Session ZC16: CFD: Immersed Boundary Methods II

High-Order Embedded Boundary Methods for Direct Numerical Simulations

Cut-cell methods for unsteady flow problems can greatly simplify the grid generation process and allow for high-fidelity simulations on complex geometries. Historically, however, cut-cell methods have been limited to low orders of accuracy. It is the conjecture of the authors that this has been driven, by the variety of procedures typically introduced to evaluate derivatives in a stable manner near the highly irregular embedded geometry. Indeed, even on a uniform mesh, it is non-trivial to derive high-order numerical boundary schemes to be used near the wall. Pursuing a finite-differences based cut-cell approach (also referred to as an "embedded boundary" or "Cartesian grid" method), allowed for formulating the small cell problem encountered by cut-cell methods as an optimization problem. Coupled with a truncation error matching idea, the optimization strategy has been successfully applied to cut-cells resulting in conservative 4th order approximations to hyperbolic problems without the addition of numerical dissipation as well as 8th order approximations for elliptic and parabolic systems. Generalizing our previous work to the Navier-Stokes equations highlights the need for optimizing interpolation stencils in addition to the derivative stencils. In the present work, we will discuss a unified framework to accomplish this. Several test cases will be showcased to highlight the stability of the resulting method and provide a path forward for pursuing higher-order stencils for general governing equations.   

A high-order immersed interface method for elliptic PDEs with variable or discontinuous coefficients

We present a high-order immersed interface method for discretizing elliptic PDEs on complex domains with variable or discontinuous coefficients. High-order accuracy is achieved by combing fourth and sixth-order dimension-split finite difference schemes with a high-order weighted least-squares reconstruction of the solution near domain boundaries and material interfaces. The approach does not require the derivation of jump conditions at material interfaces, which greatly facilitates the construction of discretization up to sixth order. We evaluate the spectra and conditioning of the resulting non-symmetric linear systems, and discuss appropriate matrix-based and matrix-free preconditioners that can be used when solving these systems iteratively. We also evaluate the accuracy of surface quantities on domain boundaries and material interfaces, particularly normal gradients that represent surface tractions or viscous fluxes. We conclude by applying the discretization to to scalar and vector elliptic PDEs that appear in simulations of incompressible flows with fluid-structure interaction.   

A third-order Immersed Interface Method for the velocity-pressure Navier-Stokes equations on collocated grids

Combining a finite-difference discretization with an immersed interface method (IIM) has been shown to yield high-order spatial accuracy for solving simple PDEs on complex domains. Representing the quantities on collocated grids enables both a third-order or higher discretization error, but also retains the simplicity and scalability associated with explicit finite-difference stencils. Extending this IIM approach to the incompressible Navier-Stokes equations while achieving high order spatial and temporal accuracy is an open challenge. In this talk we present a third-order Runge-Kutta based algorithm to solve the incompressible Navier-Stokes equations using a third-order IIM boundary discretization on collocated grids. We present and analyze convergence and stability results for flows with stationary and moving boundaries in 2D, though the algorithm should readily translate to 3D as well.   

Towards an efficient immersed boundary projection method for wall-bounded flows with complex surface and sub-surface structures

High-fidelity simulation tools that can predict the fully coupled dynamics between wall-bounded turbulent flows and advanced structures will play a key role in understanding the fluid-metamaterial interaction (FMI) and developing design paradigms that utilize these structures in effective passive flow control. Immersed boundary methods are a natural choice to couple the flow-metamaterial interplay, as these approaches do not require complex body-conforming meshes that would have to adapt in time as the structure moved. In this vein, we present an immersed boundary projection framework that facilitates the simulation of these complex FMI dynamics. We focus on novel features of our formulation, including a new and versatile time stepping methodology based on a Runge-Kutta Chebyshev treatment of stiff viscous terms. Results of this novel formulation will be presented for validation and to demonstrate its potential for complex FMI flows of interest.   

Fast Parallel Stability and Resolvent Analysis for Multi-Resolution External Flows

IBLGF-AMR is an incompressible external flow solver combining Lattice Green's Function (LGF), Immersed Boundary (IB) Method, and Adaptive Mesh Refinement (AMR). This method successfully yielded a range of flow simulations, including fully 3D turbulent flow past a sphere, span-wise periodic turbulent flow past a cylinder, and high-fidelity simulations of starting vortex generated by a flat plate undergoing translational and rotational motions. The success of IBLGF-AMR inspired us to leverage the same techniques to devise an efficient formulation of the discretized linearized Navier-Stokes equations (LNSE), which is foundational to a plethora of flow simulation and analysis methods. In this presentation, we will introduce how we combined IB, LGF, AMR, and Fast Multiple Method (FMM) to create a sparse representation of the LNSE. We applied our method to conduct stability and resolvent analysis of a range of problems to demonstrate the accuracy and effectiveness of this method. In the future, we also plan to apply this sparse linearization to solve external incompressible flows using the time-spectral method.   

Layered Immersed Boundary Method for General CFD Applications

In this work, we present a novel Layered Immersed Boundary Method (LIBM) designed for general CFD applications. The proposed method overcomes the limitations of traditional immersed boundary methods by introducing immersed boundary layers that are aligned with the boundary geometry. These layers enhance resolution in the normal direction while stretching the grid in the lateral and longitudinal directions, effectively reducing computational costs. LIBM exhibits exceptional accuracy in resolving boundary layers, even at high Reynolds numbers, where conventional IBMs tend to fail or become prohibitively expensive. This makes it particularly well-suited for simulating turbulent and multiphase flows involving moving and/or morphing structures. Additionally, immersed boundary layers ensure a continuous and consistent description of the geometry which ensures the integrity of the numerical simulation and facilitates the implementation of Neumann and Robin boundary conditions. Unlike traditional direct forcing IBMs, the proposed method does not suffer from mass-conservation issues, freshly-vacated cells, and the stair-step problem since LIBM separates points of action of hydrodynamic forces from those of virtual forces. Overall, the versatility and accuracy of the LIBM make it a valuable tool for a wide range of CFD simulations, particularly those involving complex flows with intricate boundary geometries and high Reynolds numbers.   

An explicit and non-iterative immersed boundary method for incompressible flows with scalar fields

Incompressible flows with scalar fields are very common in natural and engineering applications, and are often accompanied by complex motion and deformation boundary, such as fluid-structure interaction, finite-size particle flows, and bubbly flow. To solve such problems, a numerical method based on the moving least squares immersed boundary method is proposed. We reveal that the boundary error of the baseline method follows a narrow peak distribution, and propose an explicit and non-iterative correction method. This method can be used to correct the volume force of velocity field and scalar field, including the Dirichlet and Neumann boundary conditions. Validation calculations are performed for the flows over the stationary sphere and the flexible flag, the buoyant sphere and the mixed convection around an iso-heat-flux rotating sphere. The results demonstrate that the proposed method can achieve low boundary error similar to the iterative method, but with less computing time. This method can be further combined with the iterative methods to achieve smaller error and may also be applied to other types of diffusion-type immersed boundary methods.