:

(1) Limited applicability. Reduced forms of the N–S equations are valid with additional assumptions, such as the F-P equations are valid for the irrotational, isentropic and inviscid flow while the B-L and the parabolized N–S equations are merely suitable for the attached viscous flow.

(2) Programming complexity. Different governing equations are treated in different ways with different working variables, and special treatment of information exchange between different zones must be performed so as to eliminate the jumps of different mathematical models.

(3) Difficulties in partitioning. The majority of zonal techniques usually subdivide the computational domain into regular shapes by experience and physical considerations, which may not define rational zones.

To circumvent these disadvantages, flows in different zones should be described by the governing equations capable of sharing working variables as well as solution methods. The Euler, the N–S and the RANS equations would be a potential combination. The reasons are threefold: firstly, they are suitable for inviscid, laminar and turbulent flows, respectively, without any extra assumption; secondly, they employ common conservative variables and all can be solved by the time-marching technique [11] for compressible flows; thirdly, smooth transitions between them can be achieved readily by measuring the order of magnitude of their different terms.

At present, the time-marching technique is in common use to solve steady compressible flows in the computational fluid dynamics [11] since it is valid for flows over a wide range of Mach and Reynolds numbers. This technique utilizes unsteady governing equations, starting from arbitrarily assumed initial conditions, and advances in time to attain an asymptotic limit at large time as the steady-state solution. The indispensable time-dependent term makes the mathematical problem hyperbolic with respect to time in the full speed range of compressible flows and allows the solution of both the subsonic and supersonic regions simultaneously with the same numerical technique [19]. Nevertheless, it suffers from efficiency issues, requiring more computational effort as compared with alternative methods like the downstream-marching procedure. Fortunately, in our previous work [20], it is found that there is much worthless computational effort being performed throughout the conventional global update method (GUM) that updates the solution in all cells of a preassigned computational domain (PCD), e.g., in [21], [22]. With observations on some convergence histories of the time-marching technique, we revealed the characteristics of the time-marching solution process for steady compressible inviscid flows. Therefore, in accordance with the boundary-layer theory, features of the time-marching solution process for steady compressible viscous flows can be deduced as follows.

(1) Disturbances are produced from where the steady-state governing equations cannot be satisfied. For the case initialized from the freestream conditions, wall conditions lead to discontinuities in flow properties such that disturbances develop from the wall.

(2) Disturbances would propagate gradually to the surrounding flow with finite wave speeds as the time evolves, and flow properties in the region that has not been disturbed yet would maintain at the initial states.

(3) In compressible flows, the upstream region would not converge slower than the downstream since the flow speed is orders of magnitude equal to or larger than the speed of sound.

(4) For viscous flows, viscous effects are merely dominant in a finite zone near the surface, no matter whether the flow is attached or separated.

The former three features can be utilized by the disturbance region update method (DRUM) [20] proposed by us, which solves governing equations in a dynamic computational domain (DCD) and updates the DCD with the propagation of disturbances to reduce the computational effort. The last feature is exactly the cornerstone of the zonal technique.

An attempt is made in the present work to extend DRUM to steady compressible viscous flows by combining with the zonal technique. The new acceleration methodology is named the zonal disturbance region update method (zDRUM), motivated by the desire to eliminate worthless computation from the time-marching solution process as much as possible. Compared with DRUM, zDRUM utilizes two kinds of DCDs, i.e., the advective and the viscous, to treat inviscid and viscous flows separately; zDRUM also adopts a new data structure inspired by the pin art to store DCDs more efficiently. The zDRUM is distinguished from GUM and conventional zonal techniques because of its distinctive evolution of DCDs, in which DCDs would build from where disturbances are generated, and extend with the propagation of disturbances, and then contract as the solution converges.

The remainder of this paper is organized as follows. In Section 2, currently implemented mathematical models and numerical methods are introduced briefly. Principle and algorithms of zDRUM are described in Section 3. In Section 4, numerical results attained by zDRUM and GUM are compared to validate zDRUM’s accuracy, efficiency and robustness. The last section summarizes the concluding remarks and discusses future work.

Section snippets

Governing equations and turbulence models

The zDRUM employs the Euler equations for inviscid flows, the N–S equations for laminar flows and the density-weighted RANS equations for turbulent flows, respectively. The conservative form of the compressible N–S equations can be written as ddtΩWdΩ+Ω(FcFv)dS=ΩQdΩwhere t represents the time variable; a structured control volume is denoted with Ω, the boundary of Ω with Ω, and a surface element on Ω with dS. The conservative variables W, convective fluxes Fc and viscous fluxes Fv are

Basic principles

The zDRUM is constructed in accordance with observations on viscous flows past over obstacles and on convergence histories of the time-marching solution process, aimed at reducing the worthless computational effort. The time-marching solution process for steady flows can be viewed as roughly reproducing the evolution of the flowfield around a body placed suddenly in an assumed flow [20]. Disturbances carrying information about the presence of the body attempt to be transmitted to the

Validation and discussion

Six relevant problems, including laminar and turbulent flows in different dimensions and at a wide range of speeds, simulated with various spatial and temporal discretization schemes, are reported in this section to demonstrate the capacity of zDRUM, as listed in Table 2. The zDRUM is implemented in Fortran 95 on Windows, running on an Intel dual-core CPU at 2.9 GHz. The 3-D test case F is paralleled by using OpenMP. All norms employed in zDRUM are taken to be the root mean square or L2 norm.

Conclusion and discussions

The present work introduces the zonal disturbance region update method (zDRUM) suitable for solving steady compressible viscous flow problems. The methodology is constructed based on the cell-centered finite volume technique and characterized by advancing in time with two high-flexible, dynamic computational domains (DCDs), in which the advective DCD retains solely the disturbed cells with non-convergent solutions while the viscous DCD covers the region dominated by viscous effects. A new data

Acknowledgments

This work was supported by grants from the National Natural Science Foundation of China (Nos. 11372028, 11721202), the Aerospace Science & Technology Joint Foundation of China Aerospace Science & Technology Corporation (No. 6141B06220405), and the Fundamental Research Funds for the Central Universities of China .

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