Boo Cheong Khoo

An immersed interface method for viscous incompressible flows involving rigid and flexible boundaries

We present an immersed interface method for the incompressible Navier-Stokes equations capable of handling both rigid and flexible boundaries. The immersed boundaries are represented by a number of Lagrangian control points. In order to ensure that the no-slip condition on the rigid boundary is satisfied, singular forces are applied on the fluid. The forces are related to the jumps in pressure and the jumps in the derivatives of both pressure and velocity, and are interpolated using cubic splines. The strength of the singular forces at the rigid boundary is determined by solving a small system of equations at each timestep. For flexible boundaries, the forces that the boundary exerts on the fluid are computed from the constitutive relation of the flexible boundary and are applied to the fluid through the jump conditions. The position of the flexible boundary is updated implicitly using a quasi-Newton method (BFGS) within each timestep. The Navier-Stokes equations are discretized on a staggered Cartesian grid by a second order accurate projection method for pressure and velocity and the overall scheme is second order accurate.

A conservative interface method for compressible flows

In this work, we present a conservative interface method for both multi-fluid and complex boundary problems, in which the standard finite volume scheme on Cartesian grids is modified by considering computational cells being cut by interface. While the discretized governing equations are updated conservatively, the method treats the topological changes naturally by combining interface description and geometric operations with a level set technique. Extensive tests in 1D are carried out, and 2D examples suggest that the present scheme is able to handle multi-fluid and complex (static or moving) boundary problems in a straightforward way with good robustness and accuracy.

An extended level set method for shape and topology optimization

In this paper, the conventional level set methods are extended as an effective approach for shape and topology optimization by the introduction of the radial basis functions (RBFs). The RBF multiquadric splines are used to construct the implicit level set function with a high level of accuracy and smoothness and to discretize the original initial value problem into an interpolation problem. The motion of the dynamic interfaces is thus governed by a system of coupled ordinary differential equations (ODEs) and a relatively smooth evolution can be maintained without reinitialization. A practical implementation of this method is further developed for solving a class of energy-based optimization problems, in which approximate solution to the original Hamilton-Jacobi equation may be justified and nucleation of new holes inside the material domain is allowed for. Furthermore, the severe constraints on the temporal and spatial discretizations can be relaxed, leading to a rapid convergence to the final design insensitive to initial guesses. The normal velocities are chosen to perform steepest gradient-based optimization by using shape sensitivity analysis and a bi-sectioning algorithm. A physically meaningful and efficient extension velocity method is also presented. The proposed method is implemented in the framework of minimum compliance design and its efficiency over the existing methods is highlighted. Numerical examples show its accuracy, convergence speed and insensitivity to initial designs in shape and topology optimization of two-dimensional (2D) problems that have been extensively investigated in the literature.

A Real Ghost Fluid Method for the Simulation of Multimedium Compressible Flow

In the previous ghost fluid methods (GFMs) developed, the focus is on the definition of ghost fluid states while the pressure and velocity in the real fluid sides are taken for granted, except for the correction made to the density at the real fluid nodes next to the interface to overcome the possible problems related to overheating. It has been found that such GFMs encounter many difficulties when applied to shock impedance matching (-like) problems due to the inability of accurately imposing interfacial conditions. By predicting the flow states for the real fluid nodes just next to the interface and the ghost fluid nodes using the Riemann problem solver, a more accurate interface boundary condition can be imposed and the said difficulties are mitigated to a large extent. This leads to the development of a proposed real-GFM in this work. A simple yet efficient extension of the present method to multidimensions is also introduced. In order to overcome issues associated with the severe bunching of level set contours due to the large flow velocity gradient, an extension (artificial) velocity field is constructed in the computation of the level set function. The present method is applied to various one- and two-dimensional problems involving strong shock-interface interaction and complex flow physics.

Experimental and numerical study of transient bubble-elastic membrane interaction

A study of the interaction between a membrane and a submerged oscillating bubble is presented. Though the behavior of such a bubble near an elastic (relatively thick) boundary has been studied by several authors, much less attention is focused on the behavior of such a bubble near a (thin) elastic membrane. For membranes, it is the curvature and not the deflection that is responsible for a pressure buildup in the fluid close to the bubble. Due to this difference in physics, it is not a certainty if the dynamics of bubbles near a deformable elastic boundary vis-a-vis a membrane would exhibit any similarity. Our intent is a systematic study on the latter, which can be exploited in future work (e.g., in biomedical applications where elastic membranes are often involved). Experimental observations of transient bubble interaction with a thin elastic membrane are presented and the dynamics of the bubble in the vicinity of the membrane are compared to the boundary element method simulations. The bubble is genera...

A numerical study for the performance of the Runge-Kutta discontinuous Galerkin method based on different numerical fluxes

Runge-Kutta discontinuous Galerkin (RKDG) method is a high order finite element method for solving hyperbolic conservation laws employing useful features from high resolution finite volume schemes, such as the exact or approximate Riemann solvers serving as numerical fluxes, TVD Runge-Kutta time discretizations, and limiters. In most of the RKDG papers in the literature, the Lax-Friedrichs numerical flux is used due to its simplicity, although there are many other numerical fluxes which could also be used. In this paper, we systematically investigate the performance of the RKDG method based on different numerical fluxes, including the first-order monotone fluxes such as the Godunov flux, the Engquist-Osher flux, etc., and second-order TVD fluxes, with the objective of obtaining better performance by choosing suitable numerical fluxes. The detailed numerical study is mainly performed for the one dimensional system case, addressing the issues of CPU cost, accuracy, non-oscillatory property, and resolution of discontinuities. Numerical tests are also performed for two dimensional systems.

Microchannel flows with superhydrophobic surfaces: Effects of Reynolds number and pattern width to channel height ratio

Superhydrophobic surfaces are widely adopted for reducing the flow resistance in microfluidic channels. The structures on the superhydrophobic surfaces may consist of longitudinal grooves, transverse grooves, posts, holes, etc. In this paper their effective slip performances are systematically studied and compared in detail through numerical simulations. The numerical results show that channel wall confinement effects have a positive influence on the effective slip length for square posts and longitudinal grooves, and a negative influence for square holes and transverse grooves. Square posts, holes, and transverse grooves all exhibit deteriorating effective slip performances at higher Reynolds numbers, while the effective slip performance of longitudinal grooves remains independent of the Reynolds number. For small pattern width to channel height ratios and at low Reynolds numbers, for low shear-free fractions, the effective slip length of square posts is equivalent of that of transverse grooves, and both...

Flow around spheres by dissipative particle dynamics

The dissipative particle dynamics (DPD) method is used to study the flow behavior past a sphere. The sphere is represented by frozen DPD particles while the surrounding fluids are modeled by simple DPD particles (representing a Newtonian fluid). For the surface of the sphere, the conventional model without special treatment and the model with specular reflection boundary condition proposed by Revenga et al. [Comput. Phys. Commun. 121–122, 309 (1999)] are compared. Various computational domains, in which the sphere is held stationary at the center, are investigated to gage the effects of periodic conditions and walls for Reynolds number (Re)=0.5 and 50. Two types of flow conditions, uniform flow and shear flow are considered, respectively, to study the drag force and torque acting on the stationary sphere. It is found that the calculated drag force imposed on the sphere based on the model with specular reflection is slightly lower than the conventional model without special treatment. With the conventional...

Interaction of lithotripter shockwaves with single inertial cavitation bubbles

The dynamic interaction of a shockwave (modelled as a pressure pulse) with an initially spherically oscillating bubble is investigated. Upon the shockwave impact, the bubble deforms non-spherically and the flow field surrounding the bubble is determined with potential flow theory using the boundary-element method (BEM). The primary advantage of this method is its computational efficiency. The simulation process is repeated until the two opposite sides of the bubble surface collide with each other (i.e. the formation of a jet along the shockwave propagation direction). The collapse time of the bubble, its shape and the velocity of the jet are calculated. Moreover, the impact pressure is estimated based on water-hammer pressure theory. The Kelvin impulse, kinetic energy and bubble displacement (all at the moment of jet impact) are also determined. Overall, the simulated results compare favourably with experimental observations of lithotripter shockwave interaction with single bubbles (using laser-induced bubbles at various oscillation stages). The simulations confirm the experimental observation that the most intense collapse, with the highest jet velocity and impact pressure, occurs for bubbles with intermediate size during the contraction phase when the collapse time of the bubble is approximately equal to the compressive pulse duration of the shock wave. Under this condition, the maximum amount of energy of the incident shockwave is transferred to the collapsing bubble. Further, the effect of the bubble contents (ideal gas with different initial pressures) and the initial conditions of the bubble (initially oscillating vs. non-oscillating) on the dynamics of the shockwave-bubble interaction are discussed.

An oscillating bubble near an elastic material

A method is presented to describe the behavior of an oscillating bubble in a fluid near a second elastic (biological) fluid. The elasticity of the second fluid is modeled through a pressure term at the interface between the two fluids. The Laplace equation is assumed to be valid in each of the fluids, and a difference in the respective densities is allowed. A relationship between the two velocity potentials just above and below the fluid-fluid interface can be found. The boundary integral method is then used to solve for the unknown normal velocities at both the bubble interface and fluid-fluid interface. These said normal velocities are subsequently utilized to update the position of the interface(s) for the next time step. For bubbles oscillating near a second nonelastic fluid, the bubbles can develop a jet towards or away from the fluid-fluid interface (depending on the distance of the bubble from the fluid-fluid interface and the density ratios of the two fluids). This behavior can be greatly modified...