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THREE-DIMENSIONAL IMPLICIT LAMBDA METHODS

IP.com Disclosure Number: IPCOM000149113D
Original Publication Date: 1899-Dec-30
Included in the Prior Art Database: 2007-Apr-12

Publishing Venue

Software Patent Institute

Related People

Napolitano, M.: AUTHOR [+3]

Abstract

M. Napolitano Institute for Computer Applications in Science and Engineering and Istituto di Macchine, via Re David 200, 70125 Rari, Italy A. Dadone Istituto di Macchine, via Re David 200, 70125 Rari., Italy ABSTRACT This paper derives the three-dimensional lambda-formulation equations f r a general orthogonal curvilinear coordinate sys tern and provides vari ous hlock- explicit and block-implicit methods for solving them, numerically. Three model problems, characterized by subsonic, supersonic and transonic flow conditions, are used to assess the reliability and compare the efficiency of the proposed methods. *Research supported in part by the Consiglio Nazionale delle Ricerche and in part hy the National Aeronautics and Space Administration under NASA Contract No. NAS1-17070 whi1.e the first author was in residence at the ICASE, NASA Langley Research Center, Hampton, Va 23665. Among the many theoretical models employed in the numerical simulation of compressible inviscid flows the so-call ecl lambda-f ormulation has received considerable interest (see, e. g. , [l-81) : the time-dependent Euler equations are recast into compatibility conditions of bicharacteristic variables along the corresponding hicharacteristic lines and discretized using windward differences, in order to account for the direction of wave propagation phenomena, correctly. Such an approach has many nice properties: it provides very accurate numerical results, even with rather coarse meshes (see, e.p., [2J, 131, f61); it requires only the physical boundary conditions, so that there is no need for any additional numerical boundary treatments, which are frequently the cause of numerical instability f91; it handles in a most automatic and physically-souncl way mixed supersonic-subsonic flow fields; and finally, it has a well documented, a1 thouah controversial, capahi 1 ity of capturing shocks without any additional dissipation [2-6 J . For these reasons, in spite of the fact that the "captured shocks" are only isentropic approximations to correct weak solutions of the Euler equations and do not correctly move within the flow field unless properly fitted 141, the lambda- formulation is considered to be a very useful and reliable tool for predictinr compressible flow fields and, therefore, very worthy of further studies and improvements; and in fact, in the last two years, for the cases of quasi-one dimensional and two dimensional flows, the development of various kinds of implicit integration schemes 15-81 has removed the only major limitation of previous lambda methods, namely, the CFL stability restriction associated with their explicit inteeration procedures.

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                   M. Napolitano
Institute for Computer Applications in Science and Engineering
and
Istituto di Macchine, via Re David 200, 70125 Rari, Italy

A. Dadone

Istituto di Macchine, via Re David 200, 70125 Rari., Italy

                           ABSTRACT
This paper derives the three-dimensional lambda-formulation equations f ~ r a general orthogonal curvilinear coordinate sys tern and provides vari ous hlock-
explicit and block-implicit methods for solving them, numerically. Three
model problems, characterized by subsonic, supersonic and transonic flow
conditions, are used to assess the reliability and compare the efficiency of
the proposed methods.

*Research supported in part by the Consiglio Nazionale delle Ricerche and
in part hy the National Aeronautics and Space Administration under NASA
Contract No. NAS1-17070 whi1.e the first author was in residence at the
ICASE, NASA Langley Research Center, Hampton, Va 23665.

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Page 2 of 24

[This page contains 1 picture or other non-text object]

Page 3 of 24

    Among the many theoretical models employed in the numerical simulation of
compressible inviscid flows the so-call ecl lambda-f ormulation has received
considerable interest (see, e. g. , [l-81) : the time-dependent Euler equations
are recast into compatibility conditions of bicharacteristic variables along
the corresponding hicharacteristic lines and discretized using windward
differences, in order to account for the direction of wave propagation
phenomena, correctly. Such an approach has many nice properties: it provides
very accurate numerical results, even with rather coarse meshes (see, e.p., [2J, 131, f61); it requires only the physical boundary conditions, so that
there is no need for any additional numerical boundary treatments, which are
frequently the cause of numerical instability f91; it handles in a most
automatic and physically-souncl way mixed supersonic-subsonic flow fields; and
finally, it has a well documented, a1 thouah controversial, capahi 1 ity of
capturing shocks without any additional dissipation [2-6 J . For these reasons,
in spite of the fact that the "captured shocks" are only isentropic
approximations to correct weak solutions of the Euler equations and do not
correctly move within the flow field - unless properly fitted 141, the lambda-
formulation is considered to be a very useful and reliable tool for predictinr
compressible flow fields and, therefore, very worthy of further studies and
improvements; and in fact, in the last two years, for the cases of quasi-one
dimensional and two dimensional flows, the development of various kinds of
implicit integration schemes 15-81 has removed the only major limitation of
previous lambda methods, namely, the CFL stability restriction associated with
their explicit inteeration procedures.

    It now appears very timely and worthwhile to develop efficient numerical
meth...