Two-dimensional Unsteady Incompressible Navier-Stokes Equations Research Articles

A high-order upwind compact difference scheme for solving the streamfunction-velocity formulation of the unsteady incompressible Navier–Stokes equations

In this paper, we propose an upwind compact difference method with fourth-order spatial accuracy and second-order temporal accuracy for solving the streamfunction-velocity formulation of the two-dimensional unsteady incompressible Navier–Stokes equations. The streamfunction and its first-order partial derivatives (velocities) are treated as unknown variables. Three types of compact difference schemes are employed to discretize the first-order partial derivatives of the streamfunction. Specifically, these schemes include the fourth-order symmetric scheme, the fifth-order upwind scheme, and the sixth-order symmetric scheme derived by combining the two parts of the fifth-order upwind scheme. As a result, the fourth-order spatial discretization schemes are established for the Laplacian term, the biharmonic term, and the nonlinear convective term, along with the Crank–Nicolson scheme for the temporal discretization. The unconditional stability characteristic of the scheme for the linear model is proved by discrete von Neumann analysis. Moreover, six numerical experiments involving three test problems with the analytic solutions, and three flow problems including doubly periodic double shear layer, lid-driven cavity flow, and dipole-wall interaction are carried out to demonstrate the accuracy, robustness, and efficiency of the present method. The results indicate that the present method not only has good numerical performance but also exhibits quite efficiency.

Numerical Simulation of Non-Newtonian Power-Law Fluid Flow in a Lid-Driven Skewed Cavity

Non-Newtonian laminar fluid flow in a lid-driven skewed cavity has been studied numerically using power-law viscosity model. The governing two-dimensional unsteady incompressible Navier–Stokes equations were initially non-dimensionalized using appropriate transformation, and then the dimensionless form is transformed to generalized curvilinear coordinates to simulate complex geometry. The transformed equations are discretized using finite volume method with the collocated grid arrangement. The code is first validated against the existing benchmark results for two-dimensional lid-driven square cavity problem considering both Newtonian and non-Newtonian fluids. The validation has also been carried out for a lid-driven skewed cavity in the case of a Newtonian fluid. Then the code is applied to the skewed cavity problem involving non-Newtonian fluid flow which can be described by the power-law viscosity model. Moreover, grid independence test has been performed for a skewed cavity for different values of power-law index. In the present case, the skewness of the geometry has been changed by changing the skew angle for both shear-thinning and shear-thickening fluids. The consequent numerical results are presented in terms of the velocity as well as streamlines for the different values of the power-law index $$n=0.5$$ , 1 and 1.5, Reynolds number $$Re = 100, 200, 300$$ and 500 as well as for the different angles of the skewed cavity ( $$\alpha =15^{\circ }$$ to $$165^{\circ }$$ ).

Numerical Study of Hydrothermal Wave Suppression in Thermocapillary Flow Using a Predictive Control Method

Hydrothermal waves in flows driven by thermocapillary and buoyancy effects are suppressed by applying a predictive control method. Hydrothermal waves arise in the manufacturing of crystals, including the “open boat” crystal growth process, and lead to undesirable impurities in crystals. The open boat process is modeled using the two-dimensional unsteady incompressible Navier–Stokes equations under the Boussinesq approximation and the linear approximation of the surface thermocapillary force. The flow is controlled by a spatially and temporally varying heat flux density through the free surface. The heat flux density is determined by a conjugate gradient optimization algorithm. The gradient of the objective function with respect to the heat flux density is found by solving adjoint equations derived from the Navier–Stokes ones in the Boussinesq approximation. Special attention is given to heat flux density distributions over small free-surface areas and to the maximum admissible heat flux density.

A Comparison between the Reduced Differential Transform Method and Perturbation-Iteration Algorithm for Solving Two-Dimensional Unsteady Incompressible Navier-Stokes Equations

In this work, approximate analytical solutions to the lid-driven square cavity flow problem, which satisfied two-dimensional unsteady incompressible Navier-Stokes equations, are presented using the kinetically reduced local Navier-Stokes equations. Reduced differential transform method and perturbation-iteration algorithm are applied to solve this problem. The convergence analysis was discussed for both methods. The numerical results of both methods are given at some Reynolds numbers and low Mach numbers, and compared with results of earlier studies in the review of the literatures. These two methods are easy and fast to implement, and the results are close to each other and other numerical results, so it can be said that these methods are useful in finding approximate analytical solutions to the unsteady incompressible flow problems at low Mach numbers.

Numerical investigation into the effects of stroke trajectory on the aerodynamic performance of insect hovering flight

Various stroke trajectories may be observed in insect hovering flight in nature; however their influences on the flight performance of insect are not well estimated. In this study, a numerical investigation into the effects of stroke trajectories on the aerodynamic performance of insect hovering flight is carried out through the solution of the two-dimensional unsteady incompressible Navier-Stokes equations. An insect wing model with ellipse cross section in hovering flight is considered for the purpose and four types of idealized trajectories (Named linear, oval, figure-eight and double-eight) which possess different deviation characteristics from the stroke plane are examined. The influences of the deviation amplitude of trajectory, the attack angle of wing and the inclined angle of stroke plane on the aerodynamic characteristics of hovering wing are systematically analyzed. The results show that in the case of the wing in a normal hovering flight where the stroke plane is horizontal, the trajectory deviation from the stroke plane weakens the aerodynamic performance for each trajectory case considered, and this deteriorative effect becomes more serious as the amplitude of deviation increases. With regard to the influence of the angle of attack, the results show that the time-averaged drag force and power consumption increase monotonically with the angle, whereas the time-averaged lift force and the lifting efficiency increase first and then decrease as the angle increases further. In the case of a hovering flight with an inclined stroke plane, distinctly different trends from a normal hovering flight are obtained.

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 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.

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.

A p-multigrid spectral difference method for two-dimensional unsteady incompressible Navier–Stokes equations

This paper presents the development of a 2D high-order solver with spectral difference method for unsteady incompressible Navier–Stokes equations accelerated by a p-multigrid method. This solver is designed for unstructured quadrilateral elements. Time-marching methods cannot be applied directly to incompressible flows because the governing equations are not hyperbolic. An artificial compressibility method (ACM) is employed in order to treat the inviscid fluxes using the traditional characteristics-based schemes. The viscous fluxes are computed using the averaging approach (Sun et al., 2007; Kopriva, 1998) [29,12]. A dual time stepping scheme is implemented to deal with physical time marching. A p-multigrid method is implemented (Liang et al., 2009) [16] in conjunction with the dual time stepping method for convergence acceleration. The incompressible SD (ISD) method added with the ACM (SD-ACM) is able to accurately simulate 2D steady and unsteady viscous flows.

Numerical simulation of flow past row of square cylinders for various separation ratios

A systematic study for the flow around a row of five square cylinders placed in a side-by-side arrangement and normal to the oncoming flow at a Reynolds number of 150 is carried out through the numerical solution of the two-dimensional unsteady incompressible Navier–Stokes equations. Special attention is paid to investigate the effect of the spacing between the five cylinders on the wake structure and vortex shedding mechanism. The simulations are performed for the separation ratios (spacing to size ratio) of 1.2, 2, 3 and 4. Depending on the separation ratio the following flow patterns are observed: a flip-flopping pattern, in-phase and anti-phase synchronized pattern and non-synchronized pattern. These flow patterns are supposed to be a consequence of the interaction between two types of frequencies viz. the vortex shedding (primary) and the cylinder interaction (secondary) frequencies. At small separation ratio the flow is predominantly characterized by the jet in the gaps between successive cylinders and the secondary frequencies play a role in the resulting chaotic flow. On the contrary, at higher separation ratio the secondary frequencies almost disappear and the resulting flow becomes more synchronized dominated by the primary frequency.

Control of a Simplified Unsteady Film-Cooling Flow Using Gradient-Based Optimization

Maximization of film-cooling effectiveness in an idealized film-cooling flow is accomplished using a gradient-based control strategy. The state equations are the two-dimensional unsteady incompressible Navier-Stokes equations and the temperature equation. The control is a spatially and temporally varying normal-velocity boundary condition subject to a constraint of zero mass and energy flux. The control that maximizes the objective function is found using a conjugate gradient method, in which 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 over which the control is obtained by the optimizer. Drastic improvements in the film-cooling effectiveness are obtained for certain choices of the objective function and event horizon. Although not a specific target of the optimizer, one result of the control is that the wall jet remains attached to the wall. The numerical issues involved with finding the control are discussed and the features of the resulting control are analyzed, with the goal of understanding the mechanisms that affect the film-cooling effectiveness.

A numerical study on mechanism of S-starts of northern pike ( Esox lucius)

Northern pike is regarded as a specialist in swimming acceleration. The force production mechanism of northern pike, Esox lucius, during its predation S-starts was numerically studied in this article. The problem was reasonably simplified to a loose-coupling problem of fish swimming dynamics and hydrodynamics just in the swimming direction. The approach involved the simulation of the flow by solving the two-dimensional unsteady incompressible Navier-Stokes equations and decribing the fish motion dynamics based on Newton's Second Law. Visualizations of flow fields and vortex structures were performed. The results show that the large acceleration is obtained mainly in the first undulatory cycle in which the amplitude increases. In the second cycle, a couple of vortices are generated and induce a jet. In the third cycle, the jet is strengthened by the mergence of the vortices in the same direction. Through discussing the effects of various controllable factors on the swimming performance, it is found that the actual locomotion mode of the northern pike in nature is just the best choice.

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.

A cell boundary element method applied to laminar vortex shedding from circular cylinders

The two-dimensional unsteady incompressible Navier–Stokes equations are solved for flows around arrangements of circular cylinders at Reynolds number 100 and 200. A hybrid boundary element/finite element method is used to discretise the spatial domain together with a second order implicit finite difference approximation in time. The numerical scheme of study is validated for a uniform stream past an isolated circular cylinder by comparing findings with experimental and numerical studies. Both steady state and time dependent solutions were predicted with good agreement. The numerical approach, known here as the cell boundary element method (cell BEM), was also used to solve flows around two cylinders of equal diameter side by side and in tandem. It was found that no modifications to the method were needed to compute the flow field for these connected domains. In-phase and anti-phase vortex shedding modes were found to exist in the flow simulation. These simulations were in excellent agreement with phenomena observed in experiments. Particle simulations, generated from the cell BEM velocity fields, were found to have great similarity with smoke visualisations from experiment.

The unsteady structure of two-dimensional steady laminar separation

The two-dimensional unsteady incompressible Navier–Stokes equations, solved by a fractional time-step method, were used to investigate separation due to the application of an adverse pressure gradient to a low-Reynolds number boundary layer flow. The inviscid pressure distribution of Gaster [AGARD CP 4, 813 (1966)] was applied in the present computations to study the development of a laminar separation bubble. In all cases studied, periodic vortex shedding occurred from the primary separation region. The shed vortices initially lifted from the boundary layer and then returned towards the surface downstream. The shedding frequency nondimensionalized by the momentum thickness was found to be independent of Reynolds number. The value of the nondimensional Strouhal number, however, was found to differ from the results of Pauley et al. [J. Fluid Mech. 220, 397 (1990)], indicating that the shedding frequency varies with the nondimensional pressure distribution, Cp. The computational results were time averaged over several shedding cycles and the results were compared with Gaster. The numerical study accurately reproduced the major characteristics of the separation found in Gaster’s study such as the separation point, the pressure plateau within the upstream portion of the separation bubble, and the reattachment point. The similarity between the experimental results and the time-averaged two-dimensional computational results indicates that the low-frequency velocity fluctuations detected by Gaster are primarily due to the motion of large vortex structures. This suggests that large-scale two-dimensional structures control bubble reattachment and small-scale turbulence contributes a secondary role.