A computational strategy for simulating heat transfer and flow past deformable objects

https://doi.org/10.1016/j.ijheatmasstransfer.2007.11.055Get rights and content

Abstract

Simulations of flow and heat transfer around deforming objects require the accurate resolution of the moving interface. An approach that combines the Hybrid Immersed Boundary Method (HIBM) for handling complex moving boundaries and the Material Point Method (MPM) for resolving structural stresses and the movement of the deformable body is presented here. In the HIBM, a fixed Eulerian, curvilinear grid is generally defined, and the variable values at grid points adjacent to a curvilinear boundary are interpolated to satisfy the boundary conditions. The MPM is used to solve equations of the solid structure (stresses and deflection) and communicates with the flow equations through appropriate interface-boundary conditions. As a validation of the new approach for heat transfer problems, flow and heat transfer past a rigid and deforming isothermal sphere is simulated. Predictions agree well with published results of Nusselt number for flow past a rigid sphere.

Introduction

In many applications involving heat transfer, surfaces that deform under the action of the fluid flow are encountered. These problems which involve fluid–structure interaction (FSI) require specific treatment in the vicinity of the interface. Examples include applications in thermal sprays, injection molding, and polymer processing.

Numerical approaches for solving FSI problems are broadly classified as: fixed-grid (Eulerian) or moving-grid (Lagrangian or Arbitrary Lagrangian–Eulerian) methods [1]. Fixed-grid methods generally embody a surface-capturing strategy [2], and the interface has a non-zero thickness and is diffuse [1]. Moving-grid methods belong to the surface-tracking family, since with these approaches the interface is maintained to be sharp with an essentially zero thickness. Popular moving-grid methods for solving FSI problems are the Lagrangian [3] and Arbitrary LagrangianEulerian (ALE) methods [4], [5]. A purely Lagrangian method was employed by Belytschko and Kennedy [6], and Donea et al. [7] to study hydro-structural interactions. However, ALE methods are more popular, since they use a moving-grid that follows the deforming boundaries and allows the resolution needed near the boundary [4], [5], [8]. However, due to the need for the mesh to conform to the body at all times, they are inherently limited to problems with moderate body deformations.

Fixed-grid approaches have been widely used due to the ease of generating a fixed-grid. Different strategies with a fixed-grid have been proposed. In the Cut-Cell Method [9], [10], [11], [12], [13], [14], [15] the boundary cells and fluxes adjacent to the complex interface are redefined at each step. In the Volume of Fluid Method (VOF) [16], [17] the interface is reconstructed from the fractional volume of fluid content in each cell through special surface functions which are used to distinguish one fluid from another. Level Set Methods (LSM) were introduced by Osher and Sethian [18] and rely on an implicit formulation of the interface whose zero-level set always gives the location of the propagating interface through a solution of the time-dependent initial-value problem. These methods are attractive because they admit a convenient description of topologically complex interfaces and are quite simple to implement [19], [20]. The Fictitious Domain Method (FDM) was introduced by Saul’ev [21] and has been primarily applied to the interaction of fluid with rigid body particles by Glowinski et al. [22], [23]. The main idea of FDM consists of coupling of moving rigid particles with fluid by using a Lagrangian multiplier. A new method combining the fictitious domain [22] and the mortar element [24] methods for the computational analysis of fluid–structure interaction of a Newtonian fluid with slender bodies was developed by Baaijens [25]. An extension of this approach was used to describe the motion of a large leaflet and its interaction with the surrounding fluid [26].

The Immersed Boundary Method (IBM) was introduced by Peskin [27] to study the flow in a heart valve. The idea was very useful to solve FSI problems that included the free movement of a structure through a fluid domain. The interaction between the fluid and a deformable body was realized through nodal forces at selected grid points incorporated in the momentum equations. These external terms were spread over the computational domain through smoothed approximation of the Dirac delta function and satisfied the boundary conditions on the surface. This approach is associated with some disadvantages inherent to the diffuse-interface methods. For example, IBM is only first-order accurate in space and the boundary spreads over 3–5 grid nodes. A short review of Immersed Boundary and Cartesian Cut Methods for flows with moving boundaries was recently published by Mittal and Iaccarino [28].

The Immersed Interface Method (IIM) was designed by LeVeque and Li [29] to further develop the IBM of Peskin [30]. Instead of using a smooth approximation of the delta function, the IIM used approximations of the delta function with discontinuity across the boundary (jump conditions). Thus, the IIM is classified as a sharp-interface method. In [31], [32] it was shown that IIM has second-order accuracy and is free from shortcomings of IBM. The Immersed Finite Element Method was developed by Zhang et al. [33] where the fluid and solid body are modeled with the finite element method. To avoid expensive grid regeneration, a fixed Eulerian grid for the fluid was used. The connection between the Lagrangian solid body and the fluid was implemented as in the IBM, but instead of the Dirac delta function, the higher-order reproducing kernel particle method (RKPM) [34] delta function was used.

A variant of the classical IBM [27] approach that does not require the explicit addition of discrete forces to the governing equations was developed by Mohd-Yusof [35] and Fadlun et al. [36]. This approach treats the solid boundary as a sharp interface. The specific values of various flow variables at the near-boundary nodes are calculated by interpolating linearly along an appropriate grid line between the nearest interior node, where flow variables are available from the solution of the governing equations, and the point where the grid line intersects the solid boundary, where physical boundary conditions are known. This approach can be classified as a Hybrid Cartesian-Immersed Boundary (HCIB) approach [36].

A new HCIB formulation applicable to three-dimensional flows with arbitrarily complex immersed boundaries moving with prescribed motion was developed in [37]. This methodology maintains a sharp fluid-body interface by discretizing the body surface using an unstructured, triangular mesh. The nodes of this mesh constitute a set of Lagrangian control points, which are used to track the motion and reconstruct the instantaneous shape of the moving immersed boundary. The reconstruction of the solution at the near-boundary nodes is carried out using interpolation along the normal to the body [38].

The most common strategy for solving structural deformation in problem involving fluid–structure interaction is the finite element method (FEM) [39]. In contrast to these studies, the material point method (MPM) [40] has certain advantage over standard FEM including the ability to handle large structural deformations. While the MPM has been demonstrated in a number of studies to be an effective strategy for solid objects, its use in resolving FSI is limited to that of York et al. [41] who utilized an MPM approach both the fluid and solid.

The purpose of this study is to develop a numerical method to simulate heat transfer problems for deformable bodies moving and interacting with the surrounding fluid. Our method combines HCIB method [37] to resolve the flow around a body with complex shape and MPM [41] to solve for the deformation of the solid structure moving under forces from the surrounding fluid. Instead of HCIBM which uses a Cartesian grid, we use the term HIBM to emphasize that we have realized this method in general curvilinear grid [42], [43]. The method is validated by solving for sphere falling in a channel/box under the action of gravitational forces. The predicted data is in excellent agreement with experimental data of Cate et al. [44]. Because a full system of HIBM & MPM equations was solved (on the assumption that solid body has a high rigidity) this test validates FSI algorithm. For the validation of the heat transfer problem, a steady flow past a hot sphere was considered and predictions were compared with data of Bagchi et al. [45]. The calculated Nusselt numbers are in good agreement with the cited data. The HIBM & MPM approach was finally applied to a deforming sphere and the difference in heat transfer between a rigid and deforming sphere are presented and discussed.

The focus of this paper is to demonstrate the applicability of the HIBM and MPM strategy for FSI problems involving heat transfer. To our knowledge, this is the first application of such an approach for heat transfer problems.

Section snippets

Governing equations

The specific aim of the present paper is to develop and validate an efficient numerical method for simulating unsteady, three-dimensional flows and heat transfer for complex and deformable bodies. The unsteady, 3D, incompressible Navier–Stokes (NS) equations are solved using an efficient finite-difference method that is second-order accurate both in space and time. A hybrid approach that combines curvilinear grids, and the immersed boundary method was used to develop a powerful and very general

Solution of the fluid equations

The discrete equations are integrated in time via a second-order accurate dual time-stepping, artificial compressibility iteration scheme. To solve the system of governing equations (1), (2), (3), a pressure-based Residual Smoothing Operator, Multistage Pseudocompressibility Algorithm developed by Sotiropoulos and Constantinescu [47] was used. This approach incorporates the idea of combining pressure-based method [48] and the Artificial Compressibility [49] method to get an efficient diagonal

Validation tests

The goal of the present paper is to validate the combined HIBM & MPM as a strategy for solving FSI problems with strong structural deflections and heat transfer between the deformable body and fluid. In this section we will present several validation studies and compare with available data/theory/published simulations to demonstrate the accuracy of the approach.

Heat transfer from a solid and deformable sphere

As a final example involving heat transfer, we consider a hot deformable sphere falling in a channel under the action of a gravitational force. A number of practical examples that involve moving hot spheres include spray cooling, plasma sprays, quenching of ball bearings, etc. For simplicity, we consider here the case of an isothermal sphere, and examine the Nusselt number around the sphere with deformations. While we have considered isothermal sphere in this paper, conjugate heat transfer

Concluding remarks

A numerical method for simulating Fluid–Structure Interaction problems with heat transfer has been presented. The method developed uses the Immersed Boundary Method for resolving complex boundaries for the fluid flow, and couples this with the Material Point Method for the structural stresses and deformation. The methodology is ideally suited for flow problems with initially complex geometries where the surfaces undergo large structural deformation. The methodology is demonstrated on a number

Acknowledgements

This work was supported by a Health Excellence Grant to the Biological Computation and Visualization Center (BCVC) by the State of Louisiana. Their support is greatly appreciated.

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