Session E26: Computational Fluid Dynamics: Immersed Boundary Methods II

A Distributed Element Roughness Model for Deterministic Roughness Morphologies using the Double Averaged Navier Stokes Equations

Design for cooling effectiveness in internal flow systems relies on accurate models for dynamic losses and heat transfer. In these systems (e.g., gas turbine blades, intercoolers), thousands of individual passages of varying configuration and roughness morphology can be present, and this can render roughness-element resolved CFD methods impractical. Alternatively, a volumetric roughness modeling approach, such as distributed element roughness modeling (DERM) has a computational cost orders of magnitude lower.

In this approach, which draws on elements of Eulerian two-fluid modeling, the detailed geometry of the roughness is not resolved, and the surface is represented by volume fraction and volume fraction gradient distributions. Attendant forces due to interfacial drag and spatial dispersion are imparted on the flow to represent the roughness field.  

In this work, a DERM model based on the Double Averaged Navier-Stokes (DANS) equations is presented. Unique to this formulation of DERM is the specific treatment of the spatially averaged Reynolds stresses, the treatment of wall normal facing surfaces, and the spatially varying DERM sectional drag coefficient. Comparisons between this DERM model and resolved CFD are shown for a suite of academic roughness configurations.   

Higher-order EB method for the Incompressible Navier-Stokes Equations.

We present a higher-order embedded boundary (EB) method to solve partial  differential equations in the presence of complex geometries for structured meshes. 

We focus on the development and analysis of a new EB method that creates a water-tight grid via the divergence theorem, and uses cell/faces averages to achieve high order accuracy and stability for finite volume operators in the incompressible Navier-Stokes equations. 

The method uses a novel weighted least squares approach to find the appropriate stencil coefficients for several operators to ensure conservation, accuracy, and stability in the presence of smooth and non-smooth geometries. We present results that verify the accuracy of the method, and use an eigenvalue analysis to demonstrate linear operator stability even with arbitrarily-small cut cells. Finally, we show the difference in accuracy between low-order and high-order results for simple and complex flow fields.   

High Order Cut-Cell Method 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.  Recently, our novel approach to the design of finite-difference

  cut-cell stencils allowed for stable discretizations of elliptic, parabolic and hyperbolic,

  systems using non-dissipative schemes in the interior of the domain.

  This was accomplished without the need for any ad-hoc stabilization by

  constructing the discrete operators based on two simple and intuitive design principles.

  These principles, and an a-priori optimization process, allowed for

  the construction of stable 8th order approximations to elliptic and

  parabolic problems and stable and conservative 4th order

  approximations to hyperbolic problems.  In this work, the extension of 

  the method to more general boundary conditions will be presented.   

The method of immersed layers, with application to internal and external flows

The immersed boundary method (IBM) has served for several decades as a versatile tool for simulating and studying flows with complex moving boundaries on stationary grids. In this work, we show that the IBM is a special case of a broader and more powerful approach in which the interior and exterior of any surface can be distinguished from one another. To show this, we make use of generalized functions on level sets to reformulate the incompressible Navier--Stokes equations and related partial differential equations so that jumps in surface quantities are incorporated exactly into the equations. In this manner, we can enforce any well-posed boundary or interface condition in a straightforward manner and readily compute the associated surface forces. Because of their resemblance to single and double layers in the Poisson equation, we denote these jump terms as `immersed layers'. We discretize the equations with second-order differencing operators on a staggered Cartesian grid. These operators' mimetic properties in clearly-defined inner product spaces---coupled with the use of the lattice Green's function for inverting the Laplace operator---ensure that various continuous identities are exactly preserved in the discrete sense and that other results are simple to derive. We highlight an open-source, extendable framework of tools that implement the operators and methods, and demonstrate the framework's use on various flows, including past multiple moving bodies and through internal ducts.   

A novel mass and momentum conserving immersed boundary method based on volume-filtering.

We present a novel approach towards solving for complex boundaries within a flow using the

immersed boundaries method in a conservative manner without the need for body-fitting meshes.

This method uses the volume-filtering approach by taking the boundary conditions at the

interfaces and turning them into body forces that apply to the right-hand side of the mass and

momentum equations. This rigorous mathematical and physical volume-filtering framework, allows to

explicitly express the immersed boundary forcing terms, without assuming any

particular discretization scheme. The derivation of this method is based on selecting a filter

kernel that is symmetric in nature, and one that integrates to unity. The usage of the filter kernel

creates new filtered equations, similar in style to the LES approach. These equations include the

body forces and the sub-filter scale terms which are closed using various methods specific to the

term. To show the accuracy of the method, 3 canonical test cases are run: (1) static cylinder at

Reynolds number of 100, (2) oscillating cylinder at Reynolds number of

100, and (3) flow past a sphere with increasing Reynolds number. The results are compared with previous experiments or

simulations with body-fitted meshes. The accuracy of different filter kernels (Box, triangle,

cosine, Roma-Peskin and parabolic) at different filter widths compared to the grid spacing are

tested.   

Optimal control integrated immersed boundary multiphase flow framework for simulating wave energy converter (WEC) devices

In this presentation, we present a novel  computational fluid dynamics (CFD) framework that couples the fictitious domain Brinkman penalization (FD/BP) method, a robust multiphase flow solver, and a model predictive control (MPC)-based optimal controller to simulate and maximize the energy conversion of  WEC devices. Our framework is capable of accurately resolving the complex wave structure interactions (WSI) involved in the wave energy conversion process, wherein the nonlinear Navier-Stokes equations and the dynamics of the converter are solved on locally refined Cartesian grids. The MPC controller solves an optimization problem over a time horizon of one to two wave periods into the future, given the current state of the device and the past wave elevation data; the latter information is obtained from the CFD framework. In particular, the past wave elevation data  collected by a sensor located at a pre-calculated distance in front of the WEC is used to predict the future wave elevations using an auto-regressive (AR) model. Future wave excitation forces required by the MPC are estimated using the predicted wave elevations. Path constraint on displacement, velocity and control input is also implemented. Various penalty terms are added to the objective function to: (i) smooth the actuator forces; and to (ii) restrict the flow of power flow from the device to the electric grid. Our results show that the conventional linear potential theory-based solvers overpredict the WEC dynamics and the power absorbed by the device, whereas the CFD framework provides realistic estimates of the power.   

Study of viscoelastic behavior in an immersed boundary-Lagrangian mesh model

Many biological settings involve complex fluids that have non-Newtonian mechanical responses that arise from suspended microstructures. In this study, we compare Lagrangian mesh and Oldroyd-B formulations of non-Newtonian fluid-structure interaction in an immersed boundary framework. Significantly, we find remeshing, as well as fluid and/or mesh refinement increase the stability and accuracy of Lagrangian mesh simulations and produce results highly comparable with that of the Oldroyd-B model for test cases such as planar Poiseuille flow and Stokes' problems. Further investigations revealed the trade-offs between discretization error and CPU time consumption at various regridding frequencies and determined an optimal regridding interval. In addition, cases are compared at low Reynolds number and a fixed set of parameters including the Weissenberg number and the viscosity ratio. The form of the mesh could be regarded as a coarse grain model of the physical structure of the viscoelastic component of the fluid and, in some contexts, distortion of the mesh may have a physical interpretation and regridding would be unnecessary. Modifications to the models are proposed with viscoelastic links and comparisons with properties of viscoelastic media such as mucus, biological tissues, and biofilms.   

A versatile immersed boundary method for multiphase flows

A novel immersed boundary method (IBM) is proposed with the aim to apply it to incompressible particle-laden flows. The new Hybrid method (HyBM) relies on a regularization of the transfer function, allowing both symmetrical and non-symmetrical interpolation/spreading supports to be used. This allows to locally restrain the momentum transfer to the inside of the particle, hereby avoiding competing contributions from multiple particles when particles come into close contact, as well as to safely and accurately interact with domain boundaries.

The present methodology is validated and compared to the literature for several configurations in which the Reynolds number and the particle volume fraction are varied. These simulations are performed for fixed spheroidal particles on Cartesian grids, in our fully-coupled pressure-velocity algorithm solver. The results show that the HyBM accurately predicts near wall/particle-particle interactions without loss of accuracy of the overall method.   

A hybrid level-set / embedded boundary method applied to solidification-melt problems

In this talk, we introduce a novel way to represent the interface for two-phase flows with phase change. We combine a level-set method with a Cartesian embedded boundary method and take advantage of both. This is part of an effort to obtain a numerical strategy relying on Cartesian grids allowing

the simulation of complex boundaries with possible change of topology while retaining a high-order representation of the gradients on the interface and

the capability of properly applying boundary conditions on the interface. This leads to a two-fluid conservative second-order numerical method. The ability

of the method to correctly solve Stefan problems, onset dendrite growth with and without anisotropy is demonstrated through a variety of test cases. Finally, we take advantage of the two-fluid representation to model a Rayleigh--B\'enard instability with a melting boundary.