發展一耦合拉格朗日-尤拉粒子法與隱式外力沉浸邊界法以模擬具有複雜幾何的三維不可壓縮黏性流場

Development of a Coupled Lagrangian-Eulerian Particle and Implicit Forcing Immersed Boundary (IMLE-IFIB) Method to Simulate Three-Dimensional Incompressible Viscous Flow with Complex Geometries

柳冠碩(Rex Kuan-Shuo Liu)

指導教授 ： 許文翰

國立臺灣大學/工學院/工程科學及海洋工程學研究所/博士(2019年)

https://doi.org/10.6342/NTU201900043

摘要

本論文主要發展了兩個新的數值模型以高精度地求解具有複雜幾何(流場中有複雜外型的剛體運動或是不規則的計算空間)的不可壓縮黏性流場問題。在第一個部分中，發展了一個同時具有網格法以及無網格法各自優點的改良型混合拉格朗日與尤拉法(improved mixed Lagrangian-Eulerian method, IMLE)，以求解不可壓縮黏性流場的相關問題。IMLE方法的主要內涵是在求解不可壓縮黏性流場的控制方程式時，時間微分項是在全微分(total derivative)的型式下求解，而其餘的空間微分項則是在卡式座標(均勻分佈且正交的尤拉網格系統)下求解。此一計算方法的優點在於不需要離散傳統網格法所需要面對的對流項(convection term)，因此得以避免因對流項所造成的數值不穩定現象，同時也可以避免在使用上風法(upwinding scheme)時所引入的相位誤差(dispersion/phase error)。另一方面，在卡式座標上計算空間微分項時，可以採用具有高精度的緊緻有限差分法(combined compact difference scheme, CCD)來近似控制方程式中的壓力梯度項(pressure gradient)、速度擴散項(velocity diffusion)以及壓力波以松方程式(pressure Poisson equation, PPE)。由於MLE方法同時採用了卡式網格(Cartesian grids)以及拉式粒子(Lagrangian particles)兩種儲存變數的系統，因此在兩系統之間的資料交換是採用了內插的近似，二階精度的線性內插以及三階精度的移動最小方差法(moving least squares, MLS)分別應用在卡式網格內插至拉式粒子以及拉式粒子內插至卡式網格的過程中。而隨著時間的演進，拉式粒子會在計算空間中隨著速度而到處地移動，因此需要透過連結串列(linked list)來管理粒子。如此一來，便能快速地將拉式粒子與周遭鄰近的卡式網格交換資料，節省搜尋鄰近粒子的時間。因此可以發現在MLE方法中，拉式粒子位置上的空間微分項的精度會受限於線性內插的精度，以及當粒子從一個網格移動至另一個網格時需要更新連結串列，這將造成數值解的精度較低以及計算效能變差的問題，因此在後續提出的IMLE方法中同時解決了這兩個問題。由於不管是在MLE還是IMLE方法中都需要使用MLS內插將拉式粒子位置上的速度內插至卡式網格上，然而在IMLE方法中，在進行MLS內插後同時也將拉式粒子重置到卡式網格上，因此在下一個時間步的計算時，就不需再使用線性內插，如此一來可增加在拉式粒子上的速度擴散項的計算精度，同時也可以避免使用連結串列。

本論文的第二個部分則是利用沉浸邊界法(immersed boundary method, IB)來處理流場中具有複雜外型的靜止或移動剛體及計算域為非長方形或非長方體的問題。在現有的IB方法中，計算網格依然是採用卡式座標網格，而物體幾何則是用面網格(surface mesh)來描述，並透過在動量方程式中加入一個外力項(forcing term)，使得在物體附近或是於物體內部的網格點的速度值可以滿足速度邊界條件。然而，使用投影法(projection method)或是分步法(fractional step method)求解速度與壓力耦合的控制方程式時，為了在最後一個步驟時能滿足質量守恆方程式，因此速度邊界條件僅在中間步被滿足，如此一來便會造成在整數時間步時速度邊界條件不能被滿足的情況，進而發生流線穿透物體的現象。因此本研究提出一個外力項與PPE耦合的計算方法，即隱式外力沉浸邊界法(implicit forcing immersed boundary method, IFIB)，利用疊代的方式，使得PPE的源項(source term)包含中間步的速度散度(velocity divergence)以及疊代求解過程中的外力散度(forcing divergence)。如此一來，當疊代達到收斂後便可以得到一個同時滿足質量守恆方程式以及速度邊界條件的速度場。由於疊代的過程較為耗費計算時間，故引入一超鬆弛係數(over-relaxation factor)來加速疊代計算至收斂。

關鍵字

耦合拉格朗日-尤拉粒子法 ； 隱式外力沉浸邊界法 ； 複雜幾何 ； 三維不可壓縮黏性流場

並列摘要

In this thesis, two major research tasks have been attended with success. In the first part, a new high order incompressible viscous flow solver is developed within the mixed Lagrangian-Eulerian (MLE) framework followed by its improved version called IMLE method. The key contributions of the MLE method are that the total derivative term shown in the incompressible Navier-Stokes (NS) equations is solved under the Lagrangian sense to get rid of problem of convective instability and the spatial derivative terms, such as the pressure gradient term and the velocity diffusion term, are approximated under the Eulerian sense to get a higher accuracy order in comparison with some conventional particle methods. The sixth order accurate combined compact difference (CCD) scheme is adopted to approximate the above mentioned spatial derivative terms on the Eulerian grids. Since particles keep moving within the computational domain, a customized linked list is proposed to manage particles in searching procedure. On the other hand, a linear interpolation is needed to interpolate the approximated spatial derivative terms from Eulerian grids to Lagrangian particles. In order to further improve solution accuracy and computational efficiency, the IMLE method is developed which avoids using linked list to manage particles and linear interpolation to interpolate solutions from the Eulerian grids to Lagrangian particles because a particle reinitialization procedure is adopted in each time step. The second part includes the development of an implicit forcing immersed boundary (IFIB) method which successfully resolves the problem arising in the original continuous IB method that the velocity inside a rigid body does not satisfy the velocity boundary condition. The IFIB method involves iteration to keep refining the numerical solutions of velocity, pressure and forcing terms until convergence is reached. One key difference between the proposed IFIB and the conventional IB methods is that the forcing term is a part of the source term in the pressure Poisson equation (PPE). In this way, both the velocity boundary condition and the continuity equation can be satisfied simultaneously.

並列關鍵字

Coupled Lagrangian-Eulerian Particle Method ； Implicit Forcing Immersed Boundary Method ； Three-Dimensional Incompressible Viscous Flow ； Complex Geometries

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