The Role of Momentum Interpolation Mechanism of the Roe. Scheme in the Shock Instability

Transcription

1 The Role of Momentum Interolation Mechanism of the Roe Scheme in the Shock Instabilit Xue-song Li Ke Laborator for Thermal Science and Power Engineering of Ministr of Education, Deartment of Thermal Engineering, Tsinghua Universit, Beijing , PR China Abstract: The shock instabilit henomenon is a famous roblem for the shock-caturing scheme. B subdividing the numerical dissiation of the Roe scheme, the term of ressure-difference-driven modification for the cell face velocit is regarded as a version of the momentum interolation method (MIM), which is necessar for incomressible flows to suress the ressure checkerboard roblem. Through the analsis and odd-even decouling test, it is discovered that MIM las the most imortant role on the shock instabilit. In fact, for non-linear flows MIM should be comletel removed, but unexected MIM is activated on the cell face nearl arallel to the flow for high Mach number flows or low Mach number flows in shock. For such conditions, two coefficients are designed based on local Mach number and a shock detector, resectivel, and then the imroved Roe scheme is roosed, which gives consideration to requirement of MIM for incomressible and comressible flows and is validated for good erformance of avoiding odd-even decouling. Therefore, the aim of decreasing rather than increasing numerical dissiation to cure the shock instabilit can be achieved. Ke word: shock caturing, Roe scheme, momentum interolation method, shock instabilit, odd-even decouling * Corresonding author. Tel.: ; fax: address: xs-li@mail.tsinghua.edu.cn (X.-S. Li).

2 1. Introduction Over the ast decades, the Roe scheme [1], which is one of the most imortant shock-caturing schemes, has obtained great success for simulating comressible flows and automatic caturing shock. However, the Roe scheme also meets some disastrous failings for certain roblems, such as low-mach number incomressible flows [2-4] and shock instabilit henomena of hersonic flows comutation. The shock instabilit has different forms such as famous carbuncle henomenon for suersonic flows around a blunt bod, the kinked Mach stem for a double-mach reflection flow, and the odd-even decouling for a lanar moving shock. Quirk [5] concluded these shock instabilit forms and roosed a curing method. In fact, all these different forms have the same inherent mechanism, and a method which fails or succeeds in one form will fails or succeeds in other forms. The curing method for the shock instabilit can be generall categorized into three grous. The first grou is to add dissiation to the scheme. More dissiation seems alwas hel to dam out surious oscillation. Quick [5] suggest combining a dissiative scheme, such as HLLE, and a less dissiative more accurac scheme, such as Roe, b a switch sensor. Entro fix is another classical method of addition of extra dissiation b limiting the minimum sstem eigenvalue and has man imroved versions [6, 7]. Qu and Yan et al. [8] roose the RoeMAS scheme which is efficient for curing the shock instabilit b increasing basic uwind dissiation. The second grou considers that the shock instabilit is due to the contradiction between the multi-dimensional characteristic of the comressible flow and one-dimensional grid-aligned nature of the 2

3 scheme. Therefore, multi-dimensional Riemann solvers were develoed, such as rotated Roe-te schemes [9, 10]. The last grou exlains the reason of shock instabilit as ressure difference in the numerical mass flux [11]. Kim [12] also analzed the effect of the ressure difference through a linear erturbation analsis, and then roosed an imroved Roe version named as the RoeM scheme. Extending the shock-caturing scheme to all-seed flows is another imortant subject for all schemes such as Roe-te [2-4], HLL-te [13, 14] and AUSM-te [15, 16] schemes. While studing an all-seed Roe scheme, Ref. [17] discovered that the Roe scheme has an inherent mechanism to revent the checkerboard roblem, which is classical roblem for comutation of incomressible flows and means chess-like ressure jum because of ressure-velocit decouling. The inherent mechanism is identified as the ressure-difference-driven modification for the numerical cell face velocit, which is just the term discussed in Ref. [11, 12], and can be regarded as a version of the momentum interolation method (MIM hereafter) [18], where MIM is the classical method to suress the ressure checkerboard and has man forms [19-21]. Ref. [4] further discussed the mechanism and showed that the different versions of MIM can be relaced each other as the term of ressure-difference modification for cell face velocit. Since MIM can be regarded as a art of the scheme and is indisensable for low-mach number incomressible flows, it is interesting what role of MIM in the shock instabilit and how to balance the existence of MIM between shock and incomressible flows. This question motivates this aer to understand the mechanism of MIM on the 3

4 shock instabilit and then roose a curing method with reaching the aim of decreasing rather than increasing numerical dissiation. The outline of this aer is as follows. Chater 2 gives the governing equations and the Roe scheme. Chater 3 discusses the effect of MIM on the shock instabilit and rooses an imrovement. Chater 4 validates the analsis and imrovement b the numerical examle of odd-even decouling test. Finall, Chater 5 closes the aer with some concluding remarks. 2. Governing Equations and the Roe Scheme 2.1 Governing Equations The governing three-dimensional Navier-Stokes (N-S) equations can be written as follows: Q F G H F v G v H v, (1) t x z x z where u Q v is the vector of conservation variables; w E u 2 u F uv, uw uh v uv vw vh 2 G v, w uw H vw are the vectors of Euler fluxes; 2 w wh v F, v G, and v H are the vectors of viscous fluxes which are not given in detail for simlicit; is fluid densit; is ressure; E is total energ; H is total enthal, and u, v, w are the velocit comonents in the Cartesian coordinates x,, z, resectivel. 4

5 2.2 The Roe Scheme The classical Roe scheme can be exressed as the following general sum form of a central term and a numerical dissiation term: F F F, (2) c d where F c is the central term and F d the numerical dissiation term. For a cell face of the finite volume method, 1 Fc, 12 FL F R, 2 (3) 0 u n x F U v n, w n z H 0 (4) where n x, n, and n z are the comonents of the face-normal vector, and U nxu nv nzw is the normal velocit on the cell face. Following Ref. [2], a scale uniform framework for the shock-caturing scheme is roosed [23], which is ver simle, low comutation cost, and eas to be analzed and imroved: F d 0 u n x u 1 = v u n U u U v, (5) 2 w n w z E U H where the five terms on the right side have exlicit hsical meaning: the first term is basic uwind dissiation, which is just low-mach Roe scheme [24]; the terms u and are velocit-difference-driven and ressure-difference-driven modifications for the cell face ressure, resectivel; and the terms Uu and U are 5

6 velocit-difference-driven and ressure-difference-driven modifications for the cell face velocit, resectivel. For the Roe scheme, 1, (6) U, (7) u 1 U 2 c 5 4 U, (9) 2 c 5 4, (8) u 5 4 U c, (10) where c is the sound seed, and the eigenvalues of the sstem are defined as follows: U, 4 U c, 5 U c. (11) Eqs. (6) - (11) are equivalent to the original form of the Roe scheme with Roe average [23], which make the following assumtion tenable [2]:, (12) where reresents one of the fluid variables. Eqs. (6) - (11) can also be further simlified for easier analsis as follows: max 0, c U U, (13) u sign U min U, c, c (14) U Uu sign U min U, c, c (15) U max 0, c U. (16) 2 c 3. Analsis of MIM Role on Shock and Imrovement In the following text, the term is focused. When U 0, Eq. (16) becomes U U, (17) c and can roduce results with weak checkerboard [4]. In fact, the reasonable interval of 6

7 the order of coefficient of in U is c, c. Decreasing the coefficient to 1 0 lower than 1 c, the comutational instabilit will occur because of uncontrolled ressure checkerboard [4]. Based on the ke idea of MIM, Ref. [18] derived the time-marching MIM, for which two-dimensional stead form can be exressed as follows: x x i x i 1 i 1 2 i i1 i1 2 U t n x n 2i 2i 1 i1 2 2i 2i 1 i1 2 n1. (18) Eq. (18) introduces a third ressure-derivative term into the cell face velocit. Comared with Eq. (17) and (18), the have same osition in the scheme, similar mechanism of ressure difference and effect on suressing the checkerboard roblem. Therefore, Eq. (17) can be regarded as a version of MIM, which is indisensable for low-mach number incomressible flows with the collated-grid method. For high-mach number comressible flows, however, the decouling of ressure and velocit disaear and then MIM is not necessar. More imortant, MIM is bad for comressible comutation, which is analzed as follows. From the ersective of results, MIM suress the checkerboard roblem, i.e. the ressure jum, and then make ressure field smooth. However, the shock is also a kind of ressure jum in fact. Therefore, the shock ma be smoothed and destroed b MIM. From the ersective of mechanism, the ke idea of MIM is to obtain cell face velocit b calculating the ressure gradient and interolation of other terms. Then, the calculated value and interolation of ressure difference have matching error for cell 7

8 face velocit. The error is insignificant from the angle of amount of numerical dissiation, and is helful for ressure and velocit couling when the flow field is linear. For the nonlinear flow, however, the error ma become too large to roduce hsical results. From this exlanation, it can also be deduced that the staggered-grid method, which is another famous method curing the checkerboard roblem, cannot be adoted to cature shock, because it saves calculated ressure and velocit in different osition and the matching errors cannot be avoided. From the ersective of the Roe scheme itself, it holds MIM for low-mach number flows, and tends to automaticall remove MIM for high-mach number flows. For the suersonic one-dimensional flows, the U term is equal to zero according to Eq. (16). However, because the Roe scheme is develoed from one-dimensional Riemann roblem and onl be simlified extended for multi-dimensional flows, MIM that should disaear survives on the grid face aroximatel arallel to flow direction as shown in Fig.1 (a), where U 0 although the flow seed V ma suersonic. It should also be noticed that there ma exist cells of low Mach number in shock as shown in Fig.1 (b). Flows of such cells are strong nonlinear and should be treated as shock rather than incomressible flows although their Mach numbers are low, and then MIM also should disaear. For this condition, the traditional judging criterion of Mach number becomes invalid, and a shock detector is needed to identif this condition. 8

9 MIM survives 1.5 cell V Mach number 1 Low Mach number cells in shock 0.5 MIM survives (a) Cell face arallel to flow direction 0 (b) Low Mach number cell in shock Fig.1 Conditions of unexected existence of MIM According to above discussions, two rules can be roosed to avoid harmful MIM: (1) For high Mach number flows, U 0 under an circumstances; (2) For low Mach number flows in shock, U should also be equal to zero. For the first rule, a coefficient for U can be designed as follows: 8 s1 1 f M, (19) where the Mach number M u v w c 2 2 2, and the function f is roosed [25] as follows: f min,1 2 1, (20) where reresents an of variables. As shown in Fig. 2, the function 8 f remains nearl zero when Mach number is lower than 0.3, and has a smooth transition near the sound seed. However, s 1 is based on the Mach number, and then cannot satisf the second rule. Therefore, another coefficient is roosed as follows: 8 s2 f t, (21) where t is a shock detector. A simle and effective design of t [8, 12], which 9

10 searches ressure jums among the cell face and its neighbor faces, is shown as two-dimensional form as follows: t min P, P, P, P, P, (22) i1 2, j i1 2, j i1, j1 2 i1, j1 2 i, j1 2 i, j1 2 P i1 2, j i, j i1, j min, i 1, j i, j. (23) Fig. 2 The function of f and f 8 Therefore, the MIM term U Eq. (16) can be imroved b multiling the coefficients s 1 and s 2 : U s1s2max 0, c U. (24) 2 c For the imroved Roe scheme with Eq. (24), the numerical dissiation alwas not increases and sometimes decreases for weakening MIM comared with the original Roe scheme. Therefore, the imroved Roe scheme rovides a wa to cure the shock instabilit b decreasing numerical dissiation. 4. Odd-Even Decouling Test 10

11 4.1 Comutational Object and Method Quick [5] design this test case in which a lanar shock moves in a duct where a centerline grid is odd-even erturbed as shown in Eq. (25): Y i, j,mid Yj,mid Y, for i even,, (25) Yj,mid Y, for i odd. where larger value of makes the odd-even decouling more seriousl. This test case is imortant because an scheme that does not survive it meets more or less shock-instabilit roblems for other classical cases The initial conditions are given as,, u, v,,,0 L and,, u, v 1.4,1,0,0, which roduce a normal shock with moving Mach number of R 6. The comutational mesh has 20*800 cells in Y and X directions, resectivel, with Y 1 and X 1. The schemes are adoted with first order accurac unless otherwise secified. Besides the original Roe scheme and its imrovement with Eq. (24), two other conditions are also considered as follows. The following entro fix Eqs. (26)-(27) is also adoted as comarison because it is usuall used. i, i h, i 2 1, (26) i h, i h, 2 h h max, (27) i where is a constant with usuall adoted value of 0.05 to 0.2. When the value of is zero, it means the Roe scheme without the entro fix. According to the discussion of two rules in Chater 3, for this test case the U 11

12 term can be directl set to zero 0. (28) U Eq. (28) can be used to test the ultimate effect of MIM on shock instabilit, and can be used to validate the imrovement of Eq. (24). Because increasing of can amlif the odd-even decouling henomenon and make comutation more difficult, the three conditions of 0, 0.1 are investigated as follows. 10 4, and 4.2 The Condition of ε =0 When 0, which means no odd-even decouling, the densit contours b the Roe scheme with 0, i.e. the entro fix is not adoted, is given in Fig. 3. The shock in the Fig. 3 is located at X 620 and shows good, which means that the odd-even decouling henomenon does not occur. The reason is that MIM on the cell face arallel to the flow is not activated because 0. Although MIM is also roduced on the cell face erendicular to the flow of the low Mach number cells in shock as shown in Fig. 1(b), because U is large for theses faces the effect of MIM is suressed b the numerical dissiation term in Eq. (5). In fact, large value of can remarkabl imroves daming rate of densit erturbation and then hel of avoiding odd-even decouling even MIM exists [8]. 12

13 Fig.3 The densit contours b the Roe scheme without entro fix The Condition of 4 ε =10 When 10 4, the unexected MIM is activated, and then the Roe scheme without the entro fix fails. The shock is deformed from middle art at X 220 as shown in Fig. 4(a) and is comletel destroed as shown in Fig. 4(b). (a) Begin of shock deformation (b) Comlete smoothed of shock Fig. 4 The Roe scheme without entro fix 0 Adoting the entro fix, the results onl obtain sight imrovement as shown in Fig. 5 (a) with common value of 0.05 for. Increasing value of 0.05 u to 0.2, the shock recovers normal as shown in Fig. 5 (b). (a)

14 (b) 0.2 Fig. 5 The Roe scheme with entro fix Decreasing the numerical dissiation of MIM, the Roe scheme can obtain correct results as shown in Fig. 6 and Fig. 7 without adding an extra dissiation. Eq. (24) is reliminaril validated because it has the same effect as Eq. (28) Fig. 6 The imroved Roe scheme with Eq. (28) U 0 Fig. 7 The imroved Roe scheme with Eq. (24) U s s max 0, c U c The Condition of ε = 0.1 When the value of is increased to 0.1, the comutation becomes more difficult. As shown in Fig. 8, even 0.2, the shock is also seriousl deformed. Removing MIM comletel b Eq. (28) U 0, the shock is also twisted to a great extent as shown in Fig. 9. It indicates that MIM is not onl factor for the shock instabilit. Even 14

15 so, MIM can be regarded as the most imortant factor, because removing it is obviousl better than the large entro fix. The result in Fig. 10 b the imroved Roe scheme with Eq. (24) is ver similar to that in Fig. 9, which further validates the Eq. (24). Some differences between Fig. 9 and Fig. 10 ma due to few activations of MIM in smoothed area of shock, which makes the shock detector invalid. Fig. 8 The Roe scheme with entro fix 0.2 Fig. 9 The imroved Roe scheme with Eq. (28) U 0 Fig. 10 The imroved Roe scheme with Eq. (24) U s s max 0, c U c If necessar, the term in Eq. (5) (6) can be increased as roosed in Ref. [8] for difficult cases. For the aim of decreasing dissiation as far as ossible, MUSCL [26] reconstruction with three-order interolation and minmod limiter is adoted for second order accurac. The result in Fig. 11 (a) shows that the shock instabilit is almost comletel avoided comared with Fig. 10. The reason ma be due to much fewer 15

16 nodes in shock. Comared with about ten nodes in shock for the first scheme as shown in Fig. 1 (b), onl three nodes are needed to form the shock in Fig. 11 (b). Obviousl, fewer nodes make less ossibilit of inducing shock instabilit. Therefore, combining the imroved Roe scheme with Eq. (24) and high order reconstruction, a shock-stable result can be obtained b decreasing rather than increasing numerical dissiation even for this difficult case. (a) Densit Contours (b) Mach number distribution Fig. 11 The imroved Roe scheme with Eq. (24) and MUSCL-TVD reconstruction 16

17 5. Conclusions The numerical dissiation term U of the Roe scheme is regarded as a version of MIM, and then the role of MIM is investigated for the shock instabilit roblem. The research discovers that MIM las the most imortant role, leading to the shock instabilit because of unexected activation on the cell face arallel to the flow when Mach number is high or low but in shock. Therefore, following rules should be satisfied: (1) For high Mach number flows, U 0 ; (2) For low Mach number flows in shock, U 0 ; (3) For low Mach number incomressible flows, U of the Roe scheme should be ket. According to the rules, two coefficients are roosed based on the Mach number and a shock detector, resectivel. Multiling U b two coefficients, an imroved Roe scheme can be obtained and has well erformance to avoid the shock instabilit. Eseciall, with high order reconstruction, the imroved Roe scheme can give satisfing results even for difficult cases. Therefore, the aim of decreasing numerical dissiation to cure the shock instabilit can be achieved. Acknowledgments This work is suorted b Project of the National Natural Science Foundation of China. 17

18 References [1] P.L. Roe, Aroximate Riemann Solvers: Parameter Vectors and Difference Schemes, Journal of Comutational Phsics 43 (1981) [2] H. Guillard, C. Viozat, On the Behaviour of Uwind Schemes in the Low Mach Number Limit, Comuters and Fluids 28 (1999) [3] D.G. Huang, Preconditioned Dual-Time Procedures and its Alication to Simulating the Flow with Cavitations, Journal of Comutational Phsics 223 (2007) [4] X.S. Li, C.W. Gu, Mechanism of Roe-te Schemes for All-Seed Flows and Its Alication, Comuters and Fluids 86 (2013) [5] J.J. Quirk, A Contribution to the Great Riemann Solver Debate, International Journal for Numerical Methods in Fluids 18 (1994) [6] H.C. Lin, Dissiation Addition to Flux-Difference Slitting, Journal of Comutational Phsics 117 (1995) [7] M.J. Kermani, E.G. Plett, Modified Entro Correction Formula for the Roe Scheme, AIAA Paer (2001). [8] F. Qu, C. Yan, D. Sun, Z. Jiang, A New Roe-te Scheme for All Seeds, Comuters & Fluids (2015), doi: htt://dx.doi.org/ /j.comfluid [9] Y.X. Ren, A Robust Shock-Caturing Scheme Based on Rotated Riemann Solvers, Comuters & Fluids 32 (2003) [10] H. Nishikawa, K. Kitamura, Ver Simle, Carbuncle-Free, Boundar-Laer-Resolving, Rotated-Hbrid Riemann Solvers, Journal of Comutational Phsics 227 (2008) [11] M.S. Liou, Mass Flux Schemes and Connection to Shock Instabilit, Journal of Comutational 18

19 Phsics 160 (2000) [12] S. Kim, C. Kim, O.H. Rho, S.K. Hong, Cures for the Shock Instabilit: Develoment of A Shock-Stable Roe Scheme, Journal of Comutational Phsics 185(2003) [13] S.H. Park, J.E. Lee, J.H. Kwon. Preconditioned HLLE Method for Flows at All Mach Numbers. AIAA Journal 44 (2006) [14] H. Luo, J.D. Baum, Extension of Harten-Lax-van Leer Scheme for Flows at All Seeds, AIAA Journal 43 (2005) [15] M.S. Liou, A Sequel to AUSM, Part II: AUSM+-u for All Seeds, Journal Comutational Phsics 214 (2006) [16] E. Shima, K. Kitamura, Parameter-Free Simle Low-Dissiation AUSM-famil Scheme for All Seeds AIAA Journal 49 (2011) [17] X.S. Li, C.W. Gu, An All-Seed Roe-Te Scheme and Its Asmtotic Analsis of Low-Mach-Number Behaviour, Journal of Comutational Phsics 227 (2008) [18] X.S. Li, C.W. Gu, The Momentum Interolation Method Based on the Time-Marching Algorithm for All-Seed Flows, Journal of Comutational Phsics 229 (2010) [19] C.M. Rhie, W.L. Chow, Numerical Stud of the Turbulent Flow Past An Airfoil with Trailing Edge Searation, AIAA Journal 21 (1983) [20] A. Pascau, Cell Face Velocit Alternatives in A Structured Colocated Grid for the Unstead Navier Stokes Equations, International Journal for Numerical Methods in Fluids 65 (2011) [21] Y. Moguen, T. Kousksou, P. Bruel, J. Vierendeels, E. Dick, Pressure-Velocit Couling Allowing Acoustic Calculation in Low Mach Number Flow, Journal of Comutational Phsics 231 (2012) 19

20 [22] J.M. Weiss, W.A. Smith, Preconditioning Alied to Variable and Const Densit Flows, AIAA Journal 33 (1995) [23] X.S. Li, Uniform Algorithm for All-Seed Shock-Caturing Schemes, International Journal of Comutational Fluid Dnamics 28 (2014) [24] X.S. Li, J,Z. Xu and C.W. Gu. Preconditioning Method and Engineering Alication of Large Edd Simulation, Science in China Series G: Phsics, Mechanics & Astronom, 51 (2008) [25] X.S. Li, C.W. Gu, J.Z. Xu, Develoment of Roe-Te Scheme for All-Seed Flows Based on Preconditioning Method, Comuters and Fluids 38 (2009) [26] B. Van Leer, Towards the Ultimate Conservative Difference Scheme. V. A Second-Order Sequel to Godunov s Method, Journal of Comutational Phsics 32 (1979)