Nonlinear dynamics of a buoyancy-induced turbulent fire

We conduct a numerical study on the dynamic behavior of a buoyancy-induced turbulent fire from the viewpoints of symbolic dynamics, complex networks, and statistical complexity. Here, we consider two classes of entropies: the permutation entropy and network entropy in $\epsilon$-recurrence networks, both of which evaluate the degree of randomness in the underlying dynamics. These entropies enable us to capture the significant changes in the dynamic behavior of flow velocity fluctuations. The possible presence of two important dynamics, low-dimensional deterministic chaos in the near field dominated by the motion of large-scale vortices and high-dimensional chaos in the far field forming a well-developed turbulent plume, is clearly identified by the multiscale complexity-entropy causality plane.

Figure 1

Instantaneous flow velocity field on the $x-z$ plane and the extracted temporal evolution of the centerline streamwise velocity $w$ as a function of height $z$. (a) $z=0.5$ m, (b) $z=1.0$ m, and (c) $z=3.0$ m.

Figure 2

Spatial distribution of (a) the permutation entropy $H_p$ and (b) the root mean square $w_{\mathrm{rms}}^{\prime }$ for the temporal evolution of streamwise velocity fluctuations $w^{\prime }$ on the $x-z$ plane during a turbulent fire.

Figure 3

Variations in the permutation entropy $H_p$ and network entropy $H_n$ along the centerline of the fire source as a function of height $z$.

Figure 4

Variations in (A) the permutation entropy $H_p$ and (B) the Jensen-Shannon statistical complexity $C_{JS}$ of streamwise velocity fluctuations $w^{\prime }$ in terms of the delay time $\tau$ and height $z$. (a) $z=0.5$ m, (b) $z=1.0$ m, and (c) $z=3.0$ m.

Figure 5

Complexity-entropy causality planes of streamwise velocity fluctuations $w^{\prime }$ at different heights $z$. (a) $z=0.5$ m, (b) $z=1.0$ m, and (c) $z=3.0$ m.

Figure 6

Variations in SDLE $\lambda \left(\epsilon \right)$ as a function of $\epsilon$ at different heights $z$. (a) $z=0.5$ m, (b) $z=1.0$ m, and (c) $z=3.0$ m.

Figure 7

Complexity-entropy causality planes for second-order moment of streamwise velocity fluctuations $W^{\prime }$ at different heights $z$. (a) $z=0.5$ m, (b) $z=1.0$ m, and (c) $z=3.0$ m.

Figure 8

Variations in the delay time ${\tau }_{Cmax}$ for streamwise velocity fluctuations $w^{\prime }$ and second-order moment fluctuations $W^{\prime }$ as a function of height $z$.