Fully HOC Scheme For Incompressible Viscous Flow And Heat Transfer Enhancement

Higher-order accuracy and higher-resolution have become a decisive factor to improve the reliability and effectiveness in numerical simulation. To solve the separated flow and simulate the physical problems in multi-scale, higher-order scheme can capture the small changes accurately and then distinguish small-scale structure of the flow. Traditional methods with higher-order need more based grid points to deal with the boundary points and the points near the boundary. However, compact difference schemes can achieve higher-order solution with fewer points and require less CPU time compared with the traditional methods.General speaking, the higher-order compact (HOC) difference scheme we usually mentioned is fourth-order accuracy only on internal points and it is very difficult to achieve fourth-order accuracy on boundary points. Even the boundary points can obtain fourth-order accuracy, the numbers of nodes used are beyond 3x3 based points, then it can not be considered as a really compact scheme. We present a general way to construct a fully higher-order compact (FHOC) finite difference scheme for the following equation The FHOC scheme for Eq.(4) is Unlike other compact solution procedure, the FHOC scheme is fully compact and fully higher-order accuracy not only on interior points but also on boundary points. The governing equations of stream function, vorticity and temperature can be written in the same form as Eq.(4). Introducing pseudo-time derivatives to Eq.(5), we have the following form: We solve Eq.(6) using ADI method until the solutions converge to the steady state. The established scheme is proved unconditionally stable and the coefficient matrix is strictly diagonally dominant which can be solved by Thomas algorithm. Using FHOC method, we directly simulate the classical lid-driven cavity flow, flow around a cylinder, arbitrary triangular cavity flow, mixed convection and heat transfer enhancement, and we obtain a lots of theoretical and practical results:(1) During numerical simulation in lid-driven cavity, FHOC scheme can achieve the same numerical results under the same conditions using fewer grid points compared with the general fourth order difference scheme. Then, FHOC scheme can save CPU time and can improve the efficiency of the computation;(2) For the incompressible viscous flow in the form of stream function and vorticity, we can easily construct second-order central (SOC) difference scheme. We note that FHOC scheme has the same form with SOC difference scheme and the only difference between them is the coefficients. Therefore, any iterative numerical method (such as SOR, ADI) used to solve SOC difference scheme can also be easily applied to the FHOC scheme;(3) By a simple coordinate transformation, FHOC difference scheme on uni-form grids can be used to simulate the complex flow around a cylinder and ar-bitrary triangular cavity lid-driven flow. Without local refinement of grids and without using non-uniform grids (or non-structural grids), FHOC scheme can deal with the boundary layer perfectly;(4) Through the numerical simulations for heat transfer enhancement with nanofluids in all kinds of lid-driven cavity, we compare the results with the lit-eratures and verify that our algorithm is valid. And we further confirm that nanofluid can enhance the heat transfer effectively with FHOC finite difference methods.