Unsteady Incompressible Navier-Stokes Equations Research Articles

Effect on measurements of anemometers due to a passing high-speed train

The three-dimensional unsteady incompressible Reynolds-averaged Navier-Stokes equations and k-<TEX>${\varepsilon}$</TEX> double equations turbulent model were used to investigate the effect on the measurements of anemometers due to a passing high-speed train. Sliding mesh technology in Fluent was utilized to treat the moving boundary problem. The high-speed train considered in this paper was with bogies and inter-carriage gaps. Combined with the results of the wind tunnel test in a published paper, the accuracy of the present numerical method was validated to be used for further study. In addition, the difference of slipstream between three-car and eight-car grouping models was analyzed, and a series of numerical simulations were carried out to study the influences of the anemometer heights, the train speeds, the crosswind speeds and the directions of the induced slipstream on the measurements of the anemometers. The results show that the influence factors of the train-induced slipstream are the passing head car and tail car. Using the three-car grouping model to analyze the train-induced flow is reasonable. The maxima of horizontal slipstream velocity tend to reduce as the height of the anemometer increases. With the train speed increasing, the relationship between <TEX>$V_{train}$</TEX> and <TEX>$V_{induced\;slipstream}$</TEX> can be expressed with linear increment. In the absence of natural wind conditions, from the head car arriving to the tail car leaving, the induced wind direction changes about <TEX>$330^{\circ}$</TEX>, while under the crosswind condition the wind direction fluctuates around <TEX>$-90^{\circ}$</TEX>. With the crosswind speed increasing, the peaks of <TEX>$V_X,{\mid}V_{XY}-V_{wind}{\mid}$</TEX> of the head car and that of <TEX>$V_X$</TEX> of the tail car tend to enlarge. Thus, when anemometers are installed along high-speed railways, it is important to study the effect on the measurements of anemometers due to the train-induced slipstream.

Isogeometric Analysis of the Navier–Stokes equations with Taylor–Hood B-spline elements

This paper presents our numerical results of the application of Isogeometric Analysis (IGA) to the velocity–pressure formulation of the steady state as well as to the unsteady incompressible Navier–Stokes equations. For the approximation of the velocity and pressure fields, LBB compatible B-spline spaces are used which can be regarded as smooth generalizations of Taylor–Hood pairs of finite element spaces. The single-step θ-scheme is used for the discretization in time. The lid-driven cavity flow, in addition to its regularized version and flow around cylinder, are considered in two dimensions as model problems in order to investigate the numerical properties of the scheme.

CFD Analysis of Oscillating Blades for Small Horizontal Axis Wind Turbines in Dynamic Stall Condition

Unsteady aerodynamics and specifically the dynamic stall phenomenon significantly affect the aerodynamic performance of the wind turbine blades. This paper presents a 2D computational investigation on the aerodynamic characteristics of three specific airfoils for small horizontal axis wind turbines subjected to unsteady viscous flow. The unsteady incompressible Navier-Stokes equations are considered, and ANSYS-Fluent CFD code is used for the flow simulation. The simulation method is validated by calculating the aerodynamic coefficients of a pitching NACA23012 airfoil and comparing the results with the corresponding published experimental data. A complete set of dynamic simulations is then performed to find lift and drag forces acting on the airfoils in different unsteady conditions such as various Reynolds numbers and different amplitude and frequency of pitching oscillations. The results of this study emphasize that the curvature of leading edge and the thickness of the airfoil have major roles in early flow separation and dynamic stall of a pitching airfoil. The dynamic stall has wide negative effects on the performance of the blades such as lift reduction, drag increment and delay in the flow reattachment that are discussed in this paper.

A Scalable Numerical Method for Simulating Flows Around High-Speed Train Under Crosswind Conditions

AbstractThis paper presents a parallel Newton-Krylov-Schwarz method for the numerical simulation of unsteady flows at high Reynolds number around a high-speed train under crosswind. With a realistic train geometry, a realistic Reynolds number, and a realistic wind speed, this is a very challenging computational problem. Because of the limited parallel scalability, commercial CFD software is not suitable for supercomputers with a large number of processors. We develop a Newton-Krylov-Schwarz based fully implicit method, and the corresponding parallel software, for the 3D unsteady incompressible Navier-Stokes equations discretized with a stabilized finite element method on very fine unstructured meshes. We test the algorithm and software for flows passing a train modeled after China’s high-speed train CRH380B, and we also compare our results with results obtained from commercial CFD software. Our algorithm shows very good parallel scalability on a supercomputer with over one thousand processors.

On preconditioned iterative methods for unsteady incompressible Navier–Stokes equations

When the implicit finite difference scheme and linearization technique in conjunction with the artificial compressibility method are employed to discretize the unsteady incompressible Navier-Stokes equations, in each pseudo-time step, a structured system of linear equations is obtained and needs to be solved numerically. In this paper, preconditioning techniques are applied to solve such linear systems. Numerical example is given to illustrate the effectiveness of the proposed method. We also find that a combination of the linearization technique with the preconditioned GMRES method is quite efficient for the implicit finite difference discretization of the incompressible Navier-Stokes equations.

Simulation of free surface flows using volume of fluid method and genetic algorithm

In this paper, details of a numerical model development for simulation of fluid flows with moving free surface are presented. The unsteady incompressible Navier–Stokes equations on a fixed grid system are used to obtain velocity and pressure values in the computational domain and volume of fluid (VOF) method is used to determine free surface location. In order to reduce numerical smearing and increase accuracy of numerical modeling of fluid flow with moving free surface, a new free surface-tracking method is proposed. The proposed method is a combination of genetic algorithm and free surface tracking method based on donor and acceptor scheme. The specification of the new combinational method can be summarized in determining orientation vector and plane constant to represent the free surface orientation in each cell. The proposed algorithm can be easily used in any unstructured grids. In this method, the fluid flow equations are explicitly discretized with the finite volume method and the projection method is used to determine the velocity and pressure magnitude in computational domain. Validity of the new solution algorithm is demonstrated through its application to the dam break and the bore motion examples.

Moving least squares simulation of free surface flows

In this paper a Moving Least Squares method (MLS) for the simulation of 2D free surface flows is presented. The emphasis is on the governing equations, the boundary conditions, and the numerical implementation. The compressible viscous isothermal Navier–Stokes equations are taken as the starting point. Then a boundary condition for pressure (or density) is developed. This condition is applicable at interfaces between different media such as fluid–solid or fluid–void. The effect of surface tension is included. The equations are discretized by a moving least squares method for the spatial derivatives and a Runge–Kutta method for the time derivatives. The computational frame is Lagrangian, which means that the computational nodes are convected with the flow. The method proposed here is benchmarked using the standard lid driven cavity problem, a rotating free surface problem, and the simulation of drop oscillations. A new exact solution to the unsteady incompressible Navier–Stokes equations is introduced for the rotating free surface problem.

A Conceptual Model of Flapping-Wing Mechanism Inspired from the Results of Numerical Study

This paper investigates the aerodynamic forces of several plunging wing models by means of computational fluid dynamics. A finite volume method was used to solve the two-dimensional unsteady incompressible Navier-Stokes equations. The forces and power efficiency have been calculated and compared between sets of different models. Current work found that the nonsymmetrical moving can enhance the lift and thrust forces. The numerical results also prove that the flexible wing model can be use to improve the efficiency and reduce the input. Additionally, a new conceptual model for flapping wing mechanism with active deformation and adaptive nonsymmetrical driving motion is proposed base on the numerical results.

Study on Deloading Properties of Perforated Sound Barriers

Based on the 3D unsteady incompressible Navier-Stokes equation and the turbulent model ofk-εtwo equations, the processes of electric multiple units CRH380B passing by sound barriers installed on viaducts at the speed of 350 km/h were numerically simulated by finite volume method. The aerodynamic impulse pressure on sound barriers was analyzed. The distribution of impulse pressure on perforated sound barriers was compared with that on those with no holes. Effects of the shape, the size, the density, and tilt angle of the holes on deloading properties of perforated sound barriers were investigated. The numerical results show that the deloading properties of perforated sound barriers with circular holes are similar to those with square holes. The holistic distribution of impulse pressure on perforated sound barriers is similar to those with no holes. The density increases in two different directions has almost the same influence on the deloading properties. Deloading properties of perforated sound barriers become worse when tilt angle of holes increases.

An octree-based solver for the incompressible Navier–Stokes equations with enhanced stability and low dissipation

The paper introduces a finite difference solver for the unsteady incompressible Navier–Stokes equations based on adaptive cartesian octree grids. The method extends a stable staggered grid finite difference scheme to the graded octree meshes. It is found that a straightforward extension is prone to produce spurious oscillatory velocity modes on the fine-to-coarse grids interfaces. A local linear low-pass filter is shown to reduce much of the bad influence of the interface modes on the accuracy of numerical solution. We introduce an implicit upwind finite difference approximation of advective terms as a low dissipative and stable alternative to semi-Lagrangian methods to treat the transport part of the equations. The performance of method is verified for a set of benchmark tests: a Beltrami type flow, the 3D lid-driven cavity and channel flows over a 3D square cylinder.

A parallel fully coupled implicit domain decomposition method for numerical simulation of microfluidic mixing in 3D

A parallel fully coupled implicit fluid solver based on a Newton–Krylov–Schwarz algorithm is developed on top of the Portable, Extensible Toolkit for Scientific computation for the simulation of microfluidic mixing described by the three-dimensional unsteady incompressible Navier–Stokes equations. The popularly used fractional step method, originally designed for high Reynolds number flows, requires some modification of the inviscid-type pressure boundary condition in order to reduce the divergence error near the wall. On the other hand, the fully coupled approach works well without any special treatment of the boundary condition for low Reynolds number microchannel flows. A key component of the algorithm is an additive Schwarz preconditioner, which is used to accelerate the convergence of a linear Krylov-type solver for the saddle-point-type Jacobian systems. As a test case, we carefully study a three-dimensional passive serpentine micromixer and report the parallel performance of the algorithm obtained on a parallel machine with more than one hundred processors.

Resolution of Unsteady Navier-stokes Equations with the C a,b Boundary condition

in this work, we introduce the unsteady incompressible Navier-Stokes equations with a new boundary condition, generalizes the Dirichlet and the Neumann conditions. Then we derive an adequate variational formulation of time-dependent Navier- Stokes equations. We then prove the existence theorem and a uniqueness result. A Mixed finite-element discretization is used to generate the nonlinear system corresponding to the Navier-Stokes equations. The solution of the linearized system is carried out using the GMRES method. In order to evaluate the performance of the method, the numerical results are compared with others coming from commercial code like Adina system.

Numerical Investigation of Vortex Shedding Modes of Flow past Two Circular Cylinders in Side-by-Side Using Immersed Boundary Method

The vortex shedding modes of flow past two circular cylinders in side-by-side arrangement are investigated numerically in this paper. The simulations are carried out using a ghost cell immersed boundary method which imposes the boundary condition through reconstruction of the local velocity field near the immersed boundary. The two-dimensional unsteady incompressible Navier-Stokes equations are solved using an implicit fractional step method based on cell-center, collocated arrangement of the primary variables. Vorticity contours of the flow around the cylinders and force time histories are presented. Anti-phase and in-phase vortex shedding modes were found to exist in the flow simulation. These results of simulations were in agreement with phenomena observed in experiment and numerical results of previous researchers.

Simulating 3D Flows Passing Wind Turbine Rotors with a Domain Decomposition Method on a Moving Domain

Accurate numerical simulations of flows around wind turbines play an important role in rotor blades design and operation control. The computation is very challenging because of the large size of the blades, the large fluids domain, the complex moving geometry, and the high Reynolds number. A popular method is the blade element momentum method which is relatively easy to implement, but has low fidelity. More recently, some high fidelity methods have been developed for the wind turbine simulations, such as Reynolds averaged Navier-Stokes methods, large-eddy simulations and direct numerical simulations (DNS). In this paper, we study DNS for flows passing 3D wind turbine rotors. The flow in the moving domain is modeled by the 3D unsteady incompressible Navier-Stokes equations in the arbitrary Lagrangian-Eulerian form. A stabilized bi-linear moving mesh finite element method, based on the unstructured tetrahedron mesh, is introduced to discretize the problem in space, and an implicit scheme is used to discretize the temporal variable. A parallel Newton-Krylov method with a domain decomposition type preconditioner is applied to solve the fully coupled nonlinear algebraic system at each timestep. We mainly focus on the performance of the domain decomposition preconditioner. To understand the efficiency of the algorithm, we test the software on a supercomputer for the simulation of a NREL 5MW wind turbine rotor. The numerical results show that the newly developed algorithm is scalable with thousands of processors for problems with tens of millions of unknowns.

Fluid–structure interactions of photo-responsive polymer cantilevers

A new class of photomechanical liquid crystal networks (LCNs) has emerged, which generate large bending deformation and fast response times that scale with the resonance of the polymer films. Here, a numerical study is presented that describes the photomechanical structural dynamic behavior of an LCN in a fluid medium; however, the methodology is also applicable to fluid–structure interactions of a broader range of adaptive structures. Here, we simulate the oscillation of photomechanical cantilevers excited by light while simultaneously modeling the effect of the surrounding fluid at different ambient pressures. The photoactuated LCN is modeled as an elastic thin cantilever plate, and gradients in photostrain from the external light are computed from the assumptions of light absorption and photoisomerization through the film thickness. Numerical approximations of the equations governing the plate are based on cubic B-spline shape functions and a second order implicit Newmark central scheme for time integration. For the fluid, three dimensional unsteady incompressible Navier–Stokes equations are solved using the arbitrary Lagrangian–Eulerian (ALE) method, which employs a structured body-fitted curvilinear coordinate system where the solid–fluid interface is a mesh line of the system, and the complicated interface boundary conditions are accommodated in a conventional finite-volume formulation. Numerical examples are given which provide new insight into material behavior in a fluid medium as a function of ambient pressure.

Turbulence Models Performance Assessment for Pressure Prediction during Cylinder Water Entry

Numerical simulations of two-dimensional cylinder free droping into water are presented based on volume of fluid (VOF) method and dynamic mesh technique. Solutions with a time-accurate finite-volume method (FVM) were generated based on the unsteady compressible ensemble averaged Navier-Stokes equations for the air and the unsteady incompressible ensemble averaged Navier-Stokes equations for the water. Computed pressure histories of the cylinder were compared with experimentally measured values. The performance of various turbulence models for pressure prediction was assessed. The results indicate that Realizable k-epsilon model with Enhanced Wall Treatment is the best choice for engineering practice.

Flow control for VATT by fixed and oscillating flap

The present study is aimed to explore the potential to improve Vertical Axis Tidal Turbine (VATT) energy harnessing efficiency by using modified blades with fixed and oscillating flap. The fixed flap concept is borrowed from its application in aerodynamics area for reaching a high lift force at low flying speed. Oscillating flap is motivated by our relevant biomimetic studies on the flapping wing propulsion or energy extraction. Present investigation is concentrated on a VATT with NACA 0018 profile blade as its baseline turbine. Numerical simulations are carried out by solving incompressible Unsteady Navier–Stokes equations at turbulence flow condition. Computed results show that under certain optimal flap geometry and flow conditions, turbine power coefficient reaches 28% enhancement as compared to the conventional blade turbine. Detailed analysis on the flow structure demonstrates that this is related to the effective flow separation suppression and vortex control by applying a fixed and oscillating flap.

Maximization of mass flow in a channel obstructed by an infinitely thin plate using a gradient-based control strategy

The problem of maximizing the mass flow in a two-dimensional channel partially blocked by an infinitely thin flat plate is addressed using a gradient based control strategy. The state equations are the two-dimensional unsteady incompressible Navier–Stokes equations. The control is a spatially and temporally varying wall-normal velocity boundary condition subject to spatial and temporal constraints of zero mass flux, which are chosen to model realistic Micro Electronic Mechanical Systems based on the movement of small plungers or diaphragms. The control which maximizes the mass flow is found using a conjugate gradient method, where the gradient of the objective function with respect to the control variables is obtained from solving a set of adjoint equations. The effectiveness of choices for the objective function are examined along with the effect of the event horizon and the temporal constraint. Significant increases in the mass flow are obtained for certain choices of these parameters. In many cases, the increase in the mass flow was obtained with a net energy savings. The numerical issues involved with finding the control are discussed and the features of the resulting control are analyzed with the goal of understanding how it affects the mass flow.

Existence of global weak solutions to finitely extensible nonlinear bead–spring chain models for dilute polymers with variable density and viscosity

We prove the existence of global-in-time weak solutions to a general class of coupled bead–spring chain models that arise from the kinetic theory of dilute solutions of nonhomogeneous polymeric liquids with noninteracting polymer chains, with finitely extensible nonlinear elastic (FENE) spring potentials. The class of models under consideration involves the unsteady incompressible Navier–Stokes equations with variable density and density-dependent dynamic viscosity in a bounded domain in Rd, d=2 or 3, for the density, 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 and a nonlinear density-dependent drag coefficient. We require no structural assumptions on the drag term in the Fokker–Planck equation; in particular, the drag term need not be corotational. With initial density ρ0∈[ρmin,ρmax] for the continuity equation, where ρmin>0; a square-integrable and divergence-free initial velocity datum u˜0 for the Navier–Stokes 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 certain regularization parameters, the existence of a global-in-time 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), such that t↦ρ(t)∈[ρmin,ρmax], t↦u˜(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 Gibbs measure dν:=M(q˜)dq˜dx˜) over any time interval [0,T], T>0.

Navier–Stokes simulations in gappy PIV data

Velocity measurements conducted with particle image velocimetry (PIV) often exhibit regions where the flow motion cannot be evaluated. The principal reasons for this are the absence of seeding particles or limited optical access for illumination or imaging. Additional causes can be laser light reflections and unwanted out-of-focus effects. As a consequence, the velocity field measured with PIV contains regions where no velocity information is available, that is gaps. This work investigates the suitability of using the unsteady incompressible Navier–Stokes equations to obtain accurate estimates of the local transient velocity field in small gaps; the present approach applies to time-resolved two-dimensional experiments of incompressible flows. The numerics are based on a finite volume discretization with partitioned time-stepping to solve the governing equations. The measured velocity distribution at the gap boundary is taken as time-varying boundary condition, and an approximate initial condition inside the gap is obtained via low-order spatial interpolation of the velocity at the boundaries. The influence of this I.C. is seen to diminish over time, as information is convected through the gap. Due to the form of the equations, no initial or boundary conditions on the pressure are required. The approach is evaluated by a time-resolved experiment where the true solution is known a priori. The results are compared with a boundary interpolation approach. Finally, an application of the technique to an experiment with a gap of complex shape is presented.