The stability of steady and time-dependent plane Poiseuille flow

Grosch, C. E.; Salwen, Harold

doi:10.1017/s0022112068001837

Cited by 125 publications

(51 citation statements)

References 13 publications

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“…It is found that the sinusoidally pulsating flow is more stable than the steady plane Poiseuille flow for a range of frequencies greater than about Wo = 12. Lower or much higher frequencies were found to make the flow unstable, in contrast with the results of Grosch & Salwen (1968). The perturbation analysis also confirms the result obtained by Hall (1975) that the growth rate depends quadratically on small pulsating amplitudes.…”

Section: Literature Reviewsupporting

confidence: 84%

“…However, at low frequencies, the perturbation energy may vary by several orders of magnitude within each cycle. These authors confirm findings by von Kerczek (1982) and suspect that those by Grosch & Salwen (1968) are underresolved. They are also probably the first to attempt a nonlinear simulation.…”

Section: Literature Reviewsupporting

confidence: 80%

“…Surprisingly, apart from a few recent computations of turbulent periodic flows, the nonlinear regime has not yet attracted much theoretical or numerical attention. Grosch & Salwen (1968) were among the first to address the linear stability of timedependent plane Poiseuille flow, by expanding the disturbance streamfunction on a small set of basis functions. They found that for weak pressure gradient modulations, the resulting modulated flow was more stable than the steady flow, while a rather drastic destabilization was observed at higher velocity modulations.…”

Section: Literature Reviewmentioning

confidence: 99%

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Linear and nonlinear dynamics of pulsatile channel flow

Pier

Schmid

2017

J. Fluid Mech.

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The dynamics of small-amplitude perturbations as well as the regime of fully developed nonlinear propagating waves is investigated for pulsatile channel flows. The timeperiodic base flows are known analytically and completely determined by the Reynolds number Re (based on the mean flowrate), the Womersley number Wo (a dimensionless expression of the frequency) and the flowrate waveform. This paper considers pulsatile flows with a single oscillating component and hence only three non-dimensional control parameters are present. Linear stability characteristics are obtained both by Floquet analyses and by linearized direct numerical simulations. In particular, the long-term growth or decay rates and the intracyclic modulation amplitudes are systematically computed. At large frequencies (mainly Wo 14), increasing the amplitude of the oscillating component is found to have a stabilizing effect, while it is destabilizing at lower frequencies; strongest destabilization is found for Wo ≃ 7. Whether stable or unstable, perturbations may undergo large-amplitude intracyclic modulations; these intracyclic modulation amplitudes reach huge values at low pulsation frequencies. For linearly unstable configurations, the resulting saturated fully developed finite-amplitude solutions are computed by direct numerical simulations of the complete Navier-Stokes equations. Essentially two types of nonlinear dynamics have been identified: "cruising" regimes for which nonlinearities are sustained throughout the entire pulsation cycle and which may be interpreted as modulated Tollmien-Schlichting waves, and "ballistic" regimes that are propelled into a nonlinear phase before subsiding again to small amplitudes within every pulsation cycle. Cruising regimes are found to prevail for weak baseflow pulsation amplitudes, while ballistic regimes are selected at larger pulsation amplitudes; at larger pulsation frequencies, however, the ballistic regime may be bypassed due to the stabilizing effect of the base-flow pulsating component. By investigating extended regions of a multi-dimensional parameter space and considering both two-dimensional and threedimensional perturbations, the linear and nonlinear dynamics are systematically explored and characterized.

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Section: Literature Reviewsupporting

confidence: 84%

Section: Literature Reviewsupporting

confidence: 80%

Section: Literature Reviewmentioning

confidence: 99%

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Linear and nonlinear dynamics of pulsatile channel flow

Pier

Schmid

2017

J. Fluid Mech.

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“…According to the results of figure 13, the inner mode can be stabilized, albeit inappreciably, in two regimes: (i) steady, high-amplitude streaks which cause a stabilizing modification to the mean flow, and (ii) low-amplitude, higher-frequency streaks. The enhanced stability for high-frequency, low-amplitude streaks can be compared to the work of Grosch & Salwen (1968). They studied the stability of timedependent plane Poiseuille flow.…”

Section: The Inner Modementioning

confidence: 99%

Stability of zero-pressure-gradient boundary layer distorted by unsteady Klebanoff streaks

2011

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The secondary instability of a zero-pressure-gradient boundary layer, distorted by unsteady Klebanoff streaks, is investigated. The base profiles for the analysis are computed using direct numerical simulation (DNS) of the boundary-layer response to forcing by individual free-stream modes, which are low frequency and dominated by streamwise vorticity. Therefore, the base profiles take into account the nonlinear development of the streaks and mean flow distortion, upstream of the location chosen for the stability analyses. The two most unstable modes were classified as an inner and an outer instability, with reference to the position of their respective critical layers inside the boundary layer. Their growth rates were reported for a range of frequencies and amplitudes of the base streaks. The inner mode has a connection to the TollmienSchlichting (T-S) wave in the limit of vanishing streak amplitude. It is stabilized by the mean flow distortion, but its growth rate is enhanced with increasing amplitude and frequency of the base streaks. The outer mode only exists in the presence of finite amplitude streaks. The analysis of the outer instability extends the results of Andersson et al. (J. Fluid Mech. vol. 428, 2001, p. 29) to unsteady base streaks. It is shown that base-flow unsteadiness promotes instability and, as a result, leads to a lower critical streak amplitude. The results of linear theory are complemented by DNS of the evolution of the inner and outer instabilities in a zero-pressure-gradient boundary layer. Both instabilities lead to breakdown to turbulence and, in the case of the inner mode, transition proceeds via the formation of wave packets with similar structure and wave speeds to those reported by Nagarajan, Lele & Ferziger (J. Fluid Mech., vol. 572, 2007, p. 471).

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“…To that end, much study has been carried out in the past on the instability of confined flows such as Couette flow and/or Poiseuille flow for both Newtonian and non-Newtonian fluids alike. [2][3][4][5][6][7][8][9][10]) Surprisingly, however, instability of unbounded flows (e.g., Blasius flow) has not been addressed to the same extent [11][12][13][14][15] , and this is particularly so for viscoelastic fluids. The apparent lack of interest in studying instability of boundary layer flows may have originated from the fact that boundary layer problems are not mathematically so wellposed.…”

Section: Introductionmentioning

confidence: 99%

Hydromagnetic Instability of Viscoelastic Fluids in Blasius Flow

Khabazi

Sadeghy

2009

J. Soc. Rheol., Jpn

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The effect of a fluid's elasticity is investigated on the instability of Blasius flow at the presence of a transverse magnetic field. To determine the critical Reynolds number as a function of the elasticity and magnetic numbers, a twodimensional linear temporal stability analysis is used assuming that the viscoelastic fluid of interest obeys Walters' B model as its constitutive equation. Neglecting terms nonlinear in the perturbation quantities, an eigenvalue problem is obtained which is solved numerically using the Chebychev collocation method. Based on the results obtained in this work, fluid's elasticity is predicted to have a destabilizing effect on Blasius flow. In contrast, the effect of the magnetic field is predicted to be always stabilizing in Blasius flow, at least in the range of parameters used in this work.

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The stability of steady and time-dependent plane Poiseuille flow

Cited by 125 publications

References 13 publications

Linear and nonlinear dynamics of pulsatile channel flow

Linear and nonlinear dynamics of pulsatile channel flow

Stability of zero-pressure-gradient boundary layer distorted by unsteady Klebanoff streaks

Hydromagnetic Instability of Viscoelastic Fluids in Blasius Flow

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