Governing Navier-Stokes Equations Research Articles

Numerical Analysis of Pulsatile Blood Flow in a Stented Human Coronary Artery with a Flow Divider

Shear stress is known to play a central role in restenosis formation and is sensitive to stent geometry. Local flow alterations created by a different stents without and with flow divider were studied to compare the hemodynamic effects of Stent design properties on restenosis in stented human coronary artery. Blood pressure and shear stress values were computed in three different sites, including stented arterial segment, pre-stent and post-stent regions using computational fluid dynamics. Blood flow was assumed as pulsatile, incompressible and Newtonian flow. Rigid boundary conditions were assumed for all models. The governing Navier-Stokes equations were solved using commercial software package (Fluent V6.0.12). Stents are assumed with real structure and modeled using the commercial software package (Gambit, V2.0). The arterial wall shear stress distribution was investigated in three major regions and critical sites were located. It is concluded that the wall shear stress between stent struts was sensitive to strut spacing, profile of strut, number of struts and curvature. Our 3D computational fluid dynamics modeling demonstrate that with increasing the angle between two sides of the stent strut the percentage of intrastrut area that exposed to critical value of WSS decreases. By application of a flow divider, the wall shear stress in stented segment increases markedly and so is the pressure gradient in stented segment. Flow divider influences the blood flow pattern in proximal of stented segment. In this section, the WSS increases with application of the flow divider. The results for different diameters of flow divider show that optimum diameter for flow divider is D/3.

Simulations of fibre orientation in dilute suspensions with front moving in the filling process of a rectangular channel using level-set method

The simulation of fibre orientation in dilute suspension with front moving is carried out using the projection and level-set methods. The motion of fibres is described using the Jeffery equation, and the contribution of fibres to the flow is accounted for by the configuration-field method. The dilute suspension of short fibres in Newtonian fluids is considered. The governing Navier–Stokes equation for the fluid flow is solved using the projection method with finite difference scheme, while the fibre-related equations are directly solved with the Runge–Kutta method. In the present study for fibres in dilute suspension flow for injection molding, the effects of various flow and material parameters on the fibre orientation, the velocity distributions and the shapes of the leading flow front are found and discussed. Our findings indicate that the presence of fibre motion has little influence on the front shape in the ranges of fibre parameters studied at the fixed Reynolds number. Influence of changing fibre parameters only causes variation of front shape in the region near the wall, and the front shape in the central core area does not vary much with the fibre parameters. On the other hand, the fibre motion has strong influence on the distributions of the streamwise and transverse velocities in the fountain flow. Fibre motion produces strong normal stress near the wall which leads to the reduction of transversal velocity as compared to the Newtonian flow without fibres, which in turn, leads to the increased streamwise velocity near the wall. Thus, the fibre addition to the flow weakens the strength of the fountain flow. The Reynolds number has also displayed significant influence on the distribution of the streamwise velocity behind the flow front for a given fibre concentration. It is also found that the fibre orientation is not always along the direction of the velocity vector in the process of mold filling. In the region of the fountain flow, the fibre near the centreline is more oriented across the streamwise direction compared to that in the region far behind the flow front. This leads to the fact that the fibre near the centreline in the region of fountain flow is more extended along the transverse direction. As the fibre orientation in the suspension flow and the shape of the flow front have great bearing on the quality of the product made from injection molding, this study has much implications for engineering applications. These results can also be useful in other fields dealing with fibre suspensions.

In-duct orifice and its effect on sound absorption

A numerical investigation of sound absorption by an in-duct orifice with and without flow was carried out using a sixth-order finite difference direct numerical simulation (DNS) scheme with explicit fourth-order time marching to solve the governing Navier–Stokes equation. The DNS scheme has previously been validated against benchmark aeroacoustic problems and good agreement was obtained. Thus, it was applied to simulate the acoustic impedance of a circular orifice with different openings and a laminar flow through the same orifice. Both discrete frequency and broadband excitations were studied. When the in-duct orifice is exposed to discrete frequency sound wave in the absence of flow, alternate vortex shedding on both sides of the orifice is observed. The strength of shed vortices is stronger at low frequencies and thus the reduction of sound energy is higher. These vortices dissipate while moving away from the orifice. Therefore, the process provides a mechanism for adsorption of incident sound. The numerical results of broadband excitation indicate that small orifice opening is a more efficient sound absorber whereas a large opening is more or less transparent to the incident wave. The absorption, reflection and transmission coefficients of the in-duct orifice are calculated by a transfer function method. It is found that the sound coefficients are strongly dependent on the orifice opening size and frequency. In the presence of a flow, only alternate vortex shedding on one side of the orifice is observed. In spite of this, the results show that sound absorption behavior is very similar to the no flow case, i.e., sound absorption is more effective with small orifice.

Rotational oscillations of a cylinder in cross-flow

The laminar unsteady periodic flow motion found in the wake of a rotationally oscillating circular cylinder is investigated numerically at a Reynolds number of 855 over a wide range of oscillation amplitude and frequency ratio. The governing Navier–Stokes equations are integrated to determine the flow field structure for large values of the time using an accurate spectral-finite difference method, but with the boundary vorticity calculated using global vorticity conditions rather than local finite-difference approximations. The lock-on phenomenon has been predicted and its effect on the flow hydrodynamics has been determined. The numerical scheme is verified by applying it to the special case of a steadily rotating cylinder (no forced oscillations) and good qualitative agreements are found.

Particle deposition onto micropatterned charge heterogeneous substrates: Trajectory analysis

A trajectory analysis of particles near a micropatterned charged substrate under radial impinging jet flow conditions is presented to investigate the effect of surface charge heterogeneity on particle trajectory and deposition efficiency. The surface charge heterogeneity is modeled as concentric bands of specified width and pitch having positive and negative surface potentials. The flow distribution is obtained using finite element analysis of the governing Navier–Stokes equations. The particle trajectory analysis takes into consideration the hydrodynamic interactions, gravity, van der Waals and electrostatic double layer interactions. The presence of surface charge heterogeneity on the substrate gives rise to an oscillating particle trajectory near the collector surface due to repulsive and attractive forces. As a result of the coupled effects of hydrodynamic and colloidal forces, the particle trajectories and deposition efficiencies are increasingly affected by surface charge heterogeneity as one moves radially away from the stagnation point. The results indicate that it is possible to render collectors with up to 50% favorable surface fraction completely unfavorable by modifying the ratio of the radial to normal fluid velocity. Utilizing the real favorable area fraction of the collector, the patch model expression for calculating the deposition efficiency is modified for impinging jet flow geometry.

Turbulence particle models for tracking free surfaces

Two numerical particle models, the Smoothed Particle Hydrodynamics (SPH) and Moving Particle Semi-implicit (MPS) methods, coupled with a sub-particle scale (SPS) turbulence model, are presented to simulate free surface flows. Both SPH and MPS methods have the advantages in that the governing Navier-Stokes equations are solved by Lagrangian approach and no grid is needed in the computation. Thus the free surface can be easily and accurately tracked by particles without numerical diffusion. In this paper different particle interaction models for SPH and MPS methods are summarized and compared. The robustness of two models is validated through experimental data of a dam-break flow. In addition, a series of numerical runs are carried out to investigate the order of convergence of the models with regard to the time step and particle spacing. Finally the efficiency of the incorporated SPS model is further demonstrated by the computed turbulence patterns from a breaking wave. It is shown that both SPH and MPS models provide a useful tool for simulating free surface flows.

A 2D compact fourth-order projection decomposition method

A 2D fourth-order compact direct scheme projection decomposition method for solving incompressible viscous flows in multi-connected rectangular domains is devised. In each subdomain, the governing Navier–Stokes equations are discretized by using fourth-order compact schemes in space and second-order scheme in time. The coupling between subdomains is based on a direct non-overlapping multidomain method: it allows to solve each Helmholtz/Poisson problem resulting of a projection method in complex geometries. The major difficulty of the Poisson–Neumann problem solvability is addressed and correctly treated. The present numerical method is checked through some classical numerical experiments. First, the second-order accuracy in time and the fourth-order accuracy in space are shown by matching with the analytical solution of the Taylor problem. The method is also tested by simulating the flow in a 2D lid-driven cavity. The utility of the compact scheme projection decomposition method approach is further illustrated by two other benchmark problems, viz., the flow over a backward-facing step and the laminar flow past a square prism. The present results are in good agreement with the experimental data and other numerical solutions available in the literature.

CFD studies of dynamic gauging

Dynamic gauging is a non-contact technique for measuring the thickness of soft deposit layers on solid surfaces immersed in liquid environments, in situ and in real time. The technique works by inducing a flow into a nozzle located close to, and normal to, the deposit surface; the relationship between pressure drop and mass flow rate yields a measure of the distance between the nozzle and the deposit, whence the thickness of the deposit can be deduced. Computational fluid dynamics (CFD) studies were performed to illuminate the fluid dynamics of this technique, with particular focus on the flow patterns and on the stresses imposed on the surface. The governing Navier–Stokes equations were solved using the augmented Lagrangian method implemented by the commercial partial differential equation solver, Fastflo TM . The code was first tested successfully against previous studies in the literature featuring confined, slow Homann flows, where fluid flowed out of a nozzle. Then, simulations of gauging flows, where fluid enters a nozzle from a confined entry region, were compared with experimental data; good agreement was observed. Laminar Newtonian flows have been investigated, with Reynolds number at the nozzle throat in the range 0< Re t <2200. The shear and normal stresses on the gauged surface were predicted using the output from the CFD simulations. An initial comparison of experimental results for power-law fluids (aqueous carboxy-methyl-cellulose solutions) demonstrated the versatility of the technique and implied its applicability to more complex fluids, which would be useful for industrial application. The success of this study will enable (i) use of the gauge to measure the strength of deposits, (ii) optimization of the shape of the nozzle for different tasks and (iii) extension of the technique to power-law fluids.

Numerical analysis of fluid flow through a cylinder array using a lattice Boltzmann model

In this paper we present a detailed computational study of an incompressible Newtonian fluid flow across a periodic array of two-dimensional cylinders which is a simplest non-trivial representation of a porous media. A two-dimensional Lattice Boltzmann Method is used to solve the governing Navier–Stokes equation taking into account of viscous dissipation effects and influence of nonlinear fluid drag. Both the flow fields and the Darcy–Forchheimer drag coefficient as a function of the solid volume fraction are calculated for a wide range of flow Reynolds numbers. The predictions were compared with the results from conventional numerical and empirical models for verification. Apart from confirming that inertial effects can cause a significant deviation from Darcy's law for large velocities the results also show that the characteristics of the vorticity field vary considerably as the Reynolds number increases, which will have major implications to the transport of passive particulate substances within the pores and their removal rate.

Numerical studies of viscous incompressible flow between two rotating concentric spheres

The problem of determining the induced steady axially symmetric motion of an incompressible viscous fluid confined between two concentric spheres, with the outer sphere rotating with constant angular velocity and the inner sphere fixed, is numerically investigated for large Reynolds number. The governing Navier‐Stokes equations expressed in terms of a stream function‐vorticity formulation are reduced to a set of nonlinear ordinary differential equations in the radial variable, one of second order and the other of fourth order, by expanding the flow variables as an infinite series of orthogonal Gegenbauer functions. The numerical investigation is based on a finite‐difference technique which does not involve iterations and which is valid for arbitrary large Reynolds number. Present calculations are performed for Reynolds numbers as large as 5000. The resulting flow patterns are displayed in the form of level curves. The results show a stable configuration consistent with experimental results with no evidence of any disjoint closed curves.

TRANSIENT BUOYANT CONVECTION IN AN ENCLOSURE BY SUDDEN IMPOSITION OF GRAVITY

Transient buoyant convection in a rectangular cavity is considered. The vertical side walls are at two different temperatures, and the horizontal end walls are thermally insulated. At the initial state, the fluid is motionless and under zero gravity, and a conduction-controlled, horizontally linear temperature profile exists. At t =0, gravity g is imposed on the system, and the flow is initiated at large Rayleigh number. A scale analysis is performed. Numerical solutions to the governing Navier-Stokes equations are acquired. In the present setup, due to the presence of the preexisting cavity-scale horizontal temperature gradient, strong bulk flows occur in much of the interior core within a relatively short time duration. The peak value of the Nusselt number is order-of-magnitude larger than that of the conventional heat-up model. Detailed evolutionary flow patterns are illustrated. Physical rationalizations, with the aid of the scale analysis, are offered.

ENCLOSED BUOYANT CONVECTION OF A VARIABLE-VISCOSITY FLUID UNDER TIME-PERIODIC THERMAL FORCING

Finite-volume numerical solutions are obtained for buoyant convection of a fluid with temperature-dependent viscosity in an enclosed space. At one vertical wall the temperature is constant, and at the other vertical wall the temperature is time-periodic. Solutions to the governing Navier-Stokes equations are acquired for a fixed Rayleigh number and over wide ranges of viscosity contrast, which measures the ratio of viscosities at the cold side wall and hot side wall. The effects of variable viscosity on the time-mean value as well as the amplitude of fluctuation of instantaneous heat transfer rate are delineated. The extensive results reveal the existence of resonance and show that resonance becomes more distinctive for large viscosity contrast in the case of the hot-wall temperature oscillation. As the viscosity contrast increases, the upward convective motion is invigorated during the relative heating phase due to the lowered viscosity in the thermal boundary layer near the hot vertical side wall. This is also reflected in the augmentation of the cycle-averaged heat transfer rate. When the temperature oscillation is imposed on the cold wall, the flow is less sensitive to the viscosity variation. Physical interpretations of the overall flow and heat transfer are offered.

Numerical simulation of 2D square driven cavity using fourth-order compact finite difference schemes

Fourth-order compact finite difference schemes are employed with multigrid techniques to simulate the two-dimensional square driven cavity flow with small to large Reynolds numbers. The governing Navier-Stokes equation is linearized in streamfunction and vorticity formulation. The fourth-order compact approximation schemes are coupled with fourth-order approximations for velocities and vorticity boundaries. Numerical solutions are obtained for square driven cavity flow at high Reynolds numbers and are compared with solutions obtained by other researchers using other approximation methods.

Discrete Singular Convolution–Finite Subdomain Method for the Solution of Incompressible Viscous Flows

This paper proposes a discrete singular convolution-finite subdomain method (DSC–FSM) for the analysis of incompressible viscous flows in multiply connected complex geometries. The DSC algorithm has its foundation in the theory of distributions. A block-structured grid of fictitious overlapping interfaces is designed to decompose a complex computational geometry into a finite number of subdomains. In each subdomain, the governing Navier–Stokes equations are discretized by using the DSC algorithm in space and a third-order Runge–Kutta scheme in time. Information exchange between fictitious overlapping zones is realized by using the DSC interpolating algorithm. The Taylor problem, with decaying vortices, could be solved to machine precision, with an excellent comparison against the exact solution. The reliability of the proposed method is tested by simulating the flow in a lid-driven cavity. The utility of the DSC–FSM approach is further illustrated by two other benchmark problems, viz., the flow over a backward-facing step and the laminar flow past a square prism. The present results compare well with the numerical and experimental data available in the literature.

The effect of surfactant on the stability of a liquid thread

The surface-tension-driven motion of a surfactant-coated liquid thread in inviscid surroundings is investigated using linear stability theory as well as one-dimensional nonlinear approximations to the governing Navier–Stokes equations. Examination of analytic limits of the linear dispersion relationship demonstrates that surfactant acts as a distinct mechanism for long-wavelength cut-off, instead of inertia, if the surfactant effects exceed a critical value, β = ½, where β is a dimensionless surface-tension gradient. Two different long-wavelength regimes can be identified, depending on the degree of tangential stress, with β = 1 characterizing a transition from extensionally dominated inertial flow to shear-dominated viscous flow. One-dimensional nonlinear models are formulated which capture the changes in behaviour with variation of β by inclusion of the necessary high-order terms. Scaling close to breakup shows that surfactant is swept away from the pinching region and then has little effect.

Finite element analyses of flow in a cavity with internal blockages

Although studies of flows in cavities without internal blockages are extremely common, those in which blockages arise are correspondingly rare. The present work describes a numerical study of one such flow using two complementary finite element approximations to the governing Navier-Stokes equations. Two formulations are adopted: the first employs a streamfunction-vorticity representation that has proven effective in elucidating fine flow detail in related cavity flows while the second is expressed in terms of primitive variables. The advantages and disadvantages of each formulation in relation to ease of application of boundary conditions and efficiency of numerical solution are compared and contrasted. In both cases, results for the chosen test problem are found to be in excellent agreement and reveal that previously reported data for this problem are in error. Moreover, results are presented that demonstrate the rich variety of flow patterns, recirculation regions and stagnation points that can arise as the cavity geometry and external wall speeds are varied.

Deposition of Particles in the Impinging-Jet Cell for the High Coverage Regime

Particle deposition in the radial impinging-jet cell for the high coverage regime was studied theoretically and experimentally. A detailed description of the flow distribution in the cell was attained by solving the governing Navier–Stokes equation numerically. The macroscopic flow pattern was decomposed into simpler local flows. It was demonstrated that for tangential distances r/R>0.5 the overall flow at the interface was dominated by the simple shear. The intensity of this flow was calculated numerically as a function of the Reynolds number and the distance from the cell center. Knowing the fluid velocity field the convective diffusion equation was formulated describing a two-dimensional transport of particles. As a result of nonlinearity of the boundary condition this equation was solved in an exact manner for low coverage only. For higher coverage, approximate methods were proposed exploiting the random sequential adsorption (RSA) approach. The validity of the theoretical predictions was verified experimentally using the direct microscope observation method and polystyrene latex particles of the size 0.87 μm. Particle coverage distribution was studied in detail as a function of the Re number governing the local shear rate. It was demonstrated that for low Re number (Re<4) uniform particle monolayers of high coverage can be attained. On the other hand, for Re>8 the particle coverage distribution became nonuniform as a result of the hydrodynamic scattering. This effect, leading to an apparent kinetic saturation of the surface at coverages of a few percents, was quantitatively interpreted in terms of the theoretical model.

Slip flow past a stretching surface

The slip-flow of a Newtonian fluid past a linearly stretching sheet is considered. The partial slip is controlled by a dimensionless slip factor, which varies between zero (total adhesion) and infinity (full slip). An exact analytical solution of the governing Navier-Stokes equation is found, which is formally valid for all Reynolds numbers.

Helicopter flight around a ship's' superstructure

A computational fluid dynamics model of a hovering helicopter main rotor is developed to examine air flow in the presence of ship structures and side winds. An illustration of the problem is given. The rotor is modelled by modifying the governing Navier-Stokes equations in the region of the disc. The extra terms added to the governing equations apply a downward force to the fluid; these forces are independent of the flow around the rotor and are equal to the helicopter weight. The helicopter rotor model and the ship model are combined to yield one flow solution, which, due to the severe non-linearities of the problem, cannot be achieved by superposition. The resultant flow yields valuable data about the induced velocities at the rotor, which ultimately determine the control pitch and power required to maintain the hover in a given location. Indeed, the interactions between the rotor downwash and ship air flow are known to produce unexpected and adverse flight dynamic behaviour of the aircraft.

Numerical simulation of the unsteady flow over an elliptic cylinder at different orientations

AbstractA numerical method is developed for investigating the two‐dimensional unsteady viscous flow over an inclined elliptic cylinder placed in a uniform stream of infinite extent. The direction of the free stream is normal to the cylinder axis and the flow field unsteadiness arises from two effects, the first is due to the flow field development following the start of the motion and the second is due to vortex shedding in the wake region. The time‐dependent flow is governed by the full conservation equations of mass and momentum with no boundary layer approximations. The parameters involved are the cylinder axis ratio, Reynolds number and the angle of attack. The investigation covers a Reynolds number range up to 5000. The minor–major axis ratio of the elliptic cylinder ranges between 0.5 and 0.6, and the angle of attack ranges between 0° and 90°. A series truncation method based on Fourier series is used to reduce the governing Navier–Stokes equations to two coupled infinite sets of second‐order differential equations. These equations are approximated by retaining only a finite number of terms and are then solved by approximating the derivatives using central differences. The results reveal an unusual phenomenon of negative lift occurring shortly after the start of motion. Various comparisons are made with previous theoretical and experimental results, including flow visualizations, to validate the solution methodology. Copyright © 2001 John Wiley &amp; Sons, Ltd.