Influence of inlet flow profiles on swirling flow dynamics in a finite-length pipe

The influence of various inlet swirling flow profiles on the manifold of steady axisymmetric states of flows in a straight, long circular pipe of a finite length $L$ and on flow dynamics is investigated. The study is motivated by the bifurcation diagrams originally studied by Leclaire and Sipp [B. Leclaire and D. Sipp, J. Fluid Mech. 645, 81 (2010)] and explores, in all of their inflow cases, the relationship between bifurcation and flow evolution. Steady circumferential and axial velocities and azimuthal vorticity are prescribed at the inlet. A parallel flow state is set at the outlet. The outlet state of the steady problem may be determined by solutions of the axially independent Squire-Long equation. For each of the incoming flows studied, the solutions include the base columnar flow solution, a decelerated flow along the centerline, an accelerated flow along the centerline, a vortex-breakdown solution, and a wall-separation solution. These solutions correspond to respective steady states in the pipe. Branches of higher-order noncolumnar states that bifurcate at a sequence of critical swirl levels are also presented. The theoretical predictions are numerically realized by unsteady and inviscid flow simulations. The simulations clarify the base flow stability and the dynamics of initial perturbations to various attracting states. Results demonstrate that, depending on the inlet flow profile, the global minimum state of the energy function $\mathcal{E}$ of the problem turns at a certain critical swirl level, denoted by ${\omega }_0+\epsilon \left(L\right)$ [where $0<\epsilon \left(L\right)\ll 1$ and tends to zero with an increase of $L]$, from the columnar state to become a centerline-decelerated flow state, a vortex-breakdown state, or a wall-separation state. The jump of the global minimum state of $\mathcal{E}$ at $\omega ={\omega }_0+\epsilon \left(L\right)$ together with the critical swirl of the columnar state at $\omega ={\omega }_1$ govern the evolution of pipe vortex flows. This analysis provides understanding of all possible axisymmetric flow evolution processes for various inlet profiles and swirl levels, and the domain of attraction of crucial states.

Figure 1

Case 1: $b=1.6978,d=1.6263$, and $L=10$. Here ${\omega }_B={\omega }_0=3.7900$ and ${\omega }_0+\epsilon \left(L\right)={\omega }_1=3.7925$. (a) Bifurcation diagrams of solutions ${\psi }_{cy}\left(0\right),{\psi }_{sy}(L,0)$, and ${\psi }_y(L,0,t\gg 1)$ as a function of $\omega$. (b) Bifurcation diagram of $y_0$ and $y_w$ as a function of $\omega$. The solid black lines represent solutions of the columnar Squire-Long (ODE) problem. The yellow circles represent the columnar critical state at ${\omega }_B$, and the states at ${\omega }_{BD}$ for the first appearance of vortex breakdown and at ${\omega }_{WS}$ for the first appearance of wall separation according to the columnar ODE problem. The red lines represent Leclaire and Sipp [24] solutions of the Squire-Long (PDE) problem, where solid lines are asymptotically stable solutions and the dashed line is the unstable solution. The red circles represent the critical state at ${\omega }_1$, the state for the first appearance of vortex breakdown (BD) at ${\omega }_{BD}+{\epsilon }_{BD}\left(L\right)$ and the state for the first appearance of wall separation (WS) at ${\omega }_{WS}+{\epsilon }_{WS}\left(L\right)$ according to Leclaire and Sipp [24]. The open circles represent time-asymptotic results from the numerical simulations of flow evolution. The black dashed lines represent unstable states according to the weakly nonlinear theory of Rusak et al. [23].

Figure 2

Case 1: $b=1.6978,d=1.6263$, and $L=10$. (a) Contours of the time-asymptotic solution of $\psi (x,y,t\gg 1)$ of the vortex breakdown state at $\omega =3.81$. (b) Plot of ${\psi }_c\left(y\right)$ (solid line) and $\psi (L,y,t\gg 1)$ (circles) as a function of $y$ of the vortex breakdown state at $\omega =3.81$. (c) Contours of the time-asymptotic solution of $\psi (x,y,t\gg 1)$ of the wall-separation state at $\omega =3.81$. (d) Plot of ${\psi }_c\left(y\right)$ (solid line) and $\psi (L,y,t\gg 1)$ (circles) as a function of $y$ of the wall-separation state at $\omega =3.81$. In (a) and (c) flow runs from left to right, the left side is the pipe inlet and the right side is the outlet, the bottom side is the centerline and the top side is the wall, and there are 26 equispaced lines from 0 to 0.5. In (b) and (d) the results are close to each other.

Figure 3

Case 1: $b=1.6978,d=1.6263$, and $L=10$. (a) Lines of axial velocity along the centerline ${\psi }_y(x,0,t)$ vs $x$ at various times during the evolution of the flow from a perturbed columnar state (with $\delta <0)$ to a vortex breakdown state at $\omega =3.81$. (b) Lines of axial velocity along the centerline ${\psi }_y(x,0,t)$ vs $x$ at various times during the evolution of the flow from a perturbed columnar state (with $\delta >0)$ to a wall-separation state at $\omega =3.81$. The arrows in both panels show the direction of evolution in time.

Figure 4

Bifurcation diagrams of solutions ${\psi }_{cy}\left(0\right),{\psi }_{sy}(L,0)$, and ${\psi }_y(L,0,t\gg 1)$ as a function of $\omega$ for case 2: $b=6,d=4.386$, and $L=10$. Here ${\omega }_0=11.4390,{\omega }_B=12.2758$, and ${\omega }_1=12.2796$. The solid black lines represent solutions of the columnar Squire-Long (ODE) problem. The yellow circles represent the columnar critical state at ${\omega }_B$ and the state at ${\omega }_{BD}$ for the first appearance of vortex breakdown according to the columnar ODE problem. The red lines represent Leclaire and Sipp [24] solutions of the Squire-Long (PDE) problem, where solid lines are asymptotically stable solutions and the dashed line is the unstable solution. The red circles represent the critical state at ${\omega }_1$ and the state for the first appearance of vortex-breakdown at ${\omega }_{BD}+{\epsilon }_{BD}\left(L\right)$ according to Leclaire and Sipp [24]. The open circles represent time-asymptotic results from the numerical simulations of flow evolution. The black dashed lines represent unstable states according to the weakly nonlinear theory of Rusak et al. [23].

Figure 5

Bifurcation diagrams of solutions ${\psi }_{cy}\left(0\right),{\psi }_{sy}(L,0)$, and ${\psi }_y(L,0,t\gg 1)$ as a function of $\omega$ for case 3: $b=6,d=5.417$, and $L=10$. Here ${\omega }_0=12.198,{\omega }_B=12.5349$, and ${\omega }_1=12.5378$. See the caption of Fig. 1 for details about the various lines and points in this figure.

Figure 6

Bifurcation diagrams of solutions ${\psi }_{cy}\left(0\right),{\psi }_{sy}(L,0)$, and ${\psi }_y(L,0,t\gg 1)$ as a function of $\omega$ for case 4: $b\to 0,d=1.636$, and $L=10$. Here ${\omega }_0=1.6539,{\omega }_B=1.9093$, and ${\omega }_1=1.9106$. See the caption of Fig. 1 for details about the various lines and points in this figure.