A Preliminary Study of the Burgers Equation with Symbolic ...
A Preliminary Study of the Burgers Equation with Symbolic ...
A Preliminary Study of the Burgers Equation with Symbolic ...
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SYMBOLIC COMPUTATION OF THE BURGERS EQUATION 229<br />
FIG. 4. Amplitudes <strong>of</strong> 1st through 10th harmonics at each <strong>of</strong> seven discrete time steps in symbolic computation<br />
<strong>of</strong> <strong>the</strong> <strong>Burgers</strong> equation for (a) Re = 500 and (b) infinite Re. The artifice <strong>of</strong> continuous curve fitting between discrete<br />
harmonics helps to clarify harmonic growth in time.<br />
mix, are involved in <strong>the</strong> multiplication. Their inclusion is not essential for <strong>the</strong> discussion<br />
at hand. Three possibilities exist for a nonlinear interaction between two harmonics represented<br />
by <strong>the</strong> ordered pair, (n1, n2): (i) n1 > n2, in which case <strong>the</strong> outcome <strong>of</strong> backscatter<br />
has a positive sign, (ii) n1 < n2, backscatter is negative in sign, and (iii) n1 = n2, which represents<br />
a self- interaction <strong>of</strong> a harmonic (or <strong>the</strong> fundamental), producing outscatter but no<br />
backscatter. Outscatter is identical for cases (i) and (ii). Each nonlinear interaction between<br />
a given pair (n1, n2) produces both outscatter and backscatter, <strong>with</strong> <strong>the</strong> exception <strong>of</strong> case
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