Extremely rare collapse and build-up of turbulence in stochastic models of transitional wall flows

This paper presents a numerical and theoretical study of multistability in two stochastic models of transitional wall flows. An algorithm dedicated to the computation of rare events is adapted on these two stochastic models. The main focus is placed on a stochastic partial differential equation model proposed by Barkley. Three types of events are computed in a systematic and reproducible manner: (i) the collapse of isolated puffs and domains initially containing their steady turbulent fraction; (ii) the puff splitting; (iii) the build-up of turbulence from the laminar base flow under a noise perturbation of vanishing variance. For build-up events, an extreme realization of the vanishing variance noise pushes the state from the laminar base flow to the most probable germ of turbulence which in turn develops into a full blown puff. For collapse events, the Reynolds number and length ranges of the two regimes of collapse of laminar-turbulent pipes, independent collapse or global collapse of puffs, is determined. The mean first passage time before each event is then systematically computed as a function of the Reynolds number $r$ and pipe length $L$ in the laminar-turbulent coexistence range of Reynolds number. In the case of isolated puffs, the faster-than-linear growth with Reynolds number of the logarithm of mean first passage time $T$ before collapse is separated in two. One finds that $ln\left(T\right)=A_{p}r-B_p$, with $A_p$ and $B_p$ positive. Moreover, $A_p$ and $B_p$ are affine in the spatial integral of turbulence intensity of the puff, with the same slope. In the case of pipes initially containing the steady turbulent fraction, the length $L$ and Reynolds number $r$ dependence of the mean first passage time $T$ before collapse is also separated. The author finds that $T\asymp exp\left[L\right(Ar-B\left)\right]$ with $A$ and $B$ positive. The length and Reynolds number dependence of $T$ are then discussed in view of the large deviations theoretical approaches of the study of mean first passage times and multistability, where $ln\left(T\right)$ in the limit of small variance noise is studied. Two points of view, local noise of small variance and large length, can be used to discuss the exponential dependence in $L$ of $T$. In particular, it is shown how a $T\asymp exp\left[L(A^{\prime }R-B^{\prime })\right]$ can be derived in a conceptual two degrees of freedom model of a transitional wall flow proposed by Dauchot and Manneville. This is done by identifying a quasipotential in low variance noise, large length limit. This pinpoints the physical effects controlling collapse and build-up trajectories and corresponding passage times with an emphasis on the saddle points between laminar and turbulent states. This analytical analysis also shows that these effects lead to the asymmetric probability density function of kinetic energy of turbulence.

Figure 1

Sketch of the conceptual phase space of a transitional wall flow, including basin boundary, minimal seed of turbulence, the turbulent coherent structures (1 to $n$ puffs) and the events linking these metastable states: turbulence collapse, puff splitting, build-up of turbulence from the laminar base flow under vanishing perturbations (possibly joining optimal trajectory). $L$ stands for laminar base flow.

Figure 2

Streamwise velocity $u$ and turbulence intensity $q$ as a function of space $x$ in two situations: (a) isolated puff, (b) steady turbulent fraction. (c) Space and time average of turbulence intensity $\mathcal{Q}$ and streamwise velocity deficit $1-\mathcal{U}$ as a function of the Reynolds number $r$ in the transitional range. The range of laminar-turbulence coexistence, where $0<1-\mathcal{U}<1$, is indicated by black dashed lines.

Figure 3

(a) Sketch of first passages and reactive trajectories in phase space. (b) Sketch of the principle of the adaptive multilevel splitting algorithm, in the case of three clones. The sketch indicates the starting state $\mathcal{A}$, the arrival state $\mathcal{B}$, levels of reactions coordinates $\varphi$, and two examples of branching.

Figure 4

Spatiotemporal diagrams of reactive trajectories in the SPDE model computed using AMS. (a) Collapse of an isolated puff at $r=1$. (b) Splitting of a puff at $r=0.95$. (c) Collapse of a domain of length $L=1200$, initially with steady turbulent fraction, at $r=1.1$. (d) Collapse of a domain of length $L=1200$, initially with steady turbulent fraction, at $r=1.2$. (e) Color levels of the logarithm of turbulence intensity for a build-up of a turbulent puff out of noise of vanishing variance at $r=1.15$ and $\mathrm{\Sigma }=25000$.

Figure 5

Normalized distribution of duration of collapses of isolated puffs in pipes of lengths $200\le L\le 1600$ at $r=1.0$, sampled by mean of AMS and DNS, compared to a normalized Gumbel distribution.

Figure 6

(a) Decimal logarithm of the mean first passage time of several events as a function of the Reynolds number $r$ in a pipe of length $L=800$: collapse fixed puffs (lines), collapse of near equilibrium puffs (red lozenges), as well as splitting of near equilibrium puff (red dots). The vertical dashed lines indicate the range of laminar turbulent coexistence [Fig. 2]. (b) Measured slopes $A_p$ (blue asterisks) and ordinate at the origin $B_p$ (red lozenges) for equilibrium puffs at different Reynolds numbers as a function of the total intensity of turbulence of said puffs $Q$. The dashed lines indicate the affine fits of these data.

Figure 7

(a) Logarithm of the mean first passage time before collapse of turbulence as a function of length at $r=1.15$ in pipes initially containing either the steady turbulent fraction or an isolated puff. The black dashed line is the linear fit of $ln\left[T\right(L\left)\right]$. (b) Logarithm of the mean first passage time before collapse of turbulence as a function of length for increasing $r$ in pipes initially containing steady turbulent faction. (c) Slope $f\left(r\right)$ of the linear fit of the logarithm of the mean first passage time before collapse of turbulence as a function of the Reynolds number. (d) Logarithm $ln\left(\tau \right)$ of duration of reactive trajectories of collapse of laminar-turbulent pipes as a function of pipe length for increasing Reynolds number. The error bars are determined using the variance of the estimate of $\tau$ over several realizations of AMS computations.

Figure 8

Rate function $I$ of the mean first passage time before collapse of a laminar-turbulent domain as a function of $\beta$ [the inverse variance of the multiplicative noise of Eq. (3)], calculated by two manners: ratio of $ln\left(T\right)/\beta$ and slope of the best linear fir of $ln\left(T\right)\left(\beta \right)$. The computation is performed using a domain length of $L=200$ and a Reynolds number of $r=1.3$.

Figure 9

Rescaled probability density function $\rho$ of the kinetic energy of turbulence $e$ sampled in numerical simulations of plane Couette flow (from the data of [3]).

Figure 10

Bifurcation diagram of the deterministic two degrees of freedom model, as a function of Reynolds number $R$, including both $X_1$ and $X_2$. The continuous lines indicate the stable fixed points. The dashed lines indicate the unstable fixed points. The green dot indicates the saddle node bifurcation.

Figure 11

(a) Color levels of the probability density function $\rho (X_1,X_2)$ obtained by solving the Fokker-Planck equation (26) with $a=0$ (additive noise), $\overline{\beta }=200$, $R=2.1$. (b) ${min}_{X_1}[-(1/\overline{\beta })ln\left(\rho \right)]$ for increasing values of $\overline{\beta }$ at $R=2.1$ and $a=0$ (additive noise), along with asymptotic quasipotential. (c) Asymptotic quasipotentials calculated analytically [Eqs. (32), (34), and (36)] for $R=4$ and all three noise cases (labeled “asy,” $a=0$, additive, and $a=1$, $a=\frac{1}{2}$ multiplicative). On top of it is added the quasipotential calculated numerically by solving the Fokker-Planck equation (labeled “FP,” $a=0$) and the quasipotentials evaluated numerically by integration of the SDEs for all three noise cases (labeled “num”).

Figure 12

Reactive trajectories in the two degrees of freedom model at $R=2.5$, showing build-up of turbulence under additive noise ($\overline{\beta }=10000$), collapse of turbulence under additive noise ($\overline{\beta }=1500$), the starting and ending points of trajectories, the relaxation paths from the saddle point $X_{1,2}^{+}$ to the nodes $X_{1,2}^{-}$ and 0 and the $X_1$ nullcline $X_1=[X_2/(X_2-1/R)]$.

Figure 13

(a) Logarithm of the mean first passage time before exit of state $X_{1,2}^{-}$ as a function of $\overline{\beta }$ in the two degrees of freedom model for increasing Reynolds number $R$ in the three noise cases. (b) Growth rate $\overline{f}\left(R\right)$ of the mean first passage time before exit of state $X_{1,2}^{-}$ as a function of Reynolds number $R$ in the two degrees of freedom model for additive noise.

Figure 14

(a) Probability ${\langle \alpha \rangle }_o$ of observing a collapse estimated after a sum over the number of independent repetitions of AMS as a function of $o$ at $r=0.85$, for several $N$. (b) Estimate of the probability of observing a collapse $\alpha$ (red curve) as a function of the number of clones used in AMS computations, with large $o$. The blue curve indicates the probability as estimated by DNS ${\alpha }_{\mathrm{DNS}}$. The error bars indicate the 95% confidence interval computed using the variance over repetitions. The dashed blue lines indicate the 95% confidence interval for estimation of ${\alpha }_{\mathrm{DNS}}$ by DNS. (c) Relative variance of the estimate of the probability of observing a collapse or a splitting as a function of $N$ for several $r$. (d) Relative difference between the estimate of $\alpha$ by AMS and DNS for collapse and splitting events as a function of $N$ for several $r$.

Figure 15

Normalized variance $\langle {(\mathfrak{Q}-\langle \mathfrak{Q}\rangle )}^2\rangle Ldt$ sampled in direct numerical simulations of the model (while no collapse occurred) as a function of the domain size, for several Reynolds numbers.

Figure 16

(a) Time series of $X_1,X_2$ obtained from numerical integration of the coupled SDEs in the $a=\frac{1}{2}$ case (square root multiplicative noise) at $\overline{\beta }=5$ and $R=3$. (b) Color levels of probability density function ${\rho }_{a=\frac{1}{2}}(X_1,X_2)$ sampled numerically by integration of the SDEs for $R=3$ and $\overline{\beta }=5$.