Finite Volume Method for Calculation of Flow Field

FINITE VOLUME METHOD FOR CALCULATION OF FLOW FIELD 

The discussion in the previous section assumed that the velocity field required to calculate the necessary coefficients of the discretized equations was somehow known. However, generally, the velocity field needs to be calculated as part of the overall solution procedure by solving momentum conservation equations. The governing equations are discussed in Chapters 2 to 5. The basic momentum transport equations governing laminar flow are considered here to illustrate the application of the finite volume method to calculation of the flow field. The governing equations can be written  

The momentum and continuity equations can be combined to derive an equation for pressure. For example, for constant density and viscosity fluid, the continuity equation can be used to simplify the divergence of the momentum equation (Eq. (6.35)) to yield an equation for pressure  

One of the popular methods proposed by Patankar and Spalding (1972) is called SIMPLE (semi-implicit method for pressure linked equations). In this method, discretized momentum equations are solved using the guessed pressure field. The discretized form of the momentum equations can be written  

The discretized versions of the momentum equations and Eq. (6.38) lead to discretized equations in terms of velocity and pressure correction  

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In one of the earliest attempts, Schuetz and Piesche [73] calculated the flow field in a stirred tank first to determine the local energy dissipation rate and then solved the PBEs using the finite volume method [74] to predict the local aggregate size distribution. Heath and Koh [75] have solved the population balances as scalar equations in the commercial CFD software CFX for simulating flocculation of suspensions by polymers. They employed 35 discrete sectional equations to represent the aggregate size distribution. 

Understanding the free surface flow of viscoelastic fluids in micro-channels is important for the design and optimization of micro-injection molding processes. In this paper, flow visualization of a non-Newtonian polyacrylamide (PA) aqueous solution in a transparent polymethylmethacrylate (PMMA) channel with microfeatures was carried out to study the flow dynamics in micro-injection molding. The transient flow near the flow front and vortex formation in microfeatures were observed. Simulations based on the control volume finite element method (CVFEM) and the volume of fluid (VOF) technique were carried out to investigate the velocity field, pressure, and shear stress distributions. The mesoscopic CONNFFESSIT (Calculation of Non-Newtonian How Finite Elements and Stochastic Simulation Technique) method was also used to calculate the normal stress difference, the orientation of the polymer molecules and the vortex formation at steady state. 

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