On the intrinsic instability of the advective-diffusive transport
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On the intrinsic instability of the advective-diffusive transportData
2004Resumo
[Abstract] A new theory for the advective–diffusive phenomenon is described in this study and the causes for the failure of the conventional numerical methods for this problem are investigated.
It is shown that Fick’s law —the constitutive equation of the transport problem— is
the cause of the appearance of oscillations in the numerical solutions of predominantly
advective problems. Fick’s law leads to the unreasonable result that mass can propagate
at an infinite speed.
We propose a new formulation for the advective–diffusive problem by using a constitutive
equation derived by M. Carlo Cattaneo in 1958 for thermodynamic and pure–diffusion
problems. This new approach overcomes the problem of mass propagation at an infinite
speed. It is also shown that the advective–diffusive problem is a wave–like problem. Hence,
a pollutant diffuses like a wave in a fluid.
A detailed analysis of the new equations shows an important conclusion: a critical
fluid velocity exists for each advective–diffusive problem. When fluid velocity is greater
or equal than this critical speed the steady state problem is not anymore a well–posed
problem and the transient problem is as well ill–posed if it is stated as a bounduary value
problem. In this case we should formulate the advective–diffusive problem as an initial
value problem. Furthermore, we propose stability conditions for the steady state advective–
diffusive problem.
Several problems have been solved to check the good behaviour of the numerical solution
of the new equations and the proposed stability conditions.
Palabras chave
Advection-diffusion
Constitutive equation
Numerical instability
Cattaneo's equation
Constitutive equation
Numerical instability
Cattaneo's equation
Descrición
European Congress on Computational Methods in Applied Sciences and Engineering ECCOMAS 2004, Jyväskiylä, Finland, 24–28 July 2004