Rigid Lid Research Articles

Rigid lid limit in shallow water over a flat bottom

AbstractWe perform the so‐called rigid lid limit on different shallow water models such as the abcd Bousssinesq systems or the Green–Naghdi equations. To do so, we consider an appropriate nondimensionalization of these models where two small parameters are involved: the shallowness parameter and a parameter which can be interpreted as a Froude number. When the parameter tends to zero, the surface deformation formally goes to the rest state, hence the name rigid lid limit. We carefully study this limit for different topologies. We also provide rates of convergence with respect to and careful attention is given to the dependence on the shallowness parameter .

Large-amplitude internal waves and turbulent mixing in three-layer flows under a rigid lid

We consider a nonlinear long-wave Boussinesq-type model describing the propagation of breaking internal solitary waves in a three-layer flow between two rigid boundaries. The Green–Naghdi-type equations govern the fluid flow in the top and bottom homogeneous layers. In the intermediate hydrostatic layer, the fluid is non-homogeneous, and its flow is described by the depth-averaged shallow water equations for shear flows. The velocity shear in the outer layers can lead to the development of the Kelvin–Helmholtz instability and turbulent mixing. To take this into account, we propose a simple law of vertical mixing, which governs the interaction of these layers. Stationary solutions and non-stationary calculations show the effect of mixing (or breaking) for waves of sufficiently large amplitude. We construct steady-state soliton-like solutions of the three-layer model adjacent to a given constant flow. The obtained theoretical profiles of breaking solitary waves are consistent with laboratory experiments.

Weakly nonlinear models of stochastic wave propagation in two-layer hydrodynamic systems

The paper discusses three-dimensional models of the propagation of stochastic internal waves in hydrodynamic systems: ’half-space - half-space’, ’half-space - layer with rigid lid’, and ’layer with solid bottom - layer with rigid lid’. In constructing the models, the layers are considered to be ideal fluids separated by a contact surface. The main objective of the modeling is to obtain a dynamic equation for the stochastic amplitude of surface waves. A comparative analysis of the obtained results has been conducted. In order to control the contribution of nonlinear terms, a dimensionless non-numerical parameter has been introduced. The models are distinguished by boundary conditions that determine the general form of solutions. As a result, a dynamic equation for the stochastic amplitude of internal waves has been derived. After ensemble averaging of the amplitudes, the dynamic equation is formulated in integral form using Fourier-Stieltjes integrals. The dynamic equation reveals two-wave and three-wave interactions, as well as the contribution of dispersion to wave dynamics. An investigation of the boundary case of the transition of internal waves in the ’half-space - half-space’ system to surface waves in the absence of an upper liquid layer confirms the validity of the results.

The wind farm pressure field

Abstract. The disturbed atmospheric pressure near a wind farm arises from the turbine drag forces in combination with vertical confinement associated with atmospheric stability. These pressure gradients slow the wind upstream, deflect the air laterally, weaken the flow deceleration over the farm, and modify the farm wake recovery. Here, we describe the airflow and pressure disturbance near a wind farm under typical stability conditions and, alternatively, with the simplifying assumption of a rigid lid. The rigid lid case clarifies the cause of the pressure disturbance and its close relationship to wind farm drag. The key to understanding the rigid lid model is the proof that the pressure field p(x,y) is a harmonic function almost everywhere. It follows that the maximum and minimum pressure occur at the front and back edge of the farm. Over the farm, the favorable pressure gradient is constant and significantly offsets the turbine drag. Upwind and downwind of the farm, the pressure field is a dipole given by p(x,y)≈Axr-2, where the coefficient A is proportional to the total farm drag. Two derivations of this law are given. Field measurements of pressure can be used to find the coefficient A and thus to estimate total farm drag.

A high-order nonlinear Boussinesq-type model for internal waves over a mildly-sloping topography in a two-fluid system

With a rigid lid assumption on the upper boundary in a two-fluid system, a new high-order Boussinesq-type model for internal waves over a mildly-sloping topography is derived. The model is formulated in terms of computational velocity vectors defined at mid-water depth in each fluid. Stokes-type expansions are used to theoretically analyze the linear and nonlinear properties of the new equations. High accuracy in linear, nonlinear, shoaling and kinematic aspects is achieved and the model is applicable from deep to shallow water. For an equivalent water depth in upper and lower fluid, the model can be applied up to kh2 ≈ 10 (where k is the wave number and h2 is the water depth of the lower fluid) in linear dispersion, up to kh2 ≈ 7.0 in the second nonlinear property within 2% error, and up to 0 < kh2 < 10 in the linear shoaling property. Compared with most existing Boussinesq-type models, the accuracy of the horizontal and vertical velocity profiles along the water column is improved significantly and is applicable to up to kh2 ≈ 4.48.

Boundary integral relations for submerged bodies and free-surface piercing ships and offshore structures

The linear flow model called rigid waterplane flow model, in which a body (ship, offshore structure) that pierces the free surface is treated as the smooth limit of a body that is submerged at a small depth, is considered. The flow within the thin layer between the horizontal rigid lid that closes the (open) body surface and the waterplane above the lid is analyzed. This analysis shows that the seemingly different boundary integral relations obtained by applying Green’s basic identity to the boundary value problems associated with the rigid waterplane flow model or a body that pierces the free surface are equivalent. Additionally, the rigid waterplane flow model yields three alternative weakly singular boundary integral relations, which only involve distributions of sources and weak dipoles that are continuous at the body surface. One of these three relations is notable because it does not involve a waterline integral, and only involves the flow potential (not its derivatives along directions tangent to the body surface) and smooth ordinary functions that are easily evaluated and integrated.

Internal solitary waves generated by a moving bottom disturbance

The strongly nonlinear Miyata–Choi–Camassa model under the rigid lid approximation (MCC-RL model) can describe accurately the dynamics of large-amplitude internal waves in a two-layer fluid system for shallow configurations. In this paper, we apply the MCC-RL model to study the internal waves generated by a moving body on the bottom. For the case of the moving body speed $U=1.1c_{0}$ , where ${c_0}$ is the linear long-wave speed, the accuracy of the MCC-RL results is assessed by comparing with Euler's solutions, and very good agreement is observed. It is found that when the moving body speed increases from $U=0.8c_{0}$ to $U=1.241c_{0}$ , the amplitudes of the generated internal solitary waves in front of the moving body become larger. However, a critical moving body speed is found between $U=1.241c_{0}$ and $U=1.242c_{0}$ . After exceeding this critical speed, only one internal wave right above the body is generated. When the moving body speed increases from $U=1.242c_{0}$ to $U=1.5c_{0}$ , the amplitudes of the internal waves become smaller.

Unstructured grid dynamical modeling of planetary atmospheres using planetMPAS: The influence of the rigid lid, computational efficiency, and examples of Martian and Jovian application

We present a new planetary global circulation model, planetMPAS, based on the state-of-the-art NCAR MPAS General Circulation Model. Taking advantage of the cross compatibility between WRF and MPAS, planetMPAS includes most of the planetWRF physics parameterization schemes for terrestrial planets such as Mars and Titan. PlanetMPAS also includes a set of physics that represents radiative transfer, dry convection, moist convection and its associated microphysics for the Jovian atmosphere. We demonstrate that, despite the rigid-lid approximation, planetMPAS is suitable to simulate the climate systems in Martian and Jovian atmospheres with potential application to slow-rotating planets such as Titan. Simulations using planetMPAS show that the new model can reproduce many aspects of the observed features on Mars and Jupiter, such as the seasonal CO2 cycle, polar argon enrichment, zonal mean temperature, and qualitative dust opacity on Mars, as well as the equatorial superrotation and banded zonal wind patterns on Jupiter.

Fluid-Fluid Interaction Problems at High Reynolds Numbers: Reducing the Modeling Error with LES-C

We consider a fluid-fluid interaction problem, where two flows (with high Reynolds numbers for one or both of these flows) are coupled through a joint interface. A nonlinear coupling equation, known as the rigid lid condition, creates an extra level of difficulty, typical for atmosphere-ocean problems. We propose a novel turbulence model, NS- -C, from the recently introduced family of LES-C (large eddy simulation with correction) models. Combining it with the so-called geometric averaging (GA) partitioning method, we obtain the NS- -C-GA model that is shown to possess several key properties. First, the preexisting solvers for the subdomains can be used, which is critical, e.g., for atmosphere-ocean applications. Second, the LES-C turbulence models use defect correction to efficiently reduce the modeling error of the corresponding LES models; we demonstrate numerically that the NS- -C model outperforms its LES counterpart, the NS- model. It has also been shown recently that it is favorable for an LES model to have the nonfiltered velocity in the interface terms. The NS- -C-GA model possesses this important property; we also show it to be stable and have optimal convergence properties.

Liouville-type results for the time-dependent three-dimensional (inviscid and viscous) water wave problem with an interface

Under consideration here are time-dependent three-dimensional stratified water flows of finite depth with a free surface and an interface (playing the role of an internal wave and separating two layers of constant and different densities). The main outcome of the paper is that, under the assumption that the vorticity vectors in the two layers are constant, non-vanishing and parallel to each other, we prove that bounded solutions to the three-dimensional equations are essentially two-dimensional. More precisely, the free surface, the interface, the pressure and the velocity field present no variations in the direction orthogonal to the direction of motion. In addition, one of the horizontal components of the velocity field is constant throughout the flow and the vorticity also points in the direction orthogonal to the wave propagation direction. The rigid lid case is analyzed from the perspective of relaxed assumptions concerning the vorticity vectors. We also consider the viscous water wave problem case with normal and tangential stress conditions at the free surface and find that under similar conditions on the vorticity vectors, the conclusion about the two-dimensionality remains in place. Along the way, we also provide an explicit solution for the time-dependent three-dimensional inviscid water wave problem with an interface.

Integrative Study on Flow Characteristics and Impact of Non-Submerged Double Spur Dikes on the River System.

The flow characteristics around non-submerged spur dikes continuously placed in the channel on the same side with orthogonal angle to the wall were investigated by numerical simulations and experimental measurements. Three-dimensional (3D) numerical simulations with the standard k-ε Model for incompressible viscous flow based on finite volume method and the rigid lid assumption for free surface treatment were conducted. A laboratory experiment was applied to verify the numerical simulation. The experimental data indicated that the developed mathematical model can effectively predict 3D flow around non-submerged double spur dikes (NDSDs). The flow structure and turbulent characteristics around them were analyzed and it was found that a distinct cumulative effect of turbulence exists between the dikes. By examining the interaction rules of NDSDs, the judgment criterion of spacing threshold was generalized as whether velocity distributions at the cross-sections of NDSDs along the main flow approximately coincided or not. It can be used to investigate the impact scale of the spur dike groups on the straight and prismatic channels and it is of great significance for artificial scientific river improvement and the assessment of river system health under human activities.

Comparison of the Characteristics of Internal Wave Propagation under Two Surface-Boundary Conditions in Continuously Density-Stratified Oceans

Zhang, H.-s.; He, L.; Zhang, Z.-h., and Zhang, Y., 2023. Comparison of the characteristics of internal wave propagation under two surface boundary-conditions in continuously density-stratified oceans. Journal of Coastal Research, 39(2), 303–316. Charlotte (North Carolina), ISSN 0749-0208. Complete theoretical solutions were developed for the propagation of internal waves (IWs) with small amplitudes in a continuously density-stratified ocean under the free surface-boundary condition and rigid lid assumption. Additionally, a numerical model was established for the propagation of IWs under the free surface-boundary condition. In the proposed model, the Euler equations are employed as the governing equations and are discretized with an improved SIMPLE algorithm. In waters with a constant depth, a comparison between the numerical and theoretical results proved the reasonability and accuracy of the proposed method. Additionally, the numerical results under the free surface-boundary condition were compared with those under the rigid lid assumption to analyse the differences between the calculated velocities and density fields. Herein, the influences of characteristic parameters, including the period and amplitude of the incident waves and the coefficient of density and the mode number on the velocity fields under the two surface-boundary conditions, are discussed thoroughly. In the waters with nonuniform varied depths, numerical simulations on the IWs propagation were performed under the two surface-boundary conditions. Differences between the calculated physical fields of IWs in front of and behind the submerged dike are discussed. The detailed analyses showed that differences exist between the calculation results under the two surface-boundary conditions. In particular, they were obvious for the cases with high mode number and variable topography. The results are significant considering the propagation mechanism of IWs under different surface-boundary conditions in a continuously density-stratified ocean.

Clonal microplant production of Corymbia maculata: effect of chemical sterilisation, plant growth regulator, gas exchange, activated charcoal and lighting

Corymbia maculata stands out for its resistance to biotic and abiotic stress and for its high-density wood, which is ideal for sawmills, sleepers, posts, firewood and charcoal. In vitro culture techniques can be used for large-scale clonal microplant production, given the difficulty in adventitious rooting via cutting or mini-cutting techniques. The aim of the work was to evaluate the in vitro multiplication, elongation and adventitious rooting stages of Corymbia maculata based on chemical sterilisation of the culture medium, plant growth regulators, gaseous exchange, activated charcoal and growth room lighting. Supplementation of active chlorine in culture medium reduced fungal and bacterial contamination. The best result for the in vitro multiplication stage was observed in the absence of active chlorine and 0.5 mg l−1 benzylaminopurine (BAP). In vitro shoot elongation in culture medium supplemented with 0.5 mg l−1 α-naphthaleneacetic acid (NAA), 0.001% or 0.003% active chlorine, and vessels sealed with a rigid polypropylene lid with one or three membranes was the most suitable. Glass flasks with porous membrane lids allowed gaseous exchange and favoured elongation and adventitious rooting. Microshoots treated with indole-3-butyric acid (IBA) for 48 h (pulse effect) in the absence of light and activated charcoal in the culture medium is recommended for in vitro adventitious rooting of Corymbia maculata.

Secular Evolution of Continents and the Earth System

AbstractUnderstanding of secular evolution of the Earth system is based largely on the rock and mineral archive preserved in the continental lithosphere. Based on the frequency and range of accessible data preserved in this record, we divide the secular evolution into seven phases: (a) “Proto‐Earth” (ca. 4.57–4.45 Ga); (b) “Primordial Earth” (ca. 4.45–3.80 Ga); (c) “Primitive Earth” (ca. 3.8–3.2 Ga); (d) “Juvenile Earth” (ca. 3.2–2.5 Ga); (e) “Youthful Earth” (ca. 2.5–1.8 Ga); (f) “Middle Earth” (ca. 1.8–0.8 Ga); and (g) “Contemporary Earth” (since ca. 0.8 Ga). Integrating this record with knowledge of secular cooling of the mantle and lithospheric rheology constrains the changes in the tectonic modes that operated through Earth history. Initial accretion and the Moon forming impact during the Proto‐Earth phase likely resulted in a magma ocean. The solidification of this magma ocean produced the Primordial Earth lithosphere, which preserves evidence for intra‐lithospheric reworking of a rigid lid, but which also likely experienced partial recycling through mantle overturn and meteorite impacts. Evidence for craton formation and stabilization from ca. 3.8 to 2.5 Ga, during the Primitive and Juvenile Earth phases, likely reflects some degree of coupling between the convecting mantle and a lithosphere initially weak enough to favor an internally deformable, squishy‐lid behavior, which led to a transition to more rigid, plate like, behavior by the end of the early Earth phases. The Youthful to Contemporary phases of Earth, all occurred within a plate tectonic framework with changes between phases linked to lithospheric behavior and the supercontinent cycle.

Computational fluid dynamics simulation of rough bed open channels using openFOAM

With increased flood risk due to climate change, population expansion and urbanisation; robust waterway design and management are critical. One common type of waterway used to gather and transport ground water is the open channel. Most simulations do not account for the physical roughness of the bed, instead using a roughness coefficient. This means that only the turbulent energy content can be modelled whilst physical turbulent eddies and vortices cannot. Furthermore, many past studies assume the free surface is a rigid lid. This could affect the way that turbulent structures near the free surface behave. Computational Fluid Dynamics simulation of an open channel with a rough bed and rigid lid are conducted using OpenFOAM. Results show good correlation with experimental tests. It can be visually observed that turbulent structures generated from the rough bed do interact with the free surface and thus a rigid lid is perhaps not a great approximation. This is supported by an apparent decrease in the Reynolds shear stress from the free surface and 30% of the flow depth immediately beneath.

Complex wave loads and hydroelastic responses of a very large tourism floating platform ‘Ocean Diamond'

Potential fluid In order to meet the needs of tourism development in the South China Sea, this paper designs a heart-shaped very large floating tourism platform, which integrates the comprehensive functions of cruise ship docking and entertainment. The platform has the characteristics of low flexibility, discontinuous multi-free-surface water and complex phenomena such as water resonance. Firstly, the three-dimensional finite element model of the platform is established, and the complex vibration modes in vacuum are analyzed. It is found that the heart-shaped platform has complex vertical and torsional coupled vibration modes. Secondly, the irregular frequency and water resonance of the elastic structure of the platform are studied by using the three-dimensional frequency domain hydroelastic method based on the extended boundary integral method. The irregular frequencies of the floating body are removed through one rigid lid effectively, and the resonant points of internal free water is clearly revealed. Then, the laws of motions, elastic deformation responses and structural sectional loads are analyzed, and the effects of water resonance point and structural elastic resonance point on motions, structural deformation responses and sectional loads are investigated. Finally, the extreme load prediction is carried out, and the dangerous load conditions and dangerous structural areas are identified.

Free Surface Double Actuator Disc Theory and Double Multiple Streamtube model for in-stream Darrieus hydrokinetic turbines

According to the literature, the last improvement to the Actuator Disc Theory for hydrokinetic turbines in channel flows included blockage ratio and Froude number as independent variables in a single-disc formulation. The authors attempted to solve these equations in a Blade Element Momentum model for axial-flow turbines, ensuring satisfactory results.To our best knowledge, no Blade Element Momentum code was developed for Darrieus rotors including an extended double-disc theory so far. In this perspective, a Free Surface double-disc LMADT is implemented to solve the governing equations in a Double Multiple Streamtube model, facilitating the assessment of In-Stream Darrieus hydrokinetic turbines. Rigid Lid equations, derived for higher-depth applications, are included for completeness. A further numerical procedure allows to set up the channel flow from downstream and update the inflow data due to the resistance induced by the turbine operation in a subcritical flow.In addition to corrections for flow curvature, shading of the downstream flow due to the shaft, and losses due to the mechanical struts, this work encloses an effective sub-model accounting for the turbine installation depth. The Double Multiple Streamtube model is finally validated with previous experimental data of a Darrieus turbine deployed in a narrow channel.

A Constant Pressure Upper Boundary Formulation for Models Employing Height-Based Vertical Coordinates

Abstract For the numerical simulation of atmospheric flows that extend as high as the thermosphere, it is more appropriate to represent the upper boundary of the model domain as a material surface at constant pressure rather than one characterized by a rigid lid. Consequently, in adapting the Model for Prediction Across Scales (MPAS) for geospace applications, a modification of the height-based vertical coordinate is presented that permits the coordinate surfaces at upper levels to transition toward a constant pressure surface at the model’s upper boundary. This modification is conceptually similar to a terrain-following coordinate at low levels, but now modifies the coordinate surfaces at upper levels to conform to a constant pressure surface at the model top. Since this surface is evolving in time, the height of the upper boundary is adaptively adjusted to follow a designated constant pressure upper surface. This is accomplished by applying the hydrostatic equation to estimate the change in height along the boundary that is consistent with the vertical pressure gradient at the model top. This alteration in the vertical coordinate requires only minor modifications and little additional computational expense to the original height-based time-invariant terrain-following vertical coordinate employed in MPAS. The viability of this modified vertical coordinate formulation has been verified in a 2D prototype of MPAS for an idealized case of upper-level diurnal heating. Significance Statement Most atmospheric numerical models that use a height-based vertical coordinate employ a rigid lid at the top of the model domain. While a rigid lid works well for applications in the troposphere and stratosphere, it is not well suited for applications extending into the thermosphere where significant vertical expansion/contraction occurs due to deep heating/cooling of the atmosphere. This paper develops and tests a simple modification to the height-based coordinate formulation that allows the height of the upper boundary to adaptively follow a constant pressure surface. This added flexibility in the treatment of the upper domain boundary for height-based models may be particularly beneficial in facilitating their transition to a deep atmosphere configuration without significant retooling of the model numerics.

Trapped modes in a fluid with three layers topped by a rigid lid

We consider trapping of linear water waves by a submerged horizontal cylinder in a three‐layer fluid topped by a rigid lid. Trapped modes correspond to time harmonic oscillations with finite energy of the fluid surrounding a submerged structure and can be found as eigenfunctions of a certain spectral boundary‐value problem. Our main result is a geometric condition relating the cross sections of the submerged parts of the obstacles and the line integrals along the parts of the interfaces pierced by the obstacles and guaranteeing the existence of trapped modes: This follows from variational techniques applied to a suitable operator formulation of the problem. Several examples of structures (piercing or not the interfaces between the fluid layers) satisfying the condition and supporting trapped modes are given.

Approximate deconvolution models for a fluid-fluid interaction problem with high Reynolds numbers

We propose and investigate three approaches to a fluid-fluid interaction problem with a nonlinear rigid lid condition on the joint boundary when one or both flows are at a high Reynolds number. To do so, we try to combine the Approximate Deconvolution Model of turbulence with a partitioned method of [1], called the Geometric Averaging. The nonlinear interface condition poses an extra difficulty, as there are different approaches to modeling the Reynolds stresses on the joint boundary of the two flows. We investigate three such approaches; all three models are shown to be quantitatively similar when tested on a model with a manufactured solution. Two of them are proven to be unconditionally stable, and yet it is the third model that clearly outperforms the others in the most computationally appealing case when high Reynolds number flows are modeled on a coarse mesh with large filtering width.