The numerical solution of the transient two‐phase flow in rigid pipelines

Consideration is given in this paper to the numerical solution of the transient two‐phase flow in rigid pipelines. The governing equations for such flows are two coupled, non‐linear, hyperbolic, partial differential equations with pressure dependent coefficients. The fluid pressure and velocity are considered as two principle dependent variables. The fluid is a homogeneous gas–liquid mixture for which the density is defined by an expression averaging the two‐component densities where a polytropic process of the gaseous phase is admitted. Instead of the void fraction, which varies with the pressure, the gas–fluid mass ratio (or the quality) is assumed to be constant, and is used in the mathematical formulation. The problem has been solved by the method of non‐linear characteristics and the finite difference conservative scheme. To verify their validity, the computed results of the two numerical techniques are compared for different values of the quality, in the case where the liquid compressibility and the pipe wall elasticity are neglected. Copyright © 1999 John Wiley & Sons, Ltd.     

Numerical Flow Simulation for Bingham Plastics in a Single-Screw Extruder

Numerical simulations have been performed concerning the operation of a single-screw extruder, pumping a Bingham plastic under isothermal, developed flow conditions. Under the assumption of sufficiently low Reynolds numbers, inertia effects are neglected. The singular rheological behavior of the Bingham plastic is considered as the limiting case within a class of generalized Newtonian liquids with smooth constitutive equations. The validation of this regularization process is shown for a related flow problem where the Bingham solution is known analytically. A mixed finite-element method is applied to the flow in the screw-extruder to reduce the equations of motion, the continuity equation, and the regularized constitutive equation to a set of nonlinear algebraic equations, which are solved using a Newton method. In particular, the pumping characteristics of a given screw geometry are extracted from the finite-element calculations, i.e., the dependence of the volumetric flow rate and of the power requirement on the axial pressure drop, on the screw speed, and on the rheological parameters. Calculated flow fields clearly show the size and position of regions in the extruder channel where the Bingham plastic behaves like a solid. Received: 12 December 1995 and accepted 12 November 1996     

Transient natural convection along vertical cylinder with Heat and Mass transfer

A numerical solution for the transient natural convection flow over a vertical cylinder under the combined buoyancy effect of heat and mass transfer is presented. The velocity, temperature and concentration profiles, local and average skin-friction, Nusselt number and Sherwood number are shown graphically. It is observed that time taken to reach steady state increases with Schmidt number and decreases as combined buoyancy ratio parameter N increases. Stability and convergence of the finite difference scheme are established. Received on 8 July 1997     

THE TRANSIENT ELLIPTIC FLOW OF POWER-LAW FLUID IN FRACTAL POROUS MEDIA

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Experimental and numerical heat transfer investigation of an impingement jet array with V-ribs on the target plate and on the impingement plate

The secondary vortex structure of an impingement jet system is enhanced by V-ribs on both the impingement and target plates. Numerical and experimental investigations are conducted to study the flow field and heat transfer resulting from V-rib turbulators in an impingement cooling configuration. Three different cases are tested: V-ribs on both the impingement and target plates (V-rib), V-ribs only on the impingement plate (V-rib-impingement) and V-ribs only on the target plate (V-rib-target). The experiment is carried out on a 9 by 9 inline impingement array test facility. For the transient measurements, narrow band thermochromic liquid crystals (TLC) and thermocouples are applied to obtain the local heat transfer distribution. Pressure taps are used to measure the pressure loss. The numerical simulation is carried out with ANSYS CFX 14, using a steady state Reynolds-Averaged Navier-Stokes (RANS) approach and the Shear Stress Transport (SST) turbulence model. All studies are done for a Reynolds number range of 15,000 to 35,000. There is a good overall agreement between the experimental and numerical results for the cases studied. The detailed flow field from the numerical simulation is used to understand and complement the phenomena observed in the experiment. The evaluation of the flow field confirms that the V-ribs enhance the secondary flow structure in the impingement system and induce a positive heat flux ratio compared to the baseline case. Both experimental and numerical results show a Nusselt number increase for the V-rib-impingement and V-rib configuration, with a highest Nusselt number ratio of 1.16. Notice that the experiment cannot take the rib part into account due to the invalid 1D semi-infinite wall assumption there, while the CFD simulation allows for the consideration of heat transfer on the rib surface and thus complements the heat flux data on the target plate. Depending on the configuration, the CFD simulation shows a heat flux ratio of 1.06–1.34. The pressure loss of the system is comparable to the case with a smooth impingement plate and a smooth target plate.     

Effects of Temperature-Dependent Viscosity on Forced Convection Inside a Porous Medium

Considering the exponential viscosity–temperature relation, effect of temperature-dependent viscosity on forced convection of a liquid through a porous medium, bounded by isoflux parallel plates, is investigated numerically based on the general model of momentum transfer. Local effects of viscosity variation on the distribution of velocity and temperature are analyzed. Moreover, global aspects of the problem are investigated where corrections are proposed for total pressure drop and the fully developed Nusselt number, in the form of out/in viscosity ratio. Results are obtained over a wide range of permeabilities from clear (of solid material) fluid to very low permeability, where for constant properties one expects a nearly slug flow.     

Finite-element analysis of fluid flow and heat transfer for staggered bundles of cylinders in cross flow

The two-dimensional Navier-Stokes equations and the energy equation governing steady laminar incompressible flow are solved by a penalty finite-element model for flow across finite depth, five-row deep, staggered bundles of cylinders. Pitch to diameter ratios of 1·5 and 2·0 are considered for cylinders in equilateral triangular and square arrangements. Reynolds numbers studied range from 100 to 400, and a Prandtl number of 0·7 is used. Velocity vector fields, streamline patterns, vorticity, pressure and temperature contours, local and average Nusselt numbers, pressure and shear stress distributions around the cylinder walls and drag coefficients are presented. The results obtained agree well with available experimental and numerical data.     

A numerical study based on a weakly compressible formulation for thermosolutal convection in vertical cavities

Laminar thermosolutal convection in cavities with uniform, constant temperature and mass fraction profiles at the vertical side is studied numerically. The study is conducted in the case where an inert carrier gas (species “1”) present in the cavity is not soluble in species “2”, and do not diffuse into the walls. A mass flux of species “2” into the cavity occurs at the hot vertical wall and a mass flux out of the cavity occurs at the opposite cold wall. The weakly compressible model proposed in this work was used to investigate the flow fields, and heat and mass transfer in cavities filled with binary mixtures of ideal gases. The dimensionless form of the seven governing equations for constant thermophysical properties, except density, show that the problem formulation involves ten dimensionless parameters. The results were validated against numerical results published in the literature for purely thermal convection, and thermodynamic predictions for transient thermosolutal flows. A parametric study has been performed to investigate the effects of the initial conditions, molecular weight ratio, Lewis number, and aspect ratio of the cavity for aiding or opposing buoyancy forces. For the range of parameters considered, the results show that variations in the density field have larger effects on mass transfer than on heat transfer. For opposing buoyancy forces, the numerical simulations predict complex flow structures and possible chaotic behavior for rectangular vertical cavities according to the value of the molecular weight ratio.     

One-dimensional nonequilibrium air flow

A solution is found for the problem of steady quasi-one-dimensional air flow in a stream tube with nonequilibrium chemical reactions, ionization reactions, and nonequilibrium excitation of the vibrational degrees of freedom in the molecular components. We consider the inverse problem: for a given pressure distribution find the distributions of all the other gas-thermodynamic quantities and the streamtube sections. The use of an implicit scheme for approximating the equations makes it possible to carry out the calculations over the entire range of variation in the degree of nonequilibrium — from the frozen state to equilibrium. We discuss the nature of the variation in temperature, vibrational energies, and component concentrations along the stream tube. A numerical analysis is made of the transition to equilibrium flow.     

Numerical analysis of transient laminar forced convection of nanofluids in circular ducts

In this study, forced convection heat transfer characteristics of nanofluids are investigated by numerical analysis of incompressible transient laminar flow in a circular duct under step change in wall temperature and wall heat flux. The thermal responses of the system are obtained by solving energy equation under both transient and steady-state conditions for hydro-dynamically fully-developed flow. In the analyses, temperature dependent thermo-physical properties are also considered. In the numerical analysis, Al₂O₃/water nanofluid is assumed as a homogenous single-phase fluid. For the effective thermal conductivity of nanofluids, Hamilton–Crosser model is used together with a model for Brownian motion in the analysis which takes the effects of temperature and the particle diameter into account. Temperature distributions across the tube for a step jump of wall temperature and also wall heat flux are obtained for various times during the transient calculations at a given location for a constant value of Peclet number and a particle diameter. Variations of thermal conductivity in turn, heat transfer enhancement is obtained at various times as a function of nanoparticle volume fractions, at a given nanoparticle diameter and Peclet number. The results are given under transient and steady-state conditions; steady-state conditions are obtained at larger times and enhancements are found by comparison to the base fluid heat transfer coefficient under the same conditions.     

On the numerical stability of mixed finite-element methods for viscoelastic flows governed by differential constitutive equations

Mixed finite-element methods for computation of viscoelastic flows governed by differential constitutive equations vary by the polynomial approximations used for the velocity, pressure, and stress fields, and by the weighted residual methods used to discretize the momentum, continuity, and constitutive equations. This paper focuses on computation of the linear stability of the planar Couette flow as a test of the numerical stability for solution of the upper-convected Maxwell model. Previous theoretical results prove this inertialess flow to be always stable, but that accurate calculation is difficult at high De because eigenvalues with fine spatial structure and high temporal frequency approach neutral stability. Computations with the much used biquadratic finite-element approximations for velocity and deviatoric stress and bilinear interpolation for pressure demonstrate numerical instability beyond a critical value of De for either the explicitly elliptic momentum equation (EEME) or elastic-viscous split-stress (EVSS) formulations, applying Galerkin's method for solution of the momentum and continuity equations, and using streamline upwind Petrov-Galerkin (SUPG) method for solution of the hyperbolic constitutive equation. The disturbance that causes the instability is concentrated near the stationary streamline of the base flow. The removal of this instability in a slightly modified form of the EEME formulation suggests that the instability results from coupling the approximations to the variables. A new mixed finite-element method, EVSS-G, is presented that includes smooth interpolation of the velocity gradients in the constitutive equation that is compatible with bilinear interpolation of the stress field. This formulation is tested with SUPG, streamline upwinding (SU), and Galerkin least squares (GLS) discretization of the constitutive equation. The EVSS-G/SUPG and EVSS-G/SU do not have the numerical instability described above; linear stability calculations for planar Couette flow are stable to values of De in excess of 50 and converge with mesh and time step. Calculations for the steady-state flow and its linear stability for a sphere falling in a tube demonstrate the appearance of linear instability to a time-periodic instability simultaneously with the apparent loss of existence of the steady-state solution. The instability appears as finely structured secondary cells that move from the front to the back of the sphere.Financial support for this research was given by the National Science Foundation, the Office of Naval Research, and the Defense Research Projects Agency. Computational resources were supplied by a grant from the Pittsburgh National Supercomputer Center and by the MIT Supercomputer Facility.     

Heat transfer study in rotating cascade using IR thermography and CFD analyses

This paper presents an experimental and CFD numerical study of convective heat transfer in a rotating cascade. Infrared thermography was used to measure surface temperature distribution on a rotating hollow blade, heated internally by secondary air. A CFD numerical model was made according to the actual test rig geometry and operating conditions. Tests were carried out in an iposonic flow regime at relatively low fluid temperatures, with the rotational and Reynolds numbers varied and hot-to-cool air mass flow ratio kept constant. Experimental and numerical results for the blade pressure side are compared in terms of surface temperature 2D distribution and Nusselt number one-dimensional distribution along the blade midspan, providing a reasonable agreement.     

Unsteady numerical simulation of double diffusive convection heat transfer in a pulsating horizontal heating annulus

A numerical study is conducted on time-dependent double-diffusive natural convection heat transfer in a horizontal annulus. The inner cylinder is heated with sinusoidally-varying temperature while the outer cylinder is maintained at a cold constant temperature. The numerical procedure used in the present work is based on the Galerkin weighted residual method of finite-element formulation by incorporating a non-uniform mesh size. Comparisons with previous studies are performed and the results show excellent agreement. In addition, the effects of pertinent dimensionless parameters such as the thermal Rayleigh number, Buoyancy ratio, Lewis number, and the amplitude of the thermal forcing on the flow and heat transfer characteristics are considered in the present study. Furthermore, the amplitude and frequency of the heated inner cylinder is found to cause significant augmentation in heat transfer rate. The predictions of the temporal variation of Nusselt and Sherwood numbers are obtained and discussed.     

Mixed convection boundary layer flow along vertical moving thin needles with variable heat flux

The problem of steady laminar mixed convection boundary layer flow of an incompressible viscous fluid along vertical moving thin needles with variable heat flux for both assisting and opposing flow cases is theoretically considered in this paper. The governing boundary layer equations are first transformed into non-dimensional forms. The curvature effects are incorporated into the analysis whereas the pressure variation in the axial direction has been neglected. These equations are then transformed into similarity equations using the similarity variables, which are solved numerically using an implicit finite-difference scheme known as the Keller-box method. The solutions are obtained for a blunt-nosed needle (m = 0). Numerical calculations are carried out for various values of the dimensionless parameters of the problem, which include the mixed convection parameter λ, the Prandtl number Pr and the parameter a representing the needle size. It is shown from the numerical results that the skin friction coefficient, the surface (wall) temperature and the velocity and temperature profiles are significantly influenced by these parameters. The results are presented in graphical form and are discussed in detail.     

Free (open) boundary condition: some experiences with viscous flow simulations

The free (or open) boundary condition (FBC, OBC) was proposed by Papanastasiou et al. (A new outflow boundary condition, International Journal for Numerical Methods in Fluids, 1992; 14:587–608) to handle truncated domains with synthetic boundaries where the outflow conditions are unknown. In the present work, implementation of the FBC has been tested in several benchmark problems of viscous flow in fluid mechanics. The FEM is used to provide numerical results for both cases of planar and axisymmetric domains under laminar, isothermal or non‐isothermal, steady‐state conditions, for Newtonian fluids. The effects of inertia, gravity, compressibility, pressure dependence of the viscosity, slip at the wall, and surface tension are all considered individually in the extrudate‐swell benchmark problem for a wide range of the relevant parameters. The present results extend previous ones regarding the applicability of the FBC and show cases where the FBC is inappropriate, namely in the extrudate‐swell problem with gravity or surface‐tension effects. Particular emphasis has been given to the pressure at the outflow, which is the most sensitive quantity of the computations. In all cases where FBC is appropriate, excellent agreement has been found in comparisons with results from very long domains. The formulation for Picard‐type iterations is given in some detail, and the differences with the Newton–Raphson formulation are highlighted regarding some computational aspects. Copyright © 2011 John Wiley & Sons, Ltd.     

A low‐dimensional spectral approach for transient axisymmetric free‐surface flow inside thin cavities of arbitrary shape

A low‐dimensional spectral method is used to solve the transient axisymmetric free surface flow inside thin cavities of arbitrary shape. The flow field is obtained on the basis of the lubrication equations, which are expanded in terms of orthonormal functions over the cavity gap. The formulation accounts for nonlinearities stemming from inertia and front location. The work is of close relevance to the filling stage during die casting, and injection molding, or the flow inside annular (extrusion) dies. Both flows under an imposed flow rate, and an imposed pressure at the cavity entrance are examined. The influence of inertia, aspect ratio, gravity, and wall geometry on the evolution of the front, flow rate, and pressure is assessed particularly in the early stage of flow, when a temporal behavior of the ‘boundary‐layer’ type develops. The multiple‐scale method is applied to obtain an approximate solution at small Reynolds number, Re. Comparison with the exact (numerical) solution indicates a wide range of validity for the multiple‐scale approach, including the moderately small Re range. Copyright © 2002 John Wiley & Sons, Ltd.     

Effect of Discrete Heating on Natural Convection in a Rectangular Porous Enclosure

The main objective of this article is to study the effect of discrete heating on free convection heat transfer in a rectangular porous enclosure containing a heat-generating substance. The left wall of the enclosure has two discrete heat sources and the right wall is isothermally cooled at a lower temperature. The top and bottom walls, and the unheated portions of the left wall are adiabatic. The vorticity–stream function formulation of the governing equations is numerically solved using an implicit finite difference method. The effects of aspect ratio, Darcy number, heat source length, and modified Rayleigh number on the flow and heat transfer are analyzed. The numerical results reveal that the rate of heat transfer increases as the modified Rayleigh number and the Darcy number increases, but decreases on increasing the aspect ratio. The average heat transfer rate is found to be higher at the bottom heater than at the top heater in almost all considered parameter cases except for ε = 0.5. Also, the maximum temperature takes place generally at the top heater except for the case ε = 0.5, where the maximum temperature is found at the bottom heater. Further, the numerical results reveal that the maximum temperature decreases with the modified Rayleigh number and increases with the aspect ratio.     

Modelling the three-dimensional flow of a semi-dilute polymer solution in microfluidics—on the effect of aspect ratio

The flow of polymer solutions in microfluidic devices is inherently three-dimensional, especially in the non-linear flow regime, and often results in flow phenomena that might not even be encountered in macro-devices. Using a multi-mode Phan-Thien–Tanner model, three-dimensional (3-D) simulations of a semi-dilute polyethylene oxide (PEO) solution through 8:1 planar contraction micro-channels with various depths have been carried out to systematically study the effect of the aspect ratio on the flow fields. Vortex dynamics in the upstream flow section and excess pressure drop are quantified in detail. A transition from a salient-corner vortex mechanism to a lip vortex mechanism is observed as the aspect ratio is varied from 1 to 1/4, which corresponds to the elasticity numbers El = 36.3 to 48.3. The numerical results show that varying the aspect ratio of microfluidic channels has similar effects to varying other parameters, such as fluid properties, which influence the elasticity number. Thus, our results support the view that vortex growth mechanism is determined by the elasticity number, which is fixed for a given fluid and geometry. The principle is of significance to the design of new microfluidic chips for a wide range of applications.     

两相正激波松弛流的数值研究

对气、固两相正激波松弛流动进行了数值研究,方程中考虑了压力梯度及非匀速运动引起的附加质量力。给出了完整的计算方法,对三种不同固相容积比、不同固相颗粒直径和不同马赫数分别进行了数值计算,所得计算结果给出了气、固两相的速度、温度、压力、密度、固相容积比及熵值沿流向分布情况。文章对气、固两相参数在松弛区中的变化规律进行了讨论。