Lattice Boltzmann simulations of heat transfer in fully developed periodic incompressible flows

Flow and heat transfer in periodic structures are of great interest for many applications. In this paper, we carefully examine the periodic features of fully developed periodic incompressible thermal flows, and incorporate them in the lattice Boltzmann method (LBM) for flow and heat transfer simulations. Two numerical approaches, the distribution modification (DM) approach and the source term (ST) approach, are proposed; and they can both be used for periodic thermal flows with constant wall temperature (CWT) and surface heat flux boundary conditions. However, the DM approach might be more efficient, especially for CWT systems since the ST approach requires calculations of the streamwise temperature gradient at all lattice nodes. Several example simulations are conducted, including flows through flat and wavy channels and flows through a square array with circular cylinders. Results are compared to analytical solutions, previous studies, and our own LBM calculations using different simulation techniques (i.e., the one-module simulation vs. the two-module simulation, and the DM approach vs. the ST approach) with good agreement. These simple, however, representative simulations demonstrate the accuracy and usefulness of our proposed LBM methods for future thermal periodic flow simulations.

Figure 1

Schematic illustration of the periodic flow passage. The vertical dashed lines are plotted to divide the flow passage into individual periodic modules. The coordinate system is set with the $x$ direction in the flow direction and the $y$ direction in the transverse direction.

Figure 2

Schematic illustrations of the modified periodic boundary treatment for thermal field. The inlet nodes are displayed as squares and the outlet nodes are shown as circles, with the color changing from black to white along the flow direction. The numbered arrows are used to show how a density distribution leaving the outlet can be adopted to specify the incoming distribution at the inlet. Please refer the text for detailed method description.

Figure 3

(a) The simulated temperature field and (b) transverse profiles for flow through the flat channel with CWT condition on the channel surfaces. In (b) the symbols are our LBM results and the underlying curves are from the analytical solution in Appendix pp2.

Figure 4

The relative errors (a) $E_2$ and (b) $E_{Nu}$ for flows through CWT flat channels with different channel height $H$. The straight lines are liner fittings of the LBM data points (symbols) in the log-log plots, and the line slopes are displayed in the figure labels.

Figure 5

[(a) and (${\mathrm{a}}^{\prime }$)] The simulated flow and [(b) and (${\mathrm{b}}^{\prime }$)] temperature fields and transverse velocity and temperature profiles [(c) and (${\mathrm{c}}^{\prime }\left)\right]$ for the streamwise velocity; and [(d) and (${\mathrm{d}}^{\prime }\left)\right]$ for the temperature) at two locations $x=0$ and $x=L/2$ for the flows through a wavy channel with Reynolds number $\text{Re}=25$ [(a)–(d)] and $\text{Re}=100$ [(${\mathrm{a}}^{\prime }$)–(${\mathrm{d}}^{\prime }$)].

Figure 6

Distributions of (a) the normalized wall vortex ${\zeta }_w$ and (b) local Nusselt number $\text{Nu}$ along the channel wall.

Figure 7

Result comparison of the simulations for the cooling process of flow around cylinder with one or two modules included in the simulation domain. In (a) for the temperature field, the color patches are from the two-module simulation, and the isotherm lines are from the one-module simulation. The temperature profiles at $x/L=0$, 3/8, 1, and 11/8 (indicated by labels) are displayed in (b), with the symbols from the one-module calculation and curves from the two-module calculation.

Figure 8

Distributions of temperature difference (a) in the left $L\times L$ domain between the one-module and two-module simulations, and (b) that between the left ($x\in [0,L]$) and right ($x\in [L,2L]$) half domain from the two-module simulation. In (b) the temperature in the right half domain has been scaled up by a factor of $e^{{\lambda }_{L}L}$ for direct comparison. (c) shows the temperature difference from (a) and (b) at two representative $x$ locations.

Figure 9

Result comparison of the simulations for the thermal flows around cylinder using the DM or ST approaches for the [(a) and (b)] CWT and [(${\mathrm{a}}^{\prime })$ and (${\mathrm{b}}^{\prime }$)] SHF wall conditions. In the temperature fields [(a) and (${\mathrm{a}}^{\prime }$)], the color patches are from the DM approach, and the isotherm lines are from the ST approach. The temperature profiles at $x/L=0$ and 3/8 (indicated by labels) are displayed in (b) and (${\mathrm{b}}^{\prime }$), with the symbols from the DM approach and curves from the ST approach.

Figure 10

Distributions of temperature difference between simulations using the ST and DM approaches for the [(a) and (b)] CWT and [(${\mathrm{a}}^{\prime }$) and (${\mathrm{b}}^{\prime }$)] SHF wall conditions. The temperature difference from (a) and (${\mathrm{a}}^{\prime }$) at two representative locations $x/L=0$ (black solid line) and 3/8 (blue dashed line) are shown in (c).