Two-dimensional Incompressible Navier-Stokes Equations Research Articles

Development of a Dispersion Relation-Preserving Upwinding Scheme for Incompressible Navier–Stokes Equations on NonStaggered Grids

ABSTRACT In this article a scheme which preserves the dispersion relation for convective terms is proposed for solving the two-dimensional incompressible Navier–Stokes equations on nonstaggered grids. For the sake of computational efficiency, the splitting methods of Adams-Bashforth and Adams-Moulton are employed in the predictor and corrector steps, respectively, to render second-order temporal accuracy. For the sake of convective stability and dispersive accuracy, the linearized convective terms present in the predictor and corrector steps at different time steps are approximated by a dispersion relation-preserving (DRP) scheme. The DRP upwinding scheme developed within the 13-point stencil framework is rigorously studied by virtue of dispersion and Fourier stability analyses. To validate the proposed method, we investigate several problems that are amenable to exact solutions. Results with good rates of convergence are obtained for both scalar and Navier–Stokes problems.

Numerical Analysis of Flow Loading on Bluff Bodies

Today, Computational Fluid Dynamics (CFD) offers to engineers a promising way to understand, to test and to design large bridges. Several attempts have been made during the past ten years to introduce such techniques. The purpose of this paper is to present comparative results between numerical simulations and experimental measurements of the fluid loadings on three typical fixed or moving bluff bodies. The numerical approach is based on an efficient finite-element solver using an Arbitrary Lagrangian-Eulerian formulation for solving the two-dimensional incompressible Navier-Stokes equations with moving domains. Experimental tests consist of unsteady pressure measurements performed during sectional model tests. The Proper Orthogonal Decomposition applied to the wall pressure distributions is used in addition to spectral analyses in order to explain complex flow signature and motion-induced mechanisms. Effects of body shapes and dynamic characteristics (reduced wind speed, motion amplitude) on flow response and resulting loads are also examined. Those studies highlight that the direct resolution of the Navier-Stokes equations (with a laminar strategy without turbulence models) is sufficiently performing to reproduce the physics of the large vortex scales which dominate the signature and aeroelastic loadings, but can badly estimate reattachment areas and mean loading effects.

Two-dimensional simulation of a thermally stratified reservoir with high sediment-laden inflow

A two-dimensional laterally averaged reservoir model is developed to study the density currents generated in a thermally stratified reservoir due to inflow with sediment concentration. The two-dimensional incompressible Navier-Stokes equations are used as the governing equations with the assumptions of hydrostatic pressure distribution and the Boussinesq approximation. The governing equations are solved using the Galerkin's weighted residual finite element method, with an arbitrary Lagrangian-Eulerian scheme to accommodate the free surface movement. The model developed is validated with the experimental results obtained for gravitational underflows of the Generalized Reservoir Hydrodynamics (GRH) flume at the US Army Corps of Engineers. Then the model is applied to simulate the field scale hydrodynamic and temperature structures for 1 -year annual cycle of the Te-Chi reservoir located at central Taiwan. Numerical results depicting the different regimes of density currents such as the turbidity current, underflow and interflow are predicted for sediment-laden inflow during the days of heavy rainfall due to storm. The numerical predictions of water levels and temperature profiles compare favorably well with the measured field data.

Analysing fluid loadings on moving bluff bodies using proper orthogonal decomposition

Numerical simulations and experimental measurements of the fluid loadings on three typical fixed or moving bluff bodies are presented and analysed. The numerical approach is based on an efficient finite-element solver using an Arbitrary Lagrangian–Eulerian formulation for solving the two-dimensional incompressible Navier–Stokes equations with moving domains. Experimental tests consist of unsteady pressure measurements performed during sectional model tests. The Proper Orthogonal Decomposition applied to the wall pressure distributions is used in addition to spectral analyses in order to explain complex flow signature and motion-induced mechanisms. Effects of body shapes and dynamic characteristics (reduced wind speed, motion amplitude) on flow response and resulting loads are then examined.

Stabilized finite-element method for the stationary Navier-Stokes equations

A stabilized finite-element method for the two-dimensional stationary incompressible Navier-Stokes equations is investigated in this work. A macroelement condition is introduced for constructing the local stabilized formulation of the stationary Navier-Stokes equations. By satisfying this condition, the stability of the Q1−P0 quadrilateral element and the P1−P0 triangular element are established. Moreover, the well-posedness and the optimal error estimate of the stabilized finite-element method for the stationary Navier-Stokes equations are obtained. Finally, some numerical tests to confirm the theoretical results of the stabilized finite-element method are provided.

Reynolds number dependence of two-dimensional laminar co-rotating vortex merging

The paper reports unsteady Navier–Stokes calculations of laminar two-dimensional co-rotating vortex merging for various Reynolds numbers. The unsteady, incompressible two-dimensional Navier–Stokes equations were solved with fourth-order Runge–Kutta temporal discretization and fourth-order symmetric compact schemes for spatial discretization. Calculations of the unsteady Taylor vortex benchmark showed that fourth-order accurate solutions for all primitive variables were indeed achieved. Calculations for a pair of equal-strength co-rotating vortices show good agreement with reported direct numerical simulation and experiments for the evolution of the separation distance and core radius. It is found that the time required for merging is inversely proportional to the square root of the Reynolds number. According to previous experimental research, it was also found that complete merging in laminar regime undergoes four stages with physical meaning. The physical mechanism responsible for the merging process is investigated and it is found that the antisymmetric vorticity dynamics plays an important role until full merging.

Unsteady viscous flow over elliptic cylinders at various thickness with different reynolds numbers

Two-dimensional incompressible Navier-stokes equations ale solved using SIMPLER method in the intrinsic curvilinear coordinates system to study the unsteady viscous flow physics over two-dimensional ellipses Unsteady viscous flows over various thickness-to-chord ratios of 0.6, 0.8, 10, and 1.2 elliptic- cylinders are simulated at different Reynolds numbers of 200, 400, and 1,000 This study is focused on the understanding the effects of Reynolds number and elliptic cylinder thickness on the drag and lift forces. The present numeric-a i solutions are compared with available experiment and numerical results and show a good agreement Through this study, it is observed that the Reynolds number and the cylinder thickness affect significantly the frequencies of the force oscillations as well as the mean values and the amplitudes of the drag and lift forces.

Numerical analysis on the propulsive performance and vortex shedding of fish-like travelling wavy plate

Numerical analysis is carried out to investigate viscous flow over a travelling wavy plate undergoing lateral motion in the form of a streamwise travelling wave, which is similar to the backbone undulation of swimming fish. The two-dimensional incompressible Navier–Stokes equations are solved using the finite element technique with the deforming-spatial-domain/stabilized space–time formulation. The objective of this study is to elucidate hydrodynamic features of flow structure and vortex shedding near the travelling wavy plate and to get into physical insights to the understanding of fish-like swimming mechanisms in terms of drag reduction and optimal propulsive performance. The effects of some typical parameters, including the phase speed, amplitude, and relative wavelength of travelling wavy plate, on the flow structures, the forces, and the power consumption required for the propulsive motion of the plate are analysed. These results predicted by the present numerical analysis are well consistent with the available data obtained for the wave-like swimming motion of live fish in nature. Copyright © 2005 John Wiley & Sons, Ltd.

On the reinitialization procedure in a narrow-band locally refined level set method for interfacial flows

A local level set algorithm for simulating interfacial flows described by the two-dimensional incompressible Navier–Stokes equations is presented. The governing equations are solved using a finite-difference discretization on a Cartesian grid and a second-order approximate projection method. The level set transport and reinitialization equations are solved in a narrow band around the interface using an adaptive refined grid, which is reconstructed every time step and refined using a simple uniform cell-splitting operation within the band. Instabilities at the border of the narrow band are avoided by smoothing the level set function in the outer part of the band. The influence of different PDE-based reinitialization strategies on the accuracy of the results is investigated. The ability of the proposed method to accurately compute interfacial flows is discussed using different tests, namely the advection of a circle of fluid in two different time-reversed vortex flows, the advection of Zalesak's rotating disk, the propagation of small-amplitude gravity and capillary waves at the interface between two superposed viscous fluids in deep water, and a classical test of Rayleigh–Taylor instability with and without surface tension effects. The interface location error and area loss for some of the results obtained are compared with those of a recent particle level set method. Copyright © 2005 John Wiley & Sons, Ltd.

A pure-compact scheme for the streamfunction formulation of Navier–Stokes equations

A pure-streamfunction formulation is introduced for the numerical simulation of the two-dimensional incompressible Navier–Stokes equations. The idea is to replace the vorticity in the vorticity-streamfunction evolution equation by the Laplacian of the streamfunction. The resulting formulation includes the streamfunction only, thus no inter-function relations need to be invoked. A compact numerical scheme, which interpolates streamfunction values as well as its first order derivatives, is presented and analyzed. A number of numerical experiments are presented, including driven and double driven cavities, where the Reynolds numbers are sufficiently large, leading to symmetry breaking of asymptotic solutions.

Flow past a cylinder: shear layer instability and drag crisis

AbstractFlow past a circular cylinder for Re=100 to 107 is studied numerically by solving the unsteady incompressible two‐dimensional Navier–Stokes equations via a stabilized finite element formulation. It is well known that beyond Re ∼ 200 the flow develops significant three‐dimensional features. Therefore, two‐dimensional computations are expected to fall well short of predicting the flow accurately at high Re. It is fairly well accepted that the shear layer instability is primarily a two‐dimensional phenomenon. The frequency of the shear layer vortices, from the present computations, agree quite well with the Re0.67 variation observed by other researchers from experimental measurements. The main objective of this paper is to investigate a possible relationship between the drag crisis (sudden loss of drag at Re ∼ 2 × 105) and the instability of the separated shear layer. As Re is increased the transition point of shear layer, beyond which it is unstable, moves upstream. At the critical Reynolds number the transition point is located very close to the point of flow separation. As a result, the shear layer eddies cause mixing of the flow in the boundary layer. This energizes the boundary layer and leads to its reattachment. The delay in flow separation is associated with narrowing of wake, increase in Reynolds shear stress near the shoulder of the cylinder and a significant reduction in the drag and base suction coefficients. The spatial and temporal power spectra for the kinetic energy of the Re=106 flow are computed. As in two‐dimensional isotropic turbulence, E(k) varies as k−5/3 for wavenumbers higher than energy injection scale and as k−3 for lower wavenumbers. The present computations suggest that the shear layer vortices play a major role in the transition of boundary layer from laminar to turbulent state. Copyright © 2004 John Wiley & Sons, Ltd.

Vortex formation and force characteristics of a foil in the wake of a circular cylinder

A numerical simulation is performed to investigate the vortex structure and force characteristics of a foil placed in the wake of a circular cylinder by solving the two-dimensional incompressible Navier–Stokes equations with a finite element method. The objective of this study is to reveal the vortex structures in the near region of the foil when the vortices shed from the circular cylinder are incident upon the foil, and to deal with the forces acting on the cylinder and the foil. Based on the relative position of the foil and the cylinder, three basic types of interference are classified; correspondingly, some basic patterns of vortex shedding are identified in the near region of the foil. According to our calculated results, it is found that a preferred vortex shedding frequency of the foil is synchronized with that of the cylinder due to the interference between the cylinder and the foil. Further, the influence of the relative position between the cylinder and foil and the incidence angle of the foil on the vortex formation, force character, and the frequency of the vortex shedding is analyzed and discussed.

A numerical and analytical study on a tail-flapping model for fish fast C-start

The force production physics and the flow control mechanism of fish fast C-start are studied numerically and theoretically by using a tail-flapping model. The problem is simplified to a 2-D foil that rotates rapidly to and fro on one side about its fixed leading edge in water medium. The study involves the simulation of the flow by solving the two-dimensional unsteady incompressible Navier-Stokes equations and employing a theoretical analytic modeling approach. Firstly, reasonable thrust magnitude and its time history are obtained and checked by fitting predicted results coming from these two approaches. Next, the flow fields and vortex structures are given, and the propulsive mechanism is interpreted. The results show that the induction of vortex distributions near the trailing edge of the tail are important in the time-averaged thrust generation, though the added inertial effect plays an important role in producing an instant large thrust especially in the first stage. Furthermore, dynamic and energetic effects of some kinematic controlling factors are discussed. For enhancing the time-averaged thrust but keeping a favorable ratio of it to time-averaged input power within the limitations of muscle ability, it is recommended to have a larger deflection amplitude in a limited time interval and with no time delay between the to-and-fro strokes.

Liquid convection effects on the pushing-engulfment transition of insoluble particles by a solidifying interface: Part II. Numerical calculation of drag and lift forces on a particle in parabolic shear flow

In this work, lift and drag forces acting on a particle in the close vicinity of a wall are calculated by numerically solving the incompressible two-dimensional Navier-Stokes equations. The flow field computations are done using the well-known Marker and Cell (MAC) method on a staggered grid. A parabolic shear flow at the inlet is assumed. The particle is assumed to be nonrotating and Magnus forces are not considered. The numerical results are compared to those obtained from analytical/empirical expressions for drag and lift forces from different theoretical models. Reasonably good agreement has been found between the two approaches.

Development of a fully coupled control‐volume finite element method for the incompressible Navier–Stokes equations

AbstractThis paper proposes and investigates fully coupled control‐volume finite element method (CVFEM) for solving the two‐dimensional incompressible Navier–Stokes equations. The proposed method borrows many of its features from the segregated CVFEM described by Baliga et al. Thus finite‐volume discretization is employed on a colocated grid using either the MAW or the FLO schemes and an element‐by‐element assembling procedure is applied for the construction of the discretizations equations. In this paper, and unlike the case for most fully coupled formulations available in the literature, the Poisson pressure equation has been retained from the segregated approach. The use of a pressure equation leads to an unfavourable size increase of the fully coupled linear system, but significantly improves the system's conditioning. The fully coupled system obtained is solved using an ILUT preconditioned GMRES algorithm. The other important element in this paper is the proposal of a Newton linearization of the convection terms in lieu of the common Picard iteration procedure. A systematic comparison between two segregated and four fully coupled fomulations has been presented which has allowed for an evaluation of the individual benefits and strengths of the coupling and linearization procedure by studying lid‐driven cavity problems and flows past a circular cylinder. All coupled formulations have proven to be significantly superior both in robustness and efficiency, as compared with the segregated formulation. In some circumstances, the coupled methods yield a converged solution of the system of discretized equations constructed using the FLO scheme, while the segregated formulations diverge. Compared to Picard's linearization, Newton's linearization is more efficient at reducing the number of iterations needed to converge, but requires more computational effort per iteration from the linear equation solver. Furthermore, the Jacobian matrix should include contributions from the nonlinearity appearing at both the governing‐equation level and the interpolation‐scheme level to ensure Newton's method convergence. The key element in guaranteeing successful, fully coupled solutions lies in the use of an efficient linear equation solver and preconditioner. Copyright © 2004 John Wiley & Sons, Ltd.

A ‘poor man's Navier–Stokes equation’: derivation and numerical experiments—the 2‐D case

AbstractWe present a systematic derivation of a discrete dynamical system directly from the two‐dimensional incompressible Navier–Stokes equations via a Galerkin procedure and provide a detailed numerical investigation (covering more than 107 cases) of the characteristic behaviours exhibited by the discrete mapping for specified combinations of the four bifurcation parameters. We show that this simple 2‐D algebraic map, which consists of a bilinearly coupled pair of logistic maps, can produce essentially any (temporal) behaviour observed either experimentally or computationally in incompressible Navier–Stokes flows as the bifurcation parameters are varied in pairs over their ranges of stable behaviours. We conclude from this that such discrete dynamical systems deserve consideration as sources of temporal fluctuations in synthetic‐velocity forms of subgrid‐scale models for large‐eddy simulation. Copyright © 2004 John Wiley & Sons, Ltd.

Application of mixed differential quadrature method for solving the coupled two-dimensional incompressible Navier-stokes equation and heat equation

The traditional differential quadrature method was improved by using the upwind difference scheme for the convective terms to solve the coupled two-dimensional incompressible Navier-stokes equations and heat equation. The new method was compared with the conventional differential quadrature method in the aspects of convergence and accuracy. The results show that the new method is more accurate, and has better convergence than the conventional differential quadrature method for numerically computing the steady-state solution.

Propulsive performance and vortex shedding of a foil in flapping flight

The propulsive performance and vortex shedding of an oscillating foil, which mimics biological locomotion, are investigated based on a computational fluid dynamics analysis. The objectives of this study are to investigate unsteady forces, in particular a thrust force, for the foil in pitching and plunging motion, and to deal with the relations of the propulsive performance with leading-edge vortex structure and vortex shedding in the near wake. The two-dimensional incompressible Navier–Stokes equations in the vorticity and stream-function formulation are solved by fourth-order essentially compact finite difference schemes for the space derivatives and a fourth-order Runge-Kutta scheme for the time advancement. To reveal the mechanism of the propulsive performance, the unsteady forces and the shedding of the leading- and trailing- edge vortices of the foil in the pitching and plunging motion are analyzed. Based on our calculated results, three types of the leading-edge vortex shedding, which have an effective influence on the vortex structures in the wake of the oscillating foil, are identified. The effects of some typical factors, such as the frequency and amplitude of the oscillation, the phase difference between the pitching and plunging motions, and the thickness ratio of the foil, on the vortex shedding and the unsteady forces are discussed.

An optimal nonlinear Galerkin method with mixed finite elements for the steady Navier‐Stokes equations

AbstractAn optimal nonlinear Galerkin method with mixed finite elements is developed for solving the two‐dimensional steady incompressible Navier‐Stokes equations. This method is based on two finite element spaces XH and Xh for the approximation of velocity, defined on a coarse grid with grid size H and a fine grid with grid size h ≪ H, respectively, and a finite element space Mh for the approximation of pressure. We prove that the difference in appropriate norms between the solutions of the nonlinear Galerkin method and a classical Galerkin method is of the order of H5. If we choose H = O(h2/5), these two methods have a convergence rate of the same order. We numerically demonstrate that the optimal nonlinear Galerkin method is efficient and can save a large amount of computational time. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 762–775, 2003.

Discrete Noise Prediction of Variable Pitch Cross-Flow Fans by Unsteady Navier-Stokes Computations

The unsteady viscous flow fields of a cross-flow fan are computed by time-accurately solving the two-dimensional incompressible Navier-Stokes equations with the unstructured triangular mesh solver algorithms. Based on pressure fluctuation data acquired at the surfaces of 35 rotating blades and stabilizer, acoustic pressures are predicted by the Ffowcs Williams-Hawkings equation. The aerodynamic noise sources of the cross-flow fan are also identified by correlating the acoustic pressure fluctuations with the unsteady flow characteristics during one revolution of the impeller. The present method is applied to the uniform and random pitch fans to investigate their performance and aeroacoustic noise characteristics, especially the frequency modulation of the tonal noise at the blade passing frequency (BPF).