**Lagrangian Difference Approximations for Fluid Dynamics**

**Local PDF:** ADA382488.pdf

AD Number: ADA382488

Subject Categories: FLUID MECHANICS

Corporate Author: LOS ALAMOS NATIONAL LAB NM

Title: Lagrangian Difference Approximations for Fluid Dynamics

Personal Authors: Fromm, Jacob E.

Report Date: 14 JUN 1961

Pages: 68 PAGES

Report Number: LA-2535

Contract Number: W-7405-ENG-36

Monitor Acronym: XF

Monitor Series: XD

Descriptors: *FLUID DYNAMICS, *LAGRANGIAN FUNCTIONS, APPROXIMATION(MATHEMATICS)
, KINETIC ENERGY, HYDRODYNAMICS, HUGONIOT EQUATIONS, RANKINE CYCLE.

Abstract: Various procedures are given for writing explicit difference
approximations to the one-dimensional lagrangian hydrodynamics equations.
Computational comparisons are made among systems of equations with timing
modifications. These comparisons lead to experimentally superior differencing
forms. Stability analyses of these difference forms show the reasons for the
superiority of one form over another. Of greater importance, the stability
criteria obtained show the function of an artificially introduced diffusion term
required in the treatment here given to shocks. The stability criterion in each
case involves the familiar Courant condition and **a** term which corresponds to the
stability criterion of the diffusion equation. Upper limits to the magnitude of
the coefficient of the diffusion term are established as **a** function of Courant
number. While lower limits are also indicated, they require modification when
shocks are involved. Alternate differencing schemes are considered in which the
previously-used total energy calculation is replaced by an internal energy
calculation. It is shown that care must be taken that the kinetic and internal
energies are expressible in terms of local quantities. That is, in addition to
the equations being conservative in **a** gross sense, they must also be locally
conservative. This is necessary in order that the energy condition of the
Rankine-Hugoniot equations be satisfied when shocks arise. Finally discussion is
given to errors resulting from the replacement of shocks by **a** shock layer, that
is, errors connected with the artificially inserted diffusion term. These errors
are manifested in distortions of profiles at material discontinuities through
which shocks have passed and in rarefactions associated with such occurrences.
The errors in turn effect stability in the vicinity of the material
discontinuities.

Limitation Code: APPROVED FOR PUBLIC RELEASE

Source Code: 211350

Citation Creation Date: 18 OCT 2000