System Of Flow Equations Research Articles

Research on a Simulation and Prediction Method of Traffic Flow Information Based on PINN

Traffic flow prediction can help intelligent transportation effectively be carried out. This paper employs the Physics-Informed Neural Network (PINN) as a deep learning method and partial differential equation (PDE) multivariable methods to approximate solutions for the Lighthill-Whitham-Richards (LWR) traffic flow equation system. The relationship between traffic flow density function and time/displacement is obtained by setting different function conditions and initial boundary constraints. Furthermore, a series of three-dimensional graphics are generated during the research process through dynamic comparisons and evaluations of independent variables to intuitively display and analyze traffic flow trends and gradients. Additionally, before the commencement of the study, the authors propose specific initial condition combinations corresponding to real-world application scenarios to enhance the practicality of the research. Moreover, through the interpretation and reflection on the obtained images, the study identifies inconsistencies between actual situations and predictions, leading to necessary adjustments. Furthermore, the research explores the compatibility of the u(x) function and hard constraint conditions in different scenarios, aiming to accurately predict traffic information and ultimately enhance the utilization of traffic resources. By leveraging PINN and PDE technologies, this study contributes to advancing traffic flow analysis and optimization, potentially influencing urban planning and traffic management strategies positively.

Streamline Simulation of Compositional Nonisothermal Flows of Viscoplastic Oils

The aim of this study is to develop a numerical method providing the possibility of computing nonisothermal compositional flows faster than traditional finite-volume methods. A plane problem of oil, water, and gas flows is considered. The oil phase is represented by two components: the light fraction and the heavy fraction, which, similarly to water, can be gasified. The study takes into account both the nonlinearity of the oil flow law and the temperature dependence of the parameters of this law. This formulation of the problem is relevant for the simulation of the development of high-viscosity oilfields. To reduce the computational complexity of the problem, the streamline method with splitting with respect to physical processes is used. It separates the convective transport directed along the flow’s propagation from processes related to the heat conduction and gravity, which directions do not coincide with the convective flow. A distinctive feature of the proposed method is the joint solution of the pressure, energy-balance, and mass-component equations both on streamlines and on the initial grid. This feature allows us to correctly calculate oil flows with a complex temperature-dependent rheology. Numerical solutions of the system of flow equations on a two-dimensional grid and on streamlines are obtained by the IMPEC method. For the presented streamline method, we propose an algorithm accounting for the thermal conductivity and transition criteria between calculations on streamlines and on a two-dimensional grid. The developed software is verified by comparison with the analytical solutions and with the results of calculations by finite-volume methods on five-point and nine-point finite-difference stencils.

English

This article presents a novel system of flow equations that models the pressure deficit of a reservoir considered as a triple continuous medium formed by the rock matrix, vugular medium and fracture. In non- conventional reservoirs, the velocity of the fluid particles is altered due to physical and chemical phenomena caused by the interaction of the fluid with the medium, this behaviour is defined as anomalous. A more exact model can be obtained with the inclusion of the memory formalism concept that can be expressed through the use of fractional derivatives. Using Laplace transform of the Caputo fractional derivative and Bessel functions, a semi-analytical solution is reached in the Laplace space.

Accelerating multiscale simulation of complex geomodels by use of dynamically adapted basis functions

A number of different multiscale methods have been developed as a robust alternative to upscaling and as a means for accelerated reservoir simulation of high-resolution geomodels. In their basic setup, multiscale methods use a restriction operator to construct a reduced system of flow equations on a coarser grid, and a prolongation operator to map pressure unknowns from the coarse grid back to the original simulation grid. The prolongation operator consists of basis functions computed numerically by solving localized flow problems. One can use the resulting multiscale solver both as a CPR-preconditioner in fully implicit simulators or as an efficient approximate iterative linear solver in a sequential setting. The latter approach has been successful implemented in a commercial simulator. Recently, we have shown that you can obtain significantly faster convergence if you instead of using a single pair of prolongation-restriction operators apply a sequence of such operators, where some of the operators adapt to faults, fractures, facies, or other geobodies. Herein, we present how you can accelerate the convergence even further, if you also include additional basis functions that capture local changes in the pressure.

Effect of Fluid Viscoplasticity on the Constructal Design of Elliptic Tubes Subjected to Forced Convection

The main purpose of this work is to study the effects of flow on the Herschel–Bulkley number (HB), a parameter that should account for fluid viscoplasticity, in the optimal designs of elliptical section tubes subjected to forced convection of viscoplastic fluids. The constructal design was associated with exhaustive search to obtain the ellipse aspect ratios that maximize the Nusselt number and minimize dimensionless pressure drop, for fixed ellipse area and distance between tubes.We employed a modified Herschel–Bulkleymodel to predict viscoplastic behavior, and a fixed flow index equal to 0.4. To solve the system of differential equations for flow and heat transfer, we used a finite volume method code. We investigated the HB ranging from 1 to 100. The maximum dimensionless heat transfer increased with the increase of the HB. Still, the HB did not affect the ellipse aspect ratio that maximized dimensionless heat transfer. For all values of HB, the optimal aspect ratio was equal to 0.96. The Herschel–Bulkley number strongly affected dimensionless pressure drop. The minimum pressure drop increased with the increase of HB, and the optimal geometry was slender as HB increased.

A Three-Dimensional Model of Two-Phase Flows in a Porous Medium Accounting for Motion of the Liquid–Liquid Interface

A new three-dimensional hydrodynamic model for unsteady two-phase flows in a porous medium, accounting for the motion of the interface between the flowing liquids, is developed. In a minimum number of interpretable geometrical assumptions, a complete system of macroscale flow equations is derived by averaging the microscale equations for viscous flow. The macroscale flow velocities of the phases may be non-parallel, while the interface between them is, on average, inclined to the directions of the phase velocities, as well as to the direction of the saturation gradient. The last gradient plays a specific role in the determination of the flow geometry. The resulting system of flow equations is a far generalization of the classical Buckley–Leverett model, explicitly describing the motion of the interface and velocity of the liquid close to it. Apart from propagation of the two liquid volumes, their expansion or contraction is also described, while rotation has been proven negligible. A detailed comparison with the previous studies for the two-phase flows accounting for propagation of the interface on micro- and macroscale has been carried out. A numerical algorithm has been developed allowing for solution of the system of flow equations in multiple dimensions. Sample computations demonstrate that the new model results in sharpening the displacement front and a more piston-like character of displacement. It is also demonstrated that the velocities of the flowing phases may indeed be non-collinear, especially at the zone of intersection of the displacement front and a zone of sharp permeability variation.

Constructal design applied to elliptic tubes in convective heat transfer cross-flow of viscoplastic fluids

The objective of this work was to obtain geometries that maximize the heat transfer and minimize pressure drop for viscoplastic fluids in cross flow around elliptical section tubes. The Constructal Design associated with exhaustive search was employed to obtain the ellipse aspect ratios that maximize the Nusselt number and to minimize dimensionless pressure drop, for fixed parameters such as ellipse area, Reynolds (Re=1), Prandtl (Pr=1) and Herschel-Bulkley (HB=1) numbers. The viscoplastic fluid behavior was modeled using the Herschel-Bulkley constitutive equation. The power-law index, n, was a parameter to predict fluid shear thinning. The system of differential equations for flow and heat transfer was solved numerically by the finite volume method. The aspect ratio to maximize Nu‾, rq,opt, and the one to minimize Δp̃, rp,opt, were searched for different values of n, ranging from 0.4 to 1. The rq,opt was found to be very close to 1 for all n’s, which corresponds to the tube of circular cross section. Low aspect ratio tubes lead to the formation of greater unyielded zones as compared to high aspect ratio tubes. This may be a key to understand the increase in heat transfer from the tube as the aspect ratio increases to one. The aspect ratio that led to the lowest pressure drop was not the same for all fluids, contrarily to the behavior noticed for thermal purpose. It was noticed that as n decreases, the rp,opt, also decreases, meaning that to reduce pressure drop for more shear thinning fluids the tube must be slender. This may lead to the conclusion that elliptic tubes might be a good alternative in applications of heat transfer with mild viscoplastic fluids (low HB) only when there is the need to reduce pressure drop or pumping power.

Use of Multiple Multiscale Operators To Accelerate Simulation of Complex Geomodels

Summary Multiscale methods have been developed as a robust alternative to upscaling and to accelerate reservoir simulation. In their basic setup, multiscale methods use both a restriction operator to construct a reduced system of flow equations that can be solved on a coarser grid and a prolongation operator to map pressure unknowns from the coarse grid back to the original simulation grid. When combined with a local smoother, this gives an iterative solver that can efficiently compute approximate pressures to within a prescribed accuracy and still provide mass-conservative fluxes. We present an adaptive and flexible framework for combining multiple sets of such multiscale approximations. Each multiscale approximation can target a certain scale; geological features such as faults, fractures, facies, or other geobodies; or a particular computational challenge such as propagating displacement and chemical fronts, wells being turned on or off, and others. Multiscale methods that fit the framework are characterized by three features. First, the prolongation and restriction operators are constructed by use of a nonoverlapping partition of the fine grid. Second, the prolongation operator is composed of a set of basis functions, each of which has compact support within a support region that contains a coarse gridblock. Finally, the basis functions form a partition of unity. These assumptions are quite general and encompass almost all existing multiscale (finite-volume) methods that rely on localized basis functions. The novelty of our framework is that it enables multiple pairs of prolongation and restriction operators—computed on different coarse grids and possibly also by different basis-function formulations—to be combined into one iterative procedure. Through a series of numerical examples consisting of both idealized geology and flow physics as well as a geological model of a real asset, we demonstrate that the new iterative framework increases the accuracy and efficiency of the multiscale technology by improving the rate at which one converges the fine-scale residuals toward machine precision. In particular, we demonstrate how it is possible to combine multiscale prolongation operators that have different spatial resolution and that each individual operator can be designed to target, among others, challenging grids, including faults, pinchouts, and inactive cells; high-contrast fluvial sands; fractured carbonate reservoirs; and complex wells.

Primordial perturbations in multi-scalar inflation

Multiple field models of inflation exhibit new features than single field models. In this work, we study the hierarchy of parameters based on Hubble expansion rate in curved field space and derive the system of flow equations that describe their evolutions. Then we focus on obtaining derivatives of number of e-folds with respect to scalar fields during inflation and at hypersurface of the end of inflation.

Successful application of multiscale methods in a real reservoir simulator environment

For the past 10 years or so, a number of so-called multiscale methods have been developed as an alternative approach to upscaling and to accelerate reservoir simulation. The key idea of all these methods is to construct a set of prolongation operators that map between unknowns associated with cells in a fine grid holding the petrophysical properties of the geological reservoir model and unknowns on a coarser grid used for dynamic simulation. The prolongation operators are computed numerically by solving localized flow problems, much in the same way as for flow-based upscaling methods, and can be used to construct a reduced coarse-scale system of flow equations that describe the macro-scale displacement driven by global forces. Unlike effective parameters, the multiscale basis functions have subscale resolution, which ensures that fine-scale heterogeneity is correctly accounted for in a systematic manner. Among all multiscale formulations discussed in the literature, the multiscale restriction-smoothed basis (MsRSB) method has proved to be particularly promising. This method has been implemented in a commercially available simulator and has three main advantages. First, the input grid and its coarse partition can have general polyhedral geometry and unstructured topology. Secondly, MsRSB is accurate and robust when used as an approximate solver and converges relatively fast when used as an iterative fine-scale solver. Finally, the method is formulated on top of a cell-centered, conservative, finite-volume method and is applicable to any flow model for which one can isolate a pressure equation. We discuss numerical challenges posed by contemporary geomodels and report a number of validation cases showing that the MsRSB method is an efficient, robust, and versatile method for simulating complex models of real reservoirs.

Modeling and simulation of surfactant–polymer flooding using a new hybrid method

Chemical enhanced oil recovery by surfactant–polymer (SP) flooding has been studied in two space dimensions. A new global pressure for incompressible, immiscible, multicomponent two-phase porous media flow has been derived in the context of SP flooding. This has been used to formulate a system of flow equations that incorporates the effect of capillary pressure and also the effect of polymer and surfactant on viscosity, interfacial tension and relative permeabilities of the two phases. The coupled system of equations for pressure, water saturation, polymer concentration and surfactant concentration has been solved using a new hybrid method in which the elliptic global pressure equation is solved using a discontinuous finite element method and the transport equations for water saturation and concentrations of the components are solved by a Modified Method Of Characteristics (MMOC) in the multicomponent setting. Numerical simulations have been performed to validate the method, both qualitatively and quantitatively, and to evaluate the relative performance of the various flooding schemes for several different heterogeneous reservoirs.

Von Neumann Stability Analysis of a Segregated Pressure-Based Solution Scheme for One-Dimensional and Two-Dimensional Flow Equations

The goal of this paper is to derive the von Neumann stability conditions for the pressure-based solution scheme, semi-implicit method for pressure-linked equations (SIMPLE). The SIMPLE scheme lies at the heart of a class of computational fluid dynamics (CFD) algorithms built into several commercial and open-source CFD software packages. To the best of the authors' knowledge, no readily usable stability guidelines appear to be available for this popularly employed scheme. The Euler equations are examined, as the inclusion of viscosity in the Navier–Stokes (NS) equation serves to only soften the stability limits. First, the one-dimensional (1D) Euler equations are studied, and their stability properties are delineated. Next, a rigorous stability analysis is carried out for the two-dimensional (2D) Euler equations; the analysis of the 2D equations is considerably more challenging as compared to analysis of the 1D form of equations. The Euler equations are discretized using finite differences on a staggered grid, which is used to achieve equivalence to finite-volume discretization. Error amplification matrices are determined from the stability analysis, stable and unstable regimes are identified, and practical stability limits are predicted in terms of the maximum allowable Courant–Friedrichs–Lewy (CFL) number as a function of Mach number. The predictions are verified using the Riemann problem, and very good agreement is obtained between the analytically predicted and the “experimentally” observed CFL values. The successfully tested stability limits are presented in graphical form, as compared to complicated mathematical expressions often reported in published literature. Since our analysis accounts for the solution scheme along with the full system of flow equations, the conditions reported in this paper offer practical value over the conditions that arise from analysis of simplified 1D model equations.

A point implicit time integration technique for slow transient flow problems

We introduce a point implicit time integration technique for slow transient flow problems. The method treats the solution variables of interest (that can be located at cell centers, cell edges, or cell nodes) implicitly and the rest of the information related to same or other variables are handled explicitly. The method does not require implicit iteration; instead it time advances the solutions in a similar spirit to explicit methods, except it involves a few additional function(s) evaluation steps. Moreover, the method is unconditionally stable, as a fully implicit method would be. This new approach exhibits the simplicity of implementation of explicit methods and the stability of implicit methods. It is specifically designed for slow transient flow problems of long duration wherein one would like to perform time integrations with very large time steps. Because the method can be time inaccurate for fast transient problems, particularly with larger time steps, an appropriate solution strategy for a problem that evolves from a fast to a slow transient would be to integrate the fast transient with an explicit or semi-implicit technique and then switch to this point implicit method as soon as the time variation slows sufficiently. We have solved several test problems that result from scalar or systems of flow equations. Our findings indicate the new method can integrate slow transient problems very efficiently; and its implementation is very robust.

Mathematical Analysis of Hall Effect on Transient Hartman Flow about a Rotating Horizontal Permeable Surface in a Porous Medium under Inclined Magnetic Field.

This paper proposes the exact solution for unsteady flow of a viscous incompressible electrically conducting fluid past a impulsively started infinite horizontal surface which is rotating with an angular velocity embedded in a saturated porous medium under the influence of strong magnetic field with hall effect. Our study focuses on the change of direction of the external magnetic field on the flow system which leads to change in the flow behavior and skin frictional forces at the boundary. Systems of flow equations are solved using Laplace transform technique. The impacts of control parameters Hartman number, rotation of the system, hall effect, inclination of the magnetic field, and Darcy number on primary and secondary velocities are shown graphically, skin friction at horizontal boundary in tabular form. For validating our results, in the absence of permeability of the porous medium and inclination of the magnetic field the results are in good agreement with the published results.

Hydromagnetic stagnation point flow of a micropolar viscoelastic fluid towards a stretching/shrinking sheet in the presence of heat generation

In this article, the boundary layer two-dimensional Hiemenz flow and heat transfer for a micropolar, viscoelastic electrically conducting fluid over a shrinking/stretching sheet with heat generation is investigated. The governing system of flow equations is first reduced to coupled nonlinear ordinary differential equations by means of similarity transformations. A purely analytical series solutions of these equations is obtained in which domain (0 ≤ η ≤ ∞) using an analytic technique, namely, the homotopy analysis method. The convergence of the series solutions is discussed explicitly. The influences of an emerging parameters on the fluid velocity, microrotation velocity and temperature field are shown graphically and discussed in detail. The comparison of present results with the existing results is also given and found in excellent agreement.

Hyperbolic conservation laws for continuous two-phase flow without mass exchange

It is well known that classic two-phase flow equation systems have complex characteristic roots and, therefore, constitute an ill-posed initial-value problem. Here we suggest that ill-posedness is due to working with two different material derivatives for the phases, which have varying velocities, but employing the same position vector for both operators. There follows an analysis of the conditions required for a global treatment of both phases, but using only one material derivative for both phases, now coherent with only one position vector. Consequently, new global mass- and momentum-conservation equations for a two-phase flow without mass exchange are derived by strictly following the classic Reynolds’ transport theorem. The new global mass-conservation equation proposed would only be valid if the ‘zero-net-mass-flux’ condition, another independent equation, was fulfilled. We also found that the new equation system is well-posed, i.e. its two characteristic roots are real if a new relation between velocities and densities is satisfied. Finally, we have highlighted the strong connections of new conservation laws with classic treatments, and also shown that minor modifications of the current equation system would turn it into a hyperbolic one, thus easing the computational solution of this complex problem.

Wykorzystanie metod optymalizacji heurystycznej do minimalizacji kosztów bilansowania systemu elektroenergetycznego

The paper presents results of the particle swarm method application to solve the SCOPF task that minimizes power balancing costs in the electric power system. Although the principal element of the calculation process is multiple iteration of the power flow equation system solution, owing to the apt application of the particle swarm method the whole calculation process exhibits good convergence and repeatability of the determined optimal state. (Heuristic optimization methods applied to minimize the electric power system balancing costs). Slowa kluczowe: koszty bilansowania systemu elektroenergetycznego, heurystyka, optymalizacja.

FlowPy—A numerical solver for functional renormalization group equations

FlowPy is a numerical toolbox for the solution of partial differential equations encountered in Functional Renormalization Group equations. This toolbox compiles flow equations to fast machine code and is able to handle coupled systems of flow equations with full momentum dependence, which furthermore may be given implicitly. Program summaryProgram title: FlowPyCatalogue identifier: AEPB_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEPB_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: GNU General Public License, version 3No. of lines in distributed program, including test data, etc.: 4078No. of bytes in distributed program, including test data, etc.: 46,609Distribution format: tar.gzProgramming language: Python, C.Computer: PC or workstation.Operating system: Unix.RAM: approx. 40 MBClassification: 4.12, 11.1.External routines: Python, libpython, SciPy, NumPy, python-simpleparse.Nature of problem:In the study of functional renormalization group equations non-local integro-differential equations arise which furthermore can contain singular coefficient functions for the highest derivative and may only be given implicitly. Solving these equations beyond the simplest cases thus provides a numerical challenge.Solution method:A combination of numerical differentiation, integration, interpolation, and ODE solving.Restrictions:Due to the nature of FRG problems, computational effort (run time) will scale quadratically with the number of discretization points. Using more than at most a few hundred discretization points may be impractical.Running time:For the SUSY_QM example: ∼10 s for 10 support points, ∼5 min for 100 discretization points. For the momentum_dependent_wavefunction example: ∼40 min for 5 discretization points.

Stability Analysis of Flow in Thin Liquid Film Under Heat Effect

This research is dedicated for analyzing the stability of system of flow equations for a delicate chip of a fluid exposed for medium heat ,where we found the growth of disturbance exist or not after making the system linearization it was shown through the results of analysis that these equations were in stable state when the real part of the wave velocity(α) is a negative quantity and it is unstable state when the real part of this velocity is positive quantity , We get stability and this happens when the Reynold's number is positive quantity i.e. R>0 . and when α>0 we get unstable state and this happens when the heat conductivity is a positive quantity.

Dynamical modelling of concentration–age–discharge in watersheds

AbstractThere is now a wide literature on the use of tracer age and transit time distributions to diagnose transport in environmental systems. Theories have been proposed using idealized tracer age modelling for ocean ventilation, atmospheric circulation, soil, stream and groundwater flow. Most approaches assume a steady flow regime and stationarity in the concentration (tracer) distribution function for age, although recent work shows that this is not a necessary assumption. In this paper, dynamic model for flow, concentration, and age in volume‐averaged and a spatially distributed watershed system are derived in terms of the moments of the underlying distribution function for tracer age, time, and position. Several theoretical and practical issues are presented: (1) The low‐order moments of the age distribution function are sufficient to construct a dynamical system for the mean age and concentration under steady or transient flow conditions. (2) Solutions to the coupled system of equations for flow, concentration and age show that ‘age’ of solutes stored within the watershed or leaving the watershed is a dynamic process which depends on flow variations as well as the solute or tracer dynamics. (3) Intermittency of wetting and drying cycles leads to an apparent increase in the tracer age in proportional to the duration of the ‘dry’ phase. (4) The question of how mobile/immobile flow may affect the age of solutes is examined by including a low permeable, passive store that relaxes the well‐mixed assumption. (5). A spatially distributed advective and dispersive transport solution for age evolution over a simple 1‐D hillslope is developed to demonstrate the age theory for a distributed source of water and tracer, and the solution is shown to have very similar input–output behaviour when compared to the volume‐average model for comparable parameters. Copyright © 2010 John Wiley & Sons, Ltd.