Deformable ellipsoidal bubbles in Taylor-Couette flow with enhanced Euler-Lagrange tracking

In this work we present numerical simulations of $10^5$ sub-Kolmogorov deformable bubbles dispersed in Taylor-Couette flow (a wall-bounded shear system) with rotating inner cylinder and outer cylinder at rest. We study the effect of deformability of the bubbles on the overall drag induced by the carrier fluid in the two-phase system. We find that an increase in deformability of the bubbles results in enhanced drag reduction due to a more pronounced accumulation of the deformed bubbles near the driving inner wall. This preferential accumulation is induced by an increase in the resistance on the motion of the bubbles in the wall-normal direction. The increased resistance is linked to the strong deformation of the bubbles near the wall which makes them prolate (stretched along one axes) and orient along the stream-wise direction. A larger concentration of the bubbles near the driving wall implies that they are more effective in weakening the plume ejections which results in stronger drag reduction effects. These simulations which are practically impossible with fully resolved techniques are made possible by coupling a sub-grid deformation model with two-way coupled Euler-Lagrangian tracking of sub-Kolmogorov bubbles dispersed in a turbulent flow field which is solved through direct numerical simulations. The bubbles are considered to be ellipsoidal in shape and their deformation is governed by an evolution equation which depends on the local flow conditions and their surface tension

The motion of a deforming capsule through a corner

A three-dimensional deformable capsule convected through a square duct with a corner is studied via numerical simulations. We develop an accelerated boundary integral implementation adapted to general geometries and boundary conditions. A global spectral method is adopted to resolve the dynamics of the capsule membrane developing elastic tension according to the neo-Hookean constitutive law and bending moments in an inertialess flow. The simulations show that the trajectory of the capsule closely follows the underlying streamlines independently of the capillary number. The membrane deformability, on the other hand, significantly influences the relative area variations, the advection velocity and the principal tensions observed during the capsule motion. The evolution of the capsule velocity displays a loss of the time-reversal symmetry of Stokes flow due to the elasticity of the membrane. The velocity decreases while the capsule is approaching the corner as the background flow does, reaches a minimum at the corner and displays an overshoot past the corner due to the streamwise elongation induced by the flow acceleration in the downstream branch. This velocity overshoot increases with confinement while the maxima of the major principal tension increase linearly with the inverse of the duct width. Finally, the deformation and tension of the capsule are shown to decrease in a curved corner

IMMERSED BOUNDARY CONDITIONS METHOD FOR COMPUTATIONAL FLUID DYNAMICS PROBLEMS

This dissertation presents implicit spectrally-accurate algorithms based on the concept of immersed boundary conditions (IBC) for solving a range of computational fluid dynamics (CFD) problems where the physical domains involve boundary irregularities. Both fixed and moving irregularities are considered with particular emphasis placed on the two-dimensional moving boundary problems. The physical model problems considered are comprised of the Laplace operator, the biharmonic operator and the Navier-Stokes equations, and thus cover the most commonly encountered types of operators in CFD analyses. The IBC algorithm uses a fixed and regular computational domain with flow domain immersed inside the computational domain. Boundary conditions along the edges of the time-dependent flow domain enter the algorithm in the form of internal constraints. Spectral spatial discretization for two-dimensional problems is based on Fourier expansions in the stream-wise direction and Chebyshev expansions in the normal-to-the-wall direction. Up to fourth-order implicit temporal discretization methods have been implemented. The IBC algorithm is shown to deliver the theoretically predicted accuracy in both time and space. Construction of the boundary constraints in the IBC algorithm provides degrees of freedom in excess of that required to formulate a closed system of algebraic equations. The ‘classical BBC formulation’ works by retaining number boundary constraints that are just sufficient to form a closed system of equations. The use of additional boundary constraints leads to the ‘over-determined formulation’ of die IBC algorithm. Over- determined systems are explored in order to improve the accuracy of the IBC method and to expand its applicability to more extreme geometries. Standard direct over-determined solvers based on evaluation of pseudo-inverses of the complete coefficient matrices have been tested on three model problems, namely, the Laplace equation, the biharmonic equation and the Navier-Stokes equations. In all cases tested the over-determined m formulations based on standard solvers were found to improve the accuracy and the range o f applicability o f the IBC method. Efficient linear solvers suitable for the spectral implementation of the IBC method have been developed and tested in the context of two-dimensional steady and unsteady Stokes flow in the presence of fixed boundary irregularities. These solvers can work with the classical as well as the over-determined formulations of the method. Significant acceleration of the computations as well as significant reduction of the memory requirements have been accomplished by taking advantage of the structure of the coefficient matrix resulting from the implementation of the IBC algorithm. Performances o f the new solvers have been compared with the standard direct solvers and are shown to be of up to two orders of magnitude better. It has been determined that the new methods are by at least an order of magnitude faster than the iterative methods while removing restrictions based on the convergence criteria and thus expanding the severity of the geometries that can be dealt with using the IBC algorithm. The performance of the IBC method combined with the new solvers has been compared with the performance of a method based on the generation of the boundary conforming grids, and is found to be better by at least two orders of magnitude. Application of the new solvers to the unsteady problems also results in performance improvement of up to two orders of magnitude. The specialized solvers applied to the over-determined formulation is shown to be at least two orders of magnitude faster than their standard counterparts while capable of extending the range of applicability of the IBC algorithm by 50%-70% for the Stokes flow problem. The concept of the specialized solvers has been extended to solve two-dimensional moving boundary problems described by the Navier-Stokes equations, where the new solver has been shown to result in a significant acceleration of computations as well as substantial reduction in memory requirements. The conceptual aspects of extending the IBC algorithm for solving three-dimensional problems have been presented using the vorticity-velocity formulation of the three- dimensional Navier-Stokes equations. Test results on the implementation of the IBC algorithm for three-dimensional problems are discussed in the context of heat diffusion IV problems in the presence of fixed as well as moving boundaries. The algorithm is shown to be spectrally-accurate in space and capable of delivering theoretically predicted accuracy in time for the different test problems. Given a potentially large size of the resultant linear algebraic system, various methods that take advantage of the special structure of the coefficient matrix have been explored in search for an efficient solver, including two versions of the specialized direct solver as well as serial and parallel iterative solvers. Both versions of the specialized direct solver have been shown to be more computationally efficient than the other solution methods. Possible applications of the IBC algorithm for analyzing physical problems have also been presented. The advantage of using IBC algorithm is illustrated by considering its application to two physical problems, which are - i) analysis of the effects of distributed roughness on friction factor and ii) analysis of traveling wave instability in wavy channels. These examples clearly show the attractiveness of the IBC algorithm for studying effects of a large array of boundary geometries on the flow field

IMMERSED BOUNDARY CONDITIONS METHOD FOR COMPUTATIONAL FLUID DYNAMICS PROBLEMS

This dissertation presents implicit spectrally-accurate algorithms based on the concept of immersed boundary conditions (IBC) for solving a range of computational fluid dynamics (CFD) problems where the physical domains involve boundary irregularities. Both fixed and moving irregularities are considered with particular emphasis placed on the two-dimensional moving boundary problems. The physical model problems considered are comprised of the Laplace operator, the biharmonic operator and the Navier-Stokes equations, and thus cover the most commonly encountered types of operators in CFD analyses. The IBC algorithm uses a fixed and regular computational domain with flow domain immersed inside the computational domain. Boundary conditions along the edges of the time-dependent flow domain enter the algorithm in the form of internal constraints. Spectral spatial discretization for two-dimensional problems is based on Fourier expansions in the stream-wise direction and Chebyshev expansions in the normal-to-the-wall direction. Up to fourth-order implicit temporal discretization methods have been implemented. The IBC algorithm is shown to deliver the theoretically predicted accuracy in both time and space. Construction of the boundary constraints in the IBC algorithm provides degrees of freedom in excess of that required to formulate a closed system of algebraic equations. The ‘classical IBC formulation’ works by retaining number boundary constraints that are just sufficient to form a closed system of equations. The use of additional boundary constraints leads to the ‘over-determined formulation’ of the IBC algorithm. Over- determined systems are explored in order to improve the accuracy o f the IBC method and to expand its applicability to more extreme geometries. Standard direct over-determined solvers based on evaluation of pseudo-inverses of the complete coefficient matrices have been tested on three model problems, namely, the Laplace equation, the biharmonic equation and the Navier-Stokes equations. In all cases tested the over-determined formulations based on standard solvers were found to improve the accuracy and the range o f applicability o f the IBC method. Efficient linear solvers suitable for the spectral implementation of the IBC method have been developed and tested in the context of two-dimensional steady and unsteady Stokes flow in the presence of fixed boundary irregularities. These solvers can work with the classical as well as the over-determined formulations of the method. Significant acceleration of the computations as well as significant reduction of the memory requirements have been accomplished by taking advantage of the structure of the coefficient matrix resulting from the implementation of the IBC algorithm. Performances of the new solvers have been compared with the standard direct solvers and are shown to be of up to two orders of magnitude better. It has been determined that the new methods are by at least an order of magnitude faster than the iterative methods while removing restrictions based on the convergence criteria and thus expanding the severity of the geometries that can be dealt with using the IBC algorithm. The performance of the IBC method combined with the new solvers has been compared with the performance of a method based on the generation of the boundary conforming grids, and is found to be better by at least two orders of magnitude. Application of the new solvers to the unsteady problems also results in performance improvement of up to two orders of magnitude. The specialized solvers applied to the over-determined formulation is shown to be at least two orders of magnitude faster than their standard counterparts while capable of extending the range of applicability of the IBC algorithm by 50%-70% for the Stokes flow problem. The concept of the specialized solvers has been extended to solve two-dimensional moving boundary problems described by the Navier-Stokes equations, where the new solver has been shown to result in a significant acceleration of computations as well as substantial reduction in memory requirements. The conceptual aspects of extending the IBC algorithm for solving three-dimensional problems have been presented using the vorticity-velocity formulation of the three- dimensional Navier-Stokes equations. Test results on the implementation of the IBC algorithm for three-dimensional problems are discussed in the context of heat diffusion IV problems in the presence of fixed as well as moving boundaries. The algorithm is shown to be spectrally-accurate in space and capable of delivering theoretically predicted accuracy in time for the different test problems. Given a potentially large size of the resultant linear algebraic system, various methods that take advantage of the special structure of the coefficient matrix have been explored in search for an efficient solver, including two versions of the specialized direct solver as well as serial and parallel iterative solvers. Both versions of the specialized direct solver have been shown to be more computationally efficient than the other solution methods. Possible applications of the IBC algorithm for analyzing physical problems have also been presented. The advantage of using IBC algorithm is illustrated by considering its application to two physical problems, which are - i) analysis of the effects of distributed roughness on friction factor and ii) analysis of traveling wave instability in wavy channels. These examples clearly show the attractiveness of the IBC algorithm for studying effects of a large array of boundary geometries on the flow field

Direct numerical simulation of turbulent flow over a rough surface based on a surface scan

Typical engineering rough surfaces show only limited resemblance to the artificially constructed rough surfaces that have been the basis of most previous fundamental research on turbulent flow over rough walls. In this article flow past an irregular rough surface is investigated, based on a scan of a rough graphite surface that serves as a typical example for an irregular rough surface found in engineering applications. The scanned map of surface height versus lateral coordinates is filtered in Fourier space to remove features on very small scales and to create a smoothly varying periodic representation of the surface. The surface is used as a no-slip boundary in direct numerical simulations of turbulent channel flow. For the resolution of the irregular boundary an iterative embedded boundary method is employed. The effects of the surface filtering on the turbulent flow are investigated by studying a series of surfaces with decreasing level of filtering. Mean flow, Reynolds stress and dispersive stress profiles show good agreement once a sufficiently large number of Fourier modes are retained. However, significant differences are observed if only the largest surface features are resolved. Strongly filtered surfaces give rise to a higher mean-flow velocity and to a higher variation of the streamwise velocity in the roughness layer compared with weakly filtered surfaces. In contrast, for the weakly filtered surfaces the mean flow is reversed over most of the lower part of the roughness sublayer and higher levels of dispersive shear stress are found

Flows in Vibrating Channels

A spectral algorithm based on the immersed boundary conditions (IBC) concept has been developed for the analysis of flows in channels bounded by vibrating walls. The vibrations take the form of travelling waves of arbitrary profile. The algorithm uses a fixed computational domain with the flow domain immersed in its interior. Boundary conditions enter the algorithm in the form of constraints. The spatial discretization uses a Fourier expansion in the stream-wise direction and a Chebyshev expansion in the wall-normal direction. Use of the Galileo transformation converts the unsteady problem into a steady one. An efficient solver which takes advantage of the structure of the coefficient matrix has been used. It is demonstrated that the method can be extended to more extreme geometries using the over-determined formulation. Various tests confirm the spectral accuracy of the algorithm. Pressure losses in these types of channels have been analyzed. Mechanisms of drag generation have been studied. Analytical solutions have been determined in the limit of long wavelength waves and small amplitude waves in order to simplify identification of these mechanisms. The numerical algorithm has also been validated with the help of analytical solutions. Detailed analyses of different cases, i.e. wave propagation along one wall and both walls have been carried out. Different wave profiles have been considered in order to find forms of waves which minimize pressure losses in vibrating channels. The results show dependence of the pressure losses on the phase speed of the waves, with the waves propagating in the downstream direction reducing the pressure gradient required to maintain a fixed flow rate. A drag increase is observed when the waves propagate with a phase speed similar to the flow velocity

Convection in Corrugated Slots

This thesis consists of two parts. The first part deals with the development of proper methodology, i.e. a spectrally accurate algorithm suitable for analysis of convection problems in corrugated slots. The second part is devoted to the study of natural convection in corrugated slots. The algorithm uses the immersed boundary conditions (IBC) concept to deal with the irregular form of the solution domain associated with the presence of corrugated plates. The field equations are discretized on a regular domain surrounding the flow domain using Fourier expansions in the horizontal direction and Chebyshev expansions in the vertical direction. The boundary conditions are expressed in the form of constraints and the spectrally accurate discretization of these constraints has been proposed. The buoyancy forces associated with the temperature difference between isothermal plates drive the natural convection. This temperature difference is expressed in terms of the Rayleigh number Ra and the analysis is limited to its subcritical values where no secondary motion takes place in the absence of corrugation. Corrugations have a sinusoidal form characterized by the wave number a, the upper and lower amplitudes and the phase difference between the upper and lower corrugation systems. They create horizontal temperature gradients which lead to the formation of vertical and horizontal pressure gradients which drive the motion regardless of the intensity of the heating. Presence of corrugations affects the conductive heat flow and creates the convective heat flow. The increase of the heat flow induced by the corrugations has been determined. The convection is qualitatively similar for all Prandtl numbers with the intensity of convection increasing for smaller Pr’s and with the heat transfer augmentation increasing for larger Pr’s

Flow Control Using Traveling Waves

Flow separation is the detachment of boundary layer from a surface, which is associated with aerodynamic loss. In this work, the feasibility of controlling flow separation by backward (toward downstream) traveling waves is studied using large-eddy simulations (LES) of traveling waves (1) within an incompressible turbulent channel to investigate the impact of traveling wave parameters such as wave speed and wave steepness, (2) on an inclined plate and suction side of a NACA0018 airfoil at stall angle of attack where the flow is massively separated, and (3) within a compressible wavy turbulent channel. For (3), an LES framework for compressible flow is developed and combined with the curvilinear immersed boundary (CURVIB) method. Both incompressible and compressible frameworks are validated. The incompressible framework is validated for a fully developed turbulent channel, a pitching airfoil, two-dimensional inclined plate, and an NREL PHASE VI wind turbine. The compressible framework is validated by performing simulations for isotropic decay, subsonic and supersonic turbulent channel, and shock diffraction by a cylinder. The results of the simulations of actuated airfoil and plate reveal that low-amplitude backward traveling waves can postpone stall. Moreover, it is found for the first time that traveling waves are more effective than other types of oscillations, e.g., standing waves and pitching motion, in delaying stall because traveling waves can directly increase the axial momentum of the fluid in addition to triggering boundary layer instabilities, which occurs in all type of flow control with periodic excitation. In addition to the axial momentum, the traveling waves increase the lateral velocity of the fluid near the surface, which tends to separate the flow. The scalings of axial force and lateral velocity, which depend on amplitude, wavelength and frequency, were derived analytically using elongated body theory (EBT). Based on the scalings and the results, in contrast to common belief, wave speed (the main parameter for the axial force) is not the only parameter for flow reattachment, and amplitude, wavelength and frequency individually can impact flow separation by triggering instabilities or increasing the lateral velocity