A conservative coupling algorithm between a compressible flow and a rigid body using an Embedded Boundary method

This paper deals with a new solid-fluid coupling algorithm between a rigid body and an unsteady compressible fluid flow, using an Embedded Boundary method. The coupling with a rigid body is a first step towards the coupling with a Discrete Element method. The flow is computed using a Finite Volume approach on a Cartesian grid. The expression of numerical fluxes does not affect the general coupling algorithm and we use a one-step high-order scheme proposed by Daru and Tenaud [Daru V,Tenaud C., J. Comput. Phys. 2004]. The Embedded Boundary method is used to integrate the presence of a solid boundary in the fluid. The coupling algorithm is totally explicit and ensures exact mass conservation and a balance of momentum and energy between the fluid and the solid. It is shown that the scheme preserves uniform movement of both fluid and solid and introduces no numerical boundary roughness. The effciency of the method is demonstrated on challenging one- and two-dimensional benchmarks

Rayleigh-B\'enard convection with a melting boundary

We study the evolution of a melting front between the solid and liquid phases of a pure incompressible material where fluid motions are driven by unstable temperature gradients. In a plane layer geometry, this can be seen as classical Rayleigh-B\'enard convection where the upper solid boundary is allowed to melt due to the heat flux brought by the fluid underneath. This free-boundary problem is studied numerically in two dimensions using a phase-field approach, classically used to study the melting and solidification of alloys, which we dynamically couple with the Navier-Stokes equations in the Boussinesq approximation. The advantage of this approach is that it requires only moderate modifications of classical numerical methods. We focus on the case where the solid is initially nearly isothermal, so that the evolution of the topography is related to the inhomogeneous heat flux from thermal convection, and does not depend on the conduction problem in the solid. From a very thin stable layer of fluid, convection cells appears as the depth -- and therefore the effective Rayleigh number of the layer increases. The continuous melting of the solid leads to dynamical transitions between different convection cell sizes and topography amplitudes. The Nusselt number can be larger than its value for a planar upper boundary, due to the feedback of the topography on the flow, which can stabilize large-scale laminar convection cells.Comment: 36 pages, 16 figure

A modified equation analysis for immersed boundary methods based on volume penalization: applications to linear advection-diffusion and high-order discontinuous Galerkin schemes

The Immersed Boundary Method (IBM) is a popular numerical approach to impose boundary conditions without relying on body-fitted grids, thus reducing the costly effort of mesh generation. To obtain enhanced accuracy, IBM can be combined with high-order methods (e.g., discontinuous Galerkin). For this combination to be effective, an analysis of the numerical errors is essential. In this work, we apply, for the first time, a modified equation analysis to the combination of IBM (based on volume penalization) and high-order methods (based on nodal discontinuous Galerkin methods) to analyze a priori numerical errors and obtain practical guidelines on the selection of IBM parameters. The analysis is performed on a linear advection-diffusion equation with Dirichlet boundary conditions. Three ways to penalize the immerse boundary are considered, the first penalizes the solution inside the IBM region (classic approach), whilst the second and third penalize the first and second derivatives of the solution. We find optimal combinations of the penalization parameters, including the first and second penalizing derivatives, resulting in minimum errors. We validate the theoretical analysis with numerical experiments for one- and two-dimensional advection-diffusion equations

Characteristic-Based Volume Penalization Method for Compressible Flow Simulations on Unstructured Meshes

The Characteristic-Based Volume Penalization (CBVP) method for numerical simulations of compressible flow over solid obstacles on unstructured meshes is presented. The approach belongs to the class of immersed boundary methods and is not relying on body-fitted meshes. Characteristic penalization terms, added to the compressible Navier-Stokes equations, are used to impose Dirichlet and Neumann boundary conditions on solid-fluid interface with an a priori defined accuracy. The details of numerical implementation, utilizing hybrid finitevolume method with high order edge-based reconstruction schemes in the flow region and loworder finite-difference approximation inside of the obstacle, are discussed. The developed algorithm provides the ability to perform calculations on grids of arbitrary type, including fully unstructured meshes. The efficiency of the characteristic based volume penalization method and its numerical implementation is demonstrated for shock wave reflection, acoustic pulse reflection and Couette flow problems. The results of CBVP simulations are compared with the numerical solutions of the same problems using Brinkman volume penalization method

Modeling Particle-Laden Compressible Flows with an Application to Plume-Surface Interactions

During planetary descent, rocket plumes fluidize and eject surface granular matter. Consequently, ejected matter has been shown to obscure the landing site and even collide with the lander, causing serious damage. Given the high risk and cost of space exploration, the challenges associated with plume-surface interactions (PSI) are capable of jeopardizing future missions. The erosion, fluidization, and ejecta of granular matter during PSI occurs under transonic/supersonic, high Reynolds number conditions. These flow conditions pose significant challenges in both experimental and numerical analyses. To date, accurate and predictive physics-based models of PSI at relevant landing conditions do not exist. The objective of this project is to develop high-fidelity simulation capabilities to model compressible gas-particle flows at conditions relevant to PSI. To start, a rigorous derivation of the volume-filtered (locally averaged) compressible Navier--Stokes equations is presented for the first time. This derivation reveals many unclosed terms, for which models are either non-existent or not valid under the regimes of interest. To this end, key terms including pseudo-turbulent kinetic energy and pseudo-turbulent Reynolds stresses, are isolated and modeled via a transport equation in a new high-order finite difference Eulerian-Lagrangian framework. A new immersed boundary method is introduced to generate highly resolved, multi-particle simulations for model closure development. Using the proposed immersed boundary method and the Eulerian--Lagrangian framework, high-fidelity PSI simulations are performed. Single-phase jet impingement on flat surfaces is first shown for validation of the flow conditions. The work is then extended to PSI over a granular bed. For this case, it is shown that that ejected particles can exceed sonic speeds at high particle Reynolds numbers while the majority of the granular bed experiences subsonic particle Mach numbers. In addition, granular temperature is found to be most prevalent in region of high shear during crater formation. The uniqueness of this work lies in the combination of first principles physics and numerics to generate a modeling framework to improve predictions of plume-surface interactions for future missions involving entry, descent, and landing on planetary and satellite bodies.PHDMechanical EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/169969/1/grshall_1.pd

High-Order Flux Reconstruction Based on Immersed Boundary Method

In the last decade, high-order methods for Computational Fluid Dynamics (CFD) are becoming attractive for unsteady scale-resolving-simulations in industrial CFD applications, due to their advantages of low numerical dissipation, high efficiency on modern architectures and quasi mesh-independence. However, the generation of body-fitted mesh for high-order methods is still a significant bottleneck and often determines the overall quality of the solution. To avoid the complexity of mesh generation, the present work combines the numerical advantages of the high-order Flux Reconstruction (FR) method and the simplicity of the mesh generation based on Immersed Boundary Method (IBM) that allows solving flow past obstacles on a non body-fitted mesh. The volume penalization method is selected for its ease of implementation and robustness. The proposed method is validated by several test cases, including flow past a cylinder and NACA0012 airfoil for static and moving boundaries. Good agreement with body-fitted simulation is reported

Volume penalization for inhomogeneous Neumann boundary conditions modeling scalar flux in complicated geometry

We develop a volume penalization method for inhomogeneous Neumann boundary conditions, generalizing the flux-based volume penalization method for homogeneous Neumann boundary condition proposed by Kadoch et al. [J. Comput. Phys. 231 (2012) 4365]. The generalized method allows us to model scalar flux through walls in geometries of complex shape using simple, e.g. Cartesian, domains for solving the governing equations. We examine the properties of the method, by considering a one-dimensional Poisson equation with different Neumann boundary conditions. The penalized Laplace operator is discretized by second order central finite-differences and interpolation. The discretization and penalization errors are thus assessed for several test problems. Convergence properties of the discretized operator and the solution of the penalized equation are analyzed. The generalized method is then applied to an advection-diffusion equation coupled with the Navier-Stokes equations in an annular domain which is immersed in a square domain. The application is verified by numerical simulation of steady free convection in a concentric annulus heated through the inner cylinder surface using an extended square domain.Comment: 32 pages, 19 figure

Implementation of a level-set based volume penalization method for solving fluid flows around bluff bodies in OpenFOAM

A volume penalization-based immersed boundary technique is developed and thoroughly validated for fluid flow problems, specifically flow over bluff bodies. The proposed algorithm has been implemented in an Open Source Field Operation and Manipulation (OpenFOAM). For capturing the fluid-solid interface more accurately, the grid is refined near the solid surface using topoSetDict and refineMeshDict utilities in OpenFOAM. In order to avoid any numerical oscillation, the present volume penalization method (VPM) is integrated with a signed distance function, which is also referred to as a level-set function. Benchmark problems, such as flows around a cylinder and a sphere, are considered and thoroughly validated with the results available in the literature. For the flow over a stationary cylinder, the Reynolds number is varied so that it covers from a steady 2D (two-dimensional) flow to an unsteady 3D (three-dimensional) flow. The capability of the present solver has been further verified by considering the flow past a vibrating cylinder in the cross-stream direction. In addition, a flow over a sphere, which is inherently three-dimensional due to its geometrical shape, is validated in both steady and unsteady regimes. The results obtained by the present VPM show good agreement with those obtained by a body-fitted grid using the same numerical scheme as that of the VPM, and also with those reported in the literature. The present results indicate that the VPM-based immersed boundary technique can be widely applicable to scientific and engineering problems involving flow past stationary and moving bluff bodies of arbitrary geometry