Bistable Flows Research Articles

Reynolds number effects on the bistable flows over a wavy circular cylinder

The wake of wavy cylinder has been shown to exhibit bistability. Depending on the initial condition, the final state of the wake can either develop into steady flow (state I), or periodic shedding (state II). In this paper, we perform numerical simulations to reveal the Reynolds number effects (Re=30 – 300) on these two wake states. With increasing Reynolds number, the steady vortical structures in state I wake sways back and forth in the spanwise direction, resulting in low-frequency fluctuations in drag forces, but not in lift. For state II, the increase in Reynolds number is associated with the emergence of another spectral peak in the lift coefficient. The secondary frequency is associated with highly three-dimensional vortical structures in the wake. For both states, the wakes transition to turbulent flows at higher Reynolds numbers, with the development of small-scale vortices. We further study the streamwise gust flows over the wavy cylinder. The time-varying inflow velocity results in a wide range of instantaneous Reynolds number spanning from the absolutely unstable flow regime to the bistable regime. Depending on the period of the inflow velocity variation, the wake perturbations grown at the absolutely unstable flow regime can be damped out in state I wake, or grow large enough to trigger the transition to state II, resulting in loss of flow control efficacy. The above analyses reveal novel flow physics of the bistable states at unexplored Reynolds numbers, and showcase the complex transition behavior between the two states in unsteady flows. The insights gained from this study improve the understanding of the wake dynamics of the wavy cylinder.

Blockage influence on bi-stable flows of a notchback bluff body

The expected flow asymmetry behind a three-dimensional notchback Ahmed body is numerically investigated using large-eddy simulations with different blockage ratios of 0%, 5%, 10%, and 20%. The focus of the study is on the natural bi-stable flow influenced by the blockage ratio. Although the wake asymmetry can be observed under the blockage ratio within 0%−10%, the significant blockage influence on the sensitive bi-stability is indicated by the wake structures, pressure gradients, and wake dynamics achieved with the proper orthogonal decomposition. The higher blockage ratio increases the turbulence kinetic energy, velocity, and negative pressure in the near-wake region, resulting in the decrease in the asymmetry degree. The consistency of the asymmetric wake is found with blockage ratios between 0% and 5%. However, a 20% blockage ratio symmetrizes the bi-stable wake. Several existing drag corrections for the blockage influence are discussed. The wind tunnel experiment verifies the bi-stable flow with low blockage ratios and the wake symmetrization with a 20% blockage ratio.

On state instability of the bi-stable flow past a notchback bluff body

The wake of a notchback Ahmed body presenting a bi-stable nature is investigated by performing wind tunnel experiments and large-eddy simulations. Attention is confined to the Reynolds number ( $Re$ ) influence on the wake state instability within $5\times 10^{4}\leq Re \leq 25\times 10^{4}$ . Experimental observations suggest a wake bi-stability with low-frequency switches under low $Re$ . The wake becomes ‘tri-stable’ with the increase of $Re$ with the introduction of a new symmetric state. The higher presence of the symmetric state can be considered as a symmetrization of the wake bi-stability with an increasing $Re$ . The wake symmetry under high $Re$ attributed to the highly frequent switches of the wake is extremely sensitive to small yaw angles, showing the feature of bi-stable flows. The wake asymmetry is confirmed in numerical simulations with both low and high $Re$ . The wake asymmetries are indicated by the wake separation, the reattachment and the wake dynamics identified by the proper orthogonal decomposition. However, the turbulence level is found to be significantly higher with a higher $Re$ . This leads to a higher possibility to break the asymmetric state, resulting in highly frequent switches showing symmetry.

Transition of wake flows past two circular or square cylinders in tandem

This paper presents the dependence of flow transitions and flow regimes on Reynolds number Re (≤ 105) and scaled cylinder center-to-center spacing S* (≤ 10) for two tandem cylinders. Both circular and square cross-sectional geometries are considered, and a comparison of results is made between the two geometries. To do so, data available from the literature are collected, and supplementary numerical simulations are done to complement the literature data so as to provide clear-cut flow maps in the Re–S* domain. We identified the borders of Re- and S*-dependent flows, including steady, unsteady, transition, slender/extended body, alternating reattachment, bistable, and co-shedding flows. The flow interference between two tandem cylinders with S* < S*cr is susceptible to stabilize the flow that requires a higher Re for the onset of vortex shedding than the isolated cylinder counterpart, where S*cr is the critical spacing for the onset of co-shedding flow. Besides, the flow interference generally postpones and advances the onset of secondary vortex shedding associated with the transition from two- to three-dimensional vortex shedding for S* < S*cr and S* > S*cr, respectively. Two flow maps are made for the two geometries, both yielding fundamental contributions toward a better understanding of the flow interference effect on various flow transitions. The flow maps determine the regions of a dearth of knowledge, which future studies may pay attention to. Finally, the drag inversion, hysteresis, and flow patterns in the proximity of the onset of vortex shedding for tandem square cylinders (1.5 ≤ S* ≤ 7) are discussed.

Bistable fully developed mixed convection flow with viscous dissipation in a vertical channel.

It is shown that unstable dual solutions in fully developed mixed convection flow in a vertical channel disappear in the presence of relatively strong thermal radiation. In this case, we have a unique stable flow at each pressure gradient. When the effect of thermal radiation is weak another branch of stable solutions is created, resulting in bistable flows. In this case, the flow exhibits hysteresis with variation of the pressure gradient. Optically, a thin radiation model is used.

Numerical simulations of bistable flows in precessing spheroidal shells

Numerical simulations of bistable flows in precessing spheroidal shells

On the chaotic nature of bistable flows

Evidences from experimental measurements in tube banks submitted to turbulent cross flow in quadrangular and triangular arrangements show the presence of instabilities, being associated to the phenomenon of bistability, known in the simplified case of the flow on two cylinders side-by-side. Although not completely understood the study of the dynamic process of bistability can reveal new features about the chaotic behavior of these time series. As chaotic time series are observed routinely in experiments on physical systems, a study of the characteristics of bistability can be performed with the aid of techniques already established. This work presents a study about the determination of the Lyapunov exponents from experimental time series of bistable flows on the simplified geometry of two parallel circular cylinders and on a set of rows of a triangular tube bank. Time series of flow velocity and velocity fluctuations were obtained by means of the constant hot-wire anemometry technique in an aerodynamic channel. Discrete wavelet transform was used to make a multilevel decomposition of the series in several bandwidth values, accordingly with a selected decomposition level. Thus, the turbulence can be dissociated from the original signals, allowing a more accurate study to be conducted on the obtained state spaces. With these filtered time series, a state space reconstruction was performed by the method of time delays or Takens’ method, while the percentage of false neighbors together with the embedding dimension were useful for the choice of some of the analysis parameters. The Rosenstein’s method was applied to calculate the largest Lyapunov exponent of the time series. Easy to implement, this method is robust with respect to changes in most immersion parameters. Results show that the flow after two circular cylinders placed side-by-side and after two rows tube bank in triangular arrangement present positive largest Lyapunov exponents. This means that bistability has a clear chaotic behavior. In contrast, the flow after a five row tube bank is random.

Global-in-time Gevrey regularity solution for a class of bistable gradient flows

In this paper, we prove the existence and uniqueness of a Gevrey regularity solution for a class of nonlinear bistable gradient flows, where with the energy may be decomposed into purely convex and concave parts. Example equations include certain epitaxial thin film growth models and phase field crystal models. The energy dissipation law implies a bound in the leading Sobolev norm. The polynomial structure of the nonlinear terms in the chemical potential enables us to derive a local-in-time solution with Gevrey regularity, with the existence time interval length dependent on a certain $H^m$ norm of the initial data. A detailed Sobolev estimate for the gradient equations results in a uniform-in-time-bound of that $H^m$ norm, which in turn establishes the existence of a global-in-time solution with Gevrey regularity.

Bistable flows in precessing spheroids

Precession driven flows are found in any rotating container filled with liquid, when the rotation axis itself rotates about a secondary axis that is fixed in an inertial frame of reference. Because of its relevance for planetary fluid layers, many works consider spheroidal containers, where the uniform vorticity component of the bulk flow is reliably given by the well-known equations obtained by Busse (1968 J. Fluid Mech. 33 739–51). So far however, no analytical result for the solutions is available. Moreover, the cases where multiple flows can coexist have not been investigated in detail since their discovery by Noir et al (2003 Geophys. J. Int. 154 407–16). In this work we aim at deriving analytical results for the solutions, aiming in particular at first estimating the ranges of parameters where multiple solutions exist, and second studying quantitatively their stability. Using the models recently proposed by Noir and Cébron (2013 J. Fluid Mech. 737 412–39), which are more generic in the inviscid limit than the equations of Busse, we analytically describe these solutions, their conditions of existence, and their stability in a systematic manner. We then successfully compare these analytical results with the theory of Busse (). Dynamical model equations are finally proposed to investigate the stability of the solutions, which describe the bifurcation of the unstable flow solution. We also report for the first time the possibility that time-dependent multiple flows can coexist in precessing triaxial ellipsoids. Numerical integrations of the algebraic and differential equations have been efficiently performed with the dedicated script FLIPPER (supplementary material).

A new phenomenon of bi-stability: experimental and numerical evidence

Bistable flows around circular cylinders typically occur in the critical range of the Reynolds number (Re), due to the formation of a laminar separation bubble and its subsequent collapse on one side of the cylinder only. In turbulent flows, as well as on a rough cylinder at slightly higher Re, bistable flows tend to disappear. However, this paper gives experimental evidence of a new type of bistable flow, which is induced around a circular cylinder of finite length even at moderately high Re. Stiffening rings, applied along the height of a circular cylinder at a distance smaller than the diameter (d = 2/3D in the case of study), were found to be responsible for the phenomenon. This was demonstrated and then cross-checked by wind tunnel experiments on a H/D = 6.7 circular cylinder. The experiments were performed in two different wind tunnels, at WiSt (Ruhr University Bochum) and at CRIACIV (University of Florence). A numerical study of the latter experimental test case is also presented in this paper. A finite volume approach based on the iterative solution of the unsteady Reynolds-averaged Navier–Stokes equations has been used. Turbulence has been modelled using the SST model by Menter. The numerical results confirm the physical fundament of the phenomenon, which likely finds its explanation in the key role of the flow over the cylinder tip.

Dynamo efficiency controlled by hydrodynamic bistability.

Hydrodynamic and magnetic behaviors in a modified experimental setup of the von Kármán sodium flow-where one disk has been replaced by a propeller-are investigated. When the rotation frequencies of the disk and the propeller are different, we show that the fully turbulent hydrodynamic flow undergoes a global bifurcation between two configurations. The bistability of these flow configurations is associated with the dynamics of the central shear layer. The bistable flows are shown to have different dynamo efficiencies; thus for a given rotation rate of the soft-iron disk, two distinct magnetic behaviors are observed depending on the flow configuration. The hydrodynamic transition controls the magnetic field behavior, and bifurcations between high and low magnetic field branches are investigated.

A new type of bistable flow around circular cylinders with spanwise stiffening rings

The bistable flow condition around a single circular cylinders is a well-known fluid dynamic phenomenon in the critical range of Re. It is sensitive not only to small variations of the Reynolds number, but also to turbulence of the incoming flow and to surface roughness of the cylinder. Bistable flows are also common for side-by-side cylinders, depending on their centre-to-centre transverse pitch ratio.The paper reveals – through wind tunnel tests – the existence of a new type of bistable flow, induced around a single circular cylinder with a free-end by the presence of spanwise rings. There are some analogies with the aforementioned bistable phenomena, but the conditions of occurrence are profoundly different. The peculiarity of this phenomenon is that it does not disappear at moderately high Re. Its existence is confirmed by a cross-check of results in two different wind tunnels.In order to characterize such a bistable pressure field, the pressure is modeled in the paper as the output of a non-linear conservative system with asymmetric potential wells forced by a proper stochastic process. The identification of the system parameters is performed by the comparison between the theoretical distribution of the oscillator response and the histogram of the recorded pressure.

THE DENSITY DISTRIBUTION IN TURBULENT BISTABLE FLOWS

We numerically study the volume density probability distribution function (n-PDF) and the column density probability distribution function (Sigma-PDF) resulting from thermally bistable turbulent flows. We analyze three-dimensional hydrodynamic models in periodic boxes of 100pc by side, where turbulence is driven in the Fourier space at a wavenumber corresponding to 50pc. At low densities (n <= 0.6cm^-3) the n-PDF, is well described by a lognormal distribution for average local Mach number ranging from ~0.2 to ~5.5. As a consequence of the non linear development of thermal instability (TI), the logarithmic variance of the distribution for the diffuse gas increases with M faster than in the well known isothermal case. The average local Mach number for the dense gas (n >= 7.1cm^-3) goes from ~1.1 to ~16.9 and the shape of the high density zone of the n-PDF changes from a power-law at low Mach numbers to a lognormal at high M values. In the latter case the width of the distribution is smaller than in the isothermal case and grows slower with M. At high column densities the Sigma-PDF is well described by a lognormal for all the Mach numbers we consider and, due to the presence of TI, the width of the distribution is systematically larger than in the isothermal case but follows a qualitatively similar behavior as M increases. Although a relationship between the width of the distribution and M can be found for each one of the cases mentioned above, these relations are different form those of the isothermal case.

Two interacting cylinders in cross flow

Cylindrical structures in a group are frequently seen on land and in the ocean. Mutual flow interaction between the structures makes the wake very excited or tranquil depending on the spacing between the structures. The excited wake-enhancing forces in some cases cause a catastrophic failure of the structures. This paper presents results of an experimental investigation of Strouhal number (St), time-mean, and fluctuating forces on, and flow structures around, two identical circular cylinders at stagger angle α = 0°-180° and gap-spacing ratio T/D=0.1-5, where T is the gap width between the cylinders, and D is the diameter of a cylinder. While forces were measured using a load cell, St was from spectral analysis of fluctuating pressures measured on the side surfaces of the cylinders. A flow visualization test was conducted to observe flow structures around the cylinders. Based on forces, St, and flow structures, 19 distinct flow categories in the ranges of α and T/D investigated are observed, including one quadristable flow, three kinds of tristable flows, and four kinds of bistable flows. The quadristable, tristable, and bistable flows ensue from instabilities of the gap flow, shear layers, vortices, separation bubbles, and wakes, engendering a strong jump or drop in forces and St of the cylinders. The two cylinders interact with each other in six different mechanisms, namely interaction between boundary layer and cylinder, shear layer or wake and cylinder, shear layer and shear layer, vortex and cylinder, vortex and shear layer, and vortex and vortex. While the interaction between vortex and cylinder results in a very high fluctuating drag, that between vortex and shear layer results in a high fluctuating lift. On the other hand, the interaction between shear layer or wake and cylinder weakens mean and fluctuating forces and flow unsteadiness. A mutual discussion of forces, St, and flow structures is presented in this paper.

DENSITY POWER SPECTRUM IN TURBULENT THERMALLY BISTABLE FLOWS

In this paper we numerically study the behavior of the density power spectrum in turbulent thermally bistable flows. We analyze a set of five three-dimensional simulations where turbulence is randomly driven in Fourier space at a fixed wave-number and with different Mach numbers M (with respect to the warm medium) ranging from 0.2 to 4.5. The density power spectrum becomes shallower as M increases and the same is true for the column density power spectrum. This trend is interpreted as a consequence of the simultaneous turbulent compressions, thermal instability generated density fluctuations, and the weakening of thermal pressure force in diffuse gas. This behavior is consistent with the fact that observationally determined spectra exhibit different slopes in different regions. The values of the spectral indexes resulting from our simulations are consistent with observational values. We do also explore the behavior of the velocity power spectrum, which becomes steeper as M increases. The spectral index goes from a value much shallower than the Kolmogorov one for M=0.2 to a value steeper than the Kolmogorov one for M=4.5.

The Density and the Velocity Power Spectrum in Thermally Bi-stable Flows

AbstractWe present numerical results concerning the behavior of the density and the velocity power spectrum in turbulent thermally bistable flows for different Mach numbers.

The Pressure Distribution in Thermally Bistable Turbulent Flows

We present a systematic numerical study of the effect of turbulent velocity fluctuations on the thermal pressure distribution in thermally bistable flows. The turbulent fluctuations are characterized by their rms Mach number M (with respect to the warm medium) and the energy injection (forcing) wavenumber kfor = 1/l, where l is the injection size scale in units of the box size L = 100 pc. The numerical simulations employ random turbulent driving generated in Fourier space rather than starlike heating, in order to allow for precise control of the parameters. Our range of parameters is 0.5 ≤ M ≤ 1.25 and 2 ≤ kfor ≤ 16. Our results are consistent with the picture that as either of these parameters is increased, the local ratio of turbulent crossing time to cooling time decreases, causing transient structures in which the effective behavior is intermediate between the thermal-equilibrium and adiabatic regimes. As a result, the effective polytropic exponent γe of the simulations ranges between ~0.2 and ~1.1, and the mean pressure of the diffuse gas is generally reduced below the thermal equilibrium pressure Peq, while that of the dense gas is increased with respect to Peq. The fraction of high-density zones (n > 7.1 cm-3) with P > 104 cm-3 K increases from roughly 0.1% at kfor = 2 and M = 0.5 to roughly 70% for kfor = 16 and M = 1.25. A preliminary comparison with the recent pressure measurements of Jenkins in C I favors our case with M = 0.5 and kfor = 2. In all cases, the dynamic range of the pressure in any given density interval is larger than one order of magnitude, and the total dynamic range, summed over the entire density range, typically spans 3-4 orders of magnitude. The total pressure histogram widens as the Mach number is increased, and moreover develops near-power-law tails at high (low) pressures when γe ≲ 0.5 (γe ≳ 1), which occurs at kfor = 2 (kfor = 16) in our simulations. The opposite side of the pressure histogram decays rapidly, in an approximately lognormal form. This behavior resembles that of the corresponding density histograms, in spite of the large scatter of the pressure in any given density interval. Our results show that turbulent advection alone can generate large pressure scatters, with power-law high-P tails for large-scale driving, and provide validation for approaches attempting to derive the shape of the pressure histogram through a change of variable from the known form of the density histogram, such as that performed by Mac Low et al.

Resonant bistable flows driven by colored noise

The escape times of an overdamped Brownian particle confined to a quartic bistable potential is known to be extremely sensitive to the fluctuation correlation time. In the presence of an external Gaussian noise source with fixed variance, the escape times develop a minimum for a certain value of the correlation time—a behavior reminiscent of the so-called resonant activation phenomenon. The optimal escape conditions are investigated for a wide range of the noise variance.

Resonant bistable flows driven by colored noise

AbstractThe escape times of an overdamped Brownian particle confined to a quartic bistable potential is known to be extremely sensitive to the fluctuation correlation time. In the presence of an external Gaussian noise source with fixed variance, the escape times develop a minimum for a certain value of the correlation time—a behavior reminiscent of the so‐called resonant activation phenomenon. The optimal escape conditions are investigated for a wide range of the noise variance.

Modelling of aerodynamic loads on a downstream cylinder caused by bistable flow between two circular cylinders

The paper deals with physical and mathematical modelling of wind load on a downstream cylinder located in the region of bistable flows between two circular cylinders. The bases of the consideration are dimensional analysis of the problem and quasisteady theoretical approach to the construction of models of wind load on slender structures. Corresponding to the interferential wind load components on the downstream cylinder, the nonlinear and linear forms of the set of equations of cylinder motion were determinated and analysed. Moreover, the reduced critical velocity of self-induced vibrations of the downstream cylinder was evaluated.