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Nov 2011

Volume 23, Issue 11, Articles (11xxxx)

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Phys. Fluids 23, 111701 (2011); http://dx.doi.org/10.1063/1.3651269 (4 pages)

A. Mashayek and W. R. Peltier
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back to top Instability and Transition

Resonant three–wave interaction of Holmboe waves in a sharply stratified shear flow with an inflection–free velocity profile

S. M. Churilov

Phys. Fluids 23, 114101 (2011); http://dx.doi.org/10.1063/1.3657093 (19 pages)

Online Publication Date: 9 November 2011

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Within the context of the well-known interpretation in terms of the wave interaction [P. G. Baines and H. Mitsudera, J. Fluid Mech. 276, 327 (1994); J. R. Carpenter et al., Phys. Fluids 22, 054104 (2010)], instability of sharply stratified (so that the vertical scale ℓ of density variation is much smaller than the scale Λ of velocity shear) flows with inflection-free velocity profiles should be treated as Holmboe’s instability. In such flows with a relatively weak stratification (when the bulk Richardson number J < (ℓ/Λ)3/2), eigenoscillations (i.e., Holmboe waves) have much the same phase velocities in a broad spectral range. This creates favorable conditions for a wide variety of three-wave interactions, in contrast to the homogeneous boundary layers where subharmonic resonance is the only effective three-wave process. In the paper, evolution equations are derived which describe three-wave interactions of Holmboe waves and have the form of nonlinear integral equations. Analytical and numerical methods are both used to find their solutions in different cases, and it is shown that at the nonlinear stage disturbances increase, as a rule, explosively. Some possible relations of the results obtained with those of numerical simulations and laboratory experiments are briefly discussed.
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47.55.Hd Stratified flows
02.30.Hq Ordinary differential equations
02.30.Rz Integral equations
02.60.Nm Integral and integrodifferential equations
47.11.-j Computational methods in fluid dynamics
47.20.Ft Instability of shear flows (e.g., Kelvin-Helmholtz)

Faraday instability in a vessel with a well: A numerical analysis

E. Louis, J. A. Miralles, G. Chiappe, A. Bazán, J. P. Adrados, and P. Cobo

Phys. Fluids 23, 114102 (2011); http://dx.doi.org/10.1063/1.3657801 (9 pages)

Online Publication Date: 11 November 2011

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Simulations of the Faraday instability in a rectangular-shaped vessel with well, filled with a viscous fluid, are presented. Oscillations promoted by applying a vertical vibration of a given frequency and amplitude show the following features: (i) unstable waves become increasingly localized in the well as the amplitude of vibration increases, (ii) the threshold amplitude for an arbitrary well width is bounded by the thresholds of a vessel with no well and liquid layers thicknesses equal to those in either the plateau or in the well region, and (iii) below threshold, a weak horizontal component triggers harmonic oscillations. Experiments carried out in a vessel filled with ethanol allowed to observe wave localization and, below threshold, the harmonic wave. Below threshold, the harmonic wave had been previously observed as the only possible wave in a square vessel with an immersed concentric square well. Novel theoretical tools are developed to investigate this system: A generalized Mathieu equation is used to handle the case without well, whereas a numerical transfer matrix method is applied to the case with well.
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47.20.Gv Viscous and viscoelastic instabilities
47.20.Ky Nonlinearity, bifurcation, and symmetry breaking
47.35.-i Hydrodynamic waves
47.11.-j Computational methods in fluid dynamics
47.15.Fe Stability of laminar flows
47.15.Rq Laminar flows in cavities, channels, ducts, and conduits

Effect of base cavities on the stability of the wake behind slender blunt-based axisymmetric bodies

E. Sanmiguel-Rojas, J. I. Jiménez-González, P. Bohorquez, G. Pawlak, and C. Martínez-Bazán

Phys. Fluids 23, 114103 (2011); http://dx.doi.org/10.1063/1.3658774 (11 pages) | Cited 1 time

Online Publication Date: 18 November 2011

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We extend our previous research on the instability properties of the laminar incompressible flow of density ρ and viscosity μ, which develops behind a cylindrical body with a rounded nose and length-to-diameter ratio L/D = 2, aligned with a free-stream of velocity w∞ [E. Sanmiguel-Rojas et al., Phys. Fluids 21, 114102 (2009); P. Bohorquez et al., J. Fluid Mech. 676, 110 (2011)]. In particular, we analyze the effects of a cylindrical base cavity of length h and diameter Dc on both critical Reynolds number, Rec = ρw∞D/μ, and drag coefficient, CD, combining experiments, three-dimensional direct numerical simulations, and global linear stability analyses. The direct numerical simulations and the global stability results predict with precision the stabilizing effect of the cavity on the stationary, three-dimensional bifurcation in the wake as h/D increases. In fact, it is shown that, for a given value of Dc/D, the critical Reynolds number for the steady bifurcation, Recs, increases monotonically as h/D increases, reaching an asymptotic value which depends on Dc/D, at h/D ≈ 0.7. Likewise, for a fixed value of h/D, we have studied the effect of the cavity diameter Dc/D on the critical Reynolds number. No effect on Recs is observed over the range 0 ≤ Dc/D≲0.6, but Recs shows a monotonic growth for 0.6≲Dc/D<1. On the other hand, for steady flows, the drag coefficient decreases with the length of the cavity reaching an asymptotic minimum for h/D>rsim0.5 and Dc/D → 1. Similar behavior with the cavity length has been observed experimentally and numerically for the second, oscillatory bifurcation, and its associated critical Reynolds number, Reco.
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47.15.Fe Stability of laminar flows
47.15.Tr Laminar wakes
47.32.C- Vortex dynamics
47.11.-j Computational methods in fluid dynamics

On the early evolution in the transient Bénard–Marangoni problem

M. Weidenfeld and I. Frankel

Phys. Fluids 23, 114104 (2011); http://dx.doi.org/10.1063/1.3660249 (10 pages)

Online Publication Date: 22 November 2011

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We focus on the early evolution of small (linear) perturbations following the sudden (step-function) exposure of a liquid layer to a cold adjacent atmosphere. On a time scale short relative to that characterizing thermal relaxation across the liquid layer, the temperature distribution is nonlinear and highly transient. Thus, the conduction reference state may not be regarded quasi steady. We accordingly consider the initial-value problem and obtain a Volterra-type integral equation governing the evolution of surface-temperature perturbations. Assuming an O(1) Biot number we study the effects on perturbations evolution of the wavenumber, the Prandtl number (Pr) and the effective Marangoni number (math, which, from the dominant balance in the thermal perturbation equation, is based on the current values of the width of the thermal boundary layer and the temperature difference across it, respectively). Explicit results are first presented in the limit of Pr>>1 wherein the hydrodynamic perturbation problem is quasi steady. The dominant perturbations correspond to large wavenumbers and convection effectively confined to the thermal boundary layer. Typical of the results is the nonmonotonic temporal evolution of perturbations which initially diminish and only after some finite delay time start to grow. These trends are rationalized in terms of the Marangoni mechanism by observing that while the magnitude of the reference-state temperature gradients remain essentially constant, they extend over a widening thermal boundary layer allowing for enhanced convective effects. Thus, since perturbations may be introduced at all positive times, those evolving on the favourable background of further developed (wider) thermal boundary layers may take over perturbations introduced earlier. This suggests the existence of a nonzero “optimal” appearance time of perturbations which will eventually dominate the instability process. We study the effects of finite O(1) values of Pr on the evolution of the hydrodynamic perturbation problem which is no longer quasi steady. Increasing Pr at a constant math is destabilizing which reflects the dual role of liquid viscosity in the transient Bénard-Marangoni problem. It is further demonstrated that with increasing wavenumber convergence to the asymptotic Pr>>1 limit is practically achieved at smaller Pr.
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47.55.pf Marangoni convection
47.20.Dr Surface-tension-driven instability
47.11.-j Computational methods in fluid dynamics
02.30.Rz Integral equations

Resonant phenomenon of elliptical cylinder flows in a subcritical regime

Shih-Sheng Chen and Ruey-Hor Yen

Phys. Fluids 23, 114105 (2011); http://dx.doi.org/10.1063/1.3662003 (13 pages)

Online Publication Date: 22 November 2011

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The resonant phenomena in the wake behind a transversely vibrating elliptical cylinder with different axis ratios from Ar = 0.01 to Ar = 2.0 in the subcritical regime is numerically investigated. Navier-Stokes equations are solved by a spectral element code with a triangular mesh. Reynolds numbers range from 15 to 60 and the Roshko numbers range from 0.5 to 8 for different elliptical cylinders. Both the velocity and pressure responses in the wake are measured and analyzed. The investigations of the drag coefficients and the wake streamlines indicate that the cylinder’s axis ratio has a minor effect on the resonant frequency, Ron. However, the cylinder’s axis ratio is found to have a prominent effect on the resonant amplitude; namely, the smaller the cylinder’s axis ratio, the stronger the occurrence of resonant amplitude. The investigations of resonant responses of both the velocity and pressure and the probe locations may provide information for designing a flow meter based on pressure responses in the subcritical regime. It shows that the ratio of velocity and pressure responses poses a great linear relationship against the probe distance behind the vibrating cylinder. Moreover, a resonant method based on the different resonant frequencies at different probed locations in the subcritical regime to predict the critical conditions is examined and verified for different elliptical cylinders. Finally, based on the critical values found, a reduced Reynolds number and a reduced Roshko number are proposed to unify the different linear relationships resulting from different elliptical cylinder flows. The result indicates that the effect of axis ratio can be stripped off in the reduced plane, which may be applied to a more generalized cylinder shape.
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47.80.Cb Velocity measurements
47.80.Fg Pressure and temperature measurements
02.60.-x Numerical approximation and analysis
06.30.Gv Velocity, acceleration, and rotation
47.10.ad Navier-Stokes equations
47.15.Tr Laminar wakes

Transient growth of secondary instabilities in parallel wakes: Anti lift-up mechanism and hyperbolic instability

Sabine Ortiz and Jean-Marc Chomaz

Phys. Fluids 23, 114106 (2011); http://dx.doi.org/10.1063/1.3659158 (15 pages) | Cited 1 time

Online Publication Date: 28 November 2011

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This paper investigates the three-dimensional temporal instabilities and the transient growth of perturbations on a Von Kármán vortex street, issuing from the development of the primary instability of a parallel Bickley velocity profile typical of a wake forming behind a thin flat plate. By solving iteratively the linearized direct Navier Stokes equations and its adjoint equations, we compute the optimal perturbations that exhibit the largest transient growth of energy between the initial instant and different time horizons. At short time horizons, optimal initial perturbations are concentrated on the points of maximal strain of the base flow. The optimal gain of energy and the mechanism of instability are well predicted by local theories that describe the lagrangian evolution of a perturbation wave packet. At time of order unity, hyperbolic region leads the dynamics. Only at large time (t ≥ 20), the growth is led by the most amplified eigenmode. This eigenmode evolves, when the wavenumber increases, from perturbation centred in the core of the vortices, to perturbations localised on the stretching manifold of the hyperbolic points. At finite and large time, the gain in energy is initially associated with a mechanism reminiscent to the anti lift-up mechanism described by Antkowiak and Brancher [J. Fluid Mech. 578, 295 (2007)] in the context of an axisymmetric vortex. Presently, the optimal initial condition (the adjoint modes at large time) corresponding to streamwise streaks localised on the contracting manifold of the hyperbolic point induces streamwise vortices aligned with the stretching manifold of the hyperbolic point (the direct modes). The localisation on distinct manifolds of direct and adjoint eigenmodes is more pronounced when the Reynolds number is increased. An interpretation is proposed based on a balance between diffusion and stretching effects that predicts the thickness of the energy containing region for the adjoint and the direct mode decreasing as 1/math. The extra gain of energy due to non normal effects grows, since direct and adjoint modes are localised in different regions of space, i.e., the stretching and contracting manifold, a novel effect of the so called convective non normality associated with the transport of the perturbation by the base flow.
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47.20.Lz Secondary instabilities
47.32.ck Vortex streets
47.10.ad Navier-Stokes equations

Nonlinear effects in the combined Rayleigh-Taylor/Kelvin-Helmholtz instability

Britton J. Olson, Johan Larsson, Sanjiva K. Lele, and Andrew W. Cook

Phys. Fluids 23, 114107 (2011); http://dx.doi.org/10.1063/1.3660723 (10 pages) | Cited 1 time

Online Publication Date: 28 November 2011

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The combined Rayleigh-Taylor/Kelvin-Helmholtz (RT/KH) instability is studied in the early nonlinear regime. Specifically, the effect of adding shear to a gravitationally unstable configuration is investigated. While linear stability theory predicts that any amount of shear would increase the growth rate beyond the Rayleigh-Taylor value, numerical (large eddy) simulations show a more complex and non-monotonic behavior where small amounts of shear in fact decrease the growth rate. A velocity scale for the combined instability is proposed from linear stability arguments and is shown to effectively collapse the growth rates for different configurations. The specific amount of shear that minimizes the peak growth rate is identified and the physical origins of this non-monotonic behavior are investigated.
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47.20.Ma Interfacial instabilities (e.g., Rayleigh-Taylor)
47.27.Cn Transition to turbulence
47.11.-j Computational methods in fluid dynamics
47.27.ep Large-eddy simulations
47.27.ek Direct numerical simulations
02.60.Cb Numerical simulation; solution of equations

The instability of the boundary layer over a disk rotating in an enforced axial flow

Z. Hussain, S. J. Garrett, and S. O. Stephen

Phys. Fluids 23, 114108 (2011); http://dx.doi.org/10.1063/1.3662133 (13 pages) | Cited 1 time

Online Publication Date: 30 November 2011

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We consider the convective instability of stationary and traveling modes within the boundary layer over a disk rotating in a uniform axial flow. Complementary numerical and high Reynolds number asymptotic analyses are presented. Stationary and traveling modes of type I (crossflow) and type II (streamline curvature) are found to exist within the boundary layer at all axial flow rates considered. For low to moderate axial flows, slowly traveling type I modes are found to be the most amplified, and quickly traveling type II modes are found to have the lower critical Reynolds numbers. However, near-stationary type I modes are expected to be selected due to a balance being struck between onset and amplification. Axial flow is seen to stabilize the boundary layer by increasing the critical Reynolds numbers and reducing amplification rates of both modes. However, the relative importance of type II modes increases with axial flow and they are, therefore, expected to dominate for sufficiently high rates. The application to chemical vapour deposition (CVD) reactors is considered.
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47.32.Ef Rotating and swirling flows
81.15.Gh Chemical vapor deposition (including plasma-enhanced CVD, MOCVD, ALD, etc.)
47.15.Fe Stability of laminar flows
47.20.Ib Instability of boundary layers; separation
47.27.Cn Transition to turbulence
47.27.nb Boundary layer turbulence
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