Unsteady Bernoulli Equation Research Articles

Numerical estimation of wave breaking forces on seawalls

ABSTRACT This paper presents a computationally efficient numerical scheme to estimate wave forces from plunging breakers on vertical rigid seawalls. The approximated velocity equation, an asymptotical solution of the Green-Naghdi equation in the Camassa-Holm scaling, is utilized to obtain the evolution of the free surface and the depth-averaged horizontal velocity over time, leading to the formation of a plunging breaker. The uniform dynamic pressure along the depth is obtained by utilizing the unsteady Bernoulli equation. The wave forces on seawalls are then determined using Froude-Krylov theory, with adjustments for diffraction effects. The results, while being comparable to some existing formulations, notably differ with some others. This highlights the limitations imposed by the inherent assumptions in the existing empirical formulations and underlines the importance of the proposed methodology that is founded on an accurate modelling of the wave breaking process to estimate wave breaking forces on seawalls.

Flow of water out of a funnel

The unsteady Bernoulli equation is used to numerically determine the surface height and velocity distribution of water flowing out of a conical tube as a function of time. The speed is found to interpolate between freefall for a cylindrical pipe of constant radius and Torricelli’s law for a funnel having a small exit hole. In addition, the applied force needed to hold the conical vessel in place is calculated using Newton’s second law including the rocket thrust due to the water flowing out of the funnel. A comparison is made with the analogous expressions for the flow through and holding force on a right cylindrical tank having a hole in its bottom face. The level of presentation is appropriate for an undergraduate calculus-based physics course in mechanics that includes a module on fluid dynamics.

Aerodynamic Stability Derivative Calculations Using the Compressible Source and Doublet Panel Method

This work presents the development of closed-form expressions for the aerodynamic stability derivatives of wings and aircraft flying at subsonic compressible conditions using the unsteady subsonic 3D source and doublet panel method (SDPM). The stability derivatives are obtained directly from the frequency domain version of the nonlinear unsteady subsonic Bernoulli equation and can be calculated for negligible additional computational cost around the true geometry of the wing or aircraft for each frequency of interest. The methodology is validated by applying it to an experimental test case of a swept wing oscillating in yaw and sideslip in the wind tunnel, demonstrating that the values of the aerodynamic stability derivatives obtained from the SDPM are in good agreement with the experimental data at low and moderate angles of attack. Then, the methodology is applied to a blended-wing–body unmanned aerial vehicle configuration and the SDPM stability derivative predictions are compared to estimates obtained from both steady computational fluid dynamics calculations and the USAF DATCOM methodology. It is shown that the SDPM approach is accurate and can yield more information about the stability of the aircraft than the DATCOM, given that the latter was developed for conventional aircraft configurations.

A theoretical model for predicting the startup performance of pumps as turbines

Using the unsteady Bernoulli equation for the piping system and the angular momentum equation for the rotor, derives here a theoretical model to predict the startup performance of a pump as turbine (PAT). This model is effective for predicting the instantaneous evolution characteristics of the main performance parameters of PAT during startup, and these changings are initially faster and then slowly as a whole. The effect of the rotor moment of inertia and the final stabilized rotational speed of PAT on evolution characteristics of parameters is opposite. The rotational speed, head, hydraulic power, and conversion efficiency show a upward rising trend with the startup time, whereas the flow rate and hydraulic head loss display a downward trend.

Velocity field and cavity dynamics in drop impact experiments.

Drop impact experiments allow to model a wide variety of natural processes, from raindrop impacts to planetary impact craters. In particular, interpreting the consequences of the planetary impacts requires an accurate description of the flow associated with the cratering process. In our experiments, we release a liquid drop above a deep liquid pool to investigate simultaneously the dynamics of the cavity and the velocity field produced around the air-liquid interface. Using particle image velocimetry, we analyse quantitatively the velocity field using a shifted Legendre polynomials decomposition. We show that the velocity field is more complex than considered in previous models, in relation to the non-hemispherical shape of the crater. In particular, the velocity field is dominated by the degrees 0 and 1, with contributions from the degree 2, and is independent of the Froude and the Weber number when these numbers are large enough. We then derive a semi-analytical model based on the Legendre polynomials expansion of an unsteady Bernoulli equation coupled with a kinematic boundary condition at the crater boundary. This model explains the experimental observations and can predict the time evolution of both the velocity field and the shape of the crater, including the initiation of the central jet.

Mechanistic-mathematical modeling of intracranial pressure (ICP) profiles over a single heart cycle. The fundament of the ICP curve form

The intracranial pressure (ICP) curve with its different peaks has been comprehensively studied, but the exact physiological mechanisms behind its morphology has not been revealed. If the pathophysiology behind deviations from the normal ICP curve form could be identified, it could be vital information to diagnose and treat each single patient. A mathematical model of the hydrodynamics in the intracranial cavity over single heart cycles was developed. A Windkessel model approach was generalized but the unsteady Bernoulli equation was utilized for blood flow and CSF flow. This is a modification of earlier models using the extended and simplified classical Windkessel analogies to a model that is based on mechanisms rooted in the laws of physics. The improved model was calibrated with patient data for cerebral arterial inflow, venous outflow, cerebrospinal fluid (CSF), and ICP over one heart cycle from 10 neuro-intensive care unit patients. A priori model parameter values were obtained by considering patient data and values taken from earlier studies. These values were used as an initial guess for an iterated constrained-ODE (ordinary differential equation) optimization problem with cerebral arterial inflow data as input into the system of ODEs. The optimization routine found patient-specific model parameter values that produced model ICP curves that showed excellent agreement with clinical measurements while model venous and CSF flow were within a physiologically acceptable range. The improved model and the automated optimization routine gave better model calibration results compared to previous studies. Moreover, patient-specific values of physiologically important parameters like intracranial compliance, arterial and venous elastance, and venous outflow resistance were determined. The model was used to simulate intracranial hydrodynamics and to explain the underlying mechanisms of the ICP curve morphology. Sensitivity analysis showed that the order of the three main peaks of the ICP curve was affected by a decrease in arterial elastance, a large increase in resistance to arteriovenous flow, an increase in venous elastance, or a decrease in resistance to CSF flow in the foramen magnum; and the frequency of oscillations were notably affected by intracranial elastance. In particular, certain pathological peak patterns were caused by these changes in physiological parameters. To the best of our knowledge, there are no other mechanism-based models associating the pathological peak patterns to variation of the physiological parameters.

손상 선박 기름 유출량 추정을 위한 수치해석과 이론식의 비교 연구

This paper provides the results of numerical and theoretical predictions of oil outflows from damaged single-hull and double-hull ships.Theoretical equations derived from the unsteady Bernoulli equation and a CFD method for multi-phase flow analysis were used to estimate the oil outflow rate from cargo tank. The predicted oil outflow rate from a single-hull cargo tank damaged due to grounding and collision accidents showed a good agreement with the available experimental results in both numerical and theoretical analyses. However, in the case of the double-hull conditions, the time variation of the amount of water and oil mixture inside the ballast tank predicted by the theoretical equation showed some different behavior from the numerical results. The reason was that the interaction of the oil flow with the water inflow in the ballast tank was not reflected in the theoretical equations. In the problems of the initial pressure condition in the cargo and ballast tanks, the oil outflow and water inflow were delayed at the pressure condition that the tanks were sealed. When the flow interaction between the oil and water in the ballast tank was less complicated, the theoretical and the numerical results showed a good agreement with each other.

Understanding the dynamics of fluid–structure interaction with an Air Deflected Microfluidic Chip (ADMC)

A deformable microfluidic system and a fluidic dynamic model have been successfully coupled to understand the dynamic fluid–structure interaction in transient flow, designed to understand the dentine hypersensitivity caused by hydrodynamic theory. The Polydimethylsiloxane thin sidewalls of the microfluidic chip are deformed with air pressure ranging from 50 to 500 mbar to move the liquid meniscus in the central liquid channel. The experiments show that the meniscus sharply increased in the first 10th of second and the increase is nonlinearly proportional to the applied pressure. A theoretical model is developed based on the unsteady Bernoulli equation and can well predict the ending point of the liquid displacement as well as the dynamics process, regardless of the wall thickness. Moreover, an overshooting and oscillation phenomenon is observed by reducing the head loss coefficient by a few orders which could be the key to explain the dentine hypersensitivity caused by the liquid movement in the dentine tubules.

Symmetry-breaking self-sustained oscillation in nonlinear two-phase flow

The nonlinear dynamics of two-phase flow advances our understanding of limit-cycle phenomena and informs the control of fluid oscillation. Here, we introduce nonlinear fluid dynamics of a high-temperature steam jet as a class of two-phase flow with steam-liquid coupling. Transition of steam plume from continuous expansion to rupture contraction is revealed as a nonequilibrium symmetry-breaking phenomena in high-frequency fluid oscillation. We establish a versatile platform of high-temperature steam jet to explore self-sustained two-phase flow oscillator dynamics. A generalized nonlinear steam-liquid coupled dynamic model derived from isentropic flow, the unsteady Bernoulli equation and interface mechanics gives theoretical predictions and serves as rationale for our experimental observations. Our theory about the oscillation mechanism and propagation law of the steam-liquid flow of high-temperature steam jets points toward exciting directions for future research, from understanding hydrovolcanic eruption to propagation and control of high-temperature condensation.

Filling and emptying a tank of liquid

A right cylindrical tank open to the atmosphere is being filled by a laminar jet of incompressible inviscid liquid falling onto its free surface. At the same time, fluid is escaping through a hole centered in the bottom of the tank. Newton’s second law for variable mass and the unsteady Bernoulli equation are combined to find the time dependence of the liquid height in the tank. The level of analysis is suitable for an introductory undergraduate course in fluid dynamics.

Stochastic response determination of U-OWC energy harvesters: a statistical linearization solution treatment accounting for intermittent wave excitation

A novel statistical linearization technique is developed for determining approximately the response statistics and the power output of U-Oscillating Water Column (U-OWC) energy harvesting systems. In this regard, first, the governing equations are derived by employing the unsteady Bernoulli equation. Note that the intermittent, i.e., non-stationary, nature of the wave excitation, occurring in severe sea states due to uncovering of the U-OWC inlet, is explicitly accounted for in the herein proposed model. This is done by multiplying the excitation process with a Heaviside function dependent on the instantaneous free surface displacement. Next, the resulting coupled system of nonlinear integro-differential stochastic equations is solved approximately by relying on a statistical linearization technique. Specifically, the original system of nonlinear equations is replaced by an equivalent linear one, whose parameters and response first- and second-order statistics are obtained by minimizing the mean square error between the two systems. A significant novel aspect of the technique relates to the fact that the Heaviside function is replaced in the equivalent linear system by an “equivalent excitation” coefficient to be determined as part of the statistical linearization solution scheme. Further, compared with other relevant solution schemes adopted in earlier research efforts in the literature, it is shown that the developed technique can be construed as a direct generalization that exhibits an enhanced accuracy degree. The U-OWC installed in the Civitavecchia harbor (Rome, Italy) is considered as an illustrative numerical example, where the reliability of the approximate technique is demonstrated by comparisons with pertinent Monte Carlo simulation data.

Subsonic source and doublet panel methods

This work presents a source and doublet surface panel method for solving unsteady subsonic flows around wing and bodies. The approach is based on Morino’s theory but features three modifications. The impermeability boundary condition is defined as zero mass flux normal to the surface instead of zero normal velocity, a new methodology is introduced to calculate the aerodynamic influence coefficient matrices and the pressure distribution around the surface is evaluated using a nonlinear unsteady Bernoulli equation. The method is demonstrated on three experimental test cases from the literature; it is shown that it can predict more accurately the steady and oscillating pressure distribution around the surface than Morino’s original linear approach. Crucially, the modified approach can represent asymmetries in the pressure distribution due to both asymmetric motion and shape. While higher fidelity alternatives exist for solving such flows, the present method has very low computational cost and would be suitable for optimization procedures during preliminary design, as long as the flow remains subcritical and attached at all times.

Wind Loading of Photovoltaic Panels Installed on Hip Roofs of Rectangular and L-Shaped Low-Rise Buildings

Many residential houses in Japan have hip roofs with pitches ranging from 20° to 30°. Recently, roof-mounted photovoltaic (PV) panels have become popular all over the world for environmental conservation. The design of PV systems in Japan is usually based on the Japanese Industrial Standard (JIS) C 8955 (2017). However, the standard does not provide wind force coefficients for PV panels installed near roof edges (up to 0.3 m from the edge) because flow separation at the roof edges causes large up-lift forces on such panels. In this paper, we investigated the wind force coefficients for designing PV panels installed on hip roofs of rectangular and L-shaped low-rise buildings. The roof pitch was set to 25° as a typical value. Rectangular panels were installed almost over the whole roof, including the edge zones. Because the thickness of PV panels and the distance between PV panels and the roof are both as small as several centimeters, it is difficult to make wind tunnel models of PV systems with the same geometric scale as that for buildings. We focused on a numerical simulation using the unsteady Bernoulli equation to estimate the pressures in the space between PV panels and the roof. In the simulation, we used the time histories of wind pressure coefficients on the bare roof, which were measured in a turbulent boundary layer. We propose installing PV panels with small gaps between them along their short sides. The gaps may reduce the wind loads not only on the PV panels but also on the roofing due to pressure equalization. We discuss the optimum gap width from the viewpoint of wind load reduction.

Application of a Numerical Simulation to the Estimation of Wind Loads on Photovoltaic Panels Installed Parallel to Sloped Roofs of Residential Houses

Many residential houses with sloped roofs are equipped with photovoltaic (PV) systems. In Japan, PV systems are generally designed based on JIS C 8955, which specifies wind force coefficients for designing PV panels. However, no specification is provided to the PV panels installed near the roof edges where high suctions are induced. When installing PV panels in such high-suction zones, we need to evaluate the wind loads on the PV panels appropriately, usually by performing a wind tunnel experiment. However, it is difficult to make wind tunnel models of PV panels with the same geometric scale as that for the building, e.g., 1/100, because the thickness of PV panels and the distance between PV panels and a roof are both several centimeters. Therefore, in the present paper a numerical simulation is applied to the estimation of pressures in the space between the lower surface of PV panels and the roof surface, called “layer pressures”, using the unsteady Bernoulli equation and the time histories of external pressure coefficients obtained from a wind tunnel experiment. An assumption of the weak compressibility of the air and an adiabatic condition is made for predicting the layer pressures from the flow speed through the gaps. The simulation method is validated by a wind tunnel experiment using a model of square-roof building.

The Benefit of Horizontal Photovoltaic Panels in Reducing Wind Loads on a Membrane Roofing System on a Flat Roof

The present paper proposes a measure for improving the wind-resistant performance of photovoltaic systems and mechanically attached single-ply membrane roofing systems installed on flat roofs by combining them together. Mechanically attached single-ply membrane roofing systems are often used in Japan. These roofing systems are often damaged by strong winds, because they are very sensitive to wind action. Recently, photovoltaic (PV) systems placed on flat roofs have become popular. They are also often damaged by strong winds directed onto the underside, which cause large wind forces onto the PV panels. For improving the wind resistance of these systems, we proposed to install PV panels horizontally with gaps between them. Such an installation may decrease the wind forces on the PV panels due to the pressure equalization effect as well as on the waterproofing membrane due to the shielding effect of the PV panels. This paper discusses the validity of such an idea. The pressure on the bottom surface of a PV panel, called the “layer pressure” here, was evaluated by a numerical simulation based on the unsteady Bernoulli equation. In the simulation, the time history of the external pressure coefficients, measured at many points on the roof in a wind tunnel, was employed. It was found that the wind forces, both on the PV panels and on the roofing system, were significantly reduced. The reduction was large near the roof’s corner, where large suction pressures were induced in oblique winds. Thus, the proposed method improved the wind resistance of both systems significantly.

Vessel Drainage Under the Influence of Gravity

The physical problem of a body of water in a tank that drains through a hole in the base is a classical problem that has been studied since at least the time of Torricelli. To fixate this in a student’s mind, one could ask them to visualize a bathtub that is being drained through the plughole or a bottle being drained through a tap. This problem is an ideal one to propose at the secondary school level, since it provides a nice example of a physically relevant elementary ODE that can be studied by students with some calculus skills, and it also provides an opportunity to start introducing some concepts from fluid dynamics, as well as general mathematical modelling skills. Like the brachistochrone, the draining vessel and variants of it have been studied many times in the literature. We will focus on a simple pedagogical example, but there are many other possibilities. For example, Castro de Oliveira et al. performed experiments to measure the speed of water flowing through a pinhole at the bottom of a cylindrical bottle as a function of time and the level of water. In the process, they made an empirical correction to Torricelli’s law. Cross studied three different arrangements whereby air and water can flow into or out of two small holes in the side of a water bottle. Hong studied the unsteady draining problem for an incompressible, non-viscous fluid to obtain an exact solution of the unsteady Bernoulli equation, showing that once again Torricelli’s law must be modified in this situation. In a more mathematically involved treatment, Digilov showed that one can obtain an analytic solution for time evolution of a liquid column draining under gravity through a capillary tube in terms of a special function called the Lambert W function. More recently, Durbin studied vessel drainage of fluids of different viscosities through tubes of varying lengths.

Fragmentation of a liquid metal droplet falling in a water pool

This paper focuses on the experimental investigation of breakup regimes of a molten fusible metallic droplet in water, at intermediate Weber numbers with emphasis on mass and energy balance. The experiment consists in impacting perpendicularly a molten drop onto the interface of a deep water pool, at a controlled temperature. Using a drop-on-demand device and high-speed shadowgraph, a single drop can be visualized during its evolution. There is a noticeable velocity jump when the droplet crosses the interface that can be modeled using an unsteady Bernoulli equation. As observed for liquid–gas systems, the drop experiences different regimes of fragmentation, depending on its Weber number: oscillations, bag oscillations, prolate drop stretching breakup, and then bowl-shaped bag breakup. However, opposite to the gas–liquid case, a Rayleigh–Taylor instability mechanism seems to be absent and this seems related to the bowl-shaped bag breakup mechanism when compared to the dome-shaped gas–liquid case. Statistics of the daughter droplets are then given, using either image analysis for large droplets size distribution or sieving and weighting of the solidified fragments for measurement of the Sauter mean diameter and surface energy creation. Finally, a simple relation between the Sauter mean diameter and the Weber number is presented based on the energy and mass balances. When comparing with previous higher Weber number results, a viscous transition corresponding to a strong increase in the energy loss is also shown to occur for the higher Weber number.

Liquid oscillating in a U-tube of variable cross section

A liquid is set into oscillations in a vertical manometer of smoothly varying cross-sectional area open at both ends to the atmosphere. Ignoring viscous damping, the displacement of either free surface and the pressure at any point in the liquid can be calculated using the unsteady Bernoulli equation which is shown to be consistent with conservation of mechanical energy. If the cross section is constant, the equations are analytically solvable; simple harmonic motion is found for the surface displacement as a function of time, and the pressure variation as a function of depth is in accordance with Newton’s second law. More generally, however, numerical solution of the equations is necessary, as illustrated for the example of a U-tube whose radius is linearly tapered from one end to the other. The level of presentation is appropriate for an upper level undergraduate course in fluid mechanics.

SIMULATION OF AIRLESS MODULATED 2D SMALL-SCALE/MEMS ATOMIZER USING VISCOUS POTENTIAL THEORY WITH GENERALIZED PRESSURE-CORRECTION

The accurate prediction of multiphase flow phenomena is associated with high computational costs. Among many different numerical approaches that have been employed to reduce computational resources is the so-called Viscous Potential Theory (VPT) pioneered by D.D. Joseph [Joseph (2006)]. The present method builds on this theory and extends it to include a generalized form of the viscous pressure correction for viscous potential flow, thereby providing more accurate results. In addition, an efficient and accurate numerical algorithm has been developed allowing for the analysis of complex flows with complex boundary conditions. The proposed method is based on a variant of the unsteady Bernoulli equation, which includes the full effect of viscosity through the VPT combined together with a boundary element formulation and a Runge-Kutta scheme to solve the governing equations. The method is currently implemented for two-dimensional (2D) flows and was previously shown to produce comparable results to those from general purpose multiphase simulation methods, such as volume of fluid or smoothed-particle-hydrodynamics, despite the underlying ir-rotationality assumption of the flow. Here, we use the new method to investigate the performance of a small-scale [microelectromechanical system (MEMS)-type], vibrating 2D slit-nozzle atomizer. Neglecting the edge effects of the discharging planar liquid film, its disintegration into cylindrical ligaments is fully resolved by the present method. To arrive at a droplet size prediction, breakdown of the 2D ligaments into droplets is simply modeled and estimated based on the Rayleigh-Plateau instability. The simulations provide preliminary insight into how frequency and amplitude of the imposed sinuous and/or dilation modulations affect nonlinear film rupture and droplet size distribution in the device.

Dense core response to forced acoustic fields in oxygen-hydrogen rocket flames

Oscillatory combustion representative of thermo-acoustic instability in liquid rockets is simulated by experiment and LES calculation to investigate the flame behavior in detail. In particular, we focus on how the velocity and pressure fluctuations affect the behavior of the dense oxygen jet, or ‘LOx core’. The test case investigated is a high pressure, multi-injector, oxygen-hydrogen combustor with a siren for acoustic excitation. First, the LES calculation is validated by the resonant frequencies and average flame topology. A precise frequency correction is conducted to compare experiment with LES. Then an unforced case, a pressure fluctuation case, and a velocity fluctuation case are investigated. LES can quantitatively reproduce the LOx core shortening and flattening that occurs under transverse velocity excitation as it is observed in the experiments. On the other hand, the core behavior under pressure excitation is almost equal to the unforced case, and little shortening of the core occurs. The LOx core flattening is explained by the pressure drop around an elliptical cylinder using the unsteady Bernoulli equation. Finally, it is shown that the shortening of the LOx core occurs because the flattening enhances combustion by mixing and increase of the flame surface area.