A sufficient condition for free-stream preserving in the nonlinear conservative finite difference schemes on curvilinear grids
FFree-stream preserving metrics and Jacobian for the conservative finitedifference method on curvilinear grids
Hongmin Su , Yixuan Lian , Jinsheng Cai, Kun Qu ∗ , Shucheng Pan Department of Fluid Mechanics, School of Aeronautics, Northwestern Polytechnical University, Xi’an 710072, China
Abstract
For the compressible flows simulations, the conservative finite difference based on the upwind schemes, i.e.the linear upwind and WENO, is widely used for their simplicity and conservative property. However, thismethod loses the geometric conservation law (GCL) identity due to the upwind dissipation when applied onthe curvilinear grids. In this paper, we suggest a free-stream preserving metrics and Jacobian in the upwinddissipation to maintain the free-stream preserving identity. This technique avoids destroying the accuracyand the standard forms of the upwind schemes as far as possible. Therefore, this technique is convenient tooperate in conservative finite difference method. Some verifications are conducted to show the accuracy inthe smooth flow filed and the robustness in the discontinuous regions of the present free-stream preservingmethod, such as the isotropic vortex problem, the double Mach reflection problem, the transonic flow pastNACA0012 airfoil and ONERA M6 wing, etc..
Keywords:
Conservative finite difference, geometric conservative law, free-stream preserving,linear-upwind scheme, WENO scheme
1. Introduction
The geometric conservation law (GCL) [1], including volume conservation law (VCL) and surface conser-vation law (SCL) [2, 3, 4], is very important in computational fluid dynamics, especially for the high-accuracysimulations. Unlike the finite volume method (FVM) and finite element method (FEM), the conservativefinite difference method (FDM) binds the physical quantities and geometric metrics together during theprocess of flux reconstruction in the computational space such that the GCL is not easy to be satisfied dueto the discretization errors of grid metrics for the upwind dissipation. The violation of the GCL will yieldlarge errors, ossilations and instabilities for the simulations [1, 5, 6], and even lead to the non-conservation
Co-first authors ∗ Corresponding author
Email addresses: [email protected] (Hongmin Su), [email protected] (Yixuan Lian), [email protected] (Jinsheng Cai), [email protected] (Kun Qu), [email protected] (Shucheng Pan)
Preprint submitted to Elsevier September 9, 2020 a r X i v : . [ phy s i c s . c o m p - ph ] S e p f the governing equations [7, 4]. On the stationary grids, the VCL identity is satisfied naturally but theSCL is not. Therefore, some efforts should be made to the numerical method to achieve the SCL identity.Many achievements have been put forward to maintain the free-stream preservation, especially for thelow order schemes, as concluded in Ref. [8, 9]. In the high-accuracy FDM, the symmetry conservativemetric method(SCMM) [10], inspired by the conservative metrics in Ref.[1], have been an efficient techniqueto fulfill this identity under the sufficient condition of evaluting the derivatives of the grid metrics and theconvection fluxes with the unique scheme given by Deng et al. [11] and Abe et al. [2]. However, the sufficientcondition is only acceptable for linear central schemes to guarantee the free-stream preservation. For theupwind schemes which are very important in the numerical simulations of compressible flows, it is not easyto be achieved due to the inconsistent differential operators applied for the grid metrics and fluxes [8]. Atpresent, there are mainly two ideas to deal with the free-stream conservation problem for high-order upwindschemes. The first one is to consider the independent interpolation for flow variables and metrics, suchas WCNS [12, 8, 11], alternative finite-difference form of WENO (AWENO) [13, 14, 15], etc.. In thoseschemes, the dissipation is handled by the finite volume method and then obtain their derivatives by centralschemes. The second method is to separate the central part from the standard upwind shcemes and employthe high-order central schemes to it. Then, employs the finite-volume-like schemes, i.e. freezing metricseither for the entire stencil [9] or for the local difference form partially [16], or replacing the transformedconservative variables with the original one [17] to the dissipative part. Among the above schemes, eitherthe standard WENO scheme is modified or its metrics are frozen in the local stencil, which results in moreor less additional complications or metrics frozen errors.In this study, we propose a simple, efficient and non-frozen high-order technique to achieve the free-streampreserving identity for the standard linear upwind and WENO schemes. The present method replaces thediscretized metrics and Jacobian with a free-stream preserving one in the dissipation part. This techniquepossesses at least two advantages, that is, it destroys the accuracy and the standard forms of the standardlinear upwind and WENO schemes as less as possible. As a result, it is convenient to operate this techniquein the standard upwind schemes. The outline is organized as follows. In section 2, we introduce the governingequations, SCL, and the upwind schemes in conservative finite difference method. In Section 3, the free-stream preserving metrics and Jacobian are explained in detail. Next, several validations and numericalexamples are given in Section 4. Finally, a brief conclusion is given in Section 5.2 . Governing equations and metrics on stationary curvilinear coordinates The compressible Navier-Stokes equations on curvilinear grids are given by ∂ Q ∂t + ∂∂ξ (cid:18) ξ x F + ξ y G + ξ z H J (cid:19) + ∂∂η (cid:18) η x F + η y G + η z H J (cid:19) + ∂∂ζ (cid:18) ζ x F + ζ y G + ζ z H J (cid:19) − ∂∂ξ (cid:18) ξ x F v + ξ y G v + ξ z H v J (cid:19) − ∂∂η (cid:18) η x F v + η y G v + η z H v J (cid:19) − ∂∂ζ (cid:18) ζ x F v + ζ y G v + ζ z H v J (cid:19) = (1)with Q = (cid:16) ρ ρu ρu ρu ρE (cid:17) T , (2) F = (cid:16) ρu ρu u + p ρu u ρu u ( ρE + p ) u (cid:17) T , (3) G = (cid:16) ρu ρu u ρu u + p ρu u ( ρE + p ) u (cid:17) T , (4) H = (cid:16) ρu ρu u ρu u ρu u + p ( ρE + p ) u (cid:17) T , (5) F v = (cid:16) τ τ τ u i τ i − ˙ q (cid:17) T (6) G v = (cid:16) τ τ τ u i τ i − ˙ q (cid:17) T (7) H v = (cid:16) τ τ τ u i τ i − ˙ q (cid:17) T (8)where ξ, η, ζ are the transformed coordinates on a uniform computational domain, and J is the transformedJacobian. u i , i = 1 , , F , G , H and F v , G v , H v represent the inviscid andviscous flux vectors in x, y and z direction, respectively. ρ , p and E are the density, pressure and the totalspecific energy. t is the physical time. τ ij is the shear stress tensor τ ij = 2 µ ( S ij − δ ij S kk , (9) S ij = 12 ( ∂u i ∂x j + ∂u j ∂x i ) , (10)and ˙ q i is the heat flux in direction i ˙ q i = − λ ∂T∂x i , (11)where µ and λ is the shear viscosity and thermal conductivity.The equation of state for ideal gas is p = ( γ − ρe, (12)where the specific heat ratio γ = 1 .
4. Generally, the fluxes in computational space are written as ˜ Q = Q J (13)3 F = ξ x F + ξ y G + ξ z H J ˜ G = η x F + η y G + η z H J ˜ H = ζ x F + ζ y G + ζ z H J (14) ˜ F v = ξ x F v + ξ y G v + ξ z H v J ˜ G v = η x F v + η y G v + η z H v J ˜ H v = ζ x F v + ζ y G v + ζ z H v J (15) Imposing the free-stream condition to the Navier-Stokes equations gives I x = (cid:18) ξ x J (cid:19) ξ + (cid:16) η x J (cid:17) η + (cid:18) ζ x J (cid:19) ζ = 0 ,I y = (cid:18) ξ y J (cid:19) ξ + (cid:16) η y J (cid:17) η + (cid:18) ζ y J (cid:19) ζ = 0 ,I z = (cid:18) ξ z J (cid:19) ξ + (cid:16) η z J (cid:17) η + (cid:18) ζ z J (cid:19) ζ = 0 . (16)These equations are regarded as the SCL by Zhang et al. [4] because they delineate the consistence ofvectorized computational cell surfaces in finite volume method [18]. Theoretically, Eq. (16) are strictlysatisfied while the discretizd errors of the metrics can easily destroy this identity.The SCMM is widely used to satisfy the SCL in high accuracy finite difference numerical simulationsunder the sufficient condition of Deng et al. [11] and Abe et al. [2] that the operators within the symmetricconservative metrics are unique with that of the fluxes discretization. The symmetric conservative metricsare expressed as ξ x J = 12 (cid:104) ( y η z ) ζ − ( y ζ z ) η + ( yz ζ ) η − ( yz η ) ζ (cid:105) ,ξ y J = 12 (cid:104) ( xz η ) ζ − ( xz ζ ) η + ( x ζ z ) η − ( x η z ) ζ (cid:105) ,ξ z J = 12 (cid:104) ( x η y ) ζ − ( x ζ y ) η + ( xy ζ ) η − ( xy η ) ζ (cid:105) ,η x J = 12 (cid:104) ( y ζ z ) ξ − ( y ξ z ) ζ + ( yz ξ ) ζ − ( yz ζ ) ξ (cid:105) ,η y J = 12 (cid:104) ( xz ζ ) ξ − ( xz ξ ) ζ + ( x ξ z ) ζ − ( x ζ z ) ξ (cid:105) ,η z J = 12 (cid:104) ( x ζ y ) ξ − ( x ξ y ) ζ + ( xy ξ ) ζ − ( xy ζ ) ξ (cid:105) ,ζ x J = 12 (cid:104) ( y ξ z ) η − ( y η z ) ξ + ( yz η ) ξ − ( yz ξ ) η (cid:105) ,ζ y J = 12 (cid:104) ( xz ξ ) η − ( xz η ) ξ + ( x η z ) ξ − ( x ξ z ) η (cid:105) ,ζ z J = 12 (cid:104) ( x ξ y ) η − ( x η y ) ξ + ( xy η ) ξ − ( xy ξ ) η (cid:105) , (17)4nd 1 J = 13 (cid:34)(cid:18) x ξ x J + y ξ y J + z ξ z J (cid:19) ξ + (cid:16) x η x J + y η y J + z η z J (cid:17) η + (cid:18) x ζ x J + y ζ y J + z ζ z J (cid:19) ζ (cid:35) . (18)The geometrical metrics and Jacobian are usually discretized with central schemes so that it is not easyfor the upwind schemes to satisfy the SCL preserving sufficient condition given by Deng et al. [11] and Abeet al. [2]. The conservative finite difference method [19, 20, 21] is explained briefly to discrete the Navier-Stokesequations. The key thought of this method is to reconstruct the high-order consistent numerical fluxes ateach cell-face. Without loss of generality, we choose ξ direction ordered by i , shown in Fig. 1, to delineatehow to reconstruct the cell-face fluxes. The fluxes ˜ F i at cell i is regarded as an average of a primitivefunction ˆ H ( ξ ) ˜ F i = 1∆ ξ (cid:90) i +1 / i − / ˆ H ( ξ ) dξ (19)Then we can exactly obtain the derivative of ˜ F i , (cid:32) ∂ ˜ F ∂ξ (cid:33) i = ˆ H ( i + 1 / − ˆ H ( i − / ξ . (20)Therefore, the derivative of the convective fluxes can be approximated by the reconstructed cell-face fluxes i+2i-1i-2 i-1/2 i+1/2 i+3/2i-3/2 i i+1 Figure 1: An illustration of discretization stencil cells ˜ F i +1 / (cid:32) ∂ ˜ F ∂ξ (cid:33) i = ˜ F i +1 / − ˜ F i − / ∆ ξ + O ( (cid:52) ξ k − ) , (21)where ˜ F i +1 / is the approximation of the primitive function value at cell-face ˆ H i +1 / , which can be recon-structed by unwind schemes from the cell fluxes ˜ F i − k +1 , · · · , ˜ F i + k − to keep the (2 k − For the purpose of improving the robustness of the simulations, the fluxes and conservative variables aretransformed into the characteristic space and then a flux vector splitting scheme, such as local Lax-Friedrichssplitting, is applied, ˜ F ± m = 12 L i +1 / · (cid:16) ˜ F m ± λ i +1 / ˜ Q m (cid:17) , m = i − k + 1 , · · · , i + k, (22)5here λ is the diagonal matrix of the eigenvalues of the local linearized Roe-average Jacobian matrix A i +1 / = (cid:16) ∂ ˜ F /∂ ˜ Q (cid:17) i +1 / . L i +1 / is the left matrix composed of the corresponding eigenvectors of A i +1 / .Then the cell-face fluxes are given by ˜ F i +1 / = R i +1 / · (cid:16) ˜ F + i +1 / + ˜ F − i +1 / (cid:17) , (23)where R i +1 / is the inverse matrix of L i +1 / .For the smooth and continuous flow field, ˜ F ± i +1 / can be reconstructed by the 5th-order linear upwindscheme, ˜ F + i +1 / = 160 (cid:16) ˜ F + i − − ˜ F + i − + 47 ˜ F + i + 27 ˜ F + i +1 − ˜ F + i +2 (cid:17) ˜ F − i +1 / = 160 (cid:16) − ˜ F − i − + 27 ˜ F − i + 47 ˜ F − i +1 − ˜ F − i +2 + 2 ˜ F − i +3 (cid:17) (24)With respect to the flow field containing noncontinuous zones, we choose the classical 5th-order WENOscheme [22] to obtain the cell-face flux by (cid:101) f ± i +1 / = (cid:88) k =0 ω ± k q ± k , (25)where (cid:101) f ± denotes one of the component of ˜ F ± . Taking the positive fluxes as an example, there 3rd-orderapproximations for the different sub-stencils are formulated as q +0 = 13 (cid:101) f + i − − (cid:101) f + i − + 76 (cid:101) f + i ,q +1 = − (cid:101) f + i − + 56 (cid:101) f + i + 13 (cid:101) f + i +1 ,q +2 = 13 (cid:101) f + i + 56 (cid:101) f + i +1 − (cid:101) f + i +2 . (26)The non-linear weight ω + k in Eq.(25) is evaluated by ω + k = C k (cid:0) β + k + (cid:15) (cid:1) n / (cid:88) r =0 C r (cid:0) β + k + (cid:15) (cid:1) n , (27)where C = 110 , C = 35 , C = 310 are the optical weights and (cid:15) = 1 . × − , n = 2. The smooth indicatorsare determined by β +0 = 14 (cid:16) (cid:101) f + i − − (cid:101) f + i − + 3 (cid:101) f + i (cid:17) + 1312 (cid:16) (cid:101) f + i − − (cid:101) f + i − + (cid:101) f + i (cid:17) ,β +1 = 14 (cid:16) − (cid:101) f + i − + (cid:101) f + i +1 (cid:17) + 1312 (cid:16) (cid:101) f + i − − (cid:101) f + i + (cid:101) f + i +1 (cid:17) ,β +2 = 14 (cid:16) − (cid:101) f + i + 4 (cid:101) f + i +1 − (cid:101) f + i +2 (cid:17) + 1312 (cid:16) (cid:101) f + i − (cid:101) f + i +1 + (cid:101) f + i +2 (cid:17) . (28)
3. Free-stream preserving metrics and Jacobian for the upwind schemes
In this section, the free-stream preserving metrics and Jacobian are deduced to give a novel, simple, non-frozen and high-order strategy on free-stream preserving for the upwind schemes. Without loss of generality,6he 5th-order linear upwind and WENO scheme are considered to reconstruct the cell-face fluxes with thissuggested free-stream preserving method.First, the SCMM [9] is applied to discrete the geometric metrics and Jacobian, as shown in Eq. (29).Therefore, the errors generated by the metrics discretization are effectively decreased if the unique centralscheme is applyed to the discretization of the fluxes, due to the sufficient condition of Deng et al. [11] andAbe et al. [2]. x i +1 / = 160 ( x i − − x i − + 37 x i + 37 x i +1 − x i +2 + x i +3 ) , (cid:18) ∂x∂ξ (cid:19) i = x i +1 / − x i − / . (29)In the followings, the metrics and Jacobian in the local reconstruction stencil discretizd by the SCMMwith the 6th-order central scheme are denoted by g / , g i − , · · · , g i +3 . The proposed free-stream preservingmetrics and Jacobian are represented by g ∗ / , g ∗ i − , · · · , g ∗ i +3 . Then, a sufficient condition to maintainthe free-stream preserving identity which is proved in Appendix A is given as Theorem 1.
During the 5th-order linear upwind and WENO reconstruction procedures, if the cell-facemetrics and Jacobian g ∗ / reconstructed in each sub-stencil share the unique value, that is, g ∗ i − − g ∗ i − + 116 g ∗ i = g ∗ i +1 / , − g ∗ i − + 56 g ∗ i + 13 g ∗ i +1 = g ∗ i +1 / , g ∗ i + 56 g ∗ i +1 − g ∗ i +2 = g ∗ i +1 / , g ∗ i +1 − g ∗ i +2 + 13 g ∗ i +3 = g ∗ i +1 / , (30) the free-stream preserving identity can be satisfied for their upwind dissipations. Moreover, if g ∗ / equals g / , where g i +1 / = 160 ( g i − − g i − + 37 g i + 37 g i +1 − g i +2 + g i +3 ) , (31) the free-stream preserving identity can be satisfied for their central parts. According to this theorem, we suggest the free-stream preserving metrics and Jacobian as g ∗ i − = 76 g ∗ i − − g ∗ i + g ∗ i +1 / , g ∗ i − = 56 g ∗ i + 13 g ∗ i +1 − g ∗ i +1 / ,g ∗ i = g i ,g ∗ i +1 / = g i +1 / ,g ∗ i +1 = g i +1 , g ∗ i +2 = 56 g ∗ i +1 + 13 g ∗ i − g ∗ i +1 / , g ∗ i +3 = 76 g ∗ i +2 − g ∗ i +1 + g ∗ i +1 / . (32)7here g i +1 / is calculated by Eq. (31)It is obvious to see that the proposed free-stream preserving metrics and Jacobian g ∗ m ( m = i + 1 / , i − · · · , i + 3) are reconstructed by the 3rd-order scheme from the original 6th-order central one g m ( m = i, i + 1 / , i + 1). The 3rd-order scheme is the same with the fluxes reconstruction in the sub-stencil ofWENO5 scheme, shown in Eq.(26). Therefore, the proposed free-stream preserving metrics and Jacobian g ∗ i , g ∗ i +1 and g ∗ i +1 / maintain the 6th-order accuracy while g ∗ i − , g ∗ i − , g ∗ i +2 and g ∗ i +3 retain only 3rd-orderaccuracy, as shown in Appendix B.Next, we adopt g ∗ m and g m to compute the cell-averaged fluxes and the conservative variables in trans-formed space, respectively. For example, ˜ F ∗ m = F m (cid:18) ξ x J (cid:19) ∗ m + G m (cid:18) ξ y J (cid:19) ∗ m + H m (cid:18) ξ z J (cid:19) ∗ m , ˜ Q ∗ m = Q m (cid:18) J (cid:19) ∗ m , (33)and ˜ F m = F m (cid:18) ξ x J (cid:19) m + G m (cid:18) ξ y J (cid:19) m + H m (cid:18) ξ z J (cid:19) m , ˜ Q m = Q m (cid:18) J (cid:19) m . (34)After that, we reconstruct the cell-face fluxes ˜ F ∗ i +1 / from the cell-averaged fluxes ˜ F ∗ m by the specificupwind schemes, say WENO5, ˜ F ∗ i +1 / = W EN O LF (cid:16) ˜ F ∗ i − , · · · , ˜ F ∗ i +3 , ˜ Q ∗ i − , · · · , ˜ Q ∗ i +3 (cid:17) , (35)where W EN O LF stands for the operator of the characteristic WENO5 scheme coupled with Lax-Friedrichsflux splitting.Unfortunately, the fluxes ˜ F ∗ i +1 / only achieve a 3rd-order accuracy due to applying the 3rd-order metricsand Jacobian. Nevertheless, as proved in Appendix B, we can realize a fact that Theorem 2.
The 3rd-order accurate free-stream metrics and Jacobian given in Eq. (32) do not change the5th-order accuracy of the upwind dissipations of the cell-face fluxes reconstructed by the 5th-order linearupwind or WENO scheme.
Therefore, we suggest replacing the central part of ˜ F ∗ i +1 / , denoted by ˜ F (3) i +1 / , with the 6th-order one,denoted by ˜ F (6) i +1 / . Specifically, ˜ F (3) i +1 / = 160 (cid:16) ˜ F ∗ i − − ˜ F ∗ i − + 37 ˜ F ∗ i + 37 ˜ F ∗ i +1 − ˜ F ∗ i +2 + ˜ F ∗ i +3 (cid:17) , ˜ F (6) i +1 / = 160 (cid:16) ˜ F i − − ˜ F i − + 37 ˜ F i + 37 ˜ F i +1 − ˜ F i +2 + ˜ F i +3 (cid:17) , ˜ F i +1 / = ˜ F ∗ i +1 / + ˜ F (6) i +1 / − ˜ F (3) i +1 / . (36)8inally, the new fluxes ˜ F i +1 / can approach the 5th-order accuracy. If we choose a linear upwind schemein Eq.(35), the free-stream preserving identity can be satisfied as well without destroying the convergenceorder of this scheme.
4. Numerical tests on curvilinear grids
Several verifications, such as the isotropic vortex convection, the double Mach reflection problem, thetransonic flow past the ONERA M6 wing, etc. are conducted to evaluate the accuarcy and the capabilityin shock capturing of the proposed free-stream preserving method on curvilinear grids. If not otherwisespecified, the local Lax-Friedrichs flux splitting and the 3rd-order TVD Runge-Kutta scheme [23] are utilizedfor the simulations. For the viscous issues, the 6th-order central scheme is adopted to discrete the viscousterms. In the following, WENO5/WENO7 stand for the standard 5th/7th-order WENO schemes of Shu [24],WENOZ is the standard improved 5th-order WENO scheme of Borges et al. [25] and WENO5-Present,WENO7-Present, WENOZ-Present are the free-stream preserving schemes suggested in the present paper.
The wavy and randomized grids, as shown in Fig. 2, are considered to verify the proposed free-streampreserving scheme. The wavy grid is defined in the domain ( x, y ) ∈ [ − , × [ − ,
10] by x i,j = x min + ∆ x (cid:20) ( i −
1) + A x sin (cid:18) n xy π ( j − y L y (cid:19)(cid:21) y i,j = y min + ∆ y (cid:20) ( j −
1) + A y sin (cid:18) n yx π ( i − x L x (cid:19)(cid:21) (37)where L x = L y = 20, x min = − L x / y min = − L y / A x ∆ x = 0 . A y ∆ y = 0 .
6, and n xy = n yx = 8.The randomized grid is disturbed randomly with 20% grid spacing of the uniform Cartesian grid. The gridresolution of the two grids are both 21 × M = 0 . u = 0 . , v = 0 , p = 1 , ρ = γ (38)where γ = 1 . L -norm and L ∞ -norm errors of the velocity components v forthe two grids are estimated at t = 20. In Table 1, compared with the standard WENO5/WENO7 scheme,the proposed free-stream preserving method effectively decreases the geometrically induced errors, whichare both below 1 × − and close to the machine zero for the double-precision computations. The two-dimensional moving isotropic vortex problems on the wavy and randomized grids are investigatedto evaluate the accuracy and vortex preserving capability of the present free-stream preserving schemes.9 a) Wavy grid (b) Randomized grid
Figure 2: Illustration of the wavy and randomized grids.Table 1: The L and L ∞ errors of the v component on the wavy and randomized grids. Method Wavy grid Random grid L error L ∞ error L error L ∞ errorWENO5 2 . × − . × − . × − . × − WENOZ 6 . × − . × − . × − . × − WENO7 1 . × − . × − . × − . × − WENO5-Present 6 . × − . × − . × − . × − WENOZ-Present 1 . × − . × − . × − . × − WENO7-Present 5 . × − . × − . × − . × − γ = 1 .
4. An isotropic vortex centered at( x c , y c ) = (0 ,
0) is superposed to a uniform flow with Mach 0.5. Specifically, the perturbations of the velocity,temperature and entropy are expressed by:( δu, δv ) = (cid:15)τ e α (1 − τ ) ( sinθ, − cosθ ) δT = − ( γ − (cid:15) αγ e α (1 − τ ) δS = 0 (39)where α = 0 . τ = r/r c and r = (cid:112) ( x − x c ) + ( y − y c ) . r c = 1 . (cid:15) = 0 .
02 denote the vortex core lengthand strength, respectively. T = p/ρ is the temperature and S = p/ρ γ is the entropy. The periodic boundarycondition is imposed and the results are estimated when the vortex moving back to the original position at t = 40.The numerical vorticity magnitude contours on the two wavy and random grids at a resolution of 21 × ×
21, 41 ×
41, 81 ×
81, 161 ×
161 and 321 ×
321 are considered to evaluate theconvergence rate of the proposed schemes. The time step ∆ t respect to each grid decreases until the L and L ∞ errors are invariant to eliminate the errors by the 3rd-order time integration, as proposed in Ref. [9] and[16]. The L and L ∞ errors of the v component and their corresponding convergence rates on those wavygrids, listed in Table 2, indicate that the WENO5-Present, WENOZ-Present and WENO7-Present schemescan maintain the theoretical convergence orders. The double Mach problem [26] is carried out to demonstrate the shock-capturing capability of the presentfree-stream preserving WENO scheme. The initial condition is( ρ, u, v, p ) = (1 . , , , . x − ytan ( π/ > / , (8 . , . , − . , . x − ytan ( π/ < / . (40)The computation is conducted up to t = 0 . × able 2: The L and L ∞ errors of the v component and their corresponding convergence rates on the wavy grids. Method Grid size L error Convergence rate L ∞ error Convergence rateWENO5 21 ×
21 2 . × − - 5 . × − -41 ×
41 2 . × − . × − ×
81 1 . × − . × − ×
161 3 . × − . × − ×
321 4 . × − . × − ×
21 7 . × − - 2 . × − -41 ×
41 8 . × − . × − ×
81 3 . × − . × − ×
161 1 . × − . × − ×
321 3 . × − . × − ×
21 2 . × − - 1 . × − -41 ×
41 4 . × − . × − ×
81 1 . × − . × − ×
161 5 . × − . × − ×
321 1 . × − . × − ×
21 2 . × − - 1 . × − -41 ×
41 5 . × − . × − ×
81 1 . × − . × − ×
161 5 . × − . × − ×
321 1 . × − . × − ×
21 2 . × − - 1 . × − -41 ×
41 4 . × − . × − ×
81 3 . × − . × − ×
161 3 . × − . × − ×
321 3 . × − . × − c) WENOZ(a) WENOZ (d) WENOZ-Present(b) WENOZ-Present Figure 3: The vorticity magnitude distributions ranging from 0.0 to 0.006 of the 2D moving vortex on the wavy and randomizedgrids (21 ×
21) for the WENOZ scheme. (f) WENO7-Present(c) WENO7-Present(a) WENO5(d) WENO5 (b) WENO5-Present(e) WENO5-Present
Figure 4: The vorticity magnitude distributions ranging from 0.0 to 0.006 of the 2D moving vortex on the wavy and randomizedgrids (21 ×
21) for the WENO5 and WENO7 schemes. b) WENO7, uniform Cartisian (c) WENO5 (d) WENO7 (a) WENO5, uniform Cartisian (g) WENO5-Present (h) WENO7-Present (e) WENOZ (f) WENOZ-Present Figure 5: The density contours of the double Mach reflection problem ranging from 1.25 to 21.5 with 5% randomization. the flow field smoothly. When the grid is randomized up to 20% of the uniform grid spacing, the densitycontours in Fig 6 indicate that the proposed free-stream preserving WENO schemes can still reduce thegeometry errors. Some disturbances are observed in the result of WENO7-Present scheme, but they areessentially improved on such a highly distorted grid, compared with the standard WENO7 schemes.
The case is from Ref [2] to demonstrate the shock-capture capabilities and high-resolution of the schemes.The length and width of the wind tunnel are 3 units and 1 unit, respectively. The step in the bottom ofthe wind tunnel is located at 0.6 units from the left boundary with a height of 0.2 units. The initial flowfield is given by a right-going Mach 3 flow with P = 1 and ρ = γ = 1 .
4. The in-flow and out-flow boundarycondition are implied to the left and right boundary, and the reflective boundary conditions are considered14 b) WENO7(c) WENO5-Present (d) WENO7-Present (a) WENO5
Figure 6: The density contours of the double Mach reflection problem ranging from 1.25 to 21.5 with 20% randomization. along the walls of the tunnel. The computational domain is discretized by two grids with a randomizationof 5% and 20%, respectively, under a resolution of ∆ x = 1 /
200 units. The global Lax-Friedrichs dissipationis considered. The computed results at t = 4 in Fig. 7 indicate that either WENO5-Present or WENO7-Present achieve the free-stream preserving identity on the severely randomized grids. The reflective shocksaround the wall of the wind tunnel are captured correctly and the vortexes generated in the shear layer areresolved significantly, which show that the present schemes have been improved a lot compared with thestandard WENO schemes. The supersonic flow past a cylinder [22] is solved to examine the shock capturing capability of thefree-stream preserving schemes on the inhomogeneous curvilinear grid. The M = 2 supersonic flow movestowards the cylinder and the slip wall boundary condition is imposed to the wall and supersonic inflow andoutflow boundary condition are assigned to the left boundary and others, respectively. The grid is given by: x = ( R x − ( R x − η (cid:48) ) cos ( θ (2 ξ (cid:48) − y = ( R y − ( R y − η (cid:48) ) sin ( θ (2 ξ (cid:48) − ξ (cid:48) = ξ − i max − , ξ = i + 0 . φ i η (cid:48) = η − j max − , η = j + 0 . (cid:113) − φ i (41)where θ = 5 π/ R x = 3, R y = 6 and φ i is a random number distributed between [0 , i max = 61 and j max = 81. The free stream pressure and density are p = 1 and ρ = γ , respectively.The computational results with the global Lax-Friedrichs dissipation are evaluated after t = 25. In Fig. 8,15 a) Grid with 5% randomization (b) Grid with 20% randomization(c) WENO5-Present, 5% randomization (d) WENO5-Present, 20% randomization(e) WENO7-Present, 5% randomization (f) WENO7-Present, 20% randomization Figure 7: The density contours of the flow in wind tunnel with a step. Totally 31 contours from 0.2 to 6.5. the pressure distributions around the cylinder calculated by the standard WENO schemes are significantlydisturbed by the unphysical oscillations. However, the results from the the present free-stream preservingWENO schemes are very smooth and the pressure distributions are well resolved.
In this section, the inviscid transonic flows past a NACA0012 airfoil with Mach number M = 0 . AOA = 1 . ◦ (case 1) and Mach number M = 0 .
85 and angle of attack
AOA =1 . ◦ (case 2) are simulated by the present free-stream preserving scheme. A coarse grid discretized with160 ×
32 cells in circumferential and radial, respectively, is chosen to demonstrate the accuracy of the presentscheme. The reference simulations are conducted by the FVM (ROE scheme coupled with 2nd-order MUSCLreconstruction) on this coarse grid and the fine grid with a resolution of 1280 × d) WENO5-Present (e) WENO7-Present(b) WENO5(a) Grid (c) WENO7 Figure 8: The pressure distributions from 1.2 to 5.4 of the supersonic flow past a cylinder. (a) (b)
Figure 9: The pressure coefficient distributions along the wall of the NACA0012 airfoil. a) Case1 (b) Case 2 Figure 10: The Mach number contours from 0.172 to 1.325 of the transonic flow past the NACA0012 airfoil.
The three-dimensional transonic flow pass the ONERA M6 wing is considered in this test case. Thegeometry is very simple but the transonic flow features are complicated. The simulation is conducted at aMach number M = 0 .
84 and an angle of attack
AOA = 3 . ◦ with a Reynold number of Re l = 1 . × based on the mean aerodynamic chord of l = 0 . m . The computational grid consists of 12 blocks and294912 cells in total, as illustrated in Fig. 11. In this case, the viscous effects are taken into consideration andthe SA turbulence model [27] is adopted. The lower-upper symmetric-Gauss-Seidel implicit method (LUSGS)is employed for the time marching. The numerical pressure contours around the surface of M6 wing andthe symmetry plane drawn in Fig. 12 show that the transonic flows around the 3D wing can be resolvedsmoothly by the WENO5-Present scheme. Fig. 13 compares the simulated pressure coefficient distributionswith experimental data at six spanwise stations. They are in good agreement with the experimental resultsexcept at y/b = 0 .
8, which is because the ideal syemetry boudary of the middle plane in simulation can notexactly reproduce the flow physics of the half-wing in wind tunnel [28].
5. Concluding remarks
In this paper, we give a sufficient condition on preserving the free-stream identity for the upwind dissi-pations. Based on this sufficient condition, the free-stream preserving metrics and Jacobian are proposedfor the upwind dissipation of the linear upwind and WENO schemes. Coupled with the high-order accuratecentral part, this technique avoids destroying the accuracy and the forms of the standard upwind schemes as18 a) (b)
Figure 11: The grid illustration of ONERA M6 wing. (a) MUSCL-ROE-SA (b) WENO5-Present-SA
Figure 12: The pressure contours around the surface and the symmetry pane of the ONERA M6 wing. Totally 61 contoursfrom 130 Kpa to 490 Kpa. a) y/b=0.20 (b) y/b=0.44(c) y/b=0.65 (d) y/b=0.80(e) y/b=0.90 (f) y/b=0.95 Figure 13: The pressure coefficient distributions at six stations along the wall of the ONERA M6 wing.
Acknowledgements
This work was supported by the National Natural Science Foundation of China (Grant Nos. 11902271and 91952203) and 111 project on Aircraft Complex Flows and the Control (Grant No. B17037).
Appendix A
The standard WENO5 scheme can be divided into a central part and a dissipation part [9, 16], ˜ F i +1 / = ˜ F + i +1 / + ˜ F − i +1 / = (cid:88) s R si +1 / f s, + i +1 / + (cid:88) s R si +1 / f s, − i +1 / = 160 (cid:16) ˜ F i − − ˜ F i − + 37 ˜ F i + 37 ˜ F i +1 − ˜ F i +2 + ˜ F i +3 (cid:17) − (cid:88) s R si +1 / (cid:104)(cid:0) ω +0 − (cid:1) ˆ f s, + i, − (cid:0) ω +0 + 10 ω +1 − (cid:1) ˆ f s, + i, + ˆ f s, + i, (cid:105)(cid:124) (cid:123)(cid:122) (cid:125) D + + 160 (cid:88) s R si +1 / (cid:104)(cid:0) ω − − (cid:1) ˆ f s, − i, − (cid:0) ω − + 10 ω − − (cid:1) ˆ f s, − i, + ˆ f s, − i, (cid:105)(cid:124) (cid:123)(cid:122) (cid:125) D − (42)where ˆ f s, + i,r +1 = (cid:101) f s, + i + r +1 − (cid:101) f s, + i + r + 3 (cid:101) f s, + i + r − − (cid:101) f s, + i + r − , r = 0 , ,
2= 12 L si +1 / (cid:16) ˜ F i + r +1 − ˜ F i + r + 3 ˜ F i + r − − ˜ F i + r − (cid:17) + 12 λ s L si +1 / (cid:16) ˜ Q i + r +1 − ˜ Q i + r + 3 ˜ Q i + r − − ˜ Q i + r − (cid:17) (43)ˆ f s, − i,r +1 = (cid:101) f s, − i − r +3 − (cid:101) f s, − i − r +2 + 3 (cid:101) f s, − i − r +1 − (cid:101) f s, − i − r , r = 0 , ,
2= 12 L si +1 / (cid:16) ˜ F i − r +3 − ˜ F i − r +2 + 3 ˜ F i − r +1 − ˜ F i − r (cid:17) − λ s L si +1 / (cid:16) ˜ Q i − r +3 − ˜ Q i − r +2 + 3 ˜ Q i − r +1 − ˜ Q i − r (cid:17) (44)First, taking the dissipation terms D + as examples, the upwind dissipations are rearranged as D + = D F + + D Q + , (45)21here D F + = − (cid:88) s R si +1 / (cid:20) (cid:0) ω +0 − (cid:1) L si +1 / (cid:16) ˜ F i +1 − ˜ F i + 3 ˜ F i − − ˜ F i − (cid:17) − (cid:0) ω +0 + 10 ω +1 − (cid:1) L si +1 / (cid:16) ˜ F i +2 − ˜ F i +1 + 3 ˜ F i − ˜ F i − (cid:17) + 12 L si +1 / (cid:16) ˜ F i +3 − ˜ F i +2 + 3 ˜ F i +1 − ˜ F i (cid:17)(cid:21) , (46)and D Q + = − (cid:88) s R si +1 / λ s (cid:20) (cid:0) ω +0 − (cid:1) L si +1 / (cid:16) ˜ Q i +1 − ˜ Q i + 3 ˜ Q i − − ˜ Q i − (cid:17) − (cid:0) ω +0 + 10 ω +1 − (cid:1) L si +1 / (cid:16) ˜ Q i +2 − ˜ Q i +1 + 3 ˜ Q i − ˜ Q i − (cid:17) + 12 L si +1 / (cid:16) ˜ Q i +3 − ˜ Q i +2 + 3 ˜ Q i +1 − ˜ Q i (cid:17)(cid:21) . (47)In the following discussions, we mainly focus on D F + because D Q + has the similar form. After simplifying D F + , we obtain D F + = − (cid:88) s R si +1 / (cid:16) − ˜ F i − + 5 ˜ F i − − ˜ F i + 10 ˜ F i +1 − ˜ F i +2 + ˜ F i +3 (cid:17)(cid:124) (cid:123)(cid:122) (cid:125) linear dissipation +60 (cid:0) C − ω +0 (cid:1) (cid:18) ˜ F i − − ˜ F i − + 116 ˜ F i (cid:19)(cid:124) (cid:123)(cid:122) (cid:125) sub-stencil 1 +60 (cid:0) C − ω +1 (cid:1) (cid:18) − ˜ F i − + 56 ˜ F i + 13 ˜ F i +1 (cid:19)(cid:124) (cid:123)(cid:122) (cid:125) sub-stencil 2 +60 (cid:0) C − ω +2 (cid:1) (cid:18) ˜ F i + 56 ˜ F i +1 − ˜ F i +2 (cid:19)(cid:124) (cid:123)(cid:122) (cid:125) sub-stencil 3 , (48)where the linear dissipation part can be reformulated as − ˜ F i − + 5 ˜ F i − − ˜ F i + 10 ˜ F i +1 − ˜ F i +2 + ˜ F i +3 = − (cid:18) ˜ F i − − ˜ F i − + 116 ˜ F i (cid:19)(cid:124) (cid:123)(cid:122) (cid:125) sub-stencil 1 − (cid:18) − ˜ F i − + 56 ˜ F i + 13 ˜ F i +1 (cid:19)(cid:124) (cid:123)(cid:122) (cid:125) sub-stencil 2 + 9 (cid:18) ˜ F i + 56 ˜ F i +1 − ˜ F i +2 (cid:19)(cid:124) (cid:123)(cid:122) (cid:125) sub-stencil 3 +3 (cid:18) ˜ F i +1 − ˜ F i +2 + 13 ˜ F i +3 (cid:19)(cid:124) (cid:123)(cid:122) (cid:125) sub-stencil 4 . (49)Eq. (48) and Eq. (49) can be extended to D Q + directly. They indicate that the positive dissipations arecomposed of the linear part and the non-linear part reconstructed in the three sub-stencils. Furthermore,22hether the linear part or the non-linear part is a combination of the 3rd-order reconstruction in the sub-stencil. Therefore, when the free-stream condition is imposed, the upwind dissipations can not satisfy thefree-stream preserving identity due to conflicting with the sufficient condition of Deng et al. [11] and Abeet al. [2].However, if the Eqs. (30) in Theorem 2 are satisfied, ignoring the flow variables because they are allconstant vectors under the free-stream condition, the values of the metrics and Jacobian reconstructed inthe four sub-stencils in Eq. (49) and (48) are unique. Then, it is obvious to see that the linear dissipationscancel each other in Eq. (49), and the accumulation of the three non-linear parts in Eq. (48) is zero as wellunder the relations C + C + C = ω + ω + ω = 1 . (50)Finally, the central part can be rearranged as160 (cid:16) ˜ F i − − ˜ F i − + 37 ˜ F i + 37 ˜ F i +1 − ˜ F i +2 + ˜ F i +3 (cid:17) = 120 (cid:18) ˜ F i − − ˜ F i − + 116 ˜ F i (cid:19)(cid:124) (cid:123)(cid:122) (cid:125) sub-stencil 1 + 310 (cid:18) − ˜ F i − + 56 ˜ F i + 13 ˜ F i +1 (cid:19)(cid:124) (cid:123)(cid:122) (cid:125) sub-stencil 2 + 320 (cid:18) ˜ F i + 56 ˜ F i +1 − ˜ F i +2 (cid:19)(cid:124) (cid:123)(cid:122) (cid:125) sub-stencil 3 + 120 (cid:18) ˜ F i +3 − ˜ F i +2 + 116 ˜ F i +1 (cid:19)(cid:124) (cid:123)(cid:122) (cid:125) sub-stencil 4 + 310 (cid:18) − ˜ F i +2 + 56 ˜ F i +1 + 13 ˜ F i (cid:19)(cid:124) (cid:123)(cid:122) (cid:125) sub-stencil 3 + 320 (cid:18) ˜ F i +1 + 56 ˜ F i − ˜ F i − (cid:19)(cid:124) (cid:123)(cid:122) (cid:125) sub-stencil 2 . (51)Similarly, ignoring the flow variables, if the Eqs. (30) in Theorem 2 are satisfied, the values of the metricsand Jacobian reconstructed in the four sub-stencils and their combination in Eq. (51) are all g i +1 / whichmeans that it equals to obtaining the fluxes by the 6th-order central scheme. As a result, the free-streampreerving identity is satisfied because of the sufficient condition given by Deng et al. [11] and Abe et al. [2].For the 5th-order linear upwind scheme, as proposed in Ref. [16], it can be written as ˜ F i +1 / = 160 (cid:16) ˜ F i − − ˜ F i − + 37 ˜ F i + 37 ˜ F i +1 − ˜ F i +2 + ˜ F i +3 (cid:17) + 160 (cid:88) s R si +1 / λ s L si +1 / (cid:16) ˜ Q i − − ˜ Q i − + 10 ˜ Q i − ˜ Q i +1 + 5 ˜ Q i +2 − ˜ Q i +3 (cid:17)(cid:124) (cid:123)(cid:122) (cid:125) D i +1 / . (52)23imilarly, we rearrange this form to ˜ F i +1 / = 160 (cid:16) ˜ F i − − ˜ F i − + 37 ˜ F i + 37 ˜ F i +1 − ˜ F i +2 + ˜ F i +3 (cid:17) + 160 (cid:88) s R si +1 / λ s L si +1 / (cid:18) ˜ Q i − − ˜ Q i − + 116 ˜ Q i (cid:19)(cid:124) (cid:123)(cid:122) (cid:125) sub-stencil 1 +9 (cid:18) − ˜ Q i − + 56 ˜ Q i + 13 ˜ Q i +1 (cid:19)(cid:124) (cid:123)(cid:122) (cid:125) sub-stencil 2 − (cid:18) ˜ Q i + 56 ˜ Q i +1 − ˜ Q i +2 (cid:19)(cid:124) (cid:123)(cid:122) (cid:125) sub-stencil 3 − (cid:18) ˜ Q i +1 − ˜ Q i +2 + 13 ˜ Q i +3 (cid:19)(cid:124) (cid:123)(cid:122) (cid:125) sub-stencil 4 (53)Obviously, the above analyses can be applied to the 5th-order linear upwind scheme to verify its free-streampreserving identity. Appendix B
The 5th-order linear upwind scheme
The proposed free-stream preserving metrics and Jacobian g ∗ i − , g ∗ i − , g ∗ i +2 and g ∗ i +3 are computed by the3rd-order reconstruction from three 6th-order ones g i , g i +1 / and g i +1 . Therefore, they maintain 3rd-orderaccuracy at least. In smooth regions, Taylor expansion of Eq. (32) gives, respectively,∆ g ∗ i − = g ∗ i − − g i − = 52 R (cid:48)(cid:48)(cid:48) ∆ ξ − R (4) ∆ ξ + 103240 R (5) ∆ ξ + O (cid:0) ∆ ξ (cid:1) , ∆ g ∗ i − = g ∗ i − − g i − = 12 R (cid:48)(cid:48)(cid:48) ∆ ξ + 120 R (4) ∆ ξ + 11240 R (5) ∆ ξ + O (cid:0) ∆ ξ (cid:1) , ∆ g ∗ i = g ∗ i − g i = 0 , ∆ g ∗ i +1 = g ∗ i +1 − g i +1 = 0 , ∆ g ∗ i +2 = g ∗ i +2 − g i +2 = − R (cid:48)(cid:48)(cid:48) ∆ ξ − R (4) ∆ ξ − R (5) ∆ ξ + O (cid:0) ∆ ξ (cid:1) , ∆ g ∗ i +3 = g ∗ i +3 − g i +3 = − R (cid:48)(cid:48)(cid:48) ∆ ξ − R (4) ∆ ξ − R (5) ∆ ξ + O (cid:0) ∆ ξ (cid:1) , (54)where R (cid:48)(cid:48)(cid:48) = R (cid:48)(cid:48)(cid:48) ( ξ ) and R (4) = R (4) ( ξ ) are the third and fourth derivatives at i of the primary function R ( ξ ), g i = 1∆ ξ (cid:90) i +1 / i − / R ( ξ ) dξ. (55)If the free-stream preserving metrics and Jacobian are adopted to the dissipation part of the 5th-orderlinear upwind scheme shown in Eq. (52), we can obtain ˜ Q ∗ m = ˜ Q m + (cid:16) ˜ Q ∗ m − ˜ Q m (cid:17) , m = i − , i + 3= ˜ Q m + Q m ∆ g ∗ m , (56)24nd D i +1 / = (cid:16) ˜ Q ∗ i − − ˜ Q ∗ i − + 10 ˜ Q ∗ i − ˜ Q ∗ i +1 + 5 ˜ Q ∗ i +2 − ˜ Q ∗ i +3 (cid:17) = (cid:16) ˜ Q i − − ˜ Q i − + 10 ˜ Q i − ˜ Q i +1 + 5 ˜ Q i +2 − ˜ Q i +3 (cid:17) + (cid:16) R (cid:48)(cid:48)(cid:48) Q (cid:48)(cid:48) i +1 / + 5 R (4) Q (cid:48) i +1 / + R (5) Q i +1 / (cid:17) ∆ ξ + O (cid:0) ∆ ξ (cid:1) , (57)where g refers to the Jacobian 1 /J . Similarly, D i − / = (cid:16) ˜ Q i − − ˜ Q i − + 10 ˜ Q i − − ˜ Q i + 5 ˜ Q i +1 − ˜ Q i +2 (cid:17) + (cid:16) R (cid:48)(cid:48)(cid:48) Q (cid:48)(cid:48) i − / + 5 R (4) Q (cid:48) i − / + R (5) Q i − / (cid:17) ∆ ξ + O (cid:0) ∆ ξ (cid:1) . (58)The additional terms retain 5th-order accuracy in the conservative finite difference scheme because D i +1 / − D i − / ∆ ξ = (cid:104)(cid:16) ˜ Q (5) i +1 / − ˜ Q (5) i − / (cid:17) + 10 R (cid:48)(cid:48)(cid:48) (cid:16) Q (cid:48)(cid:48) i +1 / − Q (cid:48)(cid:48) i − / (cid:17) +5 R (4) (cid:16) Q (cid:48) i +1 / − Q (cid:48) i − / (cid:17) + R (5) (cid:16) Q i +1 / − Q i − / (cid:17)(cid:105) ∆ ξ + O (cid:0) ∆ ξ (cid:1) = (cid:16) ˜ Q (6) + 10 R (cid:48)(cid:48)(cid:48) Q (cid:48)(cid:48)(cid:48) + 5 R (4) Q (cid:48)(cid:48) + R (5) Q (cid:48) (cid:17) ∆ ξ + O (cid:0) ∆ ξ (cid:1) (59)Obviously, even the 3rd-order metrics and Jacobian are applied to the dissipation of the 5th-order linearupwind scheme, only the extra O (cid:0) ∆ ξ (cid:1) terms are added to the standard terms such that the analytic con-vergence order of the proposed upwind dissipation still maintains 5th-order accuracy. With this dissipation,if the central part in Eq. (52) is obtained by the 6th-order metrics and Jacobian g m , then the linear upwindscheme achieves the 5th-order accuracy. The 5th-order WENO scheme
According to Ref. [22], the WENO5 scheme is a convex combination of the 3rd-order reconstruction ofall the candidate sub-stencils˜ f i +1 / = q ( ˜ f i − , · · · , ˜ f i +2 ) + (cid:88) k =0 ( ω k − C k ) q k (cid:16) ˜ f i + k − , ˜ f i + k − , ˜ f i + k (cid:17) , (60)and under the condition of Eq. (50). As given by Borges et al. [25], the conservative finite difference schememaintains the 5th-order accuracy if the non-linear weights ω k and q k in sub-stencils satisfy the followings, ω k = C k + O (∆ ξ ) , (61) q k = ˜ h i +1 / + O (∆ ξ ) , (62) (cid:88) k =0 A k (cid:16) ω k − ω (cid:48) k (cid:17) = O (∆ ξ ) , (63)where ω (cid:48) k denotes the non-linear weight for ˜ f i − / . It should be noted that Eq. (63) is ignored in the standardWENO5 scheme of Jiang and Shu [22] due to the large (cid:15) = 1 × − .25f the proposed metrics and Jacobian g ∗ m and the 6th-order one g m are adopted to calculate cell-averagedfluxes and apply the fluxes splitting, respectively, we can obtain the relation between ˜ F ∗ + and ˜ F + as ˜ F ∗ + = ˜ F + + L (cid:26) F (cid:20)(cid:18) ξ x J (cid:19) ∗ − (cid:18) ξ x J (cid:19)(cid:21) + G (cid:20)(cid:18) ξ y J (cid:19) ∗ − (cid:18) ξ y J (cid:19)(cid:21) + H (cid:20)(cid:18) ξ z J (cid:19) ∗ − (cid:18) ξ z J (cid:19)(cid:21) + λQ (cid:20)(cid:18) J (cid:19) ∗ − (cid:18) J (cid:19)(cid:21)(cid:27) = ˜ F + + O (∆ ξ ) . (64)Therefore, the reconstructed cell-face fluxes q ∗ + k in the sub-stencil by ˜ f ∗ + m which is one of the component of ˜ F ∗ + m can still maintain 3rd-order accuracy because of Eq. (54). To simplify notation, we drop the superscript+ for ˜ f ∗ + , q ∗ + and β ∗ + in the followings. In details, q ∗ = 13 ˜ f ∗ i − −
76 ˜ f ∗ i − + 116 ˜ f ∗ i = (cid:18)
13 ˜ f i − −
76 ˜ f i − + 116 ˜ f i (cid:19) + O (∆ ξ )= ˜ h i +1 / + O (∆ ξ ) (65)Similarly, q ∗ = ˜ h i +1 / + O (∆ ξ ) (66) q ∗ = ˜ h i +1 / + O (∆ ξ ) . (67)To investigate the accuracy of the non-linear weights, we define˜ f ∗ m = ˜ f m + (cid:16) ˜ f ∗ m − ˜ f m (cid:17) = ˜ f m + f m ∆ g ∗ m , m = i − , · · · , i + 2 , (68)and f m ∆ g ∗ m = l s (cid:20) F m ∆ (cid:18) ξ x J (cid:19) ∗ + G m ∆ (cid:18) ξ y J (cid:19) ∗ + H m ∆ (cid:18) ξ z J (cid:19) ∗ + λ s Q m ∆ (cid:18) J (cid:19) ∗ (cid:21) , (69)which represents for the difference between ˜ f ∗ m and ˜ f m . Then, the smoothness indicators can be given by β ∗ = 14 (cid:104)(cid:16) ˜ f i − − f i − + 3 ˜ f i (cid:17) + (cid:0) f i − ∆ g ∗ i − − f i − ∆ g ∗ i − + 3 f i ∆ g ∗ i (cid:1)(cid:105) + 1312 (cid:104)(cid:16) ˜ f i − − f i − + ˜ f i (cid:17) + (cid:0) f i − ∆ g ∗ i − − f i − ∆ g ∗ i − + f i ∆ g ∗ i (cid:1)(cid:105) , (70) β ∗ = 14 (cid:104)(cid:16) ˜ f i − − ˜ f i +1 (cid:17) + (cid:0) f i − ∆ g ∗ i − − f i +1 ∆ g ∗ i +1 (cid:1)(cid:105) + 1312 (cid:104)(cid:16) ˜ f i − − f i + ˜ f i +1 (cid:17) + (cid:0) f i − ∆ g ∗ i − − f i ∆ g ∗ i + f i +1 ∆ g ∗ i +1 (cid:1)(cid:105) , (71) β ∗ = 14 (cid:104)(cid:16) f i − f i +1 + ˜ f i +2 (cid:17) + (cid:0) f i ∆ g ∗ i − f i +1 ∆ g ∗ i +1 + f i +2 ∆ g ∗ i +2 (cid:1)(cid:105) + 1312 (cid:104)(cid:16) ˜ f i − f i +1 + ˜ f i +2 (cid:17) + (cid:0) f i ∆ g ∗ i − f i +1 ∆ g ∗ i +1 + f i +2 ∆ g ∗ i +2 (cid:1)(cid:105) . (72)26n smooth regions, Taylor expansion of Eq.(70) at i gives, β ∗ = ˜ f (cid:48) ∆ ξ + (cid:18) f (cid:48)(cid:48) −
23 ˜ f (cid:48) ˜ f (cid:48)(cid:48)(cid:48) + 12 ˜ f (cid:48) R (cid:48)(cid:48)(cid:48) f (cid:19) ∆ ξ − (cid:18)
136 ˜ f (cid:48)(cid:48) ˜ f (cid:48)(cid:48)(cid:48) −
12 ˜ f (cid:48) ˜ f (4) −
134 ˜ f (cid:48)(cid:48) R (cid:48)(cid:48)(cid:48) f + 3 ˜ f (cid:48) R (cid:48)(cid:48)(cid:48) f (cid:48) + 1120 ˜ f (cid:48) R (4) f (cid:19) ∆ ξ + O (∆ ξ ) , (73) β ∗ = ˜ f (cid:48) ∆ ξ + (cid:18) f (cid:48)(cid:48) + 13 ˜ f (cid:48) ˜ f (cid:48)(cid:48)(cid:48) −
12 ˜ f (cid:48) R (cid:48)(cid:48)(cid:48) f (cid:19) ∆ ξ + (cid:18) f (cid:48)(cid:48) R (cid:48)(cid:48)(cid:48) f + 12 ˜ f (cid:48) R (cid:48)(cid:48)(cid:48) f (cid:48) −
120 ˜ f (cid:48) R (4) f (cid:19) ∆ ξ + O (∆ ξ ) , (74) β ∗ = ˜ f (cid:48) ∆ ξ + (cid:18) f (cid:48)(cid:48) −
23 ˜ f (cid:48) ˜ f (cid:48)(cid:48)(cid:48) + 12 ˜ f (cid:48) R (cid:48)(cid:48)(cid:48) f (cid:19) ∆ ξ + (cid:18)
136 ˜ f (cid:48)(cid:48) ˜ f (cid:48)(cid:48)(cid:48) −
12 ˜ f (cid:48) ˜ f (4) − f (cid:48)(cid:48) R (cid:48)(cid:48)(cid:48) f + ˜ f (cid:48) R (cid:48)(cid:48)(cid:48) f (cid:48) + 920 ˜ f (cid:48) R (4) f (cid:19) ∆ ξ + O (∆ ξ ) . (75)Therefore, applying the 3rd-order metrics and Jacobian to calculate the smoothness indicators β ∗ k doesnot violate the convergence orders of them. Furthermore, it is straightforward to see that(1) for the WENO5-Present scheme, we obtain β ∗ k = (cid:16) ˜ f (cid:48) ∆ ξ (cid:17) (cid:0) O (∆ ξ ) (cid:1) ˜ f (cid:48) (cid:54) = 01312 (cid:16) ˜ f (cid:48)(cid:48) ∆ ξ (cid:17) (1 + O (∆ ξ )) ˜ f (cid:48) = 0 , (76)with k = 0 , ,
2, which is the same with the standard WENO5 scheme proposed by Jiang and Shu [22];(2) for WENOZ-Present scheme, we obtain τ ∗ = | β ∗ − β ∗ | = (cid:18)
133 ˜ f (cid:48)(cid:48) ˜ f (cid:48)(cid:48)(cid:48) − ˜ f (cid:48) ˜ f (4) −
133 ˜ f (cid:48)(cid:48) R (cid:48)(cid:48)(cid:48) f i + 4 ˜ f (cid:48) R (cid:48)(cid:48)(cid:48) f (cid:48) + ˜ f (cid:48) R (4) f (cid:19) ∆ ξ + O (∆ ξ ) , (77)whose truncation error is the same order with the standard WENOZ scheme suggested by Borges et al. [25].Then, Eqs. (76) and (77) indicate that the orders of the non-linear weights are retained by applying thepresent free-stream preserving metrics and Jacobian.Finally, we conclude that the sufficient conditions given in Eqs. (61) ∼ (63) are all satisfied in thepresent free-stream preserving schemes. Considering that the present linear upwind scheme achieves 5th-order accuracy as well, therefore, the dissipation parts of the present 5th-order WENO reconstruction retain5th-order accuracy in the non-critical points as the same as the standard WENO5 and WENOZ scheme.27 ppendix C For the WENO7-Present scheme, the free-stream preserving metrics and Jacobian are represented as g ∗ / , g ∗ i − , · · · , g ∗ i +4 . We define the free-stream preserving metrics and Jacobian as g ∗ i − = 1312 g ∗ i − − g ∗ i − + 2512 g ∗ i − g ∗ i +1 / , g ∗ i − = 512 g ∗ i − − g ∗ i − g ∗ i +1 + g ∗ i +1 / ,g ∗ i − = g i − ,g ∗ i = g i ,g ∗ i +1 = g i +1 ,g ∗ i +2 = g i +2 , g ∗ i +3 = − g ∗ i +1 − g ∗ i +2 + 512 g ∗ i +3 + g ∗ i +1 / , g ∗ i +4 = 2512 g ∗ i +2 − g ∗ i +3 + 1312 g ∗ i +4 − g ∗ i +1 / . (78)where g ∗ i +1 / = 112 ( − g i − + 7 g i + 7 g i +1 − g i +2 ) . (79)Specially, the above g ∗ i +1 / is calculated by the 4th-order scheme rather than the 8th-order one, shownin Eq. (79). Therefore, it only makes the upwind dissipation rather than the central part satisfying thefree-stream preserving identity. This special treatment is to reduce the approximation from the 6th-ordercell-averaged metrics and Jacobian to the 4th-order ones. Specifically, only g i − , g i − , g i +3 , g i +4 need to beapproximated by the 4th-order g ∗ i − , g ∗ i − , g ∗ i +3 , g ∗ i +4 , while g i − , g i , g i +1 and g i +2 are unnecessary to bereplaced.The free-stream preserving identity and the accuracy of the central part is achieved by the followings.We replace the central part of ˜ F ∗ i +1 / with a 8th-order one ˜ F (4) i +1 / = 1840 (cid:16) − ˜ F ∗ i − + 29 ˜ F ∗ i − − ˜ F ∗ i − + 533 ˜ F ∗ i + 533 ˜ F ∗ i +1 − ˜ F ∗ i +2 + 29 ˜ F ∗ i +3 − ˜ F ∗ i +4 (cid:17) , ˜ F (8) i +1 / = 1840 (cid:16) − ˜ F i − + 29 ˜ F i − − ˜ F i − + 533 ˜ F i + 533 ˜ F i +1 − ˜ F i +2 + 29 ˜ F i +3 − ˜ F i +4 (cid:17) , ˜ F i +1 / = ˜ F ∗ i +1 / + ˜ F (8) i +1 / − ˜ F (4) i +1 / , (80)where ˜ F ∗ m and ˜ F m are the cell-averaged fluxes calculated by the 4th- and 8th-order metrics and Jacobian,respectively. 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