The entropy split method is based on the physical entropies of the thermally perfect gas Euler equations.The Euler flux derivatives are approximated as a sum of a conservative portion and a non-conservative portion in...The entropy split method is based on the physical entropies of the thermally perfect gas Euler equations.The Euler flux derivatives are approximated as a sum of a conservative portion and a non-conservative portion in conjunction with summation-by-parts(SBP)difference boundary closure of(Gerritsen and Olsson in J Comput Phys 129:245-262,1996;Olsson and Oliger in RIACS Tech Rep 94.01,1994;Yee et al.in J Comp Phys 162:33-81,2000).Sj?green and Yee(J Sci Comput)recently proved that the entropy split method is entropy conservative and stable.Stand-ard high-order spatial central differencing as well as high order central spatial dispersion relation preserving(DRP)spatial differencing is part of the entropy stable split methodol-ogy framework.The current work is our first attempt to derive a high order conservative numerical flux for the non-conservative portion of the entropy splitting of the Euler flux derivatives.Due to the construction,this conservative numerical flux requires higher oper-ations count and is less stable than the original semi-conservative split method.However,the Tadmor entropy conservative(EC)method(Tadmor in Acta Numerica 12:451-512,2003)of the same order requires more operations count than the new construction.Since the entropy split method is a semi-conservative skew-symmetric splitting of the Euler flux derivative,a modified nonlinear filter approach of(Yee et al.in J Comput Phys 150:199-238,1999,J Comp Phys 162:3381,2000;Yee and Sj?green in J Comput Phys 225:910934,2007,High Order Filter Methods for Wide Range of Compressible flow Speeds.Proceedings of the ICOSAHOM09,June 22-26,Trondheim,Norway,2009)is proposed in conjunction with the entropy split method as the base method for problems containing shock waves.Long-time integration of 2D and 3D test cases is included to show the com-parison of these new approaches.展开更多
The variable high-order multiblock overlapping(overset)grids method of Sj¨ogreen&Yee[CiCP,Vol.5,2009]for a perfect gas has been extended to nonequilibrium flows.This work makes use of the recently developed h...The variable high-order multiblock overlapping(overset)grids method of Sj¨ogreen&Yee[CiCP,Vol.5,2009]for a perfect gas has been extended to nonequilibrium flows.This work makes use of the recently developed high-order well-balanced shock-capturing schemes and their filter counterparts[Wang et al.,J.Comput.Phys.,2009,2010]that exactly preserve certain non-trivial steady state solutions of the chemical nonequilibrium governing equations.Multiscale turbulence with strong shocks and flows containing both steady and unsteady components is best treated by mixing of numerical methods and switching on the appropriate scheme in the appropriate subdomains of the flow fields,even under the multiblock grid or adaptive grid refinement framework.While low dissipative sixth-or higher-order shock-capturing filter methods are appropriate for unsteady turbulence with shocklets,second-and thirdorder shock-capturing methods are more effective for strong steady or nearly steady shocks in terms of convergence.It is anticipated that our variable high-order overset grid framework capability with its highly modular design will allow for an optimum synthesis of these new algorithms in such a way that the most appropriate spatial discretizations can be tailored for each particular region of the flow.In this paper some of the latest developments in single block high-order filter schemes for chemical nonequilibrium flows are applied to overset grid geometries.The numerical approach is validated on a number of test cases characterized by hypersonic conditions with strong shocks,including the reentry flow surrounding a 3D Apollo-like NASA Crew Exploration Vehicle that might contain mixed steady and unsteady components,depending on the flow conditions.展开更多
Flows containing steady or nearly steady strong shocks on parts of the flow field,and unsteady turbulence with shocklets on other parts of the flow field are difficult to capture accurately and efficiently employing t...Flows containing steady or nearly steady strong shocks on parts of the flow field,and unsteady turbulence with shocklets on other parts of the flow field are difficult to capture accurately and efficiently employing the same numerical scheme,even under the multiblock grid or adaptive grid refinement framework.While sixthorder or higher-order shock-capturing methods are appropriate for unsteady turbulence with shocklets,third-order or lower shock-capturing methods are more effective for strong steady or nearly steady shocks in terms of convergence.In order to minimize the short comings of low order and high order shock-capturing schemes for the subject flows,a multiblock overlapping grid with different types of spatial schemes and orders of accuracy on different blocks is proposed.The recently developed single block high order filter scheme in generalized geometries for Navier Stokes and magnetohydrodynamics systems is extended to multiblock overlapping grid geometries.The first stage in validating the high order overlapping approach with several test cases is included.展开更多
基金The support of the DOE/SciDAC SAP grant DE-AI02-06ER25796 is acknowledgedFinancial support from the NASA Aerosciences/RCA program for the second author is gratefully acknowledgedWork by the fifth author was performed under the auspices of the U.S.Department of Energy at Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344
基金support from the NASA TTT/RCA program for the second author is grate-fully acknowledged.
文摘The entropy split method is based on the physical entropies of the thermally perfect gas Euler equations.The Euler flux derivatives are approximated as a sum of a conservative portion and a non-conservative portion in conjunction with summation-by-parts(SBP)difference boundary closure of(Gerritsen and Olsson in J Comput Phys 129:245-262,1996;Olsson and Oliger in RIACS Tech Rep 94.01,1994;Yee et al.in J Comp Phys 162:33-81,2000).Sj?green and Yee(J Sci Comput)recently proved that the entropy split method is entropy conservative and stable.Stand-ard high-order spatial central differencing as well as high order central spatial dispersion relation preserving(DRP)spatial differencing is part of the entropy stable split methodol-ogy framework.The current work is our first attempt to derive a high order conservative numerical flux for the non-conservative portion of the entropy splitting of the Euler flux derivatives.Due to the construction,this conservative numerical flux requires higher oper-ations count and is less stable than the original semi-conservative split method.However,the Tadmor entropy conservative(EC)method(Tadmor in Acta Numerica 12:451-512,2003)of the same order requires more operations count than the new construction.Since the entropy split method is a semi-conservative skew-symmetric splitting of the Euler flux derivative,a modified nonlinear filter approach of(Yee et al.in J Comput Phys 150:199-238,1999,J Comp Phys 162:3381,2000;Yee and Sj?green in J Comput Phys 225:910934,2007,High Order Filter Methods for Wide Range of Compressible flow Speeds.Proceedings of the ICOSAHOM09,June 22-26,Trondheim,Norway,2009)is proposed in conjunction with the entropy split method as the base method for problems containing shock waves.Long-time integration of 2D and 3D test cases is included to show the com-parison of these new approaches.
文摘The variable high-order multiblock overlapping(overset)grids method of Sj¨ogreen&Yee[CiCP,Vol.5,2009]for a perfect gas has been extended to nonequilibrium flows.This work makes use of the recently developed high-order well-balanced shock-capturing schemes and their filter counterparts[Wang et al.,J.Comput.Phys.,2009,2010]that exactly preserve certain non-trivial steady state solutions of the chemical nonequilibrium governing equations.Multiscale turbulence with strong shocks and flows containing both steady and unsteady components is best treated by mixing of numerical methods and switching on the appropriate scheme in the appropriate subdomains of the flow fields,even under the multiblock grid or adaptive grid refinement framework.While low dissipative sixth-or higher-order shock-capturing filter methods are appropriate for unsteady turbulence with shocklets,second-and thirdorder shock-capturing methods are more effective for strong steady or nearly steady shocks in terms of convergence.It is anticipated that our variable high-order overset grid framework capability with its highly modular design will allow for an optimum synthesis of these new algorithms in such a way that the most appropriate spatial discretizations can be tailored for each particular region of the flow.In this paper some of the latest developments in single block high-order filter schemes for chemical nonequilibrium flows are applied to overset grid geometries.The numerical approach is validated on a number of test cases characterized by hypersonic conditions with strong shocks,including the reentry flow surrounding a 3D Apollo-like NASA Crew Exploration Vehicle that might contain mixed steady and unsteady components,depending on the flow conditions.
基金This work performed under the auspices of the U.S.Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344。
文摘Flows containing steady or nearly steady strong shocks on parts of the flow field,and unsteady turbulence with shocklets on other parts of the flow field are difficult to capture accurately and efficiently employing the same numerical scheme,even under the multiblock grid or adaptive grid refinement framework.While sixthorder or higher-order shock-capturing methods are appropriate for unsteady turbulence with shocklets,third-order or lower shock-capturing methods are more effective for strong steady or nearly steady shocks in terms of convergence.In order to minimize the short comings of low order and high order shock-capturing schemes for the subject flows,a multiblock overlapping grid with different types of spatial schemes and orders of accuracy on different blocks is proposed.The recently developed single block high order filter scheme in generalized geometries for Navier Stokes and magnetohydrodynamics systems is extended to multiblock overlapping grid geometries.The first stage in validating the high order overlapping approach with several test cases is included.
