Axisymmetric Pipe Research Articles

On the Complex Flow Dynamics of Shear Thickening Fluids Entry Flows

Due to their nature, using shear thickening fluids (STFs) in engineering applications has sparked an interest in developing energy-dissipating systems, such as damping devices or shock absorbers. The Rheinforce technology allows the design of customized energy dissipative composites by embedding microfluidic channels filled with STFs in a scaffold material. One of the reasons for using microfluidic channels is that their shape can be numerically optimized to control pressure drop (also known as rectifiers); thus, by controlling the pressure drop, it is possible to control the energy dissipated by the viscous effect. Upon impact, the fluid is forced to flow through the microchannel, experiencing the typical entry flow until it reaches the fully developed flow. It is well-known for Newtonian fluid that the entrance flow is responsible for a non-negligible percentage of the total pressure drop in the fluid; therefore, an analysis of the fluid flow at the entry region for STFs is of paramount importance for an accurate design of the Rheinforce composites. This analysis has been numerically performed before for shear-thickening fluids modeled by a power-law model; however, as this constitutive model represents a continuously growing viscosity between end-viscosity plateau values, it is not representative of the characteristic viscosity curve of shear-thickening fluids, which typically exhibit a three-region shape (thinning-thickening-thinning). For the first time, the influence of these three regions on the entry flow on an axisymmetric pipe is analyzed. Two-dimensional numerical simulations have been performed for four STFs consisting of four dispersions of fumed silica nanoparticles in polypropylene glycol varying concentrations (7.5–20 wt%).

A Mesh Convergence Study for 2D Axisymmetric Pipe Wall Thickness

A meshing size study is essential for any simulation, as it affects the accuracy of the results by approximating the real-world geometry. This paper presents a detailed mesh convergence study for a 2D axisymmetric pipe model. The model incorporates a one-quarter cross-section due to the axisymmetric nature of the pipe. Four pipe wall thicknesses were investigated: 3.40 mm, 7.11 mm, 10.97 mm, and 18.26 mm. The results revealed a significant advantage for simulations involving thin pipes. They achieved convergence, a state with stable solution values, by utilizing a less dense mesh, leading to reduced computational time. This trend was exemplified by the fact that a thinner pipe with thickness, t = 3.40 mm has converged with only 8 mesh elements, whereas a significantly thicker pipe, t = 18.26 mm necessitated 14 elements for convergence. This suggests that more complex geometries, with intricate details, may require a larger and denser mesh for convergence, leading to increased computational time. In conclusion, the mesh convergence study confirms that the finite element analysis (FEA) model has achieved a converged solution.

Newtonian flow with slip and pressure-drop predictions in hyperbolic confined geometries

We study theoretically the steady Newtonian flow in confined and hyperbolic long tubes (symmetric channels and axisymmetric pipes) considering slip along the walls. Using a stream function formulation, and the extended (or high-order) lubrication method in terms of the square of the aspect ratio of the tube, ε, the solution for the stream function is found analytically up to twentieth order in ε. At the classic lubrication limit, i.e. i.e. for a vanishing small aspect ratio, and for perfect slip conditions, the analysis predicts a plug-like velocity profile and a constant strain-rate on the midplane/axis of symmetry of the tube. A constant strain-rate is also predicted for the non-slip case. Furthermore, the high order asymptotic results for the stream function and fluid velocity are post-processed with an acceleration technique to investigate the convergence and accuracy of the solution. The results reveal the existence of a boundary layer at the inlet of the tube, the influence of which diminishes in a very short distance from the entrance. We discuss the effect of the contraction ratio of the tube and the dimensionless slip coefficient on the midplane/centerline and wall (slip) velocities, as well as on the average pressure-drop, required to maintain a constant flow-rate. The acceleration of converge technique on the solution for the pressure-drop revealed a remarkable convergence at a value slightly larger (∼1 %) than the value predicted by the classic lubrication theory. Finally, we comment on the common practice in the literature for approaching the velocity profile with the velocity profile at the classic lubrication limit, and we compare the high-order results for the strain rate at the midplane/centerline with the effective strain rate previously derived in the literature by Housiadas & Beris, J. Rheology, 68(3), 327–339, 2024.

The analytical solution for the flow of a Papanastasiou fluid in ducts with variable geometry

We solve the isothermal, steady and creeping flow of the regularized Papanastasiou model in ducts, such as symmetric channels and axisymmetric pipes, with varying rigid walls. The classic lubrication approximation is invoked to simplify the governing equations and to produce an exact analytical solution for the flow field with the aid of the Lambert W function. For specific suitable values of the parameters of the Papanastasiou model, the solution reduces to the corresponding exact analytical solutions for the ideal yield-stress Bingham and Newtonian fluids. The analytical solutions derived here are used to calculate the average pressure drop required to maintain the constant volumetric flowrate through the duct. Results are provided for symmetric undulating or linearly varying channels with the emphasis put on the effects of the Bingham and regularization numbers, and the amplitude of the undulation or the wall variation, respectively. For Bn→0 the yield surface approaches the axis of symmetry, σ(z)→0, i.e., no unyielded region appears in the flow. In this case, the last term in the formula for v is well-defined at y=0 provided that the numerator is zero. This implies that p′=c/h4 where c is constant. With the aid of the total mass balance, we find that c=−16, and thus p′→−16/h4 as Bn→0. Thus, Eqs. (34) and (37) reduce to the corresponding formulas for a Newtonian fluid given in Eq. (25). Noting that f(y,z)>0, and substituting the solution for γ˙ into inequality (52) gives the trivial W(f(y,z))≥0 which holds for any value of y and z because of the properties of the Lambert function. Eq. (53) is used in conjunction with Eq. (8) to find the unknown pressure gradient, p′(z). First, we integrate by parts Eq. (8) and apply the no-slip condition at the upper wall to find ∫0h(z)y2γ˙dy=2. Then, by substituting Eq. (53) into the latter, and performing the integration with respect to y from the axis of symmetry (y=0) to the wall (y=h(z)), gives: Using properties of the Lambert function and denoting with w=w(z) the value of W function at f(y=h(z),z), i.e.: we find the integral I(z) in Eq. (54) in terms of W and w. Then, we set Φ(z)≡−p′(z)h(z)/2>0, rearrange the resulting equation and upon simplification, we derive the strongly non-linear algebraic equation for Φ: The axial velocity component, u=u(y,z), can be found by integrating Eq. (53) with respect to y between y and h(z) and applying the no-slip condition at the wall:

Granular flow through a vertical axisymmetric pipe

Granular flow through a vertical axisymmetric pipe

On the large-Weissenberg-number scaling laws in viscoelastic pipe flows

This work explains a scaling law of the first Landau coefficient of the derived Ginzburg–Landau equation in the weakly nonlinear analysis of axisymmetric viscoelastic pipe flows in the large-Weissenberg-number ($Wi$) limit, recently reported in Wanet al.(J. Fluid Mech., vol. 929, 2021, A16). Using an asymptotic method, we derive a reduced system, which captures the characteristics of the linear centre-mode instability near the critical condition in the large-$Wi$limit. Based on the reduced system we then conduct a weakly nonlinear analysis using a multiple-scale expansion method, which readily explains the aforementioned scaling law of the Landau coefficient and some other scaling laws. Particularly, the equilibrium amplitude of disturbance near linear critical conditions is found to scale as$Wi^{-1/2}$, which may be of interest to experimentalists. The current analysis reduces the numbers of parameters and unknowns and exemplifies an approach to studying the viscoelastic flow at large$Wi$, which could shed new light on the understanding of its nonlinear dynamics.

Colloidal exchange flow in ducts

Buoyancy-driven exchange flow, i.e. drainage of a heavy colloidal suspension by an immiscible light pure fluid in a vertical duct, is studied analytically. A lubrication model is developed for two-dimensional channel and axisymmetric pipe geometries. The outcome transport equations are solved numerically to capture the effect of important governing dimensionless parameters: Péclet number, precursor film thickness, Reynolds number, mixtures’ viscosity ratio, initial volume fraction of hard-sphere particles, particles concentration within the precursor film, and capillary number. Recent study of colloidal film flow over a flat surface reveals interface thinning over time due to diffusion impact on suspension viscosity. However, the current model suggests that there is no such thinning phenomenon observed for colloidal flows within confined geometry unless in the absence of surface tension. The extent of the mixed particle zone along the duct grows as the Péclet number is decreased. Precursor film thickness is found to decrease the capillary ridge height forming at heavy suspension front. Either an increase in the Reynolds number or a decrease in light-to-heavy-fluid viscosity ratio extends the interpenetration rate of mixtures. As the bulk volume fraction of particles is increased close to 50 %, interesting secondary fronts emerge within the flow that coincide with the range where the diffusion coefficient is minimized. The heavy suspension layer drains slightly faster when the gradient in particle concentration between the bulk and precursor film regions is maximum. Finally, the non-monotonic impact of surface tension on the extent of mixtures’ exchange zone is uncovered via simulations conducted at various capillary numbers.

Sound power radiated from acoustically thick, fluid loaded, axisymmetric pipes excited by a central monopole

The aim of this study is to determine how the breakout noise of infinite length cylindrical shells excited by a central internal point source differ between acoustically thin and thick walls. This will further our understanding of acoustic radiation from axisymmetric pipes with thick walls, whose breakout noise excited by an internal monopole has not been studied previously. To accomplish this, pipes filled with air and immersed in an external fluid are investigated. This is performed numerically using the semi analytical finite element method to generate predictions above the critical frequency of the duct. It is observed that the maxima in the breakout noise occur when a certain class of eigenmodes with sound energy lying predominantly in the structure veer away from those where the sound energy lies predominantly in the fluid. For lightly fluid loaded pipes with thin walls this veering and associated increase to breakout noise is observed at the ring frequency and above the critical frequency. However for thick walled pipes, no corresponding increase in breakout noise is observed at the critical frequency for a pipe immersed in air. Instead the increase in breakout noise is observed only at the ring frequency and above. However, if the pipe is immersed in water then the increase in breakout noise is observed to occur below the ring frequency.

Численное исследование влияния неизотермичности на характеристики течения степенной жидкости в трубе с резким расширением

In this paper, a steady laminar non-isothermal flow of a power-law fluid in an axisymmetric sudden pipe expansion is numerically simulated. The rheological behavior of the fluid is described by the modified Ostwald-de Waele law; the apparent viscosity is an exponential function of temperature. The equations are written in terms of dimensionless stream function - vortex - temperature. No-slip conditions and zero temperature are used on the solid wall. At the inlet boundary, the velocity and temperature profiles correspond to a one-dimensional steady non-isothermal flow of the considered fluid. “Soft” boundary conditions are assigned at the outlet boundary. The formulated problem is solved using the finite-difference method. The structure of the flow through a sudden pipe expansion is shown to include one- and two-dimensional flow zones with a recirculation region occurring in the inner corner vicinity. The variation in the two-dimensional flow zone length is analyzed with respect to a power-law index and dimensionless criteria of the problem. Distributions of the velocity, temperature, and apparent viscosity are presented at various Peclet and Reynolds numbers for dilatant and pseudoplastic fluids.

Subcritical and supercritical bifurcations in axisymmetric viscoelastic pipe flows

Axisymmetric viscoelastic pipe flow of Oldroyd-B fluids has been recently found to be linearly unstable by Garget al.(Phys. Rev. Lett., vol. 121, 2018, 024502). From a nonlinear point of view, this means that the flow can transition to turbulence supercritically, in contrast to the subcritical Newtonian pipe flows. Experimental evidence of subcritical and supercritical bifurcations of viscoelastic pipe flows have been reported, but these nonlinear phenomena have not been examined theoretically. In this work, we study the weakly nonlinear stability of this flow by performing a multiple-scale expansion of the disturbance around linear critical conditions. The perturbed parameter is the Reynolds number with the others being unperturbed. A third-order Ginzburg–Landau equation is derived with its coefficient indicating the bifurcation type of the flow. After exploring a large parameter space, we found that polymer concentration plays an important role: at high polymer concentrations (or small solvent-to-solution viscosity ratio$\beta \lessapprox 0.785$), the nonlinearity stabilizes the flow, indicating that the flow will bifurcate supercritically, while at low polymer concentrations ($\beta \gtrapprox 0.785$), the flow bifurcation is subcritical. The results agree qualitatively with experimental observations where critical$\beta \approx 0.855$. The pipe flow of upper convected Maxwell fluids can be linearly unstable and its bifurcation type is also supercritical. At a fixed value of$\beta$, the Landau coefficient scales with the inverse of the Weissenberg number ($Wi$) when$Wi$is sufficiently large. The present analysis provides a theoretical understanding of the recent studies on the supercritical and subcritical routes to the elasto-inertial turbulence in viscoelastic pipe flows.

Numerical analysis of laminar viscoelastic fluid hammer phenomenon in an axisymmetric pipe

Numerical analysis of laminar viscoelastic fluid hammer phenomenon in an axisymmetric pipe

CFD study of Convective Heat Transfer of Water Flow Through Micro-Pipe with Mixed Constant Wall Temperature and Heat Flux Wall Boundary Conditions

The dissipation of heat in tiny engineering systems can be achieved with fluid flow through micro pipes. They have the advantage of less volume to large surface ratio convective heat transfer. There are deep-rooted analytical relations for convective heat transfer available for fluid flow through macro size pipes. But differences exist between the convective heat transfer for fluid flow through macro and micro pipes. Therefore, there is a good scope of work in micro convection heat transfer to study the mechanism of fundamental flow physics. There have been studies with either constant heat flux wall boundary conditions or constant wall temperature boundary conditions with constant and variable property flows. In this article, first, the numerical simulations are validated with the experimental data for 2D axisymmetric conventional pipe with pipe diameter of 8 mm is taken with laminar, steady, and single-phase water flows with constant wall heat flux boundary condition of 1 W/cm2. The computed Nusselt number is compared to the experimental results at different Reynolds numbers of 1350, 1600 and 1700. In the next study, three-dimensional micropipe laminar flow is studied numerically using water with an inlet velocity of 3 m/s and pipe diameter of 100 µm. The mixed wall boundary conditions with upper half pipe surface subjecting to constant wall temperature of 313 K and lower half surface subjecting to 100 W/cm2 are used in the simulations. The focus of research would be to consider the effect of temperature-dependent properties like thermal conductivity, viscosity, specific heat, and density (a combined effect we call it as variable properties) on micro-pipe flow characteristics like Nusselt number at mixed wall boundary conditions and compare it with the constant property flows. The conventional pipe showed no significant difference with variable and constant property flows with different Reynolds numbers. On contrary the flow through 3D micropipe shows that the Nusselt number with variable property flows is less as compared to the constant property flows.

Developing Turbulent Flow in Pipes and Analysis of Entrance Region

Turbulent flows have complex structures due to its nature and its’ analyses are hard either by numerical or experimental means. Hydrodynamic development of turbulent flow is also complex. In this study, velocity distribution in hydrodynamic entrance length of pipes is investigated depending on axial and radial locations. Literature was surveyed for a single empirical expression that provides velocity profile directly according to Reynolds number, radial and axial locations. Requisite for computational fluid dynamics in hydrodynamic entry length of pipes is stressed by assessing turbulence magnitudes in radial and axial directions. Evaluation of the region and its properties are conducted from heat transfer perspective. An axisymmetric pipe entrance region was analyzed by means of a commercial CFD code with nondimensional parameters. Four different Reynolds numbers that are 5x103, 1x104, 5x104, 1x105 were used in calculations. k-ϵ turbulence model and standard wall functions were used for turbulence modeling. Hydrodynamic entry length, velocity profiles and turbulence indicator parameters results are presented by means of axial and radial profiles. According to the obtained results, radial velocity component values exist that would lead to radial thermal convection in hydrodynamic entrance length. It is found that simultaneous development of velocity profiles and turbulence quantities leads to characteristic velocity profiles. Also, it is seen that a good resolution in hydrodynamic entrance length can be easily achieved by computational fluid dynamics.

Fluid Dynamics and Pressure Drop Prediction of Two-Phase Flow Through Sudden Contractions

Abstract The aim of the present study is to investigate fluid dynamics and pressure drop across sudden contractions in a two-dimensional, axisymmetric pipe carrying a two-phase mixture of air (secondary phase) and water (primary phase), using the Eulerian–Eulerian model of the multiphase flow physics to solve the mass, momentum, volume fraction and turbulent quantities with relevant boundary conditions in a finite volume framework. The realizable per-phase k-ε and Reynolds stress models have been used as the closure for turbulent quantities along with enhanced wall function for the near-wall treatment. The effects of various parameters such as mass flux, mass flow quality, area ratio (0.056–0.619), flow directions (horizontal/vertical), and system pressure on the two-phase pressure drops due to a contraction in the pipe have been quantified. For both the single and two-phase flows, it has been observed that the pressure drop decreases with area ratio, and increases with mass flux and mass flow quality of two-phase flow. The vena contracta for a single-phase flow was found. But for two-phase flow, neither the vena contracta nor the recirculation zone has been observed, as the mass quality exceeds above 50%. A higher pressure drop has been observed for vertical pipes as compared to horizontal pipes. The present numerical results have also been validated with published experimental results, believed to be one of the alternatives to the costly experimental methods for predicting the flow dynamics and pressure drop.

Mechanical response of buried and covered pipes under water hammer

The classical water hammer model omits bending and inertial effects in the pipe wall. The validity of such assumptions for buried and covered pipes is assessed by developing a one-way coupled water hammer model based on an axisymmetric thin shell pipe finite element that captures bending and inertial effects. A finite/infinite pipe element formulation is developed to capture the stiffness contribution of the surrounding soil (or pipe cover) by adopting a decay function derived from plane strain analysis of the medium surrounding the pipe. Continuous and discrete elastic longitudinal restraint effects are also incorporated into the formulation. The model is then used to investigate the effects of viscous damping, elastic longitudinal restraints, presence of intermediate longitudinal anchors, and the elastic modulus of the surrounding soil. The pipe response is found insensitive to viscous damping. The present model shows that, while classical model predictions are valid for cases where the valve end of the pipe is fully (or nearly-fully) restrained in the longitudinal direction, this is not the case where the valve end is longitudinally unrestrained. Intermediate longitudinal restraints are found to divide the pipe into two segments with distinct responses. The ability of the proposed finite/infinite element to predict the response of buried pipe subjected to water hammer is illustrated through numerical examples.

ЧИСЛЕННОЕ МОДЕЛИРОВАНИЕ ТЕЧЕНИЯ СТЕПЕННОЙ ЖИДКОСТИ В КАНАЛЕ С ДВОЙНЫМ СУЖЕНИЕМ

Flow fields of non-Newtonian fluids in cylindrical channels with obstacles are of great interest for researchers studying the mechanics of fluids. In engineering techniques, such channels represent component parts of heat exchangers and hydraulic systems of various types. In this work, the numerical simulation of a steady non-Newtonian fluid flow in an axisymmetric pipe with two overlaps, whose geometry is described by a cosine function, is carried out. Mathematical formulation of the problem is written in terms of vortex and stream function variables. The Ostwald – de Waele model is used to describe rheological properties of the medium. The solution to a system of differential equations is obtained numerically using the false transient method. To determine the pressure field, the Poisson equation for pressure is solved. Three media are considered in the paper: Newtonian, pseudoplastic, and dilatant fluids. The influence of the Reynolds number, power-law index in the rheological model, and geometric parameters of the channel on the flow characteristics is shown as streamline distributions and functional curves.

Dynamic soil resistance to vertical vibration of pipe pile

Pile-soil interaction affects the pile dynamic characteristics and its response to dynamic loads. The one-dimensional rod model widely used to analyze the response of pipe pile is inaccurate, especially for large-diameter pipe piles. An axisymmetric pipe pile-soil interaction model is proposed to account for vertical and radial displacements of both the pile and soil medium. Analytical solutions are deduced for vertical vibration of open-end and closed-end pipe piles under harmonic excitation. Based on the obtained solutions, soil resistance factor (SRF) is defined to evaluate the dynamic soil resistance along the pile shaft. The model is verified by comparing its predictions with those obtained from existing solutions. The verified model is then used to conduct an extensive parametric study to evaluate the influence of different parameters on the dynamic soil resistance of the pipe pile supported by inner and outer soils. The results show that the inner SRF is much smaller compared to outer SRF. It is also found that stiffness factors of SRFs of inner and outer soils are almost independent of each other. However, the damping of inner SRF is greatly influenced by properties of the outer soil.

Simulation of vibration piercing of pipe blank at press

In some cases, the processes of piercing or expanding pipe blanks involve the use of high-frequency active vibrations. However, due to insufficient knowledge, these processes are not widely used in the practice of seamless pipes production. In particular, the problems of increasing the efficiency of the processes of piercing or expanding a pipe blank at a piercing press using high-frequency vibrations are being solved without proper research and, as a rule, by experiments. The elaboration of modern technological processes for the production of seamless pipes using high-frequency vibrations is directly related to the choice of rational modes of metal deformation and the prediction resistance indicators of technological tools and the reliability of equipment operation. The creation of a mathematical model of the process of vibrating piercing (expansion) of an axisymmetric pipe blank at a piercing press of a pipe press facility is an actual task. A calculation scheme for the process of piercing a pipe plank has been elaborated. A dependence was obtained characterizing the speed of front of plastic deformation propagation on the speed of penetration of a vibrated axisymmetric mandrel into the pipe workpiece being pierced. The dynamic characteristics of the occurrence of wave phenomena in the metal being pierced under the influence of a vibrated tool have been determined, which significantly complements the previously known ideas about the stress-strain state of the metal in the deformation zone. The deformation fields in the zones of the disturbed region of the deformation zone were established, taking into account the high-frequency vibrations of the technological tool. It has been established that the choice of rational parameters (amplitude-frequency characteristics) of the vibration piercing process of a pipe blank results in significant increase in the efficiency of the process, the durability of the technological tool and the quality of the pierced blanks.

Non-Newtonian fluid flow through a sudden pipe contraction under non-isothermal conditions

• Simulation of a non-isothermal viscoplastic fluid flow through a sudden contraction. • Investigation of the flow structure with unyielded regions and rigid static zones. • Extension of two-dimensional flow zones in the contraction plane vicinity. • Effect of the governing parameters on the local resistance coefficient. A steady-state laminar flow of a viscoplastic fluid through an axisymmetric sudden pipe contraction under non-isothermal conditions has been studied numerically. Mathematical formulation of the problem is given using stream function – vorticity – temperature variables. Rheological properties of the fluid are described by the Herschel–Bulkley model, which implies an unyielded region formation in the flow. Viscous dissipation effects and temperature dependence of the viscosity are taken into account when simulating the process. An asymptotic time solution of unsteady flow equations is used to obtain a steady-state solution to the initially formulated problem. The finite-difference method based on the alternative directions scheme is applied to solve the problem numerically. Regularization of the rheological equation is performed to provide a through computation of the flow with unyielded regions. A parametric investigation of the flow structure, which includes dead zones and yielded/unyielded regions, is implemented in order to show the impact of main dimensionless parameters.

Effect of viscous dissipation on the flow structure and pressure losses for a viscoplastic fluid flow through a sudden pipe contraction

A viscoplastic fluid flow through an axisymmetric sudden pipe contraction is studied numerically in a laminar regime under non-isothermal conditions. Governing equations are written in terms of stream function – vorticity – temperature variables. A modified Schwedoff -Bingham rheological model is used to describe characteristics of the fluid. The problem is solved by the finite-difference method based on the alternative directions scheme. Simulation of the process involves taking into account viscous dissipation effects and temperature dependence of viscosity. Distributions of flow characteristics and flow structures are shown and compared with those obtained in the case when viscous dissipation is neglected. A local resistance coefficient is calculated and presented as a function of the Brinkman number.