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- Numerical (error) issues on compressible multicomponent flows using a high-order differencing scheme: Weighted compact nonlinear scheme

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<ul><li><p>Taku Nonomura , Seiichiro Morizawa , Hiroshi Terashima , Shigeru Obayashi , Kozo Fujiia JAXA/Institute of Space and Astronautical Science, Yob Tohoku University, Katahira 2-1-1, Aoba-ku Sendai, Mc The University of Tokyo, Hongo 7-3-1, Bunkyo, Tokyo</p><p>a r t i c l e i n f o</p><p>Article history:Received 27 August 2011Received in revised form 20 December 2011Accepted 29 December 2011Available online 10 January 2012</p><p>rocket plume is different from ambient. Several researchers conducted numerical simulations of rocket plumes under anassumption of single-species ows, to predict acoustic environments for satellite or launch elds, even though the gas ofrocket plumes is different from that of ambient [1,2]. The effects of mixing of different gas components on acoustic wavesfrom rocket plume possibly affect the generation of acoustic waves. However, simulations for multi-component ows have</p><p>0021-9991/$ - see front matter 2012 Elsevier Inc. All rights reserved.</p><p> Corresponding author. Tel./fax: +81 42 759 8259.E-mail address: nonomura@ab.isas.jaxa.jp (T. Nonomura).</p><p>Journal of Computational Physics 231 (2012) 31813210</p><p>Contents lists available at SciVerse ScienceDirect</p><p>Journal of Computational Physicsdoi:10.1016/j.jcp.2011.12.035considered to be better choice for compressible multicomponent ows in the framework ofWCNS. Meanwhile better choice of fully or quasi-conservative form depends on a situationbecause the error due to fully conservative form can be suppressed by smoothed interfaceand because quasi-conservative form eliminates all the numerical oscillation but has poormass conservation.</p><p> 2012 Elsevier Inc. All rights reserved.</p><p>1. Introduction</p><p>Mixing of multiple gases in compressible owelds including shock waves is of importance in various engineering elds.For example, the accurate prediction of acoustic waves from a rocket plume is required for rocket launching, where the gas ofKeywords:Weighted compact nonlinear schemeMulti-component compressible owResolutionshinodai 3-1-1 1813, Sagamihara, Kanagawa, Japaniyagi, Japan, Japan</p><p>a b s t r a c t</p><p>A weighted compact nonlinear scheme (WCNS) is applied to numerical simulations of com-pressible multicomponent ows, and four different implementations (fully or quasi-con-servative forms and conservative or primitive variables interpolations) are examined inorder to investigate numerical oscillation generated in each implementation. The resultsshow that the different types of numerical oscillation in pressure eld are generated whenfully conservative form or interpolation of conservative variables is selected, while quasi-conservative form generally has poor mass conservation property. The WCNS implementa-tion with quasi-conservative form and interpolation of primitive variables can suppressthese oscillations similar to previous nite volume WENO scheme, despite the presentscheme is nite difference formulation and computationally cheaper for multi-dimensionalproblems. Series of analysis conducted in this study show that the numerical oscillationdue to fully conservative form is generated only in initial ow elds, while the numericaloscillation due to interpolation of conservative variables exists during the computations,which leads to signicant spurious numerical oscillations near interfaces of different com-ponent of uids. The error due to fully conservative form can be greatly reduced bysmoothing interface, while the numerical oscillation due to interpolation of conservativevariables cannot be signicantly reduced. The primitive variable interpolation is, therefore,Numerical (error) issues on compressible multicomponent ows usinga high-order differencing scheme: Weighted compact nonlinear scheme</p><p>a, b c b a</p><p>journal homepage: www.elsevier .com/locate / jcp</p></li><li><p>3182 T. Nonomura et al. / Journal of Computational Physics 231 (2012) 31813210several problems discussed later, and it is important to clarify the characteristics of schemes for such ows. We are now con-sidering application of a weighted compact nonlinear scheme, which is adopted in the study by Nonomura and Fujii [1,2], tosuch problems.</p><p>Here, WCNS is a variation of weighted-essentially-nonoscillatory (WENO) [3,4] type scheme, and has three advantagescompared with the original WENO: (1) WCNS has higher resolution than WENO [5,6]; (2) various ux evaluation methodscan be applied [5], e.g., Roes ux difference splitting method (FDS) [7]; and (3) freestream and vortex preservation proper-ties are very good on a wavy grid [8,9]. These advantages are basically owing to that the variable interpolation used in nitevolume method can be used in the WCNS procedure despite WCNS is a nite difference method. Considering these advan-tages, high-order WCNS was developed by Nonomura et al. [10] and by Zhang et al. [6], independently.</p><p>Numerical methods for compressible multicomponent ows can be decomposed into two types: (1) sharp interfacemethod and (2) diffused interface method. In the sharp interface method, immiscible ows are assumed. Interfaces betweendifferent uids (hereafter we call simply interface) are captured by volume of uid method [11], level-set method [1214]or front tracking method [1517] and each uid is solved independently. Here, boundary conditions at interface are given byghost-uid method [16,18] or application of exact Riemann solver at the interface [17,19]. Although interface can be resolvedsharply, it is difcult to apply these methods to the complex shape of interface. On the other hand, diffused interface meth-ods [2024] assumes that the uid is miscible with introducing equations of mass fraction, and the interface is not treateddirectly. Due to the easiness of implementation and complicated interfaces expected to be generated, diffused interfacemethod is adopted in the framework of WCNS for the application to multicomponent ows.</p><p>Abgrall [20] applied diffused interface methods to convection problems of uids with different specic heat ratio, and hereported that the numerical oscillation is observed when a fully conservative form of the governing equations was used. Thisproblem is because pressure equilibrium is violated at the interface, owing to mismatch of evaluations of energy and specicheat ratio. Abgrall and other researchers [20] showed that this numerical oscillation can be eliminated by nonconservativeform of mass fraction equation (or energy equation [21]). This type of formulation is so-called a quasi-conservative form.Recently, Johnsen and Colonius [22] extended this idea to a higher-order nite volume WENO scheme. They demonstratedthat the characteristic interpolation of primitive variables is a key for eliminating numerical oscillations in compressiblemulticomponent ows, while the general WENO uses the conservative variables. This technique cannot be applied to thenite difference WENO because nite difference WENO does not include the variable interpolation, but conservative uxinterpolation. With regard to applications of high-order scheme to compressible multicomponent ows, Kawai and Terashi-ma [24] conducted compressible multi-component ow simulations with high-order compact scheme and localized articialdiffusivity. They demonstrated that the scheme works effectively for reducing spurious oscillations at interfaces, showingthe good mass conservation property due to the use of the fully conservation form. They also suggested that the use ofsmooth initial interface is effective to eliminate the startup error and the subsequent spurious oscillations related to com-pressible multicomponent ow simulations.</p><p>The nite volume WENO by Johnsen and Colonius [22] seems to be a good way while it has following two problems; (1)computationally expensive due to its multi-dimensional reconstruction inherently adopted in nite volume scheme and (2)mass conservation of each species is not fully satised due to quasi-conservative formulation [25]. With regard to the formerproblem, WCNS is able to use variable interpolation despite nite difference formulation. Thus, the characteristic interpola-tion of primitive variable adopted in Johnsen and Colonius [22] can be implemented in nite difference formulation, and theerror indicated by Johnsen and Colonius can be eliminated while keeping computational costs cheaper, because the nitedifference scheme has lower computational costs than nite volume scheme, which is reported in Ref. [26,27] and shortlydiscussed in Appendix B. On the other hand, the latter problem relates to the selection of fully conservative or quasi-conser-vative forms. Although there is a trade-off between this problem and the error indicated by Abgrall [20] the discussionincluding this trade-off has not been conducted in detail. It should be noted that there are limited studies on implementationof high-order scheme to compressible multi-component ows.</p><p>More recently, Johnsen and his colleague [28] proposed the formulation for keeping the pressure, velocity, and temperatureequilibriumwith the fully conservative equations. In their formulation, the fully conservative equations and specic heat ratioare simultaneously solved, and the specic heat ratio is used for the estimation of pressure, though this method considers theoverestimation system because the specic heat ratio can be also estimated by themass fractionwhich is solved in the conser-vative form. Although the method by Johnsen is very attractive for keeping the equilibrium and conservation, it still has thedisadvantage of high computational cost and further discussion should be needed for the disadvantage of using overestimationsystem. Therefore, we do not introduce this technique for the discussion in this paper, while it should be noted that this tech-nique also can be used for the WCNS without losing its advantage of the nite difference formulation.</p><p>Also, Shukla et al. [29] introduced the articial compression technique for the diffused interface method. Their resultshows that this method is good to capture the interface. It should be noted that this technique is also applicable to the WCNSwhile maintaining the advantage of WCNS to WENO, but it is not discussed in this paper because it is out of scope of thispaper discussed below.</p><p>The objective of the present study is to understand the preferable implementation of WCNS to compressible multi-component ows; especially to understand effects of the choice of fully conservative or quasi-conservative forms and thechoice of characteristic interpolation of conservative variables or primitive variables. To investigate effects and trade-offrelationship of these choices, four different implementations of WCNS is evaluated throughout various test cases. Theimplementation with quasi-conservative forms and characteristic interpolation of primitive variable is expected to be the</p></li><li><p>schemof init</p><p>2. Num</p><p>In</p><p>characcharacacteri</p><p>insteawavesinterfa</p><p>In t</p><p>follow</p><p>wheretively</p><p>A f</p><p>T. Nonomura et al. / Journal of Computational Physics 231 (2012) 31813210 3183@t @xj</p><p>where Y1 is the mass fraction of gas 1. The mass fraction of gas 0 is written as follows:</p><p>Y0 1 Y1: 3On the other hand, a quasi-conservative form means the following nonconservative form of the mass fraction equations isused instead of Eq. (2)</p><p>@Y1@t</p><p> @ujY1@xj</p><p> Y1 @ui@xi</p><p> 0: 4</p><p>It is noted that, although the standard nonconservative form of mass fraction equation is written as</p><p>@Y1@t</p><p> u1 @Yi@xi</p><p> 0; 5</p><p>Eq. (4) is employed to obtain the same convection velocity for density, energy and mass fraction, as discussed in Johnsen andColonius [22]. Assuming temperature equilibrium at the interface, the specic heat ratio of mixture is dened as follows:</p><p>c 1Y1c11</p><p>Y0c01</p><p> 1; 6</p><p>where c, c0, and c1 are specic heat ratio of mixed gas, gas 0 and gas 1, respectively. The ideal gas equation of state is writtenas follows, using the specic heat ratio of mixed gas:</p><p>qe pc 1</p><p>12quiui: 7</p><p>Table 1 also summarizes equations employed in each implementation.@t j</p><p>@xj 0; 1</p><p>q, ui, p, e, xi and t are the density, velocity vector, pressure, energy per unit mass, position vector, and time, respec-. Here, dij denotes Kronecker delta.ully conservative form means that a following conservative form of equation of mass fraction is added to Eq. (1):</p><p>@qY1 @qujY1 0; 2@t @xj@qui@t</p><p> @quiuj pdij@xj</p><p> 0;@qe @qe pus:</p><p>@q @quj 0;mass fraction equations are solved as the governing equations. A conservative form of the compressible Euler equations is as</p><p>his section, details of implementations for two-component ows are presented. The compressible Euler equations and2.1. Governing equationsd of componentwise interpolation because componentwise interpolation generates spurious oscillation near shock. It should be noted that componentwise interpolation is almost the same as characteristic interpolation for capturingce of two-components in our a priori test.acteristic interpolation of primitive variable is named Qprim. In all the implementations, we use characteristic interpolationteristic interpolation of conservative variable is named Fcnsv. The implementation of fully conservative from withteristic interpolation of primitive variable is named Fprim. The implementation of quasi-conservative from with char-stic interpolation of conservative variable is named Qcnsv. The implementation of quasi-conservative from with char-tive forms and the choice of characteristic interpolation of conservative variable or primitive variable are investigated. Theseimplementations are constructed on the well-validated in-house code. The implementation of fully conservative from witherical methods</p><p>this study, four possible implementations of WCNS are conducted. The choice of fully conservative or quasi-conserva-addition, we demonstrate the high resolution of WCNS compared with conventional upwind scheme and the high efciency(low computational costs) of WCNS compared with nite volume WENO in Appendixes A and B, respectively.e. This is simply analyzed based on formulations and computationally veried through the test cases. Also, the effectsial condition (smoothness of interface) for each error are investigated as in the study by Kawai and Terashima [24]. Inerror-free implementation like the nite volume WENO adopted by Johnsen and Colonius despite it is nite difference</p></li><li><p>2.2. Fi</p><p>whereresoluHartetail. T</p><p>Henode j</p><p>wherethe strepresinterp</p><p>Thfollow</p><p>5 15 5 1</p><p>3184 T. Nonomura et al. / Journal of Computational Physics 231 (2012) 31813210Next, qLj1=2;m is computed as weighted averaging of qL;kj1=2;m:</p><p>qLj1=2;m w1;mqL;1j1=2;m w2;mqL;2j1=2;m w3;mqL;3j1=2;m w4;mqL;4j1=2;m; 13where wk,m is nonlinear weight computed as</p><p>wk;m ak;mPlal;m</p><p>; 14</p><p>ak;m CkISk;m e2: 15</p><p>Here,</p><p>C ;C ;C ;C 1 ;21 ;35 ; 7 </p><p>16are op</p><p>wheresimplqLj1=2qL;4j1=2;m 16 qj;m 16 qj1;m 16 qj2;m 16 qj3;m: 12qj1=2;m 16 qj3;m 16 qj2;m 16 qj1;m 16 qj;m; 9</p><p>qL;2j1=2;m 116</p><p>qj2;m 516</p><p>qj1;m 1516</p><p>qj;m 516</p><p>qj1;m; 10</p><p>qL;3j1=2;m 116</p><p>qj1;m 916</p><p>qj;m 916</p><p>qj1;m 116</p><p>qj2;m; 11qjn;m lj;mQjn;m; n 3; . . . ;3; 8</p><p>qj+n,m is mth component of characteristic variable at node j + n, lj,m is the mth left eigenvector for the center node j forencil. Here...</p></li></ul>