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ABSTRACT
Mixed convection flow in a 2D rectangular cavity is simulated by a novel finite element method, namely the projection- and characteristic-based operator-splitting algorithm. In each time step, the Navier–Stokes equations are split as follows: the diffusion part, the convection part by applying operator-splitting method, and the Poisson’s equation by adopting projection method. The implicit diffusion part is solved by the preconditioned conjugate gradient (PCG) method, whereas characteristic method is applied for the convection part in a multistep explicit scheme. The characteristic Galerkin approach is used to solve the energy equation. To validate the model, lid-driven cavity flow and natural convection flow are simulated.
Nomenclature
| C | = | convection matrix |
| fnu,v,P,T | = | forcing terms at the nth time level |
| g | = | gravitational acceleration |
| G∽ | = | the first or second gradient matrix of x, y |
| h | = | the number of sub-time-step |
| Ku,v,P,T | = | diffusion matrix |
| l | = | the sequence number of sub-time-step, 1 ≤l ≤ n |
| L | = | dimensionless size of the cavity, L = 1 |
| M | = | mass matrix |
| n′ | = | normal direction |
| N | = | matrix of interpolation function |
| Nu | = | Nusselt number |
| P | = | dimensionless pressure |
| Pr | = | Prandtl number, Pr = ν/α |
| Ra | = | Rayleigh number, Ra = gβΔTL3/να |
| Re | = | Reynolds number, Re = L•ulid/α |
| Ri | = | Richardson number, Ri = Ra/Pr•Re2 |
| S | = | curve length along isothermal walls |
| Su,v,T | = | stabilization matrix |
| t | = | dimensionless time |
| t′ | = | dimensional time |
| T | = | dimensionless temperature |
| u | = | dimensionless velocity in x-direction |
| U | = | dimensional velocity in x-direction |
| ulid | = | top-wall-driven velocity |
| v | = | dimensionless velocity in y-direction |
| x, y | = | dimensionless coordinates |
| X | = | horizontal coordinate |
| X′ | = | the moving horizontal coordinate |
| = | stabilization terms of the convective part | |
| α | = | thermal diffusivity |
| β | = | volumetric coefficient of thermal expansion |
| γ | = | the boundary edges |
| Δt | = | global time step |
| δt | = | sub-time-step, δt = Δt/h |
| θ1 | = | 0 ≤ θ1 ≤ 1 |
| θ2 | = | 0 ≤ θ2 ≤ 1 |
| ν | = | fluid kinematic viscosity |
| Φ | = | arbitrary scalar variable |
| Ω | = | computational domain |
| Subscripts | = | |
| Ave | = | average Nu number |
| C | = | cold temperature |
| H | = | hot temperature |
| Lid | = | the top moving lid |
| Local | = | local Nu number |
| i, j, k | = | indicial notation, i, j, k = 1, 2 |
| u, v | = | stand for x- and y-directions |
| P, T | = | stand for pressure and temperature |
| Superscripts | = | |
| E | = | a triangular element |
| g(l) | = | time level, g(l) = n+(l − 1)/h |
| g(l + 1) | = | time level, g(l + 1) = n + l/h |
| n,n+1 | = | number of time level |
Nomenclature
| C | = | convection matrix |
| fnu,v,P,T | = | forcing terms at the nth time level |
| g | = | gravitational acceleration |
| G∽ | = | the first or second gradient matrix of x, y |
| h | = | the number of sub-time-step |
| Ku,v,P,T | = | diffusion matrix |
| l | = | the sequence number of sub-time-step, 1 ≤l ≤ n |
| L | = | dimensionless size of the cavity, L = 1 |
| M | = | mass matrix |
| n′ | = | normal direction |
| N | = | matrix of interpolation function |
| Nu | = | Nusselt number |
| P | = | dimensionless pressure |
| Pr | = | Prandtl number, Pr = ν/α |
| Ra | = | Rayleigh number, Ra = gβΔTL3/να |
| Re | = | Reynolds number, Re = L•ulid/α |
| Ri | = | Richardson number, Ri = Ra/Pr•Re2 |
| S | = | curve length along isothermal walls |
| Su,v,T | = | stabilization matrix |
| t | = | dimensionless time |
| t′ | = | dimensional time |
| T | = | dimensionless temperature |
| u | = | dimensionless velocity in x-direction |
| U | = | dimensional velocity in x-direction |
| ulid | = | top-wall-driven velocity |
| v | = | dimensionless velocity in y-direction |
| x, y | = | dimensionless coordinates |
| X | = | horizontal coordinate |
| X′ | = | the moving horizontal coordinate |
| = | stabilization terms of the convective part | |
| α | = | thermal diffusivity |
| β | = | volumetric coefficient of thermal expansion |
| γ | = | the boundary edges |
| Δt | = | global time step |
| δt | = | sub-time-step, δt = Δt/h |
| θ1 | = | 0 ≤ θ1 ≤ 1 |
| θ2 | = | 0 ≤ θ2 ≤ 1 |
| ν | = | fluid kinematic viscosity |
| Φ | = | arbitrary scalar variable |
| Ω | = | computational domain |
| Subscripts | = | |
| Ave | = | average Nu number |
| C | = | cold temperature |
| H | = | hot temperature |
| Lid | = | the top moving lid |
| Local | = | local Nu number |
| i, j, k | = | indicial notation, i, j, k = 1, 2 |
| u, v | = | stand for x- and y-directions |
| P, T | = | stand for pressure and temperature |
| Superscripts | = | |
| E | = | a triangular element |
| g(l) | = | time level, g(l) = n+(l − 1)/h |
| g(l + 1) | = | time level, g(l + 1) = n + l/h |
| n,n+1 | = | number of time level |
Acknowledgments
The authors gratefully acknowledge the support of National Science NSFC-Liaoning Joint Fund (U1508215) and the National Basic Research Program of China (No. 2011CB012900).