Regular Wave Regime Research Articles

Computational Study of Overtopping Phenomenon over Cylindrical Structures Including Mitigation Structures

Wave overtopping occurring in offshore wind renewable energy structures such as tension leg platforms (TLPs) or semi-submersible platforms is a phenomenon that is worth studying and preventing in order to extend the remaining useful life of the corresponding facilities. The behaviour of this phenomenon has been extensively reported for linear coastal defences like seawalls. However, no referenced study has treated the case of cylindrical structures typical of these applications to a similar extent. The aim of the present study is to define an empirical expression that portrays the relative overtopping rate over a vertical cylinder including a variety of bull-nose type mitigation structures to reduce the overtopping rate in the same fashion as for the linear structures characteristic of shoreline defences. Hydrodynamic interaction was studied by means of an experimentally validated numerical model applied to a non-impulsive regular wave regime and the results were compared with the case of a plain cylinder to evaluate the expected improvement in the overtopping performance. Four different types of parapets were added to the crest of the base cylinder, with different parapet height and horizontal extension, to see the influence of the geometry on the mitigation efficiency. Computational results confirmed the effectivity of the proposed solution in the overtopping reduction, though the singularity of each parapet geometry did not lead to an outstanding difference between the analysed options. Consequently, the resulting overtopping decrease in all the proposed geometries could be modelled by a unique specific Weibull-type function of the relative freeboard, which governed the phenomenon, showing a net reduction in comparison with the cylinder without the geometric modifications. In addition, the relationship between the reduced relative overtopping rate and the mean flow thickness over the vertical cylinder crest was studied as an alternative methodology to assess the potential damage caused by overtopping in real structures without complex volumetric measurements. The collection of computational results was fitted to a useful function, allowing for the definition of the overtopping discharge once the mean flow thickness was known.

Computational fluid dynamics modelling of the regular wave flow regime in air-water downwards annular flows

Although the global flow characteristics of annular gas-liquid flows have been studied experimentally for more than 50 years, the spatiotemporally-resolved details of these flows have remained relatively unexplored until recently, with data provided via advanced experimental methods based, e.g., on optical techniques. Similarly, the numerical modelling of annular flows is still an immature process. The present work aims to provide a computational fluid dynamics (CFD) model based on the volume of fluid (VOF) method for simulating annular gas-liquid flows, setting the stage for a deeper investigation of these flows at global and local scales. The work focuses on the most common downwards annular flow (DAF) flow pattern: the regular wave regime. 3-D and 2-D axisymmetric transient simulations have been performed using a commercial code (ANSYS Fluent 2021 R1). The code is validated through available experimental data regarding topological flow properties, mainly film thickness and wave statistics. The validation results suggest that 3-D simulations are needed to provide predictions that agree with the experimental data, highlighting strong 3-D features in the flow.

Viscosity effects in wind wave generation

We investigate experimentally the influence of the liquid viscosity on the problem of the generation of waves by a turbulent wind at the surface of a liquid, extending the results of Paquier, Moisy and Rabaud [Phys. Fluids {\bf 27}, 122103 (2015)] over nearly three decades of viscosity. The surface deformations are measured with micrometer accuracy using the Free-Surface Synthetic Schlieren method. We recover the two regimes of surface deformations previously identified: the wrinkles regime at small wind velocity, resulting from the viscous imprint on the liquid surface of the turbulent fluctuations in the boundary layer, and the regular wave regime at large wind velocity. Below the wave threshold, we find that the characteristic amplitude of the wrinkles scales as $\nu^{-1/2} u^{* 3/2}$ over nearly the whole range of viscosities, whereas their size are essentially unchanged. We propose a simple model for this scaling, which compares well with the data. We finally show that the critical friction velocity $u^*$ for the onset of regular waves slowly increases with viscosity as $\nu^{0.2}$. Whereas the transition between wrinkles and waves is smooth at small viscosity, including for water, it becomes rather abrupt at large viscosity. Finally, a new regime is found at $\nu > 100-200 \times 10^{-6}$~m$^2$~s$^{-1}$, characterized by a slow, nearly periodic emission of large-amplitude isolated fluid bumps.

Assessing eddy parameterization schemes in a differentially heated rotating annulus experiment

We investigate three of the most common hypotheses underpinning parameterizations of baroclinic eddy fluxes in the context of the differentially heated rotating annulus experiment. The investigation is carried out over a region of parameter space which embraces the onset of baroclinic instability, the regular wave regime and the onset of irregular flows, the latter of which is arguably most relevant to oceanic conditions. Through diagnostics from a 2D axisymmetric and a 3D eddy-resolving numerical model, it was found that the transport of heat by baroclinic eddies is not strictly an adiabatic process but that diffusive ‘ventilation’ of the flow in the thermal boundary layers is significant during the nonlinear development of the flow. Total heat transport, however, is conserved overall. Depending on the stages of flow evolution and on the region in parameter space under consideration, either heat, quasi-geostrophic potential vorticity (QGPV) or relative vorticity (QGRV) may become a suitable variable on which to parameterize baroclinic eddy fluxes in a down-gradient manner. These results raise issues for eddy parameterization schemes that rely on these assumptions in ocean and atmosphere models.

Experiments on transitions of baroclinic waves in a differentially heated rotating annulus

Abstract. Experiments of baroclinic waves in a rotating, baroclinic annulus of fluid are presented for two gap widths. The apparatus is a differentially heated cylindrical gap, rotated around its vertical axis of symmetry, cooled from within, with a free surface, and filled with de-ionised water as working fluid. The surface flow was observed with visualisation technique while thermographic measurements gave a detailed understanding of the temperature distribution and its time-dependent behaviour. We focus in particular on transitions between different flow regimes. Using a wide gap, the first transition from axisymmetric flow to the regular wave regime was characterised by complex flows. The transition to irregular flows was smooth, where a coexistence of the large-scale jet-stream and small-scale vortices was observed. Furthermore, temperature measurements showed a repetitive separation of cold vortices from the inner wall. Experiments using a narrow gap showed no complex flows but strong hysteresis in the steady wave regime, with up to five different azimuthal wave modes as potential steady and stable solutions.

The effect of sloping boundairies on baroclinic instability in two related in ernally heated, rotating fluid systems

Results have been obtained from two internally heated, rotating fluid systems with differing aspect ratios, in which a β-effect (variation in Coriolis parameter with latitude) has been simulated by the inclusion of oppositely sloping endwalls. The boundaries are arranged so that the fluid depth (D) can either increase with radius (∂D∂r > 0) or decrease with radius ∂D∂r < 0). In both systems, stable eddy features we observed over a wide range of rotation rates for each endwall arrangement. While a large number of different azimuthal wavenumber (m) modes are seen in the regular wave regime of the ∂D∂r > 0 endwall experiments, only the gravest mode (m = 1) is seen in the regular wave regime of the ∂D∂r < 0 endwall experiments. Within the former experimental arrangement, the radial scale of the eddy features isto be approximately that of the Rhines scale Lβ = π √2Uβ while in the ∂D∂r < 0 endwall experiments the eddy kinetic energy is seen to penetrate the Lβ scale and accumulate at the largest domain-filling scales. This remarkable difference in the flow structure of the two endwallgments may find a partial explanation in terms of established results on the instability of finite amplitude barotropic Rossby waves, where ∂D∂r < 0 endwalls remove the stability thresholds on wave amplitudes for all but the gravest mode.

Experiments on the structure of baroclinic waves and zonal jets in an internally heated, rotating, cylinder of fluid

In this paper we report a laboratory investigation of the motion within a rotating cylinder of fluid subject to internal heating and to cooling at the outer cylindrical sidewall. The internal heating is supplied by ohmic dissipation as an electric current passes between the outer sidewall and an axial wire. The experiments focus on the formation of eddy features and the associated zonal jets. To identify how the flow regimes generated in this continuously forced system are modified by the presence of a radial depth gradient, experiments have been performed with both horizontal (f-plane) and oppositely sloping boundaries. Endwall configurations which cause the fluid depth (D) to increase with radius (∂D/∂r&amp;gt;0) and to decrease with radius (∂D/∂r&amp;lt;0) have been studied, as the former is applicable to the terrestrial atmosphere and oceans, while the latter may be relevant to deep atmospheres such as those of the giant planets and even planetary interiors. Stable, coherent, regular eddy features are observed over a wide range of rotation rates (Ω) in both horizontal and oppositely sloping endwall experiments. In the f-plane experiments regular modes with azimuthal wavenumbers m=1 and m=2 are observed. The regular wave regime of the ∂D/∂r&amp;gt;0 endwall experiments consists of modes with azimuthal wavenumbers m=1 to 5. Only the mode m=1 is seen in the regular wave regime of the ∂D/∂r&amp;lt;0 endwall experiments. The regular eddy structures produced in the ∂D/∂r&amp;gt;0 (∂D/∂r&amp;lt;0) endwall experiments are seen to be “vertically trapped” close to the bottom (top) boundary, whilst the amplitude of the f-plane modes is found not to vary substantially with depth. Further effects of the radial depth gradient are the observed reduction in the lateral scale of the eddy features in the ∂D/∂r&amp;gt;0 endwall experiments, which leads to the formation of between two and three independent trains of eddies within the lateral domain at sufficiently high values of Ω. Within the non-axisymmetric flow regimes of all three endwall configurations the number of zonal jets observed in the lateral domain of the experiment is larger than expected from the form of the background thermal forcing. The radial scale of the multiple zonal jets seen in the oppositely sloping boundary experiments is, however, much larger than a barotropic Rhines scale Lβ, suggesting that Lβ cannot be used to predict the radial wavenumber of the zonal mean flow in this continuously forced system.

Wave interactions and the transition to chaos of baroclinic waves in a thermally driven rotating annulus

A series of laboratory experiments is presented investigating regular and chaotic baroclinic waves in a high–Prandtl number fluid contained in a rotating vessel and subjected to a horizontal temperature gradient. The study focuses on nonlinear aspects of mixed–mode states at moderate values of the forcing parameters within the regular wave regime. Frequency entrainment and phase locking of resonant triads and sidebands were found to be widespread. Cases were analysed in phase space reconstructions through a singular value decomposition of multi–variate time series. Four forms of mixed–mode states were found, each in well–defined regions of parameter space: (1) a nonlinear interference vacillation associated with strong phase locking through higher harmonics; (2) a modulated amplitude vacillation showing strong phase coherence in triads involving the long wave; (3) an intermittent bursting of secondary modes; (4) an attractor switching flow, where the dominant wave number switched at irregular intervals between two possible wave numbers. Many of the mixed–mode states are suggested to arise from homoclinic bifurcations, whereas no secondary Hopf bifurcations were found. One of the postulated homoclinic bifurcations was consistent with a bifurcation through intermittency. The bifurcation sequences, however, were strongly affected by phase locking between different wave number components and frequency locking between drift and modulation frequencies. When all free frequencies were locked, the flow reduced to a limit cycle which subsequently became unstable through an incomplete period–doubling cascade. The only observed case of torus–doubling was also associated with strong phase locking. Most of the observed regimes were consistent with low–dimensional dynamics involving a limited number of domain–filling modes, which can be represented in phase space reconstructions and characterized by invariants such as attractor dimensions and the Lyapunov exponents. Some flows associated with a weak structural vacillation, however, were not consistent with low–dimensional dynamics. It appeared rather that they were the result of spatially localized instabilities consistent with high–dimensional dynamics, which can be parametrized as stochastic dynamics.

Characteristics of amplitude vacillation in a differentially heated rotating fluid annulus

Abstract Experimental results are presented on the characteristics and occurrence of amplitude vacillation observed in the regular wave regime of a differentially heated rotating fluid annulus. The amplitude vacillation appears as the periodic oscillation of the amplitude and frequency of the dominant wave component and its harmonics and occurs, in general, adjacent to a transition to the next lowest wavenumber. The frequency spectra of the amplitude vacillations are characterized by the presence of two distinct frequencies, one from the vacillation and one from the drift of the wave; in general the amplitude vacillations observed can be said to be doubly periodic. The wave interference mechanism of amplitude vacillation is examined but no evidence was found which could be unambiguously interpreted in its favour.

Relationships Among Eddy Fluxes of Heat, Eddy Temperature Variances and Basic-State Temperature Parameters in Thermally Driven Rotating Fluids

Relationships are examined among radial eddy fluxes of heat, eddy temperature variances and basic-state temperature parameters (e.g., radial gradients and variances of the azimuthally averaged temperature) over a broad range of dimensionless parameters in thermally driven rotating annuli of fluid. We consider, first, changes of the time-averaged eddy heat flux and of the time and azimuthally averaged radial temperature gradient which take place when we conduct experiments at different imposed temperature differences With the rotation rate held fixed, or at different rotation rates with the imposed temperature difference held fixed. It is found that the observed changes of these variables cannot be described by a positive proportionality between the eddy heat flux and some positive power of the radial temperature gradient, except perhaps over limited ranges of dimensionless-parameter space. Within the regular wave regime, not too far from marginal stability, the eddy heat flux increases and the azimuthally averaged radial temperature gradient decreases with increasing rotation rate, when the imposed temperature contrast is held constant, suggesting that the effect of the eddies is to reduce substantially the magnitude of the temperature gradient. Evaluations are made also of the time variations of the eddy heat flux and of the azimuthally averaged radial temperature gradient at fixed points in dimensionless parameter space. The lag correlation function between these two variables is found to exhibit a unique characteristic which is common to all experiments. A lag correlation function calculated by Peter Stone using atmospheric data reveals a similar characteristic. The results suggest a new time-dependent parameterization of heat flux variations in terms of the basic-state temperature gradient.

Determination of Eddy Fluxes of Heat and Eddy Temperature Variances Using Weakly Nonlinear Theory

Drazin's (1972) weakly nonlinear theory of the self interaction of a single, slightly unstable, normal mode in a viscous, baroclinic fluid with a continuous density distribution is used to determine eddy fluxes of heat and eddy temperature variances as a function of rotation and internal thermal gradients. The calculations are applied to annulus experiments in that portion of the regular wave regime in which the observed flow is dominated by a single mode. V. Barcilon's (1964) linear theory for a viscous fluid is used for the purpose of mode selection at each point in dimensionless-parameter space. The geostrophic eddy heat flux and temperature variance are then determined from Drazin's lowest order nonlinear theory (which is a direct extension of Barcilon's theory). It is found that the eddy beat flux and temperature variance depend upon internal thermal gradients at marginal stability and upon the distance of the point in dimensionless-parameter space from marginal stability for each mode. This result is different from those obtained from linear theory (e.g., by Green, 1970; Saltzman and Vernekar, 1971; Stone, 1972) but agrees in essence with that obtained by Hart (1974) from a nonlinear analysis of a two-layer model. Numerical calculations from the theoretically derived formulas show that the eddy beat flux and temperature variance corresponding to each mode increase with increasing rotation rate when the imposed temperature contrast is held constant, but that there is an abrupt drop in the magnitude of both quantities when the wavenumber changes to the next higher integral value. Experimental evidence obtained by Pfeffer et al. (1978) verifies that real fluids behave in this way. Another feature observed in laboratory experiments and predicted by the theory is that at fixed rotation rate (or Taylor number) the eddy heat flux and the eddy temperature variance may be either larger or smaller at larger values of the imposed temperature contrast, depending on the location of the experiments in dimensionless-parameter space. If we think of seasons being brought about by changes in imposed temperature contrast at constant rotation rate, this result implies that only in certain regions of dimensionless-parameter space is winter (defined as the season with the highest imposed temperature contrast) the season with the largest eddy heat flux or eddy temperature variance.

An experimental study of the effects of Prandtl number on thermal convection in a rotating, differentially heated cylindrical annulus of fluid

This paper presents the results of experimental studies of the behaviour of different fluids (mercury, Pr = 0·0246; water, Pr = 7·16; and 5 cS silicone oil, Pr = 63) when each is contained in a rotating cylindrical annulus and subjected to an imposed radial temperature difference across the annulus. The results are summarized in the form of two-parameter regime diagrams (thermal Rossby number RoTvs. Taylor number Ta) at different Prandtl numbers Pr. The fluids are in contact above and below with rigid insulating boundaries. The range of thermal Rossby and Taylor numbers surveyed is 10−3 &lt; RoT &lt; 10, 105 &lt; Ta &lt; 109.The regime diagram for water with a rigid upper lid in contact with the fluid resembles in certain respects the more familiar regime diagram for water with a free upper surface obtained by Fultz (1956) and by Fowlis &amp; Hide (1965). Significant differences do, however, occur and are discussed in a separate paper by Fein (1973). The regime diagram for silicone oil possesses no lower symmetric regime within the range of thermal Rossby and Taylor numbers surveyed; that for mercury possesses no upper symmetric regime within the range surveyed. In mercury, turbulence is observed at high Rossby numbers. An experimental traverse across the regime diagram in which the imposed temperature difference is held constant at ΔT = 5 °C and the rotation rate Ω is changed monotonically reveals a highly conductive temperature structure in mercury and a highly convective temperature structure in both water and silicone oil. The regular wave regime, which appears at all three Prandtl numbers, is found to shift towards higher Taylor and Rossby numbers with decreasing Prandtl number. As a result, a single point in two-dimensionless-parameter space (RoT = 5 × 10−1, Ta = 1 × 106) lies in the upper symmetric regime for 5 cS silicone oil (Pr = 63), in the regular wave regime for water (Pr = 7·16) and in the lower symmetric regime for mercury (Pr = 0·0246).

Baroclinic waves in a container with sloping end walls

The effects of sloping upper and lower boundaries on baroclinic waves in a differentially heated rotating fluid annulus have been studied experimentally. Measurements have been made of the internal temperature structure, the transition from axisymmetric to non-axisymmetric flow, the wave numbers and flow types, the drift rates and the heat transfer. The sloping boundaries influence the baroclinic waves in two distinct ways: if they are parallel the energy releasing mechanism is directly affected, while if they are not parallel effects similar to the latitudinal variation of coriolis parameter in a planetary atmosphere (the (3-effect) are introduced. In order to identify essentially non-linear effects the results on the transition from axisymmetric to non-axisymmetric flow, the wave numbers and the drift rates have been compared with an appropriately extended version (by Hide) of Eady’s linear perturbation theory of baroclinic instability. In the case of the transition from axisymmetric to non-axisymmetric flow, when account was taken of concomitant changes in the internal temperature structure, surprisingly good agreement with linear theory was obtained. Within the regular wave regime, the azimuthal wave numbers agreed quite well when the theoretically predicted value was 1, 2 or 3 but for higher predicted wave numbers the observed wave numbers were smaller. In the case of oppositely sloping boundaries the observed drift rates relative to the mean flow were generally 30 % less than that of the unstable waves of the linear theory. Another departure from the linear theory also occurred when the boundaries sloped in opposite senses, namely the occurrence at sufficiently rapid rotation rates of two separate wave trains of limited radial extent and differing wave numbers and drift speeds; one dominated the flow near the inner cylinder and the other dominated the flow near the outer cylinder. This result suggests that the variation of coriolis parameter with latitude can determine the radial extent of the baroclinic waves. Apart from its intrinsic interest as a fluid-mechanical study this work bears on the role played by the (3-effect and large-scale topography in natural systems such as the atmosphere and oceans. It also shows that at the higher rotation rates used the sloping eeopotentials inevitably present in previous laboratory experiments on baroclinic waves have important effects, and in particular explains the failure of previous theories to account for the position of part of the transition from axisymmetric to non-axisymmetric flow.

Wave Dispersion in a Rotating, Differentially Heated Cylindrical Annulus Of Fluid

During a traverse of the regular wave regime in a rotating, differentially heated annulus of viscous liquid (Prandtl number = 86.0), an asymmetrical wave pattern was observed in the transition region between 4 and 5 regular waves. The shape and time development of this pattern is interpreted as being primarily the result of the simultaneous presence of a 4-wave and a 5-wave pattern, and of the dispersion resulting from the relative motion of these two waves.