Showing papers on "Streamlines, streaklines, and pathlines published in 1987"
TL;DR: In this paper, a general set of 2-D equations for the conservation of mass and momentum of a two-phase system of melt in a deformable matrix is used to derive analytic solutions for the corner flow of a constant porosity melt-saturated porous medium.
440 citations
TL;DR: In this article, the authors consider the evolution of an isolated elliptical vortex in a weakly dissipative fluid and derive a simple geometrical formula relating the rate of change of the aspect ratio of a particular vorticity contour to its orientation relative to the streamlines.
Abstract: We consider the evolution of an isolated elliptical vortex in a weakly dissipative fluid. It is shown computationally that a spatially smooth vortex relaxes inviscidly towards axisymmetry on a circulation timescale as the result of filament generation. Heuristically, we derive a simple geometrical formula relating the rate of change of the aspect ratio of a particular vorticity contour to its orientation relative to the streamlines (where the orientation is defined through second-order moments). Computational evidence obtained with diagnostic algorithms validates the formula. By considering streamlines in a corotating frame and applying the new formula, we obtain a detailed kinematic understanding of the vortex's decay to its final state through a primary and a secondary breaking. The circulation transported into the filaments although a small fraction of the total, breaks the symmetry and is the chief cause of axisymmetrization.
300 citations
TL;DR: In this article, the theory describing 3D exact solutions of the Navier-Stokes equations is applied to the problem of stability of 2D viscous flow with elliptical streamlines.
Abstract: The recent theory describing 3‐D exact solutions of the Navier–Stokes equations is applied to the problem of stability of 2‐D viscous flow with elliptical streamlines. An intrinsically inviscid instability mechanism persists in all such flows provided the length scale of the disturbance is sufficiently large. Evidence is presented that this mechanism may be responsible for 3‐D instabilities in high Reynolds number flows whose vortex structures can be locally described by elliptical streamlines.
205 citations
TL;DR: In this paper, the phase-averaged velocity vector fields are discussed in terms of critical point theory and an investigation of turbulent wakes is conducted, as well as phaseaveraged and global Reynolds normal and shear stresses.
Abstract: An investigation of turbulent wakes was conducted and phase-averaged velocity vector fields are presented, as well as phase-averaged and global Reynolds normal and shear stresses. The topology of the phase-averaged velocity fields is discussed in terms of critical point theory. Here in Part 1, the vortex formation process in the cavity region of several nominally two-dimensional bluff bodies is investigated and described using phase-averaged streamlines where the measurements were made in a nominal plane of symmetry. It was found that the flows encountered were always three-dimensional and that the mean-flow patterns in the cavity region were quite different from those expected using classical two-dimensional assumptions.
153 citations
TL;DR: In this article, a finite volume methodology was developed to predict fully developed heat transfer coefficients, friction factors, and streamlines for flow in a corrugated duct, which can be adopted for other convection-diffusion problems in which two boundaries of the flow domain do not lie along the coordinate lines.
Abstract: A finite volume methodology was developed to predict fully developed heat transfer coefficients, friction factors, and streamlines for flow in a corrugated duct. The basis of the method is an algebraic coordinate transformation which maps the complex fluid domain onto a rectangle. The method can be adopted for other convection-diffusion problems in which two boundaries of the flow domain do not lie along the coordinate lines. Representative results were found for laminar flow uniform wall temperature, and for a range of Reynolds number, Prandtl number, corrugation angle, and dimensionless interwall spacing. As seen from the streamlines, the flow patterns are highly complex including large recirculation zones. The pressure drops and friction factor results are higher than the corresponding values for a straight duct. Finally, the performance of the corrugated duct was compared with the straight duct under three different constraints - fixed pumping power, fixed pressure drop, and fixed mass flow rate. There are small differences in the heat transfer rate ratios under these constraints.
116 citations
TL;DR: In this paper, the effect of the rigid sidewalls of a rectangular box on the spatial structure and the dynamical behaviour of the flow is analyzed, both conducting and adiabatic sidewalls are considered.
Abstract: Steady and oscillatory convection in a rectangular box heated from below are studied by means of a numerical solution of the three-dimensional, time-dependent Boussinesq equations. The effect of the rigid sidewalls of the box on the spatial structure and the dynamical behaviour of the flow is analysed. Both conducting and adiabatic sidewalls are considered. Calculated streamlines illustrate the three-dimensional structure of the steady flow with Prandtl numbers 0.71 and 7. The onset and the frequency of the oscillatory instability are calculated and compared with available experimental and theoretical data. With increasing Rayleigh number a subharmonic bifurcation and the onset of a quasi-periodic flow can be observed. A comparison of the different time-dependent solutions shows some interesting relations between the spatial structure and the dynamical behaviour of the confined flow.
81 citations
TL;DR: In this paper, a simple to understand and easy to implement numerical technique, based on reverse calculation of ground-water flow paths, is applied to delineation of time-related capture zones around water-supply wells.
Abstract: A simple to understand and easy to implement numerical technique, based on reverse calculation of ground -water flow paths, is applied to delineation of time-related capture zones around water-supply wells. The areal extent of a t-year capture zone in a two-dimensional regional flow domain is approximated by a large set of reverse pathlines all with t-year travel times. The numerical approach is suitable for calculation of time-related capture zones in steady, nonuniform flow with inhomogeneous, anisotropic aquifer conditions. The example applications show the sensitivity of this technique to the influence of multiple wells in a block inhomogeneous flow regime.
76 citations
TL;DR: In this paper, the Navier-Stokes equations in axisymmetric cylindrical coordinates were used for predicting detailed flow patterns and temperature profiles during natural convection heating of canned liquids.
Abstract: Amathematical model was developed for the first time for predicting detailed flow patterns and temperature profiles during natural convection heating of canned liquids. Finite difference methods were used to solve the governing Navier-Stokes equations in axisymmetric cylindrical coordinates. A vorticity-stream function formulation of the equations was used. Details of the numerical techniques used are discussed. Plots of transient isotherms, streamlines and velocities are provided. From the standpoint of food processing, the slowest heating points migrated within the bottom 15% of the can with no particular pattern of migration.
65 citations
TL;DR: In this article, a finite-element simulation of the two-dimensional Navier-Stokes equations at a Reynolds number of 110 was used to characterize the wake in terms of its critical point trajectories.
Abstract: Streamline, streakline, and material-line flow-visualization techniques have been numerically simulated in the vortex-shedding flow field from a finite-element simulation of the two-dimensional Navier-Stokes equations at a Reynolds number of 110. The results have been used (i) to characterize the wake in terms of its critical-point trajectories, and (ii) to verify that the two-dimensional Navier-Stokes model predicts the mechanism of vortex shedding experimentally observed by Gerrard (1978). A technique for determining vorticity balances in the flow field is also presented.
61 citations
TL;DR: In this paper, the average number of particles per floc varied as 〈N〉 ∝ E−1.0 with E = Q πc 3, possibly reflecting a change in shape from predominantly spherical to elongated.
61 citations
TL;DR: In this paper, an axisymmetric analogue for three-dimensional boundary layers is developed for calculating the heating rates on the Space Shuttle Orbiter and other advanced reentry configurations.
Abstract: A rapid, approximate method has been developed for calculating the heating rates on three-dimensional vehicles such as the Space Shuttle Orbiter and other advanced reentry configurations. The method is based on the axisymmetric analogue for three-dimensional boundary layers. It uses information obtained from a three-dimensional inviscid flowfield solution, such as HALIS, to calculate inviscid surface streamlines along which approximate heating rates are calculated independent of what happens along other streamlines. Three-dimensional effects are included through the metric coefficient that describes the divergence or convergence of streamlines. Boundary-layer edge properties are obtained from the inviscid flowfield solution by interpolating in the inviscid flowfield at a distance equal to the boundary-layer thickness away from the wall. This accounts, approximately, for the variable boundary-layer edge entropy. Using this method, heating calculations can be made along a typical streamline in a few seconds. This method has been used to accurately predict heating rates for simple shapes such as a spherically blunted cone and more complex shapes such as the Shuttle Orbiter for a variety of wind-tunnel and flight conditions. A unique feature of the method is its ability to accurately predict heating rates on the Shuttle Orbiter wing.
Book•
01 Jan 1987
TL;DR: In this article, the authors present an approach to the management of groundwater in the context of aquifers, using a two-dimensional model of the aqua and a three-dimensional flow model.
Abstract: 1. Introduction.- 1.1. Groundwater and Aquifers.- 1.2. Management of Groundwater.- 1.3. Groundwater Modeling.- 1.4. Continuum Approach to Porous Media.- 1.5. Horizontal Two-Dimensional Modeling of Aquifers.- 1.6. Objectives and Scope.- 2. Groundwater Motion.- 2.1. Darcy's Law and its Extensions.- 2.2. Aquifer Transmissivity.- 2.3. Dupuit Assumption.- 3. Modeling Three-Dimensional Flow.- 3.1. Effective Stress in Porous Media.- 3.2. Mass Storage.- 3.3. Fundamental Mass Balance Equation.- 3.4. Initial and Boundary Conditions.- 3.5. Complete Statement of Mathematical Flow Model.- 3.6. Modeling Soil Displacement.- 4. Modeling Two-Dimensional Flow in Aquifers.- 4.1. Aquifer Storativity.- 4.2. Fundamental Continuity Equations.- 4.3. Initial and Boundary Conditions.- 4.4. Complete Statement of Aquifer Flow Model.- 4.5. Regional Model for Land Subsidence.- 4.6. Streamlines and Stream Function.- 5. Modeling Flow in the Unsaturated Zone.- 5.1. Capillarity and Retention Curves.- 5.2. Motion Equations.- 5.3. Balance Equations.- 5.4. Initial and Boundary Conditions.- 5.5. Complete Statement of Unsaturated Flow Model.- 6. Modeling Groundwater Pollution.- 6.1. Hydrodynamic Dispersion.- 6.2. Advective, Dispersive, and Diffusive Fluxes.- 6.3. Balance Equation for a Pollutant.- 6.4. Initial and Boundary Conditions.- 6.5. Complete Statement of Pollution Model.- 6.6. Pollution Transport by Advection Only.- 6.7. Macrodispersion.- 7. Modeling Seawater Intrusion.- 7.1. The Interface in a Coastal Aquifer.- 7.2. Modeling Seawater Intrusion in a Vertical Plane.- 7.3. Modeling Regional Seawater Intrusion.- 8. Introduction to Numerical Methods.- 8.1. Analytical versus Numerical Solutions.- 8.2. Survey of Numerical Methods.- 8.3. Computer Programming.- 9. The Finite Difference Method.- 9.1. Steady Flow.- 9.2. Unsteady Flow.- 9.3. Accuracy and Stability.- 9.4. Generalizations.- 10. The Finite Element Method.- 10.1. Steady Flow.- 10.2. Steady Flow in a Confined Aquifer.- 10.3. Steady Flow with Infiltration and Leakage.- 10.4. Steady Flow through a Dam.- 10.5. Unsteady Flow in an Aquifer.- 10.6. Generalizations.- 11. Transport by Advection.- 11.1. Basic Equations.- 11.2. Semi-Analytic Solution.- 11.3. System of Wells in an Infinite Field.- 11.4. System of Wells in an Infinite Strip.- 11.5. Numerical Solution in Terms of the Piezometric Head.- 11.6. Numerical Solution in Terms of the Stream Function.- 11.7. Tracing Particles Along a Stream Line.- 12. Transport by Advection and Dispersion.- 12.1. Dispersion in One-Dimensional Flow.- 12.2. Numerical Dispersion.- 12.3. A Finite Element Model for Two-Dimensional Problems.- 12.4. Random Walk Model.- 13. Numerical Modeling of Seawater Intrusion.- 13.1. Model for Flow in a Vertical Plane.- 13.2. Basic Equations for a Regional Model of Seawater Intrusion.- 13.3. Finite Element Model for Regional Interface Problems.- Appendix. Solution of Linear Equations.- References.- Problems.- Index of Subjects.
TL;DR: In this paper, a local analysis about a point at the free surface of an incompressible viscous fluid flow at steady state by means of series expansions of the Navier-Stokes equations is made.
Abstract: A local analysis has been made about a point at the free surface of an incompressible viscous fluid flow at steady state by means of series expansions of the Navier–Stokes equations. Dividing streamlines, curvature effects, and the role of vorticity have been studied. A single dividing streamline is always perpendicular to the free surface. This means, in particular, that all steady vortices with one end of their axes attached to a free surface are perpendicular to it. Double dividing streamlines of two‐dimensional viscous fluid flows at a free surface always have an angle of 90° between them, whereas double dividing streamlines of inviscid irrotational motion on a free or solid surface are separated by an angle of 60°. For an interface between two immiscible viscous fluids a novel ‘‘refraction law’’ for dividing streamlines has been derived.
TL;DR: In this article, the dynamic and thermal properties of a laminar flow past a sinusoidal cavity are studied, where velocity and temperature fields inside and above the cavity are numerically determined from the viscous flow equations.
TL;DR: In this article, the authors proposed a conceptual model of turbulence structure in which the elementary unit of coherent structure in the buffer layer is presumed to be a horseshoe vortex and in which characteristics of the multiple structure of turbulence are shown with respect to the scale, arrangement and generating process of the vortex.
Abstract: Coherent structures of turbulent open-channel flow in the wall region of a channel bed were examined quantitatively using experimental data obtained by flow visualization. Successive pictures of flow patterns in two horizontal cross-sections at different levels near the channel bed were taken, and then were digitized and analysed by a computer.This method of flow visualization and picture processing enabled us to calculate the distributions of the three components of the velocity vectors. The distributions of velocities, streamlines, two-dimensional divergence and three components of vorticity could be calculated and are displayed as graphical output. In our numerical analyses, the idea of a two-dimensional correlation coefficient is introduced, through which the degree of similarity of turbulence structures can be better estimated than with the usual one-dimensional coefficient. Use of the data was based on the premise that the essential element in a turbulence structure is vortex motion.We propose a conceptual model of turbulence structure in which the elementary unit of coherent structure in the buffer layer is presumed to be a horseshoe vortex and in which the characteristics of the multiple structure of turbulence are shown with respect to the scale, arrangement and generating process of horseshoe vortices and longitudinal vortices. Our model clearly explains the generating mechanism and mutual relations of low-speed regions, high-speed regions, ejections, sweeps and localized free-shear layers.
Dissertation•
01 Jan 1987
TL;DR: In this paper, a quasi-static penetration of a cone penetrometer into clay can be formulated as a steady state problem by considering a steady flow of soil past a stationary cone.
Abstract: The quasi-static penetration of a cone penetrometer into clay can be formulated as a steady state problem by considering a steady flow of soil past a stationary cone. The soil velocities are estimated from the flow field of an inviscid fluid, and the incompressibility condition is achieved by adopting a stream function formulation. Emphasis is placed on obtaining an accurate velocity estimate and this is accomplished by a solution of the Navier-Stokes equations. The strain rates are evaluated from the flow field using a finite difference scheme. The clay is modelled as a homogeneous incompressible elastic-perfectly plastic material and the soil stresses are computed by integrating along streamlines from some initial stress state in the upstream region. These stresses do not in general obey the equilibrium equations, although one of the two equations can be satisfied by an appropriate choice of the mean stress. Several attempts have been made to use the remaining equilibrium equation to obtain an improved velocity estimate and three plausible iterative methods are detailed in this thesis. In a second study, a series of finite element calculations on the cone penetration problem is performed. In modelling the penetration process, the cone is introduced in a pre-formed hole and some initial stresses assumed in the soil, incremental displacements are then applied to the cone until a failure condition is reached. Although the equilibrium condition is satisfied very closely in the finite element calculations, it is extremely difficult to achieve a steady state solution. In a third series of computations, the stresses evaluated by the strain path method are used as the starting condition for the finite element analysis. This is believed to give the most realistic solution of the cone penetration problem because both the steady state and equilibrium conditions are approximately satisfied. Numerically derived cone factors are presented and these are found to depend on the rigidity index of the soil and the in situ stresses. The pore pressure distribution in the soil around the penetrometer is estimated using Henkel's empirical equation. The dissipation analysis is based on Terzaghi's uncoupled consolidation theory. The governing equation is formulated in the Alternating-Direction-Implicit finite difference scheme. This formulation is unconditionally stable and variable time steps are used to optimise the solution procedure. The dissipation curves are found to be significantly affected by the rigidity index of the soil and a dimensionless time factor is proposed to account for this effect.
TL;DR: In this paper, an exact, closed-form analytic solution is presented for three-dimensional steady-state flow due to a source disc, which is used to model a circular recharge area at the upper boundary of a confined aquifer.
TL;DR: In this article, the authors considered the steady self-propagation with respect to the fluid at infinity of two equal symmetrically shaped vortices in a compressible fluid and formulated the nonlinear free-boundary problem for the vortex-pair flow in the hodograph plane of compressible-flow theory.
Abstract: We consider the steady self-propagation with respect to the fluid at infinity of two equal symmetrically shaped vortices in a compressible fluid. Each vortex core is modelled by a region of stagnant constant-pressure fluid bounded by closed constant-pressure, constant-speed streamlines of unknown shape. The external flow is assumed to be irrotational inviscid isentropic flow of a perfect gas. The flow is therefore shock free but may be locally supersonic. The nonlinear free-boundary problem for the vortex-pair flow is formulated in the hodograph plane of compressible-flow theory, and a numerical solution method based on finite differences is described. Specific results are presented for a range of parameters which control the flow, namely the Mach number of the pair translational motion and the fluid speed on each vortex bounding streamline. Perturbation-theory predictions are developed, valid for vortices of small core radius when the pair Mach number is much less than unity. These are in good agreement with the hodograph-plane calculations. The numerical and the perturbation-theory results together confirm the recently discovered (Barsony-Nagy, Er-El & Yungster 1987) existence of continuous shock-free transonic compressible flows with embedded vortices. For the vortex-pair geometry studied, solution branches corresponding to physically acceptable flows that could be calculated using the present hodograph-plane numerical method were found to be terminated when either the flow on the streamline of symmetry separating the vortiqes tends to become superonic or when limiting lines appear in the hodograph plane giving a locally multivalued mapping to the physical plane.
TL;DR: In this paper, a nonlinear integral constitutive equation, novel streamlined finite elements, Galerkin's method of weighted residuals, and Newton iteration are analyzed by means of a two-dimensional viscoelastic liquid with memory.
Abstract: Steady, two-dimensional flows of viscoelastic liquid with memory are analyzed by means of a nonlinear integral constitutive equation, novel streamlined finite elements, Galerkin's method of weighted residuals, and Newton iteration. By their relation to streamlines the new elements allow, for the first time, full Newton iteration of the algebraic equation set to which the governing integrodifferential system reduces. Streamlines are computed simultaneously with velocity components and pressure, the primary unknowns; to track the history of liquid particles the deformation equations are solved analytically in a Protean system of coordinates that conforms to the streamlines. In the cases studied the Newton iteration converges quadratically to a creeping flow state of Weissenberg number (product of upstream wall shear rate and relaxation time of the liquid) up to about 20 in channel flow, 10 in film flow, and 2 in die-swell flow. Shear-thinning shrinks the domain of convergence; it also reduces die-swell. A slip boundary condition in the vicinity of the contact line extends the domain of convergence.
TL;DR: In this article, it was shown that the standard formula giving the shape of the streamlines involves geometric elements rather than orbital elements, when the oblateness of the planet is taken into account.
TL;DR: In this article, the problem of creeping flow of a newtonian fluid around and through a permeable sphere that is moving towards an impermeable wall with constant velocity is solved in terms of the stream function and the pressure.
Abstract: The problem of creeping flow of a newtonian fluid around and through a permeable sphere that is moving towards an impermeable wall with constant velocity is solved in terms of the streamfunction and the pressure. The permeability of the sphere is assumed to be continuous, uniform and isotropic, The flow in the sphere is modelled with Darcy's law, and a Beavers-Joseph-Saffman slip-flow boundary condition is assumed at the boundary. Sample streamlines and isobars are calculated. The hydrodynamic correction factor to Stoke's law, ƒ is calculated as a function of the dimensionless permeability, κ, the dimensionless slip factor, β, and the dimensionless gap length, δ. As expected, for typical κ and β values the values of ƒ are substantially smaller than those for an impermeable sphere. An important result is that as δ decreases the value of ƒincreases much more slowly than it does for an impermeable sphere; furthermore, ƒ is finite as δ→0. For moderate and large κ values the ƒ vs δ curve has a maximum.
TL;DR: In this paper, the existence of infinite sets of exact solutions for the flow in the geometry of an orthogonal rheometer in which the above non-torsional oscillations are superposed on the disks, is established.
TL;DR: In this article, the problem of calculating nonlinear two-dimensional free-surface potential flow about a circular cylinder rising to a free surface is solved numerically, and the deeplysubmerged circular cylinder is accelerated smoothly from rest to a uniform vertical velocity.
Abstract: The problem of calculating nonlinear two-dimensional free-surface potential flow about a circular cylinder rising to a free surface is solved numerically. The deeplysubmerged circular cylinder is accelerated smoothly from rest to a uniform vertical velocity. A boundary/integral-equation method is used to obtain free-surface elevations and streamlines about the rising cylinder for several final speeds. Results, including pressure forces, are compared with a cylinder rising to a rigid wall and a cylinder moving in an infinite fluid.
TL;DR: In this paper, an expression for the integral of the equations of plane-parallel viscous incompressible flow expressing conservation of the Bernoulli function along a certain family of lines which, as the viscosity tends to zero, go over into the streamlines.
Abstract: An expression is obtained for the integral of the equations of plane-parallel viscous incompressible flow expressing conservation of the Bernoulli function along a certain family of lines which, as the viscosity tends to zero, go over into the streamlines. These lines also determine the direction of transfer of the vorticity of the viscous flow.
TL;DR: In this paper, the effect of finiteness of the length of a circular cylinder on the translational motion of a small sphere along its centerline is investigated. But the analysis is based on the Stokes equations and the authors assume that the fluid occupying that region is viscous and incompressible.
Abstract: A theoretical study is made on the effect of finiteness of the length of a circular cylinder on the translational motion of a small sphere along its centerline. Fluid occupying that region is assumed viscous and incompressible, and the analysis is based on the Stokes equations. The method of reflections is employed and first-order correction of the force experienced by the sphere is obtained. For longer cylinders, end-effects on the force are important only when the sphere is very close to the endwalls, while for cylinders whose length is the same order of magnitude as its radius, end-and sidewalls play equally significant roles. Among cylindrical boxes containing the same amount of fluid volume, additional resistance force at the center becomes minimum for a cylinder with its length/diameter ratio of about 0.56. Streamlines are also shown.
TL;DR: In this article, several models (including that of an inviscid fluid, that of a Coulomb plastic with and without the condition of coaxiality, and one which is purely kinematic) have been applied to the annular region of a spouted bed in an attempt to determine the streamlines of the solids flow.
TL;DR: In this paper, the die-swell problem is reconsidered by using the concept of stream tubes for the incompressible axisymmetric case, and it is possible to study the influence of the singularity at the corner where the free surface forms, in addition to the upstream and downstream boundary conditions.
Abstract: The die-swell problem is reconsidered by using the concept of stream tubes for the incompressible axisymmetric case. Using some assumptions and analytical equations, it is possible to study the influence of the singularity at the corner where the free surface forms, in addition to the upstream and downstream boundary conditions. A minimization technique is used for the determination of the coefficients related to the analytical equations of the streamlines. This enables us to compute the flow field in the jet for a Newtonian fluid, the swelling ratio of which is found to be 12%.
TL;DR: In this article, the upwind influence is detected at Froude number larger than 1.5 and the empirically determined height is, on average, greater than that obtained by Sheppard's dividing streamline formula, which could provide a lower limit.
Abstract: Atmospheric stable airflow over a mesoscale mountain chain has been considered in order to provide an evaluation, based on isentropic analysis, of the base height of the upwind flow layer passing over the mountains. The trend of this height as a function of the Froude number suggests that in the real atmosphere the upwind influence is detectable at Froude number larger than 1.5. The empirically determined height is, on average, greater than that obtained by Sheppard's dividing-streamline formula, which could provide a lower limit. The wind shear produces a downward deflection of the streamlines, which can be simply accounted for as far as strong stratification cases are concerned. The three-dimensionality of the flow is discussed in order to assess the structure of the region below this dividing height.
TL;DR: In this article, it is shown that two massive bodies are 'pushed together' by their common gravitational field by means of momentum current density field lines, and computer sketches of such streamlines in the common field of the Earth and the Moon are presented.
Abstract: In gravitation an action-at-a-distance description of the interaction between two bodies is still in use. However, the momentum current picture presents a local-causes description of this interaction. The suggested approach can be used to visualise and quantitatively sketch the stress distribution in a weak static gravitational field by means of momentum current density field lines. Computer sketches of such streamlines in the common field of the Earth and the Moon are presented. It is shown that two massive bodies are 'pushed together' by their common gravitational field.