Global stability of ${180}^{\circ }$-bend pipe flow with mesh adaptivity

The global stability of the flow in a spatially developing ${180}^{\circ }$-bend pipe with curvature $\delta =R/R_c=1/3$ is investigated by performing direct numerical simulations to understand the underlying transitional mechanism. A unique application of the adaptive mesh refinement technique is used during the stability analysis for minimizing the interpolation and quadrature errors. Independent meshes are created for the direct and adjoint solutions, as well as for the base flow extracted via selective frequency damping. The spectrum of the linearized Navier-Stokes operator reveals a pair of complex conjugate eigenvalues, with frequency $f\approx 0.233$. Therefore, the transition is attributed to a Hopf bifurcation that takes place at ${\text{Re}}_{b,\text{cr}}=2528$. A structural sensitivity analysis is performed by extracting the wavemaker. We identify the primary source of instability located on the outer wall, $\theta \approx {15}^{\circ }$ downstream of the bend inlet. This region corresponds to the separation bubble on the outer wall. We thus conclude that the instability is caused by the strong shear resulting from the backflow, similar to the ${90}^{\circ }$-bend pipe flow. We believe that understanding the stability mechanism and characterizing the base flow in bent pipes is crucial for studying various biological flows, like blood vessels. Hence, this paper aims to close the knowledge gap between a ${90}^{\circ }$-bend and toroidal pipes by investigating the transition nature in a ${180}^{\circ }$-bend pipe flow.

Figure 1

Sketch of the ${180}^{\circ }$-bend pipe geometry with black arrows indicating the inflow and outflow directions. The reference systems are reported in the (left) symmetry and (right) cross planes.

Figure 2

Spectral-element grid for different meshes designed via AMR technique for the (a) nonlinear base flow, the linear (b) direct, and (c) dual solutions, respectively. From top to bottom: The symmetry plane ($y=0$) and the cross planes at the inlet ($\theta =0$) and outlet ($\theta ={180}^{\circ }$) of the bend. Blue isocontours of the velocity magnitude normalized between $[0,U_b]$ for the base flow and [0,1] (i.e., the maximum) for the linear and dual solutions. In the symmetry plane view, a shortened domain is shown with inflow and outflow lengths equal to $2D$. The black arrows indicate the base flow direction.

Figure 3

Velocity magnitude field computed via nonlinear DNS after 200 convective time units ($D/U_b$) at (a) ${\text{Re}}_b=2500$ and (b) ${\text{Re}}_b=3000$. The $xz$ symmetry plane and the $xy$ cross plane at $z=0$ (bend outlet) on the left and right, respectively. The black arrows indicate the inflow and outflow directions. The inner and outer walls are marked with letters I and O

Figure 4

(a) Isocontours of the streamwise velocity $u_s$ in the symmetry plane ($y=0$) for the base flow extracted via SFD at ${\text{Re}}_b=2550$. The blue three-dimensional isosurface for $u_s\le 0$ shows the three distinct recirculation regions, symmetric with respect to the $xz$ plane. The black arrows indicate the flow direction. (b) Phase-space representation of local velocity measurements of a probe located at ($x=-1.30D,y=-0.35D,z=-4.00D$) for the nonlinear DNS of the ${180}^{\circ }$-bend pipe flow at ${\text{Re}}_b=2550$.

Figure 5

Planar perspectives of the base flow extracted via SFD at the Reynolds number ${\text{Re}}_b=2550$ (curvature $\delta =1/3$). (a) The variation of velocity magnitude in circular cross sections of the bend is illustrated at various angles, including $\theta =0^{\circ }$, ${45}^{\circ }$, ${75}^{\circ }$, ${90}^{\circ }$, ${120}^{\circ }$, and ${180}^{\circ }$ (A, B, C, D, E, and F, respectively). In addition, the in-plane streamlines are shown at ${90}^{\circ }$ (panel $D$) and ${180}^{\circ }$ (panel $F$). (b), (c) Isocontours of velocity magnitude in the $xz$ plane at $y=0$ for the ${180}^{\circ }$ and ${90}^{\circ }$ bent pipes, respectively. (d) Lastly, the velocity magnitude and in-plane streamlines for the ${180}^{\circ }$-bend (in the first row) and ${90}^{\circ }$-bend (in the second row) are shown for G, H, J, and L, respectively, at various streamwise stations $s=1D$, $s=2D$, $s=4D$, and $s=8D$.

Figure 6

Isosurfaces of the (blue) negative and (red) positive streamwise velocity component $u_s$ of the perturbation solution for ${\text{Re}}_b=2550$ of the (left) direct and (right) dual problem. The values are in the range $[-0.1,0.1]$ of the maximum for the (top) ${180}^{\circ }$- and (bottom) ${90}^{\circ }$-bend pipes. For the sake of clarity, a shortened domain is shown with inflow and outflow lengths equal to $5D$ and $3D$, respectively. The black arrows indicate the flow direction. Note that the phases of the disturbances are arbitrary and thus not directly comparable between the two cases.

Figure 7

Portion of the spectrum for the direct problem at ${\text{Re}}_b=2550$ of the (cross) ${90}^{\circ }$ and (circle) ${180}^{\circ }$ bent pipe flows.

Figure 8

The growth of the amplitude of the limit cycle. From local velocity probes, the saturation amplitude, $A$, measured as a function of Reynolds number (red line). The nonlinear DNS data (red squares) fit the function $A\sim \sqrt{{\text{Re}}_b-{\text{Re}}_{b,cr}}$. The intercept (black square) of the red line with the horizontal axis (black line) provides a critical Reynolds number approximately equal to 2538.

Figure 9

Structural sensitivity map of the ${180}^{\circ }$-bend pipe flow normalized between 0 and 1. (a) Three-dimensional isosurface of $\eta =0.65$ in a shortened domain, starting from $\theta =0^{\circ }$ with outflow length equal to $2D$. (b) Two-dimensional isocontours of $\eta$ (top) in the symmetry plane for $\theta$ between $0^{\circ }$ and ${90}^{\circ }$ and (bottom) in the cross plane at $\theta ={15}^{\circ }$. The black arrows indicate the flow direction.