以高階方法模擬浮力引導流

Abstract

本論文採用高階數值方法模擬因浮力而造成的二維流動，藉由推導其控制方程以求得數值近似解，並探討其結果。其中，「數值方法」包括五階dispersion-relation-preserving upwinding combined compact difference scheme (DRPCCD5)、四階龍格－庫塔法(Runge-Kutta method)、投影法(projection method)、三階quadratic upstream interpolation for convective kinematics scheme (QUICK)和二階中央差分法(central difference scheme)；而測試的「模型」題目包括拉穴流(lid-driven cavity)、空穴自然對流(natural convection in a cavity)、定界交換異重流(lock-exchange turbidity current)以及雙擴散空穴對流(double-diffusive convection in a cavity)；「方程」包含不可壓縮納維－斯托克斯方程(incompressible Navier-Stokes equation)和顆粒濃度、溫度及鹽度的傳遞方程。透過和對流－擴散方程(convection-diffusion equation)做實解驗證以及和前人的數值結果做一比較，以確保吾人所提數值方法的正確性與泛用性。爾後，將此一數值模型應用於求解雙擴散定界交換異重流問題(double-diffusive lock-exchange turbidity current problem)。透過改變模型內的溫度條件以瞭解溫差對異重流隨時間變化的基準量值之影響。

In this thesis, several high-order numerical schemes are used to simulate the two-dimensional flow induced by buoyancy forces, which includes the fifth-order dispersion-relation-preserving upwinding combined compact difference scheme (DRPCCD5), the fourth-order Runge-Kutta (RK4) method, the projection method, the third-order quadratic upstream interpolation for convective kinematics scheme (QUICK) and the second-order central difference scheme. Next, the governing equations are derived, the approximated solutions are obtained and the results are demonstrated and discussed. The models adopted are the lid-driven cavity, the natural convection in a cavity, the lock-exchange turbidity current and the double-diffusive convection in a cavity. Also, the incompressible Navier-Stokes equations and the solutal, thermal and saline transport equations are solved numerically. Through the verification of the convection-diffusive equation and the validation of other numerical results, it can be ensured the correctness and generality of the proposed mathematical models. Lastly, the result of the double-diffusive lock-exchange turbidity current problem is presented numerically, which is compared with the single-diffusive lock-exchange problem. Especially, the influence of the temperature on the benchmark quantities for two cases is also exhibited and discussed in detail.