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On bifurcations and traction forces on an obstacle in incompressible flow

Jakub Cach, Karel Tůma, Jan Blechta, Sebastian Schwarzacher

TL;DR

The paper investigates bifurcations in 2D incompressible Navier–Stokes flow past a confined cylinder (Schäfer–Turek benchmark) up to Re = 1000, showing that changes in steady traction profiles on the obstacle correlate with long-time transitions such as Hopf bifurcations, symmetry breaking, and the emergence of multiple steady states.Using a combination of deflated continuation to uncover multiple steady branches, linear perturbation theory to hint at instabilities, and direct simulations to identify long-time attractors, the authors establish that traction diagnostics on the upstream obstacle provide a computationally inexpensive indicator of regime changes that precede and reflect unsteady dynamics.While linear perturbation theory reliably detects the first Hopf bifurcation and predicts associated frequencies, its predictive power diminishes for later transitions, whereas traction profiles consistently reveal localized, upstream-initiated changes aligned with the observed bifurcations and attractor transitions, including the onset of a Kármán vortex street at high Re.

Abstract

We present a systematic numerical investigation of bifurcations in the two-dimensional incompressible Navier-Stokes flow past a confined circular cylinder. The results indicate that there is a qualitative correspondence between changes in the traction profiles of the steady Navier-Stokes equations and bifurcations of the long-time behavior of the unsteady Navier-Stokes equations. The bifurcations include the appearance of symmetry breaking, oscillations, and multiple steady solutions. The well-known planar Schäfer-Turek benchmark is considered with Reynolds numbers up to 1000. For the analysis of bifurcations and traction profiles, several numerical strategies are applied, including a duality method for computing traction profiles, deflation methods, and linear stability analysis. Long-time flow behavior is often explored through direct numerical simulation of the unsteady equations; an approach that is computationally demanding. The relations presented here indicate the possibility of a computationally inexpensive strategy to detect critical Reynolds numbers.

On bifurcations and traction forces on an obstacle in incompressible flow

TL;DR

The paper investigates bifurcations in 2D incompressible Navier–Stokes flow past a confined cylinder (Schäfer–Turek benchmark) up to Re = 1000, showing that changes in steady traction profiles on the obstacle correlate with long-time transitions such as Hopf bifurcations, symmetry breaking, and the emergence of multiple steady states.Using a combination of deflated continuation to uncover multiple steady branches, linear perturbation theory to hint at instabilities, and direct simulations to identify long-time attractors, the authors establish that traction diagnostics on the upstream obstacle provide a computationally inexpensive indicator of regime changes that precede and reflect unsteady dynamics.While linear perturbation theory reliably detects the first Hopf bifurcation and predicts associated frequencies, its predictive power diminishes for later transitions, whereas traction profiles consistently reveal localized, upstream-initiated changes aligned with the observed bifurcations and attractor transitions, including the onset of a Kármán vortex street at high Re.

Abstract

We present a systematic numerical investigation of bifurcations in the two-dimensional incompressible Navier-Stokes flow past a confined circular cylinder. The results indicate that there is a qualitative correspondence between changes in the traction profiles of the steady Navier-Stokes equations and bifurcations of the long-time behavior of the unsteady Navier-Stokes equations. The bifurcations include the appearance of symmetry breaking, oscillations, and multiple steady solutions. The well-known planar Schäfer-Turek benchmark is considered with Reynolds numbers up to 1000. For the analysis of bifurcations and traction profiles, several numerical strategies are applied, including a duality method for computing traction profiles, deflation methods, and linear stability analysis. Long-time flow behavior is often explored through direct numerical simulation of the unsteady equations; an approach that is computationally demanding. The relations presented here indicate the possibility of a computationally inexpensive strategy to detect critical Reynolds numbers.

Paper Structure

This paper contains 19 sections, 1 theorem, 25 equations, 27 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Problem description
  3. Schäfer--Turek benchmark
  4. Branching, multiplicity, and linear perturbation theory
  5. Relevance for the mathematical theory of the Navier--Stokes equations
  6. Critical Reynolds numbers
  7. Numerical methods
  8. Unsteady Schäfer--Turek benchmark observations
  9. Flow at low Reynolds number (Re < 7)
  10. Steady unique flow (Re < 48)
  11. The flow with different long-time and steady solutions (Re >= 48)
  12. Multiple steady solutions (Re > 315)
  13. Unsteady solution
  14. Kármán vortex street (Re >= 950)
  15. Global visualization of turning points
  16. ...and 4 more sections

Key Result

Theorem 1

Suppose that eq:laplace admits $H^{3/2+\varepsilon}$ regularity, i.e., there exist $\varepsilon>0$ and $c>0$ such that the solution operator $T\colon f\mapsto u$ for eq:laplace satisfies Then there exists $C>0$ such that where $\delta_{k,1}$ denotes the Kronecker delta and $B^s_{p,q}(\Omega)$ are the Besov spaces.

Figures (27)

  • Figure 1: The Schäfer--Turek benchmark. The lower left corner is at $(0\,\mathrm{m}, 0\,\mathrm{m})$ and the cylinder center is located at $(0.2\,\mathrm{m}, 0.2\,\mathrm{m})$ with radius $R = 0.05\,\mathrm{m}$. The domain is slightly asymmetric. The benchmark does not require use of a specific outflow condition.
  • Figure 2: The upper row illustrates the evolution of long-time flow regimes in the Schäfer--Turek benchmark as the Reynolds number increases, governed by the time-dependent Navier–Stokes equations. Transitions include the onset of periodic shedding with the formation of a vortex street, symmetry breaking of the street, and the eventual development of a Kármán street. The lower row marks the Reynolds numbers at which the traction profile of the steady solution undergoes characteristic change in either the lift or the drag on the upstream face of the cylinder. These qualitative changes in the steady profiles align strikingly with transitions in the time-dependent dynamics, suggesting that bifurcations in the (possibly unstable) steady solution correspond to global-in-time regime changes in the unsteady flow.
  • Figure 3: Traction profiles across the range of Reynolds numbers where the transition from low Reynolds number flow to flow with a vortex wake occurs
  • Figure 4: Velocity field at $\mathrm{Re}=9$. Lift profiles (shown using warp-by-scalar plot, with reversed sign for plotting convenience) at the transition from stationary vortex-free flow ($\mathrm{Re}=7$, in shades of green) to stationary flow with a vortex wake ($\mathrm{Re}=9$, in shades of gray); mind the pair of small vortices past the cylinder downstream. The turning point ($\theta = 85^\circ$, $\mathrm{Re}=7$, in yellow), where the lift is stationary with respect to $\mathrm{Re}$.
  • Figure 5: Velocity field on the downstream side of the cylinder in line integral convolution plot. A pair of stationary vortices appears at $\mathrm{Re}=7$ and grows in size with increasing $\mathrm{Re}$.
  • ...and 22 more figures

Theorems & Definitions (4)

  • Remark : Remark on generality
  • Remark : Discussion on LPT
  • Theorem : Horger, Melenk, and Wohlmuth HorgerMelenkWohlmuth2013
  • Remark : Superconvergence of DVND