Rigid Lid Research Articles

Wetting and drying of two-layer shallow-water flow in a moving vessel via semi-Lagrangian simulation

The sloshing motion of two inviscid, incompressible, immiscible, shallowwater fluid layers in a rectangular vessel with a rigid lid is considered. The vessel is forced to oscillate in a horizontal rectilinear motion, at the lowest resonant frequency of the system, such that the lower fluid hits the rigid lid causing a wetting-drying scenario. The two-layer shallow-water equations are solved using a conservative quasi-monotone semi-Lagrangian scheme which conserves the mass of the system. Two examples are considered: a low-fill example where the lower fluid does not interact with the upper-rigid lid, in order to validate the code against an existing Lagrangian particle path scheme, and a high-fill example where there is wetting and drying of the upper rigid lid. In both examples, results of the semi-Lagrangian simulations are compared to existing experimental data and good agreement is obtained.

МОДУЛЯЦІЙНА СТІЙКІСТЬ ХВИЛЬОВИХ ПАКЕТІВ У ТРИШАРОВІЙ ГІДРОДИНАМІЧНІЙ СИСТЕМІ

The article is devoted to the problem of propagation of weakly nonlinear wave-packets along contact surfaces in a three-layer hydrodynamic system "half space – layer – layer with rigid lid". The condition of solvability of the problem in the third-order approximation is obtained, the evolution equation is derived in the form of a nonlinear Schrodinger equation and the modulation stability condition for its solutions is obtained. The stability diagram and its analysis are presented for the solution which takes place in the case of the balance between dispersion and non-linearity.

Validation of the nonhydrostatic General Curvilinear Coastal Ocean Model (GCCOM) for stratified flows

While global- and basin-scale processes can be captured quite well with computationally-inexpensive hydrostatic models, smaller-scale features such as shoaling nonlinear internal waves and bores, coastal fronts, and other convective processes require the use of a nonhydrostatic model to accurately capture dynamics. Here the nonhydrostatic capabilities of the General Curvilinear Coastal Ocean Model (GCCOM) in a stratified environment are introduced. GCCOM is a three-dimensional, nonhydrostatic Large Eddy Simulation (LES), rigid lid model that has the ability to run in a fully three-dimensional general curvilinear coordinate system. This model was previously validated for unstratified flows with curvilinear coordinates. Here, recent advances of the model to simulate stratified flows are presented, focusing on sigma coordinate grids with both flat bottom geometry and a local gently sloping seamount. In particular, a suite of test cases widely used as benchmarks for assessing the nonhydrostatic capabilities for gravity-driven flows and internal waves is presented: an internal seiche in a flat bottom tank, the classic lock release and gravity current experiment, and a field-scale internal wave beam experiment consisting of an oscillating tidal flow over a topographic ridge. GCCOM shows excellent agreement with the benchmark test cases and is able to accurately resolve complex nonhydrostatic phenomena in stratified flows. Future studies will utilize the model capabilities for realistic field-scale internal wave simulations.

Sucrose Concentration, Light Intensity, and CO<sub>2</sub> Concentration Affect Growth and Development of Micropropagated Carnation

Carnation (Dianthus caryophyllus L.) ‘Purple Beauty’ at micropropagation stage 3 was cultured under two levels each of medium sucrose concentration (0 and 30 g·L-1), photosynthetic photon flux density (50 and 200 μmol·m-2·s-1 PPFD), and CO2 concentration (350 and 1,000 μmol·mol-1), in a factorial experimental design, consisting of all eight possible treatment combinations. Shoot tip explants, obtained from in vitro-grown plantlets, were cultured on 50 mL agar-solidified Murashige and Skoog medium per container. Each culture container was sealed with a rigid lid vented by a high-efficiency particulate air filter attached to a 10 mm diameter hole made in the lid center, with an estimated number of 2.8 air exchanges per hour. All cultures were maintained for four weeks at 24°C/22°C (day/night) temperatures, 70%–80% relative humidity, and a 16 h/8 h (day/night) photoperiod was provided by white light-emitting diodes. The treatment combining high light intensity (200 μmol·m-2·s-1), high CO2 concentration (1,000 μmol·mol-1), and without supplementation of sucrose to the medium (i.e., photo-autotrophic conditions) resulted in better plantlet growth and development, with higher values being observed in terms of total fresh weight, tissue water content, leaf length, leaf width, and total chlorophyll content than in plantlets grown under the other treatments.

Analysis of energy of internal waves in a three­layer semi­infinite hydrodynamic system

Energy characteristics of waves propagation along the contact surfaces in a hydrodynamics system "liquid half-space – layer –layer with a rigid lid" are explored. Based on the solutions of first approximation to a weakly non-linear model, the integral relations for wave motion energy in each layer and for the total energy of the system were obtained. An analysis of energy of wave processes revealed that an increase in wave number causes a decrease in energy of wave motion of the upper layers, and energy of wave motion of the lower half-space at some values of wave number reaches extreme values. In this case, total energy of the system is descending in nature and rather quickly approaches its limit value.The numerical values of energy for three different cases of propagation of progressive waves were obtained: only along the upper contact surface, only along the lower contact surface, along both surfaces simultaneously. Comparison of the obtained values of energy revealed that in the case of waves propagation along both contact surfaces simultaneously, the total energy of the system is close to the sum of energies of the system at waves propagation along one of the surfaces

Effect of free surface on submerged stratified shear instabilities

In this paper, we have considered the effects of the shallowness of the domain as well as the air–water free surface on the stratified shear instabilities of the fluid underneath. First, we numerically solve the non-Boussinesq Taylor–Goldstein equation for smooth velocity and density profiles of a model shear layer with a free surface. When the depth of the fluid is relatively shallow compared to the shear layer thickness, the surface gravity waves existing at the free surface come closer to the waves existing in the shear layer. This can lead to resonant wave interactions, making the flow unstable to more varieties of modal instabilities. In order to obtain a deeper understanding of the instability mechanisms, we have performed analytical studies with broken-line profiles (profiles for which vorticity and density are piecewise constant). Furthermore, reduced-order broken-line profiles have also been developed, based on which dispersion diagrams are constructed. Through these diagrams we have underpinned the resonantly interacting waves leading to each type of instability. Two new instabilities have been found; one of them, referred to as the ‘surface gravity – interfacial gravity (SG-IG) mode’, arises due to the interaction between a surface gravity wave and an interfacial gravity wave, and would therefore be absent if there is no internal density stratification. The other one – the ‘surface gravity – lower vorticity (SG-LV) mode’, which arises due to the interaction between a surface gravity wave and the lower vorticity wave, surpasses Kelvin–Helmholtz (KH) instability to become the most unstable mode, provided the system is significantly shallow. Stability boundary of the SG-LV mode is found to be quite different from that of KH. In fact, KH becomes negligible for relatively shallow flows, while SG-LV’s growth rate is significant – comparable to the growth rate of KH for unbounded domains (${\approx}0.18$). Moreover, the SG-LV mode is found to be analogous to the barotropic mode observed in two-layer quiescent flows. We have found that the effect of a free surface on the Holmboe instability is not appreciable. Holmboe in the presence of a free surface is found to be analogous to the baroclinic mode observed in two-layer quiescent flows. Except for the Holmboe instability, remarkable differences are observed in all other instabilities occurring in shallow domains when the air–water interface is replaced by a rigid lid. We infer that the rigid-lid approximation is valid for large vertical domains and should be applied with caution otherwise. Furthermore, we have also shown that if shear is absent at the free surface, our problem can be modelled using Boussinesq approximation, that is, $O(1)$ density variations in the inertial terms can still be neglected.

A new front condition for non-Boussinesq gravity currents

ABSTRACTA new front condition for gravity currents in a two-layer shallow water framework with a free surface is developed by applying the theory of the weak formulation to a first order system of four partial differential equations written in conservation form. The Rankine–Hugoniot jump conditions are applied and a relation between the velocity and the height of the shock is obtained. Laboratory experiments on lock-release non-Boussinesq gravity currents are performed and the front propagation and the free surface displacement are evaluated. Front velocities, predicted by the present mathematical model, are compared to both laboratory measurements and the predictions of the Benjamin formula, which is based on the rigid lid approximation. The new jump condition is in a good agreement with the laboratory measurements, while the Benjamin formula exhibits an overestimation of the shock velocity in the non-Boussinesq cases. In addition, the free surface displacement near the shock predicted by the new model is in agreement with the laboratory measurements.

Internal wave generation by tidal flow over a two-dimensional ridge: energy flux asymmetries induced by a steady surface trapped current

The effects of a surface trapped steady background current on internal waves generated by tidal currents oscillating over a small symmetric ridge are investigated using a two-dimensional primitive equation model. A rigid lid is used with a linearly stratified fluid and the effects of rotation are not considered. We consider uni-directional background currents $\bar{U}(z)\geqslant 0$ confined to a surface layer lying well above the ridge. The current introduces asymmetries in the generated wave field. For sufficiently narrow ridges the upstream energy flux is larger than the downstream flux while the opposite is the case for sufficiently wide ridges. The total energy flux radiating away from the ridge is not significantly affected by the current. Mean second-order currents and pressure fields are shown to make important contributions to the total energy flux. A first-order linear theory, valid for a general stratification and surface current, which accurately predicts the wave field is also developed.

Wave propagation in a three-layer semi-infinite hydrodynamic system with a rigid lid

Research into propagation and interaction of waves in a three-layer hydrodynamic system is one of the relevant problems of modern theoretical and experimental hydrodynamics. The authors studied propagation and interaction of waves along contact surfaces of the three-layer hydrodynamic system "liquid half-space – layer – layer with a rigid lid". By applying a method of large-scale approximations, the first three linear approximations of the correspondent weakly nonlinear problem were obtained. The structure of wave motions on contact surfaces was explored. Dependence of amplitudes of waves-responses on contact surfaces at various geometrical and physical parameters was analyzed. In particular, for large values of thickness of the upper layer, it was found that a change in value of the wave number leads to rapid convergence of amplitudes of waves-responses to the common limited value. The authors showed the need for a detailed study of the limited case in the absence of density jump, in which one of the solutions of dispersion equation tends to zero. Results of the present research can be used in the design of algorithms for detection of wave motions in various liquid media.

Beyond the Rigid Lid: Baroclinic Modes in a Structured Atmosphere

Abstract The baroclinic-mode decomposition is a fixture of the tropical-dynamics literature because of its simplicity and apparent usefulness in understanding a wide range of atmospheric phenomena. However, its derivation relies on the assumption that the tropopause is a rigid lid that artificially restricts the vertical propagation of wave energy. This causes tropospheric buoyancy anomalies of a single vertical mode to remain coherent for all time in the absence of dissipation. Here, the authors derive the Green’s functions for these baroclinic modes in a two-dimensional troposphere (or, equivalently, a three-dimensional troposphere with one translational symmetry) that is overlain by a stratosphere. These Green’s functions quantify the propagation and spreading of gravity waves generated by a horizontally localized heating, and they can be used to reconstruct the evolution of any tropospheric heating. For a first-baroclinic two-dimensional right-moving or left-moving gravity wave with a characteristic width of 100 km, its initial horizontal shape becomes unrecognizable after 4 h, at which point its initial amplitude has also been reduced by a factor of 1/π. After this time, the gravity wave assumes a universal shape that widens linearly in time. For gravity waves on a periodic domain the length of Earth’s circumference, it takes only 10 days for the gravity waves to spread their buoyancy throughout the entire domain.

Numerical Investigation of the Influence of Water Jumping on the Local Scour beneath a Pipeline under Steady Flow

Rigid-lid approximation is usually used to replace the free surface in scour simulation. The influence of the rigid lid assumption on the prediction precision of scour hole in steady flow is studied in this paper. Firstly, a local scour model was constructed based on the open sources Computational Fluid Dynamics (CFD) model OpenFOAM, where both the bed load and suspended load were considered. In the present model, the bed shear stress was calculated by the Newton shear stress formula, instead of the traditional calculation method with the assumption that the flow velocity in vertical direction complies with a logarithmic distribution. The Volume of Fluid (VOF) method was used to capture the free surface and a moving-mesh method was used to track the change of bed surface. Then, several experiments were chosen to validate the model, and the modeling results fitted well with the measured data. Lastly, the effect of the rigid lid assumption on surface elevation, bed shear stress and the profile of the scour hole in steady flow are studied. The result shows that the surface elevation suffers a drop above the pipeline, and the difference of surface elevation between the upstream and downstream increases with decreasing dimensionless depth. Compared with the free surface condition, the bed shear stress and scour hole depth computed with the rigid lid approximation were underestimated.

Drag Produced by Waves Trapped at a Density Interface in Nonhydrostatic Flow over an Axisymmetric Hill

Abstract Linear nonhydrostatic theory is used to evaluate the drag produced by 3D trapped lee waves forced by an axisymmetric hill at a density interface. These waves occur at atmospheric temperature inversions, for example, at the top of the boundary layer, and contribute to low-level drag possibly misrepresented as turbulent form drag in large-scale numerical models. Unlike in 2D waves, the drag has contributions from a continuous range of wavenumbers forced by the topography, because the waves can vary their angle of incidence to match the resonance condition. This leads to nonzero drag for Froude numbers (Fr) both <1 and >1 and a drag maximum typically for Fr slightly below 1, with lower magnitude than in hydrostatic conditions owing to wave dispersion. These features are in good agreement with laboratory experiments using two axisymmetric obstacles, particularly for the lower obstacle, if the effects of a rigid lid above the upper layer and friction are taken into account. Quantitative agreement is less satisfactory for the higher obstacle, as flow nonlinearity increases. However, even in that case the model still largely outperforms both 3D hydrostatic and 2D nonhydrostatic theories, emphasizing the importance of both 3D and nonhydrostatic effects. The associated wave signatures are dominated by transverse waves for Fr lower than at the drag maximum, a dispersive “Kelvin ship-wave” pattern near the maximum, and divergent waves for Fr beyond the maximum. The minimum elevation at the density-interface depression existing immediately downstream of the obstacle is significantly correlated with the drag magnitude.

The power available to tidal turbines in an open channel flow

Linear momentum actuator disc theory is extended to address the power available to a tidal turbine array spanning the cross-section of an open channel flow. A generalised formulation is presented, which relaxes constraints in previous models on the Froude number of the flow and the geometry of the turbine array, and also considers the effects of far-wake mixing on the overall power removed from the flow. In the limiting case of no free surface deformation, the rigid lid model is recovered. Blockage, the ratio of turbine frontal area to the cross-sectional area of the surrounding flow passage, has the greatest effect on available power, with the peak power coefficient increasing by 55% from 0·60 to 0·93 as the blockage ratio increases from 0·05 to 0·20. A further 3% increase in peak power coefficient is achieved as the Froude number increases from 0·05 to 0·20. The efficiency of energy extraction may be determined relative to the total power extracted from the flow, comprising the power available to the turbines, and the power dissipated in wake mixing. Higher blockage turbines operating at low thrust coefficients are shown to be more efficient than lower blockage turbines.

Lagrangian particle path formulation of multilayer shallow-water flows dynamically coupled to vessel motion

The coupled motion—between multiple inviscid, incompressible, immiscible fluid layers in a rectangular vessel with a rigid lid and the vessel dynamics—is considered. The fluid layers are assumed to be thin and the shallow-water assumption is applied. The governing form of the Lagrangian functional in the Lagrangian particle path (LPP) framework is derived for an arbitrary number of layers, while the corresponding Hamiltonian is explicitly derived in the case of two- and three-layer fluids. The Hamiltonian formulation has nice properties for numerical simulations, and a fast, effective and symplectic numerical scheme is presented in the two- and three-layer cases, based upon the implicit-midpoint rule. Results of the simulations are compared with linear solutions and with the existing results of Alemi Ardakani et al. (J Fluid Struct 59:432–460, 2015) which were obtained using a finite volume approach in the Eulerian representation. The latter results are extended to non-Boussinesq regimes. The advantages and limitations of the LPP formulation and variational discretization are highlighted.

The initiation of a planar fluid plume beneath a rigid lid

The unsteady growth of a viscous fluid plume beneath a rigid upper lid is investigated. Two-dimensional (planar) flow is assumed, through a fissure in the horizontal lower boundary. Initially, the fluid exiting the bottom is assumed to form a semi-circular region, but rises as time progresses, and spreads across the upper boundary. The problem is modelled using Boussinesq theory, and solved using a time-dependent spectral method. These numerical solutions are also compared with the results of a simpler inviscid asymptotic solution. Results are indicated for different input fluid speeds and fissure widths.

The Vertical Mode Decomposition of Surface and Internal Tides in the Presence of a Free Surface and Arbitrary Topography

AbstractThe method of decomposing surface and internal tides determines the expression for internal tide energy, energy flux, and energy conversion. The de facto standard is to define surface tides as depth-averaged pressure and horizontal velocity and internal tides as the residuals. This decomposition, which is equivalent to projecting motion onto vertical modes that obey a rigid lid, is known to produce spurious energy conversion CS through movement of the free surface. Here, motion is instead projected onto modes that obey a linear, free-surface boundary condition. The free-surface modes are shown to obey a more complicated orthogonality condition than rigid-lid modes but are still straightforward to calculate numerically. The resulting decomposition (i) completely eliminates spurious energy conversion CS and (ii) leads to a more precise expression for topographic internal tide generation C, which now depends on horizontal gradients in the vertical structure of the surface tide. Numerical simulations and rough global estimates indicate that corrections to C are a maximum of a few percent. However, CS produces spurious energy flux divergences/convergences in the open ocean, which are the same order of magnitude [O(1–10) mW m−2] as open-ocean internal tide energy dissipation.

Internal solitary waves in a two-fluid system with a free surface

Internal solitary waves in a system of two fluids, silicone oil and water, bounded above by a free surface are studied both experimentally and theoretically. By adjusting an extra volume of silicone oil released from a reservoir, a wide range of amplitude waves are generated in a wave tank. Wave profiles as well as wave speeds are measured using multiple wave probes and are then compared with both the weakly nonlinear Korteweg–de Vries (KdV) models and the strongly nonlinear Miyata–Choi–Camassa (MCC) models. As the density difference between the two fluids in the experiment is relatively small (approximately 14 %), but non-negligible, special attention is paid to the effect of the boundary condition at the top surface. The nonlinear models valid for rigid-lid (RL) and free-surface (FS) boundary conditions are considered separately. It is found that the solitary wave of the FS model for a given amplitude is consistently narrower than that of the RL model and it propagates at a slightly lower speed. Due to strong nonlinearity in the internal-wave motion, the weakly nonlinear KdV models fail to describe the measured internal solitary wave profiles of intermediate and large wave amplitudes. The strongly nonlinear MCC-FS model agrees better with the measurements than the MCC-RL model, which indicates that the free-surface boundary condition at the top surface is crucial in describing the internal solitary waves in the experiment correctly. Leaving the top surface free in the experiment allows us to observe small and relatively short wave packets on the top surface, particularly when the amplitude of the internal solitary wave is large. Once excited, the wave packet is located above the front half of the internal solitary wave and propagates with a speed close to that of the internal solitary wave underneath. A simple resonance mechanism between short surface waves and long internal waves without and with nonlinear effects is examined to estimate the characteristic wavelength of modulated short surface waves, which is found to be in good agreement with the observed wavelength when nonlinearity is taken into account. Using ray theory, the evolution of short surface waves in the presence of a background current induced by an internal solitary wave is also investigated to examine the location of the modulated surface wave packet.

The Impact of Gradient Wind Imbalance on Tropical Cyclone Intensification within Ooyama’s Three-Layer Model

Abstract This paper addresses the validity of the gradient wind balance approximation during the intensification phase of a tropical cyclone in Ooyama’s three-layer model. For this purpose, the sensitivity to various model modifications is examined, given by the inclusion of (i) unbalanced dynamics in the free atmosphere, (ii) unbalanced dynamics in the slab boundary layer, (iii) a height-parameterized boundary layer model, and (iv) a rigid lid. The most rapid intensification occurs when the model employs the unbalanced slab boundary layer, while the simulation with the balanced boundary layer reveals the slowest intensification. The simulation with the realistic height-parameterized boundary layer model exhibits an intensification rate that lies in between. Intensification is induced by a convective ring in all experiments, but a distinct contraction of the radius of maximum gradient wind only takes place with unbalanced boundary layer dynamics. In all experiments the rigid lid and the balance approximation for the free atmosphere have no crucial impact on intensification, and a linear stability analysis cannot explain the found sensitivity to intensification. Most likely the nonlinear momentum advection term plays an important role in the boundary layer. It is found on the basis of a diagnostic radial mass flux equation that the source term for latent heat provides the largest contribution to intensification and contraction. Furthermore, it turns out that the position of the convective ring inside or outside of the radius of maximum gradient wind (RMGW) is of vital importance for intensification and most likely explains the large impact of boundary layer imbalance.

Internal gravity–capillary solitary waves in finite depth

We consider a two‐dimensional inviscid irrotational flow in a two layer fluid under the effects of gravity and interfacial tension. The upper fluid is bounded above by a rigid lid, and the lower fluid is bounded below by a rigid bottom. We use a spatial dynamics approach and formulate the steady Euler equations as a Hamiltonian system, where we consider the unbounded horizontal coordinate x as a time‐like coordinate. The linearization of the Hamiltonian system is studied, and bifurcation curves in the (β,α)‐plane are obtained, where α and β are two parameters. The curves depend on two additional parameters ρ and h, where ρ is the ratio of the densities and h is the ratio of the fluid depths. However, the bifurcation diagram is found to be qualitatively the same as for surface waves. In particular, we find that a Hamiltonian‐Hopf bifurcation, Hamiltonian real 1:1 resonance, and a Hamiltonian 02‐resonance occur for certain values of (β,α). Of particular interest are solitary wave solutions of the Euler equations. Such solutions correspond to homoclinic solutions of the Hamiltonian system. We investigate the parameter regimes where the Hamiltonian‐Hopf bifurcation and the Hamiltonian real 1:1 resonance occur. In both these cases, we perform a center manifold reduction of the Hamiltonian system and show that homoclinic solutions of the reduced system exist. In contrast to the case of surface waves, we find parameter values ρ and h for which the leading order nonlinear term in the reduced system vanishes. We make a detailed analysis of this phenomenon in the case of the real 1:1 resonance. We also briefly consider the Hamiltonian 02‐resonance and recover the results found by Kirrmann. Copyright © 2016 John Wiley & Sons, Ltd.

Multi-symplectic structure of fully nonlinear weakly dispersive internal gravity waves

In this short communication, we present the multi-symplectic structure for the two-layer Serre–Green–Naghdi equations describing the evolution of large amplitude internal gravity water waves when both layers are shallow. We consider only a two-layer stratification with rigid bottom and lid for simplicity, generalisations to several layers being conceivable. This multi-symplectic formulation allows the application of various multi-symplectic integrators (such as Euler or Preissman box schemes) that preserve exactly the multi-symplecticity at the discrete level.