Two-dimensional Incompressible Navier-Stokes Equations Research Articles

A Meshless Local Radial Basis Function Method for Two-Dimensional Incompressible Navier-Stokes Equations

A meshless local radial basis function method is developed for two-dimensional incompressible Navier-Stokes equations. The distributed nodes used to store the variables are obtained by the philosophy of an unstructured mesh, which results in two main advantages of the method. One is that the unstructured nodes generation in the computational domain is quite simple, without much concern about the mesh quality; the other is that the localization of the obtained collocations for the discretization of equations is performed conveniently with the supporting nodes. The algebraic system is solved by a semi-implicit pseudo-time method, in which the convective and source terms are explicitly marched by the Runge-Kutta method, and the diffusive terms are implicitly solved. The proposed method is validated by several benchmark problems, including natural convection in a square cavity, the lid-driven cavity flow, and the natural convection in a square cavity containing a circular cylinder, and very good agreement with the existing results are obtained.

On a potential‐velocity formulation of incompressible Navier‐Stokes equations

AbstractComputational fluid dynamics has emerged as an essential investigative tool in nearly every field of technology. Despite a well‐developed mathematical theory and the existence of readily available commercial software codes, computing solutions to the governing equations of fluid motion remains challenging, especially due to the non‐linearity involved. Additionally, in the case of free surface film flows the dynamic boundary condition at the free surface complicates the mathematical treatment notably.Recently, by introduction of an auxiliary potential field, a first integral of the two‐dimensional incompressible Navier‐Stokes equations has been constructed leading to a set of equations, the differential order of which is lower than that of the original equations [1]. A useful application to free surface simulation was found in [2]. Moreover the new formulation is naturally extendible to three dimensions via tensor calculus, involving a non‐unique symmetric tensor potential. The corresponding degrees of freedom can be used in order to achieve a numerically convenient representation. Finally an efficient staggered‐grid finite difference scheme is applied to a Stokes flow problem in a 3D lid‐driven cavity to demonstrate the capabilities of the new approach. (© 2014 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Analysis of Strouhal number based equivalence of pitching and plunging airfoils and wake deflection

Numerical simulations have been used to analyze the equivalence of pitching and plunging motions found in a flapping NACA0012 airfoil. Two-dimensional incompressible Navier–Stokes equations are solved at Reynolds number of 103 over a range of Strouhal numbers. A novel criterion based on the Strouhal number is proposed which provides equivalence of pitching and plunging motions using the length scale traversed by the trailing edge in each case. Aerodynamic coefficients are found to match well for both the kinematics in temporal as well as spectral domains. Detailed analysis provides contribution of different mechanisms, such as vortex shedding, added mass, interaction of leading and trailing edge vortices, in the overall aerodynamic forces produced by a pitching or plunging airfoil. Wake deflection is observed for a plunging airfoil at high Strouhal numbers resulting in a bias in the lift coefficient. Further investigation reveals the dominance of second harmonic of the fundamental frequency in the lift spectrum emphasizing the role of quadratic nonlinearity in the observed phenomenon.

A Fast Immersed Boundary Fourier Pseudo-spectral Method for Simulation of the Incompressible Flows

Abstract The present paper is devoted to implementation of the immersed boundary technique into the Fourier pseudo-spectral solution of the vorticity-velocity formulation of the two-dimensional incompressible Navier-Stokes equations. The immersed boundary conditions are implemented via direct modification of the convection and diffusion terms, and therefore, in contrast to some other similar methods, there is not an explicit external forcing function in the present formulation. At the beginning of each time step, the solenoidal velocities (also satisfying the desired immersed boundary conditions), are obtained and fed into a conventional pseudo-spectral solver, together with a modified vorticity. The classical explicit fourth-order Runge-Kutta method is used in time integration, and the boundary conditions are set at the beginning of each sub-step, in order to increase the time accuracy. The method is employed in simulation of some different test cases, including the flow behind impulsively started circular cylinder, oscillating circular cylinder in fluid at rest and insect-like flapping wing motion. The results show accuracy and efficiency of the method.

A conservative discontinuous Galerkin scheme for the 2D incompressible Navier–Stokes equations

In this paper we consider a conservative discretization of the two-dimensional incompressible Navier–Stokes equations. We propose an extension of Arakawa’s classical finite difference scheme for fluid flow in the vorticity–stream function formulation to a high order discontinuous Galerkin approximation. In addition, we show numerical simulations that demonstrate the accuracy of the scheme and verify the conservation properties, which are essential for long time integration. Furthermore, we discuss the massively parallel implementation on graphic processing units.

Roll damping decay of a FPSO with bilge keel

The roll damping decay is investigated for a Floating Production Storage and Offloading (FPSO). For this purpose, a roll decay test of a middle section of FPSO with bilge keel is simulated by means of the numerical solution of the incompressible two-dimensional Navier–Stokes equations. The governing equations are solved using the finite volume method and the upwind Total Variation Diminishing (TVD) scheme of Roe–Sweby. The upwind TVD scheme resolves and captures the physics of the fluid dynamics without spurious oscillations in regions of the flow field with strong pressure gradients. The numerical results are compared with experimental data for validating the numerical scheme implemented. The simulations indicated the strong influence of the bilge radius in the damping coefficient of the FPSO section. Interesting results were obtained regarding the time series of the displacement of the body and vortex shedding around the bilge keel.

Derivation of Prandtl boundary layer equations for the incompressible Navier–Stokes equations in a curved domain

Abstract The proposal of this note is to derive the equations of boundary layers in the small viscosity limit for the two-dimensional incompressible Navier–Stokes equations defined in a curved bounded domain with the non-slip boundary condition. By using curvilinear coordinate system in a neighborhood of boundary, and the multi-scale analysis we deduce that the leading profiles of boundary layers of the incompressible flows in a bounded domain still satisfy the classical Prandtl equations when the viscosity goes to zero, which are the same as for the flows defined in the half space.

Metastability and rapid convergence to quasi-stationary bar states for the two-dimensional Navier–Stokes equations

Quasi-stationary, or metastable, states play an important role in two-dimensional turbulent fluid flows, where they often emerge on timescales much shorter than the viscous timescale, and then dominate the dynamics for very long time intervals. In this paper we propose a dynamical systems explanation of the metastability of an explicit family of solutions, referred to as bar states, of the two-dimensional incompressible Navier–Stokes equation on the torus. These states are physically relevant because they are associated with certain maximum entropy solutions of the Euler equations, and they have been observed as one type of metastable state in numerical studies of two-dimensional turbulence. For small viscosity (high Reynolds number), these states are quasi-stationary in the sense that they decay on the slow, viscous timescale. Linearization about these states leads to a time-dependent operator. We show that if we approximate this operator by dropping a higher-order, non-local term, it produces a decay rate much faster than the viscous decay rate. We also provide numerical evidence that the same result holds for the full linear operator, and that our theoretical results give the optimal decay rate in this setting.

Lagrangian coherent structures at the onset of hyperchaos in the two-dimensional Navier-Stokes equations

We study a transition to hyperchaos in the two-dimensional incompressible Navier-Stokes equations with periodic boundary conditions and an external forcing term. Bifurcation diagrams are constructed by varying the Reynolds number, and a transition to hyperchaos (HC) is identified. Before the onset of HC, there is coexistence of two chaotic attractors and a hyperchaotic saddle. After the transition to HC, the two chaotic attractors merge with the hyperchaotic saddle, generating random switching between chaos and hyperchaos, which is responsible for intermittent bursts in the time series of energy and enstrophy. The chaotic mixing properties of the flow are characterized by detecting Lagrangian coherent structures. After the transition to HC, the flow displays complex Lagrangian patterns and an increase in the level of Lagrangian chaoticity during the bursty periods that can be predicted statistically by the hyperchaotic saddle prior to HC transition.

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.

Numerical Study of the Control of Tollmien–Schlichting Waves Using Plasma Actuators

The use of plasma actuators for active transition control in a flat-plate boundary layer is studied numerically. The full unsteady two-dimensional incompressible Navier–Stokes equations are solved with a perturbation formulation using a finite volume discretization. The effect of the plasma actuator is modeled using generic as well as experimentally derived body force distributions. An adaptive control system based on the filtered- least-mean-squares algorithm is included within the simulations. The control system selects the optimum frequency, phase, and amplitude of the body force in order to attenuate Tollmien–Schlichting instability waves. The factors influencing the effectiveness of transition control are discussed. It is shown that the relative size of the instability wavelength and the spatial extent of the body force have a strong influence on control performance. It is also demonstrated that the experimentally derived force distributions produce suboptimal control performance when compared with generic body force distributions. This has implications for the design of plasma actuators for boundary-layer instability control.

Casimir cascades in two-dimensional turbulence

AbstractIn addition to conserving energy and enstrophy, the nonlinear terms of the two-dimensional incompressible Navier–Stokes equation are well known to conserve the global integral of any continuously differentiable function of the scalar vorticity field. However, the phenomenological role of these additional inviscid invariants, including the issue as to whether they cascade to large or small scales, is an open question. In this work, well-resolved implicitly dealiased pseudospectral simulations suggest that the fourth power of the vorticity cascades to small scales.

Fekete-Gauss Spectral Elements for Incompressible Navier-Stokes Flows: The Two-Dimensional Case

AbstractSpectral element methods on simplicial meshes, say TSEM, show both the advantages of spectral and finite element methods, i.e., spectral accuracy and geometrical flexibility. We present aTSEM solver of the two-dimensional (2D) incompressible Navier-Stokes equations, with possible extension to the 3D case. It uses a projection method in time and piecewise polynomial basis functions of arbitrary degree in space. The so-called Fekete-Gauss TSEM is employed,i.e., Fekete (resp. Gauss) points of the triangle are used as interpolation (resp. quadrature) points. For the sake of consistency, isoparametric elements are used to approximate curved geometries. The resolution algorithm is based on an efficient Schur complement method, so that one only solves for the element boundary nodes. Moreover, the algebraic system is never assembled, therefore the number of degrees of freedom is not limiting. An accuracy study is carried out and results are provided for classical benchmarks: the driven cavity flow, the flow between eccentric cylinders and the flow past a cylinder.

Vortex induced vibrations of a square cylinder at subcritical Reynolds numbers

Vortex induced vibration (VIV) of an elastically mounted square cylinder of low non-dimensional mass is simulated at subcritical Reynolds numbers (Re), i.e.,Re≤50. The cylinder is allowed to vibrate in the transverse direction to the incoming flow. Four cases are considered for understanding the behavior of VIV of the square cylinder at subcritical Re. In the first case, the non-dimensional frequency varies as 3.1875/Re. In the second case, the non-dimensional frequency is kept constant at 0.1333. In both the cases, Re is varied. In third and fourth cases, studies are conducted for Re=25, 30 and 35, and Re=80, respectively. In the third and fourth cases, the non-dimensional velocity, U⁎, is varied. It is found that maximum transverse displacement is approximately 0.15D when the non-dimensional frequency, Fn, varies with Re. The maximum transverse displacement is 0.25D when the non-dimensional frequency is constant. For the first case, VIV starts at Re as low as 23.9 and it ceases at Re∼33.5. In all these cases, it is observed that the phase difference between the lift coefficient and the transverse displacement depends upon non-dimensional mass, Re, and non-dimensional velocity. In all the cases, the lock-in phenomenon is observed. In the fourth case of supercritical Re, hysteresis is also observed and it is seen that its extent depends upon non-dimensional mass. Stabilized finite-element space–time formulations (SUPG and PSPG) are utilized to solve the two-dimensional incompressible Navier–Stokes equations together with the equations of motion of the body.

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.

Theoretical study on the constricted flow phenomena in arteries

The present study is dealt with the constricted flow characteristics of blood in arteries by making use of an appropriate mathematical model. The constricted artery experiences the generated wall shear stress due to flow disturbances in the presence of constriction. The disturbed flow in the stenosed arterial segment causes malfunction of the cardiovascular system leading to serious health problems in the form of heart attack and stroke. The flowing blood contained in the stenosed artery is considered to be non-Newtonian while the flow is treated to be two-dimensional. The present pursuit also accounts for the motion of the arterial wall and its effect on local fluid mechanics. The flow analysis applies the time-dependent, two-dimensional incompressible nonlinear Navier-Stokes equations for non-Newtonian fluid representing blood. An extensive quantitative analysis presented at the end of the paper based on large scale numerical computations of the quantities of major physiological significance enables one to estimate the constricted flow characteristics in the arterial system under consideration which deviates significantly from that of normal physiological flow conditions.

A determining form for the two-dimensional Navier-Stokes equations: The Fourier modes case

The determining modes for the two-dimensional incompressible Navier-Stokes equations (NSE) are shown to satisfy an ordinary differential equation (ODE) of the form dv/dt = F(v), in the Banach space, X, of all bounded continuous functions of the variable \documentclass[12pt]{minimal}\begin{document}$s\in \mathbb {R}$\end{document}s∈R with values in certain finite-dimensional linear space. This new evolution ODE, named determining form, induces an infinite-dimensional dynamical system in the space X which is noteworthy for two reasons. One is that F is globally Lipschitz from X into itself. The other is that the long-term dynamics of the determining form contains that of the NSE; the traveling wave solutions of the determining form, i.e., those of the form v(t, s) = v0(t + s), correspond exactly to initial data v0 that are projections of solutions of the global attractor of the NSE onto the determining modes. The determining form is also shown to be dissipative; an estimate for the radius of an absorbing ball is derived in terms of the number of determining modes and the Grashof number (a dimensionless physical parameter).

Flow characteristics of two in-phase oscillating cylinders in side-by-side arrangement

Numerical simulations of flow past two side-by-side arranged cylinders with the same diameters of D, forced to oscillate transversely in a uniform cross-flow, are presented for a fixed Reynolds number of 100. The two-dimensional incompressible Navier–Stokes equations in Arbitrary-Lagrangian–Eulerian formulation are solved by Characteristic-Based-Split finite element method using MINI triangular element. The simulations are performed for four center-to-center cylinder spacings, s, ranging from 1.2D to 4.0D, corresponding to four kinds of wake pattern for the stationary two cylinders. The cylinders are oscillated in in-phase mode and their harmonic motions are restricted to relatively smaller amplitude with wide range of frequencies. The numerical results show that there exist five different flow response states, according to the different characteristics shown by the power spectra of lift forces, phase portraits of lift force against cylinder motion, time histories of energy transferring and flow fields visualized in vorticity contours. The force characteristics are also investigated in terms of time-averaged and root-mean-square quantities, and show that the roles of the gap spacing, flow response state and excitation frequency are different for different aerodynamic force quantities.

Global Stability Analysis of Flow Past Two Side-by-Side Circular Cylinders at Low Reynolds Numbers by a POD-Galerkin Spectral Method

A proper orthogonal decomposition (POD) method is applied to the problem of a two-dimensional flow past two side-by-side circular cylinders. Based on the POD bases, which are constructed by a snapshot method, a low-dimensional model is established for representing two-dimensional incompressible Navier-Stokes equations. Coupled with the low-dimensional model, the Chiba method is used to analyze the global stability of the basic flow. Different bifurcation paths at three major regions are revealed, in good agreement with the available results by other methods. However, the computation amount in the POD method is low, which shows the availability and advantage of the POD method.

Numerical modelling of a low Reynolds number plunging airfoil flow field characteristics

Complex viscous mechanisms such as leading edge vortices play a dominant role in the generation of instantaneous force and moment in low Reynolds number flows. The dependence of the corresponding fluid flow characteristics on the governing flow and system parameters in unsteady motions, e.g. plunging, adds to the inherent complexity of the problem. The respective fluid dynamics of such a flow is investigated here via computational fluid dynamics based on a finite volume method. The governing equations are the unsteady, incompressible two-dimensional Navier–Stokes equations. The flow field and vortical patterns around a thin ellipsoidal plunging airfoil are examined in detail with and without freestream velocity, and the effects of Reynolds and Strouhal numbers on the flow characteristics are explored. It is shown that both Reynolds and Strouhal numbers increase the aerodynamic performance in nonzero freestream velocity simulations. Increasing Reynolds and Strouhal numbers causes the airfoil to generate thrust for some time intervals of the plunging period. This thrust generation is penalized with higher peaks of drag coefficient when Strouhal number increases. However, the same penalty in the Reynolds number effect simulations is negligible compared to that of the Strouhal number effects. Increasing Strouhal number causes the airfoil to experience negative pitching moment with higher peak values for longer time intervals, but Reynolds number does not change the time at which negative pitching moment is exerted on the airfoil, but the peaks of pitching moment depend on the governing Reynolds number. The lift coefficient changes noticeably versus Strouhal number, where there is significant lead/lag at the peak lift coefficient for zero-freestream velocity simulations. Reynolds number effects on the lift coefficients mostly occur around the time at which the peak lift coefficient is obtained for both zero and nonzero freestream velocity cases. All of these effects are caused by the complex vortical patterns around the airfoil, described throughout the present article.