Linear Pressure Gradient Research Articles

A Verification Suite of Test Cases for the Barotropic Solver of Ocean Models

AbstractThe development of any atmosphere or ocean model warrants a suite of test cases (TCs) to verify its spatial and temporal discretizations, order of accuracy, stability, reproducibility, portability, scalability, etc. In this paper, we present a suite of shallow water TCs designed to verify the barotropic solver of atmosphere and ocean models. These include the non‐dispersive coastal Kelvin wave; the dispersive inertia‐gravity wave; the dispersive planetary and topographic Rossby waves; the barotropic tide; and a non‐linear manufactured solution. These TCs check the implementation of the linear pressure gradient term; the linear constant or variable‐coefficient Coriolis and bathymetry terms; and the non‐linear advection terms. Simulation results are presented for a variety of time‐stepping methods as well as two spatial discretizations: a mimetic finite volume method based on the TRiSK scheme, and a high‐order discontinuous Galerkin spectral element method. The experimental procedure for conducting these numerical experiments is detailed. It underscores several key considerations that vary depending on the chosen spatial discretization method. Finally, convergence studies of every TC are conducted with refinement in both space and time, only in space, and only in time. The convergence slopes match the expected theoretical predictions.

Varying-G gravity: physical properties, asymptotic regimes, and Green’s functions, an event horizon, the vacuum energy density, and the external pressure that modifies Jeans stability

ABSTRACT We study varying-G gravity and we add the necessary proofs (general force law, asymptotic forms, and Green’s functions, vacuum and external pressures, linearization of perturbations leading to a new Jeans stability criterion, and a physical origin) to elevate this novel idea to the status of a classical theory. The theory we lay out is not merely a correction to Newtonian gravity, it is a brand-new theory of gravity that encompasses the Newtonian framework and weak-field Weyl gravity in the limit of high accelerations, as well as Modified Newtonian Dynamics (MOND) in the opposite limit. In varying-G gravity, the source of the potential of a spherical mass distribution M(x) is σ(dG/dx) + (G/x2)(dM/dx), where x is the dimensionless radial coordinate and σ(x) = M(x)/x2 is the surface density away from the center x = 0. We calculate the potential $\Phi (x)=\int {G(x)\, \sigma (x)\, dx}$ from Poisson’s equation and the radial acceleration $a(x) = G(x)\, \sigma (x)$. Furthermore, a non-linear scaling transformation of the radial coordinate $x\in (0, \infty)\longmapsto \xi \in (0, 1)$ with scale factor ξ/x ∝ 1/G produces a finite space, in which the spherical surface ξ = 1 is an event horizon. In this classical context, it is the coupling of σ(x) to the gradient dG/dx in the above source that modifies the dynamics at all astrophysical scales, including empty space (where dG/dx ≠ 0). In vacuum, the source σ(dG/dx) supports an energy density distribution that supplies a repelling pressure gradient outside of discrete isolated massive systems. Surprisingly, the same source becomes attractive in linearized perturbations, and its linear pressure gradient opposes the kinetic terms in the Jeans stability criterion.

The Numerical Investigation on Bubble Interaction Dynamics in Hydrodynamic Cavitation

Bubble-bubble interactions are of great importance for bubble dynamics. A mathematical model describing the dynamics of a cluster in hydrodynamic cavitation is presented. The interaction strength (i.e. the number density of bubbles) is introduced into Keller-Misis equation. Using this model, numerical investigations of bubble dynamics (i.e. radial motion and internal energy) of single bubble in a cluster have been made due to linear pressure gradient. With the increase of interaction strength, the times of bubble reaching the maximum and minimum radii are delayed. The more of bubbles are in a cluster, the more significant of the delay effect is. The maximum internal energy inside the bubble is closely related to interaction strength (i.e. positive correlation). Furthermore, the initial bubble radius and final recovered pressure of the orifice on bubble dynamics are quantitatively discussed. Based on numerical results, some references are put forward for optimize and manipulate of hydrodynamic cavitation reactor.

Unsteady Flow of a Dusty Gas Through a Horizontal Pipe with Time Varying Pressure Gradient

In the present paper, the unsteady flow of a dusty gas through a pipe under the effect of the linear and exponential pressure gradient is established. Firstly the equation of motion of the fluid and dust particles is considered and then converts it into the Bessel equation by introducing two different parameters. Using the solution of the Bessel equation, the velocity of the gas and dust particles is obtained and shown graphical representation.It is found that the velocities increases as one move towards the axis of the pipe. Under the linear pressure gradient, the velocity of the gas is greater than the velocity of dust particles. As time progresses velocity increases and under exponential pressure gradient the velocity of the dust particles is greater than the velocity of the gas. As time progresses velocity decreases.

Unsteady Flow of a Dusty Gas Through a Horizontal Pipe with Time Varying Pressure Gradient

In the present paper, the unsteady flow of a dusty gas through a pipe under the effect of the linear and exponential pressure gradient is established. Firstly the equation of motion of the fluid and dust particles is considered and then converts it into the Bessel equation by introducing two different parameters. Using the solution of the Bessel equation, the velocity of the gas and dust particles is obtained and shown graphical representation.It is found that the velocities increases as one move towards the axis of the pipe. Under the linear pressure gradient, the velocity of the gas is greater than the velocity of dust particles. As time progresses velocity increases and under exponential pressure gradient the velocity of the dust particles is greater than the velocity of the gas. As time progresses velocity decreases.

Unsteady Flow of a Dusty Gas Through a Horizontal Pipe with Time Varying Pressure Gradient

In the present paper, the unsteady flow of a dusty gas through a pipe under the effect of the linear and exponential pressure gradient is established. Firstly the equation of motion of the fluid and dust particles is considered and then converts it into the Bessel equation by introducing two different parameters. Using the solution of the Bessel equation, the velocity of the gas and dust particles is obtained and shown graphical representation.It is found that the velocities increases as one move towards the axis of the pipe. Under the linear pressure gradient, the velocity of the gas is greater than the velocity of dust particles. As time progresses velocity increases and under exponential pressure gradient the velocity of the dust particles is greater than the velocity of the gas. As time progresses velocity decreases.

Impact of transversal traveling surface waves in a non-zero pressure gradient turbulent boundary layer flow

The impact of non-zero pressure gradient flow in a turbulent boundary layer flow over a surface undergoing spanwise transversal traveling waves is investigated via large-eddy simulations. While it is known that in zero-pressure gradient flow spanwise surface waves can lead to drag reduction, this question still remains open for non-zero pressure gradient flows. In the present analysis, the effect of a linear pressure gradient is investigated and compared to the zero-pressure gradient flow at a momentum thickness based Reynolds number Reθ=2000 for a constant surface wave, i.e., the wave length is λ+=500, the amplitude A+=50, and the wave speed is c+=6.25. The results show a drag reduction of about 10% for the zero-pressure gradient flow, 6% for the adverse-pressure gradient, and a drag increase of 4% for the favorable-pressure gradient flow. The analysis of the velocity profiles shows a reduced gradient at the trough region for all actuated setups. At the crest, an increased gradient is obtained. Furthermore, the viscous sublayer is extended. The streamwise turbulent intensity is reduced for all configurations compared to the non-actuated reference case at the crest. At the trough, the shift off the wall is only present for the zero-pressure gradient flow and the adverse pressure gradient flow. The hypothesis based on numerous zero-pressure gradient flow investigations of a reduced wall-normal vorticity component at the crest and trough indicating drag reduction is corroborated. That is, for the adverse-pressure gradient flow the wall-normal component distribution is lowered and for the favorable pressure gradient flow, which possesses a drag increase, the distribution at the trough is similar to that of the reference non-actuated case.

Vibration and rotation during biaxial pressure algometry is related with decreased and increased pain sensations.

During palpation, the pressure intensity and direction include minor deviations suggesting that standardized variations of the pressure intensity during pressure algometry may optimize the stimulus efficacy. This study examined the perceived pain outcome and reliability of a biaxial (bidirectional) algometer exerting rotational and vibratory stimulation on top of the basic pressure. In 24 healthy subjects, pressure pain thresholds (PPTs) were recorded with a linear pressure gradient (30 kPa/s) applied by a 1-cm(2) probe bilaterally on the tibialis anterior muscle via biaxial and handheld algometers. During constant pressure stimulation (5 seconds, 75%, 100%, 125% PPT), rotational (45°, 90°, and 180°), linear vibrational (15, 25, and 50 Hz), and radial vibrational stimulations (5, 15, and 25 Hz) were applied randomly via regular and fanning rounded probes (1 cm(2)). Subjects rated perceived pain on a 10-cm visual analogue scale on two occasions separated by 1 week period. Repeated measures analysis of variance revealed enhanced effect of rotation angle (P < 0.001), probe (P < 0.001), and radial vibration frequency (P < 0.02), and suppressing effect of axial vibration frequency (P < 0.03) on pain perception, relative to basic pressure alone. PPT reliability of biaxial and handheld algometers showed averaged intraclass correlation coefficient of 0.94 and 0.945, and coefficient of variations of 15.4 and 13.5%, respectively. PPT assessment and multidirectional stimulations can be exerted reliably via biaxial algometer. Linear vibrational stimulation effect on pressure pain perception verified the inhibitory interaction between simultaneous pressure stimulation of low-threshold mechanoreceptors and nociceptors, while radial vibration and rotational stimulation showed facilitatory effects.

Heat transfer coefficient calculated using a linear pressure gradient assumption and measurement for flow boiling in microchannels

Abstract To obtain local flow boiling heat transfer coefficients, some studies used a linear pressure gradient method. In this study, this method is compared with a pressure gradient measurement method. The measurement was carried out for flow boiling of de-ionized water in single horizontal microchannels. Three single microchannels with widths of 0.5 mm, 1 mm and 1.71 mm and a depth of 0.39 mm, giving hydraulic diameters of 0.438 mm, 0.561 mm and 0.635 mm, were tested. The channel length was the same, 62 mm. The nominal mass flux used was 800 kg/m2 s and heat fluxes ranging from 222 to 685 kW/m2 were applied. An inlet fluid temperature of 98 °C and an inlet pressure of 125 kPa (abs) corresponding to an inlet subcooling of 7 K, were maintained at the channel entrance. The trends of heat transfer coefficient decrease with quality at a fixed heat flux and mass flux along the channel. However, the heat transfer mechanism was dominated by nucleate boiling for the 0.561 mm and 0.635 mm channels indicated by the increased local heat transfer coefficient with heat flux, while for the 0.438 mm channel, it was dominated by convective boiling because there was no effect of heat flux on the local heat transfer coefficient. Local heat transfer coefficients determined using the two methods were found to be significantly different, especially at high heat fluxes. When the Φ is less than 9 W/m2 Pa for the 0.438 mm channel, less than 11 W/m2 Pa for the 0.561 mm channel and less than 16 W/m2 Pa for the 0.635 mm channel, the linear pressure distribution may be used with caution.

Minor Losses During Air Flow into Granular Porous Media

Pressure gradients during uniform fluid flow in porous media are traditionally assumed to be linear. Thus, pressure loss across a sample of porous medium is assumed directly proportional to the thickness of the sample. In this study, measurements of pressure gradients inside coarse granular (2–18 mm particle size) porous media during steady gas flow were carried out. The results showed that pressure variation with distance in the porous media was nonlinear near the inlet (where pressure gradients were higher) but became linear at greater distances (with a lower gradient). This indicates that the pressure loss in porous media consists of two components: (1) a linear pressure gradient and (2) an initial pressure loss near the inlet. This initial pressure loss is also known from hydraulics in tubes as a minor loss and is associated with abrupt changes in the flow field such as narrowings and bends. The results further indicated that the minor loss depends on the particle size and particle size distribution in a manner similar to that of the linear pressure gradient. There is, thus, a close relation between these two components. In porous media, the minor loss is not instantaneous at the inlet point but happens over some distance starting upstream from the inlet and ending some distance downstream.

Examination of Behavior of Fresh Concrete Under Pressure

Transporting fresh concrete constitutes a significant part of the production process. Transferring ready-mixed concrete on-site is done using concrete pumps. Recent developments in concrete technology, and in mineral and chemical additives, have resulted in new developments in pumping techniques and the use of different concrete mixtures and equipment. These developments required further knowledge of the behavior of fresh concrete under pressure. Two criteria were determined for the pumpability of concrete: the power required to move the concrete or of the repulsive force; and the cohesion of the fresh concrete. It would be insufficient to relate pumpability to these two criteria; the values of segregation pressure, diffusion ability, water retention capacity, and side friction of the mixture are significant parameters in ensuring that concrete is pumped freely along the pipe. To solve the pumpability problem, friction stresses should be determined as a function of the linear pressure gradient, the pressure leading to segregation of the fresh concrete should be determined, and tests for the bleeding of concrete under pressure should be examined. The scope of the research is the examination of the behavior of fresh concrete under pressure. To determine the segregation pressures, a test apparatus was designed for the bleeding of concrete under pressure. The main purpose of the study is to determine whether the concrete can be pumped easily and whether it will lose its cohesion during the pumping, based on tests of concrete workability and bleeding of concrete under pressure.

Modeling of hydrodynamic cavitation reactors: a unified approach

An attempt has been made to present a unified theoretical model for the cavitating flow in a hydrodynamic cavitation reactor using the nonlinear continuum mixture model for two-phase flow as the basis. This model has been used to describe the radial motion of bubble in the cavitating flow in two geometries in hydrodynamic cavitation reactors, viz., a venturi tube and an orifice plate. Simulations of the bubble dynamics in a venturi flow demonstrate the stable oscillatory radial bubble motion due to a linear pressure gradient. Due to an additional pressure gradient due to turbulent velocity fluctuations the radial bubble motion in case of an orifice flow is a combination of both stable and oscillatory type. The results of numerical simulations have been explained on the basis of analogy between hydrodynamic cavitation and acoustic cavitation.

Microspheres Based Detoxification System: In Vitro Study and Mathematical Estimation of Filter Performance

Because of the closed plasma (secondary) circuit in the Microspheres based Detoxification System (MDS), a convective blood purification system, the same amount of filtrated plasma is backfiltrated into the blood circuit. Therefore, there is no direct way to determine the ultrafiltrate production rate, which is an important factor of efficiency. The only possible way to estimate the filtration properties of the filter is to consider pressure values. In this study the pressure distribution in the filter was investigated in vitro. To explain the results and to calculate inaccessible parameters, a mathematical model was established which also considered the asymmetric behaviour of the filter membrane. The result was a linear pressure gradient, agreement with the measurements was reasonably good (calculated primary pressure loss differs <13% from measured value when using mean measured filter resistance as model parameter). Linear pressure distribution offers the possibility of easily calculating the filtration length, a parameter which can be used to estimate the filter condition. The comparison between calculated filtration and backfiltration rates offers an instrument of control for these values.

A Model for Chemical Vapor Copolymerization of p-Xylylenes with Vinylic Comonomers: Order of Initiation and Reactivity Ratios

A model for chemical vapor homo- and copolymerization is presented in two dimensions, allowing determination of order of initiation in homopolymers and reactivity ratios in copolymers. The model assumes a linear pressure gradient along the flow direction and allows calculation of the order of initiation if thickness vs position are known, and r 1 and r 2 if the gas flow rates, final film thickness, and film composition vs distance are known. Experiments indicate the number, A, of p-xylylene molecules required to initiate polymerization : for poly(p-xylylene) and poly(p-chloroxylylene), Λ = 3 ; for poly(p-dichloroxylylene), Λ = 4. Tentative values for the reactivity ratios of poly(p-chloroxylylene) with perfluorooctyl methacrylate have been determined at different temperatures. At 20 °C, r 1 = 13.04 and r 2 ≃ 0 ; r 1 increases with increasing temperature, while r 2 remains vanishingly small.

Water Waves on Crown Breakwaters

The linear theory for waves impinging obliquely on a vertical, porous breakwater with an impermeable crown is examined. The wave‐induced flow within the breakwater is modeled by the 1972 theory of Sollitt and Cross. Using eigen‐function expansions for the velocity potentials seaward of the breakwater, beneath the crown, and behind the structure, the reflection and transmission coefficients are calculated to determine the sheltering provided by the breakwater. The results agree well with other methods, including a plane wave analysis, obtained by neglecting the evanescent wave modes. The distribution of the wave‐induced pressure along the base of the crown is also calculated, showing that the linear pressure gradient assumption usually applied for uplift forces on the breakwater crown is inaccurate.

On the mechanical interaction between a fluid-filled fracture and the earth's surface

The mechanical interaction between a fluid-filled fracture (e.g., hydraulic fracture joint, or igneous dike) and the earth's surface is analyzed using a two-dimensional elastic solution for a slit of arbitrary inclination buried beneath a horizontal free surface and subjected to an arbitrary pressure distribution. The solution is obtained by iteratively superimposing two fundamental sets of analytical solutions. For uniform internal pressure the slit behaves essentially as if it were in an infinite region if the depth-to-center is three times greater than the half-length. For shallower slits interaction with the free surface is pronounced: stresses and displacements near the slit differ by more than 10% from values for the deeply buried slit. The following changes are noted as the depth-to-center decreases: 1. (1) the mode I stress intensity factor increases for both ends of the slit, but more rapidly at the upper end; 2. (2) the mode II stress-intensity factor is significantly different from zero (except for vertical slits) suggesting propagation out of the original plane of the slit; 3. (3) displacements of the slit wall are asymmetric such that the slit gaps open more widely near the upper end. Similar changes are noted if fluid density creates a linear pressure gradient that is smaller than the lithostatic gradient. Under such conditions natural fractures should propagate preferentially upward toward the earth's surface requiring less pressure as they grow in length. If deformation near the surface is of interest, the model should account explicitly for the free surface. Stresses and displacements at the free surface are not approximated very well by values calculated along a line in an infinite region, even when the slit is far from the line. As depth-to-center of a shallow pressurized slit decreases, the following changes are noted: 1. (1) displacements of the free surface increase to the same order of magnitude as the displacements of the slit walls, 2. (2) tensile stresses of magnitude greater than the pressure in the slit are concentrated along the free surface. The relative surface displacements over a shallow vertical slit are downward over the slit and upward to both sides of this area. The tensile stress acting parallel to the free surface over a shallow vertical slit is concentrated in two maxima adjacent to a point of very low stress immediately over the slit. The solution is used to estimate the length-to-depth ratio at which igneous sills have gained sufficient leverage on overlying strata to bend these strata upward and form a laccolith. The pronounced mode II stress intensity associated with shallow horizontal slits explains the tendency for some sills to climb to higher stratigraphie horizons as they grow in length. The bimodal tensile stress concentration over shallow vertical slits correlates qualitatively with the distribution of cracks and normal faults which flank fissure eruptions on volcanoes. The solution may be used to analyze surface displacements and tilts over massive hydraulic fractures in oil fields and to understand the behavior of hydraulic fractures in granite quarries.

Behavior of Small Gas Bubbles in Accelerated Liquid

The time dependent motion of small gas bubbles in liquids subjected to a linear pressure gradient has been studied. An expression for the bubble slip, (u\Dg\N - u\Dl\N)/u\Dl\N was derived as a function of the liquid and gas densities, initial velocities, and the liquid velocity gradient. For air in water, the bubble slip asymptotically approaches zero for uniform flow, independent of initial bubble and liquid velocities, and approaches the value of 0.9967 for highly accelerating flows.