1 Discretization of Fluid Models (Navier Stokes) Dr. Farzad Ismail School of Aerospace and Mechanical Engineering Universiti Sains Malaysia Nibong Tebal Pulau Pinang Week 5 (Lecture 1 and 2)

2 Preview We have talked about various schemes to solve model problems. Would like to use the knowledge to solve real fluid models. Before we do that, first need to understand the mathematical and physical nature of fluid dynamics.

3 The Compressible Navier Stokes One of the most complete mathematical models for fluids. Includes compressibility, viscous, heat transfer, advection, pressure effects. Can also be used to account for reacting fluids

4 2D Compressible NS

5 The Compressible Navier Stokes (cont’d) In 2D, a system of 4 x 4 (3D - 5 x 5) A hybrid of hyperbolic and parabolic types for unsteady cases Elliptic in nature for steady cases Decompose NS model into inviscid (compressible) and viscous (incompressible) parts

6 2D Incompressible Navier Stokes (NS) How do you know that mass equation is numerically satisfied?

7 Pressure Poisson (1) (2) (3)(4) (1) (4) (3)

8 Pressure Poisson (cont’d) (3) Solve * and ** for incompressible NS (*) (**)

9 2D Incompressible Navier Stokes (NS) The momentum can be rewritten (***)

10 Exercise Take the gradient of Eqn (***), apply the mass equation and show that What does this equation provide? More importantly, what is the nature of this eqn And how to solve it?

11 2D Incompressible NS (cont’d) Incompressible flow has only mass and momentum equations –The energy equation drops out (3 eqns, 3 unknowns) Mass is implicitly solved through the pressure-Poisson equation –Pressure-based solver (i.e. SIMPLE, PISO) –Requires iterations to satisfy velocity divergence- expensive! Adds an elliptic nature to pde on top of the hyperbolic and parabolic natures Flow is ‘smooth’