Entire Flow Domain Research Articles

Video‐imaged surf zone wave and roller structures and flow fields

The measurement of mean water levels, roller geometries, and phase ensemble‐averaged velocity and turbulence intensity fields under spilling and plunging waves breaking in a two‐dimensional laboratory surf zone is presented. The velocities were measured using digital correlation image velocimetry, while water levels and roller geometries were determined through gray scale filtering of video images. The phase ensemble‐averaged horizontal and vertical components of velocity and turbulence intensities are measured throughout the entire flow domain, including the wave roller area, by utilizing the aerated areas as part of the flow structure. The time‐averaged horizontal velocities (undertow), turbulence intensities, and turbulent kinetic energies are determined by averaging across the wave phase. Turbulence magnitudes are found to compare favorably with existing laser Doppler anemometry measurements below the wave trough level, where such measurements have generally been confined because of aeration contamination effects. The significantly higher velocity and turbulence intensity magnitudes measured above the trough level in the present experiments highlight the novel nature of the present investigation for describing flow regimes in the surf zone.

Computational analysis of regular and chaotic mixing in a stirred tank reactor

Mixing in an unbaffled stirred tank equipped with a 6-blade radial flow impeller is examined computationally. Results demonstrate that the flow generated by constant impeller speed is partially chaotic. Under these conditions, the stretching experienced by fluid elements located within segregated toroidal regions of the flow increases slowly at a linear rate characteristic of regular (non-chaotic) flows. Within the bulk flow region, since the flow is chaotic, stretching increases at the expected exponential rate. The use of dynamic flow perturbations enhances mixing; when time-dependent RPM are applied, a globally chaotic flow is generated. We investigate computationally the effect on mixing performance of protocols in which the agitation speed oscillates between two steady values. Different frequencies of speed change and several RPM settings are compared. Under variable RPM schemes, the segregated torii are periodically relocated and fluid elements have the opportunity to abandon the regular regions. Under these conditions stretching increases exponentially throughout the entire flow domain. In general, mixing protocols with higher frequency of speed fluctuation produce the largest increase in stretching rates. Counter-intuitively, at a given RPM fluctuation frequency, stretching rates were higher for the lower RPM settings, either per revolutions or per unit of energy spent.

UNSTEADY FREE CONVECTION IN A VERTICAL CHANNEL WITH A BUILT-IN HEATED SQUARE CYLINDER

The study of natural convection past a square cylinder placed centrally within a vertical parallel plate channel has been carried out numerically using the MAC method with higher-order spatial as well as temporal discretization. The velocities are nondimensionalised using a reference velocity √gβ(T w - T∞)B. The zero Neumann boundary condition for the velocity in the direction of gravity along with vanishing transverse velocity successfully has been implemented at the inlet. The flow is found to be unstable when the Grashof number crosses the critical value of 3.0 × 10 4 . The drag coefficient on the cylinder is found to decrease while the Strouhal number ([S] nondimensional frequency) increases with the increase in Grashof number (Gr). The surface-averaged Nusselt number is seen to increase linearly with Gr. The surface-averaged Nusselt number for a square cylinder consistently shows a higher value than that for a circular cylinder, and the percentage change in heat transfer rate increases significantly when the flow becomes unsteady. The vertical velocity and temperature are correlated highly over the entire flow domain except in the recirculation region

Multispectral image analysis method to determine dynamic fluid saturation distribution in two-dimensional three-fluid phase flow laboratory experiments

The need for measuring dynamic fluid saturation distribution in multi-dimensional three-fluid phase flow experiments is hampered by lack of appropriate techniques to monitor full field transient flow phenomena. There is no conventional technique able to measure dynamic three-fluid phase saturation at several array points of the flow field at the same time. A multispectral image analysis technique was developed to determine dynamic NAPL, water and air saturation distribution in two-dimensional three-fluid phase laboratory experiments. Using a digital near-infrared camera, images of sand samples with various degrees of NAPL, water and air saturation were taken, under constant lighting conditions and within three narrow spectral bands of the visible and near-infrared spectrum. It was shown that the optical density defined for the reflected luminous intensity was a linear function of the NAPL and the water saturation for each spectral band and for any two and three-fluid phase systems. This allowed the definition of dimensionless lump reflection coefficients for the NAPL and the water phase within each spectral band. Consequently, at any given time, two images taken within two different spectral bands provided two linear equations which could be solved for the water and the NAPL saturation. The method was applied to two-dimensional three-phase flow experiments, which were conducted to investigate the migration and the distribution of LNAPL in the vadose zone. The method was used to obtain continuous, quantitative and dynamic full field mapping of the NAPL saturation as well as the variation of the water and the air saturation during NAPL flow. The method provides a non-destructive and non-intrusive tool for studying multiphase flow for which rapid changes in fluid saturation in the entire flow domain is difficult to measure using conventional techniques.

Symmetry of periodic structures in a 3D mixing cavity flow

Symmetry concepts are applied to analyze three-dimensional time-periodic mixing Stokes flows in a cubic domain. Usage of simple map algebra, developed by Franjione et al. [Phys. Fluids A 11, 1772–1783 (1989); Philos. Trans. R. Soc. London, Ser. A 338, 301–323 (1992)], allows us to reveal the symmetry in the arrangement of periodic points in such a flow. Symmetries reveal essential features of the underlying physics, and with manipulation of symmetries mixing protocols can be made efficient over the entire flow domain.

Field observation of flow in a fracture intersecting unsaturated chalk

Flow through a natural fracture crossing unsaturated chalk in an arid region was investigated in a field experiment using a specially designed experimental setup. The setup allowed complete control of the flow domain inlet and outlet. Water flux into and out of the fracture was measured in small segments of the fracture openings, and flow trajectories were identified using seven fluorobenzoic acid tracers. A 5 day percolation experiment on a 5.3 m long fracture showed significant spatial and temporal variability of the flow regime. Flow through fracture openings did not reach a steady state either in individual segments or across the entire flow domain, although the boundary conditions were kept relatively steady for the entire duration of the experiment. Flow trajectories within the fracture plane varied over time; however, most of the flow was confined to small sections of the fracture. Over 70% of the flux was transmitted through <20% of the studied fracture openings. Observations from the tracer tests suggest that flow paths can coexist near each other without water mixing, probably because the fracture fill generates unconnected flow paths across the main fracture void.

A Domain Decomposition Approach for Incompressible Flows Past Multiple Objects

A domain decomposition approach has been developed to solve for flow around multiple objects. The method combines features of mask and multigrid algorithm implemented within the general framework of a primitive variable, pseudospectral elements formulation of fluid flow problems. The computational domain consists of a global rectangular domain, which covers the entire flow domain, and local subdomains associated with each object, which are fully overlapped with the rectangular domain. There are two key steps involved in calculating flow past multiple objects. The first step approximately solves the flow field by the mask method on the Cartesian grid alone, including on those grid points falling inside an object (a fuzzy boundary between the fluid-object interface), but with the restriction that the velocity on grid points within and on the surface of an object should be small or zero. The second step corrects the approximate flow field predicted from the first step by taking account of the object surface, i.e., solving the flow field on the local body-fitted (curvilinear) grid surrounding each object. A smooth data communication between the global and local grids can be implemented by the multigrid method when the Schwarz Alternating Procedure (SAP) is used for the iterative solution between the two overlapping grids. Numerical results for two-dimensional test problems for flow past elliptic cylinders are presented in the paper. An interesting phenomenon is found that when the second elliptic cylinder is placed in the wake of the first elliptic cylinder a traction force (a negative drag coefficient) acting on the second one may occur during the vortex formation in the wake area of the first one.

Grid-free simulation of radiative transport in a participating medium

Radiative transport in a participating medium transfroms the energy equation into an integro-dierential equation which must be solved if radiation is considered as an important mechanism of heat transfer in a convective-diusive-reactive environment. In most cases, especially in combustion problems, the temperature gradients are so steep that the region where the radiative flux, which is proportional to the fourth power of temperature, is signicant in a small subset of the entire flow domain. Especially when utilizing a grid-free Lagrangian scheme to perform the convective-diusive-reactive simulations, it is desirable to develop a compatible scheme which does not require a grid to perform the computations of the radiative flux and its gradient (needed in the energy equation.) We have developed the discrete source method in which the radiative flux and irradiance are computed by summing over a collection of radiating elements distributed over the region of high temperature. In an axisymmetric domain, the irradiance and radiative flux, both are triple integrals of the radiation kernel over the elemental volume, are reduced to double integrals which are then approximated by single integrals that are evaluated semi-analytically. We demonstrate the accuracy of the scheme by comparing the results with direct numerical evaluation of the original integrals for cases relevant to combustion problems.

Inviscid separation in steady planar flows

This paper addresses the unsolved classical problem of the indeterminacy of steady, inviscid, incompressible vortical planar flows due to inviscid separation. We prove that any inviscid steady flow is “totally nonunique” in the following sense: there are a continuum of possible solutions involving recirculation zones that can be always inserted into such a flow satisfying the same boundary conditions. For sufficiently large inflow vorticity (a flow past a smoothing screen gives an example of inflow vorticity generation), the formation of a separation zone is inevitable. Various additional phenomenological conditions to achieve uniqueness and to determine the flow in the separation zone are evaluated, including the frequently used analytic continuation method. We reveal a new ‘measure paradox’ in inviscid hydrodynamics: as inflow of discontinuous vorticity increases, one streamline transforms into a stagnation zone of finite area. We focus on this new steady stagnation zone flow model with continuous velocity distribution. We prove existence and uniqueness theorems for stagnant separation zones and show that the kinetic energy of the entire flow domain is minimized when it contains a separation zone which is stagnant. We consider vortical channel flows with different inflow vorticity distributions and show that a totally quiescent stagnation zone model occurs in the limit as the inflow vorticity becomes discontinuous. For a smooth, but large-gradient vorticity distribution a quasi-stagnant zone occurs. It is shown that the analytic continuation model applied to an irrotational channel flow with a slender vortical core (the planar analog of the Rankine vortex) yields “vorticity breakdown” – a phenomenon somewhat similar to vortex breakdown in swirling flows. Nonuniqueness and hysteresis-related multiple solutions are inherent to those solutions with recirculation zones where very large (physically unrealistic) reversed flow appears. Being free from these drawbacks, the stagnation zone model appears to be superior. A numerical investigation of a vortical flow in a square domain confirms the existence of a critical value of inflow vorticity, leading to steady or permanently unsteady flow regimes.

Capillary flow in an interior corner

The design of fluids management processes in the low-gravity environment of space requires an accurate description of capillarity-controlled flow in containers. Here we consider the spontaneous redistribution of fluid along an interior corner of a container due to capillary forces. The analytical portion of the work presents an asymptotic formulation in the limit of a slender fluid column, slight surface curvature along the flow direction z, small inertia, and low gravity. The scaling introduced explicitly accounts for much of the variation of flow resistance due to geometry and so the effects of corner geometry can be distinguished from those of surface curvature. For the special cases of a constant height boundary condition and a constant flow condition, the similarity solutions yield that the length of the fluid column increases as t1/2 and t3/5, respectively. In the experimental portion of the work, measurements from a 2.2 s drop tower are reported. An extensive data set, collected over a previously unexplored range of flow parameters, includes estimates of repeatability and accuracy, the role of inertia and column slenderness, and the effects of corner angle, container geometry, and fluid properties. At short times, the fluid is governed by inertia (t[lsim ]tLc). Afterwards, an intermediate regime (tLc[lsim ]t[lsim ] tH) can be shown to be modelled by a constant-flow-like similarity solution. For t[ges ]tH it is found that there exists a location zH at which the interface height remains constant at a value h(zH, t)=H which can be shown to be well predicted. Comprehensive comparison is made between the analysis and measurements using the constant height boundary condition. As time increases, it is found that the constant height similarity solution describes the flow over a lengthening interval which extends from the origin to the invariant tip solution. For t[Gt ]tH, the constant height solution describes the entire flow domain. A formulation applicable throughout the container (not just in corners) is presented in the limit of long times.

Forced convection from a straight elliptical tube

The paper deals with the problem of two-dimensional laminar forced convection heat transfer from a straight isothermal tube of elliptic cross-section placed in a uniform stream. The study is based on numerical solutions of the conservation equations of mass, momentum, and energy which covers the entire flow domain including the wake region. The parameters influencing the heat transfer process are essentially the Reynolds number, Re, the tube geometry represented by its minor to major axis ratio, Ar, and the angle of inclination, λ. The study focuses on the effect of Re, Ar, and λ on the heat transfer process in the range of Re from 20 to 500. The study reveals that the rate of heat transfer reaches its maximum when λ=0∘ while the minimum occurs when λ=90∘. The results also show that smaller axis ratio gives higher heat transfer rate when λ=0∘. The local Nusselt number and surface vorticity distributions are plotted for a number of cases and the effect of vortex shedding on the overall rate of heat transfer is briefly discussed.

Miscible porous media displacements in the quarter five-spot configuration. Part 2. Effect of heterogeneities

Direct numerical simulations are employed to investigate the coupling between the viscous fingering instability and permeability heterogeneities for miscible displacements in quarter five-spot flows. Even moderate inhomogeneities are seen to have a strong effect on the flow, which can result in a complete bypass of the linear growth phase of the viscous fingering instability. In contrast to their homogeneous counterparts (cf. Part 1, Chen & Meiburg 1998), heterogeneous quarter five-spot flows are seen to exhibit a more uniform dominant length scale throughout the entire flow domain. In line with earlier findings for unidirectional displacements, an optimal interaction of the mobility and permeability related vorticity modes can occur when the viscous length scale is of the same order as the correlation length of the heterogeneities. This resonance mechanism results in a minimal breakthrough recovery for intermediate correlation lengths, at fixed dimensionless flow rates in the form of a Péclet number Pe. However, for a constant correlation length, the recovery does not show a minimum as Pe is varied.Confirming earlier observations, the simulations show a more rapid breakthrough as the variance of the permeability variations increases. However, this tendency is far more noticeable in some parameter regimes than in others. It is furthermore observed that relatively low variances usually cannot change the tendency for a dominant finger to evolve along the inherently preferred diagonal direction, especially for relatively small correlation lengths. Only for higher variances, and for larger correlation lengths, are situations observed in which an off-diagonal finger can become dominant. Due to the nonlinear nature of the selection mechanisms at work, a change in the variance of the heterogeneities can result in the formation of dominant fingers along entirely different channels.

Numerical and analytical investigation of vapor flow in a disk-shaped heat pipe incorporating secondary flow

This paper presents a three-dimensional numerical analysis and a pseudo-three-dimensional analytical modeling of the steady incompressible vapor flow in an asymmetrical disk-shaped heat pipe heated from the top center area. The nonlinear differential elliptical equations of motion and the continuity equation were solved numerically over the entire vapor flow domain. Discretization of the governing equations was achieved using a finite element scheme based on the Galerkin method of weighted residuals. The analytical model involves the use of the boundary layer approximation and the bifurcation of the flow field on the plane to describe the velocity profile under conditions including strong flow reversal. For both numerical and analytical studies, backflow was observed at the top entrance of the condensation zone for injection Reynolds number of 50 and higher. The three-dimensional effects and the effects of the secondary flow formation are discussed in this work. The numerical and analytical results establish that the pressure variations in the angular and transverse directions for a typical disk-shaped heat pipe are small and can be neglected. A very good agreement was found between the numerical results and the analytical results. The analytical model saves tremendous computer time compared with the numerical simulation, which requires CPU times five orders of magnitude larger than those for the analytical model.

Solution-adaptive Cartesian cell approach for viscous and inviscid flows

A Cartesian cell-based approach for adaptively refined solutions of the Euler and Navier-Stokes equations in two dimensions is presented. Grids about geometrically complicated bodies are generated automatically, by the recursive subdivision of a single Cartesian cell encompassing the entire flow domain. Where the resulting cells intersect bodies, polygonal cut cells are created using modified polygon-clipping algorithms. The grid is stored in a binary tree data structure that provides a natural means of obtaining cell-to-cell connectivity and of carrying out solution-adaptive mesh refinement. The Euler and Navier-Stokes equations are solved on the resulting grids using a finite volume formulation. The convective terms are upwinded: A linear reconstruction of the primitive variables is performed, providing input states to an approximate Riemann solver for computing the fluxes between neighboring cells. The results of a study comparing the accuracy and positivity of two classes of cell-centered, viscous gradient reconstruction procedures is briefly summarized. Adaptively refined solutions of the Navier-Stokes equations are shown using the more robust of these gradient reconstruction procedures, where the results computed by the Cartesian approach are compared to theory, experiment, and other accepted computational results for a series of low and moderate Reynolds number flows.

Numerical Flow Simulation of Viscoplastic Slurries and Design Criteria for a Tape Casting Unit

Numerical simulations have been carried out using the finite element method (FEM) for the forming flow of ceramic tapes. The flow domain encompasses both the slurry reservoir and the doctor‐blade region with free surface and is fully two‐dimensional. The material of this study is an organic‐bonded alumina slurry used in a previous experimental investigation and is modeled as a viscoplastic Bing‐ham fluid with a yield stress. For different substrate speeds, the entire flow domain is analyzed and the extent and shape of yielded/unyielded regions are found. The computed free surface profiles are in close agreement with the experimental ones. Large vortices appear in the reservoir and their intensity is computed under different operating conditions. The lubrication approximation theory (LAT) is then used to deduce a new reservoir design with tapered walls avoiding recirculation. Subsequent FEM studies show the adequacy of the new design to achieve certain criteria such as elimination of vortices and an evenly distributed unyielded region. It is proposed that a combination of LAT and FEM techniques be used for design of ceramic tape casting equipment.

Stability of coextrusion through converging dies

A linear stability analysis of multilayer flow of viscoelastic liquids through long, converging dies is performed by a rigorous two-dimensional analysis (2-D), as well as by a simplified one-dimensional analysis (1-D) on selected cross-sections along the die. The rheological behavior of liquids is represented by a modified Oldroyd-B model with a shear-rate dependent viscosity, described by the Carreau viscosity function. In the 2-D analysis, a streamlined Galerkin/finite-element method is used to discretize the domain. The resulting asymmetric generalized eigenvalue problem is large (of the order of 3000–8000), sparse, and banded, with a singular mass matrix. The leading eigenvalues of this complex problem are computed by using an iterative Arnoldi's algorithm, modified by Schur-Weilandt deflation, complex shift, and exponential preconditioning. With these series of modifications, the algorithm is now sufficiently flexible to solve any application that belongs to a generic class of large hydrodynamic stability problems. The effect of the die geometry on the neutral stability curves is investigated for various operating conditions and rheological parameters. In all the investigated cases, the critical flow-rate ratios in long converging channels are found to be independent of the shapes, and of the ratios of the thicknesses at the inlet to the outlet of the die. These results agree well with the approximate, simplified 1-D analysis, indicating that the most dangerous instability is at the inlet of the die. Thus, the analysis of the entire two-dimensional flow domain is unnecessary, at least for long channels, except for validating the 1-D analysis. The results also indicate that mesh-in-dependent eigensolutions cannot be obtained using the 2-D analysis when leading eigensolutions exist at large wavenumbers.

Evaluation of the drag force by integrating the energy dissipation rate in stokes flow for 2D domains using the FEM

AbstractThe total drag force on the surface of a body, which is the sum of the form drag and the skin friction drag in a 2D domain, is numerically evaluated by integrating the energy dissipation rate in the whole domain for an incompressible Stokes fluid. The finite element method is used to calculate both the energy dissipation rate in the whole domain as well as the drag on the boundary of the body. The evaluation of the drag and the energy dissipation rate are post‐processing operations which are carried out after the velocity field and the pressure field for the flow over a particular profile have been obtained. The results obtained for the flow over three different but constant area profiles—a circle, an ellipse and a cross‐section of a prolate spheroid—with uniform inlet velocity are presented and it is shown that the total drag force times the velocity is equal to the total energy dissipation rate in the entire finite flow domain. Hence, by calculating the energy dissipation rate in the domain with unit velocity specified at the far‐field boundary enclosing the domain, the drag force on the boundary of the body can be obtained.

Coupled space-marching method for the Navier-Stokes equations for subsonic flows

An implicit space-marching finite-difference procedure is described for solving the compressible form of the steady, two-dimensional Navier-Stokes equations in body-fitted curvilinear coordinates. The coupled system of equations is solved for the primitive variables of velocity and pressure by making multiple sweeps of the space-marching procedure. A new pressure correction method has been developed that significantly accelerates convergence of the iterative process. The scheme is used to compute incompressible flows by taking the low Mach number limit of the compressible formulation. Computed results are compared with other numerical predictions for low Reynolds number channel inlet flow, flow over a rearward-facing step in a channel, and external flow over a cylinder. NUMBER of different methods have been developed to solve the Navier-Stokes equations in primitive variables for steady, subsonic flows. Many of these algorithms are described in Anderson et al. 1 The present methods are not entirely adequate for all problems, and advanced techniques continue to be developed. New procedures and recent im- provements to existing methods are briefly reviewed in the following paragraphs. The most frequently used primitive variable algorithms for incompressible flows solve the momentum equations for the velocity components in an uncoupled (segregated) manner, holding pressure fixed. Although different in detail, the segre- gated solution schemes (e.g., Refs. 2-4) use the continuity equation indirectly in the formulation of a separate Poisson equation that is solved for the pressure. Since the velocity components and pressure are computed from different algo- rithms, rather than in a coupled manner, convergence is slowed. Furthermore, the solution of the elliptic Poisson equa- tion consumes a large part of the total computaion time for each global iteration. Van Doormaal et al.5 recently have eval- uated improved algorithms for solving the pressure Poisson equation, and Rhie4 has applied a multigrid method to acceler- ate the solution of the momentum equations and the pressure equation. Several strategies have been advanced for solving the cou- pled momentum and continuity equations for the primitive variables, including the pressure. The schemes all share the advantage that a separate procedure for imposing the continu- ity constraint and determining the pressure is not required. The choice of variables and the algebraic approach used to solve the system of nonlinear equations distinguish the direct, time- marching, and space-marching algorithms. Vanka and Leaf6 and recently Patankar et al. 7 have com- pared solutions for two-dimensional flows obtained by direct and segregated methods. A large sparse matrix solver is used to invert a system of equations spanning the entire flow domain.

Symmetries within chaos: A route to effective mixing

Experimental studies show that simple two-dimensional time-periodic flows can produce chaotic mixing. However, the mixing is not always complete; depending on the choice of the period, there can exist large dynamic structures, called islands, that stretch and compress in a time-periodic manner but remain segregated even after long times. Obviously, a flow that contains very few islands is desired. Unfortunately, in most real systems, an analytic expression for the velocity field (or the motion) does not usually exist, making theoretical prediction of the location and size of islands virtually impossible. However, using only minimal knowledge of the gross properties of the velocity field, the flow can be analyzed in terms of its symmetries. Symmetries can be detected without reference to precise mathematical equations. Large islands are located on lines of symmetry, or in pairs on opposite sides of the line. With this knowledge, it is possible to manipulate symmetries in a systematic way so as to move an island into a region of good mixing. Applying this idea repeatedly results in a recursively generated flow history that is neither periodic nor random, but is self-similar. Mixing in these flows is efficient over the entire flow domain. In addition, the idea of recursively generated flow histories might have implications for understanding the mechanisms of turbulent flow.

A Study of Flow Through a Fracture Network Described by Statistically Distributed Properties

An analysis of the confidence of flow solutions for stochastically generated hard rock formations study was carried out with the aid of a simplified synthetic model. The formation was conceptualized as a fracture network with a known geometric structure intersecting an impervious mass rock while fracture permeability was considered a stochastic process. Safety analysis of radioactive waste repositories includes prediction of travel times of possibly contaminated water particles from the repository to the biosphere. While such calculations require that rock properties, such as permeability, be known over the entire flow domain, only limited information is available in practice, and interpolation methods are called for. An a priori model was constructed as a first step, with fracture permeabilities generated according to a given probability distribution; this a priori model was considered the ''true'' formation. In a second step, a limited amount of information, similar to that obtained in reality from boreholes, was used to construct a conditioned-by-measurement model. Identical flow tests were performed on formations represented by the two models, and the flow rate ratios resulting from these tests served as the measure of confidence of the stochastically generated formation.