Fluid Flow Numerical Methods Rita Schnipke University of

Fluid Flow: Numerical Methods Rita Schnipke University of Virginia

Objectives • • • Understand application of numerical methods Domain discretization Discretization of governing equations Compare different discretization techniques Understand the process of numerical analysis Application of numerical methods in CFD July 1, 2016 Rita Schnipke © 2015 Autodesk Design Academy

Understanding Numerical Methods • Numerical methods approximate complex physics • Fluid flow equations are highly complex • Very few exact solutions • Simplifications required for numerical fluid flow • Three discretization schemes: • Finite Element Method (FEM) • Finite Volume Method (FVM) • Finite Difference Method (FDM) July 1, 2016 Rita Schnipke © 2015 Autodesk Design Academy

Domain Discretization: Part I Continuous Domain Discrete Domain Continuous Velocity Distribution Discrete Velocity Distribution 40 35 30 25 20 15 10 5 0 V(x) • Continuous domain discrete domain 0 2 4 6 x 8 10 12 Continuous domain solution everywhere July 1, 2016 Rita Schnipke 40 35 30 25 20 15 10 5 0 0 2 4 6 x 8 10 12 Discrete domain solution at grid points © 2015 Autodesk Design Academy

Domain Discretization: Part II • CFD – discretize domain into grid points July 1, 2016 Rita Schnipke © 2015 Autodesk Design Academy

Discretization of Equations - Techniques • Most common discretization techniques: • Finite Element Method (FEM) – weighted integral over elements • Finite Volume Method (FVM) – fluxes through volumes • Finite Difference (FDM) – Taylor series July 1, 2016 Rita Schnipke © 2015 Autodesk Design Academy

Discretization of Equations – Taylor Series I • Convert partial derivatives to algebraic expression • Taylor Series • So derivatives become: July 1, 2016 Rita Schnipke © 2015 Autodesk Design Academy

Discretization of Equations – Taylor Series II • Replace continuum with discrete grid points: • Approximate derivatives: July 1, 2016 Rita Schnipke © 2015 Autodesk Design Academy

Discretization of Equations – Taylor Series III • PDE with space and time • Space uses “i” – backward difference • Time uses “n” – forward difference • PDE is now algebraic equation July 1, 2016 Rita Schnipke © 2015 Autodesk Design Academy

Discretization of Equations – FDM Part I • 1 D Flow Equation: 1 2 4 • Taylor Series: July 1, 2016 Rita Schnipke © 2015 Autodesk Design Academy

Discretization of Equations – FDM Part II • July 1, 2016 Rita Schnipke © 2015 Autodesk Design Academy

FDM vs FEM vs FVM • FDM is an easy to implement, easy to understand easy to program scheme. • FDM does not show good results for unstructured meshes. • Compared to FEM and FVM, FDM is very a crude scheme. • In-house CFD codes based on FDM do exist, but most commercial software for CFD are based on either FEM or FVM. • In the next slide, differences between FEM and FVM are explored. July 1, 2016 Rita Schnipke © 2015 Autodesk Design Academy

Comparison between FEM and FVM for CFD July 1, 2016 Rita Schnipke © 2015 Autodesk Design Academy

Numerical Analysis Process • Steps to perform a numerical analysis: 1. 2. 3. 4. Establish solution domain (simplify geometry) and assumptions Discretize domain Discretize equations matrix equation for grid points Solve matrix equations July 1, 2016 Rita Schnipke © 2015 Autodesk Design Academy

CFD Process July 1, 2016 Rita Schnipke © 2015 Autodesk Design Academy

Questions for Establishing Workflow 1. What is the objective of the analysis? 2. Any simplifications of Navier-Stokes equations? • Incompressible or Compressible? • Steady or unsteady? • Inviscid, laminar or turbulent? 3. What geometry must be included? • Symmetry • Aspect Ratios 4. What are the operating and boundary conditions? 5. What material properties are needed? July 1, 2016 Rita Schnipke © 2015 Autodesk Design Academy

Using CFD Software • CFD Simulations have 3 steps: July 1, 2016 Rita Schnipke © 2015 Autodesk Design Academy

Summary • Navier-Stokes Equation is complex and difficult to solve • Replace partial derivatives in governing equations with algebraic forms – discretize equations • Discretize solution domain to accommodate discretized equations • Result in large matrix equations need computer • Discretization methods for CFD: FEM, FVM, FDM July 1, 2016 Rita Schnipke © 2015 Autodesk Design Academy