Probability Density Function Of Heights Research Articles

Prediction of equivalent sand-grain size and identification of drag-relevant scales of roughness – a data-driven approach

Despite decades of research, a universal method for prediction of roughness-induced skin friction in a turbulent flow over an arbitrary rough surface is still elusive. The purpose of the present work is to examine two possibilities; first, predicting equivalent sand-grain roughness size $k_s$ based on the roughness height probability density function and power spectrum (PS) leveraging machine learning as a regression tool; and second, extracting information about relevance of different roughness scales to skin-friction drag by interpreting the output of the trained data-driven model. The model is an ensemble neural network (ENN) consisting of 50 deep neural networks. The data for the training of the model are obtained from direct numerical simulations (DNS) of turbulent flow in plane channels over 85 irregular multi-scale roughness samples at friction Reynolds number $Re_\tau =800$ . The 85 roughness samples are selected from a repository of 4200 samples, covering a wide parameter space, through an active learning (AL) framework. The selection is made in several iterations, based on the informativeness of samples in the repository, quantified by the variance of ENN predictions. This AL framework aims to maximize the generalizability of the predictions with a certain amount of data. This is examined using three different testing data sets with different types of roughness, including 21 surfaces from the literature. The model yields overall mean error 5 %–10 % on different testing data sets. Subsequently, a data interpretation technique, known as layer-wise relevance propagation, is applied to measure the contributions of different roughness wavelengths to the predicted $k_s$ . High-pass filtering is then applied to the roughness PS to exclude the wavenumbers identified as drag-irrelevant. The filtered rough surfaces are investigated using DNS, and it is demonstrated that despite significant impact of filtering on the roughness topographical appearance and statistics, the skin-friction coefficient of the original roughness is preserved successfully.

A comparison of hydrodynamic and thermal properties of artificially generated against realistic rough surfaces

The mathematical roughness generation approaches enjoy outstanding flexibility in delivering desired roughness geometries to perform systematic research. However, whether an mathematically (artificially) generated roughness can be considered an adequate surrogate of a realistic surface in terms of its influence on the flow remains nonetheless an open question. Motivated by this, the present study discusses the possibility of reproducing flow properties over realistic roughness with artificial roughness. To this end, six types of artificial rough surfaces are generated through imitation of the realistic height probability density function (PDF) and the roughness power spectrum (PS) preserving the stochastic nature of the roughness structure. The flow properties of the artificial surfaces are assessed using direct numerical simulations (DNS) in a fully-developed turbulent channel flow at Reτ=500−2000. An excellent match in terms of global flow properties, mean velocity and temperature profiles, Reynolds stresses as well as equivalent sand grain sizes is found compared to their original counterpart with exception of a strongly anisotropic sample (surface anisotropy ratio SAR≈1.7). Additionally, some artificial surfaces are generated by matching only the PS, and it was shown that only at adequately low effective slopes this can lead to similar flow properties. Overall, the results suggest that artificial roughness generated using the employed method by mimicking realistic PDF and PS can be applied as a full-fledged surrogate for realistic roughness under the premise of surface isotropy.

Novel method for numerical reconstruction of grinding surfaces with specified roughness parameter set

Surface roughness parameters play a decisive role on grinding surface service performance. Among 26 surface roughness parameters, closely related to the surface performance and defined by ISO25178, in the 6 categories, numerical reconstruction of grinding surfaces can only control 7 height parameters and 2 spatial parameters with inability to achieve precise control of other parameter-defined features. Therefore, a novel numerical reconstruction method for grinding surfaces with specified rough parameter set (SRPS) was proposed to solve the problem of feature loss caused by other uncontrollable parameters. Combine three characteristic coefficients of height probability density function in Johnson transformation method with four characteristic coefficients of autocorrelation function expression to construct surface reconstruction coefficient set (SRCS) and build SRPS with number of 22 in consideration of the measurement error influence and industrial application frequency. Use BP neural network to establish quantitative mapping model between SRCS and SRPS for inversion. Through introducing genetic algorithm to invert SRPS corresponding SRCS, numerical reconstruction of grinding surfaces with SRPS will be realized. Experimental results show that compared with the measured grinding surfaces, the average error in SRPS of reconstructed surfaces is basically within 10%. The research provides a new means for the study of the surface performance, which solves the problem that the traditional rough surface modeling method based on random process theory could only associate 9 roughness parameters.

Direct numerical simulation-based characterization of pseudo-random roughness in minimal channels

Direct numerical simulations (DNS) are used to systematically investigate the applicability of the minimal-channel approach (Chung et al., J. Fluid Mech., vol. 773, 2015, pp. 418–431) for the characterization of roughness-induced drag on irregular rough surfaces. Roughness is generated mathematically using a random algorithm, in which the power spectrum (PS) and probability density function (p.d.f.) of the surface height can be prescribed. Twelve different combinations of PS and p.d.f. are examined, and both transitionally and fully rough regimes are investigated (roughness height varies in the range $k^+ = 25$ –100). It is demonstrated that both the roughness function ( ${\rm \Delta} U^+$ ) and the zero-plane displacement can be predicted with ${\pm }5\,\%$ accuracy using DNS in properly sized minimal channels. Notably, when reducing the domain size, the predictions remain accurate as long as 90 % of the roughness height variance is retained. Additionally, examining the results obtained from different random realizations of roughness shows that a fixed combination of p.d.f. and PS leads to a nearly unique ${\rm \Delta} U^+$ for deterministically different surface topographies. In addition to the global flow properties, the distribution of time-averaged surface force exerted by the roughness onto the fluid is calculated. It is shown that patterns of surface force distribution over irregular roughness can be well captured when the sheltering effect is taken into account. This is made possible by applying the sheltering model of Yang et al. (J. Fluid Mech., vol. 789, 2016, pp. 127–165) to each specific roughness topography. Furthermore, an analysis of the coherence function between the roughness height and the surface force distributions reveals that the coherence drops at larger streamwise wavelengths, which can be an indication that very large horizontal scales contribute less to the skin-friction drag.

Probability distribution of wind speed and wave height in Nowshahr Port using the data acquired from wave scan buoy

The present study aimed to determine probability density function (Pdf) of wind speed and wave height in Nowshahr Port using the data obtained from wave scan buoy. Three Pdfs, i.e. Weibull, Rayleigh and Lognormal, were adopted for estimation of data frequency. The shape (k) and scale (c) parameters of the Weibull function were calculated by four methods, i.e. Graphical method (Gm), Moment method (Mm), Maximum likelihood method (Mlm) and Empirical method (Em) whereas the parameters of the other two functions were estimated only through Mlm. In order to analyze the efficiency of pdfs root mean square error (Rmse), determination coefficient (R2) and chi-square test (χ2) were considered. The results revealed that Mm, Em and Mlm presented almost similar ‘k’ and ‘c’ values. However, Gm yielded smaller values. Accuracy of Weibull Pdf was generally higher than those in Rayleigh one. However, it was seen in the present study that Rayleigh Pdf is better only when Weibull parameters in wind data were measured via Gm. Lognormal Pdf was the most efficient in both wind and wave data. Finally, cumulative distribution function was estimated based upon Weibull and Lognormal probability models and the former was again found to be stronger and more accurate.

Wave identification in upward annular flow - a focus on ripple characterization

The interfacial structure of annular flow in upward inclined pipes is examined experimentally by a non-intrusive multilayer conductance liquid film sensor (LFS). In horizontal and inclined pipes, cross-sectional distribution of gas and liquid phases is asymmetric even on average. Thus, the ratio between the characteristic film thickness and the mean dominant wave parameters may vary significantly as a function of the azimuthal and inclination angles. Interface structure in upward annular flow is characterized by two kinds of waves, ripples and disturbance waves. An interface dominated by consecutive large amplitude disturbance waves may also contain numerous ripples. To differentiate between ripples and disturbance waves, a criterion based on statistical wave properties is required. Identification of individual waves is carried out using a zero-crossing procedure with a new reference value. This approach allows to identify both small and large amplitude waves at the interface, calculating the wave heights based on their individual crest and trough. The wave height probability density function (pdf) for a wave system consisting of ripples and disturbance waves is bimodal, with the first peak representing ripples and the second disturbance waves. The local minima between the two peaks in the pdf defines the boundary between ripples and disturbance waves for each flow condition and azimuthal angle. The clear differentiation of the two kinds of waves allows to focus on detailed description of the statistical properties of ripples that are presented as a function of the pipe inclinations angles (ranging from near horizontal to vertical) and for all azimuthal location across the pipe circumference.

Modelling non-Gaussian surfaces and misalignment for condition monitoring of journal bearings

In the hydrodynamic lubricated journal bearing system, the surface roughness and angular misalignment are two critical factors that affect bearing’s performance. In this paper, the coupled effects of different non-Gaussian properties and misalignments are investigated on the performance of journal bearings. Christensen’s stochastic model is extended by improving the probability density function of random roughness heights, which incorporates the Gram-Charlier expansion including skewness and kurtosis. In comparison with a Gaussian surface, the non-Gaussian rough surface has more significant influence on the bearing static performance. The negative skewness and large kurtosis increase the load capacity and decrease the friction coefficient. According to the simulations and experiments, non-Gaussian properties have more impact on the performance than misalignment when the journal bearing is operated in hydrodynamic lubrication regime based on different pressure distributions and vibration responses. These novel findings provide the basis for monitoring the conditions of hydrodynamic journal bearings.

Extreme events in a network of heterogeneous Josephson junctions.

We report intermittent large spiking events in a heterogeneous network of forced Josephson junctions under the influence of repulsive interaction. The response of the individual junctions has been inspected instead of the collective response of the ensemble, which reveals the large spiking events in a subpopulation with characteristic features of extreme events (EE). The network splits into three clusters of junctions, one in coherent libration, one in incoherent rotational motion, and another subpopulation originating EE, which resembles a chimeralike pattern. EE migrates spatially from one to another subpopulation of junctions with the repulsive strength. The origin of EE in a subpopulation and chimera pattern is a generic effect of distributed damping parameter and repulsive interaction, which we verify with another network of the Liénard system. EE originates in the subpopulation via a local riddling of in-phase synchronization. The probability density function of event heights confirms the rare occurrence of large events and the return time of EE as expressed by interevent intervals in the subgroup follows a Poisson distribution. The mechanism of the origin of such a unique clustering is explained qualitatively.

Modeling for Microslip Behavior of Lap Joints Based on Non-Gaussian Rough Surfaces

AbstractA general contact model for a lap joint interface based on non-Gaussian surfaces was proposed. The effect of surface topography parameters on microslip behavior in a lap joint interface was studied. Pearson system was applied to produce non-Gaussian surfaces. Combining the topographical-dependent Zhao–Maietta–Chang (ZMC) model with the physical-related Iwan model, the nonlinear constitutive relationship of a lap interface was constructed by using Masing hypothesis. Meanwhile, the probability density function of asperity heights of an infinitely smooth surface was mathematically proved to be a delta function, verifying that the calculated value of friction in the model conforms to the physical law. Gauss-Legendre quadrature was conducted to calculate contact relations of different Pearson distribution surfaces. Furthermore, numerical results of microslip loops under oscillating tangential forces were compared with the published experiments, indicating the present model considering non-Gaussian surfaces could agree well with the experiments.

Effects of Wall Heat Loss on Swirl-Stabilized Nonpremixed Flames With Localized Extinction

Large eddy simulation (LES) with three-dimensional conditional moment closure (CMC) subgrid model for combustion is applied to simulate a swirl-stabilized nonpremixed methane flame with localized extinction, with special focus on the effects of heat loss to the burner surface. The convective wall heat loss is modeled through introducing a source term in the conditionally filtered total enthalpy equation for the CMC cells adjacent to the wall. The mean heat flux is high on the middle surface of the bluff body, but relatively low near its edges. The turbulent heat flux based on the gradient of the resolved temperature is relatively low compared to the laminar counterpart, but increases with the turbulent intensity. The heat loss facilitates the occurrences of extinction and re-ignition for the CMC cells immediately adjacent to the wall, evidenced by comparing flame structures in the near-wall CMC cells. This can be directly linked to the increase of the mean conditional scalar dissipation near the wall in the heat loss case. Furthermore, the degree of local extinction near the bluff body measured by conditional reactedness at stoichiometry is intensified due to the wall heat loss. However, the results also show that there is negligible influence of wall heat loss on the probability density function (PDF) of the lift-off height, demonstrating the dominance of aerodynamic effects on flame stabilization. The results are in reasonable agreement with experimental measurements.

Summarising wave data for the purpose of operational planning on coastlines

The trends within various coastal hydrodynamic data sets were investigated with the aim of providing a general methodology to gather, condense and store useful wave statistics using data collected on the eastern coast of Trinidad, an island located in the Caribbean region. The lack of local comprehensive long-term data sets necessitates the use of short- to medium-term data in order to establish probabilistic descriptions to be used in long-term modelling, planning and hazard assessment exercises. While the paucity of data at certain locations initially stimulated the investigation of monthly and quarterly trends, as well as near-shore and off-shore differences of the wave parameters, it also identified a method to reduce large data sets while retaining the critical wave characteristics. The summarised data sets can be more readily used for planning and decision making, but can also be used for application to advanced scientific tools for further investigation. General methodologies for analysing the short- to medium-term data were established in order to derive basic statistical parameters, deduce persistence statistics, assess multi-modality and subsequently generate a generic wave height probability density function for the coastal location.

Numerical Prediction of Surface Wear and Roughness Parameters During Running-In for Line Contacts Under Mixed Lubrication

A stochastic model for predicting the evolutions of wear profile and surface height probability density function (PDF) of initial line contacts during running-in under mixed lubrication condition is presented. A numerical approach was developed on the basis of stochastic solution of mixed lubrication, which combined the Patir and Cheng's average flow model for calculation of the hydrodynamic pressure and the Kogut and Etsion's (KE) rough surface contact model for calculation of the asperity contact pressure. The total friction force was assumed to be the sum of the boundary friction at the contact asperities and the integration of viscous shear stress in the hydrodynamic region. The wear depth on the contact region was estimated according to the modified Archard's wear model using the asperity contact pressure. Sugimura's wear model was modified and used to link the wear particle size distribution and the variation of surface height PDF during wear. In the wear process, the variations of profile and surface height PDF of initial line contacts were calculated step by step in time, and the pressure distribution, friction coefficient, and wear rate were updated consequently. The effect of size distribution of wear particles on the wear process was numerically investigated, and the simulation results showed that the lubrication condition in which small wear particles are generated from the asperity contact region is beneficial to reduce friction coefficient and wear rate, and leads to a better steady mixed lubrication condition.

A modified Parametric Forcing Approach for modelling of roughness

Surface roughness in turbulent channel flow is effectively modelled using a modified version of the Parametric Forcing Approach introduced by Busse and Sandham (2012). In this modified approach, the model functions are determined based on the surface geometry and two model constants, whose value can be fine tuned. In addition to a quadratic forcing term, accounting for the effect of form drag due to roughness, a linear forcing term, analogous to the Darcy term in the context of porous media, is employed in order to represent the viscous drag. Comparison of the results with full-geometry resolved Direct Numerical Simulation (DNS) data for the case of dense roughness (frontal solidity ≅0.4) shows a satisfactory prediction of mean velocity profile, and hence the friction factor, by the model. The model is found to be able to reproduce the trends of friction factor with morphological properties of surface such as skewness of the surface height probability density function and coefficient of variation of the peak heights.

Finite time ruin probability and structural density properties in the presence of dependence in insurance risk model

ABSTRACTIn the present paper, we consider the classical compound Poisson risk model with dependence between claim sizes and claim inter-arrival time. We attempt to analyze the approximation of finite time ruin probability. The finite time ruin probabilities are plotted for fixed threshold value associated to the claim inter-arrival time and also for fixed dependence parameter in Nelsen (2006) copula separately. Additionally, a general form for joint density of the interclaim times and claim sizes is considered. With respect to the classical Gerber-Shiu's (1998) function, first some structural density properties of dependent collective risk model is obtained. Then the ladder height probability density function of claim sizes is computed and the dependency structure investigated for Erlang interclaim time. As the application, some dependent models of the interclaim times and claim sizes are studied.

A fast convolution based waveform model for conventional and unfocused SAR altimetry

In this paper we derive a closed-form solution to generate a backscatter power representation for both unfocused synthetic aperture radar altimetry and conventional altimetry signals in the frequency/slow-time domain. This new approach involves fast convolutions and leads to a fast numerical computational model which uses very few approximations. The transform back to the range-time/doppler frequency domain is then made numerically.Two retrackers based on this Signal Model Involving Numerical Convolution (SINC) are constructed for conventional and synthetic aperture radar altimetry and are named SINC2 and SINCS respectively. Output from retracking are the epoch t0 with respect to the tracking reference point, the significant wave height and the backscattering cross section at normal incidence σ0.Main benefits of the SINC model are its flexibility, the possibility to use the real point target response (PTR) or more complex representations of the height probability density function of scattering sea surface elements (PDF), and its fast computation speed to retrack the waveforms.The retrackers are applied to CryoSat-2 reduced SAR (RDSAR) and unfocused SAR waveforms. The RDSAR SINC2 retracker is only ten percent slower than the commonly used MLE3 retracker, which is a minor slowdown compared to the benefits of the SINC model. The SAR processing including the SINCS retracking procedure is three times slower than the RDSAR processing including retracking with SINC2.Model and algorithms are validated on both synthetic data generated by Monte-Carlo simulations and on real datasets. SAR SAMOSA+ results accessible through the ESA G-POD SARvatore service serve as a reference. Cross-validation shows higher agreement between SAR SAMOSA+ and SAR SINCS sea level anomaly and lower agreement between SAR SAMOSA+ and RDSAR SINC2, with accuracy of −0.1 cm for SAR SINCS and 1.7 cm for RDSAR SINC2. Instead, the accuracy of significant wave height, 4.9 cm and 3.9 cm for SAR and RDSAR and of backscatter coefficient, 0.16 and 0.12 dB in SAR and RDSAR, are comparable.Cross-validation results shows for sea level anomaly and significant wave height a higher precision of SAR SINCS and a lower precision of RDSAR SINC2, with precision of 0.92 cm and 6.6 cm for SAR SINCS and 1.96 cm and 12.6 cm for RDSAR SINC2 in sea level anomaly and significant wave height respectively. Instead, the precision of the backscatter coefficient is higher in RDSAR SINC2 than in SAR SINCS, and is 0.014 and 0.016 dB.In summary the new open ocean retrackers are a viable alternative recommended for both reduced and unfocused SAR processing in open ocean.

An Approximation to the Statistical Characteristics of Wind Waves in Front and from the Toe of the Structure to the Toe of the Crown of Nonovertopped Breakwaters

According to the main recommendations and technical criteria along the world, the design project of a breakwater must address the verification of the different failure modes that can affect the breakwater stability. This research focuses on the estimation of the statistical characteristics of the wind waves interacting with different breakwater types and their evolution from the toe of the structure to the toe of the crown. This knowledge is essential to calculate the reliability of the structure in its useful life. Partial standing wave patterns are likely to occur in front and along the section of the breakwater depending on their typology. Based on Rice's theory of envelope amplitude, we present an approximate solution of the total wave height distribution in front of the breakwater. The experimental results in 2D confirmed that the incident, the reflected and the total wave height in front of the breakwater followed a Rayleigh distribution in which the parameter is the root-mean-square total wave height. Its value depends on the modulus and phase of the reflection coefficient. The probability density function of the total wave height evolves from a Rayleigh to a Weibull distribution, whose scale and shape parameters vary from the toe to the crown of the structure, and depend on the breakwater type, the relative grain diameter, and the relative water depth. The largest deviation from the Rayleigh distribution occurs at the toe of the crown. These findings are weakly dependent on the incident wave steepness, and are valid for narrow-banded incident wave trains, impinging perpendicularly on nonovertopped breakwaters with a steep frontface, which ensures the rapid evolution of the wave train. Then, each mode of failure that might occur in the breakwater section can be formally checked against the same incident wave height, but verified (calculated) with the actual wave height, locally transformed by the specific typology.

Toward a Universal Roughness Correlation

The effects of several surface parameters on equivalent sand roughness (ks) in fully rough regime are investigated by means of direct numerical simulation (DNS) of flow in channels with different wall geometries at Reτ≅500. The roughness geometry is generated by randomly distributing roughness elements of random size and prescribed shape on a flat surface. The roughness generation approach allows systematic variation of moments of surface height probability density function (PDF), size distribution of roughness peaks, and surface slope. A total number of 38 cases are solved. It is understood that a correlation based on surface height skewness and effective slope (ES) can satisfactorily predict ks normalized with maximum peak-to-valley roughness height within a major part of the studied parameter space. Such a correlation is developed based on the present data points and a number of complementary data points from the literature. It is also shown that the peak size distribution can independently influence the skin friction; at fixed values of rms surface height, skewness, kurtosis, and ES, a surface with uniform size peaks causes higher skin friction compared to one with nonuniform peak sizes. Additionally, it is understood that a roughness generated by regular arrangement of roughness elements may lead to a significantly different skin friction compared to a random arrangement. A staggered and an aligned regular arrangement are examined in this paper and it is observed that the former produces significantly closer results to the corresponding random arrangement.

Detailed chemistry LES/CMC simulation of a swirling ethanol spray flame approaching blow-off

A swirling ethanol spray flame in conditions close to blow-off has been simulated using Large Eddy Simulation (LES) and the Conditional Moment Closure (CMC) combustion model aiming to further validate the capability of the LES/CMC approach to capture local extinctions in turbulent spray flames. A detailed chemical mechanism was used and a transport equation of the mixture fraction sub-grid variance, with spray interaction terms included, was solved. Numerical results are in good agreement with the experiment in terms of both instantaneous and mean flame shape and droplet velocity and size. Local extinctions were detected in the region around the bluff-body, resulting in a fluctuating lift-off of the flame there, and the probability density function of the lift-off height was in very good agreement with the experiment, suggesting that the degree of local extinction is captured quantitatively. Analysis of the CMC equation suggested that local extinction was influenced by both transport in physical space and high scalar dissipation rate. The modelling of the latter needs development in areas where the spray evaporation is strong enough to increase significantly the sub-grid mixture fraction fluctuations and their small-scale gradients, possibly leading to deviations from the present usual approach of relating the sub-grid scalar dissipation to the sub-grid mixture fraction variance.

Tropical tropopause layer cirrus and its relation to tropopause

This study examines the spatial and temporal patterns of tropical tropopause layer (TTL) cirrus clouds (i.e., clouds with bases higher than 14.5km) and their relationship to tropical tropopause including both cold point tropopause (CPT) and lapse rate tropopause (LRT). We use eight years (2006–2014) data from the Cloud–Aerosol Lidar and Infrared Pathfinder Satellite Observations (CALIPSO) and Constellation Observing System for Meteorology, Ionosphere, and Climate (COSMIC) measurements. In addition to the CALIPSO cloud layer product, the clouds included in the current CALIPSO dataset as stratospheric features have been considered by separating clouds from aerosols, which are important in the TTL cloud analysis. It is also shown that the temporal variation of the stratospheric aerosols matches well with the volcanic eruption events. The TTL cloud fraction and the tropical tropopause temperature both have pronounced annual cycles and are strongly negatively correlated both temporally and spatially. The examination of the TTL cloud height relative to tropopause from collocated CALIPSO and COSMIC observations indicates that the tropopause plays a critical role in constraining the TTL cloud top height. We show that the probability density function of TTL cloud top height peaks just below the CPT while the occurrence of TTL clouds with cloud tops above the CPT could be largely explained by observed tropopause height uncertainty associated with the COSMIC vertical resolution.

The nonlinear peak response distribution of structure–TLD systems during common wind events

Linearized equivalent mechanical models are often used to conduct preliminary tuned liquid damper (TLD) design; however, they cannot capture nonlinear behavior resulting from the coupling among sloshing modes. Therefore, these models underestimate the expected peak wave height, a quantity that is necessary to evaluate the required tank freeboard. In this study, a method is presented to estimate the nonlinear peak wave height using an equivalent mechanical model.A linearized equivalent mechanical model of the structure–TLD system is employed to determine the RMS responses of the structural displacement and the TLD fundamental sloshing mode wave height. Davenport’s derivation is used to estimate the probability density function of the peak structural displacement. Subsequently, the probability density function of the peak wave heights is derived using a Rayleigh–Stokes model, which employs an extended second order Stokes wave theory to estimate the nonlinear peak wave heights. Expressions are derived to calculate the mean-peak wave height as a function of the RMS response of the fundamental sloshing mode, and the fluid depth ratio.Structure–TLD system tests and nonlinear simulations are used to evaluate the model. Findings reveal that the Rayleigh model is in agreement with the experimental and simulated peak structural response distributions. In addition, the Rayleigh–Stokes model is in agreement with the experimental and particular simulated peak wave height distributions. Discrepancies are expected to arise when the fluid depth ratio is small, the excitation amplitude is large, and the damping screens are positioned to provide significant damping to the second sloshing mode.