High-order least-square-based finite-difference–finite-volume method for simulation of incompressible thermal flows on arbitrary grids

In this work, a high-order (HO) least-square-based finite difference-finite volume (LSFD-FV) method together with thermal lattice Boltzmann flux solver (TLBFS) is presented for simulation of two-dimensional (2D) incompressible thermal flows on arbitrary grids. In the present method, a HO polynomial based on Taylor series expansion is applied within each control cell, where the unknown spatial derivatives at each cell center are approximated by least-square-based finite difference (LSFD) scheme. Then the recently developed TLBFS is applied to evaluate the convective and diffusive fluxes simultaneously at the cell interface by local reconstruction of thermal lattice Boltzmann solutions of the density and internal energy distribution functions. The present HO LSFD-FV method is verified and validated by 2D incompressible heat transfer problems. Numerical results indicate that the present method can be effectively and flexibly applied to solve thermal flow problems with curved boundaries on arbitrary grids. Compared with the conventional low-order finite volume method, higher efficiency and lower memory cost make the present HO method more promising for practical thermal flow problems.

Figure 1

Evaluation of fluxes at cell interface.

Figure 2

Velocity and temperature profiles obtained by HO LSFD-FV method using TLBFS for porous plate problem at $\mathrm{Pr}=0.71, \mathrm{Re}=10$, and $\mathrm{Ra}=100$.

Figure 3

Accuracy of the temperature $T$ and velocity component $u$ on quadrilateral mesh (left) and triangular mesh (right) at $\mathrm{Pr}=0.71, \mathrm{Re}=10$, and $\mathrm{Ra}=100$.

Figure 4

Efficiency comparison between the HO and 2O LSFD-FV methods using TLBFS for porous plate problem on quadrilateral mesh (left) and triangular mesh (right) at $\mathrm{Pr}=0.71, \mathrm{Re}=10$, and $\mathrm{Ra}=100$.

Figure 5

Streamlines and isotherms for Rayleigh-Bénard convection problem by HO LSFD-FV method using TLBFS at Rayleigh number $\mathrm{Ra}={10}^5$.

Figure 6

Mesh used for natural convection in a square cavity by HO LSFD-FV method using TLBFS at different Rayleigh numbers: (a) $\mathrm{Ra}={10}^3$, (b) $\mathrm{Ra}={10}^4$, (c) $\mathrm{Ra}={10}^5$, and ${10}^6$.

Figure 7

Streamlines for natural convection in a square cavity by HO LSFD-FV method using TLBFS at different Rayleigh numbers: (a) $\mathrm{Ra}={10}^3$, (b) $\mathrm{Ra}={10}^4$, (c) $\mathrm{Ra}={10}^5$, (d) $\mathrm{Ra}={10}^6$.

Figure 8

Isotherms for natural convection in a square cavity by HO LSFD-FV method using TLBFS at different Rayleigh numbers: (a) $\mathrm{Ra}={10}^3$, (b) $\mathrm{Ra}={10}^4$, (c) $\mathrm{Ra}={10}^5$, (d) $\mathrm{Ra}={10}^6$.

Figure 9

Simulation results of HO LSFD-FV method using TLBFS for natural convection in a square cavity. (a) Streamlines at $\mathrm{Ra}={10}^7$, (b) streamlines at $\mathrm{Ra}={10}^8$, (c) isotherms at $\mathrm{Ra}={10}^7$, (d) isotherms at $\mathrm{Ra}={10}^8$.

Figure 10

Illustration of the setup for natural convection in a concentric annulus.

Figure 11

Mesh used for natural convection in a concentric annulus at different Rayleigh numbers. Control cells: 2740.

Figure 12

Streamlines for natural convection in a concentric annulus by HO LSFD-FV method using TLBFS at different Rayleigh numbers: (a) $\mathrm{Ra}={10}^2$, (b) $\mathrm{Ra}={10}^3$, (c) $\mathrm{Ra}={10}^4$, (d) $\mathrm{Ra}=5\times {10}^4$.

Figure 13

Isotherms for natural convection in a concentric annulus by HO LSFD-FV method using TLBFS at different Rayleigh numbers: (a) $\mathrm{Ra}={10}^2$, (b) $\mathrm{Ra}={10}^3$, (c) $\mathrm{Ra}={10}^4$, (d) $\mathrm{Ra}=5\times {10}^4$.

Figure 14

Illustration of the mixed heat transfer simulation setup from a heated circular cylinder.

Figure 15

Mesh used for mixed heat transfer from a heated circular cylinder at various Grashof numbers. Control cells: 5600.

Figure 16

Streamlines for mixed convection obtained by HO LSFD-FV method using TLBFS at $\mathrm{Re}=20$ and various Gr: (a) $\mathrm{Gr}=0$, (b) $\mathrm{Gr}=100$, (c) $\mathrm{Gr}=800$, (d) $\mathrm{Gr}=1600$.

Figure 17

Isotherms for mixed convection obtained by HO LSFD-FV method using TLBFS at $\mathrm{Re}=20$ and various Gr: (a) $\mathrm{Gr}=0$, (b) $\mathrm{Gr}=100$, (c) $\mathrm{Gr}=800$, (d) $\mathrm{Gr}=1600$.