Diffuse-interface immersed-boundary framework for conjugate-heat-transfer problems

A monolithic solver based on a diffuse-interface immersed-boundary (IB) approach for conjugate-heat-transfer (CHT) problems is presented. The IB strategy assumes that the solid which is “immersed” into the computational grid is occupied by a “virtual” fluid to facilitate construction of “unified” governing equations that are solved everywhere in the domain. A unified momentum equation is devised using the solid volume fraction that reduces to the Navier-Stokes equation outside of the solid and to the no-slip boundary condition inside of it. The “unified” energy equation is constructed in an analogous fashion reducing to a convective-diffusive equation in the fluid domain and a fully diffusive equation in the solid domain with different thermal conductivities (or diffusivities) for both domains. The resulting equations are solved in both domains simultaneously using a hybrid staggered and nonstaggered finite-volume (FV) framework for incompressible flows. The second-order accurate IB-FV solver is employed to carry out investigations for CHT problems in natural and forced convective regimes. Numerical studies for different fluid-to-solid conductivity ratios show that the monolithic IB-CHT solver is a fast, simple, and accurate framework for simulations of CHT problems for Boussinesq flows.

Figure 1

Node classification ${\mathbf{d}}_{\mathbf{s}}·\mathbf{n}>0\Rightarrow \mathrm{S}$ is solid node, ${\mathbf{d}}_{\mathbf{f}}·\mathbf{n}<0\Rightarrow$ F fluid node.

Figure 2

Cell identification.

Figure 3

Illustration of solid fraction computation for “I” cells. In this case, ${\varphi }_B=\frac{13}{25}=0.52$.

Figure 4

Time histories of force coefficients at different Re.

Figure 5

Schematic diagram of Backward facing step.

Figure 6

Comparison of velocity profile at different section with Pareschi et al.[29] at $\text{Re}=800$.

Figure 7

Comparison of temperature profile at $\frac{x}{h}=3$ with Pareschi et al. [29] for $\frac{K_s}{K_f}=10$ at $\text{Re}=800$.

Figure 8

Comparison of temperature profile at $\frac{x}{h}=3$ with Pareschi et al. [29] for $\frac{K_s}{K_f}=100$ at $\text{Re}=800$.

Figure 9

Comparison of interface temperature with Pareschi et al. [29] for $\frac{K_s}{K_f}=10$ at $\text{Re}=800$.

Figure 10

Comparison of interface temperature with Pareschi et al. [29] for $\frac{K_s}{K_f}=100$ at $\text{Re}=800$.

Figure 11

Comparison of Nusselt number profile cross the interface of solid-fluid with Pareschi et al. [29] for $\frac{K_s}{K_f}=10$ at $\text{Re}=800$.

Figure 12

Comparison of Nusselt number profile cross the interface of solid-fluid with Pareschi et al. [29] $\frac{K_s}{K_f}=100$ at $\text{Re}=800$.

Figure 13

Schematic diagram of flow past a nonhomogeneous cylinder.

Figure 14

Temperature profile along fluid-solid interface of a circular cylinder at $\text{Re}=40$.

Figure 15

Local Nusselt number distribution along fluid-solid interface of a circular cylinder at $\text{Re}=40$.

Figure 16

Local Nusselt number distribution along fluid-solid interface of a circular cylinder for different cell width at $\text{Re}=40$ and $\frac{K_s}{K_f}=20$.

Figure 17

Local Nusselt number distribution along fluid-solid interface of a circular cylinder using smoothened data at $\text{Re}=40$ and $\frac{K_s}{K_f}=20$.

Figure 18

Schematic diagram of conjugate natural convection with thermal conduction in thick vertical wall.

Figure 19

Comparison of temperature profile along fluid-solid interface with Pan et al. [12] at $\text{Gr}={10}^7$.

Figure 20

Comparison of local Nusselt distribution along fluid-solid interface with Pan et al. [12] at different value of Gr at $\frac{K_s}{K_f}=10$.

Figure 21

Schematic diagram of purely conductive problem in thick horizontal wall.

Figure 22

Comparison of temperature profile with exact solution at $\frac{K_s}{K_f}=10$.

Figure 23

Comparison of temperature profile with exact solution at $\frac{K_s}{K_f}=100$.

Figure 24

Comparison of cylinder surface temperature for purely conductive heat transfer with conformal FV solver at $\frac{K_s}{K_f}={10}^{-5}$.

Figure 25

Comparison of cylinder surface temperature with Wang et al. [31] at $\text{Ri}=1$ and $\frac{K_s}{K_f}={10}^{-5}$.

Figure 26

Comparison of cylinder surface temperature using smoothed data with Wang et al. [31] at $\text{Ri}=1$ and $\frac{K_s}{K_f}={10}^{-5}$.

Figure 27

Isotherms for mixed convective flow around an adiabatic circular cylinder in a lid-driven cavity.