Showing papers on "Pipe flow published in 1999"

Very large-scale motion in the outer layer

Abstract: Very large-scale motions in the form of long regions of streamwise velocity fluctuation are observed in the outer layer of fully developed turbulent pipe flow over a range of Reynolds numbers. The premultiplied, one-dimensional spectrum of the streamwise velocity measured by hot-film anemometry has a bimodal distribution whose components are associated with large-scale motion and a range of smaller scales corresponding to the main turbulent motion. The characteristic wavelength of the large-scale mode increases through the logarithmic layer, and reaches a maximum value that is approximately 12–14 times the pipe radius, one order of magnitude longer than the largest reported integral length scale, and more than four to five times longer than the length of a turbulent bulge. The wavelength decreases to approximately two pipe radii at the pipe centerline. It is conjectured that the very large-scale motions result from the coherent alignment of large-scale motions in the form of turbulent bulges or packets of...

The inertial lift on a spherical particle in a plane Poiseuille flow at large channel Reynolds number

Abstract: The inertial migration of a small rigid sphere translating parallel to the walls within a channel flow at large channel Reynolds numbers is investigated. The method of matched asymptotic expansions is used to solve the equations governing the disturbance flow past a particle at small particle Reynolds number and to evaluate the lift. Both neutrally and non-neutrally buoyant particles are considered. The wall-induced inertia is significant in the thin layers near the walls where the lift is close to that calculated for linear shear flow, bounded by a single wall. In the major portion of the flow, excluding near-wall layers, the wall effect can be neglected, and the outer flow past a sphere can be treated as unbounded parabolic shear flow. The effect of the curvature of the unperturbed velocity profile is significant, and the lift differs from the values corresponding to a linear shear flow even at large Reynolds numbers.

Drag reduction of Newtonian fluid in a circular pipe with a highly water-repellent wall

Abstract: Drag reduction phenomena, in which 14% drag reduction of tap water flowing in a 16 mm-diameter pipe occurs in the laminar flow range, have been clarified. Experiments were carried out to measure the pressure drop and the velocity profile of tap water and an aqueous solution of glycerin flowing in pipes with highly water-repellent walls, by using a pressure transducer and a hot-film anemometer, respectively. The same drag reduction phenomena also occurred in degassed tap water when using a vacuum tank. The velocity profile measured in this experiment gives the slip velocity at the pipe wall, and it was shown that the shear stress is directly proportional to the slip velocity.The friction factor formula for a pipe with fluid slip at the wall has been obtained analytically from the exact solution of the Navier–Stokes equation, and it agrees well qualitatively with the experimental data.The main reasons for the fluid slip are that the molecular attraction between the liquid and the solid surface is reduced because the free surface energy of the solid is very low and the contact area of the liquid is decreased compared with a conventional smooth surface because the solid surface has many fine grooves. Liquid cannot flow into the fine grooves owing to surface tension. These concepts are supported by the experimental result that drag reduction does not occur in the case of surfactant solutions.

Large-eddy simulations of a round jet in crossflow

Abstract: This paper reports on a series of large-eddy simulations of a round jet issuing normally into a crossflow. Simulations were performed at two jet-to-crossflow velocity ratios, 2.0 and 3.3, and two Reynolds numbers, 1050 and 2100, based on crossflow velocity and jet diameter. Mean and turbulent statistics computed from the simulations match experimental measurements reasonably well. Large-scale coherent structures observed in experimental flow visualizations are reproduced by the simulations, and the mechanisms by which these structures form are described. The effects of coherent structures upon the evolution of mean velocities, resolved Reynolds stresses, and turbulent kinetic energy along the centreplane are discussed. In this paper, the ubiquitous far-field counter-rotating vortex pair is shown to originate from a pair of quasi-steady ‘hanging’ vortices. These vortices form in the skewed mixing layer that develops between jet and crossflow fluid on the lateral edges of the jet. Axial flow through the hanging vortex transports vortical fluid from the near-wall boundary layer of the incoming pipe flow to the back side of the jet. There, the hanging vortex encounters an adverse pressure gradient and breaks down. As this breakdown occurs, the vortex diameter expands dramatically, and a weak counter-rotating vortex pair is formed that is aligned with the jet trajectory.

A connection between the Camassa–Holm equations and turbulent flows in channels and pipes

Abstract: In this paper we discuss recent progress in using the Camassa–Holm equations to model turbulent flows. The Camassa–Holm equations, given their special geometric and physical properties, appear particularly well suited for studying turbulent flows. We identify the steady solution of the Camassa–Holm equation with the mean flow of the Reynolds equation and compare the results with empirical data for turbulent flows in channels and pipes. The data suggest that the constant α version of the Camassa–Holm equations, derived under the assumptions that the fluctuation statistics are isotropic and homogeneous, holds to order α distance from the boundaries. Near a boundary, these assumptions are no longer valid and the length scale α is seen to depend on the distance to the nearest wall. Thus, a turbulent flow is divided into two regions: the constant α region away from boundaries, and the near wall region. In the near wall region, Reynolds number scaling conditions imply that α decreases as Reynolds number increas...

Analytical solution for fully developed channel and Pipe Flow of Phan-Thien/Tanner Fluids

Abstract: Analytical expressions are derived for the velocity vector, the stress components and the viscosity function in fully developed channel and pipe flow of Phan-Thien–Tanner (PTT) fluids; both the linearized and the exponential forms of the PTT equation are considered. The solution shows that the wall shear stress of a PTT fluid is substantially smaller than the corresponding value for a Newtonian or upper-convected Maxwell fluid, with implications for comparing predicted and measured values in a non-dimensional form.

Quasi-2d model for unsteady flow in pipe networks

Abstract: A quasi-two-dimensional model for unsteady-flow analysis in pipes and pipe networks is presented. The turbulence model is based on the mixing length hypothesis in the turbulent zone and on Newton’s law in the viscous sublayer. An expression of the mixing length in terms of the Reynolds number and an expression of the parameter of logarithmic law of the wall in terms of the friction Reynolds number are found from Nikuradse’s experimental data. An implicit numerical scheme for the integration of the equations is proposed to overcome the limitations of the explicit schemes. Uniqueness of the head and continuity of discharge are considered at the junctions. The results of both a quasi-steady 1D model and a quasi-2D model are compared with results from a laboratory network. For these experimental runs, the comparisons show that the average relative errors on the maximum head oscillations are 19.1% with the 1D model and 8.6% with the quasi-2D model; those on the minimum oscillations are 19.2% with the 1D model and 5.3% with the quasi-2D model. The latter model is in better agreement because it takes into account the velocity profile, thus allowing for a more accurate evaluation of the shear stress.

The effect of the slip boundary condition on the flow of fluids in a channel

Abstract: The assumption that a liquid adheres to a solid boundary (“no-slip” boundary condition) is one of the central tenets of the Navier-Stokes theory. However, there are situations wherein this assumption does not hold. In this paper we investigate the consequences of slip at the wall on the flow of a linearly viscous fluid in a channel. Usually, the slip is assumed to depend on the shear stress at the wall. However, a number of experiments suggests that the slip velocity also depends on the normal stress. Thus, we investigate the flow of a linearly viscous fluid when the slip depends on both the shear stress and the normal stress. In regions where the slip velocity depends strongly on the normal stress, the flow field in a channel is not fully developed and rectilinear flow is not possible. Also, it is shown that, in general, traditional methods such as the Mooney method cannot be used for calculating the slip velocity.

Extensional flow of a polystyrene Boger fluid through a 4 : 1 : 4 axisymmetric contraction/expansion

Abstract: The creeping flow of a dilute (0.025 wt%) monodisperse polystyrene/polystyrene Boger fluid through a 4:1:4 axisymmetric contraction/expansion is experimentally observed for a wide range of Deborah numbers. Pressure drop measurements across the orifice plate show a large extra pressure drop that increases monotonically with Deborah number above the value observed for a similar Newtonian fluid at the same flow rate. This enhancement in the dimensionless pressure drop is not associated with the onset of a flow instability, yet it is not predicted by existing steady-state or transient numerical computations with simple dumbbell models. It is conjectured that this extra pressure drop is the result of an additional dissipative contribution to the polymeric stress arising from a stress-conformation hysteresis in the strong non-homogeneous extensional flow near the contraction plane. Such a hysteresis has been independently measured and computed in recent studies of homogeneous transient uniaxial stretching of PS/PS Boger fluids. Flow visualization and velocity field measurements using digital particle image velocimetry (DPIV) show large upstream growth of the corner vortex with increasing Deborah number. At large Deborah numbers, the onset of an elastic instability is observed, first locally as small amplitude fluctuations in the pressure measurements, and then globally as an azimuthal precessing of the upstream corner vortex accompanied by periodic oscillations in the pressure drop across the orifice.

Velocity field for Taylor–Couette flow with an axial flow

Abstract: The flow in the gap between an inner rotating cylinder concentric with an outer stationary cylinder with an imposed pressure-driven axial flow was studied experimentally using particle image velocimetry (PIV) in a meridional plane of the annulus. The radius ratio was η=0.83 and the aspect ratio was Γ=47. Velocity vector fields for nonwavy toroidal and helical vortices show the axial flow winding around vortices. When the axially averaged axial velocity profile is removed from the velocity field in a meridional plane, the velocity field looks much like it would with no imposed axial flow except that the vortices translate axially and the distortion of the azimuthal velocity contours in meridional plane related to the vortices is shifted axially by the axial flow. The velocity vector fields for wavy vortices also show axial flow winding around the vortices. Again, removing the axial velocity profile results in a flow that appears similar to that with no axial flow. The path of the vortices is generally axial, but the vortices periodically move retrograde to the imposed axial flow due to the waviness of the vortices. The axial velocity of helical vortices, both nonwavy and wavy, is twice the rotational frequency of the inner cylinder indicating a coupling between the axial translation of the vortices and the cylinder rotation. Little fluid transport between vortices occurs for nonwavy vortices, but there is substantial transport between vortices for wavy vortex flow, much like supercritical cylindrical Couette flow with no axial flow.

High-velocity flow in a rough fracture

Abstract: We simulate high-velocity flow in a self-affine channel with a constant perpendicular opening by solving numerically the Navier–Stokes equations, and analyse the resulting flow qualitatively and quantitatively. At low velocity, i.e. vanishing inertia, the effective permeability is dominated by the narrowest constrictions measured perpendicular to the local flow direction and the flow field tends to fill the channel due to the diffusion generated by the viscous term in the Stokes equation. At high velocity (strong inertia), the high-velocity zones of the flow field resemble a narrow tube of essentially constant thickness in the direction of flow, since the transversal diffusion is weak compared to the longitudinal convection. The thickness of the flow tube decreases with Reynolds number. This narrowing in combination with mass balance results in an average velocity in the flow tube which increases faster with Reynolds number than the average velocity in the fracture. In the low-velocity zones, recirculation zones appear and the pressure is almost constant.The flow tube consists of straight sections. This is due to inertia. The local curvature of the main stream reflects the flow-tube/channel-wall interaction. A boundary layer is formed where the curvature is large. This boundary layer is highly dissipative and governs the large pressure loss (inertial resistance) of the medium. Quantitatively, vanishing, weak and strong inertial flow regimes can be described by the Darcy, weak inertia and Forchheimer flow equations, respectively. We observe a cross-over flow regime from the weak to strong inertia, which extends over a relatively large range of Reynolds numbers.

Maxwell fluid suction flow in a channel

Abstract: Two-dimensional, steady and incompressible suction flow of the upper-convected Maxwell fluid in a porous surface channel has been studied The combined effects of viscoelasticity and inertia are considered A similarity solution is assumed, resulting in a nonlinear system of ODEs that describes the relations between the two velocity components, the three deviatoric stresses and the pressure gradient This system is solved using two methods: an analytical solution, based on a power series method in terms of the transverse coordinate across the channel, and a fourth-order Runge–Kutta numerical integration scheme We first find the existing Newtonian flow solutions for suction and injection For the Maxwell fluid, the solutions of the power series and the numerical integration are in complete agreement in the range of Reynolds and Deborah numbers 0 ≤ Re ≤ 30 and 0 ≤ De ≤ 03 They show that the suction flow exhibits a flattening of the longitudinal velocity profile near the centerline and the establishment of boundary layers near the porous surfaces as Reynolds number increases It is also observed that when Deborah number increases, with a fixed Reynolds number, viscoelasticity affects the velocity profiles in the same way as inertia in a Newtonian fluid The application of the self-similar solution to the injection flow of the Maxwell fluid is also discussed

Progress in the use of non-linear two-equation models in the computation of convective heat-transfer in impinging and separated flows

Abstract: The paper considers the application of the Craft et al. [6]non-linear eddy-viscosity model to separating and impinging flows. The original formulation was found to lead to numerical instabilities when applied to flow separating from a sharp corner. An alternative formulation for the variation of the turbulent viscosity parameterc μ with strain rate is proposed which, together with a proposed improvement in the implementation of the non-linear model, removes this weakness. It does, however, lead to worse predictions in an impinging jet, and a further modification in the expression for c μ is proposed, which both retains the stability enhancements and improves the prediction of the stagnating flow. The Yap [24] algebraic length-scale correction term, included in the original model, is replaced with a differential form, developed from that proposed by Iacovides and Raisee [10]. This removes the need to prescribe the wall-distance, and is shown to lead to superior heat-transfer predictions in both an abrupt pipe flow and the axisymmetric impinging jet. One predictive weakness still, however, remains. The proposed model, in common with other near-wall models tested for the abrupt pipe expansion, returns a stronger dependence of Nusselt number on the Reynolds number than that indicated by the experimental data.

Viscoelastic flow through a planar contraction using a semi-Lagrangian finite volume method

Abstract: A semi-Lagrangian finite volume scheme for solving viscoelastic flow problems is presented. A staggered grid arrangement is used in which the dependent variables are located at different mesh points in the computational domain. The convection terms in the momentum and constitutive equations are treated using a semi-Lagrangian approach in which particles on a regular grid are traced backwards over a single time-step. The method is applied to the 4 : 1 planar contraction problem for an Oldroyd B fluid for both creeping and inertial flow conditions. The development of vortex behaviour with increasing values of We is analyzed.

Numerical solutions for unsteady gravity-driven flows in collapsible tubes: evolution and roll-wave instability of a steady state

Abstract: Unsteady flow in collapsible tubes has been widely studied for a number of different physiological applications; the principal motivation for the work of this paper is the study of blood flow in the jugular vein of an upright, long-necked subject (a giraffe). The one-dimensional equations governing gravity- or pressure-driven flow in collapsible tubes have been solved in the past using finite-difference (MacCormack) methods. Such schemes, however, produce numerical artifacts near discontinuities such as elastic jumps. This paper describes a numerical scheme developed to solve the one-dimensional equations using a more accurate upwind finite volume (Godunov) scheme that has been used successfully in gas dynamics and shallow water wave problems. The adapatation of the Godunov method to the present application is non-trivial due to the highly nonlinear nature of the pressure–area relation for collapsible tubes.The code is tested by comparing both unsteady and converged solutions with analytical solutions where available. Further tests include comparison with solutions obtained from MacCormack methods which illustrate the accuracy of the present method.Finally the possibility of roll waves occurring in collapsible tubes is also considered, both as a test case for the scheme and as an interesting phenomenon in its own right, arising out of the similarity of the collapsible tube equations to those governing shallow water flow.

An experimental study on unsteady turbulent near wake of a rectangular cylinder in channel flow

Abstract: Unsteady turbulent near wake of a rectangular cylinder in channel flow has been studied experimentally with a laser Doppler velocimetry (LDV). The time-averaged and phase-averaged statistics were measured for the cylinders having various width-to-height ratios, b/h. It is shown that the turbulent intensities on the centerline of the channel have their maxima near the rear stagnation point of a recirculation region. The contours of coherent vorticity and streamline reproduce clearly the shed vortices from the cylinder observed by the flow visualization. The characteristics of the flow field, which depends on b/h, are discussed and the significant contribution of the coherent structure to the flow field is clarified. Moreover, the turbulent kinetic energy budget has been examined.

The effect of variable viscosity on the entropy generation and pumping power in a laminar fluid flow through a duct subjected to constant heat flux

Abstract: Entropy generation and pumping power required for a laminar viscous flow in a duct subjected to constant heat flux has been investigated. The temperature dependence of the viscosity is taken into consideration. The ratio of pumping power to total heat flux decreases considerably and entropy generation increases along the duct length for viscous fluids. Therefore, it is shown that an optimum duct length may be obtained which minimizes total energy losses due to both entropy generation and pumping power. For low heat-flux conditions, entropy generation due to viscous friction becomes dominant and the dependence of viscosity on temperature must be considered in order to determine entropy generation accurately.

High-velocity laminar and turbulent flow in porous media

Abstract: We model high-velocity flow in porous media with the multiple scale homogenization technique and basic fluid mechanics. Momentum and mechanical energy theorems are derived. In idealized porous media inviscid irrotational flow in the pores and wall boundary layers give a pressure loss with a power of 3/2 in average velocity. This model has support from flow in simple model media. In complex media the flow separates from the solid surface. Pressure loss effects of flow separation, wall and free shear layers, pressure drag, flow tube velocity and developing flow are discussed by using phenomenological arguments. We propose that the square pressure loss in the laminar Forchheimer equation is caused by development of strong localized dissipation zones around flow separation, that is, in the viscous boundary layer in triple decks. For turbulent flow, the resulting pressure loss due to average dissipation is a power 2 term in velocity.

Similarity in non-rotating and rotating turbulent pipe flows

Abstract: The Lie group approach developed by Oberlack (1997) is used to derive new scaling laws for high-Reynolds-number turbulent pipe flows. The scaling laws, or, in the methodology of Lie groups, the invariant solutions, are based on the mean and fluctuation momentum equations. For their derivation no assumptions other than similarity of the Navier–Stokes equations have been introduced where the Reynolds decomposition into the mean and fluctuation quantities has been implemented. The set of solutions for the axial mean velocity includes a logarithmic scaling law, which is distinct from the usual law of the wall, and an algebraic scaling law. Furthermore, an algebraic scaling law for the azimuthal mean velocity is obtained. In all scaling laws the origin of the independent coordinate is located on the pipe axis, which is in contrast to the usual wall-based scaling laws. The present scaling laws show good agreement with both experimental and DNS data. As observed in experiments, it is shown that the axial mean velocity normalized with the mean bulk velocity um has a fixed point where the mean velocity equals the bulk velocity independent of the Reynolds number. An approximate location for the fixed point on the pipe radius is also given. All invariant solutions are consistent with all higher-order correlation equations. A large-Reynolds-number asymptotic expansion of the Navier–Stokes equations on the curved wall has been utilized to show that the near-wall scaling laws for at surfaces also apply to the near-wall regions of the turbulent pipe flow.

Numerical Modeling of Flow and Disinfection in UV Disinfection Channels

Abstract: The steady two-dimensional turbulent flow in the central region of an open channel in which vertical ultraviolet lamps are arranged in a staggered configuration is predicted using a computational fluid dynamics code incorporating a turbulence model and a boundary-fitted grid. A two-dimensional continuum model of the fate and transport of microorganisms, based on the results of flow simulations and incorporating the series-event model of disinfection kinetics also is presented. From an analysis of the governing transport equation, a dimensionless disinfection parameter that may be useful in discussions of comparative disinfection performance is identified. Model predictions are compared with measurements of flow in the laboratory and measurements of disinfection efficacy in a prototype channel. Good agreement is found between flow predictions and measurements in the flow region upstream of a model lamp/tube, but in the wake region, the measured flow exhibited greater uniformity than the numerical model. Predictions of disinfection process performance are satisfactory at high throughput rates but deteriorate at low throughput rates. Differences in predictions caused by differences in the order of the series-event model were studied and were small when the degree of disinfection was low, but became more important at greater disinfection extents.

Interfacial area density in bubbly flow

Abstract: The interfacial area per unit volume is one of the key parameters in bubbly flow. Momentum, mass and energy transfer occur through the interface between the phases. The functionality of two phase reactors with bubbly flow depends mainly on these three transfer processes. Thus, the design process of a reactor requires the prediction of interfacial area density. In the present work a simple equation for the interfacial area density is derived from the population balance, taking into account the events of coalescence and bubble break-up for each bubble fraction. The system of partial integro-differential equations is simplified. Since the integrals in these equations complicate a numerical treatment. This reduces the balance to one single partial differential equation. An approximate analytical solution is given. If the resulting equation is applied to large gas fluxes, the instability of the coalescence process causes large bubbles and gas plugs to develop. From the instability the volume fraction of the large bubbles and gas plugs may be predicted. Additives may hinder the coalescence process. Experiments show that coalescence hindrance changes the coalescence kernel only by a factor. Calculations are done for bubble columns and vertical pipe flow.

Linear stability and dynamics of viscoelastic flows using time-dependent numerical simulations

Abstract: Ultimately, numerical simulation of viscoelastic flows will prove most useful if the calculations can predict the details of steady-state processing conditions as well as the linear stability and non-linear dynamics of these states. We use finite element spatial discretization coupled with a semi-implicit θ -method for time integration to explore the linear and non-linear dynamics of two, two-dimensional viscoelastic flows: plane Couette flow and pressure-driven flow past a linear, periodic array of cylinders in a channel. For the upper convected Maxwell (UCM) fluid, the linear stability analysis for the plane Couette flow can be performed in closed form and the two most dangerous, although always stable, eigenvalues and eigenfunctions are known in closed form. The eigenfunctions are non-orthogonal in the usual inner product and hence, the linear dynamics are expected to exhibit non-normal (non-exponential) behavior at intermediate times. This is demonstrated by numerical integration and by the definition of a suitable growth function based on the eigenvalues and the eigenvectors. Transient growth of the disturbances at intermediate times is predicted by the analysis for the UCM fluid and is demonstrated in linear dynamical simulations for the Oldroyd-B model. Simulations for the fully non-linear equations show the amplification of this transient growth that is caused by non-linear coupling between the non-orthogonal eigenvectors. The finite element analysis of linear stability to two-dimensional disturbances is extended to the two-dimensional flow past a linear, periodic array of cylinders in a channel, where the steady-state motion itself is known only from numerical calculations. For a single cylinder or widely separated cylinders, the flow is stable for the range of Deborah number (De) accessible in the calculations. Moreover, the dependence of the most dangerous eigenvalue on De≡ λV / R resembles its behavior in simple shear flow, as does the spatial structure of the associated eigenfunction. However, for closely spaced cylinders, an instability is predicted with the critical Deborah number De c scaling linearly with the dimensionless separation distance L between the cylinders, that is, the critical Deborah number De L c ≡ λV / L is shown to be an O (1) constant. The unstable eigenfunction appears as a family of two-dimensional vortices close to the channel wall which travel downstream. This instability is possibly caused by the interaction between a shear mode which approaches neutral stability for De ≫ 1 and the periodic modulation caused by the presence of the cylinders. Nonlinear time-dependent simulations show that this secondary flow eventually evolves into a stable limit cycle, indicative of a supercritical Hopf bifurcation from the steady base state.