Steady solutions of the Navier-Stokes equations by selective frequency damping

Åkervik, Espen; Brandt, Luca; Henningson, Dan S.; Hœpffner, Jérôme; Marxen, Olaf; Schlatter, Philipp

doi:10.1063/1.2211705

Cited by 287 publications

(167 citation statements)

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“…(8)- (10) using the Newton-type methods, but this requires a specific solver. In order to efficiently compute the base flow by slightly revising an existing Navier-Stokes solver, Åkervik et al 39 proposed the SFD method to obtain the steady solution of the Navier-Stokes equations. In this method, a forcing term and a time derivative term are added to the momentum equation, …”

Section: B Base Flowmentioning

confidence: 99%

“…The numerical methods employed are free of any assumption or turbulence model, thus both the base flow and the perturbation characteristics are obtained following the original (linearized) Navier-Stokes equations. The base flows are obtained using an in-house finite difference code in combination with the selective frequency damping (SFD) method 39 and satisfy the steady-state Navier-Stokes equations. We perform BiGlobal linear stability analysis on the flow past a NACA0012 airfoil at AoA = 16…”

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BiGlobal linear stability analysis on low-Re flow past an airfoil at high angle of attack

Zhang

Samtaney

2016

Physics of Fluids

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CitationBiGlobal linear stability analysis on low-Re flow past an airfoil at high angle of attack 2016, 28 (4) We perform BiGlobal linear stability analysis on flow past a NACA0012 airfoil at 16• angle of attack and Reynolds number ranging from 400 to 1000. The steady-state two-dimensional base flows are computed using a well-tested finite difference code in combination with the selective frequency damping method. The base flow is characterized by two asymmetric recirculation bubbles downstream of the airfoil whose streamwise extent and the maximum reverse flow velocity increase with the Reynolds number. The stability analysis of the flow past the airfoil is carried out under very small spanwise wavenumber β = 10 −4 to approximate the two-dimensional perturbation, and medium and large spanwise wavenumbers ( β = 1-8) to account for the three-dimensional perturbation. Numerical results reveal that under small spanwise wavenumber, there are at most two oscillatory unstable modes corresponding to the near wake and far wake instabilities; the growth rate and frequency of the perturbation agree well with the two-dimensional direct numerical simulation results under all Reynolds numbers. For a larger spanwise wavenumber β = 1, there is only one oscillatory unstable mode associated with the wake instability at Re = 400 and 600, while at Re = 800 and 1000 there are two oscillatory unstable modes for the near wake and far wake instabilities, and one stationary unstable mode for the monotonically growing perturbation within the recirculation bubble via the centrifugal instability mechanism. All the unstable modes are weakened or even suppressed as the spanwise wavenumber further increases, among which the stationary mode persists until β = 4. C 2016 AIP Publishing LLC. [http://dx

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Section: B Base Flowmentioning

confidence: 99%

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confidence: 99%

BiGlobal linear stability analysis on low-Re flow past an airfoil at high angle of attack

Zhang

Samtaney

2016

Physics of Fluids

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“…As the steady solution of Navier-Stokes equations above critical Reynolds numbers might be difficult to obtain (although methods like selective frequency damping [1,6] are an option), another ways of the shiftmode computation might be considered:…”

Section: Appendix: Shift Modementioning

confidence: 99%

On the need of mode interpolation for data-driven Galerkin models of a transient flow around a sphere

Stankiewicz

Morzyński

Kotecki

et al. 2016

Theor. Comput. Fluid Dyn.

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We present a low-dimensional Galerkin model with state-dependent modes capturing linear and nonlinear dynamics. Departure point is a direct numerical simulation of the three-dimensional incompressible flow around a sphere at Reynolds numbers 400. This solution starts near the unstable steady Navier-Stokes solution and converges to a periodic limit cycle. The investigated Galerkin models are based on the dynamic mode decomposition (DMD) and derive the dynamical system from first principles, the Navier-Stokes equations. A DMD model with training data from the initial linear transient fails to predict the limit cycle. Conversely, a model from limit-cycle data underpredicts the initial growth rate roughly by a factor 5. Key enablers for uniform accuracy throughout the transient are a continuous mode interpolation between both oscillatory fluctuations and the addition of a shift mode. This interpolated model is shown to capture both the transient growth of the oscillation and the limit cycle.

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“…However, using the DNS, the residual cannot be reduced to machine zero when computing the base flow at supercritical as well as at slightly subcritical Reynolds numbers since some frequencies present in the numerical noise are amplified. In these cases, several approaches may be employed to compute the base flow based on filtering techniques 26 or on continuation methods. Here, a time-stepping continuation method has been employed.…”

Section: B Newton Procedures For the Base Flowmentioning

confidence: 99%

The effects of non-normality and nonlinearity of the Navier–Stokes operator on the dynamics of a large laminar separation bubble

2010

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. The effects of non-normality and nonlinearity of the Navier-Stokes operator on the dynamics of a large laminar separation bubble. The effects of non-normality and nonlinearity of the Navier-Stokes operator on the dynamics of a large laminar separation bubble, 2010, 22 (014102), pp.15. <10.1063/1.3276903>. The effects of non-normality and nonlinearity of the Navier-Stokes operator on the dynamics of a large laminar separation bubble The effects of non-normality and nonlinearity of the two-dimensional Navier-Stokes differential operator on the dynamics of a large laminar separation bubble over a flat plate have been studied in both subcritical and slightly supercritical conditions. The global eigenvalue analysis and direct numerical simulations have been employed in order to investigate the linear and nonlinear stability of the flow. The steady-state solutions of the Navier-Stokes equations at supercritical and slightly subcritical Reynolds numbers have been computed by means of a continuation procedure. Topological flow changes on the base flow have been found to occur close to transition, supporting the hypothesis of some authors that unsteadiness of separated flows could be due to structural changes within the bubble. The global eigenvalue analysis and numerical simulations initialized with small amplitude perturbations have shown that the non-normality of convective modes allows the bubble to act as a strong amplifier of small disturbances. For subcritical conditions, nonlinear effects have been found to induce saturation of such an amplification, originating a wave-packet cycle similar to the one established in supercritical conditions, but which is eventually damped. A transient amplification of finite amplitude perturbations has been observed even in the attached region due to the high sensitivity of the flow to external forcing, as assessed by a linear sensitivity analysis. For supercritical conditions, the non-normality of the modes has been found to generate low-frequency oscillations ͑flapping͒ at large times. The dependence of such frequencies on the Reynolds number has been investigated and a scaling law based on a physical interpretation of the phenomenon has been provided, which is able to explain the onset of a secondary flapping frequency close to transition.

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Steady solutions of the Navier-Stokes equations by selective frequency damping

Cited by 287 publications

References 10 publications

BiGlobal linear stability analysis on low-Re flow past an airfoil at high angle of attack

BiGlobal linear stability analysis on low-Re flow past an airfoil at high angle of attack

On the need of mode interpolation for data-driven Galerkin models of a transient flow around a sphere

The effects of non-normality and nonlinearity of the Navier–Stokes operator on the dynamics of a large laminar separation bubble

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