Final Steady State Flow Research Articles

Modelling the reservoir-to-tubing pressure drop imposed by multiple autonomous inflow control devices installed in a single completion joint in a horizontal well

Autonomous inflow control devices (AICDs) are used to introduce an additional pressure drop between the reservoir and the tubing of a production well that depends on the fluid phase flowing into the device: a larger pressure drop is introduced when unwanted phases such as water or gas enter the AICD. The additional pressure drop is typically represented in reservoir simulation models using empirical relationships fitted to experimental data for a single AICD. This approach may not be correct if each completion joint is equipped with multiple AICDs as the flow at different AICDs may be different. We use high-resolution numerical modelling to determine the total additional pressure drop introduced by two AICDs installed in a single completion joint in a horizontal well. The model captures the multiphase flow of oil and water through the inner annulus into each AICD. We explore a number of relevant oil-water inflow scenarios with different flow rates and water cuts. Our results show that if only one AICD is installed, the additional pressure drop is consistent with the experimentalzly-derived empirical formulation. However, if two AICDs are present, there is a significant discrepancy between the additional pressure drop predicted by the simulator and the empirical relationship. This discrepancy occurs because each AICD has a different total and individual phase flow rate, and the final steady-state flow results from a self-organising mechanism emerging from the system. We report the discrepancy as a water cut-dependent correction to the empirical equation, which can be used in reservoir simulation models to better capture the pressure drop across a single completion containing two AICDs. Our findings highlight the importance of understanding how AICDs modify flow into production wells, and have important consequences for improving the representation of advanced wells in reservoir simulation models.

Second Law Analysis of Unsteady MHD Viscous Flow over a Horizontal Stretching Sheet Heated Non-Uniformly in the Presence of Ohmic Heating: Utilization of Gear-Generalized Differential Quadrature Method.

In this article, the entropy generation characteristics of a laminar unsteady MHD boundary layer flow are analysed numerically for an incompressible, electrically conducting and dissipative fluid. The Ohmic heating and energy dissipation effects are added to the energy equation. The modelled dimensional transport equations are altered into dimensionless self-similar partial differential equations (PDEs) through suitable transformations. The reduced momentum and energy equations are then worked out numerically by employing a new hybrid method called the Gear-Generalized Differential Quadrature Method (GGDQM). The obtained numerical results are incorporated in the calculation of the Bejan number and dimensionless entropy generation. Quantities of physical interest, like velocity, temperature, shear stress and heat transfer rate, are illustrated graphically as well as in tabular form. Impacts of involved parameters are examined and discussed thoroughly in this investigation. Exact and GGDQM solutions are compared for special cases of initial unsteady flow and final steady state flow. Furthermore, a good harmony is observed between the results of GGDQM and those given previously by the Spectral Relaxation Method (SRM), Spectral Quasilinearization Method (SQLM) and Spectral Perturbation Method (SPM).

Transient Combined Convective Heat Transfer over a Stretching Surface in a Non-Newtonian Nanofluid Using Buongiorno’s Model

The present paper investigates the transient mixed convective boundary layer flow of an incompressible non-Newtonian quiescent nanofluid adjacent to a vertical stretching surface. The effects of the Brownian motion and thermophoresis are included for the nanofluid. Using appropriate non-similarity transformations the non-dimensional, coupled and highly non-linear system of equations is solved numerically using the efficient Keller-box implicit finite difference method for the whole transient from t=0 (initial state) to (final steady-state flow). The box method is unconditionally stable. Numerical results for dimensionless velocity (f’), micro-rotation (g), temperature (θ), nanoparticle volume fraction (Φ) at final steady state flow, skin friction function (), Nusselt number function () and Sherwood number function () have been presented on various parameters inform of tables and graphs. The results indicate that as Nb and Nt increase, the Nusselt number decreases whereas Sherwood number increases at initial and early state time but decreases at the final steady state time. As the K increases, the friction factor decreases whereas surface mass transfer rate and the surface heat transfer rates slightly increase. The results reveal that there is a smooth transition of flow from unsteady state to the final steady state. A special case of our results is in good agreement with an earlier published work. The study has many practical applications such as extrusion of plastic sheets, paper production, glass blowing, metal spinning and drawing plastic films.

Soret and Dufour effects on mixed convection unsteady MHD boundary layer flow over stretching sheet in porous medium with chemically reactive species

This paper studies the thermal-diffusion and diffusion thermo-effects in the hydro-magnetic unsteady flow by a mixed convection boundary layer past an impermeable vertical stretching sheet in a porous medium in the presence of chemical reaction. The velocity of the stretching surface, the surface temperature, and the concentration are assumed to vary linearly with the distance along the surface. The governing partial differential equations are transformed into self-similar unsteady equations using similarity transformations and solved numerically by the Runge-Kutta fourth order scheme in association with the shooting method for the whole transient domain from the initial state to the final steady state flow. Numerical results for the velocity, the temperature, the concentration, the skin friction, and the Nusselt and Sherwood numbers are shown graphically for various flow parameters. The results reveal that there is a smooth transition of flow from unsteady state to the final steady state. A special case of our results is in good agreement with an earlier published work.

Effects of Hall Current and Rotation on Unsteady MHD Couette Flow in the Presence of an Inclined Magnetic Field

Unsteady hydromagnetic Couette flow of a viscous incompressible electrically conducting fluid in a rotating system in the presence of an inclined magnetic field taking Hall current into account is studied. Fluid flow within the channel is induced due to impulsive movement of the lower plate of the channel. Exact solution of the governing equations is obtained by Laplace transform technique. The expression for the shear stress at the moving plate is also derived. Asymptotic behavior of the solution is analyzed for small and large values of time t to highlight (i) the transient approach to the final steady state flow and (ii) the effects of Hall current, magnetic field, rotation and angle of inclination of magnetic field on the flow-field. It is found that Hall current and rotation tend to accelerate fluid velocity in both the primary and secondary flow directions. Magnetic field has retarding influence on the fluid velocity in both the primary and secondary flow directions. Angle of inclination of magnetic field has accelerating influence on the fluid velocity in both the primary and secondary flow directions.

Influence of pump starting times on transient flows in pipes

The purpose of this study is the numerical modelling of the transient flow in a cylindrical pipe caused by the start-up of a centrifugal pump. The pump starting produced by the speed of the electric motor, which rises from zero up to the rated speed, influences the flow in the hydraulic installation and forced it to follow the starting dynamic law of the motor. The governing equations for such flows are two coupled hyperbolic partial differential equations which are the equations of continuity and motion. The head and the discharge are considered as two principal dependent variables. Under the effect of the friction force, due to the fluid viscosity, the transient flow is progressively dumped until reaching the final steady-state flow. Two boundary conditions are considered: a pump with a known motor torque at the upstream end and a constant level reservoir at the downstream end of the pipe. The motor torque is considered for different starting regimes: very rapid, rapid, medium and slow. In order to predict the dynamic behaviour of the flow during the pump start-up transient conditions, the mathematical model is solved by the method of characteristics. The computed head and discharge, caused by the starting of the pump, are presented at some cross sections of the pipe and the results show that the evolution of the hydraulic variables is well influenced by the pump starting time.

Unsteady Free Convection Flow near the Stagnation Point of a Three-dimensional Body

This study deals with the study of unsteady free convection in the stagnation point region of a three-dimensional body placed in an ambient fluid under the assumption of a step change in the surface temperature of the body. The non-linear coupled partial differential equations governing the free convection flow are solved numerically using an implicit finite-difference method for different values of the governing parameters entering these equations. Results for the flow and heat characteristics show that the transition from the initial unsteady-state flow to the final steady-state flow takes place smoothly. © 2011 Asian Network for Scientific Information.

Unsteady flow across a stretching surface

In this paper, the problem of unsteady uniform flow across a stretching surface in an arbitrary direction is studied theoretically, where the unsteadiness is caused by the impulsive motion of the stretching surface. Numerical results of the governing partial differential equations are obtained using an implicit finite-difference scheme for the whole transient from the early or initial unsteady-state flow to the final steady-state flow. The early unsteady-state flow is solved analytically. The numerical solution obtained for the reduced skin friction coefficient is compared with previously reported results and the results for velocity profiles, h and g profiles are also presented in this paper. It is found that there is a smooth transition from the small-time solution (initial unsteady flow) to the large-time solution (final steady-state flow).

HEAT AND MASS TRANSFER IN TRANSIENT FLOW BY MIXED CONVECTION BOUNDARY LAYER OVER A STRETCHING SHEET EMBEDDED IN A POROUS MEDIUM WITH CHEMICALLY REACTIVE SPECIES

An analysis is carried out to study the coupled heat and mass transfer in transient flow by a mixed convection boundary layer past an impermeable vertical stretching sheet embedded in a fluid-saturated porous medium in the presence of a chemical reaction effect. The stretching velocity, the surface temperature, and concentration are assumed to vary linearly with the distance along the surface. The flow is impulsively set into motion rest, and both the temperature and concentration at the surface are also suddenly changed from that of the ambient fluid. The governing partial differential equations are transformed into the self-similar unsteady boundary layer equations and solved by the Runge-Kutta integration scheme with shooting method for the whole transient from initial state (ξ = 0) to final steady state flow (ξ = ∞). Numerical results for the velocity, temperature, and concentration profiles as well as the variation of the local skin friction coefficient, local Nusselt and Sherwood numbers with the Darcy number, buoyancy, and chemical reaction parameters are presented graphically and discussed. This is done to elucidate the influence of the various parameters involved in the problem on the solution.

Unsteady mixed convection flow of a micropolar fluid near the stagnation point on a vertical surface

The unsteady mixed convection boundary-layer flow of a micropolar fluid near the region of the stagnation point on a double-infinite vertical flat plate is studied. It is assumed that the unsteadiness is caused by the impulsive motion of the free stream velocity and by sudden increase or sudden decrease in the surface temperature from the uniform ambient temperature. The problem is reduced to a system of non-dimensional partial differential equations, which is solved numerically using the Keller-box method. This method may present well-behaved solutions for the transient (small time) solution and those of the steady-state flow (large time) solution. It was found that there is a smooth transition from the small-time solution (initial unsteady-state flow) to the large-time solution (final steady-state flow). Further, it is shown that for both assisting and opposing cases and a fixed value of the Prandtl number, the reduced steady-state skin friction and the steady-state heat transfer from the wall (or Nusselt number) decrease with the increase of the material parameter. On the other hand, it is shown that with the increase of the Prandtl number and a fixed value of the material parameter, the reduced steady-state skin friction decreases when the flow is assisting and it increases when the flow is opposing.

Influence of grain boundaries on steady-state deformation resistance of ultrafine-grained Cu

The deformation behavior in uniaxial compression of ultrafine-grained (UFG) pure Cu produced by severe plastic deformation through equal channel angular pressing (12 passes route C) with initial spacing of large angle grain boundaries of about 350 nm is compared to that of initially coarse-grained (CG) Cu with . UFG Cu attains its steady state of deformation within a small strain interval of about 0.05. CG Cu needs much larger strains to approach the steady-state flow stress (Voce stress) terminating stage III hardening and the final steady-state flow stress . The dependences of the steady-state flow stresses on strain rate and temperature differ significantly for UFG and CG Cu. In consequence, UFG Cu becomes softer in the steady state of deformation compared to CG Cu with regard to both as well as . This is explained in terms of annihilation of lattice dislocations at large-angle grain boundaries lowering the steady-state deformation resistance. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Unsteady boundary layer flow in the region of the stagnation point on a stretching sheet

Unsteady two-dimensional stagnation point flow of an incompressible viscous fluid over a flat deformable sheet is studied when the flow is started impulsively from rest and the sheet is suddenly stretched in its own plane with a velocity proportional to the distance from the stagnation point. After a similarity transformation, the unsteady boundary layer equation is solved numerically using the Keller-box method for the whole transient from τ=0 to the steady state τ→∞. Also, a complete analysis is made of the governing equation at τ=0, the initial unsteady flow, at large times τ=∞, the steady state flow, and a series solution valid at small times τ (≪1). It is found that there is a smooth transition from the initial unsteady state flow (small time solution) to the final steady state flow (large time solution).

Unsteady mixed convection boundary layer flow near the stagnation point on a vertical surface in a porous medium

The unsteady mixed convection boundary layer flow near the region of a stagnation point on a vertical surface embedded in a Darcian fluid-saturated porous medium is studied in this paper. It is assumed that the unsteadiness is caused by the impulsive motion of the free stream velocity and by sudden increase in the surface temperature. The problem is reduced to a single partial differential equation, which is solved numerically using the Keller–Box method. The small time (initial unsteady flow) as well as the large time (final steady state flow) solutions are also included in the analysis. The asymptotic behavior of the solution for small and large values of the mixed convection parameter λ is also examined when the flow becomes steady. It is shown that there is a smooth transition from the small time solution to the large time solution. It is also shown that there is an excellent agreement between the numerical and analytical solutions. The uniqueness of this problem lies on the fact that we have been able to show that in the case of steady state flow, solutions are possible for all values of λ>0 (assisting flow) and for λ<0 (opposing flow), solutions are possible only for a limited range of λ.

Unsteady mixed convection near the forward stagnation point of a two-dimensional symmetric body

The unsteady mixed convection boundary layer flow near the forward stagnation point of a two-dimensional symmetric body resulting from an impulsive motion of the free stream velocity and by sudden increase in the surface temperature. The partial differential equations governing the flow and heat transfer have been solved numerically using Keller-box method. It is shown that there is a smooth transition from the unsteady initial flow (short time) to the final steady state flow(large time). It is also found that for the steady flow case that are dual solutions when the flow is opposing

A new algorithm for three-dimensional tidal and storm surge computations

A computational algorithm is proposed for the solution of the hydrodynamical equations appropriate for motions in shallow seas and estuaries. The surface elevation and depth-averaged velocity components are first computed using one of the standard two-dimensional algorithms for the solution of the shallow-water equations. These values are then used to compute the vertical profile of the velocity field by solving the two horizontal momentum equations. These equations form a coupled parabolic system, and a generalized Crank-Nicholson scheme is developed to solve them. In the case of constant eddy viscosity this scheme can be shown to be unconditionally stable, so the only stability restriction is the usual CFL condition on the shallow-water equations. Thus the time step is limited only in terms of the horizontal grid spacing, not by the vertical spacing. The algorithm is tested on a model problem previously used by Davies and Owen 1 involving the development of circulation in a rectangular sea under the influence of a constant surface wind-shear. The velocity profiles at various times are presented, and the final steady-state flow agrees very well with that computed by Davies and Owen. For constant eddy-viscosity the complete three-dimensional computation is only twice as slow as the two-dimensional computation that forms part of it.

Time-Dependent Laminar Backward-Facing Step Flow in a Two-Dimensional Duct

This paper presents results of numerical studies of the impulsively starting backward-facing step flow with the step being mounted in a plane, two-dimensional duct. Results are presented for Reynolds numbers of Re = 10; 368 and 648 and for the last two Reynolds numbers comparisons are given between experimental and numerical results obtained for the final steady state flow conditions. In the computational scheme, the convective terms in the momentum equations are approximated by a 13-point quadratic upstream weighted finite-difference scheme and a fully implicit first order forward differencing scheme is used to discretize the temporal derivatives. The computations show that for the higher Reynolds numbers, the flow starts to separate on the lower and upper corners of the step yielding two disconnected recirculating flow regions for some time after the flow has been impulsively started. As time progresses, these two separated flow regions connect up and a single recirculating flow region emerges. This separated flow region stays attached to the step, grows in size and approaches, for the time t → ∞, the dimensions measured and predicted for the separation region for steady laminar backward-facing flow. For the Reynolds number Re = 10 the separation starts at the bottom of the backward-facing step and the separation region enlarges with time until the steady state flow pattern is reached. At the channel wall opposite to the step and for Reynolds number Re = 368, a separated flow region is observed and it is shown to occur for some finite time period of the developing, impulsively started backward-facing step flow. Its dimensions change with time and reduce to zero before the steady state flow pattern is reached. For the higher Reynolds number Re = 648, the secondary separated flow region opposite to the wall is also present and it is shown to remain present for t → ∞. Two kinds of the inlet conditions were considered; the inlet mean flow was assumed to be constant in a first study and was assumed to increase with time in a second one. The predicted flow field for t → ∞ turned out to be identical for both cases. They were also identical to the flow field predicted for steady, backward-facing step flow using the same numerical grid as for the time-dependent predictions.

On the Effect of Trapped Air in a Liquid Conduit on the Transient Flow Rate

Trapped air in a liquid conduit has a great influence on the fluid transient. In this paper, especially its effect on the transient flow rate is studied experimentally and analytically. Transient flow velocity, following a quick opening of a valve attached at the downstream end of the test pipeline, was measured experimentally. The experiments showed that the transient flow velocity often exceeds the final steady state flow velocity. The numerical calculation by means of characteristics method showed a good agreement with the experimental results. Maximum value of the transient flow velocity depends on the volume and the position of the trapped air, and the pressure loss of the pipeline and the valve, and the pressure at the downstream side of the valve. Five dimensionless parameters, which represent above-mentioned factors respectively, are derived, and the relations between the maximum flow velocity and these five parameters are expressed graphically.

On the effect of trapped air in a liquid conduit on the transient flow rate.

Trapped air in a liquid conduit has a great influence on the fluid transient. In this paper, especially its effect on the transient flow rate is studied experimentally and analytically. Transient flow velocity, following a quick opening of a valve attached at the downstream end of the test pipeline, was measured experimentally. The experiments showed that the transient flow velocity often exceeds the final steady state flow velocity. The numerical calculation by means of characteristics method showed a good agreement with the experimental results. Maximum value of the transient flow velocity depends on the volume and the position of the trapped air, and the pressure loss of the pipeline and the valve, and the pressure at the downstream side of the valve. Five dimensionless parameters, which represent above-mentioned factors respectively, are derived, and the relations between the maximum flow velocity and these five parameters are expressed graphically.

Upwelling and Coastal Jets in a Continuously Stratified Ocean

A simple, linear, two-dimensional, f-plane model of coastal upwelling in a continuously stratified ocean is investigated. The transient response of the fluid to an impulsively applied alongshore wind stress, the nature of the approach to a steady state, and the final steady state flow are studied. Particular attention is given here to an examination, in the continuously stratified case, of the interrelation of upwelling and the phenomenon of a baroclinic coastal jet. The coastal jet is a feature which has appeared in studies of the transient response of two-layer fluid models near a coast. The stratification parameter S = (δN/f)2, where δ = II/L is the aspect ratio, H the depth, L a characteristic horizontal scale, N the Brunt-Väisälä frequency, and f the Coriolis parameter, is assumed to lie in the range O(Ev½)<S≤O(1), where λ = Ev/EH and where EH and Ev are respectively the horizontal and vertical Ekman numbers. It is also assumed that L is large enough that S≪1 and λ≫1. These are the conditions for which, during the transient flow, a baroclinc coastal jet is present in an inviscid upwelling layer of width O(S½). It also appears that these conditions are reasonable for oceanic upwelling regions. The jet development is initiated on a time scale of O(Ev−½). A steady state is reached in the O(S½) upwelling layer on a longer diffusive time scale of O(SEH−1). For Sλ<O(1), the alongshore current outside the O(S½) layer increases in magnitude with time, by diffusive spreading of the boundary effects, in an additional boundary layer of width O(λ−½). It reaches a steady state on a yet longer time scale of O(Ev−1). For Sλ = O(1), the two boundary layers merge and adjust on the same time scale. The final steady state is characterized by a coastal current which is confined to the O(S½) and O(λ−½) boundary layers.

Valve Stroking for Complex Piping Systems

A theory is developed for movement of valves in complex piping systems so that the transient ceases when the valve movement ceases. One pipe in the system is selected and the head-time relations determined so that this flow is adjusted in a controlled manner not exceeding predetermined head limits. In a complex system, the flow changes are apportioned among the branches as a linear relation of initial to final steady-state flow in each pipe. The combination of these two procedures leads to values of head and flow at each control point in the system; hence, the valves or other moving boundaries may be adjusted to cause the desired transient change to occur. Frictional effects are included in the analysis which is based on the method of characteristics solution of the transient flow equations. The theory has been verified experimentally and results are given for two cases using 4,000 ft of 0.95-in. copper tubing.