Naiver-Stokes Equations Research Articles

The global existence for the strong solution of the stochastic 3-D incompressible anisotropic Naiver-Stokes equations

This paper considers the stochastic 3-D incompressible anisotropic Navier-Stokes equations. We establish the global existence of a strong solution in probability when either the horizontal component of the initial data and external force is sufficiently small or the horizontal viscosity coefficient is sufficiently large.

Impact of ventilation efficiency on restraining hazardous toilet plume post-flushing in indoor environments

This study analyzed the hybrid ventilation efficiency of indoor toilet plumes generated post-flushing with various designs. Single-sided natural ventilation and hybrid ventilation were numerically investigated to determine the impact of vent locations commonly found in multistorey buildings. An in-situ experiment was performed to identify the size distribution of the plume with time series. A computational fluid dynamics model based on the Reynolds-averaged naiver-Stokes equations and baseline k-ω turbulence equations was used to predict aerosol pollutant transmission. The aerosol-laden particles receded gradually impacted by the convection of the ambient source. Besides flush volume, temperature and ventilation scenarios were also related to the generation and droplet transmission. The results indicated that the ventilation rates were generally increased with decreased residence time if the turbulence was not suppressed. This study suggested that appropriate ventilation design could significantly reduce the average particle residence time, in which case it could also reduce the rate of virus infection.

Vortex-induced vibration effect on the aeroelastic stability of two composite cylindrical shells using a fluid-structure interaction method

The purpose of this study, is to investigate the aeroelastic stability and effect of vortices resulting from passing air over cylindrical shells in three-dimensional condition. Here, the Naiver-Stokes equations in the integral form are considered studying the air flow based on the finite-volume method. The ANSYS Fluent software is used to study the fluid-solid interaction effect. The shear stress transport (SST) turbulence model is utilized to simulate the turbulent viscosity. Effect of different parameters such as two cylindrical shells alignment or two cylinders diameter ratio are considered. The results indicate that the vortices can play a significant role on the aeroelastic stability of the cylindrical shell.

Unconditional stability and error estimates of the modified characteristics FEMs for the Micropolar Navier–Stokes Equations

In this paper, the unconditional stability and optimal error estimate of the velocity, pressure and angular velocity for the modified characteristics FEMs of the unsteady Micropolar Naiver–Stokes Equations (MNSE) are presented. In this method, the nonlinear equation is linearized by the characteristic finite element method for dealing with the time derivative term and the convection term. Basing on the characteristic time-discrete system, the error function is split into a temporal error and a spatial error. With a rigorous analysis to the characteristic time-discrete system, we prove that the error between the numerical solution and the solution of the time-discrete system is τ-independent, where τ denotes the time stepsize. The stability results and optimal error estimates in L2 norm and H1 norm will be given. Finally, some numerical results will be provided to confirm our theoretical analysis.

Remarks on interior regularity criteria without pressure for the Navier-Stokes equations

In this note we investigate interior regularity criteria for suitable weak solutions to the 3D Naiver-Stokes equations, and obtain the solutions are regular in the interior if the LtpLxq(Q1) norm of the velocity is sufficiently small, where 1≤2p+3q<2 and 2≤p≤∞. It improves the recent result of p,q>2 by Kwon [15], and also generalizes Chae-Wolf's Lt∞Lx32+ criterion [3].

Numerical investigate of the effects of surface roughness on local scour around pipeline under steady current condition

A two-dimensional numerical model was developed to investigate the effects of the surface roughness on local scour around pipeline based on Naiver-Stokes equations. Both suspended load and bedload transport are considered for the sand motion equation. Present numerical model is verified accurately for simulating local scour around pipeline. The numerical results show that the surface roughness has a significant effect on the process of local scour, i.e. forces, Shields parameter and scour depth. For rough pipeline, the average value of the drag force coefficient on the pipeline, the RMS value of the lift coefficient, the scour depth and the scour width all decrease rapidly with increasing roughness and then change slowly.

Theoretical analysis on electrohydrodynamic instability of a low viscous electrified jet

In present work, a dispersion equation describing electrohydrodynamic (EHD) instability of a low viscous electrified jet was derived, where the Naiver-Stokes equations combined with the electrodynamic and hydrodynamic boundary conditions are utilized. The effect of operating parameters containing electric potential, initial jet velocity and electrode spacing on stability of the electrified jet was systematically examined and extensively analyzed, as well as liquid properties including liquid viscosity and surface tension. The analysis indicate that the axisymmetric surface waves dominate the breakup instability of the electrified jet at low electric potential or low flowrate. As flowrate or electric potential increases, the influence of the non-axisymmetric surface waves is gradually enhanced. The electrode spacing and initial diameter of the jet also have extremely slight effect on the surface wave growth rate. The viscosity could stabilize the electrified jet and almost has no influence on the maximum wave number. With the surface tension increasing, the stability of the jet is usually enhanced. The analytical results agree well with the experimental observation in the varicose instability. Meanwhile, the discrepancy between the predicted and experimental data in the whipping instability indicates the influence of non-linear perturbations is indispensable.

A meshless generalized finite difference scheme for the stream function formulation of the Naiver-Stokes equations

In this study, a meshless numerical scheme, which is the combination of the generalized finite difference method, the fictitious-nodes technique, and the two-step Newton-Raphson method, was proposed to solve the stream function formulation of the Navier-Stokes equations. To reduce the numbers of unknowns and partial differential equations for the two-dimensional flow field of a viscous incompressible fluid, the stream function formulation, which is a fourth-order nonlinear partial differential equation with only one unknown variable, can be derived by introducing the definition of vorticity and stream function. By applying the generalized finite difference method, the derivatives in the stream function formulation can be simply expressed as a linear combination of functional data and weighting values at several nearest nodes. To deal with the two boundary conditions for the fourth-order partial differential equations, the fictitious-nodes technique was utilized to set up the fictitious nodes outside the computational domain, and cooperate with the generalized finite difference method to form a system of nonlinear algebraic equations. The aforementioned numerical procedure can yield a square Jacobian matrix, so the Newton-Raphson method can be adopted to efficiently acquire the numerical solutions. Several numerical examples were tested to verify the practicability of the proposed meshless numerical scheme.

Computational Fluid Dynamics Modeling of Pseudotransient Perforation Erosion in a Limited-Entry Completion Cluster

Summary Hydraulic fracturing in limited-entry (LE) completion designs relies on maintaining a high bottomhole treating pressure (BHTP). LE requires high perforation friction to maintain an even distribution of the hydraulic fracturing slurry. As sand exits the perforations, the perforations start to erode. The erosional change in the perforation alters the desired perforation friction and subsequent BHTP. As operators rely on multistage hydraulic fracturing to generate economic production, the issue of perforation erosion becomes inherently repetitive from stage to stage and cumulatively a significant issue. The industry has seen how a perforation can change from a before-and-after perspective with downhole cameras and imaging techniques before and after treatments. However, a more detailed understanding of the dynamic process of perforation erosion can give a better expectation of perforation performance throughout a hydraulic fracturing treatment and not just pretreatment compared to post-treatment. Computational fluid dynamics (CFD) is a quickly emerging tool in the industry. CFD aims to model fluid flow by numerically solving the Naiver-Stokes equations within a specified domain. Along with modeling fluid systems, CFD has the capability to model dispersed particles within the fluid. Once the particles are introduced into the fluid, the domain can also be eroded away within the CFD model. By utilizing the erosional capabilities of CFD, paired with the flow of a hydraulic fracturing slurry, perforation erosion can be investigated transiently throughout an entire hydraulic fracturing stage. This work presents a better dynamic understanding of perforation erosion rather than just a “before vs. after” comparison. The CFD modeling methodology used to achieve the correct erosional pattern observed in the field is presented. Throughout this work, four different hydraulic fracturing completion parameters are investigated to determine the respective roles in perforation erosion. The four parameters include proppant size, proppant concentration, fracturing fluid viscosity, and proppant concentration ramping schedules. By investigating the impact that controlled design parameters have on perforation erosion, perforation erosion can be better anticipated to deliver improved completion results.

Multi-objective optimization of 3D micro-fins using NSGA-II

Because of the great potential for optimization of micro-fin surfaces (< 0.5 mm tall) and the lack of any optimization work encompassing three-dimensional geometries on heat exchanger surfaces, this paper proposes a robust methodology to solve the multi-objective optimization problem with a numerical model and thereby determine the flow physics that characterize the optimal geometries. The micro-fin design variables are considered as micro-fin height, angle, thickness, pitch, length of discontinuity, and number of discontinuities per micro-fin pitch. To reduce computational time, a channel with two-sided periodic domains is presented. A parallelizable numerical model is developed based on Reynolds average Naiver-Stokes equations and a realistic k-ɛ model. The simulation procedure solves convective heat transfer and turbulent flow over micro-fins and solves heat transfer for solid conduction using a constant wall temperature condition to maintain robustness for geometries that do not yield unity fin efficiencies. This study provides a robust methodology to integrate the numerical model into a multi-objective optimization algorithm named non-dominated sorting genetic algorithm II to generate the best trade-off solutions between enhancement of Nusselt number and Fanning friction factor of a micro-fin surface to a smooth surface at two Reynold numbers (Re) of 25,000 and 49,000. This study presents a unique intersection of the drag reduction approach and heat transfer enhancement by finding unique operating geometries. For example, a particular optimal design geometry at Re = 49,000 reduced drag by 8% but was also beneficial to heat transfer increasing by 21% compared to a smooth surface. The best micro-fin surface of this study at Re = 49,000 enhanced the efficiency index by 18% compared to the highest reported efficiency index for micro-fin tubes.

The effect of flow speed on the bubble dynamics: A numerical study

The Naiver-Stokes equations are utilized to investigate the bubble dynamics subjected to the flow speed in the vicinity of a rigid wall. The predictions are compared with the Gilmore method and available experimental results, and a reasonable agreement is obtained. The important flow properties, e.g., bubble morphology, collapse time, jet development, pressure wave propagation, and vorticity evolution are studied in detail. The results demonstrate that the combined effect of the flow speed and rigid wall results in the generation of oblique jet, which may contribute to the development of the fish-scale pits on the surface of the hydraulic machinery. Moreover, our results show that the impulse loads caused by the bubble collapse in moving water are lower than those in stationary water. Finally, it is indicated that the stretching term plays a major role in the vorticity both inside and outside the bubble, while the dilatation term mainly influences the vorticity inside the bubble.

An immersed boundary projection method for solving the fluid-rigid body interaction problems

We develop an immersed boundary projection method for solving the Naiver-Stokes equations and Newton-Euler equations to simulate the fluid-rigid body interactions in two and three dimensions. A novel fractional step algorithm is introduced for which fast solvers can be applied by exploiting the algebraic structure of the underlying schemes. The Navier-Stokes equations are decoupled while the Newton-Euler equations are solved simultaneously with a constraint equation of the immersed boundary force density. In contrast to previous works, the present method preserves both the fluid incompressibility and the kinematic constraint of the rigid body dynamics at a discrete level simultaneously while maintaining numerical stability. We demonstrate the numerical results of the present method involving spherical and spheroidal rigid bodies with a moderate range of density ratios, which are congruent with the results in the literature.

A Three-Dimensional Comprehensive Numerical Model of Ion Transport during Electro-Refining Process for Scrap-Metal Recycling.

A transient three-dimensional comprehensive numerical model was established to study ion transport caused by diffusion, convection, and electro-migration in the electro-refining process for scrap-metal recycling. The Poisson–Nernst–Planck equations were used to define ion transport within the electrolyte, while the Naiver–Stokes equations and the energy equation were employed to describe fluid flow and heat transfer. In addition, the Butler-Volmer formulation was used to represent the kinetics of the electrochemical reaction. The comparison between the measured and simulated data indicates the reliability of the model. Under the action of diffusion and electro-migration, the positive copper ion moves from the anode to the cathode, while the negative sulfate ion migrates in the opposite direction. The distribution of the ion concentration, however, greatly changes if the fluid flow is taken into account. The ion concentration around the anode and the rate of the electrochemical reaction that occurs at the anode surface are reduced by the fluid flow. The proposed computational framework offers a valuable basis for future research and development in the field of scrap-metal recycling technology.

Numerical study of high temperature non-equilibrium effects of double-wedge in hypervelocity flow

The high temperature non-equilibrium effects of shock wave interaction and shock wave/boundary layer interaction are important issues for hypervelocity flows. The models of thermochemical non-equilibrium gas (TCNEG), thermal non-equilibrium chemical frozen gas (TNCFG), chemical non-equilibrium gas (CNEG), and thermally perfect gas are used to simulate the double-wedge flows with a total enthalpy of 8 MJ/kg in this study. The unsteady two-temperature Naiver-Stokes equations in the laminar and turbulence flows are solved using the finite volume method. For laminar flow, the shock structures and the heat flux peak for TCNEG model at 170 μs are agreed better with the experiment result compared to reference studies. There are different size vortices in the separation zones, which causes the distributions of the wall heat flux oscillate irregularly. The thermal non-equilibrium effects are the most intense near the attached shock and detached shock, and the degree of oxygen dissociation is the strongest in the subsonic zone near the slip-line. For turbulence flow, the shock structures for the four models are close to Edney's IV interaction. The separation shock position for the TNCFG model is the most upstream, and that for the CNEG model is quite different from the TCNEG model. The intensity of the reflected shocks on the back wedge and its nearby shock interaction largely determine the peak values of the heat flux for the four models.

Effects of thermocapillary convection and shear flow on the distribution of inclusions in half-floating zones

The distribution of inclusions plays a critical role in the performance of crystalline silicon, which is an important raw material for solar panels and semiconductors. To describe the formation of the aggregated inclusions, a three-phase gas–liquid-solid system was designed, in which the gas and liquid phases interact directly through the free surface, and the inclusions interact with them in the liquid. It is worth noting that the inclusions and gas phase did not interact directly. In the gas and liquid phases, the finite volume method was used to solve the Naiver-Stokes equations, and the CLSVOF method was used to track the free surface deformation. The motion of the inclusions was tracked using the Lagrange method. The inclusions and fluid were coupled in two ways: energy and momentum exchange. The numerical results show that the spatial distributions of the inclusion volumetric concentration, number density, and characteristic radius have some features of the recirculation zone. The closer to the center of the liquid bridge, the lower the density of inclusions. When the direction of the shear flow is opposite to that of the thermocapillary convection, this phenomenon is more obvious. This indicates that the crystal quality in the middle region is higher when various parameters are appropriate.

Regularity for 3D Inhomogeneous Naiver–Stokes Equations in Vishik Spaces

In this short paper, we consider the conditional regularity for the 3D inhomogeneous incompressible Navier–Stokes equations in Vishik spaces and give regularity criterion of strong solutions.

Pointwise estimates of the solution to one dimensional compressible Naiver-Stokes equations in half space

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Effects of vortices trapped in a dead end on resistance to pore-scale flow

There are numerous dead ends in micro-scale porous media and their special expansion-contraction structure incurs vortices in pore-scale flow. These vortices involve energy dissipation and thus cause extra flow resistance. Since there are no methods available for calculating this extra resistance, this work presents an innovative method for this purpose. It involves two steps including theoretical calculation and numerical simulation. Based on the well-known Hagen-Poiseuille equation, theoretical models are established for capillary tubes with different dead ends, both for calculating intrinsic flow resistance and Reynolds number. Computational fluid dynamics (CFD) simulation, based on the well-known Naiver-Stokes equations, is employed to characterize a pore-scale flow with vortices and to obtain the total flow resistance. The extra vortex-induced resistance can be further calculated by subtracting the theoretical intrinsic resistance from the simulated total resistance. This method is performed for five base cases, including a straight one without a dead end, a square-dead-end one, a quarter-circle-dead-end one, a triangle-dead-end one and one with combination of three dead ends. CFD simulations reveal that main and secondary vortices obviously exist in dead ends. Energy dissipation caused by vortices induces extra flow resistance. The extra resistance shows significant dependence on dead end shape and size as well as number. One possible reason is that a larger space of a dead end may promote the formation of more complex vortices thus causing larger extra resistance. The existence of numerous dead ends in a real micro-scale porous medium means formation of substantial numbers of shear-driven vortices and thus induction of huge increase in flow resistance. As a result, a real Q~Δp relationship may lie far away below the theoretical Q~Δp relationship determined by the Darcy's Law, thus undermining its dominance in characterizing pressure-driven pore-scale laminar flow. This work sheds light on insight of pore-scale flow and drops a hint that we may need corrections to this law in order to improve its characterization quality.

Numerical investigation on vortex-induced vibration of an elastically mounted circular cylinder with multiple control rods at low Reynolds number

A two-dimensional numerical model was developed to investigate the problem of vortex induced vibration of a main circular cylinder with multiple small control rods by solving the Naiver-Stokes Equations at low Reynolds number. The numerical results show that the spacing ratio G / D has a significant effect on the kinematic response of the cylindrical system. Based on the vortex induced vibration response characteristics of the cylindrical group, the effect of the gap ratio on the occurrence of vortex induced vibration in the cylindrical group can be divided into three intervals. And the numerical results also show that the vibration amplitude can be reduced obviously by using small control rods in the scope of present study.

Periodic orbits exhibit oblique stripe patterns in plane Couette flow

Spatiotemporally chaotic dynamics of transitional plane Couette flow may give rise to regular turbulent-laminar stripe patterns with a large-scale pattern wavelength and an oblique orientation to the laminar flow direction. How the stripe pattern forms remains an open problem. To understand stripe pattern spatiotemporal dynamics we identify unstable periodic orbits of the 3D Naiver-Stokes equations that show oblique large-scale spatial amplitude modulation and a temporal evolution of standing waves. The unstable periodic orbits are embedded in the edge of chaos in a symmetry subspace of plane Couette flow and thereby may mediate transition to and from turbulent flows with oblique patterns.