Model for Shock Wave Chaos

We propose the following model equation, $u_t+1/2(u^2-u{u}_s{)}_x=f(x,u_s)$ that predicts chaotic shock waves, similar to those in detonations in chemically reacting mixtures. The equation is given on the half line, $x<0$, and the shock is located at $x=0$ for any $t\ge 0$. Here, $u_s(t)$ is the shock state and the source term $f$ is taken to mimic the chemical energy release in detonations. This equation retains the essential physics needed to reproduce many properties of detonations in gaseous reactive mixtures: steady traveling wave solutions, instability of such solutions, and the onset of chaos. Our model is the first (to our knowledge) to describe chaos in shock waves by a scalar first-order partial differential equation. The chaos arises in the equation thanks to an interplay between the nonlinearity of the inviscid Burgers equation and a novel forcing term that is nonlocal in nature and has deep physical roots in reactive Euler equations.

Figure 1

The long-time spatiotemporal profiles of $u(x,t)$ (color) for the periodic solution at $\alpha =4.7$ (left) and the chaotic solution at $\alpha =5.1$ (right); $\beta =0.1$. The white curves are the characteristics of Eq. (9) given by $\dot{x}=u-u_s/2$. At any fixed $t$, $u(x,t)$ generally decreases away from $x=0$.

Figure 2

The period-one and period-two limit cycles in the plane of the shock strength $u_s(t)$ vs ${\dot{u}}_s$ at two different values of $\alpha$ and $\beta =0.1$.

Figure 3

The long-time values of the local maxima, $u_s^{\max }$, of the shock strength as a function of $\alpha$.

Figure 4

The chaotic attractor in the ($u_s$, ${\dot{u}}_s$, ${\ddot{u}}_s$) space; $\alpha =5.1$.

Figure 5

The Lorenz map showing consecutive local maxima ($u_s^n$, $u_s^{n+1}$) of the shock strength, $u_s(t)$, over large times (from $t=3000$ to $t=6000$) for the chaotic case at $\alpha =5.1$.