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This volume presents the emerging applications of immersed boundary (IB) methods in computational mechanics and complex CFD calculations. It discusses formulations of different IB implementations and also demonstrates applications of these methods in a wide range of problems. It will be of special value to researchers and engineers as well as graduate students working on immersed boundary methods, specifically on recent developments and applications. The book can also be used as a supplementary textbook in advanced courses in computational fluid dynamics.



Incompressible Flow Modeling


Chapter 1. Immersed Boundary Projection Methods

Immersed boundary methods are an attractive alternative to body-fitted grids for complex geometries and fluid–structure interaction problems. The simplicity of the underlying Cartesian mesh allows for a number of useful conservation and stability properties to be embedded in the numerics, and for the resulting discrete equations to be solved efficiently and scalably. We review the immersed boundary projection method for incompressible flows, which implicitly satisfies the no-slip condition at immersed surfaces by solving a system of algebraic equations for surface traction. We discuss issues related to the smoothness of the surface stresses and solution strategies for strongly coupled fluid–structure interaction. For three-dimensional flows on unbounded domains, we discuss a fast lattice Green’s function method that provides for an adaptive domain comprising the vortical flow region and at the same time can be solved efficiently using extensions of the fast multipole method. To illustrate the methods, we present a series of benchmark simulations in two and three dimensions, ranging from inverted flag flutter, flow past spinning and inclined disks, and turbulent flow past a sphere.
Benedikt Dorschner, Tim Colonius

Chapter 2. Direct Lagrangian Forcing Methods Based on Moving Least Squares

The application of computational fluid dynamics to complex engineering flow problems necessitates the adoption of numerical algorithms that are accurate, robust and efficient at the same time. These are usually conflicting requirements and there is still debate in the scientific community when it comes to the selection of the best method for a specific field of applications. In this chapter, we will discuss a cost-efficient strategy to simulate fluid flow problems in complex configurations with large boundary displacements and/or deformations. It utilizes a direct forcing, immersed boundary formulation, where the body is represented by a Lagrangian grid, and the equations governing the fluid flow are solved on a structured or block-structured Cartesian grid. The forcing function is evaluated using moving least squares. The main advantage of this strategy compared to other direct-forcing schemes is versatility, as it decouples the local discretization from the computation of the forcing function and, therefore, can be implemented into structured or unstructured codes in a straightforward manner. In addition, it is very robust in dealing with collisions of multiple bodies. The forcing function is built and appropriately scaled based on the contributions of all bodies in the vicinity of an Eulerian point without special treatments.
Marcos Vanella, Elias Balaras

Chapter 3. Mass Conservation in Sharp Interface Immersed Boundary Method—A GPGPU Accelerated Implementation

A sharp interface immersed boundary method uses a local velocity reconstruction scheme to satisfy the boundary conditions over the complex/moving boundary. However, mass conservation is not physically ensured at the intercepted cells through this reconstruction schemes. Further, the solution can be marred with unphysical spurious oscillations in the temporal behavior of pressure when dealing with the moving boundary problems. In this chapter, a detailed discussion of a coupled MAC-SOLA solver, a simple yet effective and robust method to address the issues of improper mass conservation in IBM, is presented. Implementation of IBM to solve moving boundary problems and the acceleration and GPGPU optimization of solver using OpenACC are also discussed in detail. Further, studies on the accuracy and computational efficiency are presented.
Manish Kumar, Apurva Raj, Somnath Roy

Chapter 4. Coupling the Curvilinear Immersed Boundary Method with Rotation-Free Finite Elements for Simulating Fluid–Structure Interaction: Concepts and Applications

The sharp interface curvilinear immersed boundary (CURVIB) method coupled with a rotation-free finite element (FE) method for thin shells provides a powerful framework for simulating fluid–structure interaction (FSI) problems for geometrically complex, arbitrarily deformable structures. The CURVIB and FE solvers are coupled together on the flexible solid–fluid interfaces, which contain the structural nodal positions, displacements, velocities, and loads calculated at each time level and exchanged between the flow and structural solvers. Loose and strong coupling FSI schemes are employed, enhanced by the Aitken acceleration technique to ensure robust coupling and fast convergence, especially for low mass ratio problems. Large-eddy simulation (LES) of turbulent flow FSI problems employ the dynamic Smagorinsky subgrid scale model with a wall model for reconstructing velocity boundary conditions near the immersed boundaries. In this chapter, the CURVIB-FE FSI algorithm is reviewed and its capabilities are demonstrated via a series of examples involving thin flexible structures undergoing very large deformations. The inverted flag problem is employed to validate the method, and the problem of a tri-leaflet aortic valve in an anatomic aorta is employed to demonstrate its potential in complex cardiovascular flow applications.
Anvar Gilmanov, Henryk Stolarski, Fotis Sotiropoulos

Chapter 5. Handling Slender/Thin Geometries with Sharp Edges in Sharp Interface Immersed Boundary Approach

Immersed boundary framework offers a viable alternative to traditional body conformal computational models in simulating flow past arbitrarily complex geometries. Over the years, various immersed boundary formulations have been proposed which can be broadly classified into two approaches, namely ‘diffuse’ and ‘sharp’ interface. As the name suggests, diffuse interface impose boundary conditions not at their exact interface boundary but within a localized region around them. Sharp interface on the other hand overcomes this either by reconstructing the solution field near the surface (as in ghost cell method) such that the boundary conditions are imposed exactly at the surface or by reconstituting the boundary intercept cells into non-rectangular control volumes (cut cells) on which conservation laws are ensured. A major issue with these classes of approaches is its inconsistent handling of sharp edges in slender/thin bodies (as in flat plate, trailing edge of an airfoil). With the cut cell approach, sharp edges lead to the generation of arbitrarily small cells, complex cut cell topologies. Enforcing conservation laws for such cells give rise to numerical stability issues. In case of solution reconstruction algorithms, the nature of difficulty in handling sharp edges is twofold: one, due to inconsistent tagging of Eulerian nodes. Another difficulty is an insufficient number of points to maintain the order of accuracy of flux calculations. Several strategies have been proposed to address these issues. In case of cut cells, cell merging, cell clustering and hybrid of ghost cell and cut cell algorithms are proposed to maintain mass conservation. In case of solution reconstruction schemes, approaches like adaptive mesh refinements (AMR) are proposed. These strategies are highly complex both in terms of their formulation and implementation. This article presents a simple and robust set of procedures for efficiently handling the sharp edges. Capability of the algorithm is demonstrated successfully through a case study of dynamic stall in oscillating airfoil.
Pradeep Kumar Seshadri, Ashoke De

Chapter 6. Ghost Fluid Lattice Boltzmann Methods for Complex Geometries

Lattice Boltzmann method (LBM) is a widely recognized alternate numerical approach to simulate flow dynamics. In the conventional approach, the Navier–Stokes equations (conservation of mass, momentum and energy) are solved to obtain velocity, pressure and temperature fields. In LBM, on the other hand, the reduced version of the microscopic Boltzmann kinetic equations is solved numerically to determine particle distribution functions, which are then averaged to obtain macroscopic variables. Accurate enforcement of non-conforming and/or moving boundary conditions has been a challenge for LBM because the primary solution variables (particle distribution functions) are not the macroscopic variables on which boundary conditions are typically imposed. While macroscopic variables can be obtained from the particle distribution functions by weighted averaging, the reverse process is not as straightforward. Several researchers have developed strategies to accurately enforce boundary conditions. In this chapter, we first present an overview of the boundary issues in LBM and various approaches that have been developed to resolve them. We then discuss the implementation of immersed boundary method (IBM) on LBM, with particular emphasis on the ghost fluid approach. This technique based on ghost cells was first introduced to LBM by Tiwari and Vanka (Int J Numer Methods Fluids 69(2):481–498, 2012), who developed the so-called ghost fluid immersed boundary lattice Boltzmann method (GF-IB-LBM) based on extrapolation of particle distribution functions. The method is simple and efficient and is applicable to general boundary conditions. Another key advantage is the strict imposition of hydrodynamic conditions at the boundaries. Additionally, the method is local, thus maintains high parallelism of LBM. The application of this approach on several problems is discussed here, including its parallelization using graphical processing units (GPUs), and its coupling with molecular dynamics (MD) simulations. The chapter ends with a brief discussion on recent advances in this approach.
Arpit Tiwari, Daniel D. Marsh, Surya P. Vanka

Compressible Flow Modeling


Chapter 7. A Levelset-Based Sharp-Interface Modified Ghost Fluid Method for High-Speed Multiphase Flows and Multi-Material Hypervelocity Impact

Multi-material compressible flows are ubiquitous in industrial, defense, and environmental applications. The widely varied range of applications encompasses high-speed multiphase combustion in solid-propellant rocket motors, high-speed manufacturing processes, target–munition interactions, geological impact, and many more. Major challenges in modeling such systems lie in preserving the material interface definition while simultaneously satisfying the interfacial boundary conditions accurately. In this chapter, a levelset-based sharp-interface method is described for general classes of multi-material compressible flows problems. In this framework, the interface definition is retained sharply, while interfacial jump conditions are prescribed through a high-accuracy modified ghost fluid method. The application of the framework is demonstrated for several multi-material flow problems such as shock–particle interactions, shock–droplet interaction, shock-induced void collapse in energetic materials, and shock compaction of metallic powders. The chapter will describe the levelset-based framework for high-speed multiphase flows and shows its application to a broad spectrum of problems of engineering interest.
Pratik Das, Nirmal K. Rai, H. S. Udaykumar

Chapter 8. Development and Application of Immersed Boundary Methods for Compressible Flows

In this chapter, we discuss the development of a sharp-interface immersed boundary method for compressible, turbulent flows and its application to transonic/supersonic flows. A direct-forcing-type immersed boundary method is elucidated wherein the flow properties in the immediate neighbourhood of the immersed surface are reconstructed using inverse distance-based interpolation procedures. The flow is assumed to be locally parallel to the immersed surface, and the tangential velocity in the vicinity of the immersed surface is assumed to obey a power-law function of the local immersed surface normal. This approach helps in mimicking the energising effects of turbulent boundary layers without excessive mesh refinement near the immersed surface for suitable choices of the power-law coefficient. Temperature reconstruction is achieved from considerations of temperature variation in compressible thermal boundary layers, and density is estimated by either solving the continuity equation or by interpolation. The turbulence variables are reconstructed using law-of-the-wall-type approach. The application of the outlined immersed boundary method to the simulation of flow control devices is also discussed. Additionally, interpolation procedures for reconstructing the pressure and shear stress at the immersed surface and its application to simple cases are also presented. This information can be useful for comparison with experimental data, performing fluid–structure interaction studies, and also identifying flow-separation and re-attachment locations on the immersed surface.
Santanu Ghosh, Anand Bharadwaj S

Chapter 9. A Sharp-Interface Immersed Boundary Method for High-Speed Compressible Flows

Numerical simulation of hypersonic flows is important owing to their applications to the design of launch vehicles and re-entry capsules. We describe the development and application of a hybrid Cartesian–immersed boundary (HCIB) approach in a finite volume (FV) framework for inviscid and viscous compressible flows with a focus on the hypersonic regime. The HCIB approach employs a local one-dimensional reconstruction to obtain the near-wall solutions thereby enforcing the boundary conditions exactly on the sharp geometric interface. The role of the reconstruction on solution accuracy and discrete conservation is discussed, and the IB-FV solver is applied to several hypersonic flow problems involving inviscid as well as viscous flows. The studies demonstrate the efficacy of the HCIB approach for inviscid flows with stationary as well as moving bodies while also highlighting some of the drawbacks for heat transfer predictions in high Reynolds number flows. Some directions for future research are also outlined.
Shuvayan Brahmachary, Ganesh Natarajan, Vinayak Kulkarni, Niranjan Sahoo

Chapter 10. A Higher-Order Cut-Cell Methodology for Large Eddy Simulation of Compressible Viscous Flow Problems with Embedded Boundaries

We have developed a finite volume-based conservative cut-cell method that is up to third-order accurate for simulation of compressible viscous flow problems. A sharp representation of the embedded boundaries, described using a signed distance function, is facilitated by use of block-structured adaptive mesh refinement. A high-order reconstruction is performed by using a cell-centered piecewise polynomial approximation of flow quantities. To ensure the stability of the scheme in the presence of very low volume cut-cells, a novel cell clustering approach that preserves the design order of accuracy even locally has been developed. It is shown through numerical examples that using the proposed approach, smooth representation of flow-field quantities and their derivatives can be achieved on embedded boundaries. Smooth reconstruction of wall shear stress and a high-order accuracy makes this approach a good candidate for large eddy simulation (LES). A multi-level extension of the one-equation kinetic subgrid energy-based closure to perform LES with local refinement and embedded boundaries is presented. Results are shown for various canonical cases to demonstrate the accuracy of the approach.
Balaji Muralidharan, Suresh Menon



Chapter 11. Recent Developments on Employing Sharp-Interface Immersed Boundary Method for Simulating Fluid–Structure Interaction Problems

The objective of this brief review is to showcase recent applications of sharp-interface immersed boundary (IB) method to computationally challenging problems in fluid–structure interaction (FSI). The sharp-interface IB method is briefly reviewed and a scheme to strongly couple the flow and structural solver is discussed. The following applications have been discussed. First, the vortex-induced vibration of a cylinder has been presented. Second, FSI benchmarks of an elastic and a viscoelastic splitter attached on a cylinder involving large-scale flow-induced deformation have been presented. Third, numerical simulations of a FSI benchmark in a heated channel, demonstrating augmentation in convective heat transfer, have been reviewed. Finally, the flow-induced deformation of a thin elastic plate due to blast loading has been presented. In all of these FSI problems, the IB method has been successfully employed to simulate the coupled fluid and structural dynamics.
Rajneesh Bhardwaj

Chapter 12. Study of Momentum and Thermal Wakes Due to Elliptic Cylinders of Various Axes Ratios Using the Immersed Boundary Method

This chapter presents an application of the immersed boundary method (IBM) for simulating momentum and thermal wakes generated by elliptic cylinders. We consider elliptic cylinders of five different axis ratios (AR = 0.1, 0.4, 0.6, 0.8, 1.0) within a Reynolds number range where the flow was reported to be two-dimensional. We employ a direct forcing immersed boundary method to simulate wakes behind these cylinders. We first study the momentum wakes in terms of computing the critical Reynolds number for laminar separation and vortex shedding. Then, in the shedding regime, the wake is analyzed for large- and small-scale structures in its near and far field. We show that the low-frequency structures exist even in the near wake of an elliptic cylinder while such structures are observed only in the far wake of a circular cylinder. We then extend the momentum forcing method to the energy equation through thermal forcing and show that IBM predicts heat transfer characteristics accurately. We also note an unusual mean temperature behavior along the centerline of the wake and provide probable reasons for such a behavior. Our results prove that the IBM can be effectively used to simulate wakes of elliptic cylinders.
Immanuvel Paul, Venkatesh Pulletikurthi, K. Arul Prakash, S. Vengadesan

Chapter 13. Investigation of the Unsteady Aerodynamics of Insect Flight: The Use of Immersed Boundary Method

The study of insect flight is fascinating not only due to its underlying unsteady aerodynamic principles but also for its practical applications in the development of micro-aerial vehicles. The flapping motion of insect wings involves large translational and rotational components. Consequently, the conventional body-fitted moving-mesh computational methods face mesh tangling issues that require expensive re-meshing strategies. In addition, accomplishing such simulations involve considerable human intervention. Immersed boundary methods are ideally suited to simulate insect flight as the flow around the wings are simulated in a fixed Cartesian grid; due to which, neither mesh moving strategies nor re-meshing is required. However, the enforcement of boundary conditions on flapping wings is challenging and is achieved by an appropriate force field. In this chapter, we use the continuous forcing immersed boundary method to study the unsteady aerodynamics of an idealized flapping motion of a hovering insect. Flapping flight in the inclined stroke planes are studied to understand the associated flow dynamics. Furthermore, the effect of phase difference on the aerodynamics of insects with tandem wings (e.g., dragonfly) and the presence of ground are also considered. Results are analyzed in terms of the cycle variation of forces, vortex dynamics, and coherent structures.
Srinidhi Nagarada Gadde, Y. Sudhakar, S. Vengadesan

Chapter 14. Hybrid Lagrangian–Eulerian Method-Based CFSD Development, Application, and Analysis

In this chapter, a partitioned approach-based hybrid Lagrangian–Eulerian (HLE) method and its application for analysis of various computational fluid–structure dynamics (CFSD) problems are presented. The HLE method uses a physical law-based finite volume method (FVM) and level-set function-based immersed boundary method (LS-IBM) for computational fluid dynamics (CFD), a geometric nonlinear Galerkin finite element method (FEM) for computational structural dynamics (CSD), and an implicit coupling between CFD and CSD that involves direct implementation of fluid–solid boundary conditions. The algebraic formulation is presented by the physics-based FVM for a Cartesian control volume in CFD and the geometric nonlinear Galerkin FEM for a three-node triangular element in CSD. Further, the associated solution methodologies are presented first separately for both CFD and CSD and then together with the implicit coupling methodology. The HLE method-based applications that involve large deformation and complex geometry (of the structure) are presented here for the analysis of various types of one-way and two-way coupled CFSD problems (involving both rigid and deformable or flexible structures).
Namshad Thekkethil, Atul Sharma

Chapter 15. Immersed-Boundary Methods for Simulating Human Motion Events

The further development of an immersed-boundary method for general flow applications is outlined in this paper. A cell-classification procedure based on a signed distance to the nearest surface is used to separate the computational domain into cells outside the immersed object (field cells), cells outside but adjacent to the immersed object (band cells), and cells within the immersed object (interior cells). Interpolation methods based on laminar/turbulent boundary layer theory are used to prescribe the flow properties within the band cells. The method utilizes a decomposition of the velocity field near embedded surfaces into normal and tangential components, with the latter handled using power-law or log-law interpolations to mimic the energizing effects of turbulent boundary layers. Procedures for generating motion events using rendering technologies are described as methods for directly embedding sequences of stereo-lithography files representing frames of motion as immersed objects in the computational domain. Extensions of the methodology to zero-thickness immersed surfaces are discussed. Described applications center on human motion events, with a focus on understanding the effect of human motion on agent transport in confined environments.
Jung-Il Choi, Jack R. Edwards

Chapter 16. Immersed Boundary Method for High Reynolds Number Compressible Flows Around an Aircraft Configuration

This study focuses on the immersed boundary method (IBM) for high Reynolds number compressible flow. Using the hierarchical Cartesian grid makes it difficult to resolve the thin boundary layer of high Reynolds number flow around an object because the grid is refined equally in every direction. To alleviate the requirement for the grid size of the wall, the development of a wall boundary condition using a turbulent wall function is necessary. The IBM development is based on a hierarchical Cartesian grid generator and compressible flow solver called the University of Tokyo Cartesian-grid-based automatic flow solver (UTCart). The flow solver is validated through subsonic flow over a 2D bump. The skin friction distribution is reproduced accurately by the proposed IBM. A flow around NASA common research model is solved. A flux-based method is developed based on the balance of the numerical fluxes and used for the evaluation of the aerodynamic forces. The results exhibit favorable agreement with the existing experimental data and numerical results from the body-fitted grid.
Taro Imamura, Yoshiharu Tamaki
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