改进的3阶WENO-N3格式及其在内爆炸载荷计算中的应用
|
摘要
[目的]高分辨率激波捕捉格式对含激波流场的数值模拟具有重要意义,其不但可以降低网格的规模,而且能较好地分辨出流场中复杂的波系结构。[方法]通过在权函数公式中增加额外项的方式构造出WENO-N+3格式,并采用泰勒级数展开的方法对WENO-N3和WENO-N+3格式在极值点处的精度进行理论推导和精度数值测试。选用激波与熵波相互作用、双爆轰波碰撞、瑞利—泰勒不稳定性等问题验证改进格式WENO-N+3与WENO-JS3,WENO-Z3和WENO-N3格式的区别。最后,将改进格式WENO-N+3应用于内爆炸载荷的数值计算。[结果]研究表明,改进格式WENO-N+3相较其他格式耗散低,并且在相同的计算网格下能给出较高的冲击波峰值。[结论]该方法可用于内爆炸载荷的数值计算。
[Objectives]In this paper,an improved third-order WENO scheme named 'WENO-N + 3' isconstructed by adding a suitable term to the weights of the conventional WENO-N3 scheme.[Methods]The precision near the critical points of WENO-N3 and the WENO-N + 3 schemes is derived using theTaylor expansion,and accuracy tests are also conducted. Shock-entropy wave interaction,interactingblast waves and Rayleigh-Taylor instability problems are selected in order to verify that the improvedscheme WENO-N + 3 has less dissipation than the conventional WENO-JS3,WENO-Z3 or WENO-N3 schemes,with high-resolution flow field structures. Finally,the proposed WENO-N+3 scheme is appliedto simulations of explosion loads in a confined space.[Results]The simulation results indicate that thepresent scheme can provide a higher shockwave peak. [Conclusions]The findings of this paper canprovide valuable references for research into the numerical calculation of internal explosion loads.
[Objectives]In this paper,an improved third-order WENO scheme named 'WENO-N + 3' isconstructed by adding a suitable term to the weights of the conventional WENO-N3 scheme.[Methods]The precision near the critical points of WENO-N3 and the WENO-N + 3 schemes is derived using theTaylor expansion,and accuracy tests are also conducted. Shock-entropy wave interaction,interactingblast waves and Rayleigh-Taylor instability problems are selected in order to verify that the improvedscheme WENO-N + 3 has less dissipation than the conventional WENO-JS3,WENO-Z3 or WENO-N3 schemes,with high-resolution flow field structures. Finally,the proposed WENO-N+3 scheme is appliedto simulations of explosion loads in a confined space.[Results]The simulation results indicate that thepresent scheme can provide a higher shockwave peak. [Conclusions]The findings of this paper canprovide valuable references for research into the numerical calculation of internal explosion loads.
引文
[1]LIU X D,OSHER S,CHAN T.Weighted essentially non-oscillatory schemes[J].Journal of Computational Physics,1994,115(1):200-212.
[2]HARTEN A,ENGQUIST B,OSHER S,et al.Uniformly high order accurate essentially non-oscillatory schemes,III[J].Journal of Computational Physics,1987,71(2):231-303.
[3]JIANG G S,SHU C W.Efficient implementation of weighted ENO schemes[J].Journal of Computational Physics,1995,126(1):202-228.
[4]HENRICK A K,ASLAM T D,POWERS J M.Mapped weighted essentially non-oscillatory schemes:achieving optimal order near critical points[J].Journal of Computational Physics,2005,207(2):542-567.
[5]BORGES R,CARMONA M,COSTA B,et al.An improved weighted essentially non-oscillatory scheme for hyperbolic conservation laws[J].Journal of Computational Physics,2008,227(6):3191-3211.
[6]YAMALEEV N K,CARPENTER M H.A systematic methodology for constructing high-order energy stable WENO schemes[J].Journal of Computational Physics,2009,228(1):4248-4272.
[7]FAN P.High order weighted essentially nonoscillatory WENO-ηschemes for hyperbolic conservation laws[J].Journal of Computational Physics,2014,269:355-385.
[8]FAN P,SHEN Y Q,TIAN B L,et al.A new smoothness indicator for improving the weighted essentially non-oscillatory scheme[J].Journal of Computational Physics,2014,269:329-354.
[9]FENG H,HUANG C,WANG R.An improved mapped weighted essentially non-oscillatory scheme[J].Applied Mathematics and Computation,2014,232:453-468.
[10]SHEN Y Q,ZHA G H.Improvement of weighted essentially non-oscillatory schemes near discontinuities[J].Computers&Fluids,2014,96:1-9.
[11]KIM C H,HA Y,YOON J.Modified non-linear weights for fifth-order weighted essentially non-oscillatory schemes[J].Journal of Scientific Computing,2016,67(1):299-323.
[12]MA Y K,YAN Z G,ZHU H J.Improvement of multistep WENO scheme and its extension to higher orders of accuracy[J].International Journal for Numerical Methods in Fluids,2016,82(12):818-838.
[13]WANG R,FENG H,HUANG C.A New mapped weighted essentially non-oscillatory method using rational mapping function[J].Journal of Scientific Computing,2016,67(2):540-580.
[14]YAMALEEV N K,CARPENTER M H.Third-order energy stable WENO scheme[J].Journal of Computational Physics,2013,228(8):3025-3047.
[15]WU X S,ZHAO Y X.A high-resolution hybrid scheme for hyperbolic conservation laws[J].International Journal for Numerical Methods in Fluids,2015,78(3):162-187.
[16]WU X S,LIANG J H,ZHAO Y X.A new smoothness indicator for third-order WENO scheme[J].International Journal for Numerical Methods in Fluids,2016,81(7):451-459.
[17]ACKER F,DE R BORGES RB,COSTA B.An improved WENO-Z scheme[J].Journal of Computational Physics,2016,313:726-753.
[18]TORO E F.Riemann solvers and numerical methods for fluid dynamics:a practical introduction[M].Berlin Heidelberg:Springer,1999:87-114.
[19]SHU C W,OSHER S.Efficient implementation of essentially non-oscillatory shock-capturing schemes,II[J].Journal of Computational Physics,1989,83(1):32-78.
[20]GANDE N R,RATHOD Y,RATHAN S.Third‐order WENO scheme with a new smoothness indicator[J].International Journal for Numerical Methods in Fluids,2017,85(2):90-112.
[21]DON W S,BORGES R.Accuracy of the weighted essentially non-oscillatory conservative finite difference schemes[J].Journal of Computational Physics,2013,250:347-372.
[22]XU W Z,WU W G.An improved third-order WENO-Z scheme[J].Journal of Scientific Computing,2018,75:1808-1841.
[23]WOODWARD P,COLELLA P.The numerical simulation of two-dimensional fluid flow with strong shocks[J].Journal of Computational Physics,1984,54(1):115-173.
[24]SHI J,ZHANG Y T,SHU C W.Resolution of high order WENO schemes for complicated flow structures[J].Journal of Computational Physics,2003,186(2):690-696.
[25]HU X Y,WANG Q,ADAMS N A.An adaptive central-upwind weighted essentially non-oscillatory scheme[J].Journal of Computational Physics,2010,229(23):8952-8965.
[2]HARTEN A,ENGQUIST B,OSHER S,et al.Uniformly high order accurate essentially non-oscillatory schemes,III[J].Journal of Computational Physics,1987,71(2):231-303.
[3]JIANG G S,SHU C W.Efficient implementation of weighted ENO schemes[J].Journal of Computational Physics,1995,126(1):202-228.
[4]HENRICK A K,ASLAM T D,POWERS J M.Mapped weighted essentially non-oscillatory schemes:achieving optimal order near critical points[J].Journal of Computational Physics,2005,207(2):542-567.
[5]BORGES R,CARMONA M,COSTA B,et al.An improved weighted essentially non-oscillatory scheme for hyperbolic conservation laws[J].Journal of Computational Physics,2008,227(6):3191-3211.
[6]YAMALEEV N K,CARPENTER M H.A systematic methodology for constructing high-order energy stable WENO schemes[J].Journal of Computational Physics,2009,228(1):4248-4272.
[7]FAN P.High order weighted essentially nonoscillatory WENO-ηschemes for hyperbolic conservation laws[J].Journal of Computational Physics,2014,269:355-385.
[8]FAN P,SHEN Y Q,TIAN B L,et al.A new smoothness indicator for improving the weighted essentially non-oscillatory scheme[J].Journal of Computational Physics,2014,269:329-354.
[9]FENG H,HUANG C,WANG R.An improved mapped weighted essentially non-oscillatory scheme[J].Applied Mathematics and Computation,2014,232:453-468.
[10]SHEN Y Q,ZHA G H.Improvement of weighted essentially non-oscillatory schemes near discontinuities[J].Computers&Fluids,2014,96:1-9.
[11]KIM C H,HA Y,YOON J.Modified non-linear weights for fifth-order weighted essentially non-oscillatory schemes[J].Journal of Scientific Computing,2016,67(1):299-323.
[12]MA Y K,YAN Z G,ZHU H J.Improvement of multistep WENO scheme and its extension to higher orders of accuracy[J].International Journal for Numerical Methods in Fluids,2016,82(12):818-838.
[13]WANG R,FENG H,HUANG C.A New mapped weighted essentially non-oscillatory method using rational mapping function[J].Journal of Scientific Computing,2016,67(2):540-580.
[14]YAMALEEV N K,CARPENTER M H.Third-order energy stable WENO scheme[J].Journal of Computational Physics,2013,228(8):3025-3047.
[15]WU X S,ZHAO Y X.A high-resolution hybrid scheme for hyperbolic conservation laws[J].International Journal for Numerical Methods in Fluids,2015,78(3):162-187.
[16]WU X S,LIANG J H,ZHAO Y X.A new smoothness indicator for third-order WENO scheme[J].International Journal for Numerical Methods in Fluids,2016,81(7):451-459.
[17]ACKER F,DE R BORGES RB,COSTA B.An improved WENO-Z scheme[J].Journal of Computational Physics,2016,313:726-753.
[18]TORO E F.Riemann solvers and numerical methods for fluid dynamics:a practical introduction[M].Berlin Heidelberg:Springer,1999:87-114.
[19]SHU C W,OSHER S.Efficient implementation of essentially non-oscillatory shock-capturing schemes,II[J].Journal of Computational Physics,1989,83(1):32-78.
[20]GANDE N R,RATHOD Y,RATHAN S.Third‐order WENO scheme with a new smoothness indicator[J].International Journal for Numerical Methods in Fluids,2017,85(2):90-112.
[21]DON W S,BORGES R.Accuracy of the weighted essentially non-oscillatory conservative finite difference schemes[J].Journal of Computational Physics,2013,250:347-372.
[22]XU W Z,WU W G.An improved third-order WENO-Z scheme[J].Journal of Scientific Computing,2018,75:1808-1841.
[23]WOODWARD P,COLELLA P.The numerical simulation of two-dimensional fluid flow with strong shocks[J].Journal of Computational Physics,1984,54(1):115-173.
[24]SHI J,ZHANG Y T,SHU C W.Resolution of high order WENO schemes for complicated flow structures[J].Journal of Computational Physics,2003,186(2):690-696.
[25]HU X Y,WANG Q,ADAMS N A.An adaptive central-upwind weighted essentially non-oscillatory scheme[J].Journal of Computational Physics,2010,229(23):8952-8965.