Elliptic Mesh Generation Research Articles

Two-dimensional simulations of flow near a cavity and a flexible solid boundary

Two-dimensional fluid flow near a cavity and a flexible solid boundary is examined in this work. Stokes’ equations are used to describe the fluid flow, while the flexible solid boundary is modeled as a uniformly tensioned membrane. Equations for elliptic mesh generation are solved iteratively along with Stokes’ equations, and the equation describing the membrane is used to update its position during the iterations. Two different configurations are considered. In the first, flow passes through the gap between a moving flexible wall and a rigid cavity. This configuration is studied in order to verify the validity of a lubrication model for this flow which was developed in previous work [X. Yin and S. Kumar, Phys. Fluids 17, 063101 (2005)]. In the second, flow driven by an externally applied pressure gradient passes through the gap between a stationary rigid wall and a cavity with a flexible bottom wall that can be deformed by an external pressure. This configuration is studied in order to explore the effect of boundary deformation on the flow pattern in the cavity. The results for the first configuration indicate that the lubrication model yields good predictions of the pressure profile, position of the flexible wall, and flow rate. The comparison also confirms that the lubrication model can only predict the existence of one primary eddy in the cavity, but not multiple primary eddies or corner eddies. The results for the second configuration indicate that the flow pattern in the cavity is dramatically altered as the external pressure changes. Replacing the bottom of a cavity with a flexible wall and applying a time-periodic pressure to it may thus be a potentially useful way to improve mixing and heat/mass transport in the cavity.

Deformation and breakup of a stretching liquid bridge covered with an insoluble surfactant monolayer

The breakup of surfactant-laden drops and jets is of technological interest and fundamental scientific importance. Surfactants are routinely used to control the breakup of drops and jets in applications ranging from inkjet printing to crop spraying. Accurate computation of breakup of surfactant-laden drops and jets is often the key to the development of new applications and to providing a rational fundamental understanding of both existing and emerging applications. While highly accurate algorithms for studying the breakup of surfactant-free drops and jets are well documented and much is now known about the dynamics in such situations, little is known by contrast about the closely related problem of interface rupture when surfactant effects cannot be neglected. The deformation and breakup of a stretching liquid bridge of an incompressible Newtonian fluid whose surface is covered with an insoluble surfactant monolayer are analyzed here experimentally and computationally. In the experiments, high-speed visualization is used to capture the transient deformation of a bridge. The dynamic shapes of bridges (captive between two rods of 3.15 mm diameter) are captured and analyzed with a time resolution of 1 ms. The bridge lengths are 3.15 mm initially and about 4–7 mm at breakup, which occurs after stretching for about 0.1–0.2 s, depending on the volume and viscosity of the liquid and the surface density of spread monolayers. The dynamics of a surfactant-covered bridge is governed by the Navier-Stokes and convection-diffusion equations. First, these equations are solved with a three-dimensional, but axisymmetric, or two-dimensional (2D), finite element algorithm using elliptic mesh generation. Second, the governing set of 2D equations is reduced to a set of one-dimensional (1D) equations by means of the slender-jet approximation and the resulting set of 1D equations is solved with a 1D finite element algorithm. The presence of surfactant results not only in the lowering of surface tension and the capillary pressure, but also in surface tension gradients and Marangoni stresses, both of which affect the transient dynamics leading to breakup. In particular, the role of Marangoni stresses in delaying bridge breakup and on formation of satellite droplets is investigated as a function of the initial surface density and surface activity of the surfactant, and surface Peclet number that measures the importance of convection relative to diffusion. The predictions of the 2D algorithm are confirmed to be faithful to the physics by demonstrating that the computed results accord well with the experiments and existing scaling theories. In the pinch-off region, the surfactant is swept out of a thinning neck by strong convection. The calculations thus reveal that the scaling behavior in the presence of surfactant parallels that observed in the absence of surfactant, in accordance with recent reports by others. The 2D computations and the experiments are used in tandem to identify regions in the space of governing parameters where the 1D equations can be used with confidence.

Airfoil design via optimal control theory

In airfoil design, one problem of greatinterest is to find the target airfoil profile to achieve a giventarget velocity distribution. It can be formulated as an optimalcontrol problem, with the control being the airfoil profile andthe governing equation being the full potential equation in thetransonic regime. To discretize the problem, one approach is toemploy the finite element method. In the discretized space, adirect relationship between the objective function and the unknownprofile co-ordinates can be defined via the finite element basisfunctions. Moreover, it is advantageous to derive the gradient inthe discretized space rather than the continuous space to avoidcontamination by discretization errors. In this paper, thisapproach is studied. In particular, a new formulation is proposed.A novel decomposition of the discrete space for the potentialfunction, the gradient is derived and an efficient algorithm usingthe quasi-Newton method is described. In generating and adjustingthe mesh during iterations, the elliptic mesh generation techniqueis used.

Dynamics and breakup of a contracting liquid filament

Contraction of a filament of an incompressible Newtonian liquid in a passive ambient fluid is studied computationally to provide insights into the dynamics of satellite drops created during drop formation. This free boundary problem, which is composed of the Navier–Stokes system and the associated initial and boundary conditions that govern the evolution in time of the filament shape and the velocity and pressure fields within it, is solved by the method of lines incorporating the finite element method for spatial discretization. The finite element algorithm developed here utilizes an adaptive elliptic mesh generation technique that is capable of tracking the dynamics of the filament up to the incipience of pinch-off without the use of remeshing. The correctness of the algorithm is verified by demonstrating that its predictions accord with (a) previously published results of Basaran (1992) on the analysis of finite-amplitude oscillations of viscous drops, (b) simulations of the dynamics of contracting filaments carried out with the well-benchmarked algorithm of Wilkes et al. (1999), and (c) scaling laws governing interface rupture and transitions that can occur from one scaling law to another as pinch-off is approached. In dimensionless form, just two parameters govern the problem: the dimensionless half-length which measures the relative importance of viscous force to capillary force. Regions of the parameter space are identified where filaments (a) contract to a sphere without breaking into multiple droplets, (b) break via the so-called endpinching mechanism where daughter drops pinch-off from the ends of the main filament, and (c) break after undergoing a series of complex oscillations. Predictions made with the new algorithm are also compared to those made with a model based on the slender-jet approximation. A region of the parameter space is found where the slender-jet approximation fares poorly, and its cause is elucidated by examination of the vorticity dynamics and flow fields within contracting filaments.

Numerical and Experimental Study of Offshore Pipeline Stability in Trench

Offshore pipelines are to be stabilized against external loading caused mainly by wave and currents. When pipelines are placed in a trench, the forces acting on them are reduced considerably. To investigate the stability in various trench sections, the flow over the pipeline is numerically simulated and compared with experimental results obtained from particle image velocimetry measurement. This paper consists of two parts: single pipe and dual pipe in various trench sections. Each part compares numerical results with experimental results, which include mean flow pattern, drag-force coefficient, and lift-force coefficient. For a single pipe, Smagainsky's eddy viscosity model combined with a truncated deductive model is applied to represent the stress field in a quadrilateral structured grid system. In the numerical simulation for dual pipes, the Navier-Stokes equation and elliptic mesh generation method are used. The stability of various trench conditions is addressed in terms of mean amplitudes of oscillating lift and drag and reduction factors for each case. The results can be effectively applied to pipeline design.

A Study of Monitor Functions for Two-Dimensional Adaptive Mesh Generation

In this paper we study the problem of two-dimensional adaptive mesh generation using a variational approach and, specifically, the effect that the monitor function has on the resulting mesh behavior. The basic theoretical tools employed are Green's function for elliptic problems and the eigendecomposition of symmetric positive definite matrices. Based upon this study, a general strategy is suggested for how to choose the monitor function, and numerical results are presented for illustrative purposes.The three-dimensional case is also briefly discussed. It is noted that the strategy used here can be applied to other elliptic mesh generation techniques as well.

A computational analysis of electrohydrodynamics of a leaky dielectric drop in an electric field

Axisymmetric steady flows driven by an electric field about a deformable fluid drop suspended in an immiscible fluid are studied within the framework of the leaky dielectric model. Deformations of the drop and the flow fields are determined by solving the nonlinear free-boundary problem composed of the Navier-Stokes system governing the flow field and Laplace's system governing the electric field. The solutions are obtained by using the Galerkin finite-element method with an elliptic mesh generation scheme. Under conditions of creeping flow and vanishingly small drop deformations, the results of finite-element computations recover the asymptotic results. When drop deformations become noticeable, the asymptotic results are often found to underestimate both the flow intensity and drop deformation. By tracking solution branches in parameter space with an arc-length continuation method, curves in parameter space of the drop deformation parameter D versus the square of the dimensionless field strength E usually exhibit a turning point when E reaches a critical value Ec. Along such a family of drop shapes, steady solutions do not exist for E > Ec. The nonlinear relationship revealed computationally between D and E2 appears to be capable of providing insight into discrepancies reported in the literature between experiments and predictions based on the asymptotic theory. In some special cases with fluid conductivities closely matched, however, drop deformations are found to grow with E2 indefinitely and no critical value Ec is encountered by the corresponding solution branches. For most cases with realistic values of physical properties, the overall electrohydrodynamic behaviour is relatively insensitive to effects of finite-Reynolds-number flow. However, under extreme conditions when fluids of very low viscosities are involved, computational results illustrate a remarkable shape turnaround phenomenon: a drop with oblate deformation at low field strength can evolve into a prolate-like drop shape as the field strength is increased.

Arbitrarily oriented capillary-viscous planar jets in the presence of gravity

Numerical solutions for the steady flow of an arbitrarily oriented incompressible capillary-viscous jet in the presence of gravity are presented. A combination of the finite element method and an elliptic mesh generation technique are used to compute solutions over a wide range of the four relevant physical parameters: Reynolds number, Stokes number, capillary number, and angle of the jet nozzle with respect to gravity. It is shown that both gravity and surface-tension forces can be instrumental in the mechanism of jet turning. In addition, it is found that multiple solutions are possible when the jet nozzle is directed close to vertically upward. The linear stability of the steady solutions to two-dimensional disturbances is briefly considered.