A numerical technique for laminar swirling flow at the interface between porous and homogenous fluid domains

There have been a few recent numerical implementations of the stress‐jump condition at the interface of conjugate flows, which couple the governing equations for flows in the porous and homogenous fluid domains. These previous demonstration cases were for two‐dimensional, planar flows with simple geometries, for example, flow over a porous layer or flow through a porous plug. The present study implements the interfacial stress‐jump condition for a non‐planar flow with three velocity components, which is more realistic in terms of practical flow applications. The steady, laminar, Newtonian flow in a stirred micro‐bioreactor with a porous scaffold inside was investigated. It is shown how to implement the interfacial jump condition on the radial, axial, and swirling velocity components. To avoid a full three‐dimensional simulation, the flow is assumed to be independent of the azimuthal direction, which makes it an axisymmetric flow with a swirling velocity. The present interface treatment is suitable for non‐flat surfaces, which is achieved by applying the finite volume method based on body‐fitted and multi‐block grids. The numerical simulations show that a vortex breakdown bubble, attached to the free surface, occurs above a certain Reynolds number. The presence of the porous scaffold delays the onset of vortex breakdown and confines it to a region above the scaffold. Copyright © 2008 John Wiley & Sons, Ltd.     

Investigation of Two-Fluid Methods for Large Eddy Simulation of Spray Combustion in Gas Turbines

An extension of the large eddy simulation (LES) technique to two-phase reacting flows, required to capture and predict the behavior of industrial burners, is presented. While most efforts reported in the literature to construct LES solvers for two-phase flow focus on Euler–Lagrange formulation, the present work explores a different solution (‘two-fluid’ approach) where an Eulerian formulation is used for the liquid phase and coupled with the LES solver of the gas phase. The equations used for each phase and the coupling terms are presented before describing validation in two simple cases which gather some of the specificities of real combustion chamber: (1) a one-dimensional laminar JP10/air flame and (2) a non-reacting swirled flow where solid particles disperse (Sommerfeld and Qiu, Int. J. Multiphase Flow 19(6):1093–1127, 1993). After these validations, the LES tool is applied to a realistic aircraft combustion chamber to study both a steady flame regime and an ignition sequence by a spark. Results bring new insights into the physics of these complex flames and demonstrate the capabilities of two-fluid LES.     

Wetting and nonwetting fluid displacements in porous media

The understanding of simple laminar flow in tubes has often been used to interpret the more complicated flow in porous media. A study of the motion of two immiscible liquids in closed tubes with relatively large diameter (> 0.3 cm i.d), was conducted in order to examine the influence of wetting and nonwetting liquids on the flow behavior. The results indicate that the wetting properties of the fluids with regard to the tube wall have a major efffect on the formation and motion of long bubbles. A physically based model was used to predict the velocity and the conditions for no motion of bubbles and drops in tubes. These results were used to interpret the nature of oil and water flow in porous media. Experiments in which the wetting liquid was displaced by the nonwetting, or vice versa, were conducted by injecting the displacing liquid at a constant flux at the center of a two-dimensional chamber saturated with the displaced liquid. The influence of wetting-nonwetting characteristics on the quantity of liquid displaced, the shape of the interface between the two liquids, and the interpretation of the no motion radius in a closed tube to the case of a porous medium are discussed. It would appear that the no motion radius gives a good indication of the minimum width of a nonwetting penetrating finger and the maximum width of nonwetting ganglia left by drainage.     

Concentration distributions in a model combustor

A hot-wire concentration probe with a spatial resolution of 0.13 mm is used to measure concentration in a model cylindrical combustor. The flow inside the combustor is simulated by injecting a helium jet into a cylindrical confinement with or without swirling air flow present. Mean concentrations are essentially zero outside of the jet region, indicating complete confinement of the scalar field by the swirling flow. Consequently, concentration fluctuations are found to be relatively weak compared to velocity fluctuations, and are maximum off-axis at a point which corresponds to the interface between helium and air flows. However, in the absence of a swirling air flow, the helium diffuses quickly to fill the combustor. The resulting helium concentrations are constant and do not resemble the jet-like behavior of the velocity field.A version of this paper was presented at the ASME Winter Annual Meeting of 1984 and printed in AMD, Vol. 66     

Confined and Agitated Swirling Flows with Applications in Chemical Engineering

Turbulent, swirling flows are encountered frequently in chemical engineering practice. In this article, experiments and simulations on two classes of swirling flows, viz. agitated flows (stirred tanks), and confined swirling flows are discussed. Results of large-eddy simulations of stirred tank flow are compared with experimental data, mainly phase-resolved LDA data of the flow in the vicinity of the impeller. Next to the average velocity field, also the turbulent kinetic energy, and the anisotropy of the Reynolds stress tensor have been assessed. An important application of confined swirling flow is the cyclone separator (hydrocyclones for the separation of liquids, gas cyclones for gas-solid separation). The flow in a swirl tube geometry exhibiting many of the typical features of swirl flows (e.g. vortex breakdown) is discussed. Furthermore, a large-eddy simulation of the gas flow in a high-efficiency Stairmand cyclone separator is presented. Two examples of process modeling based on flow simulations are briefly treated: orthokinetic agglomeration of crystals in a stirred tank, and particle separation in a cyclone. This revised version was published online in July 2006 with corrections to the Cover Date.     

Heat transfer by laminar flow of an elastico-viscous liquid in a circular cylinder with linearly varying wall temperature

Summary The heat transfer by laminar flow of elastico-viscous liquids in a circular cylinder with linearly varying wall temperature has been studied by using the constitutive equation of motion for elastico-viscous liquids and energy equation. The flow phenomenon are characterized by two parameters R _c and S. The presence of the elastic elements in the viscous liquid considerably affects the velocity and temperature distributions.     

LDV measurements in complex swirling flows,their physics and a database for CFD evaluations

In an earlier paper in this journal (Mocikat et al. in Exp Fluids 34:442–448, 2003) LDV measurements in a simple geometry but for a complex flow have been provided as a database for CFD evaluation purposes. With special inflow devices swirl could now be added to the flow. By changing the exit position of the test section in order to get a non-symmetric flow field, a steady swirling flow without instability induced precessing motions could be established. This flow can be interpreted as a superposition of a swirling motion to an otherwise swirl-free flow by introducing “swirl influence factors” for various aspects of the flow field. With a modified inflow device a periodically unsteady flow with swirl emerged. The turbulence features of this flow are distinctively different from the steady flow case with swirl. For all flows under consideration the three time-averaged components of the velocity vector and all components of the Reynolds stress tensor are measured in selected cross sections and provided as a data base for CFD calculations.     

A spectral solver for the Navier–Stokes equations in the velocity–vorticity formulation

A numerical algorithm intended for the study of flows in a cylindrical container under laminar flow conditions is proposed. High resolution of the flow field, governed by the Navier–Stokes equations in velocity–vorticity formulation relative to a cylindrical frame of reference, is achieved through spatial discretisation by means of the spectral method. This method is based on a Fourier expansion in the azimuthal direction and an expansion in Chebyshev polynomials in the (nonperiodic) radial and axial directions. Several regularity constraints are used to take care of the coordinate singularity. These constraints are implemented, together with the boundary conditions at the top, bottom and mantle of the cylinder, via the tau method. The a priori unknown boundary values of the vorticity are evaluated by means of the influence-matrix technique. The compatibility between the mathematical and numerical formulation of the Navier–Stokes equations is established through a tau-correction procedure. The resolved flow field exhibits high-precision satisfaction of the incompressibility constraints for velocity and vorticity and the definition of the vorticity. The performance of the solver is illustrated by resolution of several configurations representative of generic three-dimensional laminar flows.     

Multiphase fluid dynamics and transport processes of low capillary number cavitating flows

To better understand the multiphase fluid dynamics and associated transport processes of cavitating flows at the capillary number of 0.74 and 0.54, and to validate the numerical results, a combined computational and experimental investigation of flows around a hydrofoil is studied based on flow visualizations and time-resolved interface movement. The computational model is based on a modified RNG k-ε model as turbulence closure, along with a vapor-liquid mass transfer model for treating the cavitation process. Overall, favorable agreement between the numerical and experimental results is observed. It is shown that the cavi- tation structure depends on the interaction of the water-vapor mixture and the vapor among the whole cavitation stage, the interface between the vapor and the two-phase mixture exhibits substantial unsteadiness. And, the adverse motion of the interface relates to pressure and velocity fluctuations inside the cavity. In particular, the velocity in the vapor region is lower than that in the two-phase region.     

Axisymmetric thermocapillary convection in a slowly rotating annular cylindrical container

In a slowly rotating annular cylindrical container the free liquid surface (liquid-gas interface) is subjected to a temperature gradient in radial direction. The temperature dependent surface tension creates a shear stress on the interface which is transmitting a thermocapillary convection in the bulk of the liquid. For constant temperature T ₁ of the inner and T ₂ of the outer wall a steady Marangoni convection takes place, exhibiting a double vortex ring of equal directional flow. For time-oscillatory temperatures of the walls a time-dependent thermocapillary convection appears, which will create on the free liquid surface various wave patterns. They shall, depending on the forcing frequency of the temperature, exhibit resonance peaks. The velocity distribution and the response magnitude inside the container has been determined. Received on 3 September 1997     

Order and disorder in particle–liquid flows in a pipe

The spherical expanded polystyrene particle–oil two-phase flow in a vertical pipe was used to simulate the dispersed phase distribution in laminar bubbly flows. A three-dimensional particle image tracking technique was used to track the particles in the flow to study the ordered structure of dispersed phase distribution and its transition to disorder. The ordered structures behaved as particle strings aligned in the flow direction as induced by the flow shear. The structures were quite durable in high liquid velocity flows and dispersed gradually as the liquid velocity decreased. In lower velocity flows, the particles tended to form clusters in the horizontal direction, as predicted by potential theory for spherical bubbles rising in a quiescent inviscid liquid and as observed in experiments on non-shear bubbly water flows.     

A Three-Dimensional Model of Two-Phase Flows in a Porous Medium Accounting for Motion of the Liquid–Liquid Interface

A new three-dimensional hydrodynamic model for unsteady two-phase flows in a porous medium, accounting for the motion of the interface between the flowing liquids, is developed. In a minimum number of interpretable geometrical assumptions, a complete system of macroscale flow equations is derived by averaging the microscale equations for viscous flow. The macroscale flow velocities of the phases may be non-parallel, while the interface between them is, on average, inclined to the directions of the phase velocities, as well as to the direction of the saturation gradient. The last gradient plays a specific role in the determination of the flow geometry. The resulting system of flow equations is a far generalization of the classical Buckley–Leverett model, explicitly describing the motion of the interface and velocity of the liquid close to it. Apart from propagation of the two liquid volumes, their expansion or contraction is also described, while rotation has been proven negligible. A detailed comparison with the previous studies for the two-phase flows accounting for propagation of the interface on micro- and macroscale has been carried out. A numerical algorithm has been developed allowing for solution of the system of flow equations in multiple dimensions. Sample computations demonstrate that the new model results in sharpening the displacement front and a more piston-like character of displacement. It is also demonstrated that the velocities of the flowing phases may indeed be non-collinear, especially at the zone of intersection of the displacement front and a zone of sharp permeability variation.     

The stratified flow of gas and non-newtonian liquid in horizontal pipes

Predictions of pressure drop and holdup are presented for the stratified flow of gas and non-Newtonian liquid obeying the Ostwald-de Waele power law model. The model of Taitel & Dukler (1976) for gas/Newtonian liquid flow is extended to liquids possessing either shear-thinning or shear-thickening laminar flow behaviour and computed results are given for flow behaviour indices in the range $\text{0.1 ≤ n ≤ 2}$. In particular, conditions are defined for drag reduction of the liquid flow by the presence of the gas. It is concluded that drag reduction occurs over the largest ranges of liquid and gas flow rates at the lowest $\text{n}$ values, provided that liquid flow remains laminar, but that maximum drag reduction may be expected for shear-thickening liquids with $\text{n}$ values of 2 or greater. Ratios of the liquid flow rate in the presence of gas to that for liquid flow alone under a constant pressure gradient are also presented. These ratios frequently exceed unity and are greatest for highly shear-thinning liquids.Although the Taitel & Dukler approach is consistent with experiments on gas/Newtonian liquid flow, and, in addition, appears to be valid for immiscible Newtonian liquid-liquid systems, provided that the viscosity ratio of the two phases is at least five, experiments are required to confirm its applicability for gas/non-Newtonian systems.     

A parametric study of LES on laminar-turbulent transitional flows past an airfoil

Low-Reynolds-number aerodynamic performance of small-sized air vehicles is an area of increasing interest. In this study, low-Reynolds-number flows past an SD7003 airfoil are investigated to understand important viscous features of laminar separation and transitional flow followed by the complicated behavior of the flow reattachment process. In order to satisfy the three-dimensional (3D) requirement of the code, a simple “3D wing” is constructed from a two-dimensional (2D) airfoil. A parametric study of large eddy simulation (LES) on the airfoil flows at Re = 60,000 is performed. Effects of grid resolution and sub-grid scale (SGS) models are investigated. Although 3D effects cannot be accurately captured owing to the limitation of the grid resolution in the spanwise direction, the preliminary LES calculations do reveal some important flow characteristics such as leading-edge laminar separation and vortex shedding from the primary laminar separation bubble on the low-Reynolds-number airfoil.     

Non-steady two-phase stratified laminar flow of polymeric liquids in pipes

Summary A numerical method of solution is given for the non-steady two-phase stratified. laminar flow of various non-Newtonian fluids in pipes. In particular, the Ellis fluid model is chosen to illustrate inelastic shear thinning effects. It is shown that the method can be applied to the non-steady multi-phase stratified laminar flow of some non-Newtonian fluid models. An Oldroyd six constant model is used to illustrate the fully elastic flow. It is found that the presence of two phases of the same kind of immiscible liquids tends to suppress the typically viscoelastic response to time dependent situations that the same liquids would exhibit as a single phase. Overshoot of flow rates is reduced, if not completely eliminated in both the generation and decay of steady flows. In two-phase pulsatile flows, the flow enhancement is less marked and the time dependence of the individual flow rates can be significantly different. Theoretical results are used to interpret some of the flow instabilities encountered during the capillary flow of polymeric liquids.

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Zusammenfassung Es wird eine numerische Methode für die nichtstationäre geschichtete Zwei-Phasen-Strömung verschiedener nicht-newtonscher Flüssigkeiten durch Rohre angegeben, wobei zur Veranschaulichung unelastischer Scherentzähungseffekte speziell das Ellis-Modell ausgewählt wird. Dabei zeigt sich, daß diese Methode für die Anwendung auf nicht-stationäre Mehrphasenströmungen geschichteter laminarer Strömungen nicht-newtonscher Flüssigkeiten geeignet ist. Zur Veranschaulichung des vollständigen elastischen Fließens wird ein Oldroyd-Modell mit sechs Konstanten gewählt. Es wird gezeigt, daß die Anwesenheit von zwei Phasen nicht-mischbarer Flüssigkeiten die Unterdrückung des typisch viskoelastischen Verhaltens unter zeitabhängigen Bedingungen, wie es beim Vorhandensein einer einzigen Phase beobachtet wird, zur Folge hat. Das Überschwingen der Fließgeschwindigkeit wird sowohl beim Anfahren als auch beim Anhalten einer stationären Strömung verringert, wenn nicht gar völlig verhindert. In pulsierenden Zwei-Phasen-Strömungen ist die Geschwindigkeitserhöhung weniger ausgeprägt, und die Zeitabhängigkeit der beiden Fließgeschwindigkeiten kann wesentlich verschieden sein. Die theoretischen Ergebnisse werden dazu verwendet, einige bei der Durchströmung von Polymerflüssigkeiten durch Kapillaren beobachtete Fließinstabilitäten zu interpretieren.

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With 15 figures     

Parametric study of unsteady laminar flows

Results of a parametric study of unsteady laminar flows are analyzed. Three-dimensional unsteady equations of hydromechanics for a compressible medium are solved. The range of the characteristic Reynolds number Re = 400–900 is considered. It is demonstrated that the laminar flow in a plane channel ceases to be steady at Re = 415. As the Reynolds number increases, the unsteady processes become more intense, disturbances penetrate inward the channel, and separation zones lose their stability. In the vicinity of the channel exit, however, the flow tends to stabilize, though it remains unsteady. No transition to a turbulent flow occurs in the examined range of Reynolds numbers.     

Laminar swirling power law non-newtonian fluid jet impinging on a normal plane

An analysis is presented for a steady, laminar, incompressible, swirling, Ostwald—De Waele type non-Newtonian fluid jet impinging normally over a horizontal plane with a free surface. A similarity solution is obtained in a region of radial distance away from the central stagnation point. Numerical solutions for the radial and swirling velocities have been obtained for the flow behavior index, n, varying from 0.1 to 2.5. Expressions for the free surface radial velocity, growth of the free surface and skin friction coefficient are given.     

Stability of a laminar rivulet liquid flow in a cylindrical duct in the approximation of one-dimensional waves

The problem of the stability of a viscous laminar liquid flow with a liquid free surface in an inclined duct is theoretically considered. Since the dependence of the flow rate on the free-surface height is not monotonic (the highest flow rate in a cylindrical duct is observed at H_*=1.7R), primary attention is given to the region H>H_*. It is proved that there is aw region of instability: for an arbitrarity low Reynolds number, there is a free-surface level above which the flow becomes unstable against one-dimensional disturbances. When the height of the liquid layer is close to the vertical dimension of the duct, the one-dimensional disturbances propagate mainly upstream (for moderate Reynolds numbers). Hence it follows that there is not steady regime of liquid flow from a fully filled duct with an open end. Kutateladze Institute of Thermal Physics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 3, pp. 90–96, May–June, 1999.     

Numerical predictions of the bifurcation of confined swirling flows

The bifurcation of confined swirling flows was numerically investigated by employing both the k-? and algebraic stress turbulence models. Depending upon the branch solution examined, dual flow patterns were predicted at certain swirl levels. In the lower-branch solution which is obtained by gradually increasing the swirl level from a low-swirl flow, the flow changes with increasing swirl number from the low-swirl flow pattern to a high-swirl flow pattern. In the upper-branch solution which is acquired by gradually decreasing the swirl level from a high-swirl flow, on the other hand, the flow can maintain itself in the high-swirl flow pattern at the swirl levels where it exhibits the low-swirl flow pattern in the lower branch. The bifurcation of confined swirling flows was predicted with either the k-? model or the algebraic stress model being employed. Both the k-? and algebraic stress models result in comparable and sufficiently good predictions for confined swirling flows if high-order numerical schemes are used. The reported poor performance of the k-? model was clarified to be mainly attributable to the occurrence of the bifurcation and the use of low-order numerical schemes.