Analysis and modeling of localized invariant solutions in pipe flow

Turbulent spots surrounded by laminar flow are a landmark of transitional shear flows, but the dependence of their kinematic properties on spatial structure is poorly understood. We here investigate this dependence in pipe flow for Reynolds numbers between 1500 and 5000. We compute spatially localized relative periodic orbits in long pipes and show that their upstream and downstream fronts decay exponentially towards the laminar profile. This allows us to model the fronts by employing the linearized Navier-Stokes equations, and the resulting model yields the spatial decay rate and the front velocity profiles of the periodic orbits as a function of Reynolds number, azimuthal wave number, and propagation speed. In addition, when applied to a localized turbulent puff, the model is shown to accurately approximate the spatial decay rate of its upstream and downstream tails. Our study provides insight into the relationship between the kinematics and spatial structure of localized turbulence and more generally into the physics of localization.

Figure 1

Structure of the investigated localized reflection-symmetric relative periodic solutions obtained by directly solving the Navier-Stokes equation with a Newton-Krylov method at $\mathrm{Re}=3000$ for (a) twofold (${\mathrm{LB}}_2$) and (b) threefold (${\mathrm{LB}}_3$) rotational symmetry. The central image shows isosurfaces and cross sections of the streamwise velocity disturbance (i.e., with the laminar flow subtracted). Red (blue) streaks are faster (slower) than the base flow. In order to highlight the tails of the solutions, the isovalues have been chosen to be small: $\pm 0.025U_{\mathrm{cl}}$. The axial extent of the shown isosurfaces is about $60R$. The upper right panel shows the upstream front of the same state, which appears shorter due to a different perspective.

Figure 2

Infinity norm (maximum) of the disturbance velocity components as a function of the axial position at $\mathrm{Re}=3000$ for (a) ${\mathrm{LB}}_2$ and (b) ${\mathrm{LB}}_3$.

Figure 3

Comparison of the spatial structure of the three velocity components for ${\mathrm{LB}}_3$ at $\mathrm{Re}=3000$. The color coding is analogous to Fig. 1 and the isovalues are given in parentheses: (a) $u_r$ ($\pm 5\times {10}^{-5}{U}_{\mathrm{cl}}$), (b) $u_{\theta }$ ($\pm 5\times {10}^{-6}{U}_{\mathrm{cl}}$), and (c) $u_z$ ($\pm 0.015U_{\mathrm{cl}}$).

Figure 4

Axial profiles of the infinity norm of the axial velocity disturbance for different Reynolds numbers for (a) ${\mathrm{LB}}_2$ and (b) ${\mathrm{LB}}_3$.

Figure 5

Reynolds-number dependence of the group velocity $c$ with which the localized solutions ${\mathrm{LB}}_2$ (circles) and ${\mathrm{LB}}_3$ (squares) are advected downstream. The dashed lines are to guide the eye.

Figure 6

Axial decay rate $\mu$ of $u_z$ for the localized solutions as a function of Reynolds number. Circles (${\mathrm{LB}}_2$) and squares (${\mathrm{LB}}_3$) denote DNS results. The error margins have been produced by trying several axial ranges for which the respective tail was fitted. Note that, except for the upstream tail of ${\mathrm{LB}}_3$, the error is smaller than the marker size and hence is barely visible. Dashed lines denote the decay rates obtained from the EVP (6). The downstream fronts have $m=0$ and the upstream fronts $m=4$ for ${\mathrm{LB}}_2$ and $m=6$ for ${\mathrm{LB}}_3$. Dotted lines show the result of the advection-diffusion equation (8) for the upstream front (see Sec. 4c).

Figure 7

Comparison of radial profiles of (a) and (b) $u_r$, (c) and (d) $u_{\theta }$, and (e) and (f) $u_z$ obtained from DNS (dashed lines) and the LNSE (solid lines) for ${\mathrm{LB}}_3$ at $\mathrm{Re}=3000$. The left-hand-side profiles are located upstream and the right-hand-side ones downstream.

Figure 8

Axial profiles of the infinity norm (maximum) for each individual term in the axial component of the linearized Navier-Stokes equations at $\mathrm{Re}=3000$ for ${\mathrm{LB}}_3$. Profiles are very similar for ${\mathrm{LB}}_2$ and hence are not shown here.

Figure 9

Axial profiles of the infinity norm of the three velocity components for a turbulent puff at $\mathrm{Re}=2000$ in a pipe of $360R$ in length. The orange solid lines show the spatial decay rates obtained from the LNSE, where $m=1$ and $m=0$ with $c/U_{\mathrm{cl}}=0.5$ were used to solve the EVP (6) for the upstream and downstream tails, respectively. The dotted vertical lines indicate the locations of the velocity profiles shown in Fig. 11.

Figure 10

Axial profiles of the infinity norm of the axial velocity $u_z$ and its first four Fourier modes for the puff in Fig. 9. The black dotted line denotes the axial position of the cross section in Fig. 12. Only half of the $360R$-long pipe is shown. The inset compares the decay rates of DNSs and the LNSE as a function of the azimuthal mode $m$.

Figure 11

Comparison of radial profiles of $u_z$ obtained from DNS (dashed lines) and the LNSE model (solid lines) for a puff at $\mathrm{Re}=2000$. The left-hand-side profiles are located upstream and the right-hand-side one downstream, the axial positions marked by dotted lines in Fig. 9. In the model, $m=0$ and $m=1,2$ with $c/U_{\mathrm{cl}}=0.5$ were used to solve the EVP (6) for the downstream and upstream tails, respectively.

Figure 12

Cross section of the velocity field of the puff in Fig. 9. The axial position is marked by a dashed line in Fig. 10 and is the same as where the upstream radial profile in Fig. 11 was plotted. The in-plane components are denoted by vectors and the axial velocity is shown by color.