The solution of Burgers' equation for sinusoidal excitation at the upstream boundary
- 27 Downloads
This paper generates an exact solution to Burgers' nonlinear diffusion equation on a convective stream with sinusoidal excitation applied at the upstream boundary,x=0. The downstream boundary, effectively atx=∞, is assumed to always be far enough ahead of the convective front atx=Vst that no disturbance is felt there. The Hopf-Cole transformation is applied in achieving the analytical solution, but only after integrating the equation and its conditions inx to avoid a nonlinearity in the transformed upstream boundary condition.
A very simple limiting solution valid for high Reynolds number is deduced from the exact solution. This approximate solution is found to be amenable to an elegant geometrical interpretation. This is in a style similar to Burgers' classical interpretation of the solution to the simpler problem for which the excitation is provided through the initial condition. The ‘shocks’ present in Burgers' classical solution develop with distance downstream of the excitation in the present work.
Detailed results confirming the conclusions deduced by inspection of the solution formulae are computed and presented in the form of space-time plots. Evidence of period splitting in thex-variable at lower values of the Reynolds number is found in the numerical computations. This is a characteristic indicative of the onset of aperiodic chaotic response in many nonlinear dynamical systems. However, the computations are from approximate solution formulae valid for high Reynolds number; these formulae imply complete periodicity in time, for all values of the parameters. The correct interpretation is therefore unclear at this time, although the boundary-value problem appears to have the proper structure for chaotic behavior.
Unable to display preview. Download preview PDF.
- 1.E.R. Benton and G.W. Platzmann, A table of solutions of the one-dimensional Burgers equation,Qrt. Appl. Math. 30 (1972) 195–212.Google Scholar
- 2.J.M. Burgers,The Non-Linear Diffusion Equation, Dordrecht: Reidel (1974).Google Scholar
- 5.E.T. Copson,Asymptotic Expansions, England: Cambridge University Press (1965).Google Scholar
- 6.J. Guckenheimer and P. Holmes,Non-linear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, New York: Springer-Verlag (1983).Google Scholar