Nondimensional Control Parameters Research Articles

Active chaotic mixing in a channel with rotating arc-walls

An active inline mixer, called Rotating Arc-Wall mixer (RAW), suitable for flows at low Reynolds number and high P\'eclet number, is studied. The forcing protocol, imposed by three rotating circular arc-walls, is time periodic. We use simple phenomenological arguments to estimate heuristically the mixing efficiency with two nondimensional control parameters: the frequency of the forcing protocol, and the strength of the cross flow relative to the primary flow. The validity and limitations of the proposed mixing conditions are explained by the transport mechanisms in the mixer, and the beneficial role of chaotic advection for mixing is highlighted.

Connections between nonrotating, slowly rotating, and rapidly rotating turbulent convection transport scalings

In this paper, we investigate and develop scaling laws as a function of external nondimensional control parameters for heat and momentum transport for nonrotating, slowly rotating, and rapidly rotating turbulent convection systems, with the end goal of forging connections and bridging the various gaps between these regimes. Two perspectives are considered, one where turbulent convection is viewed from the standpoint of an applied temperature drop across the domain and the other with a viewpoint in terms of an applied heat flux. While a straightforward transformation exists between the two perspectives, indicating equivalence, it is found the former provides a clear set of connections that bridge between the three regimes. Our generic convection scalings, based upon an inertial-Archimedean balance, produce the classic diffusion-free scalings for the nonrotating limit and the slowly rotating limit. This is characterized by a free-falling fluid parcel on the global scale possessing a thermal anomaly on par with the temperature drop across the domain. In the rapidly rotating limit, the generic convection scalings are based on a Coriolis-inertial-Archimedean (CIA) balance, along with a local fluctuating-mean advective temperature balance. This produces a scenario in which anisotropic fluid parcels attain a thermal wind velocity and where the thermal anomalies are greatly attenuated compared to the total temperature drop. We find that turbulent scalings may be deduced simply by consideration of the generic nondimensional transport parameters—local Reynolds Reℓ=Uℓ/ν; local Péclet Peℓ=Uℓ/κ; and Nusselt number Nu=Uϑ/(κΔT/H)—through the selection of physically relevant estimates for length ℓ, velocity U, and temperature scales ϑ in each regime. Emergent from the scaling analyses is a unified continuum based on a single external control parameter, the convective Rossby number, RoC=gαΔT/4Ω2H, that strikingly appears in each regime by consideration of the local, convection-scale Rossby number Roℓ=U/(2Ωℓ). Thus we show that RoC scales with the local Rossby number Roℓ in both the slowly rotating and the rapidly rotating regimes, explaining the ubiquity of RoC in rotating convection studies. We show in non-, slowly, and rapidly rotating systems that the convective heat transport, parametrized via Peℓ, scales with the total heat transport parameterized via the Nusselt number Nu. Within the rapidly rotating limit, momentum transport arguments generate a scaling for the system-scale Rossby number, RoH, that, recast in terms of the total heat flux through the system, is shown to be synonymous with the classical flux-based CIA scaling, RoCIA. These, in turn, are then shown to asymptote to RoH∼RoCIA∼RoC2, demonstrating that these momentum transport scalings are identical in the limit of rapidly rotating turbulent heat transfer.Received 22 February 2020Accepted 7 September 2020Corrected 30 March 2021DOI:https://doi.org/10.1103/PhysRevResearch.2.043115Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasAstrophysical fluid dynamicsGeophysical fluid dynamicsRayleigh-Bénard convectionRotating geophysical flowsScaling laws of complex systemsTurbulenceNonlinear DynamicsFluid DynamicsInterdisciplinary Physics

Machine learning strategies for path-planning microswimmers in turbulent flows.

We develop an adversarial-reinforcement learning scheme for microswimmers in statistically homogeneous and isotropic turbulent fluid flows, in both two and three dimensions. We show that this scheme allows microswimmers to find nontrivial paths, which enable them to reach a target on average in less time than a naïve microswimmer, which tries, at any instant of time and at a given position in space, to swim in the direction of the target. We use pseudospectral direct numerical simulations of the two- and three-dimensional (incompressible) Navier-Stokes equations to obtain the turbulent flows. We then introduce passive microswimmers that try to swim along a given direction in these flows; the microswimmers do not affect the flow, but they are advected by it. Two nondimensional control parameters play important roles in our learning scheme: (a) the ratio V[over ̃]_{s} of the microswimmer's bare velocity V_{s} and the root-mean-square (rms) velocity u_{rms} of the turbulent fluid and (b) the product B[over ̃] of the microswimmer-response time B and the rms vorticity ω_{rms} of the fluid. We show that the average time required for the microswimmers to reach the target, by using our adversarial-reinforcement learning scheme, eventually reduces below the average time taken by microswimmers that follow the naïve strategy.

Numerical Investigation of Heat Transfer of Non-Newtonian Power-Law Fluids Around a Triangular Prism in Time-Periodic Regime

Laminar heat transfer characteristics of non-Newtonian power-law fluids from a long (heated) triangular prism in the time-periodic regime have been explored. Momentum and energy equations were solved using finite volume methodology over the range of non-dimensional control parameters: Reynolds number (Re) = 50–150, Prandtl number (Pr) = 1–50 and power-law index (n) = 0.4–1.8. For the fixed Pr, the local and the time-averaged Nusselt numbers increase with rising Re irrespective of n. However, the local and the time-averaged Nusselt numbers decrease as the fluid nature alters from pseudo-plastic (n 1) for the fixed Re and Pr. Maximum enhancements in the values of time-averaged Nusselt numbers for Pr = 10, 20 and 50 with respect to Pr = 1 are observed to be approximately 180, 273 and 438%, respectively. Further, it is observed that for the fixed n and Pr, the Colburn heat transfer factor (or the jh factor) decreases with rising Re. Finally, the various values of the jh factor at different Re, n and Pr have been correlated via a simple expression, thus enabling its estimation in a new application.

Computational modeling of unsteady third-grade fluid flow over a vertical cylinder: A study of heat transfer visualization

Abstract The present paper aims to investigate the effect of Prandtl number for unsteady third-grade fluid flow over a uniformly heated vertical cylinder using Bejan’s heat function concept. The mathematical model of this problem is given by highly time-dependent non-linear coupled equations and are resolved by an efficient unconditionally stable implicit scheme. The time histories of average values of momentum and heat transport coefficients as well as the steady-state flow variables are displayed graphically for distinct values of non-dimensional control parameters arising in the system. As the non-dimensional parameter value gets amplified, the time taken for the fluid flow variables to attain the time-independent state is decreasing. The dimensionless heat function values are closely associated with an overall rate of heat transfer. Thermal energy transfer visualization implies that the heat function contours are compact in the neighborhood of the leading edge of the hot cylindrical wall. It is noticed that the deviations of flow-field variables from the hot wall for a non-Newtonian third-grade fluid flow are significant compared to the usual Newtonian fluid flow.

Looping tracks associated with tropical cyclones approaching an isolated mountain. Part I: Essential parameters

Essential parameters for making a looping track when a westward-moving tropical cyclone (TC) approaches a mesoscale mountain are investigated by examining several key nondimensional control parameters with a series of systematic, idealized numerical experiments, such as U/Nh, V max/Nh, U/fL x , V max/fR, h/L x , and R/L y . Here U is the uniform zonal wind velocity, N the Brunt–Vaisala frequency, h the mountain height, f the Coriolis parameter, V max the maximum tangential velocity at a radius of R from the cyclone center and L x is the halfwidth of the mountain in the east–west direction. It is found that looping tracks (a) tend to occur under small U/Nh and U/fL x , moderate h/L x , and large V max/Nh, which correspond to slow movement (leading to subgeostrophic flow associated with strong orographic blocking), moderate steepness, and strong tangential wind associated with TC vortex; (b) are often accompanied by an area of perturbation high pressure to the northeast of the mountain, which lasts for only a short period; and (c) do not require the existence of a northerly jet. The nondimensional control parameters are consolidated into a TC looping index (LI), $$ \frac{{U^{2} R^{2} }}{{V_{\hbox{max} }^{2} hL_{y} }} $$ , which is tested by several historical looping and non-looping typhoons approaching Taiwan’s Central Mountain Range (CMR) from east or southeast. It is found that LI < 0.0125 may serve as a criterion for looping track to occur.

Dynamics of a macroscopic elastic fibre in a polymeric cellular flow

We study the dynamics and transport of an elastic fibre in a polymeric cellular flow. The macroscopic fibre is much larger than the infinitesimal immersed polymer coils distributed in the surrounding viscoelastic fluid. Here we consider low-Reynolds-number flow using the Navier–Stokes/Fene-P equations in a two-dimensional, doubly periodic domain. The macroscopic fibre supports both tensile and bending forces, and is fully coupled to the viscoelastic fluid using an immersed boundary framework. We examine the effects of fibre flexibility and polymeric relaxation times on fibre buckling and transport as well as the evolution of polymer stress. Non-dimensional control parameters include the Reynolds number, the Weissenberg number, and the elasto-viscous number of the macroscopic fibre. We find that large polymer stresses occur in the fluid near the ends of the fibre when it is compressed. In addition, we find that viscoelasticity hinders a fibre’s ability to traverse multiple cells in the domain.

Linear and nonlinear dynamics of pulsatile channel flow

The dynamics of small-amplitude perturbations, as well as the regime of fully developed nonlinear propagating waves, is investigated for pulsatile channel flows. The time-periodic base flows are known analytically and completely determined by the Reynolds number $Re$ (based on the mean flow rate), the Womersley number $Wo$ (a dimensionless expression of the frequency) and the flow-rate 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 $Wo\geqslant 14$), 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 $Wo\simeq 7$. 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 base-flow 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 three-dimensional perturbations, the linear and nonlinear dynamics are systematically explored and characterized.

Atmospheric superrotation in an idealized GCM: Parameter dependence of the eddy response

Idealized Earth-like general circulation models (GCMs) have been extensively used to study superrotation on so-called “slowly rotating” bodies like Venus and Titan, however they tend to have difficulty producing superrotation if only the rotation rate is reduced to Titan- or Venus-like values. The Rossby number, RO=U/2ΩL, which characterizes the influence of rotation on the circulation, is small for Earth but large for both Venus and Titan. However, the differences in other nondimensional control parameters are often ignored in idealized planetary circulation studies. In this study we use a simplified Earth-like GCM to demonstrate the importance of the other nondimensional parameters in obtaining a superrotating flow, and identify the wave-modes responsible for generating and maintaining superrotation. We show that superrotation only emerges on a planet of slow rotation rate if the atmospheric thermal inertia is simultaneously increased; alternatively, superrotation is obtained if the only planetary radius is reduced. When only the rotation rate is reduced, a nearly axisymmetric circulation with intense Hadley cells is produced that prevents strong and persistent winds over the equator. The mechanism for generating and maintaining superrotation in the model involves a coupling between equatorial and high-latitude waves. However, the generation involves equatorial Kelvin-like waves and maintenance involves equatorial Rossby-like waves.

Numerical simulations of the flapping of a three-dimensional flexible plate in uniform flow

The self-sustained flapping of a three-dimensional flexible plate in a uniform viscous flow is numerically simulated with a fictitious domain method. The effects of the various non-dimensional control parameters including the Reynolds number, the density ratio, the dimensionless shear modulus, the length–thickness ratio, and the width–length ratio on the flapping of the plate are investigated. The results show that there exist two flapping modes: symmetrical and asymmetrical flapping about the centerline in the spanwise direction. Near the critical point a decrease in the plate width–length ratio, or the increase in the Reynolds number or the reduced velocity (a combination of the density ratio, the dimensionless shear modulus, and the length–thickness ratio) can make symmetric (or nearly symmetric) flapping become asymmetric. It is found that the flapping amplitude is mainly controlled by the density ratio and the dimensionless elastic modulus, while the frequency by the density ratio and the length–thickness ratio. In addition, the flapping amplitude and frequency are affected significantly by the confinement effect of the computational domain, and normally enhanced as the confinement effect becomes stronger. The effects of the plate width and the mass ratio (i.e., the ratio of the length–thickness and density ratios) on the critical reduced velocities are examined. The results indicate that when the fluid–plate mass ratio (or the plate length–thickness ratio) is relatively small there exist two significantly different critical velocities for the flapping instability, depending on the strength of initial plate deformation, a hysteresis phenomenon. No obvious hysteresis can be observed when the fluid–plate mass ratio (or the plate length–thickness ratio) is large.

Nondimensional control parameters governing the behavior of one‐dimensional fault slip: Effects of shear heating, inelastic pore creation, and fluid flow

We theoretically study the dynamic slip of a one‐dimensional fault considering thermoporoelastic effects including shear heating, fluid flow, and inelastic pore creation. We find that the qualitative nature of system behavior can be understood in terms of three nondimensional parameters: Su, S′u, and P*0. The parameter Su represents the relative dominance of the effect of inelastic pore creation on the fluid pressure change over that of shear heating, while S′u is associated with the dominance of the fluid flow effect over the effect of shear heating. The parameter P*0 denotes the initial fluid pressure. We observe the slip‐weakening and slip‐strengthening behaviors in the ranges Su &lt; −P*0 and Su &gt; −P*0, respectively, irrespective of the value of S′u. We can understand both fast and slow fault slips comprehensively on the basis of our model. Ordinary high‐speed fault slip can be simulated with Su slightly larger than −P*0, S′u near the lower limit, and σs0 ∼ 100 MPa. We need the conditions of Su ≫ −P*0, sufficiently large values for S′u, and sufficiently smaller values for σs0 to model much slower fault slip. Our present study also provides a model for the mechanism of negligible frictional heating on the San Andreas fault. We actually found that the temperature rise on the fault is negligibly low in the range of slip strengthening if the values of σs0 and S′u are small enough and/or the value of Su is significantly larger than a threshold.

A numerical model for coupled fluid flow and matrix deformation with applications to disequilibrium compaction and delta stability

A model is developed which couples fully saturated porous compaction to the viscous‐plastic deformation of the skeleton matrix. The Darcy fluid flow during compaction is described by an advection‐diffusion equation for the excess pressure with two source/sink terms that depend on the mechanical compressibility and viscous compaction of the pore space, the latter representing the effect of pressure solution. The incompressible deformation of the composite medium is described by a force balance equation and its rheology can be viscous, plastic, or viscoplastic (Bingham material). For the plastic and viscoplastic cases, the coupling between the compacting and plastically deforming parts of the system is through the Drucker‐Prager frictional‐plastic yield criterion modified by Terzaghi's principle, so that the yield strength depends on the effective dynamical pressure. The coupled system is solved using a two‐dimensional (2‐D) finite element method. Two problems are solved to demonstrate the behavior of our theory. The first considers compaction of a uniform sediment layer. The numerical results agree with the predictions of the nondimensional control parameters and previously published results. The second problem concerns 2‐D kinematic progradation of deltaic sediments. Substratum and delta sediments have the same compaction properties and a Bingham rheology during deviatoric deformation, such that the delta undergoes linear postyield viscous flow. For certain depositional regimes, overpressure is generated. When pore pressures approach critical values, yielding occurs and the delta front fails and becomes unstable, spreading gravitationally under its own weight. The flow velocity is limited to geological rates by the Bingham viscosity. For the range of parameter values considered, pressure solution is the most effective mechanism for generating near‐lithostatic fluid pressures that lead to initial failure, and it appears that mechanical compaction hardly contributes to the fluid overpressure at this stage.

Dynamics of Track Deflection Associated with the Passage of Tropical Cyclones over a Mesoscale Mountain

Abstract In this study, the fifth-generation Pennsylvania State University–National Center for Atmospheric Research (PSU–NCAR) Mesoscale Model (MM5) was used to simulate Supertyphoon Bilis (in 2000) and Typhoon Toraji (in 2001) in order to investigate the dynamics of track deflection caused by the Central Mountain Range (CMR) of Taiwan. The MM5 predicted the track of each storm reasonably well. Bilis was stronger and had a relatively faster forward motion, which helped make the track continuous as it crossed the CMR. The use of a “bogus” vortex in the initialization process helped produce a storm closer to the observed strength. Bilis is a classic example of a typhoon crossing Taiwan with a continuous track. For comparison, Typhoon Toraji, a typical typhoon having a discontinuous track, was also studied. Toraji was weaker and had a relatively slower forward speed, which prevented the original low center from crossing over the CMR and forced more air parcels to go around the northern tip of the CMR. As a result, it produced a vortex and a secondary low center on the lee. Potential vorticity banners on the north side of the CMR acted to organize the secondary low and the lee vortex. With time, the low-level circulation extended into the upper levels, completing the formation of the secondary center. Remnants of the initial center crossed over the CMR and were entrained into the secondary center. Nondimensional control parameters for track continuity and deflection from idealized studies are calculated for Bilis and Toraji. The results are consistent with the theory proposed in Lin et al. For tropical cyclones (TCs) approaching Taiwan from the southeast, the conceptual model proposed by Lin et al. for continuous and discontinuous tracks was applied. For continuous tracks over the CMR, the blocking effect on the outer circulation of the vortex is weak and the vorticity advection around the northern tip is strong due to an intense TC. Weak TCs tend to be totally blocked by the CMR.

Scaling Baroclinic Eddy Fluxes: Vortices and Energy Balance

Abstract The eddy heat flux generated by the statistically equilibrated baroclinic instability of a uniform, horizontal temperature gradient is studied using a two-mode f-plane quasigeostrophic model. An overview of the dependence of the eddy diffusivity D on the bottom friction κ, the deformation radius λ, the vertical variation of the large-scale flow U, and the domain size L is provided by numerical simulations at 70 different values of the two nondimensional control parameters κλ/U and L/λ. Strong, axisymmetric, well-separated baroclinic vortices dominate both the barotropic vorticity and the temperature fields. The core radius of a single vortex is significantly larger than λ but smaller than the eddy mixing length ℓmix. On the other hand, the typical vortex separation is comparable to ℓmix. Anticyclonic vortices are hot, and cyclonic vortices are cold. The motion of a single vortex is due to barotropic advection by other distant vortices, and the eddy heat flux is due to the systematic migration of hot anticyclones northward and cold cyclones southward. These features can be explained by scaling arguments and an analysis of the statistically steady energy balance. These arguments result in a relation between D and ℓmix. Earlier scaling theories based on coupled Kolmogorovian cascades do not account for these coherent structures and are shown to be unreliable. All of the major properties of this dilute vortex gas are exponentially sensitive to the strength of the bottom drag. As the bottom drag decreases, both the vortex cores and the vortex separation become larger. Provided that ℓmix remains significantly smaller than the domain size, then local mixing length arguments are applicable, and our main empirical result is ℓmix ≈ 4λ exp(0.3U/κλ).

Traveling waves in ferroelectric smectic-C liquid crystals.

This article investigates, by means of Lie point symmetries, traveling wave solutions to a dynamic equation that frequently arises in the theory of ferroelectric smectic- C liquid crystals under the influence of an electric field. The equation considered has three sinusoidal nonlinearities and possible time-dependent solutions are discussed in the context of minima and maxima of the electric energy density for these liquid crystals: solutions travel between such constant equilibrium states. Implicit solutions to an approximation of the governing dynamic equation are determined. Nondimensional control parameters that characterize changes in the availability of equilibrium states as the magnitude of the field increases are also identified.

The effect of swirl on jets and wakes: Linear instability of the Rankine vortex with axial flow

The effect of swirl on jets and wakes is investigated by analyzing the inviscid spatiotemporal instability of the Rankine vortex with superimposed plug flow axial velocity profile. The linear dispersion relation is derived analytically as a function of two nondimensional control parameters: the swirl ratio S and the external axial flow parameter a (a&amp;gt;−0.5 for jets, a&amp;lt;−0.5 for wakes). For each azimuthal wave number m, there exists a single unstable Kelvin–Helmholtz mode and an infinite number of neutrally stable inertial waveguide modes. Swirl decreases the temporal growth rate of the axisymmetric Kelvin–Helmholtz mode (m=0), which nonetheless remains unstable for all axial wave numbers. For helical modes (m≠0), small amounts of swirl lead to the widespread occurrence of direct resonances between the unstable Kelvin–Helmholtz mode and the inertial waveguide modes. Such interactions generate, in the low wave number range, neutrally stable wave number bands separated by bubbles of instability. As S increases above a critical value, all bubbles merge to give rise to unstable wave numbers throughout, but the growth rate envelope decreases continuously with increasing swirl. In the high wave number range, negative helical mode growth rates are enhanced for small swirl and decrease continuously for large swirl, while positive helical mode growth rates monotonically decrease with increasing swirl. For a given positive swirl, negative modes are more unstable than their positive counterparts, although their growth rates may not necessarily be larger than in the nonrotating case. The absolute/convective nature of the instability in swirling jets and wakes is determined in an a−S control parameter plane by numerical implementation of the Briggs–Bers criterion. In the absence of swirl, jets (a&amp;gt;−0.5) become absolutely unstable (AI) to the axisymmetric mode m=0 only for a sufficiently large external axial counterflow a&amp;lt;−0.15. AI is found to be significantly enhanced by swirl: for S&amp;gt;1.61, AI occurs, even for coflowing jets (a&amp;gt;0). As S is gradually increased, the transitional mode to AI successively becomes m=0, −1, −2, −3, etc. In the absence of swirl, wakes (a&amp;lt;−0.5) become absolutely unstable to the bending modes m=±1 only for sufficiently large counterflow 1+a&amp;gt;0.091. For S&amp;gt;0.47, AI occurs even for coflowing wakes (a&amp;lt;−1) and, as S increases, the transitional mode to AI successively becomes m=−1, −2, −3, etc. This instability analysis is found to provide a preliminary estimate of the critical Rossby number for the onset of vortex breakdown: for zero external axial flow jets (a=0), absolute/convective transition first takes place at a Rossby number Ro≡S−1∼0.62, which very favorably compares with available experimental and numerical threshold values for vortex breakdown onset.

Dynamics of a freely evolving, two-dimensional granular medium.

We consider the dynamics of an ensemble of identical, inelastic, hard disks in a doubly periodic domain. Because there is no external forcing the total energy of the system is monotonically decreasing so that this idealized granular medium is ``cooling down.'' There are three nondimensional control parameters: the coefficient of restitution r, the solid fraction \ensuremath{\nu}, and the total number of disks in the domain N. Our goal is a comprehensive description of the phenomenology of granular cooling in the (r,\ensuremath{\nu},N) parameter space. Previous studies have shown that granular cooling results in the formation of structures: both the mass and the momentum spontaneously become nonuniform. Four different regimes (kinetic, shearing, clustered, and collapsed) have been identified. Starting with the almost elastic case, in which r is just less than 1, the kinetic regime resembles a classical nondissipative gas in which there are no structures. When r is decreased (with fixed N and \ensuremath{\nu}) the system evolves into the shearing regime in which most of the energy and momentum resides in the gravest hydrodynamic shear mode. At still smaller values of r the clustered regime appears as an extended transient. Large clusters of disks form, collide, breakup, and reform. From the clustered state the gas eventually either evolves into the shearing regime or, alternatively, collapses. The collapsed regime is characterized by a dynamical singularity in which a group of particles collides infinitely often in a finite time. While each individual collision is binary, the space and time scales decrease geometrically with the cumulative number of collisions so that a multiparticle interaction occurs. The regime boundaries (i.e., the critical values of r) in the (N,\ensuremath{\nu}) plane have been delineated using event-driven numerical simulations. Analytic considerations show that the results of the simulations can be condensed by supposing that the critical values of r depend only on N and \ensuremath{\nu} through the optical depth, \ensuremath{\lambda}\ensuremath{\equiv}d\ensuremath{\surd}N\ensuremath{\pi}\ensuremath{\nu}/2 where d is the disk diameter. \textcopyright{} 1996 The American Physical Society.

Turbomachinery Committee Best 1994 Paper Award: Dynamic Control of Rotating Stall in Axial Flow Compressors Using Aeromechanical Feedback

Dynamic control of rotating stall in an axial flow compressor has been implemented using aeromechanical feedback. The control strategy developed used an array of wall jets, upstream of a single-stage compressor, which were regulated by locally reacting reed valves. These reed valves responded to the small-amplitude flow-field pressure perturbations that precede rotating stall. The valve design was such that the combined system, compressor plus reed valve controller, was stable under operating conditions that had been unstable without feedback. A 10 percent decrease in the stalling flow coefficient was obtained using the control strategy, and the extension of stable flow range was achieved with no measurable change in the steady-state performance of the compression system. The experiments demonstrate the first use of aeromechanical feedback to extend the stable operating range of an axial flow compressor, and the first use of local feedback and dynamic compensation techniques to suppress rotating stall. The design of the experiment was based on a two-dimensional stall inception model, which incorporated the effect of the aeromechanical feedback. The physical mechanism for rotating stall in axial flow compressors was examined with focus on the role of dynamic feedback in stabilizing compression system instability. As predicted and experimentally demonstrated, the effectiveness of the aeromechanical control strategy depends on a set of nondimensional control parameters that determine the interaction of the control strategy and the rotating stall dynamics.

Nonlinear dynamics of forced transitional jets: periodic and chaotic attractors

Conclusive experimental evidence is presented for the existence of a low-dimensional temporal dynamical system in an open flow, namely the near field of an axisymmetric, subsonic free jet. An initially laminar jet (4 cm air jet in the Reynolds number range 1.1 × 104[Lt ]ReD× 9.1 × 104) with a top-hat profile was studied using single-frequency, longitudinal, bulk excitation. Two non-dimensional control parameters – forcing frequencyStD(≡fexD/Ue, wherefezis the excitation frequency,Dis the jet exit diameter andUeis the exit velocity) and forcing amplitudeaf(≡u’f/Ue, whereu’fis the jet exit r.m.s. longitudinal velocity fluctuation at the excitation frequency) – were varied over the ranges 10-4&lt;af&lt; 0.3 and 0.3 &lt;StD&lt; 3.0 in order to construct a phase diagram. Periodic and chaotic states were found over large domains of the parameter space. The periodic attractors correspond to stable pairing (SP) and stable double pairing (SDP) of rolled-up vortices. One chaotic attractor, near SP in the parameter space, results from nearly periodic modulations of pairing (NPMP) of vortices. At large scales (i.e. approximately the size of the attractor) in phase space, NPMP exhibits approximately quasi-periodic behaviour, including modulation sidebands around ½fexinu-spectra, large closed loops in its Poincaré sections, correlation dimensionv∼ 2 and largest Lyapunov exponent λ1∼ 0. But investigations at smaller scales (i.e. distances greater than, but of the order of, trajectory separation) in phase space reveal chaos, as shown byv&gt; 2 and λ1&gt; 0. The other chaotic attractor, near SDP, results from nearly periodic modulations of the first vortex pairing but chaotic modulations of the second pairing and has a broadband spectrum, a dimension 2.5 [Lt ]v[Lt ] 3 and the largest Lyapunov exponent 0.2 [Lt ] λ1[Lt ] 0.7 bits per orbit (depending on measurement locations in physical and parameter spaces).A definition that distinguishes between physically and dynamically open flows is proposed and justified by our experimental results. The most important conclusion of this study is that a physically open flow, even one that is apparently dynamically open due to convective instability, can exhibit dynamically closed behaviour as a result of feedback. A conceptual model for transitional jets is proposed based on twodimensional instabilities, subharmonic resonance and feedback from downstream vortical structures to the nozzle lip. Feedback was quantified and shown to affect the exit fundamental–subharmonic phase difference ϕ – a crucial variable in subharmonic resonance and, hence, vortex pairing. The effect of feedback, the sensitivity of pairings to ϕ, the phase diagram, and the documented periodic and chaotic attractors demonstrate the validity of the proposed conceptual model.