Navier-Stokes Momentum Equations Research Articles

Effects of Nanoparticle Enhanced Lubricant Films in Dynamic Properties of Plain Journal Bearings at High Reynolds Numbers

The aim of this paper is numerical analysis of plain journal bearing with nanoparticle added lubricating oil. The fluid film bearing is studied with the modified Reynolds equation which considered time dependent inertia effects at rotating speeds with the linearized turbulence. The research is performed for numerous nanofluids include nanoparticles of CuO, TiO2, Ag and Cu and SAE 20W50 as a base fluid. The time transient governing equations include the incompressible mass conservation equation, Navier-Stokes momentum equations for thin film lubrication, and full kinematic including shaft accelerations are solved numerically. Two cases of long and short bearings are studied with Sommerfeld and Gumbel boundary conditions. Linearized force coefficients such as mass, stiffness and damping factors are obtained for an ordinary journal bearing. The ordinary plain journal bearing velocity response and the rotor displacements when sudden forces applied by rigid rotor symmetrically is achieved. The results prove that using the nanoparticle enhanced lubricant causes higher mass coefficients, damping ratios and fluid effective stiffness.

Pore level investigation of steam injection processes; visualization of oil entrapment and steam propagation

The objective of this work is to model the steam injection process and examine the displacement physics at the grain-level. The primary focus is to quantify the oil entrapment behind the swept zone and investigate the pore-level steam propagation and chamber growth. A digital two-dimensional micro-model is reconstructed and fed into a finite element VOF-based solver. The solver is developed by adding necessary source terms to the Navier Stokes equation. The phase change is simulated using the Lee phase change model assuming that mass is transferred in a constant pressure and a quasi-thermo-equilibrium state. For each phase in the multi-region model, the mass conservation, Navier-Stokes momentum, and energy equations under non-isothermal conditions are solved simultaneously. In post-processing results, steam chamber development, vapor condensation over the oil-steam interface, and oil viscosity reduction due to the heat diffusion are demonstrated properly. The simulated oil entrapment behind the swept zone as well as the temperature distribution throughout the medium are in good agreement with the experimental datasets.

A novel case study for thermal radiation through a nanofluid as a semitransparent medium via discrete ordinates method to consider the absorption and scattering of nanoparticles along the radiation beams coupled with natural convection

Natural convection in a volumetric radiant enclosure filled by a nanofluid is studied numerically for the first time by using discrete ordinates (DO) method to consider the absorption and scattering coefficients of nanoparticles on the radiation beams through the nanofluid as a semitransparent medium. Present nanofluid is a mixture of Al2O3 nanoparticles suspended in water as the base fluid. The volume concentration percentages of nanoparticles are almost small to make a semitransparent medium which means the achieved results can be used in the flat plate solar collectors. Moreover the SIMPLE algorithm of finite volume method for Navier-Stokes continuity, momentum and energy equations are solved and coupled with DO to simulate the total radiation and natural convection in a shallow inclined rectangular 2-D enclosure. This shape of enclosure is chosen due to it might represent the usual configuration of a solar collector. The enclosure inner walls are the gray diffuse emitters and reflectors. The effects of various amounts of Rayleigh number and volume concentration at different values of wavelength are investigated. The positive effect of wave length on radiation heat flux and consequently total heat flux of radiation and natural convection is observed.

Existence of global weak solutions to the kinetic Hookean dumbbell model for incompressible dilute polymeric fluids

We explore the existence of global weak solutions to the Hookean dumbbell model, a system of nonlinear partial differential equations that arises from the kinetic theory of dilute polymers, involving the unsteady incompressible Navier–Stokes equations in a bounded domain in two or three space dimensions, coupled to a Fokker–Planck-type parabolic equation. We prove the existence of large-data global weak solutions in the case of two space dimensions. Indirectly, our proof also rigorously demonstrates that, in two space dimensions at least, the Oldroyd-B model is the macroscopic closure of the Hookean dumbbell model. In three space dimensions, we prove the existence of large-data global weak subsolutions to the model, which are weak solutions with a defect measure, where the defect measure appearing in the Navier–Stokes momentum equation is the divergence of a symmetric positive semidefinite matrix-valued Radon measure.

Intra-wave-phase cross-shore profile modelling by using boundary-fitted slowly moving grid

Coastal bed profile change is described by bed load, pick-up, and settling on a boundary-fitted moving grid. Existing bed load formula is modified by changing threshold bed shear stress to reflect local bed slope. A numerical model system adopting the above function is developed to simulate cross-shore sediment transport around swash zone with steep bed slope as well as surf zone. The model system adopts a moving boundary grid which fits bed boundary slope, and other horizontal grid lines are parallel to the bed grid line. The model system is composed of flow module and sediment transport module. The flow module solves continuity equation and Reynolds-average Navier-Stokes momentum equations in intra-wave-phase manner. The flow module provides detailed flow information including near-bed fluid velocity which varies asymmetrically within a regular wave period near sea-bed and wave-phase average undertow profile of main fluid body including wave boundary layer. The sediment transport module solves sediment mass conservation equation. Exchange of sediment mass between bed itself and fluid column containing suspended sediment happens through pick-up and deposition. Wild bed level undulation is controlled by using a smoothing method. Model system is applied to two extreme cases of sand experiments by Kajima et al., and reasonable agreements between measurements and computations are obtained for both onshore-dominant and offshore-dominant cases.

A Newton–Krylov method with an approximate analytical Jacobian for implicit solution of Navier–Stokes equations on staggered overset-curvilinear grids with immersed boundaries

The explicit and semi-implicit schemes in flow simulations involving complex geometries and moving boundaries suffer from time-step size restriction and low convergence rates. Implicit schemes can be used to overcome these restrictions, but implementing them to solve the Navier–Stokes equations is not straightforward due to their non-linearity. Among the implicit schemes for non-linear equations, Newton-based techniques are preferred over fixed-point techniques because of their high convergence rate but each Newton iteration is more expensive than a fixed-point iteration. Krylov subspace methods are one of the most advanced iterative methods that can be combined with Newton methods, i.e., Newton–Krylov Methods (NKMs) to solve non-linear systems of equations. The success of NKMs vastly depends on the scheme for forming the Jacobian, e.g., automatic differentiation is very expensive, and matrix-free methods without a preconditioner slow down as the mesh is refined. A novel, computationally inexpensive analytical Jacobian for NKM is developed to solve unsteady incompressible Navier–Stokes momentum equations on staggered overset-curvilinear grids with immersed boundaries. Moreover, the analytical Jacobian is used to form a preconditioner for matrix-free method in order to improve its performance. The NKM with the analytical Jacobian was validated and verified against Taylor–Green vortex, inline oscillations of a cylinder in a fluid initially at rest, and pulsatile flow in a 90 degree bend. The capability of the method in handling complex geometries with multiple overset grids and immersed boundaries is shown by simulating an intracranial aneurysm. It was shown that the NKM with an analytical Jacobian is 1.17 to 14.77 times faster than the fixed-point Runge–Kutta method, and 1.74 to 152.3 times (excluding an intensively stretched grid) faster than automatic differentiation depending on the grid (size) and the flow problem. In addition, it was shown that using only the diagonal of the Jacobian further improves the performance by 42–74% compared to the full Jacobian. The NKM with an analytical Jacobian showed better performance than the fixed point Runge–Kutta because it converged with higher time steps and in approximately 30% less iterations even when the grid was stretched and the Reynold number was increased. In fact, stretching the grid decreased the performance of all methods, but the fixed-point Runge–Kutta performance decreased 4.57 and 2.26 times more than NKM with a diagonal and full Jacobian, respectivley, when the stretching factor was increased. The NKM with a diagonal analytical Jacobian and matrix-free method with an analytical preconditioner are the fastest methods and the superiority of one to another depends on the flow problem. Furthermore, the implemented methods are fully parallelized with parallel efficiency of 80–90% on the problems tested. The NKM with the analytical Jacobian can guide building preconditioners for other techniques to improve their performance in the future.

Mixed miscible-immiscible fluid flow modelling with incompressible SPH framework

A divergence-free Incompressible Smoothed Particle Hydrodynamics (ISPH) framework is developed for modelling multifluid immiscible/miscible flows. The numerical model considers Navier-Stokes equation along with scalar-transport equation to represent flow characteristics. Pure divergence free ISPH algorithm is used to solve Navier-Stokes momentum equation. Scalar transport equation is solved based on the nature of the fluid particles represented in terms of colour indicator (ϕ). Fixed ghost particles are utilized for simulating slip boundary. Proposed model is validated against three analytical/experimental results: (a) immiscible Rayleigh-Taylor Instability (b) lock-exchange non-Boussinesq flow (c) miscible gravity current with low density ratio. The model can capture Kelvin-Helmholtz instabilities generated at the interface. the framework is capable of capturing density-dependent flow with low to high density ratio alike. The capability of the developed model in general multifluid system is demonstrated for two cases (a) miscible multimode Rayleigh-Taylor Instability and (b) mixed miscible-immiscible fluid flow simulation.

A Robust volume conservative divergence-free ISPH framework for free-surface flow problems

This study presents a Volume Conservative approach for resolving volume conservation issue in divergence-free incompressible Smoothed Particle Hydrodynamics (ISPH). Irregular free surface deformation may introduce error in volume computation, which has a cascading effect over time. Proposed correction decreases this numerical compressibility to a minimal value. The correction is obtained directly by solving Navier-Stokes momentum equation. Consequently, the framework does not require any parametric study for mixed source/sink term or iterative solution of pressure Poisson equations. The correction is implemented on four different types of flow: (a) pressurized flow in a closed box, (b) dambreak flow, (c) flow through porous block, (d) lock-exchange flow of immiscible fluids (both free-surface and pressurized flow). All four scenarios are shown to have minimal error compared to pure divergence-free ISPH.

Performance Evaluation of Bidirectional Dry Gas Seals with Special Groove Geometry

ABSTRACTThere are a very few studies of bidirectional gas seals, particularly those with certain profiles used in the industry. A parametric study of the performance of bidirectional dry gas seals under a set of operating conditions is presented. The expounded approaches use solution of 3D Navier-Stokes momentum and continuity equations for various forms of grooved gas seals, particularly for the trapezoidal shape variety for which there has been a particular dearth of in-depth analyses. It is shown that such groove geometries enhance the load-carrying capacity of the seal through increased hydrodynamic lift. This is as the result of enhanced localized wedge flow, particularly with a reduced seal gap. Therefore, there is the opportunity of gap minimization while reducing leakage rate and power loss. For given operating loading, kinematic and thermal conditions, as well as seal geometry and topography, the operating minimum film thickness may be considered the main design parameter.

Existence of global weak solutions to compressible isentropic finitely extensible nonlinear bead–spring chain models for dilute polymers: The two-dimensional case

We prove the existence of global-in-time weak solutions to a general class of models that arise from the kinetic theory of dilute solutions of nonhomogeneous polymeric liquids, where the polymer molecules are idealized as bead–spring chains with finitely extensible nonlinear elastic (FENE) type spring potentials. The class of models under consideration involves the unsteady, compressible, isentropic, isothermal Navier–Stokes system in a bounded domain Ω in Rd, d=2, for the density ρ, the velocity u˜ and the pressure p of the fluid, with an equation of state of the form p(ρ)=cpργ, where cp is a positive constant and γ>1. The right-hand side of the Navier–Stokes momentum equation includes an elastic extra-stress tensor, which is the classical Kramers expression. The elastic extra-stress tensor stems from the random movement of the polymer chains and is defined through the associated probability density function that satisfies a Fokker–Planck-type parabolic equation, a crucial feature of which is the presence of a centre-of-mass diffusion term. This extends the result in our paper J.W. Barrett and E. Süli (2016) [9], which established the existence of global-in-time weak solutions to the system for d∈{2,3} and γ>32, but the elastic extra-stress tensor required there the addition of a quadratic interaction term to the classical Kramers expression to complete the compactness argument on which the proof was based. We show here that in the case of d=2 and γ>1 the existence of global-in-time weak solutions can be proved in the absence of the quadratic interaction term. Our results require no structural assumptions on the drag term in the Fokker–Planck equation; in particular, the drag term need not be corotational. With a nonnegative initial density ρ0∈L∞(Ω) for the continuity equation; a square-integrable initial velocity datum u˜0 for the Navier–Stokes momentum equation; and a nonnegative initial probability density function ψ0 for the Fokker–Planck equation, which has finite relative entropy with respect to the Maxwellian M associated with the spring potential in the model, we prove, via a limiting procedure on a pressure regularization parameter, the existence of a global-in-time bounded-energy weak solution t↦(ρ(t),u˜(t),ψ(t)) to the coupled Navier–Stokes–Fokker–Planck system, satisfying the initial condition (ρ(0),u˜(0),ψ(0))=(ρ0,u˜0,ψ0).

Existence of global weak solutions to compressible isentropic finitely extensible bead-spring chain models for dilute polymers

We prove the existence of global-in-time weak solutions to a general class of models that arise from the kinetic theory of dilute solutions of nonhomogeneous polymeric liquids, where the polymer molecules are idealized as bead-spring chains with finitely extensible nonlinear elastic (FENE) type spring potentials. The class of models under consideration involves the unsteady, compressible, isentropic, isothermal Navier–Stokes system in a bounded domain [Formula: see text] in [Formula: see text], [Formula: see text] or [Formula: see text], for the density [Formula: see text], the velocity [Formula: see text] and the pressure [Formula: see text] of the fluid, with an equation of state of the form [Formula: see text], where [Formula: see text] is a positive constant and [Formula: see text]. The right-hand side of the Navier–Stokes momentum equation includes an elastic extra-stress tensor, which is the sum of the classical Kramers expression and a quadratic interaction term. The elastic extra-stress tensor stems from the random movement of the polymer chains and is defined through the associated probability density function that satisfies a Fokker–Planck-type parabolic equation, a crucial feature of which is the presence of a center-of-mass diffusion term.

Inviscid Circulatory-Pressure Field Derived from the Incompressible Navier–Stokes Equations

The static-pressure field in the steady and incompressible Navier–Stokes momentum equation is decomposed into circulatory (inviscid) and dissipative (viscous) partial-pressure fields. It is shown analytically that the circulatory-pressure integral over the surface of a lifting body of thickness recovers the lift generating Kutta–Joukowski theorem in the far field, and results in Maskell’s formula for the vortex-induced drag plus an additional pressure-loss term that tends to zero for an infinitely thin wake. A Poisson equation for the circulatory-pressure field is implemented as a transport equation into the FLUENT 13 solver. Numerical examples include a circular cylinder at , the S809 airfoil at , and the ONERA M6 wing at . It is shown that the circulatory-pressure field does indeed behave as an inviscid pressure field of a fully viscous solution, and provides insight into the nature of pressure drag and its contributions to local form and vortex-induced drag.

Numerical analysis of convective dispersion of pen shell Atrina pectinata larvae to support seabed restoration and resource recovery in the Ariake Sea, Japan

In this study, numerical simulation of convective dispersion of the larvae of the pen shell Atrina pectinata was conducted to support effective seabed restoration in the Ariake Sea, Japan. A two-dimensional depth-averaged model consisting of a continuity equation and Navier–Stokes momentum equations was developed to reproduce tidal currents in the Ariake Sea. This larval transport model was then used to predict migration of larvae drifting on tidal currents. To determine effective areas for seabed restoration, 14 release points were defined in the model as potential new spawning grounds for the pen shell. The number of larvae that migrated from these points to habitable areas for pen shells (depth: 2<h<20m, grain size median diameter: >62.5μm) was calculated for each release point. Because of the anticlockwise tidal residual flow in the inner bay of the Ariake Sea, pen shell larvae also had anticlockwise trajectories. Larvae originating from the northeast area spread out over a large area, while larvae originating from the northwest area reached only the western area. Most of the larvae originating from Isahaya Bay flowed out through the mouth of the Ariake Sea. Consequently, larvae released from the northeast area (particularly from Point 13, Noku 210) reached the largest habitable area. In addition, most of the larvae ultimately drifted to Isahaya Bay, suggesting that seabed restoration in the northeast area and Isahaya Bay would be the most effective approach to recovery of pen shell resources.

Experimental investigation of the root flow in a horizontal axis wind turbine

ABSTRACTThis research investigates the flow behavior and its features in the blade's root region of a horizontal axis wind turbine by using stereoscopic particle image velocimetry (PIV) technique. Wind tunnel tests are conducted to measure the velocity field, phase‐locked with the blade motion, at different azimuth angles and at different spanwise positions. The pressure distribution is obtained from PIV velocity field by solving the Navier–Stokes momentum equations. In this paper, we aim to answer two questions: (i) How is the flow behavior in the root region? (ii) How is the evolution of the root vortex? The analysis of the velocity fields shows an outboard radial flow motion in the root region and a vorticity driven inboard motion at the bladeŠs maximum chord position. As a result of this vorticity driven flow, an increase in the axial velocity close to nacelle is measured. Wake sheets are observed and further discussed in the measured velocity and vorticity distributions. The formation and evolution of the root vortices conveyed downstream by the axial velocity are analyzed through vorticity and pressure distributions. Although the azimuthal vorticity in 3D representation is showing the trailing vorticity, the tilting of the root vortex tube is observed in the axial vorticity distribution. Moreover, the radial vorticity and azimuthal velocity from chordwise measurements show separation on the suction surface of the blade. This research concluded that the flow in the blade wake is driven by the root vortex; hence, the local effects of the root vortex cannot be ignored. Copyright © 2013 John Wiley &amp; Sons, Ltd.

Intra-aneurysmal hemodynamic alterations by a self-expandable intracranial stent and flow diversion stent: high intra-aneurysmal pressure remains regardless of flow velocity reduction

ObjectLittle is known about how much protection a flow diversion stent provides to a non-thrombosed aneurysm without the adjunctive use of coils.MethodsA three-dimensional anatomically realistic computation aneurysm model was created...

Investigation of Pressure Pulse Distribution in Porous Media

Diffusivity equation commonly used for pressure distribution prediction in porous media results from substituting equation of state and continuity equation in Navier-Stokes momentum equation. From mathematical point of view this equation format shows infinite propagation speed for pressure pulse through porous media, which is physically impossible. This issue may caused by numerous assumptions that has been implemented for developing diffusivity equation. However, if we omit two main assumptions of steady state condition and constant velocity and consider linear approximation for velocity field, the pressure propagation differential equation would be hyperbolic which is called Telegraph Equation. The propagation speed is limited for this equation. In this work, these equations are compared in prediction of pressure pulse propagation in Cartesian coordination with different parameters. The results show that the telegraph equation has minor correction in some cases as: far distances from pressure pulse source, when the fluid has high viscosity and for the rocks with low porosity and permeability; so considering common parameters in hydrocarbon reservoirs, the diffusivity equation has sufficient accuracy for reservoir engineering applications.

EXISTENCE AND EQUILIBRATION OF GLOBAL WEAK SOLUTIONS TO KINETIC MODELS FOR DILUTE POLYMERS II: HOOKEAN-TYPE MODELS

We show the existence of global-in-time weak solutions to a general class of coupled Hookean-type bead-spring chain models that arise from the kinetic theory of dilute solutions of polymeric liquids with noninteracting polymer chains. The class of models involves the unsteady incompressible Navier–Stokes equations in a bounded domain in ℝd, d = 2 or 3, for the velocity and the pressure of the fluid, with an elastic extra-stress tensor appearing on the right-hand side in the momentum equation. The extra-stress tensor stems from the random movement of the polymer chains and is defined by the Kramers expression through the associated probability density function that satisfies a Fokker–Planck-type parabolic equation, a crucial feature of which is the presence of a center-of-mass diffusion term. We require no structural assumptions on the drag term in the Fokker–Planck equation; in particular, the drag term need not be corotational. With a square-integrable and divergence-free initial velocity datum ṵ0for the Navier–Stokes equation and a non-negative initial probability density function ψ0for the Fokker–Planck equation, which has finite relative entropy with respect to the Maxwellian M, we prove, via a limiting procedure on certain regularization parameters, the existence of a global-in-time weak solution t ↦ (ṵ(t), ψ(t)) to the coupled Navier–Stokes–Fokker–Planck system, satisfying the initial condition (ṵ(0), ψ(0)) = (ṵ0, ψ0), such that t ↦ ṵ(t) belongs to the classical Leray space and t ↦ ψ(t) has bounded relative entropy with respect to M and t ↦ ψ(t)/M has integrable Fisher information (with respect to the measure [Formula: see text]) over any time interval [0, T], T&gt;0. If the density of body forces [Formula: see text] on the right-hand side of the Navier–Stokes momentum equation vanishes, then a weak solution constructed as above is such that t ↦ (ṵ(t), ψ(t)) decays exponentially in time to [Formula: see text] in the [Formula: see text]-norm, at a rate that is independent of (ṵ0, ψ0) and of the center-of-mass diffusion coefficient. Our arguments rely on new compact embedding theorems in Maxwellian-weighted Sobolev spaces and a new extension of the Kolmogorov–Riesz theorem to Banach-space-valued Sobolev spaces.

A three-dimensional PSME model for rotating flows in an annular cavity

A pseudospectral matrix-element (PSME) numerical model is described for the simulation of rotating flows in a three-dimensional annular cavity. Temporal discretisation is implemented using a second-order semi-implicit scheme. Modified compressibility is invoked to handle the coupling between velocity and pressure while maintaining the incompressibility constraint. The governing continuity and Navier–Stokes momentum equations and boundary conditions are discretised using Chebyshev and Fourier collocation formulae. The model is validated against numerical results from alternative schemes and experimental data on rotating flows in an annular cavity. A base flow regime and instability patterns are observed, in accordance with other previously published investigations. It is demonstrated that the PSME model provides an accurate representation of rotating flows in an annular cavity.

An adaptive finite element splitting method for the incompressible Navier–Stokes equations

We present an adaptive finite element method for the incompressible Navier–Stokes equations based on a standard splitting scheme (the incremental pressure correction scheme). The presented method combines the efficiency and simplicity of a splitting method with the powerful framework offered by the finite element method for error analysis and adaptivity. An a posteriori error estimate is derived which expresses the error in a goal functional of interest as a sum of contributions from spatial discretization, time discretization and a term that measures the deviation of the splitting scheme from a pure Galerkin scheme (the computational error). Numerical examples are presented which demonstrate the performance of the adaptive algorithm and high quality efficiency indices. It is further demonstrated that the computational error of the Navier–Stokes momentum equation is linear in the size of the time step while the computational error of the continuity equation is quadratic in the size of the time step.

EXISTENCE AND EQUILIBRATION OF GLOBAL WEAK SOLUTIONS TO KINETIC MODELS FOR DILUTE POLYMERS I: FINITELY EXTENSIBLE NONLINEAR BEAD-SPRING CHAINS

We show the existence of global-in-time weak solutions to a general class of coupled FENE-type bead-spring chain models that arise from the kinetic theory of dilute solutions of polymeric liquids with noninteracting polymer chains. The class of models involves the unsteady incompressible Navier–Stokes equations in a bounded domain in ℝd, d = 2 or 3, for the velocity and the pressure of the fluid, with an elastic extra-stress tensor appearing on the right-hand side in the momentum equation. The extra-stress tensor stems from the random movement of the polymer chains and is defined by the Kramers expression through the associated probability density function that satisfies a Fokker–Planck-type parabolic equation, a crucial feature of which is the presence of a centre-of-mass diffusion term. We require no structural assumptions on the drag term in the Fokker–Planck equation; in particular, the drag term needs not be corotational. With a square-integrable and divergence-free initial velocity datum ṵ0 for the Navier–Stokes equation and a non-negative initial probability density function ψ0 for the Fokker–Planck equation, which has finite relative entropy with respect to the Maxwellian M, we prove, via a limiting procedure on certain regularisation parameters, the existence of a global-in-time weak solution t ↦ (ṵ(t), ψ(t)) to the coupled Navier–Stokes–Fokker–Planck system, satisfying the initial condition (ṵ(0), ψ(0)) = (ṵ0, ψ0), such that t ↦ ṵ(t) belongs to the classical Leray space and t ↦ ψ(t) has bounded relative entropy with respect to M and t ↦ ψ(t)/M has integrable Fisher information (w.r.t. the measure [Formula: see text] over any time interval [0, T], T &gt; 0. If the density of body forces [Formula: see text] on the right-hand side of the Navier–Stokes momentum equation vanishes, then a weak solution constructed as above is such that t ↦ (ṵ(t), ψ(t)) decays exponentially in time to [Formula: see text] in the [Formula: see text] norm, at a rate that is independent of (ṵ0, ψ0) and of the centre-of-mass diffusion coefficient.