DNS of buoyancy-dominated turbulent flows on a bluff body using the immersed boundary method

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Abstract

A novel immersed boundary (IB) method has been developed for simulating multi-material heat transfer problem – a cylinder in a channel heated from below with mixed convection. The method is based on a second-order velocity/scalar reconstruction near the IB. A novel algorithm has been developed for the IB method to handle conjugate heat transfer. The fluid–solid interface is constructed as a collection of disjoint faces of control volumes associated to different material zones. Coupling conditions for the material zones have been developed such that continuity and conservation of the scalar flux are satisfied by a second-order interpolation. Predictions of the local Nusselt number on the cylinder surface show good agreement with the experimental data. The effect of the Boussinesq approximation on this problem was also investigated. Comparison with the variable density formulation suggests that, in spite of a small thermal expansion coefficient of water, the variable density formulation in a transitional flow with mixed convection is preferable.

Introduction

The immersed boundary (IB) method is a numerical technique to enforce boundary conditions on surfaces not aligned with the mesh. This method has become popular for flow problems involving very complex geometries often encountered in engineering applications. Compared to a body-fitted mesh, the IB method is advantageous in three situations. First, it provides an alternative to body-fitted mesh for very complex geometry; in this case, the amount of user interventions and turn-around time for mesh generation can be reduced by using a simpler mesh structure. The second advantage relates to problems associated with moving geometries. Since the effect of geometry is imposed in the solution step, it is relatively straightforward to include boundary motion and possible interactions between fluid flow and structures. Another advantage presents itself in multi-phase or multi-material problems. More specifically, the interface between different materials can be regarded as an immersed boundary. The IB method is then equivalent to the imposition of physical conditions at the interface.

The IB method was first introduced by Peskin [28] for computing blood flow in the cardiovascular system. Subsequently, there have been numerous efforts to enhance the accuracy, stability and range of applicability of the IB method. Readers can refer to articles by [24], [15] for information on the previous studies. So far, IB methods have been applied to a wide range of applications: compressible flows [7], [22], particulate flows [36], [38], micro-scale flows [1], interaction with solid bodies ([8], [40], among others), multi-phase flows [6], conjugate heat transfer [14], [39], environmental flows [33], bio-fluids [5], etc. However, the number of published studies which validated near-wall statistics of turbulent flows is relatively small.

In the present study, we explore a turbulent conjugate heat transfer problem where the modes of convective and conductive heat transfer are handled simultaneously. Problems with conjugate heat transfer are very common in industrial applications, thus may benefit greatly from easier mesh generation accomplished by the IB method.

In the literature, conjugate heat transfer in turbulent flows has been widely investigated. The majority of the studies have used RANS-based models ([37], [26], [13], [12], among others). Very few studies have used LES or DNS for conjugate heat transfer problems. Tiselj et al. [35] applied DNS to a conjugate heat transfer between a turbulent channel flow and a solid wall. This study yielded improved results compared to those obtained with RANS. Smirnova and Kalaev [32] applied a LES approach to a problem of crystal growth. Both studies used a body-fitted methods. Iaccarino and Moreau [14] applied the IB method to conjugate heat transfer problems using a RANS model. Yu et al. [39] applied a distributed Lagrange multiplier-based fictitious domain method to a conjugate heat transfer problem of a particulate flow at low Reynolds number. Song et al. [34] used an IB method to investigate the effect of thermal resistance of solid wall on turbulent heat transfer in a ribbed channel. Certainly, there are very few previous studies that have used an IB method for LES/DNS of a conjugate heat transfer problem.

The objective of the present study is to assess the accuracy and efficiency of the IB method for LES/DNS of a turbulent conjugate heat transfer problem. For this purpose, a novel method is developed by extending an existing IB method to multi-material problems and verified using a simple analytic solution. The present validation study is focused on a mixed, conjugate, transitional heat transfer problem around a heated cylinder in a channel heated from below [21]; we compare the present computation to the experiment for the near-wall temperature field, the transition to turbulence due to thermal instabilities, and the validity of the Boussinesq approximation.

In the next section, description of numerical schemes – the flow solver, the IB method and extension to multi-material problems – are presented. Section 3 shows a verification study to test accuracy of the developed IB method. Results from simulations of a heated cylinder are presented in Section 4. From parametric studies on flow and computational conditions, conclusions are derived in Section 5.

Section snippets

Description of the Navier–Stokes solvers

In the present study, a heat transfer problem with mixed convection is considered. The variable density formulation of the Navier–Stokes equations is written as:ρuit+ρuiujxj=-pxi+xjμuixj+ujxi+ρgi,ρt+ρuixi=0,ρht+ρujhxj=xjkTxj,where t is the time, ρ is the density, ui is the velocity, p is the pressure, μ is the molecular viscosity, and gi is the vector of the gravitational acceleration. T is the temperature, h = cpT is the enthalpy, and k is the thermal conductivity. Note

Flow around a heated sphere

A steady flow around a heated sphere was computed to validate the IB method in a heat transfer problem. When a sphere positioned in a quiescent fluid is heated to a temperature higher than the ambient, the density gradient in the fluid drives the flow upward (natural convection). The local Nusselt number on the sphere surface is compared to results from simulations with a body-fitted mesh.

The Grashof number (Gr) based on the radius (R) of the sphere is 104, the Reynolds number is Re = Gr0.5, and

A heated cylinder in a channel heated from below

In this section, the IB method is applied to a turbulent conjugate heat transfer problem investigated both experimentally and numerically by Laskowski et al. [21]. This problem involves mixed convection, transition to turbulence, and conjugate heat transfer. The most important metric for validation is the time-averaged heat flux at the interface between the fluid and the solid.

Conclusions

The immersed boundary (IB) method has become popular for flow problems involving very complex geometries and moving bodies. In the present study, we utilized its capability of handling multi-material problems and assessed the IB method as an efficient tool for a turbulent conjugate heat transfer problem.

For multi-material problems, an approach based on immersed discontinuous grids has been developed. The fluid–solid interface consists of adjoining Cartesian faces from heterogeneous material

Acknowledgment

Financial support from the Department of Energy under Advanced Simulation and Computing (DOE-ASC) program is gratefully acknowledged. We also thank the authors of Laskowski et al. [21] for their experimental data and suggestions.

References (40)

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