A novel approach to rigid spheroid models in viscous flows using operator splitting methods

Calculating cost-effective solutions to particle dynamics in viscous flows is an important problem in many areas of industry and nature. We implement a second-order symmetric splitting method on the governing equations for a rigid spheroidal particle model with torques, drag and gravity. The method splits the operators into a vector field that is conservative and one that takes into account the forces of the fluid. Error analysis and numerical tests are performed on perturbed and stiff particle-fluid systems. For the perturbed case, the splitting method greatly improves the solution accuracy, when compared to a conventional multi-step method, and the global error behaves as $\mathcal{O}(\varepsilon h^2)$ for roughly equal computational cost. For stiff systems, we show that the splitting method retains stability in regimes where conventional methods blow up. In addition, we show through numerical experiments that the global order is reduced from $\mathcal{O}(h^2/\varepsilon)$ in the non-stiff regime to $\mathcal{O}(h)$ in the stiff regime.Comment: 24 pages, 6 figures (13 if you count sub figs), all figures are in colou

An integral model based on slender body theory, with applications to curved rigid fibers

We propose a novel integral model describing the motion of curved slender fibers in viscous flow, and develop a numerical method for simulating dynamics of rigid fibers. The model is derived from nonlocal slender body theory (SBT), which approximates flow near the fiber using singular solutions of the Stokes equations integrated along the fiber centerline. In contrast to other models based on (singular) SBT, our model yields a smooth integral kernel which incorporates the (possibly varying) fiber radius naturally. The integral operator is provably negative definite in a non-physical idealized geometry, as expected from PDE theory. This is numerically verified in physically relevant geometries. We propose a convergent numerical method for solving the integral equation and discuss its convergence and stability. The accuracy of the model and method is verified against known models for ellipsoids. Finally, a fast algorithm for computing dynamics of rigid fibers with complex geometries is developed

Finite difference method in prolate spheroidal coordinates for freely suspended spheroidal particles in linear flows of viscous and viscoelastic fluids

A finite difference scheme is used to develop a numerical method to solve the flow of an unbounded viscoelastic fluid with zero to moderate inertia around a prolate spheroidal particle. The equations are written in prolate spheroidal coordinates, and the shape of the particle is exactly resolved as one of the coordinate surfaces representing the inner boundary of the computational domain. As the prolate spheroidal grid is naturally clustered near the particle surface, good resolution is obtained in the regions where the gradients of relevant flow variables are most significant. This coordinate system also allows large domain sizes with a reasonable number of mesh points to simulate unbounded fluid around a particle. Changing the aspect ratio of the inner computational boundary enables simulations of different particle shapes ranging from a sphere to a slender fiber. Numerical studies of the latter particle shape allow testing of slender body theories. The mass and momentum equations are solved with a Schur complement approach allowing us to solve the zero inertia case necessary to isolate the viscoelastic effects. The singularities associated with the coordinate system are overcome using L'Hopital's rule. A straightforward imposition of conditions representing a time-varying combination of linear flows on the outer boundary allows us to study various flows with the same computational domain geometry. {For the special but important case of zero fluid and particle inertia we obtain a novel formulation that satisfies the force- and torque-free constraint in an iteration-free manner.} The numerical method is demonstrated for various flows of Newtonian and viscoelastic fluids around spheres and spheroids (including those with large aspect ratio). Good agreement is demonstrated with existing theoretical and numerical results.Comment: 32 pages, 12 figures. Accepted at Journal of Computational Physic

Institute for Computational Mechanics in Propulsion (ICOMP) fourth annual review, 1989

The Institute for Computational Mechanics in Propulsion (ICOMP) is operated jointly by Case Western Reserve University and the NASA Lewis Research Center. The purpose of ICOMP is to develop techniques to improve problem solving capabilities in all aspects of computational mechanics related to propulsion. The activities at ICOMP during 1989 are described

Modelling Fluid Structure Interaction problems using Boundary Element Method

This dissertation investigates the application of Boundary Element Methods (BEM) to Fluid Structure Interaction (FSI) problems under three main different perspectives. This work is divided in three main parts: i) the derivation of BEM for the Laplace equation and its application to analyze ship-wave interaction problems, ii) the imple- mentation of efficient and parallel BEM solvers addressing the newest challenges of High Performance Computing, iii) the developing of a BEM for the Stokes system and its application to study micro-swimmers.First we develop a BEM for the Laplace equation and we apply it to predict ship-wave interactions making use of an innovative coupling with Finite Element Method stabilization techniques. As well known, the wave pattern around a body depends on the Froude number associated to the flow. Thus, we throughly investigate the robustness and accuracy of the developed methodology assessing the solution dependence on such parameter. To improve the performance and tackle problems with higher number of unknowns, the BEM developed for the Laplace equation is parallelized using OpenSOURCE tech- nique in a hybrid distributed-shared memory environment. We perform several tests to demonstrate both the accuracy and the performance of the parallel BEM developed. In addition, we explore two different possibilities to reduce the overall computational cost from O(N2) to O(N). Firstly we couple the library with a Fast Multiple Method that allows us to reach for higher order of complexity and efficiency. Then we perform a preliminary study on the implementation of a parallel Non Uniform Fast Fourier Transform to be coupled with the newly developed algorithm Sparse Cardinal Sine De- composition (SCSD).Finally we consider the application of the BEM framework to a different kind of FSI problem represented by the Stokes flow of a liquid medium surrounding swimming micro-organisms. We maintain the parallel structure derived for the Laplace equation even in the Stokes setting. Our implementation is able to simulate both prokaryotic and eukaryotic organisms, matching literature and experimental benchmarks. We finally present a deep analysis of the importance of hydrodynamic interactions between the different parts of micro-swimmers in the prevision of optimal swimming conditions, focusing our attention on the study of flagellated \u201crobotic\u201d composite swimmers

Magnētiskā šķidruma pilieni rotējošā laukā: teorija, eksperimenti un simulācijas

Magnētiskā šķidruma pilieni ir reizē deformējami un ietekmējami ar ārējo magnētisko lauku, kas tos padara par interesantu materiālu, kas is atradis daudzus pielietojumus mikrofluīdikā. Šajā darbā tiek aplūkota šādu pilienu dinamika rotējošā magnētiskajā laukā. Pilieni tiek pētīti, izmantojot vairākas pieejas - teorētiski, eksperimentāli un izmantojot simulācijas. Kad rotējošais magnētiskais lauks lauks ir vājš un piliena deformācija ir maza, piliena kustība tiek aprēķināta analītiski. Konstatēts, ka piliena formas attīstību laikā regulē trīs nelineāru diferenciālvienādojumu sistēma. Mazo deformāciju tuvinājumā piliena uzvedību kvalitatīvi raksturo viens parametrs, kas ir proporcionāls kapilārajam skaitlim – viskozās berzes un virsmas spraiguma spēku attiecībai. Magnētisko pilienu eksperimentālie novērojumi, kas ir iegūti fāžatdalīta magnētiskā šķidruma paraugam, kvalitatīvi atbilst analītiskajam atrisinājumam, tomēr pastāv ievērojama kvantitatīva nesakritība. Lai aprēķinātu vidēji deformētu pilienu dinamiku, tiek izstrādāta simulācija, kas balstīta uz robeželementu metodēm. Tiek kostatēts, ka ir nepieciešama laba režģa uzturēšana, lai iegūtu precīzus simulācijas rezultātus. Tiek izveidota fāžu diagramma, kas parāda pilienu dinamiku atkarībā no rotējošā lauka intensitātes un frekvences. Visbeidzot, eksperimentāli tiek novērota magnētisko pilienu kolektīvā dinamika. Pie noteikta magnētiskā lauka stipruma un frekvences pilieni veido rotējošus ansambļus ar trīsstūrveida kārtību - divdimensiju rotējošus kristālus. Mazu ansambļu dinamika tiek reproducēta, modelējot pilienus ar punktveida spēka momentiem.Due to a combination of responsiveness to external magnetic fields and their deformability, magnetic fluid droplets make an interesting material that has found many applications in microfluidics. This work explores the dynamics of such droplets in a rotating magnetic field. The droplets are examined using multiple approaches – theoretically, experimentally and using simulations. When the rotating magnetic field is weak and the droplet’s deformation is small, the droplet’s motion is calculated analytically. It is found that the droplet’s shape evolution is governed by a system of three nonlinear differential equations. In the small deformation limit, the motion of the droplet is qualitatively governed by a parameter proportional to the capillary number – the ratio of viscous drag to surface tension forces. The experimental observation of magnetic droplets obtained by the separation of a ferrofluid in two liquid phases, qualitatively follows the analytic solution, however, there is a significant quantitative discrepancy. A simulation based on the boundary element methods is developed to calculate the dynamics of the droplets up to medium deformations. It is found that good mesh maintenance is required to produce accurate simulation results. A phase diagram is produced, which shows the droplet dynamics depending on the rotating field strength and frequency. Finally, the collective dynamics of the droplets is examined experimentally. For a certain magnetic field strength and frequency, the droplets form rotating ensembles with a triangular order – two dimensional rotating crystals. The dynamics of small ensembles is reproduced by treating the droplets as point torques.Magnētiskā šķidruma pilieni ir reizē deformējami un ietekmējami ar ārējo magnētisko lauku, kas tos padara par interesantu materiālu, kas is atradis daudzus pielietojumus mikrofluīdikā. Šajā darbā tiek aplūkota šādu pilienu dinamika rotējošā magnētiskajā laukā. Pilieni tiek pētīti, izmantojot vairākas pieejas - teorētiski, eksperimentāli un izmantojot simulācijas. Kad rotējošais magnētiskais lauks lauks ir vājš un piliena deformācija ir maza, piliena kustība tiek aprēķināta analītiski. Konstatēts, ka piliena formas attīstību laikā regulē trīs nelineāru diferenciālvienādojumu sistēma. Mazo deformāciju tuvinājumā piliena uzvedību kvalitatīvi raksturo viens parametrs, kas ir proporcionāls kapilārajam skaitlim – viskozās berzes un virsmas spraiguma spēku attiecībai. Magnētisko pilienu eksperimentālie novērojumi, kas ir iegūti fāžatdalīta magnētiskā šķidruma paraugam, kvalitatīvi atbilst analītiskajam atrisinājumam, tomēr pastāv ievērojama kvantitatīva nesakritība. Lai aprēķinātu vidēji deformētu pilienu dinamiku, tiek izstrādāta simulācija, kas balstīta uz robeželementu metodēm. Tiek kostatēts, ka ir nepieciešama laba režģa uzturēšana, lai iegūtu precīzus simulācijas rezultātus. Tiek izveidota fāžu diagramma, kas parāda pilienu dinamiku atkarībā no rotējošā lauka intensitātes un frekvences. Visbeidzot, eksperimentāli tiek novērota magnētisko pilienu kolektīvā dinamika. Pie noteikta magnētiskā lauka stipruma un frekvences pilieni veido rotējošus ansambļus ar trīsstūrveida kārtību - divdimensiju rotējošus kristālus. Mazu ansambļu dinamika tiek reproducēta, modelējot pilienus ar punktveida spēka momentiem.Sponsors: Scholarship from the French Embassy in Riga, Latvia; Scholarship from SIA "Mikrotīks"; Funding from 8.2.2.0/20/I/006; Funding from lzp-2020/1-014

Complex dynamics of solid-fluid systems

The focus of this thesis was the investigation of the complex dynamics of solid-fluid systems. These systems are of great industrial importance, such as in methane clathrate formation in sub-sea pipelines. As well as being crucial to furthering our understanding of various natural phenomena, such as the rate of rain droplet formation in clouds. We began by considering the problem of the orbits tracked by ellipsoids immersed in viscous and inviscid environments. This investigation was carried out by a combination of analytical and numerical techniques: direct numerical simulations of resolved full-coupled solid-fluid systems, analysis the Kirchhoff-Clebsch equations for the case of inviscid flows, and characterising dynamics through advanced techniques such as recurrence quantification analysis. We demonstrate that the ellipsoid tracks a chaotic orbit not only in an inviscid environment but also when submerged in a viscous fluid, under specific conditions. Under inviscid environments, an ellipsoid subject to arbitrary initial conditions of linear and angular momentum demonstrates chaotic orbits when all the three axes of the ellipsoid are unequal, in agreement with the Kozlov and Onishchenko’s theorem of non-integrability of Kirchhoff’s equations and also with Aref and Jones’s potential flow solution. We then extended our methodology to understand the dynamics of a single ellipsoid tumbling in a viscous environment with the presence of both passive and viscosity coupled tracers in addition to the chaotic dynamics predicted by the Kirchhoff-Clebsch equations. Our results show that the bodies move along from viscosity gradients towards minima of the viscous stress. These bodies might become trapped in unstable minima. However, more work is needed to understand the long-term mixing of viscosity coupled tracers. Our direct numerical solver was also extended to include contact models for solid-solid interactions in the simulation domain. The validation of the contact models was presented. Finally, we expand, the theoretical framework of the Kirchhoff-Clebsch equations to account for the presence of multiple bodies. This extension was done by using Hamiltonian mechanics to extend the derivation proposed by Lamb. We present our preliminary result of simulating two solids systems using the extended Kirchhoff-Clebsch equations. The rel- ative orientations of the two solids were found to regularly switch from being correlated to anti-correlated in an otherwise chaotic system. Further work is required to understand the mechanism behind this behaviour

Fast, High-Order Accurate Integral Equation Methods and Application to PDE-Constrained Optimization

Over the last several decades, the development of fast, high-order accurate, and robust integral equation methods for computational physics has gained increasing attention. Using integral equation formulation as a global statement in contrast to a local partial differential equation (PDE) formulation offers several unique advantages. For homogeneous PDEs, the boundary integral equation (BIE) formulation allows accurate handling of complex and moving geometries, and it only requires a mesh on the boundary, which is much easier to generate as a result of the dimension reduction. With the acceleration of fast algorithms like the Fast Multipole Method (FMM), the computational complexity can be reduced to O(N), where N is the number of degrees of freedom on the boundary. Using standard potential theory decomposition, inhomogeneous PDEs can be solved by evaluating a volume potential over the inhomogeneous source domain, followed by a solution of the homogeneous part. Despite the advantages of BIE methods in easy meshing, near-optimal efficiency, and well conditioning, the accurate evaluation of nearly singular integrals is a classical problem that needs to be addressed to enable simulations for practical applications. In the first half of this thesis, we develop a series of product integration schemes to solve this close evaluation problem. The use of differential forms provides a dimensional-agnostic way of integrating the nearly singular kernels against polynomial basis functions analytically. So the problem of singular integration gets reduced to a matter of source function approximation. In 2D, this procedure has been traditionally portrayed by building a connection to complex Cauchy integral, then supplemented by a complex monomial approximation. In $3$D, the closed differential form requirement leads to the design of a new function approximation scheme based on harmonic polynomials and quaternion algebra. Under a similar framework, we develop a high-order accurate product integration scheme for evaluating singular and nearly singular volume integral equations (VIE) in complex domains using regular Cartesian grids discretization. A high-order accurate source term approximation scheme matching smooth volume integrals on irregular cut cells is developed, which requires no function extension. BIE methods have been widely used for studying Stokes flows, incompressible flows at low Reynolds' number, in both biological systems and microfluidics. In the second half of this thesis, we employ the BIE methods to simulate and optimize Stokes fluid-structure interactions. In 2D, a hybrid computational method is presented for simulating cilia-generated fluid mixing as well as the cilia-particle hydrodynamics. The method is based on a BIE formulation for confining geometries and rigid particles, and the method of regularized Stokeslets for the cilia. In 3D, we use the time-independent envelop model for arbitrary axisymmetric microswimmers to minimize the power loss while maintaining a target swimming speed. This is a quadratic optimization problem in terms of the slip velocity due to the linearity of Stokes flow. Under specified reduced volume constraint, we find prolate spheroids to be the most efficient micro-swimmer among various families of shapes we considered. We then derive an adjoint-based formulation for computing power loss sensitivities in terms of a time-dependent slip profile by introducing an auxiliary time-periodic function, and find that the optimal swimmer displays one or multiple traveling waves, reminiscent of the typical metachronal waves observed in ciliated microswimmers.PHDApplied and Interdisciplinary MathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/169695/1/hszhu_1.pd