Equations Of Fluid Motion Research Articles

Review Article A thermodynamic formulation of the equations of motion and buoyancy frequency for Earth's fluid outer core

Review Article A thermodynamic formulation of the equations of motion and buoyancy frequency for Earth's fluid outer core

Comparisons of approximate and exact techniques for convected acoustics.

In many aero‐ and environmental‐acoustic problems, convection of sound waves is handled by appropriately increasing or decreasing the local speed of sound and then solving the resulting Helmholtz equation. Such solutions are typically obtained via integral transforms or geometrical acoustics. An exact solution technique that properly incorporates the vector character of mean‐flow convection has been found [L. Nijs and C. P. A. Wapenaar, J. Acoust. Soc. Am. 87, 1987–1998 (1987)]. Unfortunately, the formal and numerical complexity of these techniques has prevented detailed comparisons of the acoustic field variables with and without the standard approximation. Results and comparisons for approximate and exact treatment of convection will be presented from a series of simple computations made directly from the linearized time‐ and space‐dependent equations of inviscid fluid motion. Specifically, the propagation of initially plane acoustic waves in a simple shear flow has been addressed to determine how: (i) wavefront orientation and location, (ii) acoustic‐particle‐velocity vector direction, and (iii) acoustic wave strength compare when mean flow convection is handled approximately and exactly. [Work supported by Ford Motor Company.]

LONG RANGE EXHAUSTION—A MATHEMATICAL MODEL FOR THE AXISYMMETRIC AIR FLOW OF A LOCAL EXHAUST VENTILATION HOOD ASSISTED BY A TURBULENT RADIAL JET

Abstract Mathematical modelling techniques are used to predict the axisymmetric air flow pattern developed by a state-of-the-art flanged exhaust hood which is reinforced by a turbulent radial jet flow. The high Reynolds number modelling techniques adopted allow the complexity of determining the hood's air flow to be reduced and provide a means of identifying and assessing the various parameters that control the air flow. The mathematical model is formulated in terms of the Stokes steam function, Ψ, and the governing equations of fluid motion are solved using finite-difference techniques. The injection flow of the exhaust hood is modelled as a turbulent radial jet and the entrained flow is assumed to be an inviscid potential flow. Comparisons made between contours of constant air speed and centre-line air speeds deduced from the model and all the available experimental data show good agreement over a wide range of typical operating conditions.

Long range exhaustion—A mathematical model for the axisymmetric air flow of a local exhaust ventilation hood assisted by a turbulent radial jet

Mathematical modelling techniques are used to predict the axisymmetric air flow pattern developed by a state-of-the-art flanged exhaust hood which is reinforced by a turbulent radial jet flow. The high Reynolds number modelling techniques adopted allow the complexity of determining the hood's air flow to be reduced and provide a means of identifying and assessing the various parameters that control the air flow. The mathematical model is formulated in terms of the Stokes steam function, Ψ, and the governing equations of fluid motion are solved using finite-difference techniques. The injection flow of the exhaust hood is modelled as a turbulent radial jet and the entrained flow is assumed to be an inviscid potential flow. Comparisons made between contours of constant air speed and centre-line air speeds deduced from the model and all the available experimental data show good agreement over a wide range of typical operating conditions.

The hydrodynamic field due to forcing by modulated acoustic waves

This theoretical investigation determines the hydrodynamic field interior and exterior to an oscillating viscous drop for the case in which the hydrodynamic field exists strictly as a result of forcing by a modulated acoustic standing wave field. Perturbation of the governing equations of fluid motion generates a hierarchy of systems of equations. At second order, this system represents the hydrodynamic field which is forced by terms quadratic in acoustic field quantities. Viscous effects are incorporated into the acoustic field. Of particular interest is the nature of the hydrodynamic field in a sublayer region which exists near drop-host interface. The boundary conditions, which are imposed at the drop-host fluid interface and which include the radial and tangential forcing due to the acoustic radiation pressure vector, complete the problem formulation.

Mesh generation and numerical simulation of fluid entering a large tube bundle

A three-dimensio nal finite element model is used to calculate fluid entering a tank containing a large array of tubes (assemblies). The time-dependent primitive equations of fluid motion are solved using equal order, isoparametric hexahedral elements. Petrov-Galerkin weighting, mass lumping, and reduced integration are utilized with explicit time marching. The tank is 7.5 m tall by 6.7 m in diameter, and contains a total of 600 tubes, consisting primarily of 10.8-cm-diam assemblies with some 2.5-cm-diam control rods. The 10.8-cm assemblies are hollow and thin shelled; the 2.5-cm control rods are solid. The fluid first flows into a plenum, then begins to drain into the assemblies, which are positioned below the plenum; the fluid subsequently discharges from the ends of the assemblies near the bottom of the tank and recirculates within the tank prior to returning to the plenum. A total of 623,040 elements was used to discretize the plenum and tank vessel. Over 30 MW of storage were required to execute the problem on a Cray Y-MP, necessitating the use of a special Cray sparse matrix solver. The numerical results predict a marked decrease in the amount of fluid available to the interior tubes.

Spin-up of a stratified magnetofluid as a model of planetary interiors

Abstract We study spin-up of a viscous electrically conducting Boussinesq fluid in a circular cylinder subject to thermal stratification and uniform axial magnetic field. The cylinder is a perfect conductor of electricity and of heat. Physical parameters are chosen to be consistent with those of planetary interiors: Em ∼ E ≤ 1, β ∼ E½, o ∼ E−1, S ∼ 1, where E, Em' β, o and S are, respectively, the Ekman number, the magnetic Ekman number, the reciprocal square of the Alfven Mach number, the Prandtl number and the stratification number [see definitions just after (2.22)]. The linearized basic equations for axisymmetric fluid motion are solved by boundary layer analysis. Three kinds of end-plate boundary layers and two kinds of side-wall boundary layers are found to exist. The non-diffusive interior flow is solved by Fourier-Bessel expansion method combined with Laplace transformation. The Laplace transform solution is inverted numerically by the use of residue theorem combined with Mathematica (Wolfram Res....

Effects of wall shear stress and fluid recirculation on the localization of circulating monocytes in a three-dimensional flow model

There is a correlation between the location of early atherosclerotic lesions and the hemodynamic characteristics at those sites. Circulating monocytes are key cells in the pathogenesis of atherosclerotic plaques and localize at sites of atherogenesis. The hypothesis that the distribution of monocyte adhesion to the vascular wall is determined in part by hemodynamic factors was addressed by studying monocyte adhesion in an in vitro flow model in the absence of any biological activity in the model wall. Suspensions of U937 cells were perfused ( Re = 200) through an axisymmetric silicone flow model with a stenosis followed by a reverse step. The model provided spatially varying wall shear stress, flow separation and reattachment, and a three-dimensional flow pattern. The cell rolling velocity and adhesion rates were determined by analysis of videomicrographs. Wall shear stress was obtained by numerical solution of the equations of fluid motion. Cell adhesion patterns were also studied in the presence of chemotactic peptide gradients. The cell rolling velocity varied linearly with wall shear stress. The adhesion rate tended to decrease with increasing local wall shear stress, but was also affected by the radial component of velocity and the dynamics of the recirculation region and flow reattachment. Adhesion was increased in the vicinity of chemotactic peptide sources downstream of the expansion site. Results with human monocytes were qualitatively similar to the U937 experiments. Differences in the adhesion rates of U937 cells occurring solely as a function of the fluid dynamic properties of the flow field were clearly demonstrated in the absence of any biological activity in the model wall.

A model for two‐phase turbulent mixing in a jet bubble column

AbstractMixing behavior of the two phase air‐water turbulent flow in a jet bubble column is examined. The time evolution of the mixing behavior of a liquid tracer in a turbulent air‐water flow within a jet bubble column is predicted using a model based on the fundamental governing equations of fluid motion. The predictions of the model are compared with experimental measurements. Measured residence time distributions (RTD) of the liquid tracer within the cone agree well with the predicted values given by the model. For the range of parameters considered in the study, lack of radial mixing and large axial mixing are evident within the cone of the jet bubble column. Use of fundamental mathematical models for the study of hydrodynamics in a two‐phase conventional bubble column has been reported earlier (Torvik, 1990; Jakobsen et al., 1993). The present paper extends the use of such models to predict the mixing characteristics in a jet bubble column.

TRANSPORT OF GAS-LIQUID FLOWS THROUGH VERTICAL COLUMNS

Abstract Applicability of a theoretical model, based on the fundamental governing equations of fluid motion is investigated to predict two-phase bubbly air-water flow structure through vertical columns. The model predictions are compared to experimental data for a wide range of flow parameters. The relative importance of various modes of interfacial momentum transfer, under different flow conditions, is also examined. It is found that correct estimation of the interfacial momentum transfer is necessary for agreement between the predictions and experimental data. The present study shows that the flow structure for air-water flows through vertical columns depends on the inlet flow conditions. This model predicts the experimentally observed trends well. In most cases the experimental data of local liquid velocity and gas volume fraction agree well with the calculated values.

Statistical mechanics of point vortices.

Thermodynamical functions and probability distributions are found in an explicit form for a multicomponent vortex gas. A key point in the derivation is the interpretation of integration in phase space as an average, with respect to some ``complex measure.'' The exact formulas for the probability distribution of a vortex gas are used to obtain the averaged equations for two-dimensional fluid motion. The exact formulas for thermodynamical functions help to clear up the relation between various definitions of entropy and temperature, which, in contrast to the classical models of statistical mechanics, are not equivalent for a vortex gas.

Solid rocket motor internal flow during ignition

A numerical procedure is presented for the analysis of the internal flow in a solid rocket motor (SRM) during the ignition transient period of operation, along with the results obtained when this computer code was applied to several motors. The purpose of this code development effort was to achieve a detailed picture of the unsteady flowfield for a SRM of arbitrary design during this period of ignition delay, propellant ignition, flame spreading, and chamber filling/pressurization. The approach was to combine an unsteady, axisymmetric solution of the equations of inviscid fluid motion (Euler equations) with simple models for the convective and radiative heat transfer to the propellant surface during the run up to ignition. An unsteady, one-dimensional heat conduction solution for the propellant grain is coupled to this unsteady flow solution in order to calculate the propellant surface temperature. This solution, together with a surface temperature ignition criterion, determines the ignition delay and flame spreading. First, data were used from a Titan 5-1/2-segment solid rocket motor static firing to fix an unknown constant in the heat transfer model. Then, the computer code was applied to two solid rocket motors, Titan 7-segment and Space Shuttle, for which time-dependent chamber pressure measurements weremore » available from static firings. Good agreement with the data was obtained. 17 refs.« less

A Nonlinear Simulation Method of 3-D Body Motions in Waves (1st Report)

A full nonlinear method to simulate three dimensional body motions in waves is presented. This is a time domain method to simulate Euler's equation of ideal fluid motion coupled with the equation of solid body motions.Introducing Prandtl's nonlinear acceleration potential, whose gradient gives acceleration of the fluid, Euler's differential equation of the ideal fluid motion is converted to the integral equation of the acceleration potential. The boundary condition of the acceleration potential on the body surface is systematically derived from the kinematic relation between the acceleration of the solid body and the acceleration of the fluid on the body surface. Since this kinematic boundary condition is a function of the body acceleration, the boundary values on the floating body can not be evaluated explicitly. To overcome this point, the unknown acceleration of the free floating body is eliminated by substituting the equation of body motion into kinematic condition, then implicit body surface boundary condition is derived. This is the kinematic and dynamic condition which guarantees dynamic equilibrium of forces between ideal fluid and the solid body at any instance. With the free-surface boundary condition of the acceleration potential, the formulation of the boundary value problem for the acceleration field is completed.Although this formulation of the acceleration field is mathematically correct, this is not appropriate to numerical computation, because Prandtl's nonlinear acceleration potential does not satisfy Laplace's equation. Therefore, the nonlinear part is shifted from the governing equation to the boundary condition, then the alternative formulation for the numerical computation is derived. The computational flow of the nonlinear simulation method based on this alternative formulation is also given. In order to show the accuracy of this new method, two dimensional numerical results are presented. They show that the conservation of mass, momentum and energy are satisfied excellently.

An investigation and comparative analysis of two- and three-dimensional turbulent natural convection in a horizontal annulus

In the present study, simulation of turbulent buoyancy-driven flow in an annulus bounded by concentric, horizontal cylinders and adiabatic end walls has been carried out. Time-averaged equations of turbulent fluid motion and heat transfer were solved using a wall function approach coupled with the standard k- ε model. Discretization of the governing equations was achieved using a finite element scheme based on the Galerkin method of weighted residuals. Using a two-dimensional analysis, results were obtained for Rayleigh numbers ranging from 10 6 to 10 9 and the effects of Prandtl number and radius ratio on the flow and heat transfer characteristics were thoroughly examined. A wide range of parameters (10 6 < Ra < 10 9, 0.01 < Pr < 5000, 1.5 < R < 11) was considered in the present study. A comprehensive comparative analysis establishing a unified treatment of previous experimental and computational results is presented. A good agreement was found between the results from this investigation and previous works. Results obtained from a three-dimensional model are also presented to describe more realistic three-dimensional flow characteristics. It is shown that, if the annulus is sufficiently long, there exists a core region over a substantial length of the cavity, which can be approximated by a two-dimensional model.

Alternative Formulation of Laterally Averaged Hydrodynamics

Longitudinal and vertical hydrodynamics and transport are described from the laterally averaged equations of fluid motion as shown by Blumberg in 1977, Edinger and Buchak in 1978 and 1980, and by Wang and Kravitz in 1980. The equations not only allow examining the vertical distribution of flows, but also allow examining flow properties introduced by buoyancy. Longitudinal and vertical hydrodynamics and transport apply to reservoirs and estuaries that are relatively narrow in comparison to their length and that are sufficiently deep so that stratified conditions can be established. An alternative formulation of the laterally averaged hydrodynamic and transport relationships is presented in terms of longitudinal and vertical fluxes rather than velocities and their products with various geometric averages. The formulation eliminates much of the spatial averaging of the geometry and leads to a basis for computations that more readily obtain perfect mass balances when developed to the numerical forms of the equations.

An algebraic multigrid solver for Navier‐Stokes problems

AbstractAn efficient numerical method is presented for solving the equations of motion for viscous fluids. The equations are discretized on the basis of unstructured finite element meshes and then solved by direct iteration. Advective fluxes are temporarily fixed at each iteration to provide a linearized set of coupled equations which are then also solved by iteration using a fully implicit algebraic multigrid (AMG) scheme. A rapid convergence to machine accuracy is achieved that is almost mesh‐independent. The scaling of computing time with mesh size is therefore close to the optimum.

Characteristics of the linearized equations of motion for a compressible fluid

The characteristics and bicharacteristics of the linearized equations of motion for compressible fluid are identical to the wave fronts and rays, respectively, calculated by Fermat’s least-time principle of geometric acoustics. The bicharacteristics emanating from the middle of a compressible shear layer curve so as to hinder communication in the axial direction.

Mean wind pressures on arched-roof buildings by computation

The calculation of mean pressure coefficients for buildings with various arched-roof configurations is described. The time-averaged equations of fluid motion in three dimensions are solved, with the effect of turbulence incorporated through the k-ε turbulence model. The geometric variables that affect the wind pressures on arched-roof buildings are the span, the rise, the length along the axis of the arch, and the height of the vertical walls below the arch. The computed mean pressure coefficients increase as the length/span ratio increases when the approach wind direction is normal to the axis of the arch. The pressure coefficients on the roof become more positive as the rise/span ration increases.

Variational derivation of the equations of motion for a perfect fluid in a rotating cylinder

The relativistic Euler equations for a compressible perfect fluid are derived from a variational principle based on angular momentum flux. Both the stationary and time-dependent cases are considered, and the variational principle is satisfactory in the sense that it is based on Eulerian variables. A comment is made for dissipative fluids.

Analysis of Flow Induced Vibration Using the Vorticity Transport Equation

Crossflow induced vibrations are the subject of this work. The analysis is two dimensional. The governing equations for fluid motion are solved using linearized perturbation theory and coupled with the equations of motion for cylinders to yield the threshold of dynamic instability for an array of cylinders. Parametric analysis is performed to determine the lowest instability threshold for a rotated square array and correlations are developed relating the dominant parameters. The results are compared with theoretical and experimental data for similar arrays and the discrepancies are discussed.