AN IMMERSED BOUNDARY METHOD TO SOLVE FLOW AND HEAT TRANSFER PROBLEMS INVOLVING A MOVING OBJECT

Ming-Jyh Chern^*,§, Dedy Zulhidayat Noor^* and Tzyy-Leng Horng^**

* Department of Mechanical Engineering, National Taiwan University of Science and Technology, Taipei, Taiwan

** Department of Applied Mathematics, Feng Chia University, Taichung, Taiwan

§Correspondence author. Fax: +886-2-27376460 Email: mjchern@mail.ntust.edu.tw

ABSTRACT

A direct-forcing immerse boundary method with both virtual force and heat source is developed here to solve Navier-Stokes and the associated energy transport equations to study some thermal flow problems caused by a moving rigid solid object within. The key point of this novel numerical method is that the solid object, stationary or moving, is first treated as fluid governed by Navier-Stokes equations for velocity and pressure, and by energy transport equation for temperature in every time step. An additional virtual force term is then introduced on the right hand side of momentum equations in the solid object region to make it act exactly as if it were a solid rigid body immersed in the fluid. Likewise, an additional virtual heat source term is applied to the right hand side of energy equation at the solid object region to maintain the solid object at the prescribed temperature all the time. The current method was validated by some benchmark forced and natural convection problems such as a uniform flow past a heated circular cylinder, and a heated circular cylinder inside a square enclosure. We further demonstrated this method by studying a mixed convection problem involving a heated circular cylinder moving inside a square enclosure. Our current method avoids the otherwise requested dynamic grid generation in traditional method and shows great efficiency in the computation of thermal and flow fields caused by fluid-structure interaction.

NOMENCLATURE

dimensionless amplitude dimensionless displacement dimensionless diameter virtual force

total force acting on a solid body total heat transfer over body surface dimensionless length

7 volume of a solid object dimensionless area

horizontal and vertical cartesian coordinate dimensionless recirculation length

Greek symbols

non-dimensional oscillation angular frequency,

oscillation angular frequency, s non-dimensional temperature uniform flow past cylinders and wind past rotating blades. Simulations of fluid-structure interactions are challenging and difficult. First of all, the configuration of a structure is often complex, so a distorted or unstructured grid is necessary. In addition, given that the solid structure moves or rotates, grids will have to be re-generated due to the movement of the solid structure. The flow and heat transfer in a jet engine is a typical fluid-structure interaction problem. Air enters to the engine and flows past a series of rotating blades and stators. Fuel is burned and combustion occurs in the engine, so heat is generated and transmitted by the products of combustion to the rotating blades. Simulations of the air flow and heat transfer in the jet engine become difficult due to the complex configuration of the passageway and fast rotating blades. That is, the fluid domain always changes and its mesh generation is costly.

To predict fluid-structure interactions accurately, a variety of computational methods have been proposed. The most common method to simulate the flow with a complicated solid boundary is to use a body-fitted technique with grids fitting and clustering along the complex boundary. Most of time, the solid object may not be at rest and it requires further technique to deal with a moving object. The Arbitrary Lagrangian Eulerian (ALE) numerical method is a popular approach to accommodate the complicated fluid-structure interface varying with time. In the Eulerian coordinate frame, fluids flow through the static computational mesh. While in the Lagrangian coordinate, the mesh moves with the solid. Arbitrary Lagrangian-Eulerian (ALE) methods introduced by Hirt et al. [1974] appear to be a reasonable compromise between Lagrangian and Eulerian approaches. The ALE method consists of several Lagrangian computational time steps followed by a mesh rezoning and a conservative quantities remapping. The mesh rezoning step smoothes the Lagrangian computational mesh and avoids its distortion. During the remapping step the conservative quantities are conservatively remapped from the old Lagrangian mesh to the new smooth one. After remapping the Lagrangian computation continues until the next rezone/remap steps which introduce the Eulerian flavor into the method allowing mass flux between computational cells. The rezone/remap steps keep the quality of the moving mesh good enough during the whole computation and are performed either regularly after fixed amount of Lagrangian time steps or when mesh quality deteriorates under some threshold. Many scholars have described ALE strategies to optimize accuracy, robustness, or computational efficiency (see Bailey et al.

2010, Kucharik et al. 2006, Bension 1992, Kjellgren and Hyvarinen 1998). Nevertheless, mesh updating or re-meshing is computationally expensive for the ALE algorithm.

In addition to the ALE algorithm, the immersed boundary method is becoming popular since it was first introduced by Peskin [1972] due to its capability to handle simulations for a moving complex boundary with lower computational cost and memory requirements than the conventional body-fitted method. In this method, a fixed Cartesian grid and a Lagrangian grid are employed for fluids and immersed solid object, respectively.

The interaction between fluids and the immersed solid boundary is linked through the spreading of the singular force from the Lagrangian grid to the Cartesian grid and the interpolation of the velocity from the Cartesian grid to the Lagrangian grid using a discrete Dirac delta function. Furthermore, some modifications and improvements of this method have been proposed by other researchers (Goldstein et al. 1993, Saiki and Biringen 1996, Lai and Peskin 2000, Su et al. 2007). This method can be categorized as a continuous forcing method in which a forcing term is added to the continuous Navier-Stokes equations before they are discretized.

Instead of using a delta function to distribute force from the Lagrangian grid to the Cartesian grid, Yusouf [1996] introduced a novel immersed boundary method, namely the direct-forcing method. This method uses a virtual forcing term determined by the difference between the interpolated velocities at the boundary points and the desired boundary velocities. This method is also known as a discrete forcing method since the forcing is either explicitly or implicitly applied to the discretized Navier-Stokes equations. The idea of the direct-forcing method has been used and developed successfully in many applications(Ye 1999, Fadlun 2000, Tseng and Ferziger 2003, Zhang and Zheng 2007, Noor et al. 2009). An impressive example is the work by Fadlun et al. [2000] who developed a model combining the immersed boundary method and the finite difference method for three dimensional complex flow simulations. Some cases including the flow inside an IC piston/

cylinder assembly at high Reynolds number were simulated successfully. The major issue of their work is the interpolation of velocity inside solids via virtual force to uphold the second-order accuracy of the scheme.

For immersed boundary methods applied to heat transfer problem, Paravento et al. [2007] introduced an immersed boundary method for heat convection problems. They solved both a hot and insulated square body located in a 2D channel. follow, this method is restricted to that the solid boundary must align. Vega et al.

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using an immersed boundary method. The momentum and energy forcing terms are imposed into momentum and energy equations followed by the projection method to uphold solve the divergence free condition in a non-staggered grid configuration. Some cases regarding heat conduction, forced and natural convection were simulated and validated with success. Pan [2006] introduced an immersed boundary method on unstructured Cartesian meshes for incompressible flows with heat transfer. The solid body is identified by a volume-of-body (VOB) function analogous to the volume-of-fluid (VOF) function. This VOB approach can also be applied to the energy equation with a Dirichlet boundary condition.

In this present work, we propose an immersed boundary method to simulate fluid-structure interactions with heat transfer via the direct-forcing method. The solid object immersed within a flow field can be denoted by the volume of solid function . A cell fully occupied by solids will be denoted as = 1, while the one fully occupied by fluids will be = 0. A cell occupied party by fluids and solids, commonly called a cut cell, will be denoted by a fractional which is the volume fraction of solids in the cell. The momentum and energy equations for fluids are solved simultaneously with extra virtual forcing and energy source terms for a solid regime to achieve desired boundary conditions on fluids imposed by solids. Numerical details will be described in the following section and several flow and heat transfer problems involving a moving body are computed to demonstrate the capability of the present scheme in handling fluid-solid interactions with heat transfer.