固液二相懸浮微粒問題之雙向耦合數值模式

Abstract

本研究主旨為開發雙向耦合之固液二相流數值模式，針對懸浮微粒的固液二相流進行模擬。在這套數值模式中，流體運動以Navier–Stokes方程式在尤拉網格上進行解析，顆粒運動則為牛頓第二運動定律輔以質點網格法來解析，並追蹤每個顆粒的動向。在處理顆粒移動時，本模式使用自行開發的顆粒傳輸系統，在尤拉網格上儲存顆粒資訊，並將顆粒的移動分成三個方向來進行，控制其在各方向的位移均不可超過一個網格的距離，故計算時僅需考慮前後各一個網格的資訊即可，使得網格間傳接顆粒及顆粒體積通量等計算流程得以大幅簡化。本研究另一主要目標為模擬有限體積顆粒之特性，為了描述顆粒體積所造成的影響，我們開發一套採用混合流體之不可壓縮性的二相映射法，修改波松方程式(Poisson's equation)的來源項以及求解方法。與現行採用流體不可壓縮性的固液模式相比，本模式更進一步捕捉到固體顆粒造成的體積效應，而為了建立完整的理論架構，我們將附加值量效應一併考慮進來。 我們以瑞利泰勒不穩定性(Rayleigh–Taylor instability)作為範例，改變顆粒直徑或初始濃度，分析混合層厚度的發展。接著將本研究的數值模式與傳統研究方法以顆粒自由沉降的模擬來進行比較，發現在高濃度的情況下，壓力耦合造成的影響十分顯著。為評估顆粒所造成的壓力是否佔有相當的分量，我們設定一特定範例，在壓力方程式中僅考慮顆粒的體積通量，並令流體保持靜止狀態。模擬結果顯示純顆粒流動引起之壓力與單相流的總動壓在同一值級內，說明顆粒體積效應的重要性。

This study presents a two-way coupled Eulerian-Lagrangian model to simulate the solid-liquid two-phase flow system with suspension of fine particles. The numerical model solves the momentum equations of carrier flow phase on the Eulerian grid. Particle motion is governed by Newton’s second law and is solved with the particle-in-cell(PIC) method. We develop a three-dimensional particle transport algorithm, in which particle information is stored in the Eulerian grid. The particle motion is split into three directions of the Cartesian coordinate system, and particle movement at each computational time step is restricted to be within one cell in each direction. The algorithm significantly simplifies the calculation of particle motion and the resulting volume flux. To include the finite-size effect of particles, we develop a two-phase projection method that takes mixture incompressibility into account. The method modified both the source term and solver of Poisson-type pressure equation in the fractional-step incompressible flow calculation. Compared to existing models that only consider incompressibility of the carrier flow phase for dilute suspensions, the present model captures the volumetric effect of solid particles. In addition, in order to have a complete physical consideration, the present model takes the added mass into account. The model is then applied to study the RT instability induced by fine suspended particles. We analyze the thickness of mixing layer and examine the effect of particle size and concentration. Comparison between the present two-phase model and traditional solid-liquid model demonstrates that the influence of pressure coupling becomes important as the concentration increases. To assess the magnitude of the pressure-coupling effect induced by particles only, special cases that only account for particle flux are simulated. The results show that the pressure field induced by the volumetric effect of particles can be of the same order of magnitude of the single-phase pressure field, which demonstrates the importance of the volumetric effect of settling particles.