Calculation Of Velocity Field Research Articles

Numerical calculation of fluid flow in a continuous casting tundish

The deposition of deoxidation and reoxidation products in continuous casting tundish nozzles results in poor surface quality of the cast product and in extreme cases, disruption of the operating schedule. Since the flow of steel in the tundish is central to the behavior of nonmetallic inclusions in the tundish, the details of the flow field are determined numerically by appropriate solutions of the equations of change. The effects of the changes of the internal configuration of the tundish and various casting parameters on the computed velocity fields, particle trajectories, and particle residence times are investigated. The conclusions drawn from the computed results are compared with the results of various independent water model studies.

Simulation of Taylor-Couette flow. Part 1. Numerical methods and comparison with experiment

We present a numerical method that allows us to solve the Navier-Stokes equation with boundary conditions for the viscous flow between two concentrically rotating cylinders as an initial-value problem. We use a pseudospectral code in which all of the time-splitting errors are removed by using a set of Green functions (capacitance matrix) that allows us to satisfy the inviscid boundary conditions exactly. For this geometry we find that a small time-splitting error can produce large errors in the computed velocity field. We test the code by comparing our numerically determined growth rates and wave speeds with linear theory and by comparing our computed torques with experimentally measured values and with the values that appear in other published numerical simulations. We find good agreement in all of our tests of the numerical calculation of wavy vortex flows. A test that is more sensitive than the comparison of torques is the comparison of the numerically computed wave speed with the experimentally observed wave speed. The agreements between the simulated and measured wave speeds are within the experimental uncertainties; the best-measured speeds have fractional uncertainties of less than 0.2%.

Investigation of experimental flow patterns for plane strain extrusion of hardening materials with slipline field methods

The paper describes a method of obtaining experimental flow patterns in extrusion of aluminium alloys through tapered dies, of constructing slipline meshes from the velocity fields and of carrying out stress analyses with hardening rules based on measured stress-strain relations in compression. The slipline meshes are patched together from local Hencky-Prandtl nets which are perturbed to obtain the best overall match between the computed velocity field and that of the experimental flow pattern, by using an iterative optimization procedure. The extrusion forces computed from the stress fields and from the plastic work are compared with the measured forces and with those predicted by the corresponding solutions for rigid-perfectly plastic material.

Minimal upper mantle temperature variations consistent with observed heat flow and plate velocities

The momentum equations applied to 5° block means are integrated from the observed surface plate velocities downward to a depth of 280 km, assuming no lateral heterogeneities in density or viscosity. It is assumed that 85% of the global heat production QG = 0.85 × 4.0 ×1013 W comes from below 280 km and that at this level the transfer is fully convective. A temperature field 7 is inferred at depth 280 km by minimizing the quantity∫|T− T0|n dS + nλ{QG−pC∫(T−T0)υrdS}, where the integrals are over the sphere, ρ is density, C is heat capacity, υr is radial velocity, T0 is a prescribed mean, and λ is a Lagrangian multiplier. Norms n, ranging from 1.5 to 2.5, are tried. The intervening temperature fields are then inferred, integrating the energy equation downward by using the previously calculated velocity field. This integration is subject to the limitations that the derivative of the temperature with respect to depth is everywhere sufficient to attain the fully convecting temperature, but never less than adiabatic. A surface heat flow based on observations plus age and tectonic setting is used. The principal inferences are: (1) the greatest lateral variations in temperature, ∼1000°C, occur with the top 20 km; (2) the greatest advection, ∼200°C/m.y., occurs within the top 20 km; (3) below 50 km, the greatest departures of temperature from the mean are negative “tongues,” reaching an extreme of about −800°C at depth 100 km; (4) below 50 km, heat transfer becomes more convective than conductive; (5) at the fully convecting level, 280 km, temperature variations are at least ±180°C about the mean. The principal defect in the entire calculation is unrealistically low temperatures arising from unrepresentatively low surface heat flows. The principal defect of the model probably arises from the assumption that all heat transfer at a depth of 280 km is representable by 5° means in velocity and temperature.

Numerical evaluation of the radiation from unbaffled, finite plates using the FFT

An iteration technique is described which numerically evaluates the acoustic pressure and velocity on and near unbaffled, finite, thin plates vibrating in air. The technique is based on Rayleigh’s integral formula and its inverse. These formulas are written in their angular spectrum form so that the fast Fourier transform (FFT) algorithm may be used to evaluate them. As an example of the technique the pressure on the surface of a vibrating, unbaffled disk is computed and shown to be in excellent agreement with the exact solution using oblate spheroidal functions. Furthermore, the computed velocity field outside the disk shows the well-known singularity at the rim of the disk. The radiated fields from unbaffled flat sources of any geometry with prescribed surface velocity may be evaluated using this technique. The use of the FFT to perform the integrations in Rayleigh’s formulas provides a great savings in computation time compared with standard integration algorithms, especially when an array processor can be used to implement the FFT.

The nonlinear behaviour of a constant vorticity layer at a wall

The so-called ‘water-bag’ method is used to study the behaviour of a two-dimensional inviscid layer of constant vorticity ω and of mean thickness δ adjacent to a wall with slip at the wall. A nonlinear initial-value equation is derived which describes the motion of the material interface separating the rotational fluid within the layer from the irrotational free stream, for the case where this interface is subject to streamwise cyclic disturbances to its undisturbed shape. A linearized solution to this equation shows that a sinusoidal disturbance of wavelength λ propagates as one mode of a neutrally stable Kelvin-Helmholtz wave with velocity ωλ[1 − exp (−4πδ/λ)]/4π relative to the fluid at infinity. Numerical solutions of the full nonlinear equation for a range of wavelengths and finite disturbance amplitudes indicate different behaviour. For sufficiently large amplitude the interface valleys evolve into long re-entrant wedges of irrotational fluid which are ‘entrained’ into the layer and which are separated from the free stream by lobes or bulges of rotational fluid. This single-mode nonlinear interfacial distortion could be generated over a broad wavelength range with no indication of preferential scaling based on δ. It is suggested that the interface behaviour bears distinct resemblance to flow features observed at the interface between turbulent and non-turbulent fluid in recent smoke-in-air flow-visualization studies of the outer part of a constant pressure turbulent boundary layer. The calculated rotational fluid lobe velocities, which are not very different from the equivalent linearized wave velocities, are found to be in reasonable agreement with the few existing measurements of the velocity of bulges at the turbulent–nonturbulent fluid interface, while the computed velocity field in the lobe is in qualitative agreement with the general flow pattern observed in experiments. In the absence of a preferred scale or range of scales for the development of the interfacial distortion, however, it is concluded that the present results cannot be interpreted as supporting the hypothesis of the presence of largescale coherent motions in the outer part of the layer.

Numerical Analysis of Flow in Sedimentation Basins

The equations governing two-dimensional, steady flow in circular and rectangular primary settling basins are formulated. Mean-flow equations expressing conservation of mass and momentum are combined with a two-equation turbulence model (k-ϵ model) to compute velocity fields in sedimentation basins. The turbulence is characterized by a turbulent kinematic eddy viscosity which is computed directly as part of the solution. A numerical solution to the mathematical model is formulated by application of the Galerkin finite element method to the equation residuals. The resulting non-linear system of equations is solved by a variation of Newton's method. The model is applied to predict the flow pattern in a rectangular settling basin. A vector plot of the basin velocity field and contour plot of the turbulent kinematic eddy viscosity are presented.

Pulsatile pressure and flow in arterial stenoses simulated in a mathematical model

This paper presents the detailed pulsatile pressure and flow velocity patterns inside an axis symmetric stenosis model with 75% constriction. The pressure and velocities have been calculated by solving the Navier-Stokes equations by the finite element method, the velocity profile in a straight tube caused by a pulsating driving pressure has been calculated first and then used as a boundary condition for the stenosis calculations. The results of the mathematical simulations of the stenosis model have been obtained in terms of velocity vectors, streamlines and isobars at 16 different instances in time, each 15 degrees apart during a cardiac cycle. The calculated velocity field shows that a vortex is developed at the wall distal to the stenosis as the velocity decreases from the peak systolic value. At the site of the vortex, a local pressure minimum is found due to the conversion of pressure to kinetic energy. When the flow is reversed, the reversal occurs first along the wall, thus forcing the vortex toward the the centre of the tube. As the reverse flow velocity increases, a vortex is also developed at the proximal site of the stenosis.

Tidal exchange through Naruto, Akashi and Kitan Straits

In order to re-evaluate the water volume exchange through Naruto Strait, we have performed a numerical experiment (nonlinear barotropic model including the actual depth of water and the details of shoreline) where trajectories of a number of labeled particles are calculated during a full cycle of the M2 tidal current. The ratio of water volume exchanged through Naruto Strait to that through Akashi Strait is found to be twice as large as the previously estimated value. The calculated water exchange rate is 104 % for Naruto Strait, 52 % for Akashi Strait and 28 % for Kitan Strait. In the case where the tide-induced residual current is excluded from the calculated velocity field (i.e. considering only the M2 current), the calculated exchange rate maintains the level of 68 % for Naruto Strait, 18 % for Akashi Strait and 11 % for Kitan Strait, respectively. The mechanism of tidal exchange through these three straits is discussed, and it is shown that a suitable exchange rate is obtained by starting the calculation of trajectories of labeled particles at the time of either a maximum ebb- or flood-current.

Numerical Modeling of Tidal Circulation in Harbors

In this study a numerical model has been developed that can be used to predict the two-dimensional tide-induced velocity fields in harbors and estuaries. The model is particularly suited to basins having a narrow entrance where, on the incoming tide, the divergence of the velocity field associated with the jet inlet gives rise to the generation of vorticity. The time-dependent nonlinear equations of motion are formulated to include the effects of bottom roughness, wind action, the earth's rotation, and a simplified version of the turbulent transfer of momentum. These equations are expressed in an alternating-direction implicit finite difference form and are solved by Gaussian elimination. The numerical model has been checked by making comparisons between the computed velocity fields and experimentally measured velocities and path lines for two hydraulic model studies involving various rectangular harbors and a circular reservoir.

The generation of feeding currents by flagellar motions

The use of flagella by sessile organisms to generate feeding currents is analysed. The organism consists of a spherical cell body (radius A) to which a smooth flagellum (radius a, length L) is attached radially. The cell body is a height H above the plane substrate to which it is rigidly attached via a stalk. The organism propagates plane sinusoidal waves (amplitude α, wavenumber k) from base to tip. The flagellum is represented by distributions of stokeslets and dipoles along its centre-line. The cell body and substrate are modelled by employing an approximate form of the Green's function for the sphere in the half space. The error terms in the model are O(a/L) and O(A2/H2). The analysis and method of solution are adapted from Higdon (1979).The mean flow rate and power consumption are calculated for a wide range of parameters. Optimal motions are determined with the criterion of minimizing the power required to achieve a given flow rate. The optimum wave has maximum slope in the range 2 < αk < 2·5 (compared to the optimum value αk = 1 for swimming). The optimum number of waves Nλ increases linearly with flagellar length for L/A > 10 and is approximately constant, Nλ = 1, for shorter flagella. The optimum flagellar length is in the range 5 < L/A < 10. There is no optimum flagellar radius a/A. For optimal efficiency, the height H should be greater than or equal to the length of the flagellum.The optimum values of the parameters are compared to the values for the choano-flagellates described by Lapage (1925) and Sleigh (1964). Excellent agreement is found between the predicted optima and the observed values. The calculated velocity field closely resembles the flow described by Sleigh and Lapage.

A study of the M2 tide in the Arctic Ocean

Numerical schemes for first and second order approximations on a 75-km grid are used to calculate amplitudes and phases of the M2 tide in the Arctic Basin. Assuming that all tidal energy enters from the Atlantic Ocean across a line approximately Tromso — Jan Mayen — Scoresby Sound, the computations yield a main amphidromic point off Prince Patrick Island at 81°30′ N, 133° W, confirming earlier assumptions. Several additional amphidromies appear to exist on the Eurasian Shelf. Frictional energy dissipation is assumed to be proportional to the square of horizontal velocity and the inverse square of water depth. Except for the region of Novaya Zemlya, observed and computed amplitudes and phases at points on the Eurasian and North American coast are in good agreement with observations. The results are shown on cotidal and corange maps, and a distribution of current ellipses. Also shown is a distribution of the maximum horizontal velocity divergence due to the M2 tide. Significant divergences, which are likely to affect the compactness of sea ice, are limited to the Greenland Sea, and the shallow waters of the Eurasian Shelf. From the computed velocity field, the amount of energy dissipated by bottom friction in the Arctic Basin averaged over one tidal period is estimated to be 5·1017 erg/s.

TURBULENT FREE CONVECTION FROM A VERTICAL, ISOTHERMAL PLATE

The temperature and velocity fields adjacent to an isothermal, vertical plate have been calculated by finite-difference methods using the time-averaged equations of conservation. Effective diffusivities for turbulent transfer were computed from differential balances for the turbulent kinetic energy and the rate of dissipation of turbulent energy, using the empirical coefficients previously evaluated for forced convection together with an additional Prandtl-number-dependent coefficient associated with the buoyant production of turbulent kinetic energy. Although the turbulent boundary layer almost approaches an asymptotic condition, the slight development requires a two-dimensional solution proceeding from the leading edge through the laminar and turbulent regimes. It was necessary to trigger the transition from laminar to turbulent motion, but the ultimate turbulent behavior was found to be independent of the point of triggering. The computed velocity fields, temperature fields, and rates of heat transfer ...

Finite element solution of the Navier-Stokes equations for two-dimensional steady flow through a section of a canine aorta model

A finite element solution to the Navier-Stokes equations for flow through a segment of the aorta has been developed, providing information on the velocity profiles, wall shear stresses and pressures. One purpose was to compare these results with experimental data on wall shear stresses, obtained in a plastic model of a canine aorta-excellent agreement was found. The patterns are found to be extremely sensitive to changes in flow splits through the branching celiac and anterior mesenteric arteries and also on the fine detail of the shapes of the flow dividers. Other interesting results include possible separation of flow beyond the anterior mesenteric as indicated by the pressure variation in that region, and the calculated velocity fields show eddy formation on the outer walls in the celiac branch. The calculations illustrate the power of the method used in that the important features of the flow have been reproduced. It also indicates the potentialities for prediction in more complex cases, such as that of pulsatile flow and distensible walls, for which experimental measurements are difficult.

An implicit finite-difference method for solving the Navier-Stokes equation using orthogonal curvilinear coordinates

Orthogonal curvilinear mesh networks are generated numerically between the wavy walls of two-dimensional peristaltic channels. A dual iterative procedure is developed to facilitate the conformal mapping, as well as to adjust mesh dimensions when necessary to fit the boundaries of the flow region. An implicit finite-difference technique is employed to obtain transient solutions of the Navier-Stokes equations. The effects of initial conditions on the flow establishment are discussed, along with considerations of numerical accuracy. The effects of certain nonconservative difference forms of the governing equations are explored. A calculated velocity field for a two-dimensional nonlinear peristaltic flow is supported by laboratory flow observation. The present method is applicable for laminar flow in a nonuniform channel with or without wall peristalsis.

Inviscid secondary motions in a tube of slowly varying ellipticity

Inviscid methods of treating secondary flows due to streamline curvature are well known in the literature. The development of secondary motions at high Reynolds number in a bifurcation characteristic of the lung airways or the cardiovascular system is considered. In particular, the secondary motions in the initial section of a bifurcation, which can be considered a tube of slowly varying ellipticity, are calculated. Providing viscous effects are confined to regions near the walls, the calculated velocity field agrees with experimental observation. Thus the structure of the secondary motions can be considered an effect of stretching of vortex lines by the tube boundary.

Wave-induced vorticity in free-surface boundary layers: application to mass transport in edge waves

The time-averaged vorticity field within the free-surface boundary layer associated with a general class of propagating gravity waves is considered. The principal results are applied in a calculation of the mass transport velocity field for edge waves.

Parameters of electron transfer in InP

Calculation of the Γ minimum nonparabolicity of InP from the band structure shows that Kane's theory, which has been used in all velocity-field calculations, underestimates the nonparabolicity and hence causes understimation of the threshold field. The velocity-field characteristic is predicted to be anisotropic. The effective mass components for the L and X valleys are calculated from the band structure. The deformation potential ΞΓL, derived from photoemission data, is large enough that the two-level model appears adequate for calculation of the threshold field.

A Mathematical Model Predicting Tidal Current Velocities in the Strait of Georgia – Juan de Fuca Strait System

A one-dimensional model for calculating the time variation of tidal current velocities throughout the Strait of Georgia – Juan de Fuca Strait system is described. It is intended to use the calculated velocity field as input to a particle transport model predicting the motion of foreign material released in the tidal stream. The velocity calculations are based on the detailed channel dimensions and on observed tidal heights at a number of stations in the system. Only the hydrodynamic equation of continuity is used, which avoids the necessity of using arbitrarily adjusted roughness coefficients. As a single boundary condition it is assumed that there is a constant zero velocity at the head of the channel, where the volume flow rate is small compared to other parts of the system. The calculated velocity field compares well with the limited number of observations that have been made.

Determination of turbulent velocity field in a rectilinear duct with non-circular cross-section

The equation of motion of a liquid flowing through a non-circular duct is solved by means of integral transformation. As a result of application of some integral transformation the problem of the turbulent flow has been brought to the solution of equation of the type describing the laminar flow. The solution of this equation is known and frequently cited in the literature of the subject On the basis of these results the analytical expression describing the relation between the ratio V T V 0 and dimensionless velocity u u ∗, of rather general nature can be constructed. To illustrate the method some numerical example is included, too. The agreement between the computed velocity field and the experimental data are found to be quite satisfactory.