Incompressible Flow Equations Research Articles

Numerical Simulation of Ocean Currents with Hermite IFinite Elements

The article deals with the numerical simulations for equations of geophysical fluids. These physical phenomena are modeled by the Navier-Stokes equations which describe the motion of the fluid, the ocean currents, the flow of water in a pipe and many other fluid flow phenomenon.These equations are very useful because of there utility. The Navier-Stokes equations for incompressible flow are nonlinear partial differential equations that drive the motion of fluids in the approximation of continuous media. The existence of general solutions to the Navier Stokes equations have already proven but in this paper we have interested to the numerical solution of the incompressible Navier-Stokes equations. We get an optimal discretization of the Navier-Stokes equations and numerical approximations of the solution are also given. The convergence and the stabilityof the approximated system are proven. The numerical resolution is based on Hermite finite elements. The numerical system was expressed in matrix form for computation of velocity and the pression fieldsapproach using MATLAB software. Numerical results for velocity field in two dimensional space of the velocity u(x,y) and pression p(x,y,) are given. And finally we give physical interpretation of the results obtained.

Numerical modeling of the interaction between submerged floating tunnel and surface waves

This paper presents a numerical model for analyzing the nonlinear interaction between the moored Submerged Floating Tunnel (SFT) and surface waves. The mechanics model of the moored floating body driven by wave forces is built, and an efficient mesh update method is employed to dynamically configure the computational meshes solving the Navier-Stokes equations for viscous and incompressible free surface flows with the volume of fluid (VOF) method. Two laboratory experiments are used for validating the numerical model in terms of surface elevations, motion responses and mooring forces of the SFT, indicating the proposed model is capable of simulating the dynamics of the moored floating body under the wave action. This hydrodynamic model is then utilized to simulate the wave-structure interaction of the prototype SFT designed for Funka Bay, Hokkaido located in Japan. A total of 49 cases are designed for the numerical simulation to investigate the characteristics of the wave-tunnel interaction for different hydrodynamic parameters, including wave height, wave period, immersion depth and buoyancy-weight ratio (BWR). The numerical experiments not only shed light on the mooring forces, as well as pitch, sway and heave responses of the SFT in different wave conditions, but also provide guidance for the choice of BWR in engineering design. A medium value of BWR is suggested to be suitable, which is useful for avoiding the happening of snap forces in mooring chains and preventing SFT from experiencing large movement under external forces in severe wave conditions. As the correlation between the motion responses and BWR is not merely linear or quadratic but parabolic with a peak value, the design of BWR should avoid the case where peak motion responses of SFT happen.

Transportation of magnetite nanofluid flow and heat transfer over a rotating porous disk with Arrhenius activation energy: Fourth order Noumerov’s method

This article explains the effect of magnetite nanofluid taking into account water as the base fluid, over a rotating disk in the presence of external magnetic field. The governing boundary layer equations of laminar and incompressible flow over rotating disk have been formulated under the simultaneous influence of thermophoresis and Brownian motion. Additionally, influence of chemical reaction between species, prompted by Arrhenius activation energy is considered into the model. The Von Karman’s transformations of the Navier-Stokes equations are used to obtain the non-dimensional forms of the governing equations, and then fourth order finite difference scheme is used to find the numerical solutions using Noumerov’s discretization. This technique has strong impact on the accuracy of finite difference scheme. The physical descriptions of fluid velocity, temperature, concentration profiles and quantities of practical interest such as skin friction coefficient, Nusselt number and Sherwood number are presented with the help of graphs. One of the significant findings of this analysis include that an intensification in the thermophoresis parameter causes a downfall in the rate of heat transfer at the surface of the disk whereas another significant outcomes of the present investigation is that there is a downfall in the behaviours of the species concentration with the strengthening of Arrhenius activation energy. The fluid flow due to rotating disks has application in mechanisms of thermal transformations for nuclear propulsion devices.

Overset meshes for incompressible flows: On preserving accuracy of underlying discretizations

This study on overset meshes for incompressible-flow simulations is motivated by accurate prediction of wind farm aerodynamics involving large motions and deformations of components with complex geometry. Using first-order hyperbolic and elliptic equation proxies for the incompressible Navier-Stokes (NS) equations, we investigate the influence of information exchange between overset meshes on numerical performance where the underlying discretization is second-order accurate. The first aspect of information exchange surrounds interpolation of solution where we examine Lagrange and point-cloud-based interpolation for creating constraint equations between overset meshes. To maintain overall second-order accuracy, higher-order interpolation is required for elliptic problems, but linear interpolation is sufficient for hyperbolic problems in first-order form. Higher-order point-cloud-based interpolation provides a pathway to maintaining accuracy in unstructured meshes, but at higher complexity. The second aspect of information exchange focuses on comparing the approaches of overset single system (OSS) and overset Additive Schwarz (OAS) for coupling the linear systems of the overlapping meshes. While the former involves a single linear system, in the latter the discrete linear systems are solved separately, and solving the global system is accomplished through outer iterations and sequential information exchange in a Jacobi fashion. For the test cases studied, accuracy for hyperbolic problems is maintained by performing two outer iterations, whereas many outer iterations are required for elliptic systems. The order-of-accuracy studies explored here are critical for verifying the overset-mesh coupling algorithms used in engineering simulations. Accuracy of these simulations themselves is, however, quantified using engineering quantities of interest such as drag, power, etc. Consequently, we conclude with numerical experiments using NS equations for incompressible flows where we show that linear interpolation and few outer iterations are sufficient for achieving asymptotic convergence of engineering quantities of interest.

Theoretical, Numerical, and Experimental Study on an Unsteady Venturi Flowmeter for Incompressible Flows

Abstract To examine the dynamic characteristics of turbomachinery and cavitation, the pulsating flow rates should be evaluated. As it is difficult to measure these pulsating flow rates quantitatively, systematic research has not been conducted on the dynamic characteristics of turbomachinery and cavitation. In this paper, an unsteady energy equation for a venturi tube has been proposed to measure pulsating flow rates. The pulsating flow rates were calculated using two methods based on the unsteady energy equation for incompressible flows. The first method calculated a pulsating flow rate by using the Euler method. The second one calculated the complex amplitude of a pulsating flow rate using a transfer function derived from the linearized unsteady energy equation. We analytically examined the order of magnitude for unsteady terms. The results indicated that the unknown unsteady loss was much smaller than the unsteady momentum. In the experiment, pulsating flows were generated by a reciprocating piston, and the given pulsating flow was measured using a hot wire anemometer. The pulsating flow rates evaluated by using the proposed methods were validated via numerical simulation and experiment. In particular, the influence of amplitudes on the evaluation of pulsating flow rates was numerically examined. Therefore, the nonlinear effect could be evaluated by using the proposed method, and the time-averaged loss coefficient was enough to evaluate the pulsating flow rate coefficient. The proposed unsteady venturi flowmeter can be applied to a wide range of research fields, such as analyzing dynamic characteristics of flows.

On the Thrust Generation from an Elliptic Airfoil in Plunging and Translating Motion at Low Reynolds Numbers

In this paper, numerical simulation elliptic airfoil model, which mimics the biological locomotion, is studied. Elliptic airfoil undergoes a combined plunging and translating at low Reynolds number is simulated by using body fitted coordinate system. The moving mesh in the physical domain is mapped to a regular fixed mesh in the computational domain through a time dependent transformation between the physical and computational co-ordinates. The governing equations of laminar incompressible flow are transformed in the computational plane by incorporating the time dependent transformation, which naturally accounts for the mesh velocities. The transformed equations are discretized on the structured, collocated, o-type elliptic grid using the finite difference methodology. The unsteady equations are marched in time by using a semi-implicit pressure correction (projection) scheme. Along with the time marching of the governing equations, utilizing the mesh velocities and the forward Eulertime integration also moves the mesh points. The effect of Reynolds number (Re) is investigated on the flapping flight propulsion is investigated. It is found that there exists a critical Reynolds number (Rec) for every frequency after which there exists a thrust force. The effect of Rec is related to transformation of neutral wake to thrust generating wake. It is also found that the optimal frequency corresponds to a reduced frequency parameter of 0.7 where a lock in exists. It is also found that this Stc is independent of Re and the mode of vortex shedding is same at Re = 100 and 200 for Stc = 0.7. Further, it is shown that the mode of vortex shedding present is always helpful in thrust generation.

Numerical investigation on hydrodynamic load of coastal bridge deck under joint action of solitary wave and wind

Coastal bridges are vulnerable to extreme surges and waves during tsunamis and hurricanes. Many of them have been significantly damaged or destroyed during past extreme weather events, such as Hurricane Katrina in 2005. Although extensive experimental work and numerical simulations have been carried out focusing on identifying the damage mechanisms of coastal bridges impacted by extreme waves, the coastal bridge deck needs to withstand the combined actions of waves and wind. Strong winds can easily deform the water surface and influence how waves impact the coastal bridge deck and the final resultant hydrodynamic load. Hence, there is a strong need for investigating the hydrodynamic characteristics of coastal bridges under the joint action of wind and surge waves. The direct effects of wind on the hydrodynamic load of a bridge deck under the impact of extreme surge waves had not been taken into account in past studies. To fill this knowledge gap, in this paper, a high-resolution numerical wave-wind tank is established by solving the governing equations of two-phase incompressible flow using the finite volume method. The hydrodynamic load of coastal bridge deck under joint action of solitary wave and wind are numerically investigated by considering the effects of several prominent factors, i.e. wind speed, wave height, submersion depth, and water depth. The results show that strong wind can greatly enhance the impacting intensity of solitary waves on bridge deck, resulting in greater horizontal and vertical hydrodynamic loads. It is believed that findings of this study can further enhance our understanding on damage mechanism of coastal bridge decks during hurricanes.

Turbulent bubbly channel flows: Effects of soluble surfactant and viscoelasticity

Interface-resolved direct numerical simulations are performed to examine the combined effects of soluble surfactant and viscoelasticity on the structure of a bubbly turbulent channel flow. The incompressible flow equations are solved fully coupled with the FENE-P viscoelastic model and the equations governing interfacial and bulk surfactant concentrations. The latter coupling is achieved through a non-linear equation of state which relates the surface tension to the surfactant concentration at the interface. The two-fluid Navier-Stokes equations are solved using a front-tracking method, augmented with a very efficient FFT-based pressure projection method that allows for massively parallel simulations of turbulent flows. It is found that, for the surfactant-free case, bubbles move toward the wall due to inertial lift force, resulting in formation of wall layers and a significant decrease in the flow rate. Conversely, a high-enough concentration of surfactant changes the direction of lateral migration of bubbles, i.e., the contaminated bubbles move toward the core region and spread out across the channel. When viscoelasticity is considered, viscoelastic stresses counteract the Marangoni stresses, promoting formation of bubbly wall-layers and consequently strong decrease in the flow rate. The formation of bubble wall-layers for combined case depends on the interplay of the inertial and elastic, and Marangoni forces.

Mathematical Analysis of the Meridian System in Traditional Chinese Medicine

Energy expenditure of the meridian system in traditional Chinese medicine (TCM) for living human bodies is one of the fundamental questions regarding the physical properties of the meridian system. As a first attempt to address this, 24 hand/foot meridians were modeled as a continuous channel with the sustainable movement of Qi and blood inside. Then, the fluid mechanics energy equation for steady and incompressible flow was applied to estimate the pumping power of meridian circulation, based on the meridian length, blood viscosity and circulation velocity. Additional findings based on the pumping power estimation results include: (1) new information to corroborate the determination of the production era of Yellow Emperor’s Inner Classics (Huang Di Nei Jing, 《黄帝内经》); (2) a new definition for work of breathing (WOB) from a TCM perspective. This paper could lead to future research aimed at modernizing the understanding of the meridian system, Qi and TCM using quantitative methods.

Assessment of Performance of Tesla Turbine in Water Distribution Systems for Energy Harvesting

Abstract This article composes a systematic design methodology to obtain optimal parameters of the Tesla turbine, which is applicable in water distribution systems, faced with the need to supply power for wireless sensors and other components used in smart water networks and the challenges associated with the feasibility of Tesla turbine in a small scale. The optimal geometric model is sought by using a theoretical model relating turbine physical properties and flow conditions to power output by solving Navier–Stokes equations for steady laminar incompressible flow between adjacent disks. The model shows that the proposed design with dimensions and flow conditions relevant to water pipe networks can achieve a power output of 1.55 mW using a 2.5-cm diameter turbine given a pressure difference of 140 Pa. At 70 Pa using a 1-cm diameter turbine, a power output of 0.17 mW can be obtained. This study shows a great potential in application of Tesla turbine for energy harvesting in water distribution systems.

Numerical investigation of monochromatic wave interaction with a vertical seawall located on a reef flat

Existing design codes for coastal defense structures built on coral reefs are scarce compared to those for traditional sandy or muddy beaches. A three-dimensional numerical wave flume is developed to investigate monochromatic wave interaction with a vertical seawall located on the reef flat. The Navier-Stokes equations for two-phase incompressible flow, combining with methods of Large Eddy Simulation (LES) for turbulence closure and Volume of Fluid (VOF) for tracking the free surface, are solved. A set of large-scale laboratory experiments are performed to validate the adopted model in terms of free surface, velocity, dynamic pressure as well as overtopping discharge around the seawall. The model is first applied to examine the cross-shore evolution of waves and flows associated with breaking wave interaction with the seawall. Subsequently, a series of numerical simulations are implemented to study the impacts of different hydrodynamic (incident wave height, incident wave period, and reef-flat wave level) as well as reef and seawall morphological factors (fore-reef slope, seawall position and seawall crest width) on the overtopping discharge over the seawall as well as the cross-shore wave impact force acting on the seawall. Finally, new empirical formulas are proposed, respectively, to predict the overtopping discharge and the wave force.

Numerical investigation on performance of submerged porous breakwater to mitigate hydrodynamic loads of coastal bridge deck under solitary wave

Coastal bridges are vulnerable to extreme surges and waves generated during hurricanes and tsunamis. Over the past few decades, extensive research works were conducted to study the damage mechanisms of coastal bridges under the impact of extreme surges and waves. However, these research efforts mainly focused on the hydrodynamic characteristics involved in surge waves impinging bridges. However, there has been little research on mitigation measures to protect coastal bridges. This study examines the installation of a submerged porous breakwater to protect coastal bridge decks from the impacts of solitary waves. As an environmentally friendly wave-reduction device, submerged porous breakwater structures have been widely implemented in coastal regions due to their ease of installation and cost effectiveness. To investigate the performance of the submerged porous breakwater on the hydrodynamic loads of a coastal bridge deck under the impact of a solitary wave, a high-resolution numerical wave tank is established based on an open-source flow solver REEF3D, which numerically solves the governing equations of incompressible two-phase flow on a staggered mesh arrangement and captures the interface between air and water using a high-resolution level-set method. The hydrodynamic characteristics of coastal bridge decks with the installations of submerged porous breakwater are systematically investigated by considering the effects of six prominent factors, i.e. wave height, submersion depth, water depth, permeability, spacing distance between breakwater and bridge, and aspect ratios of breakwaters. Results show that the submerged porous breakwater is effective in reducing the hydrodynamic loads applied to the coastal bridge decks under most of the situations. Therefore, it is believed that the findings drawn from this paper can further enhance our understanding on the damage mechanism of coastal bridges under extreme waves and should be instructive for future bridge designs.

Numerical investigation of extensional flow through axisymmetric conical geometries: Finite element method

In this article, we present the numerical investigation for incompressible Newtonian laminar flow through a conical channel. We apply the Galerkin finite element method for solving the governing equations of such a problem. Usually, the governing equations for incompressible Newtonian flows are represented by conservation laws for mass and momentum, which are given in the cylindrical coordinates (axisymmetric) in the current study. Interestingly, the pressure drop through the channel is provided under a variety of Reynolds numbers. The objective of this study is to assess the influence of various effective parameters on the level of the pressure drop. Further, the effect of the boundary maximum axial velocity that is imposed at the inlet upon the solution reveals some novel features. To evaluate the influence of the conical half-angle at the recirculation region on solutions behavior, this study is achieved with a different set of angles. We found that the conical half-angle was notably affecting the critical level of pressure drop. Moreover, the response of the fluid in both shear and extensional is quite interesting, which represents one of the more important aspects of this study.

Surface pressure and viscous forces on inclined elliptic cylinders in steady flow

Surface pressure characteristics of elliptic cylinders of various thicknesses and orientations are investigated in steady flow regime. A stabilized finite-element method has been used to discretize the conservation equations of incompressible fluid flow in two dimensions. The Reynolds number, Re, is based on the major axis of cylinder and free-stream speed. Results have been presented for $$Re\le 40$$ and $$0^\circ \le \alpha \le 90^\circ $$ , where $$\alpha $$ is the angle of attack. Cylinder aspect ratios AR considered are 0.2 (thin), 0.5 and 0.8 (thick). It is found that a decrease in AR does not significantly alter the location of minimum surface pressure for $$\alpha = 90^\circ $$ , but the value of minimum pressure decreases sharply, resulting in severe adverse pressure gradient. In contrast, for $$\alpha = 0^\circ $$ , the location travels towards the base and the minimum pressure increases, leading to delayed flow separation. In general, the magnitude of forward stagnation pressure at low Re is smaller than the maximum pressure for $$AR\le 0.5$$ . The maximum pressure occurs at the forward stagnation point as the Re and AR increase. However, in most cases, the locations of forward stagnation and maximum pressure points differ even when the pressure coefficients are very close to each other. The forward stagnation and maximum pressure coefficients of an elliptic cylinder decrease monotonically with increasing $$\alpha $$ . The drag of a circular cylinder in most cases exceeds the ones obtained for elliptic cylinders. With increasing AR, the drag increases approximately linearly for small $$\alpha $$ , lift decreases approximately linearly and moment decreases non-linearly. For a thick cylinder, while the effect of Re on lift and moment is insignificant, the drag shows a strong dependence. Roughly $$\alpha = 20^\circ $$ for $$Re = 40$$ flow represents a critical angle of attack below which a cylinder of $$AR\le 0.5$$ acts like a streamlined body and above, like a bluff body.

Numerical Investigation for the Laminar Flow Effects over Rough Surface

In this paper, the heat transfer effect of rough surface in laminar flow is introduced by fully developed fields for solving the Navier–Stokes equations of incompressible flow in assuming two-dimension using the direction splitting method. Firstly, the algorithm of the incompressible Navier–Stokes equations with pressure correct is carried out. Secondly, the effects of pressure drop and heat transfer and are investigated and discussed in different rough surface elements which are configured with triangular and rectangular element. The Reynolds number, roughness element spacing, and roughness height are also considered as the factors which affect the heat transfer. The results indicate that the parallel present method reaches the stage of basically stable state rapidly and accurately. Compared with the smooth surface, the global performance of heat transfer is improved by the roughness surface since the pressure drop is lost. The effects of triangular element roughness surface on laminar flow and heat transfer are much stronger than rectangular element roughness surface when the spacing of the rough is changed. The flow over the triangular element roughness surface becomes stronger, which contributes to enhancement heat transfer meanwhile increase the pressure drop when the roughness height is higher.

Convergence Studies on Popular Computational Schemes for Solution of Navier-Stokes Equation for Incompressible Flow

Most of the fluid flow situations may be described by non-linear partial differential Navier-Stokes equations. For incompressible flow, fluid density is assumed constant and hence equation of state cannot be utilized for coupling the continuity equation with the momentum equation. Researchers have developed and used the SIMPLE, SIMPLER and derived quantity based schemes for solution of incompressible Navier Stokes equations. Solution convergence is an important criteria for judging any numerical scheme and hence convergence studies are performed on a standard 2-D lid driven cavity problem for SIMPLE, SIMPLER and stream function-vorticity based method to solve N-S equations for incompressible flow. Convergence studies are performed and reported for two values of Reynolds number i.e. 100 and 400.

Solidification of Cascaded PCM Storages Embedded Within a Channel Under Various Forced Air Flow Strategies

In recent years, latent heat storage systems have been received considerable attention as they can provide high thermal storage capacity and optimize energy consumption. Cascaded phase change materials (PCMs) are a significant strategy to manage the gap between energy supply and demand, especially in renewable thermal energy applications. This work is devoted to investigate the unsteady freezing process of cascaded latent heat thermal energy storage systems inside a channel under different forced airflow strategies for winter conditions. Cold air flows through cascaded PCMs in five various ways and starts changing their initial liquid status to the final solid phase. Results discuss the heat transfer and phase transition features of each strategy in four various Reynolds numbers. Two-dimensional finite volume method is utilized to solve the governing equations of incompressible fluid flow in which conduction and convection mechanisms within PCM chambers are taken into account. According to numerical results, U-type and counter flow schemes lead to quick solidification in all Reynolds numbers, while in one-side channel strategy, the system takes a longer time to be completely solidified in all Reynolds numbers except Re = 300 in which parallel flow and middle inlets methods cause later freezing. The aspect ratio of PCM cavities and outlet temperature of the flowing air are also studied for various cases. Results demonstrated that the lower aspect ratio of the counter flow strategy could solidify the PCM faster and could remove the higher value of heat from cavities (higher outlet temperature).

Accuracy Verification of a 2D Adaptive Mesh Refinement Method Using Backward-Facing Step Flow of Low Reynolds Numbers

Identifying centers of vortices of fluid flow accurately is one of the accuracy measures for computational methods. After verifying the accuracy of the 2D adaptive mesh refinement (AMR) method in the benchmarks of 2D lid-driven cavity flow, this paper shows the accuracy verification by the benchmarks of 2D backward-facing step flow. The AMR method refines a mesh using the numerical solution of the Navier–Stokes equations computed on the mesh by an open source software Navier2D which implemented a vertex centered finite volume method (FVM) using the median dual mesh to form control volumes about each vertex. The accuracy is shown by the comparison between vortex center locations calculated from the linearly interpolated numerical solutions and those obtained in the benchmark. The AMR method is proposed based on the qualitative theory of differential equations, and it can be applied to refine a mesh as many times as required and used to seek accurate numerical solutions of the mathematical models including the continuity equation for incompressible fluid or steady-state compressible flow with low computational cost.

Unsteady numerical investigation of the full geometry of a horizontal axis wind turbine: Flow through the rotor and wake

Aerodynamics of the Mexico wind turbine was investigated using CFD techniques. The complete wind turbine was modelled. In order to accomplish this goal, the computational domain was discretized with a multi-block structured hexahedron grid, which was generated manually to ensure a good quality mesh and optimize the number of cells. Furthermore, a sliding mesh technique was applied to the moving mesh zone to calculate in detail the flow around the blades. All the relevant operating conditions were considered: turbulent wake state, nominal condition and stall state. The simulations were performed using the unsteady Reynolds-Averaged Navier-Stokes equations for incompressible flow and the SST k-ω turbulence model to close the governing equations. The CFD predictions were compared with the experimental data available from the MEXICO experiment: global forces and torques, pressure distributions around the blades and velocity distributions along the radial and axial traverses were all in a good agreement. The flow through the rotor, the interaction between the blades and the tower, and the development of the wake were then investigated. The CFD simulations have provided more accurate results, contributing to a deeper understanding of the wind turbine aerodynamics.

A novel solution method for unsteady incompressible Euler flow using the vorticity–Bernoulli-pressure formulation

A novel numerical formulation and time-integration method for the incompressible Euler flow equations is presented using the vorticity–Bernoulli-pressure system. Lagrangian advection of vorticity-carrying particles is adopted from classical vortex methods that are known to give accurate results for vortex-dominated flows and allow larger time steps than explicit Eulerian schemes. Solving only one scalar Poisson equation per time step instead of a vector-valued equation in vortex methods makes the new scheme theoretically more efficient on large computational grids. The equation of motion in rotation form is integrated in time under certain approximation assumptions and solved explicitly for the flow velocity. The velocity is eliminated as an unknown in the continuity equation, which is efficiently solved for the Bernoulli pressure using the predicted vorticity field obtained by Lagrangian advection. The Bernoulli pressure is efficiently obtained from the Poisson equation in finite volume formulation. An estimated velocity field is calculated from the equation of motion. To effectively counterbalance the discretization errors made by the prediction, in a corrector step the kinematic relation between vorticity and estimated velocity is reformulated and solved in a Lagrangian framework as well. The velocity field is corrected at the end of the time step. Two-dimensional cases on Cartesian grids are considered. Numerical results for three flow cases demonstrate the locally smooth and accurate behavior and the second-order spatial accuracy of the solution. The new scheme produces significantly less artificial vortex distortion than the classical vortex method. The favorable properties of the solutions with respect to accuracy and long-time stability of the overall kinetic energy and enstrophy are also emphasized.