Pipe Flow Problem Research Articles

Prediction of multiphase flows with sharp interfaces using anisotropic mesh optimisation

We propose an integrated, parallelised modelling approach to solve complex multiphase flow problems with sharp interfaces. This approach is based on a finite-element, double control-volume methodology, and employs highly-anisotropic mesh optimisation within a framework of high-order numerical methods and algorithms, which include adaptive time-stepping, metric advection, flux limiting, compressive advection of interfaces, multi-grid solvers and preconditioners. Each method is integral to increasing the fidelity of representing the underlying physics while maximising computational efficiency, and, only in combination, do these methods result in the accurate, reliable, and efficient simulation of complex multiphase flows and associated regime transitions. These methods are applied simultaneously for the first time in this paper, although some of the individual methods have been presented previously. We validate our numerical predictions against standard benchmark results from the literature and demonstrate capabilities of our modelling framework through the simulation of laminar and turbulent two-phase pipe flows. These complex interfacial flows involve the creation of bubbles and slugs, which involve multi-scale physics and arise due to a delicate interplay amongst inertia, viscous, gravitational, and capillary forces. We also comment on the potential use of our integrated approach to simulate large, industrial-scale multiphase pipe flow problems that feature complex topological transitions.

CFD Modelling of Non-Newtonian Fluid Flow in a Pipe Including Obstacle

Pipe flow problems are important in transportation of wastewater, oil lines and supply of water. In this study, a non-Newtonian fluid model is discussed and a numerical CFD solution is presented for flow geometry. The effects on velocity, pressure, dynamic viscosity and cell Reynolds number are discussed for different parameters of flow inside the pipe. Power Law function is considered in the analyses.

Analytical solutions for the incompressible laminar pipe flow rapidly subjected to the arbitrary change in the flow rate

A one-dimensional mathematical model is developed for an unsteady incompressible laminar flow in a circular pipe subjected to a rapid change in the flow rate from an initial flow with flow rate, Qi, to a final flow with flow rate, Qf, in a step-like fashion at an arbitrary time, tc. The change in the flow rate may either be an increment, Qf > Qi, or a decrement, Qf < Qi. The change time, tc, may either belong to the initial flow remaining in a temporally developing state or temporally developed state. The developed model is solved using the Laplace transform method to deduce generalized analytical expressions for the flow characteristics, viz., velocity, pressure gradient, wall shear stress, and skin friction factor, CfRe, where Re is Reynolds number based on the cross-sectional area-averaged velocity and pipe radius. Exact solutions for λa=Qi/Qf=0 and λd=Qf/Qi=0 with tc≥tsi are available in the literature and the present generalized analytical solutions fill the remaining range of parameters, 0<λa<1 and 0<λd<1 with 0<tc<tsi and tc≥tsi, where tsi is the time at which the initial flow reaches the temporally developed state. Exact solutions for canonical pipe flow problems reported in the literature are deduced as subsets of the derived generalized solutions. The parametric study reveals the effects of varying λa or λd and tc on the quantities of practical importance, viz., τs and CfRe, where τs is the time required for the final flow to reach the temporally developed state.

Distortion of pipe-flow development by boundary layer growth and unconstrained inlet conditions

This study extends the analysis of the canonical developing pipe-flow problem to realistic inlet conditions affecting emerging jets. A comparison of simulations to existing theory reveals adverse phenomena caused by the inlet: the velocity profile inversion and flow separation (vena contracta) at a sharp inlet. Beginning with the simple uniform inflow, the inversion is shown to persists at significantly higher Re (Re = 2000) than previously reported. It is found to be caused by the theory’s neglected radial velocity, resulting from the boundary layer’s displacement effect. Rescaling of the inlet axial coordinate is shown to collapse all centerline velocity curves above Re = 100, thus elucidating the known weak dependence of entrance-length on Re. The sharp-inlet separation bubble is found not to occur below Re ≅ 320 although this inlet profile overrides the boundary layer’s effect. Furthermore, the bubble’s downstream length increases rapidly with Re, whereas its upstream length grows gradually and proportionally to its thickness—here identified as its characteristic-scale. Beyond the bubble, the profile relaxes to a monotonic form—captured beyond x/(Re·R) = 0.005, if theory is modified using the bubble’s characteristic-scale. This scale also sets the threshold which differentiates between a sharp-inlet regime, accompanied by a separation bubble, and a rounded-inlet one without it. The latter regime relaxes more rapidly to the monotonic profile—captured already beyond x = 2R. Finally, the modified idealized theory is demonstrated as a useful design tool—explicitly relating nozzle length to characteristics of emerging free-surface and submerged jets.

On an analytic solution for general unsteady/transient turbulent pipe flow and starting turbulent flow

Abstract The literature already offers analytic solutions for steady-state laminar (Hagen–Poiseuille), transient laminar (Szymanski) and steady-state turbulent (Pai) incompressible pipe flows. However, no reference could be found for the most general case that encompasses all of them: the unsteady turbulent incompressible pipe flow. This study presents a General Analytic Solution (GAS) for the Reynolds–averaged Navier–Stokes equations corresponding to unsteady turbulent incompressible circular pipe flow, which is composed of the transient response of the initial mean velocity field to external interactions, the unsteady response to time-dependent pressure gradient, and the unsteady component due to the turbulence’s Reynolds stress. The particular cases of laminar, turbulent, steady-state and transient incompressible pipe flows emanate directly from such solution. A general method is proposed to obtain an analytical solution for any particular pipe flow problem for which some relevant function could be provided. That general method is exemplified in a test case, a much simplified version of the starting flow initiated in a circular pipe, after suddenly applying a constant pressure gradient. The time evolution for such starting flow, with a growing stage and a shrinking stage, is presented graphically and discussed.

An efficient preconditioner for the fast simulation of a 2D stokes flow in porous media

SummaryWe consider an efficient preconditioner for a boundary integral equation (BIE) formulation of the two‐dimensional Stokes equations in porous media. While BIEs are well‐suited for resolving the complex porous geometry, they lead to a dense linear system of equations that is computationally expensive to solve for large problems. This expense is further amplified when a significant number of iterations is required in an iterative Krylov solver such as generalized minimial residual method (GMRES). In this paper, we apply a fast inexact direct solver, the inverse fast multipole method, as an efficient preconditioner for GMRES. This solver is based on the framework of ‐matrices and uses low‐rank compressions to approximate certain matrix blocks. It has a tunable accuracy ε and a computational cost that scales as . We discuss various numerical benchmarks that validate the accuracy and confirm the efficiency of the proposed method. We demonstrate with several types of boundary conditions that the preconditioner is capable of significantly accelerating the convergence of GMRES when compared to a simple block‐diagonal preconditioner, especially for pipe flow problems involving many pores.

Conservative formulation of unsteady pipe-flow model for water in its liquid form

We consider flows in a pipe system. The problem relating to the pipe system usually occurs in real life as well as in the laboratory, such as in the distillation process. This unsteady problem can be modeled into a hyperbolic system of partial differential equations. Our goal is to propose the use of a conservative finite-volume numerical method for solving the unsteady pipe-flow model. We obtain that the method is able to solve the pipe-flow problem successfully.

On the History, Science, and Technology Included in the Moody Diagram

This paper is a historical review of the science, both experimental and theoretical, behind the iconic Moody diagram used to avoid tedious iterations choosing pumps and pipes. The large body of historical pipe flow measurements and the choice of dimensionless groups and the Buckingham-Π theorem are also discussed. The traditional use of the Moody diagram to solve common pipe flow problem is discussed. Alternatives to the Moody diagram from the literature and novel ones presented here are shown to produce a solution without iteration for any type of pipe loss problem.

Numerical Solutions to Fast Transient Pipe Flow Problems

We promote a finite volume method to solve a water hammer problem numerically. This problem is of the type of fast transient pipe flow. The mathematical model governing the problem is a system of two simultaneous partial differential equations. As the system is hyperbolic, our choice of numerical method is appropriate. In particular, we consider water flows through a pipe from a pressurized water tank at one end to a valve at the other end. We want to know the pressure and velocity profile in the pipe when the valve closes as a function of time. We find that the finite volume method is very robust to solve the problem.

Efficient parallel implementation of incompressible pipe flow algorithm based on SIMPLE

SummaryParallel semi‐implicit method for pressure‐linked equations(SIMPLE) algorithm is used to solve the 3‐D incompressible pipe flow problem. In this paper, we proposed a novel parallel SIMPLE algorithm that uses the alternate tiling technique. Firstly, a parallel SIMPLE algorithm based on domain decomposition method was established, and the implementation of domain partition and data exchange was presented. Then, we presented serial finite difference stencil algorithm based on alternate tiling. Furthermore, an iteration space parallel two‐way finite difference stencil algorithm based on alternate tiling was proposed, introducing the sequence of iterative space tiles as the sequence of execution and using time skewing technique to partition the iteration space, thus to improve the data locality of algorithm. The cache misses and the cost of communication and synchronization are reduced by reordering the tiles of iteration space. Finally, the effectiveness of the two parallel SIMPLE algorithms were compared. The results showed that the parallel SIMPLE algorithm that uses the two‐way finite difference stencil algorithm based on alternate tiling has good data locality, performance, and scalability in the Deepcomp7000 cluster computing environment. Copyright © 2013 John Wiley & Sons, Ltd.

Cedric Masey White and his solution to the pipe flow problem

Cedric Masey White is a notable British hydraulician of the twentieth century, having particularly contributed to the pipe flow problem in collaboration with his PhD student Cyril Frank Colebrook. Their solution is currently accepted universally, although some particular questions remain as yet unresolved. This biographical work further introduces the professional background of White during his stays at King's College and Imperial College, London, where he became one of the most renowned British hydraulicians mainly because he introduced the post graduate course in hydraulics after World War II. This work also describes how the author was able to find White's daughter in Canada, and how she kindly gave her support in terms of photographs and biographical details. A well-known figure in the field of hydraulics is therefore finally given acknowledgement by an adequate biography.

Analytical solutions for convective fragmentation

Fragmentation of liquid drops occurs in many processes involving convection of dispersions of liquid drops in liquid or gas carrier phases. Such fragmentation is often spatially nonhomogeneous (due, for example, to turbulence induced fragmentation). Recent work has studied a class of pipe flow problems involving convective fragmentation. In the present paper we generalize a version of the problem discussed above to allow for unsteady two-dimensional axisymmetric convective fragmentation and report a corresponding analytical solution. We connect our analytical solution to the grinding solution of Reid (1965) and demonstrate its applicability by applying it to a problem of a pulsating convective fragmentation in an unsteady axisymmetric jet, with the fragmentation characteristics being spatially dependent.

Water-art problems at Sanssouci—Euler’s involvement in practical hydrodynamics on the eve of ideal flow theory

Frederick the Great blamed Euler for the failure of fountains at his summer palace Sanssouci. However, what is regarded as an example for the proverbial gap between theory and practice, is based on dubious evidence. In this paper I review Euler’s involvement with pipeflow problems for the Sanssouci water-art project. Contrary to the widespread slander, Euler’s ability to cope with practical challenges was remarkable. The Sanssouci fountains did not fail because Euler was unable to apply hydrodynamical theory to practice, but because the King ignored his advice and employed incompetent practitioners. The hydrodynamics of the Sanssouci problem also deserves some interest because it happened on the eve of the formulation of the general equations of motion for ideal fluids. Although it seems paradoxical, the birth of ideal flow theory was deeply rooted in Euler’s involvement with real flow problems.

Temperature development in pipe flow with uniform surface heat flux condition considering thermal leakage to the ambient

A practical analytical model for the classical pipe flow problem with constant wall heat flux is presented in which leakage of the heat flux to the ambient is taken into account. The results obtained by the model show that the bulk temperature tends to a limiting value (i.e. does not rise linearly as in the classical constant heat flux problem). A value for the generating heat flux is obtained for which the temperature of the fluid and tube surface remains constant, while there is generated heat over the tube. Also, the analytical solution yields a stationary length for the tube beyond which the change in the fluid temperature is insignificant.

Exact equations for pipe-flow problems/ Des équations exactes pour les problèmes d'écoulement en conduite

The three basic and problems encountered in hydraulic engineering practice are the determination of the head loss, pipe diameter, and discharge. Out of these problems Swamee and Jain [J. Hydraul. Engng. ASCE 102 (1976) 657] gave exact solution for the discharge problem. Exact solutions of the remaining two pipe flow problems are not possible because of the implicit form of Colebrook equation that gives the friction factor for commercial pipes. For the friction factor problem a large number of approximate solutions have been proposed from time to time. Reported herein are exact equations for friction factor and pipe diameter. These exact equations can also be utilized with advantage in optimization studies of pipelines and water distribution systems.

Die-swell, splashing drop and a numerical technique for solving the Oldroyd B model for axisymmetric free surface flows

This work deals with the development of a numerical method for simulating viscoelastic axisymmetric free surface flow of an Oldroyd B fluid. A novel formulation is developed for the computation of the non-Newtonian extra-stress components on rigid boundaries and on the symmetry axis. The full free surface stress conditions are employed. The resulting governing equations are solved by finite differences on a Marker-and-cell (MAC) type grid. Validation is provided by simulating a pipe flow problem. The classical die-swell problem is solved and swelling ratios are provided. The height of the splash caused by a falling liquid drop for various Reynolds and Weissenberg numbers is then studied, and the height of the splash is shown to diminish with increasing viscoelasticity.

Numerical methods for modeling transient flow in distribution systems

The authors compared two numerical methods, one Eulerian‐based, the other Lagrangian‐based, for modeling hydraulic transients in water distribution systems. The method of characteristics (MOC), considered to be the most accurate of the Eulerian methods in its representation of the governing equations, requires a number of steps or calculations to solve a typical transient pipe flow problem. The wave characteristic method (WCM), a Lagrangian approach, tracks movement and transformation of pressure waves and computes new conditions either at fixed intervals or only at times when a change actually occurs. The WCM requires orders of magnitude fewer pressure and flow calculations, allowing large systems to be solved in an expeditious manner. Furthermore, because the method is continuous in both time and space, it is less sensitive to the structure of the network and to the length of the simulation process, resulting in improved computational efficiency. Results indicated that both the MOC and WCM provide fast and accurate results for small pipe systems of short duration transients. However, WCM was superior for transient analysis of large distribution systems because it yields accurate results in a fraction of the computational time required for MOC. Pressure transients can adversely affect the quality of treated water because of potential intrusion by pathogens associated with transient events. Accurate modeling studies can help utility managers determine adequate surge protection, strengthen the integrity of their systems, and minimize costly disruptions to service.

A promising boundary element formulation for three‐dimensional viscous flow

AbstractIn this paper, a new set of boundary‐domain integral equations is derived from the continuity and momentum equations for three‐dimensional viscous flows. The primary variables involved in these integral equations are velocity, traction, and pressure. The final system of equations entering the iteration procedure only involves velocities and tractions as unknowns. In the use of the continuity equation, a complex‐variable technique is used to compute the divergence of velocity for internal points, while the traction‐recovery method is adopted for boundary points. Although the derived equations are valid for steady, unsteady, compressible, and incompressible problems, the numerical implementation is only focused on steady incompressible flows. Two commonly cited numerical examples and one practical pipe flow problem are presented to validate the derived equations. Copyright © 2004 John Wiley & Sons, Ltd.

Parallelization of a Transient Method of Lines Navier–Stokes Code

Parallel implementation of a serial code, namely method of lines (MOL) solution for momentum equations (MOLS4ME), previously developed for the solution of transient Navier–Stokes equations for incompressible separated internal flows in regular and complex geometries, is described. *E-mail: ceme@metu.edu.tr The serial code was parallelized by using a domain decomposition strategy with overlapped boundary at the interfaces for information exchange between the sub-domains. A parallel virtual machine (PVM) and dual-processor personal computer (PC) connected over a switch are employed as the message passing software and the parallel computing platform, respectively. Performance of the parallelization strategy was first tested on a transient 1-D laminar pipe flow problem for MOL, explicit and implicit finite difference methods. Assessment of the performance of the parallel code (PARMOLS4ME) was then tested by applying it to two test cases; (i) laminar pipe flow with sudden expansion and (ii) turbulent flow in a gas turbine combustor simulator (GTCS) and comparing the results of serial and parallel codes. Comparisons show that the flow fields predicted by parallel codes agree well with those of serial codes at considerably lower execution times. Effects of load balancing are also investigated and it is seen that the load balancing has a significant effect on the execution time of the parallel code.

Comparison of method of lines and finite difference solutions of 2‐D Navier–Stokes equations for transient laminar pipe flow

AbstractPerformances of method of lines (MOL) and finite difference method (FDM) were tested from the viewpoints of solution accuracy and central processing unit (CPU) time by applying them to the solution of time‐dependent 2‐D Navier–Stokes equations for transient laminar flow without/with sudden expansion and comparing their results with steady‐state numerical predictions and measurements previously reported in the literature. Predictions of both methods were obtained on the same computer by using the same order of spatial discretization and the same uniform grid distribution. Axial velocity and pressure distribution in pipe flow and steady‐state reattachment lengths in sudden expansion flow on uniform grid distribution predicted by both methods were found to be in excellent agreement. Transient solutions of both methods for pipe flow problem show favourable comparison and are in accordance with expected trends. However, non‐physical oscillations were produced by both methods in the transient solution of sudden expansion pipe flow. MOL was demonstrated to yield non‐oscillatory solutions for recirculating flows when intelligent higher‐order discretization scheme is utilized for convective terms. MOL was found to be superior to FDM with respect to CPU and set‐up times and its flexibility for incorporation of other conservation equations. Copyright © 2001 John Wiley & Sons, Ltd.