Incompressible Flow Equations Research Articles

Performance improvement of a rotating gas separator for electrical submersible pump considering entropy generation method

The utilization of Electrical Submersible Pumps (ESPs) stands as one of the foremost methods in artificially extracting oil, primarily within the oil industry. Under typical conditions, the pump encounters a two-phase flow of oil and gas. Should the gas phase within this mixture surpass an acceptable threshold, it can lead to a significant decline in performance and inflict detrimental damage upon the pump. To address this issue, the incorporation of a gas separator unit, one of the pivotal components within ESPs, is imperative at the pump's inlet section. Enhancing the gas separator unit holds paramount importance for ensuring pump safety and optimizing performance, particularly in wells with elevated gas contents.The conventional approach to calculating hydraulic losses in turbomachinery, relying on pressure drop calculations, often falls short in pinpointing the precise locations of losses. This paper delves into enhancing gas separator performance by augmenting gas separation efficiency through the application of the entropy generation method and leveraging the second law of thermodynamics. To achieve this, pivotal entropy-generating geometrical parameters of the gas separator were identified as the design variables necessary for forming the Taguchi sample space. Numerical simulations were conducted utilizing steady three-dimensional Reynolds-Averaged Navier-Stokes (RANS) equations for incompressible fluid flow, coupled with a turbulence model. Results indicate that the gas separation efficiency (GSE) of the initial model surged from 45.6% to 88% in the improved sample, albeit with a concurrent rise in entropy generation rate (EGR) from 209 W/K to 243.1 W/K. To establish a more comprehensive criterion for analysis, the ratio of GSE to EGR was employed as an objective function. This ratio quantifies the efficiency of gas separation per unit of energy loss, and it escalated from 0.218 in the primary sample to 0.362 in the improved sample, representing a 66% increase. This signifies that despite simultaneous increases in EGR and GSE, the rate of energy loss per unit of gas separation has diminished. Furthermore, the heightened EGR in the improved sample can be attributed to an increase in total volume, leading to an expanded contact surface area and consequently an elevation in total entropy. This investigation underscores the advantages of the entropy generation method, particularly in delineating the precise location and magnitude of energy dissipation within the rotating gas separator.

Study of pulse wave phenomena associated with blood flow model in human viscoelastic artery

Cardiac output monitoring has proven to be a promising hemodynamic management tool especially for critically ill patients. Pulse wave analysis is a noninvasive method used to quantify cardiac output continuously with respect to time. In this article, we have proposed a novel methodology to quantify the contribution of pulse waves to further study the role of arterial wall relaxation with respect to time. The relaxation time can further help in the diagnosis of disease. The pulse wave velocity component is derived by transforming governing fluid flow equations into hyperbolic equations for laminar incompressible blood flow in an artery of viscoelastic walls. The viscoelastic behavior of the wall is analogously modeled by the modified Zener model that has the capability to measure creep, stress relaxation, and hysteresis. The derived model equations are solved numerically by the multiderivative Runge–Kutta implicit–explicit time integration method with a weighted, essentially non-oscillatory, discretization scheme. The results are well validated with clinical trial-based Riemann problems for the case of elastic walls. It is observed that the proposed modified Zener model is well suitable to identify the location of arterial stiffness where the pulse wave manifests various types of phenomena like discontinuity and reflection.

Investigation of the performance of a horizontal-axis dual rotor wind turbine

Recent years have seen a rise in interest in wind energy as a useful alternative to harmful energies like fossil fuels. The dual rotor wind turbine (DRWT) offers more rapid rates of wind energy extraction. The current study intends to compare the performance of the turbine with and without the addition of a second rotor. Additionally, it examines how tip speed ratio and phase shift angle will affect DRWT performance. Realizable k-shear stress transport turbulence models are used to solve the three-dimensional, turbulent, stable, and incompressible flow equations for the performance of dual-rotor wind turbines. Domain-independence tests and an impartial mesh test are run to assess the results and ensure their accuracy. The researcher relies on previous studies while constructing the single rotor wind turbine model. This model uses an S826 airfoil. The front and rear rotors are given streamlined representations using ANSYS, according to the researcher. The independent mesh test indicates that the mesh density has 11.5 million elements. The experiment's results show that the DRWT has a significant effect on the efficiency of wind energy.

Exact solutions of steady Euler equations in two dimension by functional separation

Exact solutions of two dimensional Euler equations for incompressible fluid flows are found by the functional separation of variables. More specifically, two independent variables are considered in the Cartesian coordinates, polar coordinates and also general orthogonal curvilinear coordinates which include elliptic, parabolic and hyperbolic systems. Several new solutions are obtained and other classical solutions are rediscovered.

Time‐dependent double phase dusty fluid flow over an oscillatory rotating disk

AbstractThe dispersion of solid particles in a viscous fluid leads to a two‐phase flow nature. The present study incorporates the time‐dependent three‐dimensional dusty fluid flow generated by a periodically oscillatory rotating disk. The disk is stretchable along the radial axis with time‐based sinusoidal fluctuations. The governing incompressible flow equations for two‐phase equilibrium are normalized in the form of similarity systems consisting of the fluid stage and dust phase. The whole normalized system reduces to the familiar Von Kármán similarity system for the flow configuration of rotating disk by removing the dust phase and periodic oscillations of the disk. The two‐phase flow model is then numerically solved by a built‐in method namely “pdsolve” in Maple built program. The graphical aspects of the obtained physical parameters on velocity and thermal fields of the dust particle stage are investigated to show how the oscillatory disk contributes to the dusty flow features. The wall shears and thermal rates of the fluid and dust particles are also calculated in limiting case of disk rotation. It is observed that in time‐based flow, the oscillatory profiles preserve a phase shifting phenomenon. For centrifugal forces, the particle cloud moves away from the surface along a tangential direction. The thermal field is reduced by the dust particle stages.

Numerical simulation of pulsatile blood flow through eccentric double stenosed carotid artery

Purpose This study aims to perform three-dimensional numerical computations for blood flow through a double stenosed carotid artery. Pulsatile flow with Womersley number (Wo) of 4.65 and Reynolds number (Re) of 425, based on the diameter of normal artery and average velocity of inlet pulse, was considered. Design/methodology/approach Finite volume method based ANSYS Fluent 20.1 was used for solving the governing equations of three-dimensional, laminar, incompressible and non-Newtonian blood flow. A high-quality grid with sufficient refinement was generated using ICEM CFD 20.1. The time-averaged flow field was captured to investigate the effect of severity and eccentricity on the lumen flow characteristics. Findings The results show that an increase in interspacing between blockages brings shear layer instability within the region between two blockages. The velocity profile and wall shear stress distribution are found to be majorly influenced by eccentricity. On the other hand, their peak magnitude is found to be primarily influenced by severity. Results have also demonstrated that the presence of eccentricity in stenosis would assist in flow development. Originality/value Variation in severity and interspacing was considered with a provision of eccentricity equal to 10% of diameter. Eccentricity refers to the offset between the centreline of stenosis and the centreline of normal artery. For the two blockages, severity values of 40% and 60% based on diameter reduction were permuted, giving rise to four combinations. For each combination, three values of interspacing in the multiples of normal artery diameter (D), viz. 4D, 6D and 8D were considered.

Flow development region in the boundary layer: two-component molecular viscosity and partial slip

The subject of this study is the flow development region of laminar incompressible fluid flow in the boundary layer. This flow is an example where a direct application of the Navier-Stokes equations of gradient-free laminar incompressible fluid flow, in which the molecular viscosity is assumed to be a constant value independent of spatial coordinates, leads to a redefinition of the mathematical model. It is about the fluid boundary layer in the region of flow establishment in the motion problem of a semi-infinite plane, where the pressure gradient is zero. There is a situation when the number of equations is equal to three (two equations of momentum conservation and the equation of continuity), and the number of unknowns is equal to two - the number of the speed component. As a logical solution to the obtained inconsistency, it is proposed, as was already done for the problem of stationary motion of a plane and the problem of acceleration of a plane, to depart from the false statement about the constancy of molecular viscosity in the gradient-free boundary layer of an incompressible flow and consider molecular viscosity as a function of spatial coordinates. The need to consider the variable nature of molecular viscosity led to the discovery of another flaw in the Navier-Stokes theory. This non-trivial flaw was discovered during the application of the original numerical analytical method for solving the flow development region problem. The Navier-Stokes equations are supplemented by boundary conditions. The most important condition is the condition of fluid non-slipping on the surface of a solid body, which, by the way, does not follow any physical law. As a result, on the surface of a half-plane (or a moving body), the component of the velocity, which coincides with the direction of motion, has a constant value equal to the velocity of the body. It immediately follows from the continuity equation that the normal derivative of the normal component of the velocity must be equal to zero along the surface of the plane (body), since the longitudinal derivative of the velocity becomes zero. However, it is quite obvious that the velocity component normal to the surface of the plane (body) changes across the boundary layer in the region of current development, which indicates the presence of a normal gradient (both components) of the velocity. The conflict or contradiction is overcome by moving away from the generally accepted condition of non-slipping to the condition of partial non-slipping, or essentially the presence of sliding. Even with the sudden braking of any vehicle, the complete stop does not occur instantly, but after some finite time and distance, so in the case of the motion of a body in a stationary fluid, there is not an instant sticking, but a gradual one - from complete sliding, when a particle of liquid has just met a moving plane (body), to complete non-slipping at the end (and further) of the flow development region. Research methods. This work uses purely theoretical methods based on the use of calculus of variations, laws of physics, and ideas from everyday life. Conclusions. An improved model of a viscous Newtonian fluid in the area of flow development in the boundary layer was derived. On the basis of assumptions about the variable nature of the molecular viscosity, which already has two components, and the departure from the non-slipping condition, analytical solutions for both components of the velocity and both components of the molecular viscosity were obtained. A comparison of the obtained results with the results of other studies is presented.

Three-dimensional laminar flow using physics informed deep neural networks

Physics informed neural networks (PINNs) have demonstrated their effectiveness in solving partial differential equations (PDEs). By incorporating the governing equations and boundary conditions directly into the neural network architecture with the help of automatic differentiation, PINNs can approximate the solution of a system of PDEs with good accuracy. Here, an application of PINNs in solving three-dimensional (3D) Navier–Stokes equations for laminar, steady, and incompressible flow is presented. Notably, our approach involves deploying PINNs using feed-forward deep neural networks (DNNs) without depending on any simulation or experimental data. This investigation focuses on 3D square channel flow and 3D lid-driven cavity flow. For each case, one deep neural network was trained using only the governing equations and boundary conditions. Finally, the PINNs' results were compared with the computational fluid dynamics results. The goal was to assess the ability of PINNs (with DNN architectures) to predict the solution of Navier–Stokes equations in the 3D domain without any simulation or experimental data (unsupervised learning).

Numerical study of Newtonian laminar Flow around circular and square cylinders

In this study, we present a novel algorithm designed to address the computational challenges posed by the governing equations of incompressible Newtonian fluid flow, particularly of bluff bodies of circular and square in a channel. This a significant innovation algorithm is developed based on the Taylor–Galerkin/Pressure-Correction finite element method. Notably, a pivotal element of this approach involves the implementation of the two-step Lax–Wendroff scheme to resolve the momentum equation. To assess the efficacy of our novel algorithm, we conducted a comprehensive analysis using a standard benchmark scenario; the initiation of Poiseuille flow through an axisymmetric rectangular channel encompassing bluff bodies. In this context, our investigation centered on the influence of the Reynolds number (Re) on various pertinent variables. The results obtained from our study underscore the remarkable congruence between the outcomes generated by the new algorithm and established physical principles, as well as findings from prior research, as evidenced through thorough comparisons with the existing body of literature. This research represents a significant stride toward advancing our understanding of fluid dynamics and computational methodologies in this domain.

A phase-field-based multiple-relaxation-time lattice Boltzmann method for incompressible multiphase flows with density and viscosity contrasts

In this work, we present a unified framework of phase-field-based multiple-relaxation-time lattice Boltzmann (MRT-LB) method for incompressible multiphase flows with density and viscosity contrasts where a block-lower-triangular relaxation matrix is introduced. The present framework includes the classic MRT-LB model and central-moments-based LB model (CLBM, a particular version of the MRT-LB model). The governing equations of incompressible multiphase flows can be reproduced precisely through the direct Taylor expansion method at the second-order of expansion parameters. In this work, the pressure p is re-estimated, and we prove that the present pressure expression is suitable for classic MRT model, CLBM and various lattice structures with an accuracy of O(Δt2+ΔtMa2). In addition, the model construction and analysis are all carried out in the velocity space. As a result, the present model has a similar structure to the traditional Bhatnagar-Gross-Krook (BGK) model. Therefore the process of their analysis and implementation is also similar. We then test the present model by five classic physical problems. The results show that the present model is found to be accurate and has better numerical stability over the BGK model, and has the ability to simulate multiphase flow problems with large density ratio (1000).

Application of Weighted Least Squares Meshless Method to Incompressible Flows

Meshless methods have been successfully applied to solve a large number of inviscid and viscous compressible flows in both two and three dimensional flow fields. However, the conventional methods of solution for compressible flow equations cannot be applied to incompressible flow equations. This is due to the basic difference in the nature of the pressure variable in the compressible and incompressible flows. Pressure acts as a thermodynamic variable in case of compressible flows and can be easily updated by using the equation of state. In contrast, in case of incompressible flows, the pressure is merely responsible for enforcing the incompressibility constraint and hence cannot be updated by the equation of state. This has led to the development of numerical methods based on Helmholtz Decomposition to solve for the fluid flows in the incompressible flow regimes. Here, we have applied weighted least squares meshless method to solve for the flows in the incompressible flow regime using the projection method based on Helmholtz Decomposition. The standard test cases of two dimensional lid driven cavity flow at Reynolds numbers of 100, 400 and 1000 have been solved using the weighted least squares meshless method. A good match between the results of the test case using the meshless methods and conventional CFD methods for the solution of the incompressible flows has been obtained.

Fully Developed Laminar Flow of Heat Transfer Non-Newtonian Flow in Ducts with Arbitrary Cross-Sectional Shape

The pressure drop and heat transfer properties of a laminar flow with circular, square, and triangular cross sections and a constant heat flux boundary condition are given by a numerical solution. A 3D Navier-Stokes incompressible flow and energy equation serves as the foundation for the numerical model. Results are tabulated and given visually for the velocity field, Poiseuille number, and Nusselt number product for various power-law indices (0.5 n 1.0). One of the analyzed ducts causes a greater heat transfer intensification along with a significant pressure loss. A different type of duct allows for a better balance between pressure loss and heat transfer. In order to verify the accuracy of the numerical codes created in the current study, critical comparisons with earlier findings in the literature are also carried out.

Kernel free boundary integral method for 3D incompressible flow and linear elasticity equations on irregular domains

A second-order accurate kernel-free boundary integral method is presented for Stokes flow and linear elasticity boundary value problems on three-dimensional irregular domains. It solves equations in the framework of boundary integral equations, whose corresponding discrete forms are well-conditioned and solved by the GMRES method. A notable feature of this approach is that the boundary or volume integrals encountered in BIEs are indirectly evaluated by a Cartesian grid-based method, which includes discretizing corresponding simple interface problems with a MAC scheme, correcting discrete linear systems to reduce large local truncation errors near the interface, solving the modified system by a CG method together with an FFT-based Poisson solver. No extra work or special quadratures are required to deal with singular or hyper-singular boundary integrals and the dependence on the analytical expressions of Green’s functions for the integral kernels is completely eliminated. Numerical results are given to demonstrate the efficiency and accuracy of the Cartesian grid-based method.

A mass-preserving level set method for simulating 2D/3D fluid flows with deformed interface

This research paper focuses on the development of a novel interface-capturing solver for predicting two-phase flow in 2D/3D Cartesian meshes within the framework of Eulerian approaches. The primary objective is to ensure mass conservation and accurate capture of interface topology. To achieve this, a mass-preserving level set advection equation formulated as a scalar sign-distance function is proposed. The key innovation of the Eulerian solver lies in the incorporation of a scalar speed function for robust reconstruction of the classical level set equation. The effectiveness and reliability of the proposed flow solver in solving incompressible two-phase viscous flow equations are demonstrated through several benchmark problems.

Strong solutions to the equations of the unconsolidated porous media in 3D thin domains

We prove the existence of global in time strong solutions of incompressible flow equations through isotropic granular (unconsolidated) porous media in thin domains of the form with . Also, we demonstrate that as the fluid velocity tends to vanish away from the initial time.

A comparison study of the behaviors of single-phase turbulent flow at low to moderate Reynolds numbers through a vertical pipe: 3D counters analysis

The study presented three-dimensional (3D) analysis of water's upward flowing through the vertical pipe under turbulent characteristic considerations. Both numerical constructed and improved the model of 3D for cylindrical coordinates of governing equations for incompressible turbulent flow with the Reynolds Average Navier-Stokes (RANS) model using the improved constants of the (k–ε) type. The present model is then compared with a previous study to give the feasibility of the present single-phase turbulent flow parameters. The pipe length is tested to measure how much it affected the turbulent parameters though one of the expected factors is the turbulent time scale. On the other hand, the model is numerically examined to determine the velocity profile, shear rate, and surface deformation of the water domain. While the pressure distribution, turbulent kinetic energy, and turbulent dissipation rate, these parameters are classified as the mechanic's system factors. The simulation is done with wide software used to simulate industrial is COMSOL 5.4 Multiphysics software. The results obtained increased the velocity of three inlet water velocities used ranging from (0.087, 0.105, and 0.123 m/sec) of upward flow. High fluctuation in the water flow moves along the entire pipe length and it can notice the sensitivity to any change in water properties or mechanical properties. The liquid upward flow in turbulent conditions is suffered from many characteristics such them related to liquid properties and others related to the mechanics of the application through the systems. The interaction between the fluid film (fluid boarded the pipe inner diameter) has been observed by the shear rate and liquid surface deformation

Computational Fluid Dynamics Modeling of Steam Condensation in the Presence of Noncondensable Gas in Inclined Tubes

In this paper, film condensation of steam in inclined tubes in the presence of noncondensable gases is modeled using numerical method. A single-phase fluid model incorporating a wall-function boundary condition is proposed to simulate the heat and mass transfer interaction between the liquid condensate and the gas-vapor mixture region. A user defined function is implemented to impose the proposed boundary to the governing equations for steady state incompressible flow. The performance of applying this boundary condition and subsequent treatment of the heat transfer at the liquid-mixture interface is validated against published experimental results. The calculated profiles of local and average Nusselt number for various inlet conditions compares favorably to the previous published experimental results and correlations. For all simulated cases, increases in inlet and bulk concentrations of noncondensable gases resulted in a decrease in the thermal performance of the condensing tube. Additionally, it is found that the inclination angle of the condensing tube can impact the local distribution of the heat transfer coefficient and Nusselt number, while there is only small effect on the average along the length of the entire tube.

A class of exact solutions of the Navier–Stokes equations in three and four dimensions

A few basic, intuitive, properties of the Navier–Stokes system of equations for incompressible fluid flows are discussed in this paper. We present a rephrased interpretation of the Navier–Stokes equation in a space having an arbitrary number of dimensions. We then derive spatially periodic solutions for the velocity and pressure fields that span an unbounded domain in three and four dimensions, given a smooth solenoidal initial velocity vector field. In these solutions all velocity components depend non-trivially on all coordinate directions.

Modified pressure and vorticity variables using Helmholtz decomposition for solution of the incompressible flow equations

Incompressible Navier–Stokes equations are reformulated using the Helmholtz decomposition of a velocity vector into rotational and potential components. By substituting the decomposed velocity in the time derivative term in a momentum equation, the potential component representing a gradient of a pressure-like term is combined with the gradient of the pressure modifying the physical pressure field. The rotational component representing a curl of a vortex-like vector is combined with the vorticity vector, making a non-physical vorticity vector that modifies the fluid viscosity. Thus, the unsteady Navier–Stokes equation is transformed into an explicitly steady-state form in terms of non-primitive variables. The stream function vector is governed by a parabolic equation in time, while the vorticity vector is governed by the Poisson function with a source term function of the convection stretching and time dependency of the physical flow vorticity. Therefore, the resulting system of equations is numerically independent of the cell Reynolds number stability condition that hunted the convection–diffusion Navier–Stokes equation. Numerical results are obtained for the two-dimensional driven cavity problem for Reynolds number of 1000 with 21 × 21, 41 × 41, 81 × 81, and 161 × 161 grid points. The computational grids correspond to cell Reynolds numbers 25, 12.5, 6.25, and 3.125, respectively. The computed results are smooth in all cases and validate the method.

A meshless multiscale method for simulating hemodynamics

Simulating hemodynamic quantities such as pressure and velocity are of great interest to clinicians to aid in surgical planning. To accurately simulate modifications to a region of vasculature, the entire system must be modeled. To facilitate this, a localized Radial-Basis Function (RBF) collocation Meshless flow solver is developed and tightly coupled to a 0D Lumped-Parameter Model (LPM) for solution of the peripheral circulation. The Meshless solver uses localized RBF collocations at data points that are automatically generated according to the geometry. The incompressible flow equations are updated by an explicit time-marching scheme based on a pressure-velocity correction algorithm. The inlets and outlets of the domain are tightly coupled with the LPM which contains elements that draw from a fluid-electrical analogy such as resistors, capacitors, and inductors that represent the viscous resistance, vessel compliance, and flow inertia, respectively. The localized RBF Meshless approach is well-suited for modeling complicated non-Newtonian hemodynamics due to ease of spatial discretization, ease of addition of multi-physics interactions such as fluid-structure interaction of the vessel wall, and ease of parallelization for fast computations. This work introduces the tight coupling of Meshless methods and LPMs for fast and accurate hemodynamic simulations.