Computer Aided Engineering Design

www.strath.ac.uk/engineering XTHE PLACE OF USEFUL LEARNING ENGINEERING Department of Mechanical & Aerospace Engineering Marcello Lappa: Lecture 4, Pressure-based and Density-based solvers 16429 COMPUTER AIDED ENGINEERING DESIGN: Computational Fluid Dynamics (Semester 2) Part a: Incompressible Flows The “local” or “differential” formulation of the balance equations • Using the Gauss Theorem the Integral formulation of the balance on     b d EEpvkTvFv t                        b d vvvp F t                 0 v t            2 s d o v       1 3 ss o vv vI        2 T s vv v       2 int 2 v Eu The dissipative part of the stress tensor (also know as viscous stress tensor) strain rate tensor Mass Momentum Energy   2 s d o v     2 s d o v    •Anunsteady contribution(blue box): its distinguishing mark is the presence of the time derivative: •A convective term(green box): its distinguishing mark is the presence of the divergence operator •A diffusive term(red box): its distinguishing mark is the presence of the divergence operator applied to the gradient of another quantity (often expressed as a laplacian) •A production term(black box) 0 v t             b d vvvp F t                     b d EEpvkTvFv t                    2 int 2 v Eu Characteristic numbers and simplified versions of the balance equations Approximations of the original balance equations For many applications, the balance equations for mass, momentum and energy are too complex and broad in scope. In the absence of observational information to properly constrain the model parameters, a proper reduction/simplification of the equations is beneficial from an engineering point of view. Indeed, the CFD discipline is based on a diversity of model types, ranging from simple to complex, in which various physical effects are (can be) turned onor off. This is called amodeling hierarchyand its use forms the backbone of forward progress in any sub-field of engineering. The application of this way of thinking requires the introduction of nondimensional numbers. Theoreticians and CFD experts generally communicate through this melangeof dimensionless parameters. Re, M, Pr, Sc, Bo, Ca, We, Ma, Ra, Gr, Ro, Ta, Rh, Ha….. ?? • Based on the physics of the considered fluids phenomena, mathematical models and ensuing CFD strategies can be distinguished into different categories using different criteria and nondimensional numbers. •Each variant is characterized by a different set of PDEs. Physics & Modeling Hierarchy •Re •Re •M •Pr, Sc, Gr, Ra •Ca, Bo •Ma, We •Viscous vs. inviscid • Turbulent vs. laminar •Incompressible vs. compressible • Thermal/Solutal/density effects • Single- vs. multi-phase • Free-surface flow and surface tension • Phase transition • Chemical reactions and combustion ref ref VV M a R T   Re ref ref VL VL v    3 T gTL Ra v     Liquids or Problems involving gases with M