Numerical Analysis of Transient Flow through a Pipe Orifice. 2nd Report. Step Response from Initinal Steady Flow.

The present lecture summarizes some of the most recent joint research results from the cooperation between the Federal University of Rio de Janeiro, Brasil, and the University of Miami, USA, on the transient analysis of both fluid flow and heat transfer within microchannels. This collaborative link is a natural extension of a long term cooperation between the two groups, in the context of fundamental work on transient forced convection, aimed at the development of hybrid numerical-analytical techniques and the experimental validation of proposed models and methodologies [19]. The motivation of this new phase of the cooperation was thus to extend the previously developed hybrid tools to handle both transient flow and transient convection problems in microchannels within the slip flow regime. The analysis of internal flows in the slip-flow regime recently gained an important role in association with the fluid mechanics of various microelectromechanical systems (MEMS) applications, as well as in the thermal control of microelectronics, as reviewed in different sources [10-16]. For steady-state incompressible fully developed flow situations and laminar regime within simple geometries such as circular microtubes and parallel-plate microchannels, explicit expressions for the velocity field in terms of the Knudsen number are readily obtainable, and have been widely employed in the heat transfer analysis of microsystems, such as in [17-23]. Only quite recently, attention has been directed to the analysis of transient flow in microchannels [24-33]. Unsteady one-dimensional models have been extended from classical works, and analytical solutions have been sought for fully developed flows in simple geometries. These recent works are also concerned with situations in which a simple and well-defined functional form for the pressure gradient time variation is prescribed or for the time dependence of the wall imposed velocity, in the case of a Couette flow application. Research findings are yet to be further pursued in the analytical and robust solution of more generalized models, which will accommodate more general conditions and parameter specifications, and thus offer a wider validation range for the automatic general purpose numerical codes. Mikhailov and Ozisik [34] presented a unified solution for transient one-dimensional laminar flow models, with the usual no-slip boundary condition, based on the classical integral transform method. Their solution was then specialized to two situations: step change and periodically varying pressure gradient. The knowledge in regular size channels is therefore fairly well consolidated for models that use simple functional forms for the pressure gradient variation such as for the two cases cited above. One of the objectives of this paper is to illustrate the solution of a onedimensional mathematical model for transient laminar incompressible flow in microchannels such as circular tubes and parallel-plate channels, that accounts for a source term time variation in any functional form, including electrokinetic effects for liquid flows, by making use of the Generalized Integral Transform Technique (GITT) [35-40], and thus yielding analytical expressions for the time and space dependence of the velocity fields in the fully developed region. We then demonstrate this hybrid numerical-analytical solution for transient internal slip flow, obtained employing mixed symbolic-numerical computations with the Mathematica platform [41]. The goal here is to improve and complement existing analytical solution implementations to study laminar fully developed flows in micro-ducts subjected to arbitrary source term disturbances in space and time. On the other hand, the heat transfer literature of the last decade has demonstrated a vivid and growing interest in thermal analysis of flows in micro-channels, both through experimental and analytical approaches, in connection with cooling techniques of micro-electronics and with the development of micro-electromechanical sensors and actuators (MEMS), as also pointed out in recent reviews [12-16]. Since the available analytical information on heat transfer in ducts could not be directly extended to flows within microchannels with wall slip, a number of contributions have been recently directed towards the analysis of internal forced convection in the micro-scale. In the paper by Barron et al.

The objective of the present paper is a numerical analysis of the transient flow through a 4/3 hydraulic closed center direction control valve. The real-time discharge coefficients corresponding to supply flow difference have been considered. In order to develop an intelligent control system for an electro-hydraulic system (EHS), a new controller has recently been proposed to control those parameters in a complex system. However, a mathematical model of the EHS including the transient parameters has not been clarified. The main objective of this paper is to numerically analyze the dynamic characteristics of the directional control valve and also the flow behaviors in the electro-hydraulic system control. Both the steady state and transient flow though the valve which affect to flow-pressure coefficient were numerically considered. Moreover, this paper presents numerical results, which explain the flow behaviors related with the real-time pressure drop and discharge coefficient. Both a high inlet flow and a large opening spool have determined the pressure drop increment.

Rapid transients are particularly dangerous in industrial hydro-transport systems, where solid-liquid mixtures are transported via long pressure pipelines. A mathematical description of such flow is difficult due to the complexity of phenomena and difficulties in determining parameters. The main aim of the study was to examine the influence of the simplified mixture density and wave celerity description on satisfactory reproduction of pressure characteristics during the transient flow of slurry at low concentrations. The paper reports and discusses the selected aspects of experimental and numerical analyses of transient slurry flow in a polyethylene pipe. The experiments were conducted by using the physical model of a slurry’s transportation pressure. The aim of the experiments was to determine the wave celerity during a transient flow in slurries. A low concertation of slurries, which was used during experiments, is typical for one of the biggest slurry networks in Poland. A comparison of the effects of different wave celerity descriptions was performed. The research reported that the theoretical formulas for slurry wave celerity and mixture density were not sufficiently accurate to obtain satisfactory compliance between calculations and observations. To improve the model, the experimental values of wave celerity and the concept of equivalent mixture density have been applied to indirectly consider the influence of variable mixture parameters. With such modifications, the calculated pressure characteristics in all analyzed episodes demonstrated satisfactory compliance with observations. The simplified approach proved to be effective in properly reproducing the intensity and frequency of rapid pressure changes.

Coriolis devices are continuously evolving to meet the demands of different conditions, such as wet gas flow. However, their application in wet gas flow has not yet been thoroughly explored. The impact of steady flow disturbances on Coriolis flow meters is well-documented, and empirical compensation or correction methods can be implemented accordingly. However, there has been inadequate investigation into the response of Coriolis meters under transient flow conditions and their comparison with steady flow in a wet gas. In this study, a Coriolis device was horizontally installed in a 50 mm pipe diameter. The experimental fluids consisted of air and water, with Lockhart-Martinelli (XLM) values ranging from 0.02 to 0.40. Steady and transient flow conditions at different gas and liquid flow rates were studied. The findings demonstrate the capability of standard deviation (STD) in distinguishing transient flow from steady one. Additionally, a strong correlation was observed between XLM and gas Over-Reading (OR) across various gas flow rates and XLM values. This correlation is particularly evident for XLM < 0.1. At extremely low liquid loading (XLM < 0.05), the average percentage error remains below 7 % even without the utilization of any correction models. Furthermore, the impact of different sensor installations, which had been largely overlooked in previous studies, was investigated.

In this study, a transient flow in a spool valve has been studied via numerical analysis. First, a numerical procedure is developed for non-uniform grid systems. A consistent formulation of the QUICK scheme is derived for non-uniform rectangular grid systems. A transient flow field was then investigated for the condition with several values of the spool stroke and a large pressure difference between upstream and downstream boundaries. The computational result was used to obtain a mathematical model for the spool valve flow transient in high Reynolds number condition of practical importance. Unsteady characteristics of the axial flow force acting on the spool valve was investigated based on the pressure and shearing stress distribution of the computational result. Prediction of the unsteady flow force based on the momentum theory was examined using the computational result.

Transporting natural gas and gas condensate in a long distance pipeline occurs frequently during the development of offshore or desert gas condensate and/or oil fields. However, the thermohydraulic calculation of gas-condensate pipeline, especially transient flow simulation, is hitherto a challenging issue in the pipeline industry on account of a maze of complexities of pipeline undulation, changeable properties of fluid, and transfer of momentum, mass and heat. This study is intended to predict the transient flow behavior in gas-condensate pipelines. In the paper, a hydraulic and thermodynamic (such as phase behavior and properties) model for the analysis of transient gas-condensate two-phase flow in pipelines with low liquid loading is outlined. The hydraulic model is based on simplified “No Pressure Wave” model where the constitutive relation results from the Ottens et al (2001) correlation. An implicit method, the convergence and stability of which have been verified by example calculations, is utilized to solve the transient flow model equations of gas-condensate pipelines. In the end, the transient performances of low-liquid-loading gas-condensate two-phase non-isothermal flow in undulating pipelines, which are subjected to boundary conditions of increasing or decreasing inlet flow rate and specified outlet pressure with time, are numerically investigated. The results, such as pressure and liquid holdup profiles vs. time, and time evolutions of outlet condensate flow rate and accumulated liquid content etc., show that the presented model and numerical method for analyzing gas-condensate transient flow behaviors in pipelines looks reasonable.

Horizontal wells and hydraulic fracturing are the key technologies that allow commercial production from tight oil and gas reservoirs. However, rigorous analysis of production data from these reservoirs requires incorporation of the impacts of stress-dependent permeability and multi-phase flow. Changes in the stress state of the system during production may reduce the absolute permeability. Furthermore, gas phase formation and flow in presence of supersaturated oil phase affects fluid dynamics in tight oil reservoirs. This study provides a rigorous methodology for incorporation of the effects of non-static permeability and multi-phase flow in rate transient analysis (RTA) of tight oil and gas reservoirs producing at variable rate/flowing pressures during transient linear flow period. Analytical solutions for the approximate linearized form of the flow equation have been widely used as the basis for RTA tools for conventional reservoirs during transient flow period. However, they lead to considerable error when applied to tight oil and gas reservoirs. In particular, during the transient linear flow period, the slope of the square -root-of-time plot obtained from numerical solution differs from the slope calculated by analytical methods. Efforts have been made by some researchers to obtain a correction factor from the numerical solution of the flow equation to correct the slope of the square-root-of-time plot for single phase flow of gas during transient linear flow period. In this study, an iterative method is used for evaluation of the slope correction factor in the presence of multi-phase flow and non-static permeability for constant-pressure production during transient linear flow period. Further, the correction factor is used for analysis of production data from tight oil and gas reservoirs producing at variable rate/flowing pressures. The correction factor is used in the analysis of different sets of synthetic production data for tight oil and gas reservoirs. The results show that the correction factor can reduce/eliminate the considerable errors associated with the conventional analytical methods in initial permeability estimation. For multi-phase flow cases, the producing gas -oil ratio (GOR) is used to estimate the oil saturation-pressure relationship in the reservoir, which is required for calculation pseudo-pressure and the correction factor. The method developed in this study alleviates the need for using numerical simulation models to generate empirical correlations for the correction factor for the square -root-of-time plot. The easy-to-implement iterative procedure of this method only requires the pressure dependencies of the constituent elements of the hydraulic diffusivity. Therefore, this method is applicable for analysis of production profiles for variety of reservoirs with nonlinear flow equations.

The steady and transient internal flows within an automotive fuel injector are studied by solving ensemble-averaged, three-dimensional mass and momentum conservation equations. In the analysis, the effects of operating pressure, orifice size, and needle lift on the static flow rate, and the effect of opening rate on the dynamic flow rate have been parametricly investigated. Pressure drop was varied from 100 kPa to 500 kPa; the orifice diameter and needle lift ranged between 400 μm to 700 μm and 60 μm to 120 μm, respectively. A multiblock moving mesh method was used to simulate the needle motion. Predicted static flow rates provide good agreements with the experimental measurements. Static flow increases as the orifice diameter increases. For a constant orifice diameter, static flow rate is found sensitive to the changes of the needle lift for a large nozzle while relatively insensitive for the smaller one. Dynamic flows under various needle opening rates are calculated and results are presented.

This paper simulates the 2D transient magnetohydrodynamic (MHD) flow in a rectangular duct in terms of the velocity of the fluid and the induced magnetic field by using the radial basis function (RBF) approximation.  The inhomogeneities in the Poisson’s type MHD equations are approximated using the polynomial functions (1+r) and the particular solution is found satisfying both the equations and the boundary conditions (no-slip and insulated walls).  The Euler scheme is used for advancing the solution to steady-state with a time increment and a relaxation parameter which are determined for achieving stable solution.  It is shown that, as Hartmann number increases, the fluid becomes stagnant at the center of the duct, the flow is flattened and boundary layers are developed on the Hartmann and side walls.  These are the well-known characteristics of the MHD duct flow.  The stability analysis is also carried in terms of the spectral radius of the coefficient matrix of the discretized coupled system.  Stable solutions are obtained with RBF by using quite large time increment and suitable relaxation parameters on the expense of explicit Euler time-integration scheme used.

Physics-informed deep learning for incompressible laminar flows

Numerical analysis of transient mixed convection flow in storage tank: influence of fluid properties and aspect ratios on stratification

This paper investigates the effects of large‐scale temporal velocity fluctuations, particularly changes in the direction of flow, on solute spreading in a two‐dimensional aquifer. Relations for apparent longitudinal and transverse dispersivity are developed through an analytical solution for dispersion in a fluctuating, quasi‐steady uniform flow field, in which storativity is zero. For transient flow, spatial moments are evaluated from numerical solutions. Ignored or unknown transients in the direction of flow primarily act to increase the apparent transverse dispersivity because the longitudinal dispersivity is acting in a direction that is not the assumed flow direction. This increase is a function of the angle between the transient flow vector and the assumed steady state flow direction and the ratio of transverse to longitudinal dispersivity. The maximum effect on transverse dispersivity occurs if storativity is assumed to be zero, such that the flow field responds instantly to boundary condition changes.

The incompressible Smoothed Particle Hydrodynamics (ISPH) method is derived in Eulerian form with high-order smoothing kernels to provide increased accuracy for a range of steady and transient internal flows. Periodic transient flows, in particular, demonstrate high-order convergence and accuracies approaching, for example, spectral mesh-based methods. The improved accuracies are achieved through new high-order Gaussian kernels applied over regular particle distributions with time stepping formally up to 2nd order for transient flows. The Eulerian approach can be easily extended to model free surface flows by merging from Eulerian to Lagrangian regions in an Arbitrary-Lagrangian–Eulerian (ALE) fashion, and a demonstration with periodic wave propagation is presented. In the long term, it is envisaged that the method will greatly increase the accuracy and efficiency of SPH methods, while retaining the flexibility of SPH in modelling free surface and multiphase flows.

We studied steady and transient mass flow in a localized traveling wave state of convection of binary fluid mixture in an annular cell by use of a photochromic dye. The results indicate that no mass flow occurs across the self-organized boundary in a steady state, but that a large-scale flow exists at the transient stage to establish a localized traveling state. The transient flow was found to occur in the direction to lower the local critical Rayleigh number and to stabilize the localized convection state.

The transient supersonic flow of a slightly conducting gas in an MHD channel with axial symmetry is investigated assuming that the magnetic Reynolds number and the interaction parameter are small. The solutions to the linearized equations for supersonic flow show that in the asymptotic limit as t →∞, these solutions all tend to coincide with the solutions for the corresponding steady flow. This point is remarkably different as compared with the result for the transient subsonic flow in a channel. The reason for this discrepancy is made clear. The time for the transient flow to achieve the steady state is also evaluated.