Session A09: CFD: Immersed Boundary Methods I

A well-behaved grid-size-insensitive projection immersed boundary method for fluid-structure interaction problems

The projection immersed boundary method (IBM) has been successfully adopted in the simulations of compressible and incompressible flow over stationary and moving objects. The standard approach transfers the boundary conditions into the Navier-Stokes equations directly (as body forces) resulting in differential-algebraic equation (DAE). However, the method requires that the size of the surface (Lagrangian) grid be comparable to the size of the flow (Eulerian) grid. The coupled equations are otherwise ill-conditioned if surface grid is much finer, rendering fluid-structure interaction simulations with large deformation almost impossible. Here, we propose a modified projection method that removes this limitation and allows the surface grid to be much finer than the fluid grid while keeping the system of equations well-posed. The equations for the body force are modified using Tikhonov regularization, with the graph Laplacian used to formulate the penalty term. The regularized equations are better conditioned and produce significantly smoother solutions. This improvement makes the projection IBM applicable to more complex problems without manual regriding or user intervention. Various examples will be presented to demonstrate the versatility and accuracy of the new method.   

Numerical simulation of White Blood Cell rolling on endothelial wall

Rolling of White Blood Cell (WBC) on endothelial wall plays an important role in the human immune system. The rolling dynamics of RBC contain important information on the characteristics of both WBC membrane and the endothelial bonding. In this work, we replicate this process using numerical simulations. The WBC is assumed to have an idealized spherical shape. A realistic model of blood vessel is reconstructed from in-vivo imaging data. The transport of the WBC in fluid flows is simulated using a fluid-structure interaction approach, which couples the dynamics of the cell membrane in fluid flows. The Dissipative Particle Dynamics is used to model the cellular dynamics (membrane, and cytosol). The cytoskeleton components (actin filament, intermediate filament and microtubes) are modeled using non-linear Worm-Like Chain springs. The extracellular flow is modeled with the Immersed Boundary Method, which allows the efficient simulation of fluid flows in complex vasculatures. We will report and compare the history of WBC rolling as it traverses along the endothelial wall under different flow conditions.   

Variational Multiscale Immersed Boundary Method for Laminar and Turbulent flows

The ability to solve laminar and turbulent flows around complex geometric shapes without the need of boundary fitted meshes is of great interest in engineering analysis and design. For shape design optimization where the revised design in the form of STL files can be directly embedded in the flow domain without complex remeshing of the updated geometry can substantially reduce the time and efforts in the preprocessing phase in CFD.

This talk presents an immersed boundary method for weak enforcement of Dirichlet boundary conditions on surfaces that are immersed in the stationary background discretizations. An interface stabilized form is developed by applying the Variational Multiscale Discontinuous Galerkin (VMDG) method at the immersed boundaries. The formulation is augmented with a variationally derived ghost-penalty type term. The weak form of the momentum balance equations is embedded with a residual-based turbulence model for incompressible turbulent flows. A significant contribution in this work is the variationally derived analytical expression of the Lagrange multiplier for weakly enforcing the Dirichlet boundary conditions at the immersed boundary. In addition, the analytical expression for the interfacial stabilization tensor emerges that accounts for the geometric aspects of the cut elements produced when the immersed surface geometry traverses the underlying mesh. A unique attribute of the fine-scale variational equation is that it also yields a posteriori error estimator that can evaluate the local error in weak enforcement of the essential boundary conditions at the immersed boundaries. The method is shown to work with meshes comprised of hexahedral and tetrahedral elements. Numerical test cases with increasing levels of complexity are presented to validate the method on benchmark problems, and turbulent features of flows are analyzed.

    

An Immersed Interface Method for the Simulation of Coupled Fluid-Body Problems

We present a finite-difference immersed interface method (IIM) for the simulation of coupled fluid-body problems, formulated using the velocity-vorticity form of the 2D Navier Stokes equations. Using conservative finite-difference schemes our approach can enforce circulation conservation around each separate body in the domain, even when using outflow boundary conditions. Moving boundaries and emerging points are handled naturally in the IIM framework, even with high-order timestepping. For two-way coupling problems, where the body motion is driven by hydrodynamic forces, we use a control volume approach to enforce a momentum balance. The method achieves second-order convergence for most practical simulations except those involving small density ratios, where a first-order error term related to our two-way coupling approach dominates. We show the ability of this approach to accurately and efficiently run long-time simulations with compact domains and circulation conservation on several benchmark and convergence studies.   

Elastic Instability and Buckling of Cylindrical Vessels Immersed in Fluid

In this work, we develop numerical models to study the deformation of hyperelastic cylindrical vessels immersed in an incompressible fluid. The elasticity of the vessels is described using neo-Hookean elasticity theory and the interaction of the structure and fluid is modeled using the Immersed-Boundary formulation. From numerical simulations, we observed that cylindrical shells under axial compression will experience different buckling and barrelling modes depending on the slenderness and bending rigidity of the shells. Our results can be used to compare with existing theoretical works such as the one by Goriely et al. published in 2008. To expand our study, we construct and analyze other models in which the vessels are deformed under isotropic compression or under the flow of fluid within the shells.   

A High-Order Immersed Interface Method for 3D Transport Equations

Immersed Interface Methods (IIMs) are a class of Cartesian grid-based finite difference schemes that can achieve high order accuracy while allowing for non-conforming domain boundaries or material interfaces. These schemes are particularly promising for simulations of interface-coupled multiphysics problems. However, many existing IIMs have focused on second-order methods and simple 2D geometries, and their extension to higher-order methods and complex 3D geometries remains a challenge. In this presentation we demonstrate a high-order immersed interface method for 3D simulations with smooth non-convex domain boundaries or material interfaces. The method allows for third order spatial accuracy for advection terms and up to sixth order spatial accuracy for diffusion terms. This IIM discretization is implemented in MURPHY, an open-source software framework for adaptive mesh refinement, which allows for massively parallel simulations of PDEs with complex 3D interfaces. We confirm the stability and accuracy of our IIM through simulations of 3D advection-diffusion with a variety of boundary conditions on irregular domains and material interfaces, and showcase the ability of the method to handle more complex nonlinear PDEs with nontrivial dynamics on both sides of a material interface.   

A momentum forcing wall-modeling approach for large eddy simulations with the immersed boundary method

A novel approach is presented for wall-modeled large-eddy simulations with immersed boundaries. In this work, an immersed boundary method (IBM) similar to that proposed by Ghias et al. (2007) was implemented in a GPU-accelerated compressible solver that solves the Navier-Stokes equations in anti-symmetric form using co-located Cartesian grids. An equilibrium wall model function was applied to estimate the wall shear stress in an under-resolved grid. In the current method, an estimate of the contribution of under-resolved wall-normal gradients to the viscous diffusion term is obtained at fluid points near the immersed boundary. The estimate is compared against the diffusion predicted by the wall model function, and the difference between these values is added to the momentum equation as a forcing term acting on the wall-parallel direction. This method avoids directly imposing velocity values at grid points and has the effect of adding momentum in the streamwise direction that would otherwise be lost in the under-resolved flow field. A source term is also applied to the energy equation in a consistent manner to conserve total energy. The method was tested in a turbulent channel flow and other canonical wall-bounded flows.

    

Tuning the Immersed boundary-Lagrangian mesh model for viscoelastic suspensions

Naturally occurring viscoelastic fluid media such as blood and DNA suspensions are hosts to different types of swimmers and vesicles.  The non-Newtonian mechanical responses that arise from suspended microstructures require modification to the immersed boundary framework in order to capture the underlying physics. In the current study, we have extended the Oldroyd-B and Lagrangian mesh formulations to study the viscoelastic behavior of vesicles and swimmers. The models are benchmarked with analytical results where we see that remeshing increases the stability and accuracy of Lagrangian mesh simulations and produces results highly comparable with that of the Oldroyd-B model. Depending on the context, distortion of the Lagrangian mesh can be taken as a coarse grain model of the physical structure of the viscoelastic component of the fluid and regridding would be unnecessary. Further modifications include the implementation of a penalty model that allows us to model bodies that are not neutrally buoyant along with changes in the viscoelastic links that can mimic biological tissues.

    

Velocity error and consequences of the feedback immersed boundary method

This talk presents a study on velocity error and consequences of the feedback immersed boundary method (IBM), which is implemented with the lattice Boltzmann method for solving the fluid dynamics. The IBM is implemented in non-iterative (with and without velocity prediction step) and iterative ways. A few benchmark cases will be introduced to demonstrate the performance of the three implementations. Results show that both the IBM with prediction step, the iterative IBM and one iteration IBM with proper feedback coefficients can suppress the spurious flow penetration on the solid wall. While the velocity error does not significantly affect the force production and structure deformation for external flows, reducing it significantly improves the prediction of the force distribution and structure deformation for internal flows. In addition, the iterative IBM with smaller feedback coefficient has better numerical stability.