Modeling turbulent flow is one of the most difficult problems in fluid dynamics. Solutions to the Navier–Stokes equations, the full set of partial differential equations that represent fluid flow, are unstable for turbulent fluids. A highly simplified version of the Navier–Stokes equations, yet one that contains that same type of nonlinearity, is the one-dimensional inviscid (i.e., zero viscosity) Burgers equation, given by
where is the spatial coordinate, is time, and is some conserved physical quantity. In Burgers turbulence (or "burgulence"), Burgers's equation is considered to be a stochastic partial differential equation whose solution reveals some aspects of turbulent behavior in fluid flow. In this Demonstration, Burgers's equation is solved with random initial conditions (i.e., a randomly perturbed Gaussian waveform) as a model of turbulence. The left-hand plot shows the time evolution of the waveform, while the right-hand plot shows the time evolution of the -density function (i.e., the distribution of -values at times ). The range of (i.e., [0,1]) is broken into 20 intervals. Three turbulence levels can be tested: low, medium, and high.
Since Burgers's equation is a flux-conservative hyperbolic partial differential equation, it is amenable to finite-volume methods. In this Demonstration, Burgers's equation is solved with the Harten–Lax–van Leer (HLL) method with periodic boundary conditions.
J. M. Burgers, "Mathematical Examples Illustrating Relations Occurring in the Theory of Turbulent Fluid Motion," Verhandelingen der Koninklijke Nederlandse Akademie van Wetenschappen, Afdeling Natuurkunde. Tweede Sectie, 17, 1939 pp. 1–53.
U. Frisch and J. Bec, "Burgulence," in New Trends in Turbulence, M. Lesieur, A. Yaglom, and F. David, eds., Berlin: Springer-Verlag, 2001.
W. E and E. Vanden Eijnden, "Statistical Theory for the Stochastic Burgers Equation in the Inviscid Limit," Communications on Pure and Applied Mathematics., 53, 2000 pp. 852–901.