An Upwind Finite Pointset Method (FPM) for Compressible Euler and Navier-Stokes Equations
A Lagrangian scheme for compressible fluid flows is presented. The method can be viewed as a generalized finite difference upwind scheme. The scheme is based on the classical Euler equations in fluid mechanics, which concerns mainly non viscous problems.
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1
Introduction
SPH (smoothed particle hydrodynamics) is a Lagrangian numerical particle method often used to solve problems in compressible fluid flow. The advantages of such a Lagrangian method come into play when we would like to study time dependent fluid flow processes in very complex geometrical structures, in structures rapidly changing in time, as well as processes characterized by various phases or by free surfaces. In such cases, mesh based numerical methods have certain disadvantages since they would have to perform a very time consuming mesh generation as well as, possibly, a re-meshing procedure practically after each time step. In contrast, a Lagrangian particle method neither needs mesh generation nor re-meshing procedures in principle. Let us consider problems in gas dynamics governed by the Euler equations
I·
dV
dt +
A·
where
A=
v
{)V {)X(l)
nl'
(~ n
00000 o pc2 0 00
+ B·
{)V {)X(2)
+ c·
{)V {)x(3)
= S,
(1.1)
= (p , v(1) ,v(2) ,v(3) , p)T ,
S = source terms,
B=
(H ~ ~~l' 00000 0 0 pc2 0 0
c=
(m ~ ~l 0000 1 0 0 0 pc2 I)
The classical SPH method (see [1-3,8-12]) discretizes the system of equations (1.1) in Lagrangian form, i.e. they are based on particles supposed to act M. Griebel et al. (eds.), Meshfree Methods for Partial Differential Equations © Springer-Verlag Berlin Heidelberg 2003
240
J. Kuhnert
as carriers of mass, momentum, and energy. The particles move with fluid velocity. It is also possible to incorporate some sort of heat conduction into a particle scheme, where viscous terms are treated in a simple way. The approximations strategy used in SPH is based on weighting kernels as basis functions in order to handle occurring spatial derivatives. There are two major disadvantages of the classical SPH methods, these are in fact - difficulties when incorporating boundary conditions into the scheme, - necessity of employing artificial viscosity terms in order to keep the computation stable. The present article, therefore, concentrates on two main topics stated below. - We would like to introduce a particle upwind scheme where the employment of artificial viscosity is avoided. - We would like to show how to treat boundary particles, or, more generally, how to treat particles that might not move with fluid velocity, i.e. nonLagrangian particles.
2
Upwind FPM scheme - A voiding Artificial Viscosity
The method presented here is no longer called SPH, since there are only few common items. The name preferred is Finite Pointset Method (FPM). In the FPM approach, the basic approach is quite different from the one of SPH. Still, we fill the flow domain [l with N particles being located at the positions Xi, i = L.N. The particles carry the information Pi, i = L.N, where Pi = (p, pv, pE)T contains all conservative variables. The particles (points) are moved with an approximation of the fluid velocity and the Pi are updated according to the particle movement. For the approximation of the occurring derivatives in equation (1.1), we do not employ the classical interpolating ker
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