Mass Flux Conservation Research Articles

Simulation of solute transport using a coupling model based on finite volume method in fractured rocks

The discrete version of solute transport equation for porous matrix depicted with the continuum model and the discrete fractured-network model are derived for fractured rocks with the Finite Volume Method (FVM). The two models are coupled according to the continuity condition of hydraulic head and concentration and the conservation of flow flux and mass flux in the contact plane between porous matrix and fractures. Numerical results show that the simulated concentration of the coupling model based on the FVM agrees well with that from analytical solution, which demonstrates that the coupling model can effectively be applied to the simulation of solute transport in fractured rocks.

Numerical Magnetohydrodynamics

The ideal MagnetoHydroDynamic (MHD) equations accurately describe the macroscopic dynamics of a perfectly conducting plasma. Adopting a continuum, single fluid description in terms of the plasma density ρ, velocity v, thermal pressure p and magnetic field B, the ideal MHD system expresses conservation of mass, momentum, energy, and magnetic flux. This nonlinear, conservative system of 8 partial differential equations enriches the Euler equations governing the dynamics of a compressible gas with the dynamical influence - through the Lorentz force - and evolution - through the additional induction equation - of the magnetic field B. In multi-dimensional problems, the topological constraint expressed by the Maxwell equation ▿ · B = 0, represents an additional complication for numerical MHD. Basic concepts of shock-capturing high-resolution schemes for computational MHD are presented, with an emphasis on how they cope with the thight physical demands resulting from nonlinearity, compressibility, conservation, and solenoidality.

Shoaling waves: A discussion

Surface gravity waves are commonly observed to slow down and to stop at a beach without any noticeable reflection taking place. We assume that as a consequence the waves are continuously giving up their linear and angular momenta, which they carry with them, along with energy, as they propagate into gradually decreasing mean depths of water. It takes a force to cause a time rate of decrease in the linear momentum and a torque to produce a time rate of decrease in the angular momentum. Both a force and a torque operate on the shoaling waves, due to the presence of the sloping bottom, to cause the diminution of their linear and angular momenta. By Newton’s third law, action equals reaction, an equal but opposite force and torque are exerted on the bottom. No other mechanisms for transferring linear and angular momenta are included in the model. Since the force on the waves acts over a horizontal distance during shoaling, work is done on the waves and energy flux is not conserved. Bottom friction, wave interaction with a mean flow, scattering from small-scale bottom irregularities and set-up are neglected. Mass flux is conserved, which leads to a shoreward monotonic decrease in amplitude consistent with available swell data. The formula for the time-independent force on the bottom agrees qualitatively with observations in seven different ways: four for swell attenuation and three for sediment transport on beaches. Ardhuin (2006) argues against a mean force on the bottom that is not hydrostatic, mainly by using conservation of energy flux. He also applies the action balance equation to shoaling waves. Action is a difficult concept to grasp for motion in a continuum; it cannot be easily visualized, and it is not really necessary for solving the shoaling wave problem. We prefer angular momentum because it is clearly related to the observed orbital motion of the fluid particles in progressive surface waves. The physical significance of wave action for surface waves has been described recently by showing that in deep water action is equivalent to the magnitude of the wave’s orbital angular momentum (Kenyon and Sheres, 1996). Finally, Ardhuin requires that there be a significant exchange of linear momentum between shoaling waves and an unspecified mean flow, although the magnitude and direction of the exchange are not predicted. No mention is made of what happens to the orbital angular momentum during shoaling. Mass flux conservation is not stated.

Transient radial movement of a confined leaky aquifer due to variable well flow rates

The present paper investigates aquifer horizontal movement driven by hydraulic forces. New analytic solutions are found for the velocity and displacement fields describing aquifer radial transient movement caused by well discharge and recharge within a confined leaky aquifer. The development of the new solutions is based on linear momentum of bulk mass flux, mass conservation of bulk mass flux, the Darcy-Gersevanov law, and the analytic hydraulic drawdown. If the Hantush-Jacob, Carslaw-Jaeger, and Theis analytic drawdowns are applied respectively, a corresponding aquifer transient movement is found within the new solutions. Three sets of simplified solutions based on the Carslaw-Jaeger drawdown have been applied to three schemes of well flow rates that are power, periodic and exponential functions of time. For all three schemes, the dimensionless relations of aquifer displacement versus radial distance show the zones and locations where possible tensile stress may occur and cause potential earth fissuring. For the well flow rates varying as a power or periodic function, tensile stress may be generated when inertial forces change directions based on the dimensionless relations of aquifer velocity versus time. Analysis of effects that vertical leaky flow have on aquifer horizontal movement shows that the vertical water flow plays a significant role in aquifer radial movement, and that aquifer radial velocity is much more responsive to the vertical flow than is aquifer displacement.

Calculation of Entrainment in Density Jumps

The density jump in a two-layer channel flow of miscible fluids, in which one of the layers is infinitely deep and at rest, is analyzed using the momentum flux and mass flux conservation equations. The analysis yields simple equations relating the heights upstream and downstream of the jump with the upstream Froude number and the rate of entrainment into the moving layer, as well as a relation between the maximum possible entrainment and the upstream Froude number. The analysis also shows that when the flow down-stream of the jump is controlled by an obstruction or a contraction, the entrainment rate into the jump depends solely on the dimensionless obstruction height contraction ratio and the upstream Froude number.

Theoretical analysis of surface waves shoaling and breaking on a sloping bottom. Part 2: Nonlinear waves

This is the second of two papers on the theoretical analysis of surface wave propagating on sloping bottoms. In the first paper, the linear solution for velocity potential was derived as a function of bottom slope α to the third order. In this paper, Part 2, we use the two-parameter perturbation method to develop a new mathematical derivation to describe nonlinear periodic gravity surface waves propagating over sloping bottoms. In the Eulerian coordinate system, the velocity potential is obtained as a function of the nonlinear ordering parameter ε and the bottom slope α perturbed to the fourth order ( ε m α n , m + n = 4). A time-decoupled solution is derived and the condition of the conservation of mass flux is examined in detail for the first time. Then, the solution is used to estimate the mean return current for waves progressing over the sloping bottom. The analytical solution for wave asymmetry and set-down valid up to the breaker line for an arbitrary bottom slope can also be obtained. Besides, the direct influence of nonlinearity and bottom slope upon the wave number and a new second-order dispersion relation is also derived. This nonlinear analytical solution is verified by reducing to the classical Stokes fourth-order solution of progressive waves in both the limit of deep water and of constant depth. Furthermore, by comparing the theoretical values of wave asymmetry and set-down with the experimental data, it is found that theoretical results of the present solution are in good agreement with the experimental data.

Nonlinear Magnetohydrodynamics: Numerical Concepts

The ideal MagnetoHydroDynamic (MHD) equations accurately describe the macroscopic dynamics of a perfectly conducting plasma. Adopting a continuum, single fluid description in terms of the plasma density ρ, velocity v, thermal pressure p and magnetic field B, the ideal MHD system expresses conservation of mass, momentum, energy, and magnetic flux. This nonlinear, conservative system of 8 partial differential equations enriches the Euler equations governing the dynamics of a compressible gas with the dynamical influence - through the Lorentz force - and evolution - through the additional induction equation - of the magnetic field B. In multi-dimensional problems, the topological constraint expressed by the Maxwell equation ∇ B = 0, represents an additional complication for numerical MHD. Basic concepts of shock-capturing high-resolution schemes for computational MHD are presented, with an emphasis on how they cope with the thight physical demands resulting from nonlinearity, compressibility, conservation, and solenoidality.

Sensitivity of solar off-limb line profiles to electron density stratification and the velocity distribution anisotropy

The effect of the electron density stratification on the intensity profiles of the H I Ly-$\alpha$ line and the O VI and Mg X doublets formed in solar coronal holes is investigated. We employ an analytical 2-D model of the large scale coronal magnetic field that provides a good representation of the corona at the minimum of solar activity. We use the mass-flux conservation equation to determine the outflow speed of the solar wind at any location in the solar corona and take into account the integration along the line of sight (LOS). The main assumption we make is that no anisotropy in the kinetic temperature of the coronal species is considered. We find that at distances greater than 1 Rsun from the solar surface the widths of the emitted lines of O VI and Mg X are sensitive to the details of the adopted electron density stratification. However, Ly-$\alpha$, which is a pure radiative line, is hardly affected. The calculated total intensities of Ly-$\alpha$ and the O VI doublet depend to a lesser degree on the density stratification and are comparable to the observed ones for most of the considered density models. The widths of the observed profiles of Ly-$\alpha$ and Mg X are well reproduced by most of the considered electron density stratifications, while for the O VI doublet only few stratifications give satisfying results. The densities deduced from SOHO data result in O VI profiles whose widths and intensity ratio are relatively close to the values observed by UVCS although only isotropic velocity distributions are employed. These density profiles also reproduce the other considered observables with good accuracy. Thus the need for a strong anisotropy of the velocity distribution (i.e. a temperature anisotropy) is not so clear cut as previous investigations of UVCS data suggested. ...

Spectroscopic measurement of the plasma electron density and outflow velocity in a polar coronal hole

A new spectroscopic method, aimed to derive the plasma electron density and outflow velocity in expanding solar coronal regions, is discussed in this paper. The method is based on the analysis of a pair of coronal lines emitted via collisional and radiative excitation by the same ion, such as the O VI 1032, 1037 A doublet. The merit of this technique consists in allowing us to derive at the same time electron density and outflow velocity of the coronal plasma from nearby lines detectable with the same instrument, provided the constraint on mass flux conservation along the flow tube connecting solar corona and heliosphere is taken into account. The results obtained from the analysis of the OVI emission imply that the physical conditions of a polar coronal hole plasma, observed during minimum activity, are the following. The electron density decreases from $4\times 10^5$ cm -3 at 1.7 $R_{\odot}$ to $2{-}4\times 10^4$ cm -3 at 3.1 $R_{\odot}$, whereas the outflow velocity of the oxygen ions is monotonically increasing to reach 350–500 km s -1 at 3.1 $R_{\odot}$, depending on the assumptions on the degree of anisotropy of the velocity distribution of the ions. These results of the velocity of expansion of the fast wind confirm those obtained with Doppler dimming techniques when assuming the lowest observed density values for the coronal hole plasma. This implies that, for a rarified corona, the outflow velocity of the fast solar wind in polar holes can be traced by the motion of the O VI ions at least up to 2.4 $R_{\odot}$. The analysis also shows that the degree of anisotropy of the oxygen ions, due to the acceleration of the ions across the magnetic field in a coronal hole, exhibits a steep increase and that the geometry of the flow tube diverges very rapidly low down in the inner corona/transition region. The observations of the extended corona analysed in this paper are obtained with the Ultraviolet Coronagraph Spectrometer of the SOHO space mission.

Nonlinear Magnetohydrodynamics: Numerical Concepts

The ideal MagnetoHydroDynamic (MHD) equations accurately describe the macroscopic dynamics of a perfectly conducting plasma. Adopting a continuum, single fluid description in terms of the plasma density σ, velocity v, thermal pressure p and magnetic field B, the ideal MHD system expresses conservation of mass, momentum, energy, and magnetic flux. This nonlinear, conservative system of 8 partial differential equations enriches the Euler equations governing the dynamics of a compressible gas with the dynamical influence - through the Lorentz force - and evolution - through the additional induction equation - of the magnetic field B. In multi-dimensional problems, the topological constraint expressed by the Maxwell equation [nabla]·B = 0, represents an additional complication for numerical MHD. Basic concepts of shock-capturing high-resolution schemes for computational MHD are presented, with an emphasis on how they cope with the thight physical demands resulting from nonlinearity, compressibility, conservation, and solenoidality.

Wave-current loading on a vertical slender cylinder

The present paper illustrates a study on the loading of an oblique wavecurrent field on a slender cylinder in a 3D flow frame. We use the three dimensional expressions describing the characteristics of the combined wave-current field in terms of mass, momentum and energy flux conservation equations and the parameters before the interaction of the wave-free uniform current and current-free wave. We then use the colinear wave-current nonlinear interaction model and its extension in 3D to calculate the loads imparted by the fluid on a bottom mounted slender vertical cylinder. We compare the obtained results to those obtained using three other models based on a superposition principle. Morison et a1 equation is deployed for the load computations in all cases. Comparisons of the obtained results are made.

Two‐dimensional Structure of a Polar Coronal Hole at Solar Minimum: New Semiempirical Methodology for Deriving Plasma Parameters

We develop a new technique to determine the plasma parameters in a polar coronal hole. This method makes use of the line intensities of the H I Lyα λ1215.6 line and of the O VI λλ1031.9, 1037.6 doublet, measured with the Ultraviolet Coronagraph Spectrometer (UVCS) on board the ESA-NASA solar spacecraft Solar and Heliospheric Observatory (SOHO) during 1996 August. The observed intensities are self-consistently reproduced with a two-dimensional semiempirical coronal hole model, for heliocentric distances between 1.4 and 2.6 R☉ and latitudes between 90° (north pole) and 40°. Electron densities are derived by separating the O VI doublet collisional components from those due to resonant scattering. The calculated electron density radial profiles are consistent with typical polar coronal hole data and show only a moderate increase with latitude decreasing, in regions close to the equatorial streamer. The outflow speeds of protons and O VI ions are determined by means of the Doppler dimming technique. In the Doppler dimming analysis we use kinetic temperatures Tk derived from UVCS observations of the line profiles, whenever available, or we keep Tk as a free parameter if not provided by data. Mass flux conservation along the magnetic field lines is studied adopting a simple analytical model for the geometry of the magnetic flux tubes. Our model shows that protons and O VI ions accelerate outward, but their outflow speed turns out to decrease slowly as latitude decreases. The O VI speed, initially comparable to the speed of protons, exceeds the proton speed beyond ~1.7 R☉. Anisotropic O VI kinetic temperatures, T∥ and T⊥, turn out to be necessary to ensure the consistency of the model parameters with mass flux conservation, while the H kinetic temperature distribution is kept isotropic. Results from our model are compared with those from other two-dimensional models recently developed.

Lyman α intensities in a polar coronal hole from a 2D model

We simulate the coronal H I Lyman α intensity for heliocentric distances between 1.5 and 2.5 R ⊙, and latitudes between 90° (North pole) and 30°, making use of a 2D semiempirical coronal hole model. Observations are made with the UltraViolet Coronagraph Spectrometer (UVCS) on board the ESA-NASA solar satellite SOHO (SOlar and Heliospheric Observatory). Model electron densities are derived from the collisional part of the O VI λ1037.6 Å line and the proton outflow speed is calculated from mass flux conservation along the magnetic field lines. The expansion factor and the direction of the magnetic flux tubes have been derived by adopting a simple analytical magnetic field configuration. The intensities of the Ly α line predicted by the model are compared with the observed intensities.

Shock Relations for a Detonation Regime Developing under the Action of Frequency-Modulated Laser Radiation

The oscillations of the gas dynamic parameters of plasma generated under the action of a powerful UHF-modulated laser radiation are investigated theoretically when the characteristic laser radiation intensity modulation time is significantly less than the characteristic time of variation of the gas dynamic parameters of the plasma. It is shown that in this regime the shock relations, written in the form of mass, momentum, and energy flux conservation laws, require the time derivatives of the gas dynamic quantities calculated from the corresponding differential equations to be taken into account.

A phenomenological model of coronal mass ejection

A model of coronal mass ejection (CME) is proposed, that is based on the break of equilibrium of twisted magnetic tubes (loops) emerging from the photosphere into the corona. Under the assumption of mass and axial flux conservation in the tube, a phenomenological model describes the loss of equilibrium as a result of a pressure increase in the tube due to its warming and additional magnetic heating when the twist increases in the expanding tube. A fast expansion of the tube to a new equilibrium radius will reduce the density inside the tube and, due to an additional buoyancy force, the tube will be ejected into the corona. The conditions are determined under which a pressure decrease inside a tube, due to the plasma outflow through its ends removes the onset of the eruptive instability. The upward motion of mass and the onset of the eruptive instability are related in the following way: the loops that lose a large amount of mass are not susceptible to an eruptive instability but the loops that lose only a small amount of mass experience an eruptive instability that leads to an ejection of the loop.

Gas flow and dust acceleration in a cometary Knudsen layer

An analytical model of the innermost gas–dust coma region is proposed. The kinetic Knudsen layer adjacent to the surface of the cometary nucleus, where the initially non-equilibrium velocity distribution function of gas molecules relaxes to Maxwell equilibrium distribution function and, as a result, the macro-characteristics of gas and dust flows vary several-fold, is considered. The gas phase model is based on the equations for mass, momentum and energy flux conservation, and is a natural development of the Anisimov, 1968 and Cercignani, 1981 approaches. The analytical relations between the characteristics of the gas flow on the boundaries of the non-equilibrium layer and the characteristics of the returning gas flow adsorbed by the surface are determined. These values form a consistent basis both for hydrodynamic models of the inner coma and for jet force models. Three particular models are presented: (1) sublimation of a polyatomic one-component gas; (2) sublimation of a two-component polyatomic gas mixture, in both cases from a plane surface; and (3) sublimation of water ice through a porous dust mantle. We conclude that the characteristics of the gas flow emerging from the Knudsen layer over a porous dust mantle is not very sensitive to the structure of the mantle.We also treat the expansion of dust into the coma, concentrating on the interaction between a non-equilibrium gas flow and a test particle. The dynamics of a grain of idealized shape is explored by using several simplifying assumptions for the variation of the drag force. The velocity of a particle at the exterior boundary of the Knudsen layer is thus estimated. Examining various model behaviours of the drag force inside the Knudsen layer, we show that the dust velocity is not sensitive to these variations.

On Alfvén waves in a radial flow and magnetic field

This paper considers Alfvén waves in a radially stratified medium where all background quantities, namely mass density, magnetic field strength and mean flow velocity, depend only on the distance from the centre, the latter two being assumed to lie in the radial direction. It is shown that the radial dependence of Alfvén waves is the same for two cases: (i) when the velocity and magnetic field perturbations are along parallels, in the one-dimensional case of only radial and time dependence; (ii) in the three-dimensional case with dependence on all three spherical coordinates and time, for velocity and magnetic field perturbations with components along parallels and meridians, represented by the radial components of the vorticity and electric current respectively. Elimination between these equations leads to the convected Alfvén-wave equation in the case of uniform flow, and an equation with an additional term in the case of non-uniform flow with mean flow velocity a linear function of distance. The latter case, namely that of non-uniform flow with flow velocity increasing linearly with distance, is analysed in detail; conservation of mass flux requires the mass density to decay as the inverse cube of the distance. The Alfvén-wave equation has a critical layer where the flow velocity equals the Alfvén speed, leading to three sets of two solutions, namely below, above and across the critical layer. The latter is used to specify the wave behaviour in the vicinity of the critical layer, where local partial transmission occurs. The problem has two dimensionless parameters: the frequency and the initial Alfvén number. It is shown, by plotting the wave fields relative to the critical layer, that these two dimensionless parameters appear in a single combination. This simplifies the plotting of the wave fields for several combinations of physical conditions. It is shown in the Appendix that the formulation of the equations of MHD in the original Elsässer (1956) form, often used in the recent literature, does not apply if the background mass density is non-uniform on the scale of a wavelength. The present theory, based on exact solutions of the Alfvén-wave equation for a inhomogeneous moving medium, is unrestricted as to the relative magnitude of the local wavelength and scale of change of properties of the background medium. The present theory shows that the rate-of-decay of wave amplitude is strongly dependent on wave frequency beyond the critical layer, i.e. the process of change with distance of the spectrum of Alfvén waves in the solar wind starts at the critical layer.

Steady-state Mantle-Melt Interactions in One Dimension: I. Equilibrium Transport and Melt Focusing

Mantle–melt interaction during melt transport is explored in one dimension and steady state. We reconsider the equivalence between one-dimensional steady equilibrium transport and batch melting. In the absence of diffusion and radioactivity, conservation of mass flux requires that the major and trace element compositions of melt and solid at each point are the same as is generated by batch melting the source composition at the same pressure and temperature. Energy conservation requires that temperature and extent of melting are independent of melt migration except for irreversible source terms related to viscous compaction and gravitational energy release. The equivalence of phase compositions at each pressure between steady-state equilibrium transport and batch melting simplifies melt transport calculations. We examine the effects of increasing the melt flux to simulate melt focusing by channeling or by two-dimensional flow with converging melt streamlines. Melt focusing modifies the mineralogy of both residual matrix and erupted melt. We use MELTS calculations to model the formation of dunite by this mechanism and quantify the melt flux required to exhaust orthopyroxene from the residue as a function of pressure. The model dunites are found to be similar to natural dunites observed in the mantle section of ophiolite sequences.

ROUND BUOYANT JETS INTO NONUNIFORM FLOWING ENVIRONMENT

This paper presents a mathematical model for submerged turbulent round buoyant jets into a nonuniformly flowing stratified environment. The model derived herein is based on the conservation of mass flux, momentum flux and densityd eficiency.A new entrainment function which can be decided by the model and a constanto f drag coefficient are also presented here. The model is validated by comparing prediction results and large number of experimental data publishedb y other investigators. The mathematica mlodel is then used to analyze behavior of some buoyant jets into a nonuniform current. The present model should be capable of predicting not only the three-dimensional buoyant jets into nonuniformly flowing ambient fluids, but also nonbuoyant jets and simple plumes, something an existing model can not cover.

UVCS Observations and Modeling of Streamers

We present results derived from the analysis of an equatorial streamer structure as observed by the UVCS instrument aboard SOHO. From observations of the H I Lyα and Lyβ lines we infer the density and temperature of the plasma. We develop a preliminary axisymmetric, magnetostatic model of the corona which includes the effects of gas pressure gradients on the magnetic structure. We infer a coronal plasma β > 1 in the closed field regions and near the cusp of the streamer. We add to the model a parallel velocity field assuming mass flux conservation along magnetic flux tubes. We then compute the Lyα emissivity and the line-of-sight integrals to obtain images of Lyα intensity, taking into account projection effects and Doppler dimming. The images we obtain from this preliminary model are in good general agreement with the UVCS observations, both qualitatively and quantitatively.