Finite Element Method - Non - conservative form of navier - stokes equations _appa

Finite Element Method - Non - conservative form of navier - stokes equations _appa This monograph presents in detail the novel "wave" approach to finite element modeling of transient processes in solids. Strong discontinuities of stress, deformation, and velocity wave fronts as well as a finite magnitude of wave propagation speed over elements are considered. These phenomena, such as explosions, shocks, and seismic waves, involve problems with a time scale near the wave propagation time. Software packages for 1D and 2D problems yield significantly better results than classical FEA, so some FORTRAN programs with the necessary comments are given in the appendix. The book is written for researchers, lecturers, and advanced students interested in problems of numerical modeling of non-stationary dynamic processes in deformable bodies and continua, and also for engineers and researchers involved designing machines and structures, in which shock, vibro-impact, and other unsteady dynamics and waves processes play a significant role.

Appendix A

Non-conservative form of

Navier-Stokes equations

To derive the Navier-Stokes equations in their non-conservative form, we start with the conservative form

Conservation of mass:

Conservation of momentum:

Conservation of energy:

= O (A.3)

at

Rewriting the momentum equation with terms differentiated as

and substituting the equation of mass conservation (Eq A 1) into the above equation gives the reduced momentum equation

Similarly as above, the energy equation (Eq A.3) can be written with differentiated

terms as

292 Appendix A

Again substituting the continuity equation into the above equation, we have the reduced form of the energy equation

('4.7)

Some authors use Eqs (A.l), (AS) and (A.7) to study compressible flow problems

However these non-conservative equations can result in multiple or incorrect solutions

in certain cases This is true especially for high-speed compressible flow problems with shocks The reader should note that such non-conservative equations are not suitable for simulation of compressible flow problems