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extents of the velocities being the same. We solved the problem (P) in case of Q =
]O,l[x]O,l[, which allowed us to compare in a simple case our results to those obtained by the
team of Professor J. Nihoul of the University of Liege. In such a case, the solutions of the
problem (Pl) are the following:
up = gradp = grad(cos(n 1t x)cos(m 1t y»
uq = curl (qe3) = curl(sin(n 1t x)sin(m 1t y)e3)
A = (n2 + m 2 ) 1t 2
the domain is in the conditions of regularity of the !.lemma (convex polygon) and A = ° is
not a eigen-value of (2) as Q is simply connected.
The calculi of the coefficients of the equations were done using the numerical
integration on a triangular grid of 20 x 20 points with approximation of the eigen functions by
affin piecewise functions. It could have been possible to analytically use these functions, but
the calculus is inextricable although the relative accuracy by integration is of a 10- 4 extent.
The tests carried out demonstrate that we get no more considerable change from the
eigen fourteen vectors (seven grad and seven curl). The projection of the solution on the
following eigen vector is around zero.
Let us note the calculation times compared in the same experience conditions : for a
real experience time of 10 6 seconds and a real step of 10 3 seconds, the resolution calculation
times of the differential system are the following :
Finite differences
(I.B.M. PS2) : 10 5 seconds
Spectral method
(I.B.M. PS2) : 5 103 seconds
Adjustment of the eddy conditions:
The problem is difficult as on the hand, these data are not well known and on the other
hand we get a bounded solution on [0, 00 [ if the initial conditions are close to the solution of
the stationary problem. To rise above this difficulty, we solved this problem in projection on
the only eigen vectors in curl (that is to say the same to work in divu = 0, h = cte), the
asymptotical results, that then do not depend on the values of the beginning, being used as
initial conditions.
We will note that starting with constant initial conditions of a same extent as the
stationary solution, we arrive (as we are still close to the stationary condition) to the same
extents of the velocities being the same. We solved the problem (P) in case of Q =
]O,l[x]O,l[, which allowed us to compare in a simple case our results to those obtained by the
team of Professor J. Nihoul of the University of Liege. In such a case, the solutions of the
problem (Pl) are the following:
up = gradp = grad(cos(n 1t x)cos(m 1t y»
uq = curl (qe3) = curl(sin(n 1t x)sin(m 1t y)e3)
A = (n2 + m 2 ) 1t 2
the domain is in the conditions of regularity of the !.lemma (convex polygon) and A = ° is
not a eigen-value of (2) as Q is simply connected.
The calculi of the coefficients of the equations were done using the numerical
integration on a triangular grid of 20 x 20 points with approximation of the eigen functions by
affin piecewise functions. It could have been possible to analytically use these functions, but
the calculus is inextricable although the relative accuracy by integration is of a 10- 4 extent.
The tests carried out demonstrate that we get no more considerable change from the
eigen fourteen vectors (seven grad and seven curl). The projection of the solution on the
following eigen vector is around zero.
Let us note the calculation times compared in the same experience conditions : for a
real experience time of 10 6 seconds and a real step of 10 3 seconds, the resolution calculation
times of the differential system are the following :
Finite differences
(I.B.M. PS2) : 10 5 seconds
Spectral method
(I.B.M. PS2) : 5 103 seconds
Adjustment of the eddy conditions:
The problem is difficult as on the hand, these data are not well known and on the other
hand we get a bounded solution on [0, 00 [ if the initial conditions are close to the solution of
the stationary problem. To rise above this difficulty, we solved this problem in projection on
the only eigen vectors in curl (that is to say the same to work in divu = 0, h = cte), the
asymptotical results, that then do not depend on the values of the beginning, being used as
initial conditions.
We will note that starting with constant initial conditions of a same extent as the
stationary solution, we arrive (as we are still close to the stationary condition) to the same