Brinkman Penalization Method Research Articles

Asymptotic analysis of hydrodynamic forces in a Brinkman penalization method: case of an initial flow around an impulsively started rotating and translating circular cylinder

Asymptotic analysis of hydrodynamic forces in a Brinkman penalization method: case of an initial flow around an impulsively started rotating and translating circular cylinder

An effective preconditioning strategy for volume penalized incompressible/low Mach multiphase flow solvers

The volume penalization (VP) or the Brinkman penalization (BP) method is a diffuse interface method for simulating multiphase fluid-structure interaction (FSI) problems in ocean engineering and/or phase change problems in thermal sciences and engineering. The method relies on a penalty factor (which is inversely related to body's permeability κ) that must be large to enforce rigid body velocity in the solid domain. When the penalty factor is large, the discrete system of equations becomes stiff and difficult to solve numerically. In this paper, we propose a projection method-based preconditioning strategy for solving volume penalized (VP) incompressible and low-Mach Navier-Stokes equations. The projection preconditioner enables the monolithic solution of the coupled velocity-pressure system in both single phase (uniform density and viscosity) and multiphase (variable density and viscosity) flow settings. In this approach, the penalty force is treated implicitly, which is allowed to take arbitrary large values without affecting the solver's convergence rate or causing numerical stiffness. It is made possible by including the penalty term in the pressure Poisson equation (PPE), which was not included in previous works that solved VP incompressible Navier-Stokes equations using the projection method. We show how and where the Brinkman penalty term enters the PPE by re-deriving the projection algorithm for the VP method. Solver scalability under grid refinement is demonstrated, i.e., convergence is achieved with the same number of iterations regardless of the problem size. A manufactured solution in a single phase setting is used to determine the spatial accuracy of the penalized solution. Various values of body's permeability κ are considered. Second-order pointwise accuracy is achieved for both velocity and pressure solutions for reasonably small values of κ. Error saturation occurs when κ is extremely small, but the convergence rate of the solver does not degrade. The solver converges faster as κ decreases, contrary to prior experience. Two multiphase fluid-structure interaction (FSI) problems from the ocean engineering literature are also simulated to evaluate the solver's robustness and performance (in terms of its number of iterations). The proposed solver also allows us to investigate the effect of κ on the motion of the contact line over the surface of the immersed body. It also allows us to investigate the dynamics of the free surface of a solidifying metal.

A model predictive control (MPC)-integrated multiphase immersed boundary (IB) framework for simulating wave energy converters (WECs)

In this work, we present a novel MPC-integrated multiphase IB framework that can compute the optimal energy-maximizing control force “on-the-fly” by dynamically interacting with a high-fidelity numerical wave tank (NWT). The computational model closely mimics the working setup of the device at its site of operation. Due to the requirement of solving a constrained optimization problem at each time step of the IB simulation, the MPC algorithm utilizes a low-dimensional dynamical model of the device that is based on the linear potential theory (LPT). The multiphase IB solver, on the other hand, is based on the high-dimensional fictitious domain Brinkman penalization (FD/BP) method, which fully-resolves the hydrodynamic non-linearities associated with the wave–structure interaction (WSI). A time-series forecasting auto-regressive model is implemented that predicts wave heights (from the past NWT data) to estimate the future wave excitation/Froude–Krylov forces for the MPC algorithm. Moreover, we also experiment with non-linear Froude–Krylov (NLFK) forces for the first time in an MPC formulation. The NLFK forces are computed efficiently using a static Cartesian grid, in which the WEC geometry is implicitly represented by a signed distance function. Under varying sea conditions, the predictions of the MPC-integrated multiphase IB solver are compared to the widely popular LPT-based solvers. In agitated sea conditions and/or under aggressive control, the LPT-based WSI solvers produce too optimistic (and misleading) power output values. Overall, six WSI/MPC solver combinations are compared for a heaving vertical cylinder to determine the reasons for discrepancies between high- and low-fidelity predictions. We also determine the pathway of energy transfer from the waves to the power take-off (PTO) system and verify the relationships using IB simulations. Additionally, three different sea states are simulated within the IB simulation to test the adaptive capability of MPC for WECs. MPC is demonstrated to adapt to changing sea conditions and find the optimal solution for each sea state.The interaction between the distributed-memory parallel multiphase IB solver (written in C++) and the serial MPC solver (written in MATLAB) is fully described to facilitate reproducibility. A bespoke communication layer between the two solvers is developed, which can be easily modified by the WEC community to experiment with other optimal controllers and computational fluid dynamics (CFD) solvers. All codes for this work are made open-source for pedagogical and research purposes.

Asymptotic analysis of initial flow around an impulsively started circular cylinder using a Brinkman penalization method

The initial flow past an impulsively started translating circular cylinder is asymptotically analysed using a Brinkman penalization method on the Navier–Stokes equations. The asymptotic solution obtained shows that the tangential and normal slip velocities on the cylinder surface are of the order of $1/\sqrt {\lambda }$ and $1/\lambda$ , respectively, within the second approximation of the present asymptotic analysis, where $\lambda$ is the penalization parameter. This result agrees with the estimation of Carbou & Fabrie (Adv. Diff. Equ., vol. 8, 2003, pp. 1453–1480). Based on the asymptotic solution, the influence of the penalization parameter $\lambda$ is discussed on the drag coefficient that is calculated using the adopted three formulae. It can then be found that the drag coefficient calculated from the integration of the penalization term exhibits a half-value of the results of Bar-Lev & Yang (J. Fluid Mech., vol. 72, 1975, pp. 625–647) as $\lambda \to \infty$ .

A consistent and conservative Phase-Field model for thermo-gas-liquid-solid flows including liquid-solid phase change

In the present study, a consistent and conservative Phase-Field model is developed to study thermo-gas-liquid-solid flows with liquid-solid phase change. The proposed model is derived with the help of the consistency conditions and exactly reduces to the consistent and conservative Phase-Field method for incompressible two-phase flows, the fictitious domain Brinkman penalization (FD/BP) method for fluid-structure interactions, and the Phase-Field model of solidification of pure material. It honors the mass conservation, defines the volume fractions of individual phases unambiguously, and therefore captures the volume change due to phase change. The momentum is conserved when the solid phase is absent, but it changes when the solid phase appears due to the no-slip condition at the solid boundary. The proposed model also conserves the energy, preserves the temperature equilibrium, and is Galilean invariant. A novel continuous surface tension force to confine its contribution at the gas-liquid interface and a drag force modified from the Carman-Kozeny equation to reduce solid velocity to zero are proposed. The issue of initiating phase change in the original Phase-Field model of solidification is addressed by physically modifying the interpolation function. The corresponding consistent scheme is developed to solve the model, and the numerical results agree well with the analytical solutions and the existing experimental and numerical data. Two challenging problems having a wide range of material properties and complex dynamics are conducted to demonstrate the capability of the proposed model.

The Coupled Volume of Fluid and Brinkman Penalization Methods for Simulation of Incompressible Multiphase Flows

In this work, we contribute to the development of numerical algorithms for the direct simulation of three-dimensional incompressible multiphase flows in the presence of multiple fluids and solids. The volume of fluid method is used for interface tracking, and the Brinkman penalization method is used to treat solids; the latter is assumed to be perfectly superhydrophobic or perfectly superhydrophilic, to have an arbitrary shape, and to move with a prescribed velocity. The proposed algorithm is implemented in the open-source software Basilisk and is validated on a number of test cases, such as the Stokes flow between a periodic array of cylinders, vortex decay problem, and multiphase flow around moving solids.

The variable‐extended immersed boundary method for compressible gaseous reactive flows past solid bodies

AbstractAn immersed boundary method (IBM) has been developed to handle the solid body embedded flowfield simulation for compressible reactive flows, paving the way of application for a wide range of fluid–solid interaction problems. Previously, the Brinkman penalization method (BPM), originated from porous media flows, has been successfully used for incompressible Navier–Stokes equations by adding penalization terms to momentum equations. However, it is nontrivial to solve the compressible form due to the penalized continuity equation that usually poses severe numerical stiffness. In order to circumvent this issue, an extending procedure for relevant variables from the fluid to solid domain is considered, by analyzing the ordinary differential equations remained after operator splitting. Density can be then determined with the help of an equation of state (EoS). Meanwhile, efforts of enforcing the Neumann boundary condition, for example, the adiabatic wall condition, on the fluid–solid interface can be minimized by extending temperature across the interface directly. One more advantage of the extending step lies in that it can quickly reach a steady state when performed within a narrow band around the interface. Implemented into an adaptive Cartesian grid‐based flow solver for compressible Navier–Stokes equations with chemical reaction source terms, the present variable‐extended IBM is validated by numerical examples ranging from single‐species nonreactive to multispecies detonative flows in one‐ and two‐dimensional domains. Numerical results show (1) the successful specification of slip or nonslip, adiabatic or isothermal wall condition on the fluid–solid interface and (2) loss of total energy in the original BPM being avoided and the numerical accuracy being improved especially for energy‐sensitive reactive flows.

Simulating water-entry/exit problems using Eulerian–Lagrangian and fully-Eulerian fictitious domain methods within the open-source IBAMR library

In this paper we employ two implementations of the fictitious domain (FD) method to simulate water-entry and water-exit problems and demonstrate their ability to simulate practical marine engineering problems. In FD methods, the fluid momentum equation is extended within the solid domain using an additional body force that constrains the structure velocity to be that of a rigid body. Using this formulation, a single set of equations is solved over the entire computational domain. The constraint force is calculated in two distinct ways: one using an Eulerian–Lagrangian framework of the immersed boundary (IB) method and another using a fully-Eulerian approach of the Brinkman penalization (BP) method. Both FSI strategies use the same multiphase flow algorithm that solves the discrete incompressible Navier–Stokes system in conservative form. A consistent transport scheme is employed to advect mass and momentum in the domain, which ensures numerical stability of high density ratio multiphase flows involved in practical marine engineering applications. Example cases of a free falling wedge (straight and inclined) and cylinder are simulated, and the numerical results are compared against benchmark cases in literature.

Utilization of the Brinkman Penalization to Represent Geometries in a High-Order Discontinuous Galerkin Scheme on Octree Meshes

We investigate the suitability of the Brinkman penalization method in the context of a high-order discontinuous Galerkin scheme to represent wall boundaries in compressible flow simulations. To evaluate the accuracy of the wall model in the numerical scheme, we use setups with symmetric reflections at the wall. High-order approximations are attractive as they require few degrees of freedom to represent smooth solutions. Low memory requirements are an essential property on modern computing systems with limited memory bandwidth and capability. The high-order discretization is especially useful to represent long traveling waves, due to their small dissipation and dispersion errors. An application where this is important is the direct simulation of aeroacoustic phenomena arising from the fluid motion around obstacles. A significant problem for high-order methods is the proper definition of wall boundary conditions. The description of surfaces needs to match the discretization scheme. One option to achieve a high-order boundary description is to deform elements at the boundary into curved elements. However, creating such curved elements is delicate and prone to numerical instabilities. Immersed boundaries offer an alternative that does not require a modification of the mesh. The Brinkman penalization is such a scheme that allows us to maintain cubical elements and thereby the utilization of efficient numerical algorithms exploiting symmetry properties of the multi-dimensional basis functions. We explain the Brinkman penalization method and its application in our open-source implementation of the discontinuous Galerkin scheme, Ateles. The core of this presentation is the investigation of various penalization parameters. While we investigate the fundamental properties with one-dimensional setups, a two-dimensional reflection of an acoustic pulse at a cylinder shows how the presented method can accurately represent curved walls and maintains the symmetry of the resulting wave patterns.

A penalization method for the simulation of weakly compressible reacting gas-particle flows with general boundary conditions

In this paper, the Brinkman penalization method is extended to the weakly compressible formulation of the Navier–Stokes equations to study gas-particle flows with reactions and coupled heat and mass transfer. The weakly compressible approximation removes the acoustic modes from the solution while allowing a variable density. The Brinkman volume penalization method describes the solid phase as a porous medium with a vanishing permeability. The method is validated with literature benchmarks for a fixed or moving particle. Various reaction scenarios are then investigated. First, the capability of the method to deal with intra-particle diffusion and reaction is evaluated. The impact on the conversion of the particle Reynolds number based on the slip velocity is assessed, for a range of Reynolds numbers encountered in fluidization. Next, surface reactions are focused on, with infinite or finite rate. A Dirichlet-type condition is imposed on the particle surface to treat an infinite reaction rate and a high solid thermal conductivity while a finite-rate surface reaction requires Neumann and Robin surface conditions for the temperature and species mass fractions. The impact of density changes induced by heat release and molecular weight changes is assessed in these different cases. It is shown that the combination of the weakly compressible approximation and the penalization method allows to treat the presence of the solid phase and reactions in an efficient manner and to take into account the effects of compressibility in commonly encountered situations in reaction engineering.

Iterative Brinkman penalization for simulation of impulsively started flow past a sphere and a circular disc

We present a Brinkman penalization method for three-dimensional (3D) flows using particle vortex methods, improving the existing technique by means of an iterative process. We perform simulations to study the impulsively started flow past a sphere at Re=1000 and normal to a circular disc at Re=500. The simulation results obtained for the flow past a sphere are found in qualitative good agreement with previously published results obtained using respectively a 3D vortex penalization method and a 3D vortex method combined with an accurate boundary element method. From the results obtained for the flow normal to a circular disc it is found that the iterative method enables the use of a time step that is one order of magnitude larger than required by the standard non-iterative Brinkman penalization method.

A comparative study of Brinkman penalization and direct-forcing immersed boundary methods for compressible viscous flows

This paper deals with the comparison between two methods to treat immersed boundary conditions: on the one hand, the Brinkman penalization method (BPM); on the other hand, the direct-forcing method (DFM). The penalty method treats the solid as a porous medium with a very low permeability. It provides a simple and efficient approach for solving the Navier–Stokes equations in complex geometries with fixed boundaries or in the presence of moving objects. A new approach for the penalty-operator integration is proposed, based on a Strang splitting between the penalization terms and the convection-diffusion terms. Doing so, the penalization term can be computed exactly. The momentum term can then be computed first and then introduced into the continuity equation in an implicit manner. The direct-forcing method however uses ghost-cells to reconstruct the values inside the solid boundaries by projection of the image points from the interface. This method is comparatively hard to implement in 3D cases and for moving boundaries. In the present paper, the performance of both methods is assessed through a variety of test problems. The application concerns the unsteady transonic and supersonic fluid flows. Examples include a normal shock reflection off a solid wall, transonic shock/boundary layer in a viscous shock tube, supersonic shock/cylinder interaction, and supersonic turbulent channel flow. The obtained results are validated against either analytical or reference solutions. The numerical comparison shows that, with sufficient mesh resolution, the BPM and the DFM methods yield qualitatively similar results. In all considered cases, the BPM is found to be a suitable and a possibly competitive method for viscous-IBM in terms of predictive performance, accuracy and computational cost. However, despite its simplicity, the method suffers from a lack of regularity in the very near-wall pressure fluctuations, especially for the turbulent case. This is attributed to the fact that the method requires no specific pressure condition at the fluid/solid interface.

Iterative Brinkman penalization for remeshed vortex methods

We introduce an iterative Brinkman penalization method for the enforcement of the no-slip boundary condition in remeshed vortex methods. In the proposed method, the Brinkman penalization is applied iteratively only in the neighborhood of the body. This allows for using significantly larger time steps, than what is customary in the Brinkman penalization, thus reducing its computational cost while maintaining the capability of the method to handle complex geometries. We demonstrate the accuracy of our method by considering challenging benchmark problems such as flow past an impulsively started cylinder and normal to an impulsively started and accelerated flat plate. We find that the present method enhances significantly the accuracy of the Brinkman penalization technique for the simulations of highly unsteady flows past complex geometries.

Adaptive wavelet collocation method on the shallow water model

This paper presents an integrated approach for modeling several ocean test problems on adaptive grids using novel boundary techniques. The adaptive wavelet collocation method solves the governing equations on temporally and spatially varying meshes, which allows higher effective resolution to be obtained with less computational cost. It is a general method for the solving a large class of partial differential equations, but is applied to the shallow water equations here. In addition to developing wavelet-based computational models, this work also uses an extension of the Brinkman penalization method to represent irregular and non-uniform continental boundaries. This technique is used to enforce no slip boundary conditions through the addition of a term to the field equations. When coupled with the adaptive wavelet collocation method, the flow near the boundary can be well resolved. It is especially useful for simulations of boundary currents and tsunamis, where flow and the boundary is important, thus, those are the test cases presented here.

Brinkman Penalization Method를 통한 복잡한 3D 형상 주위의 음향 전파 연구

Sound scattering in 3D complex geometry is difficult to model with body-fitted grid. Thus Brinkman Penalization method is used to compute sound scattering in 3D complex geometry. Sound propagation of monitor/TV is studied. The sound field for monitor/TV is simulated by applying Brinkman Penalization method to Linearized Euler Equation. Solid Structure and ambient air are represented as penalty terms in Linearized Euler Equation.

On the use of Brinkman penalization method for computation of acoustic scattering from complex boundaries

In this study, a Brinkman penalization method (BPM) is extended for prediction of acoustic scattering from complex geometries. The main idea of the BPM is to model the solid obstacle as a porous material with zero porosity and permeability. With the aim of increasing the spatial accuracy at the immersed boundaries, computation is carried out on the boundary-fitted Cartesian-like grid with a high-order compact scheme combined with one-side differencing/filtering technique at the boundaries, while a slip boundary condition at the wall is imposed by introducing the ‘anisotropic’ penalization terms to the momentum equations. Several test cases are considered to demonstrate the accuracy, robustness and feasibility of the BPM. Numerical results are in excellent agreement with the analytic solutions for single and two cylinder scattering problems. The present BPM is then used to solve the acoustic scattering from a three-element high-lift wing (30P30N model).

A Brinkman penalization method for compressible flows in complex geometries

To simulate flows around solid obstacles of complex geometries, various immersed boundary methods had been developed. Their main advantage is the efficient implementation for stationary or moving solid boundaries of arbitrary complexity on fixed non-body conformal Cartesian grids. The Brinkman penalization method was proposed for incompressible viscous flows by penalizing the momentum equations. Its main idea is to model solid obstacles as porous media with porosity, ϕ, and viscous permeability approaching zero. It has the pronounced advantages of mathematical proof of error bound, strong convergence, and ease of numerical implementation with the volume penalization technique. In this paper, it is extended to compressible flows. The straightforward extension of penalizing momentum and energy equations using Brinkman penalization with respective normalized viscous, η, and thermal, η T, permeabilities produces unsatisfactory results, mostly due to nonphysical wave transmissions into obstacles, resulting in considerable energy and mass losses in reflected waves. The objective of this paper is to extend the Brinkman penalization technique to compressible flows based on a physically sound mathematical model for compressible flows through porous media. In addition to penalizing momentum and energy equations, the continuity equation for porous media is considered inside obstacles. In this model, the penalized porous region acts as a high impedance medium, resulting in negligible wave transmissions. The asymptotic analysis reveals that the proposed Brinkman penalization technique results in the amplitude and phase errors of order O(( ηϕ) 1/2) and O(( η/ η T) 1/4 ϕ 3/4), when the boundary layer within the porous media is respectively resolved or unresolved. The proposed method is tested using 1- and 2-D benchmark problems. The results of direct numerical simulation are in excellent agreement with the analytical solutions. The numerical simulations verify the accuracy and convergence rates.

Hybrid wavelet collocation–Brinkman penalization method for complex geometry flows

AbstractThis paper introduces the hybrid wavelet collocation–Brinkman penalization method for solving the Navier–Stokes equations in arbitrarily complex geometries. The main advantages of the wavelet collocation method and penalization techniques are described, and a brief summary of their implementation is given. The hybrid method is then applied to a simple flow around cylinders with both periodic and non‐periodic boundary conditions with Reynolds numbers in the range from 100 to 10 000 and the results are discussed. Copyright © 2002 John Wiley & Sons, Ltd.