The effect of a small initial distortion of the basic flow on the subcritical transition in plane Poiseuille flow (original) (raw)
Related papers
The instability of oscillatory plane Poiseuille flow
Journal of Fluid Mechanics, 1982
The instability of oscillatory plane Poiseuille flow, in which the pressure gradient is time-periodically modulated, is investigated by a perturbation technique. The Floquet exponents (i.e. the complex growth rates of the disturbances to the oscillatory flow) are computed by series expansions, in powers of the oscillatory to steady flow velocity amplitude ratio, about the values of the growth rates of the disturbances of the steady flow. It is shown that the oscillatory flow is more stable than the steady flow for values of Reynolds number and disturbance wave number in the vicinity of the steady flow critical point and for values of frequencies of imposed oscillation greater than about one tenth of the frequency of the steady flow neutral disturbance. At very high and low values of imposed oscillation frequency, the unsteady flow is slightly less stable than the steady flow. These results hold for the values of the velocity amplitude ratio at least up to 0·25.
Subcritical transition to turbulence in wall-bounded flows: the case of plane Poiseuille flow
arXiv: Fluid Dynamics, 2019
In wall-bounded flows, the laminar regime remain linearly stable up to large values of the Reynolds number while competing with nonlinear turbulent solutions issued from finite amplitude perturbations. The transition to turbulence of plane channel flow (plane Poiseuille flow) is more specifically considered via numerical simulations. Previous conflicting observations are reconciled by noting that the two-dimensional directed percolation scenario expected for the decay of turbulence may be interrupted by a symmetry-breaking bifurcation favoring localized turbulent bands. At the other end of the transitional range, a preliminary study suggests that the laminar-turbulent pattern leaves room to a featureless regime beyond a well defined threshold to be determined with precision.
Stability bounds on turbulent Poiseuille flow
Journal of Fluid Mechanics, 1988
For steady-state turbulent flows with unique mean properties, we determine a sense in which the mean velocity is linearly supercritical. The shear-turbulence literature on this point is ambiguous. As an example, we reassess the stability of mean profiles in turbulent Poiseuille flow. The Reynolds & Tiederman (1967) numerical study is used as a starting point. They had constructed a class of one-dimensional flows which included, within experimental error, the observed profile. Their numerical solutions of the resulting Orr-Sommerfeld problems led them to conclude that the Reynolds number for neutral infinitesimal disturbances was twenty-five times the Reynolds number characterizing the observed mean flow. They found also that the first nonlinear corrections were stabilizing. In the realized flow, this latter conclusion appears incompatible with the former. Hence, we have sought a more complete set of velocity profiles which could exhibit linear instability, retaining the requirement that the observed velocity profile is included in the set. We have added two dynamically generated modifications of the mean. The first addition is a fluctuation in the curvature of the mean flow generated by a Reynolds stress whose form is determined by the neutrally stable Orr-Sommerfeld solution. We find that this can reduce the stability of the observed flow by as much as a factor of two. The second addition is the zero-average downstream wave associated with the above Reynolds stress. The three-dimensional linear instability of this modification can even render the observed flow unstable. Those wave amplitudes that just barely will ensure instability of the observed flow are determined. The relation of these particular amplitudes t o the limiting conditions admitted by an absolute stability criterion for disturbances on the mean flow is found. These quantitative results from stability theory lie in the observationally determined Reynolds-Tiederman similarity scheme, and hence are insensitive to changes in Reynolds number.
2017
We present a new experimental set-up that creates a shear flow with zero mean advection velocity achieved by counterbalancing the nonzero streamwise pressure gradient by moving boundaries, which generates plane Couette-Poiseuille flow. We carry out the first experimental results in the transitional regime for this flow. Using flow visualization we characterize the subcritical transition to turbulence in Couette-Poiseuille flow and show the existence of turbulent spots generated by a permanent perturbation. Due to the zero mean advection velocity of the base profile, these turbulent structures are nearly stationary. We distinguish two regions of the turbulent spot: the active, turbulent core, which is characterized by waviness of the streaks similar to traveling waves, and the surrounding region, which includes in addition the weak undisturbed streaks and oblique waves at the laminar-turbulent interface. We also study the dependence of the size of these two regions on Reynolds number...
Experimental scaling law for the subcritical transition to turbulence in plane Poiseuille flow
Physical Review E, 2012
We present an experimental study of transition to turbulence in a plane Poiseuille flow. Using a well-controlled perturbation, we analyse the flow using extensive Particule Image Velocimetry and flow visualisation (using Laser Induced Fluorescence) measurements and use the deformation of the mean velocity profile as a criterion to characterize the state of the flow. From a large parametric study, four different states are defined depending on the values of the Reynolds number and the amplitude of the perturbation. We discuss the role of coherent structures, like hairpin vortices, in the transition. We find that the minimal amplitude of the perturbation triggering transition scales like Re −1 .
Linear Instability of Plane Couette and Poiseuille Flows
—It is shown that linear instability of plane Couette flow can take place even at finite Reynolds numbers Re > Re th ≈ 139, which agrees with the experimental value of Re th ≈ 150 ± 5 [16, 17]. This new result of the linear theory of hydrodynamic stability is obtained by abandoning traditional assumption of the longitudinal periodicity of disturbances in the flow direction. It is established that previous notions about linear stability of this flow at arbitrarily large Reynolds numbers relied directly upon the assumed separation of spatial variables of the field of disturbances and their longitudinal periodicity in the linear theory. By also abandoning these assumptions for plane Poiseuille flow, a new threshold Reynolds number Re th ≈ 1035 is obtained, which agrees to within 4% with experiment—in contrast to 500% discrepancy for the previous estimate of Re th ≈ 5772 obtained in the framework of the linear theory under assumption of the " normal " shape of disturbances [2].
Instability of Plane Couette Flow
The energy gradient theory has been proposed with the aim of better understanding the mechanism of flow transition from laminar flow to turbulent flow. In this theory, it is suggested that the transition to turbulence depends on the relative magnitudes of the energy gradient amplifying the disturbance and the viscous friction damping that disturbance. For a given flow geometry and fluid properties, when the maximum of K (the ratio of the energy gradient in the transverse direction to that in the streamwise direction) in the flow field is larger than a certain critical value, it is expected that instability would occur for some initial disturbances. In this paper, using the energy analysis, the equation for calculating K for plane Couette flow is derived. It is demonstrated that the critical value of K at subcritical transition is about 370 for plane Couette flow. This value is about the same as for plane Poiseuille flow and pipe Poiseuille flow (385-389). Therefore, it is concluded ...
Nonlinear stability results for plane Couette and Poiseuille flows
Physical Review E
In this article we prove, choosing an appropriately weighted L 2energy equivalent to the classical energy, that the plane Couette and Poiseuille flows are nonlinearly stable with respect to streamwise perturbations for any Reynolds number. In this case the coefficient of time-decay of the energy is π 2 /(2Re), and it is a bound from above of the time-decay of streamwise perturbations of linearized equations. We also prove that the plane Couette and Poiseuille flows are nonlinearly stable if the Reynolds number is less then Re Orr / sin ϕ when the perturbation is a tilted perturbation, i.e. 2D perturbations with wave vector which forms an angle ϕ ∈ [0, π/2] with the direction i of the motion. Re Orr is the Orr (1907) critical Reynolds number for spanwise perturbations which, for the Couette flow is Re Orr = 177.22 and for the Poiseuille flow is Re Orr = 175.31.
Anisotropic decay of turbulence in plane Couette-Poiseuille flow
arXiv (Cornell University), 2020
We report the results of an experimental investigation into the decay of turbulence in plane Couette-Poiseuille flow using 'quench' experiments where the flow laminarises after a sudden reduction in Reynolds number Re. Specifically, we study the velocity field in the streamwise-spanwise plane. We show that the spanwise velocity containing rolls, decays faster than the streamwise velocity, which displays elongated regions of higher or lower velocity called streaks. At final Reynolds numbers above 425, the decay of streaks displays two stages: first a slow decay when rolls are present and secondly a more rapid decay of streaks alone. The difference in behaviour results from the regeneration of streaks by rolls, called the lift-up effect. We define the turbulent fraction as the portion of the flow containing turbulence and this is estimated by thresholding the spanwise velocity component. It decreases linearly with time in the whole range of final Re. The corresponding decay slope increases linearly with final Re. The extrapolated value at which this decay slope vanishes is Re az ≈ 656±10, close to Re g ≈ 670 at which turbulence is self-sustained. The decay of the energy computed from the spanwise velocity component is found to be exponential. The corresponding decay rate increases linearly with Re, with an extrapolated vanishing value at Re Az ≈ 688 ± 10. This value is also close to the value at which the turbulence is self-sustained, showing that valuable information on the transition can be obtained over a wide range of Re.