A hybrid kinetic WENO scheme for compressible flow simulations

Transcription

1 Tenth Internatonal onference on omputatonal Flud Dynamcs (IFD10), Barcelona, Span, July 9-13, 2018 IFD A hybrd knetc WENO scheme for compressble flow smulatons Hongwe Lu *, hangpng Yu, Xnlang L *orrespondng author: hlu@mech.ac.cn State Key Laboratory of Hgh Temperature Gas Dynamcs, Insttute of Mechancs, hnese Academy of Scences, Bejng, hna Abstract: The hybrd knetc Weghted Essentally Non-Oscllatory (WENO) scheme proposed n Int. J. Numer. Meth. Fluds 2015, 79: s further studed n ths paper. In a conventonal WENO framework, flux vector splttng (FVS) technques are usually used for the numercal flux evaluaton at cell nterfaces, and t s well known that the numercal dsspaton of FVS methods s relatvely large due to the free transfer mechansm of gas molecules, n order to reduce the dsspaton n the procedure of flux calculatons, a hybrd knetc WENO scheme was ntroduced n the above-mentoned paper, where the 5th order scheme employng the proposed hybrd knetc approach has been valdated. In ths paper, we further test the 7th order scheme usng the hybrd knetc method. It s ndcated that the 7th order hybrd knetc WENO scheme s more accurate and less dsspatve than that adoptng conventonal FVS technques and has a good shockcapturng capablty, some examples ncludng both one-dmensonal and twodmensonal cases are presented. Keywords: Gas Knetc Theory, WENO Method, Hybrd Numercal Flux, Numercal Dsspaton. 1 Introducton Essentally Non-Oscllatory (ENO) schemes were started wth the classc paper of Harten et al.[1] and further effcently mplemented n [2, 3] for hyperbolc conservaton laws. Later Weghted ENO (WENO) schemes were developed [4, 5], usng a convex combnaton of all canddate stencls nstead of just one as n the orgnal ENO dea. The WENO reconstructon s very effectve n both controllng numercal oscllatons and restorng smooth dstrbutons, whch has been wdely used n many practcal applcatons. In recent years, the development of gas-knetc schemes for compressble flow smulatons has attracted much attenton and become mature, such as the Knetc Flux Vector Splttng () methods[6, 7, 8], the varous algorthms based on the Bhatnagar-Gross-Krook (BGK) model[9, 10, 11, 12, 13], and many others. The gas-knetc schemes use varous knetc equatons to model the dynamc processes around a cell nterface and can provde robust and accurate numercal solutons for varous compressble flows. The combnaton of the WENO reconstructon and gas-knetc flux formulaton has been recently studed by some researchers [14, 15, 16]. In [24], a hybrd knetc WENO scheme was proposed based on the hybrdzaton of two types of knetc fluxes,.e. the free transfer flux and the collsonrelated flux, both evaluated from the WENO reconstructon technque. The 5th order scheme employng the proposed hybrd knetc approach has been valdated n [24]. In ths paper, we wll 1

2 further test the 7th order scheme usng the hybrd knetc method. It s ndcated that the 7th order hybrd knetc WENO scheme s more accurate and less dsspatve than that adoptng conventonal FVS technques and has a good shock-capturng capablty, many examples ncludng both onedmensonal and two-dmensonal cases wll be presented. 2 Hybrd knetc WENO scheme The fnte volume hybrd knetc WENO scheme proposed n [24] wll be brefly presented n ths secton. The WENO reconstructon technques proposed n [5, 25] wll be used n the proposed hybrd knetc scheme. Some mproved smoothness ndcators have been proposed and nvestgated recently[17, 18, 19], our numercal experments ndcate that these smoothness ndcators can also work well for the proposed scheme. Next we wll descrbe the hybrd knetc WENO scheme [24] for solvng the Euler equatons. For the sake of smplcty, only the one-dmensonal case s presented and t can be easly extended to multdmensonal cases n a dmenson by dmenson manner [5]. The 1D Euler equatons can be wrtten as U F( U) 0, (1) t x where ue,, U and F T s gven by F u u 2 p u E p,, ( ) T. For a unform grd wth the cell center x, the cell nterface x +1/2 and the cell sze Δx, a fnte volume method can be wrtten as du 1 F ˆ ˆ 1/2 F 1/2 dt x. (2) The thrd-order TVD Runge-Kutta method proposed n [2] wll be used to ntegrate n tme. Therefore we only need to specfy the constructon of the numercal flux 2.1 onventonal WENO scheme 1/2 for a fnte volume scheme. As shown n [5], the numercal flux F n Eq. (2) can be dvded nto two parts ˆ ˆ ˆ F 1/2 F 1/2 F 1/2, (3) ˆ1/2 where F 1/2 s the flux along x-postve drecton and 1/2 be the numercal flux based on the cell averaged value s the flux along x-negatve drecton. Let U. In order to evaluate 1/2 n Eq. (3), frst we splt the numercal flux nto two parts F F F, (4) whch can be acheved by many flux splttng approaches, such as the Lax-Fredrchs [5] or Steger- Warmng[20] flux splttng method. Then the numercal flux F ˆ 1/2 n Eq. (3) can be obtaned from F by the WENO reconstructon technque, and ˆ F 1/2 can be calculated from F by a symmetrc procedure wth respect to x +1/2. The underlyng physcal prncple for such a conventonal WENO algorthm s the collsonless free transfer of gas molecules. The knetc flux vector splttng () technque [6, 7] can also be used n Eq. (4), and the numercal flux F ˆ 1/2 n Eq. (3) calculated by the approach s denoted by F ˆ, and a 1/2 conventonal 5th-order/7th-order WENO algorthm[5, 25] based on the technque wll be called the W5-/W7- scheme hereafter n ths paper. 2.2 Hybrd knetc WENO scheme 2

3 In the followng we wll present the constructon of a hybrd knetc numercal flux, whch ncludes the effects of both the free transfer and the collson of gas molecules. The hybrd knetc flux can be wrtten as ˆ 1 ˆ F F, (5) where 1/2 1/2 1/2 1/2 s the collsonless -type numercal flux, 1/2 s the numercal flux due to molecule collson effects, and s a parameter n the range 0 1 and wll be hereafter called the jump ndcator. It should be ponted out that ths knd of hybrd numercal fluxes has been used n some knetc schemes, see for example [21, 22, 23], and to name just a few. Snce the evaluaton of the collsonless -type flux F ˆ n Eq. (5) s the same as that descrbed n the precedng subsecton, therefore n order to use Eq. (5) to get the hybrd numercal flux, we only need to determne the collson-related knetc flux The collson-related knetc flux evaluatng s to calculate the flux 1/2 1/2 1/2 1/2 and the jump ndcator. can be constructed as follows. The basc dea of based on the collson-related state Uˆ 1/2 constructed at the cell nterface x +1/2. In order to get the collson-related state, frst we splt the cell averaged conservatve varable nto two parts where U Then the collson-related state where from Uˆ U 1/2 s determned from Uˆ U U 1/2 U U U U, (6) K R v 0 K R v 0 g U, v, dvd. (7) g U, v, dvd at the cell nterface can be obtaned by ˆ ˆ ˆ 1/2 1/2 1/2 by a symmetrc procedure wth respect to x +1/2. U U U, (8) by the WENO reconstructon technque, and The prncple to construct the jump ndcator α s that the contrbuton of the flux Uˆ 1/2 ˆ s calculated F should 1/2 be domnant around strong shock waves and small n smooth regons. Ths s because that the collsonless flux F ˆ s more dsspatve than the collson-related flux. Smlar to the way n 1/2 [21, 22, 23], n the present study we use the local pressure jump around the cell nterface to determne the jump ndcator, p p 1 1 exp, (9) p 1 p where s an emprcal postve constant, t can be seen that for the same local pressure jump, a larger value of results n a larger value of and therefore more -type contrbuton n the hybrd knetc flux whch makes the scheme more dsspatve. Fortunately, our numercal experments as well as those n [21, 22, 23] ndcate that the numercal results are not senstve to the chosen value of, for the present hybrd knetc scheme, we fnd that =10 s a qute good choce based on plenty of numercal experments, therefore t s used for all the numercal tests n ths paper, and the same value of has been adopted n [22]. From Eq. (5) we can see that both F ˆ and F ˆ need to be calculated for each cell nterface 1/2 except for α=0 or α=1. In order to mprove the effcency of the proposed scheme, we ntroduce the followng cut-off type of hybrd flux, 1/2 1/2 3

4 where /2 0 1/2 ˆ F 1 1 1/2 1/2 F 1/2 ˆ 1 1 s a parameter to control the cut-off range of α. Based on a large number of 0.02 trals we fnd that s a satsfactory choce after consderng the overall performance, such as the accuracy and effcency, of the proposed scheme, and therefore t s adopted n ths paper. Up to now, we have evaluated all the unknowns n Eq. (10) whch can be used to get numercal fluxes for the hybrd knetc WENO scheme [24]. Although the component by component verson of the ntroduced hybrd knetc WENO scheme s effectve and works reasonably well for many problems, n ths paper we wll use the more costly, but much more robust characterstc decomposton technque[5] n order to test some demandng problems. The proposed 5th-order/7thorder hybrd knetc WENO algorthm wll be called the W5-HK/W7-HK scheme hereafter n ths paper. 3 Numercal experments In [24], the 5th order scheme employng the proposed hybrd knetc approach has been valdated. In ths secton, we wll test the 7th order hybrd knetc WENO scheme n both one-dmensonal and two-dmensonal cases. The unform mesh s used for both 1D and 2D test problems. 3.1 Accuracy test Ths example s to test the accuracy of the 7th order hybrd knetc WENO scheme. We solve the 1D Euler equatons wth the followng ntal data: ( x,0) 1 0.2sn x, u( x,0) 0.7, p( x,0) 1. (11) The perodc boundary condton s used, and the computatonal doman s taken as [0, 2]. We compute the soluton up to t=2. The errors and convergence orders of densty are shown n Table 1. Ths table shows that the 7th order convergence rate can be obtaned by both W7- and W7-HK. Moreover, the W7-HK has smaller absolute errors than the W7- gven the same cell sze, ths means that the former s more accurate and less dsspatve than the latter. Table 1: 1D accuracy test N Scheme L error order L 1 error order L 2 error order 8 W7-2.37E E E-3 -- W7-HK 2.36E E E W7-8.35E E E W7-HK 4.14E E E W7-1.48E E E W7-HK 5.69E E E W7-1.44E E E W7-HK 5.10E E E W7-1.19E E E W7-HK 4.09E E E (10) 4

5 3.2 Blast wave problem The blast wave problem was orgnally proposed n [26], whch s a challengng test case due to the complex flow structures. The ntal flow feld s gven by 1, 0, 1000, 5 x 4, u, p 1, 0, 0.01, 4 x 4. (12) 1, 0, 100, 4 x 5 The computatonal doman s [-5, 5] wth a reflectng boundary condton on both ends. Snce the exact soluton s unknown for ths problem, the reference soluton obtaned by the W5-HK wth cells s used for comparson. The numercal results wth Δx=0.025 at t = 0.38 are shown n Fgure 1. From the fgure, we can see that the W7-HK/W7- performs much better than the W5-HK/W5-. Moreover, the W7-HK performs slghtly better than the W7-, especally for the complex flow regons. 3.3 Shock acoustc-wave nteracton Next we solve the problem proposed by Shu and Osher[3] whch descrbes the nteracton of an entropy sne wave wth a Mach number 3 rght-movng shock. The ntal condtons are gven by , , , x 4, u, p (13) 1 0.2sn(5 x), 0, 1, x 4 The computatonal doman s [-5, 5], the output tme s t=1.8. The reference soluton for comparson s obtaned by the W5-HK wth cells. The numercal results wth Δx=0.025 are shown n Fgure 2. From the fgure, we can see that the 7th order W7-HK gves much better numercal results than the W5-HK. 3.4 Double Mach reflecton problem Ths problem was extensvely studed n [26] and later by many others. The computatonal doman s taken as [0, 4] [0,1]. Fgure 3 dsplays the close-up of densty contours wth 30 equally spaced contour lnes at t=0.2 wth a 1920X480 unform grd. Frst, t s clearly seen that the 7th order W7- HK/W7- captures the nstablty and roll-ups of the slp lne better than the 5th order W5- HK/W5-. Moreover, W7-HK/W5-HK performs slghtly better than W7-/W5-, whch ndcates that W7-HK/W5-HK s less dsspatve than W7-/W5-. 5

6 6 5 Densty X W5- W5-HK W7- W7-HK Reference soluton Fgure 1: Blast wave problem, the bottom s an enlarged vew of the top. 6

7 4.5 4 Densty W5-HK W7-HK Reference soluton Fgure 2: Shock acoustc-wave nteracton problem, the bottom s an enlarged vew of the top. X 7

8 Fgure 3: Double Mach reflecton problem, Δx=Δy=1/ equally spaced densty contours from 2 to 22. Top left: W5-; top rght: W5-HK; bottom left: W7-; bottom rght: W7-HK. 4 onclusons The hybrd knetc Weghted Essentally Non-Oscllatory (WENO) scheme proposed n [24] s further studed n ths paper. In [24], the 5th order scheme employng the proposed hybrd knetc approach has been valdated, therefore n ths paper, we further test the 7th order scheme usng the hybrd knetc method. Numercal experments have demonstrated that the present 7th order method (W7-HK) s more accurate and less dsspatve than the conventonal scheme wth the technque (W7-) for smooth flows, furthermore the former can provde shaper dscontnuty transton than the latter for flows wth dscontnutes. The present study ndcates that the collsonless technque s ntrnscally very dsspatve, the cautous consderaton of the nfluences of molecule collsons can really reduce the numercal dsspaton of a hgh-order scheme. References [1] Harten A, Engqust B, Osher S and hakravarthy S. Unformly hgh order essentally nonoscllatory schemes III. J. omput. Phys. 1987; 71: [2] Shu -W and Osher S. Effcent mplementaton of essental nonoscllatory shock capturng schemes. J. omput. Phys. 1988; 77:

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