Coherent structures in dissipative particle dynamics simulations of the transition to turbulence in compressible shear flows

We present simulations of coherent structures in compressible flows near the transition to turbulence using the dissipative particle dynamics method. The structures we find are remarkably consistent with experimental observations and direct numerical simulations (DNS) simulations of incompressible flows, despite a difference in Mach number of several orders of magnitude. The bifurcation from the laminar flow is bistable and shifts to higher Reynolds numbers when the fluid becomes more compressible. This work underlines the robustness of coherent structures in the transition to turbulence and illustrates the ability of particle-based methods to reproduce complex nonlinear instabilities.

Figure 1

(Color online) A pair of streaks in compressible plane Couette flow at $\mathrm{Re}=1400$ and $\mathrm{Ma}=8$. Color contours denote the streamwise deviation from the laminar flow, vector fields denote the average in-plane motion. (a) Streamwise-direction view, showing $x$-averaged velocities. (b) Flow gradient-direction view at $y=0$. The fast streak shows a clear sinusoidal inflection.

Figure 2

(Color online) Snapshots of streaky pipe flow averaged over the pipe length. The simulation box has pipe diameter $d=60$ and periodic length $l=126$. DPD parameters are $(a,\gamma ,kT)=(5,1,0.2)$, density $\rho =4$, $N=1425026$, yielding $\nu =0.2$, $c\simeq 4.2$, and $\mathrm{Re}∕\mathrm{Ma}=1260$. (a) A four-streak configuration at $\mathrm{Re}=1700$. (b)–(d) Three different snapshots at $\mathrm{Re}=2700$. These states and the stochastic hopping between them are similar to those observed in experiments and in DNS of incompressible flows [9, 12].

Figure 3

(Color online) Mean streamwise velocity $U\left(y\right)$ and density $\rho \left(y\right)$ for the flow snapshot shown in Fig. 1 (solid), with the initial $t=0$ state (dotted) shown for comparison. (a) The mean velocity $U\left(y\right)$ develops a sinusoidal deviation from the linear profile as coherent structures develop. (b) As a result of the compressibility of the fluid, the density profile $\rho \left(y\right)$ shows a slight bulge at the center of the cell.

Figure 4

(Color online) Bistability in the bifurcation from laminar flow at $\mathrm{Re}∕\mathrm{Ma}=190$. Shown is the maximum deviation $A$ of the mean profile $U\left(y\right)$ from linearity, $A={\max }_{\mathrm{y}}\mid (U\left(y\right)∕U-y∕h)\mid$. Approaching from the laminar state, a smooth (forward) transition towards a two-pair mode is observed, which becomes unstable at $\mathrm{Re}\gtrsim 1200$. A stable one-pair mode is found by initial forcing of a pair of rolls. Decreasing the driving rate results in a jump (subcritical) transition back to the laminar state.

Figure 5

(Color online) Relaminarization in a series of runs with increasing compressibility. The box size is $48\times 16\times 32$. One-pair modes are initialized at regular intervals in Re, with the sound velocities ranging from $c\simeq 5.5$ (red) to $c\simeq 2.8$ (blue) and tracked down by slowly decreasing the driving rate. The resulting curves show that the compressibility decreases the amplitude of the turbulent state and the abruptness of the transition.