Azimuthal Wavenumber Research Articles

Temperature Gradient Effects on Stator Boundary-Layer Stability in a Rotor–Stator Cavity

A comprehensive approach, combining theoretical analysis and direct numerical simulation, is employed in this study to investigate the influence of temperature gradient on the stability phenomenon of the stator boundary layer in a rotor–stator cavity. In contrast to previous studies, a temperature term is introduced to account for centrifugal buoyancy within the cavity. The focus is on analyzing the transitional behavior and the effects of centrifugal buoyancy on the boundary layers of the stationary disk under operating conditions characterized by a Reynolds number of Re=2.5×104. The investigation reveals that this temperature gradient significantly affects the base flow and alters the instability governing the boundary-layer transition on the stationary disk. Specifically, the centrifugal buoyancy induced by the higher temperature on the stationary side weakens the spiral mode perturbations without inducing changes in the azimuthal wavenumber of the spiral mode. However, when the centrifugal buoyancy effect exceeds a certain threshold, it directly suppresses the generation of the spiral mode and induces the formation of low-radius circular waves, thereby promoting a more stable boundary layer. This research emphasizes the importance of considering temperature variations in the rotor–stator cavity for improved control of stability within the boundary-layer flow.

Floquet-based analysis on three-dimensional non-axisymmetric instabilities in oscillatory-driven Taylor–Couette flows and their low-frequency asymptotic behavior using Wentzel–Kramers–Brillouin method

This paper revisits the linear stability analysis of oscillatory-driven flows between two oscillating cylinders against non-axisymmetric disturbances. This study is motivated by the lack of a sufficiently reliable theoretical analysis giving insight into the experimentally observed spiral-like non-axisymmetric patterns when the cylinders are counter-oscillating. A new generalized time-dependent algebraic eigenvalue problem is constructed from the linearized set of the three-dimensional Navier–Stokes equations around the purely azimuthal basic state. Numerical evaluation of the critical eigenvalues combining both Floquet theory and spectral method reveals the existence of frequency ranges where this basic state becomes unstable against three-dimensional non-axisymmetric disturbances before it does so for two-dimensional axisymmetric ones. Indeed, as the oscillation frequency of the cylinders increases, the azimuthal wave number of the critical eigensolution is found to change from 0 to 2 to 1 and then back to 0. The primary bifurcation exchange between two instability modes with different azimuthal wave numbers occurs via different types of codimension-2 bifurcation points giving rise to discontinuities in the critical axial wave number where reversing and non-reversing non-axisymmetric Taylor vortex flows are identified. In addition, by extending our numerical calculations to the co-oscillating case, we show that the axisymmetric disturbances are the most unstable confirming thus existing experimental findings. Furthermore, a Wentzel–Kramers–Brillouin (WKB) analysis is performed to shed light on the asymptotic behavior of these time-dependent flows in the low-frequency limit when the cylinders are slowly oscillating.

Focusing Highly Squinted FMCW-SAR Data Using the Modified Wavenumber-Domain Algorithm

Three-dimension acceleration, azimuth dependence, and real-time processing are the main issues to be solved in highly-squinted frequency-modulation continuous-wave (FMCW) SAR with a curved trajectory. To overcome these issues, a modified wavenumber-domain algorithm is developed in this paper. The proposed modified wavenumber-domain algorithm mainly includes the following six aspects. Firstly, a modified range model considering three-dimension acceleration for a curved trajectory is established. Then, the spectrum compression and rotation operations are performed to guarantee further processing. Afterward, residual phase terms introduced by acceleration are bulk compensated for. Moreover, the azimuth dependence is eliminated by the azimuth resampling to achieve uniform focusing. Subsequently, the range-azimuth coupling is removed by the modified Stolt interpolation. Finally, to avoid image aliasing, the azimuth wavenumber domain is selected to focus final image. The experimental results of both numerical and raw-measured FMCW-SAR data demonstrate the superiority of the proposed method.

On the preferred flapping motion of round twin jets

Linear stability theory (LST) is often used to model the large-scale flow structures in the turbulent mixing region and near pressure field of high-speed jets. For perfectly expanded single round jets, these models predict the dominance of azimuthal wavenumbers $m=0$ and $m = 1$ helical modes for the lower frequency range, in agreement with empirical data. When LST is applied to twin-jet systems, four solution families appear following the odd/even behaviour of the pressure field about the symmetry planes. The interaction between the unsteady pressure fields of the two jets also results in their coupling. The individual modes of the different solution families no longer correspond to helical motions, but to flapping oscillations of the jet plumes. In the limit of large jet separations, when the jet coupling vanishes, the eigenvalues corresponding to the $m=1$ mode in each family are identical, and a linear combination of them recovers the helical motion. Conversely, as the jet separation decreases, the eigenvalues for the $m=1$ modes of each family diverge, thus favouring a particular flapping oscillation over the others and preventing the appearance of helical motions. The dominant mode of oscillation for a given jet Mach number $M_j$ and temperature ratio $T_R$ depends on the Strouhal number $St$ and jet separation $s$ . Increasing both $M_j$ and $T_R$ independently is found to augment the jet coupling and modify the $(St,s)$ map of the preferred oscillation mode. Present results predict the preference of two modes when the jet interaction is relevant, namely varicose and especially sinuous flapping oscillations on the nozzles’ plane.

Swirling instability of coaxial liquid jet in gas surroundings

Linear instability analysis of an inviscid coaxial swirling jet is carried out by deriving an analytical dispersion relation of perturbation growth. The azimuthal Rankine vortex and the axial discontinuous velocity distribution are utilized as the jet basic flow. Due to the existence of double interfaces, the instability mechanisms of the coaxial swirling jet are much more complex than those of the single-layered swirling jet. The effects of control parameters (including the swirling ratio, the Weber number, the jet radius ratio, the velocity ratios between different fluids, and the azimuthal velocity jump at the inner interface) on the temporal instability of coaxial swirling jet with different azimuthal modes are studied. By comparing the growth rate of different azimuthal modes, the predominant mode that determines the jet breakup is identified. The results indicate that an increase in the swirling ratio, the Weber number, and the radius ratio can lead to predominant mode transition to larger azimuthal wavenumbers. The velocity ratio between the inner jet and the annular jet and that between the surrounding fluid and the annular jet mainly affect the axial Kelvin–Helmholtz (KH) instability. An enhancement of the KH instability leads to the jet breakup with smaller azimuthal wavenumbers. The azimuthal velocity jump affects the azimuthal KH instability, the centrifugal instability, and the Coriolis instability simultaneously, thus leading to a multiple influence on modes transition. The phase-diagram of the predominant modes is further given, showing that the relative importance between the centrifugal force and the interfacial tension plays a significant role on the transition of predominant modes.

Global stability analysis of axisymmetric boundary layer on a rotating circular cylinder

In this manuscript, we comprehensively investigated the global stability of the incompressible axisymmetric boundary layer (ABL) developed on a rotating circular cylinder. A finite length of a rotating cylinder was considered in an axial stream of incompressible fluid. The rotation of a cylinder produces centrifugal force in the radial direction, which produces a pressure gradient (PG) in the radial direction. The axial flow (U∞) develops shear in the BL which has a stabilization effect on the BL. Thus, the Reynolds number (Re) and rotation rate (S) are two important control parameters of the BL and they are calculated based on a cylinder radius. The spectral collocation method has been used to discretize the linearized Navier–Stokes equations (LNS) and Arnold’s iterative algorithm for the numerical solution of the 2D eigenvalues problem. The computations were performed for Reynolds numbers Re=2600, 5200, and 20,800, rotation rates S=0.5, 1.0, and 2.0 for azimuthal wave-numbers N=0, 1, 2, and 3. Thus, the effects of S, Re, and N were studied on the stability of the rotating ABL. The temporal and spatial properties of the small disturbances were studied to understand the effect of shear and centrifugal force by varying S and Re. The increased rotation of a cylinder makes the BL unstable while increased axial flow stabilizes the ABL. The small disturbances are found globally unstable for N=1 at a small Re with an even small S.

Large-scale streaks in a turbulent bluff body wake

A turbulent circular disk wake database (Chongsiripinyo & Sarkar, J. Fluid Mech., vol. 885, 2020) at Reynolds number $Re = U_\infty D/\nu = 5 \times 10^{4}$ is interrogated to identify the presence of large-scale streaks – coherent elongated regions of streamwise velocity. The unprecedented streamwise length – until $x/D \approx 120$ – of the simulation enables investigation of the near and far wakes. The near wake is dominated by the vortex shedding (VS) mode residing at azimuthal wavenumber $m=1$ and Strouhal number $St = 0.135$ . After filtering out the VS structure, conclusive evidence of large-scale streaks with frequency $St \rightarrow 0$ , equivalently streamwise wavenumber $k_x \rightarrow 0$ in the wake, becomes apparent in visualizations and spectra. These streaky structures are found throughout the simulation domain beyond $x/D \approx 10$ . Conditionally averaged streamwise vorticity fields reveal that the lift-up mechanism is active in the near as well as the far wake, and that ejections contribute more than sweeps to events of intense $-u'_xu'_r$ . Spectral proper orthogonal decomposition is employed to extract the energy and the spatiotemporal features of the large-scale streaks. The streak energy is concentrated in the $m=2$ azimuthal mode over the entire domain. Finally, bispectral mode decomposition is conducted to reveal strong interaction between $m=1$ and $St = \pm 0.135$ modes to give the $m=2$ , $St \rightarrow 0$ streak mode. Our results indicate that the self-interaction of the VS mode generates the $m=2$ , $St \rightarrow 0$ streamwise vortices, which leads to streak formation through the lift-up process. To the authors’ knowledge, this is the first study that reports and characterizes large-scale low-frequency streaks and the associated lift-up mechanism in a turbulent wake.

Multipoint observations of compressional Pc5-6 pulsations in the dawn side magnetosphere: A case study

We present a case study of compressional Pc5-6 pulsations observed by THEMIS-D and -E in the dawn sector of the magnetosphere following 1100 UT on January 22, 2010. The pulsations were observed during a moderate substorm triggered by the IMF discontinuity and an accompanying change in the solar wind dynamic pressure at 1045 UT. The event can be considered a stormtime Pc5-6 event. We determine the characteristics of the waves, including their velocity, azimuthal wave number m and wavelength. The arrival of particles injected by the substorm caused favorable conditions for the development of the mirror-mode instability and therefore initiated the pulsations. A decrease in the solar wind density and dynamic pressure at 1250 UT was seen in the dayside magnetosheath, corresponding to diminished dayside magnetospheric magnetic fields strengths and terminated the compressional Pc5-6 pulsations. We deduce that the pulsations observed by THEMIS might be internally initiated and externally terminated.

Subvortices within a Numerically Simulated Tornado: The Role of Unstable Vortex Rossby Waves

Abstract Multiple subvortices corresponding to suction vortices in observations are obtained within a simulated tornado for the EF4 tornado case of Funing, China, on 23 June 2016. Within the simulation, the tornado evolves from a one-cell structure with vorticity maximum at its center to a two-cell structure with a ring of vorticity maximum. Five well-defined subvortices develop along the ring. The radial profile of tangential wind across the vorticity ring satisfies the necessary condition of barotropic instability associated with phase-locked, counterpropagating vortex Rossby waves (VRWs) along the ring edges. The phased-locked waves revolve around the parent vortex at a speed less than the maximum azimuthal-mean tangential velocity, agreeing with theoretically predicted VRW phase speed. The radii within which the wave activities are confined are also correctly predicted by the VRW theory where radial group velocity approaches zero. Several other characteristics related to the simulated subvortices agree with VRW theories also. The most unstable azimuthal wavenumber depends on the width and the relative magnitude of vorticity of the vortex ring. Their values estimated from the simulation prior to subvortex formation correctly predict wavenumber 5 as the most unstable. The largest contribution to wave kinetic energy is diagnosed to be from the radial shear of azimuthal wind term, consistent with barotropic instability. Vorticity diagnostics show that vertical vorticity stretching is the primary vorticity source for the intensification and maintenance of the simulated subvortices. Significance Statement Multiple subvortices or suction vortices in tornadoes can produce extreme damage but their cause is not well understood. An intense tornado from China that developed five strong subvortices, along a vorticity ring a distance from the tornado vortex center, was successfully simulated. By examining the propagation and other characteristics of these subvortices and comparing them with theoretical models of vortex Rossby waves (VRWs) that have been studied mostly in the context of typhoons/hurricanes, it is believed that nonlinear growth of unstable VRWs associated with barotropic instability is the primary reason for the development of subvortices within the tornado. The conclusion is further supported by analyses of the primary source of wave growth energy. Vertical vorticity stretching is the main vorticity source for intensifying and maintaining the subvortices at their development and mature stages. The unstable growth of VRWs as the cause of tornado suction vortices has not been analyzed in detail for realistic tornadoes until now.

Mode selection in concentric jets: the steady–steady 1 : 2 resonant mode interaction with O(2) symmetry

The linear and nonlinear stability of two concentric jets separated by a duct wall is analysed by means of global linear stability and weakly nonlinear analysis. Three governing parameters are considered, the Reynolds number based on the inner jet, the inner-to-outer jet velocity ratio ( $\delta _u$ ) and the length of the duct wall ( $L$ ) separating the jet streams. Global linear stability analysis demonstrates the existence of unsteady modes of inherent convective nature, and symmetry-breaking modes that lead to a new non-axisymmetric steady state with a single or double helix. Additionally, we highlight the existence of multiple steady states, as a result of a series of saddle-node bifurcations and its connection to the changes in the topology of the flow. The neutral lines of stability have been computed for inner-to-outer velocity ratios within the range $0 < \delta _u < 2$ and duct wall distances in the interval $0.5 < L < 4$ . They reveal the existence of hysteresis, and mode switching between two symmetry-breaking modes with azimuthal wavenumbers $1:2$ . Finally, the mode interaction is analysed, highlighting the presence of travelling waves emerging from the resonant interaction of the two steady states, and the existence of robust heteroclinic cycles that are asymptotically stable.

Effects of Cold Plasma on the Excitation of Internally Driven ULF Waves by Ring Current Ions Based On the Magnetosphere‐Ionosphere Coupled Model

AbstractInternally driven Pc4‐5 waves are excited in the plasmaspheric drainage plume (PDP) and near the plasmapause. The excitation of ultralow frequency (ULF) waves was investigated by using the magnetosphere‐ionosphere coupled model between Geospace Environment Modeling System for Integrated Studies‐Ring Current (GEMSIS‐RC) and GEMSIS‐POTential solver (GEMSIS‐POT). In order to investigate the effects of cold plasma on the wave excitation, the simulation code to describe the dynamics of cold plasma was included in the model. The model can reproduce the shrink of the plasmasphere on the nightside and the formation of the PDP on the dayside. First, fundamental Pc5 waves are excited through the drift resonance on the dayside. The waves are caused by positive energy gradient of ion phase space density (PSD) at 50–130 keV. Second harmonic waves (drift‐bounce resonance) are generated outside the plasmapause. These two types of ULF waves are also seen in the case of constant density. Unlike the case of constant density, localized eastward propagating Pc4 waves (drift resonance) are seen on the dawnside associated with the azimuthal density gradient. The azimuthal wave number of Pc4 waves is about 70 and anti‐earthward gradient of PSD about 10 keV contributes to wave growth. We also detect fundamental Pc4‐5 waves near the plasmapause on the nightside in the drift resonance with 100–150 keV ions. Simulation results suggest that the plasmapause has an effect to sustain the excitation of Pc4‐5 ULF wave through the drift resonance.

Modal Decomposition Study of the Non-Reactive Flow Field in a Dual-Swirl Combustor

The modal decomposition study of the non-reactive flow field in a dual-swirl combustor is investigated through the large eddy simulation. The formation mechanism and function of various recirculation zones are elaborated by analyzing the time-averaged and instantaneous velocity contours of the center section. The precessing vortex core (PVC) is first visualized by the pressure iso-surface, and the evolution process is presented. Different dimensionality reduction methods are adopted to identify the coherent structures from the flow field. The most energetic spatial structure corresponding to the PVC and its second-order harmonic structure is extracted by the classical proper orthogonal decomposition (POD). The coherent structures with high frequency have relatively low energy content. In addition, a spectral proper orthogonal decomposition (SPOD) method, which can implement spatial-temporal decomposition simultaneously, is introduced to obtain the energy-based spatial structures at all characteristic frequencies. A triple-helix with azimuth wave number m = 3 and a quadruple-helix with azimuth wave number m = 4 are discovered as the third-order and the fourth-order harmonics of single-helix, respectively.

Dynamic mode decomposition of thermocapillary convection in GaAs melt liquid bridge between unequal ends

The effects of the radius ratio on the stability of thermocapillary convection in the GaAs melt (Pr=0.068) liquid bridge, situated between unequal ends, are studied by three-dimensional direct numerical simulation. The simulated results reveal that, for the first time, the GaAs melt flow transforms from a stable axisymmetric flow into a three-dimensional steady flow through the first bifurcation at 0.5≤Γr≤0.9. The bifurcation phenomenon was previously common in the range of small Prandtl numbers(0.001≤Pr≤0.057). Additionally, we observe that the stability of the thermocapillary liquid bridge is initially weakened and then slowly improved as the radius ratio Γr decreases. The second bifurcation occurs with the further increase of Ma, and the flow becomes oscillatory. Remarkably, two different oscillation forms are identified, depending on the difference in the degree of symmetry of the flow field structure during its oscillation process. Once the flow reaches a stable periodic oscillation, the Dynamic Mode Decomposition (DMD) technique is used to study the inherent spatio-temporal evolution of the temperature field. By combining the spectrum obtained from the Fast Fourier Transform (FFT), we provide evidence for the existence of mixed forms with different azimuthal wavenumbers from a dynamic perspective, intuitively displaying the different components of the oscillation and examining the contribution of different modes to the oscillation. The dynamic model analysis demonstrates that oscillation form I is a mixed form with m = 1 and m = 2, while oscillation form II belongs to the form where m = 2 is absolutely dominant.

Static and dynamic stability of pendant drops

Despite the widespread occurrence of pendant drops in nature, there is still a lack of combined studies on their dynamic and static stability. This study focuses on the dynamic and static stability of elongated drops with either a free or pinned contact line on a plane. We first examine static stability for both axisymmetric and non-axisymmetric perturbations subject to volume or pressure constraints. The stability limits for volume and pressure disturbances (axisymmetric) correspond to the maximum volume and pressure of the drops, respectively. Drops with free contact lines are marginally stable to non-axisymmetric perturbations because of their horizontal translational invariance, whereas pinned drops are stable. The linear dynamic stability is then investigated numerically through a boundary element model, restricted to volume disturbances. Results show that when the stability limit is reached, the first zonal mode has a zero frequency, suggesting that the thresholds for static and dynamic stability are essentially equivalent. Furthermore, natural frequencies experience sharp changes as the stability limit is approached. Another zero frequency mode associated with the horizontal motion of the centre of mass is also revealed by the numerical results, reflecting the horizontal translational invariance of drops with free contact lines. Finally, the frequency spectrum modified by gravity is explored, resulting in the identification of five gravity-induced frequency shift patterns. The frequency shifts break the spectral degeneracy for hemispherical drops with free contact lines, leading to various spectral orderings according to polar and azimuthal wavenumbers.

Effect of heat transfer through an interface on convective melting dynamics of phase change materials

The presence of thermocapillary (Marangoni) convection in microgravity may help to enhance the heat transfer rate of phase change materials (PCMs) in space applications. We present a three-dimensional numerical investigation of the nonlinear dynamics of a melting PCM placed in a cylindrical container filled with n-octadecane and surrounded by passive air. The heat exchange between the PCM and ambient air is characterized in terms of the Biot number, when the air temperature has a linear profile. The effect of thermocapillary convection on heat transfer and the topology of the melting front is studied by varying the applied temperature difference between the circular supports and the heat transfer through the interface. The evolution of Marangoni convection during the PCM melting leads to the appearance of hydrothermal instabilities. A new mathematical approach for the nonlinear analysis of emerging hydrothermal waves (HTWs) is suggested. Being applied for the first time to the examination of PCMs, this procedure allows us to explore the nature of the coupling between HTWs and heat gain/loss through the interface, and how it changes over time. We observe a variety of dynamics, including standing and travelling waves, and determine their dominant and secondary azimuthal wavenumbers. Coexistence of multiple travelling waves with different wavenumbers, rotating in the same or opposite directions, is among the most fascinating observations.

Observing Tropical Cyclone Morphology Using RADARSAT-2 and Sentinel-1 Synthetic Aperture Radar Images

Abstract Since the 1960s, meteorological satellites have been able to monitor tropical cyclones and typhoons. Their images have been acquired by passive remote sensing instruments that operate in the visible and infrared bands, where they only display the cloud-top structure of tropical cyclones and make it a challenge to study the air–sea interaction near the sea surface. On the other hand, active remote sensors, such as spaceborne microwave scatterometers and synthetic aperture radars (SARs), can “see” through clouds and facilitate observations of the air–sea interaction processes. However, SAR acquires images and provides the wind field at a much higher resolution, where the eye of a tropical cyclone at surface level can be identified. The backscattered signals received by the SAR can be processed into a high-resolution image and calibrated to represent the normalized radar cross section (NRCS) of the sea surface. In this study, 33 RADARSAT-2 and 102 Sentinel-1 SAR images of Atlantic and Indian Ocean tropical cyclones and Pacific typhoons from 2016 to 2021, which display eye structure, have been statistically analyzed with ancillary tropical cyclone intensity information. To measure the size of the eye, a 34-kt (∼17 m s−1) contour is defined around it and the amount and size of pixels within the eye is utilized to provide its area in square kilometers. Additionally, an azimuthal wavenumber for each shape of the eye was assigned. Results showed that eye areas increase with decreasing wind speed and increasing wavenumber and demonstrate that SAR-derived data are useful for studying tropical cyclones at the air–sea interface and provide results of these behaviors closely to data derived from best track archives.

Direct numerical simulation of slender cones with variable nose bluntness based on graphics processing unit computation

Direct numerical simulation with up to 10×109 scale grid points based on graphics processing unit computation is carried out to investigate the bluntness effect on the hypersonic boundary-layer transition over a slender cone with zero angle of attack at Mach 6. Four cases with the nose radii of 1, 10, 20, and 40 mm are conducted, and the corresponding Reynolds number based on the nose radius varies from 1.0×104 to 4.0×105. Random disturbances with a broad spectrum of frequencies and a wide range of azimuthal wavenumbers were applied to the wall to simulate disturbances caused by wall roughness. The numerical results show that as the nose tip radius increases, the transition position gradually moves downstream with increased transition region. For the case with a nose radius of 1 mm, the flow transition and entropy swallowing occur almost simultaneously, while for other cases, the transition takes place earlier than the entropy swallowing. In consequence, the disturbance amplitude upstream of the transition in the 1 mm case is much larger than that of other cases. To further study the mechanism of the transition, the frequency spectrum analysis is carried out. It is found that all cases exhibit two characteristic frequencies within the transition region, i.e., the high frequency and extremely low frequency. Owing to the influence of the entropy layer, the characteristic high frequency of the 1 mm case is significantly higher than that of other cases. With the increase in the nose radius, the characteristic frequency of the high frequency decreases gradually.

Temporal instability of the liquid film formed by a liquid jet impinging on a curved cylindrical wall

The linear temporal instability of the confined curved liquid film formed by a liquid jet impinging on a curved wall has been investigated. The corresponding dispersion relation between the non-dimensional wave growth rate and wave number was derived. The dispersion relation further involves the influences of the local radius of curvature and the azimuthal angle of the liquid film. The dispersion relation could be verified with previous researches, and the optimal wavelength has been verified with experiments. Temporal stability analysis shows that the effects of gas-to-liquid velocity ratio, gas-to-liquid density ratio, the relative thickness of the liquid phase to the gas phase, and azimuthal wave number are similar to previous researches, but the unstable ranges are smaller than that in previous researches. This paper further studied the effects of the local radius of curvature and azimuthal angle of the liquid film. As the local radius of curvature decreases, the instability of confined curved films increases for the given conditions. And both the growth rate and instability range increase with the increase of the azimuthal angle. The solutions would be helpful to understand the atomization process of the liquid film formed by a liquid jet impinging on a curved wall.

The complex shape of the outer edge of Saturn’s B ring, as observed in Cassini occultation data

We determine the time-variable shape of the outer edge of Saturn’s B ring using the complete set of Cassini radio and stellar occultation data obtained between mid-2005 and the End-of-Mission in late 2017, considerably expanding the range and number of individual ring edge measurements used in previous analyses (Spitale and Porco, 2010; Nicholson et al., 2014a). During this 12-year interval, the dominant m=2 pattern driven by the Mimas 2:1 inner Lindblad resonance completed just over two rotations relative to Mimas, with a circulation period of 5.362 yr, while its radial amplitude varied from a minimum of 4 km to a maximum of 71 km. This circulation pattern has remained essentially unchanged over the full period of the observations. We confirm the existence of four additional perturbations with azimuthal wavenumbers m=1, 3, 4 and 5 and mean amplitudes ranging from 5 to 24 km, which we interpret as normal or edge modes, possibly triggered by viscous overstabilities in the dense B ring (Borderies et al., 1985; Longaretti, 2018). Fits of a simple WKB model to the observed pattern speeds of the edge modes with m≠1 suggest an average surface mass density in the outer 30 km of the B ring of ∼100 g cm−2, somewhat greater than the 50–70 g cm−2 inferred from density and bending waves in most other regions of this ring (Hedman and Nicholson, 2016). The m=1 mode, which extends further into the B ring, yields a more typical value of 60 g cm−2. Surprisingly, all four of these modes exhibit significant librations in their amplitudes and phases, with periods between 2.3 and 8.6 yr and amplitudes of 1.6 to 7.4 km. The origin of these librations is unknown and it is unclear if they are truly periodic and will maintain their amplitudes, periods, and phases over timescales of centuries. Their frequencies do not match those expected for interference between edge modes with varying numbers of radial nodes. Instead, they may represent periodic oscillations in the amplitudes of individual normal modes or nonlinear, non-resonant coupling between normal modes with different values of m, leading to long-term quasi-periodic variations in the mode amplitudes.

Stability characteristics of the spiral Poiseuille flow induced by inner or outer wall rotation

In this paper we reveal, that the stability behavior of the spiral Poiseuille flow in an annular gap with rotating inner or outer cylinder exhibits a striking similarity at low and intermediate rotation rates, while both cases differ significantly at higher rotation rates. Based on the work of Vasanta Ram (2019) we formulate the flow problem by means of the inner swirl parameter (Si), the outer swirl parameter (So) and the curvature parameter (ϵ). Six different values of the curvature parameter {0.005; 0.025; 0.111; 0.25; 0.333; 0.785} and a swirl range of 10−5<Si;So<105 are investigated. We reveal that for the rotating outer cylinder case three characteristic jumps of the critical axial and azimuthal wavenumbers can be identified. Similar features can be identified for the inner rotation case. Based on this, we define three regimes delimited for both the rotating inner and rotating outer cylinder which enables us to perform a systematic comparison. The first regime transition occurs at low values of the swirl parameter (Si; So>10−4) and is accompanied by a rapid decrease of the critical Reynolds number. Our computations reveal that this regime transition does not occur for a small curvature parameter of ϵ= {0.005; 0.025} in the outer rotation case. The second wavenumber jump occurs at higher swirl parameters of around Si>2.5 and So>1.6. While the critical Reynoldsnumber decreases monotonously in the inner cylinder rotating case, we show that in the case of rotating outer cylinder a rapid increase of Rec is observed as So increases further. To gain an insight into the mechanisms involved that induce such wavenumber jumps, a detailed comparative study on the associated Reynolds shear stress distributions is performed. For larger ϵ it is shown that the distributions of the Reynolds shear stresses change significantly over the first jump exhibiting similar features for both flow cases, leading to the conclusion that in both cases the same centrifugal instability is triggered here. Instead, the distributions of the Reynolds shear stresses before and after the second jump are fundamentally different for inner and outer rotation. While the Reynolds shear stresses spread over the whole gap height for the inner rotation case, they are nonzero only close to a small regime near the inner cylinder for the outer rotation case at high rotation rates. Overall, analyzing the shear stresses, we reveal that a similar instability arises in the Spiral Poiseuille flow with inner and outer cylinder rotation at intermediate swirls, but at higher swirl outer rotation seems to give rise to an instability associated with a critical layer that is inherently different to the inner rotation case.