Fluent Lecture Dr. Thomas J. Barber engr.uconn/~barbertj

Fluent LectureDr. Thomas J. Barberwww.engr.uconn.edu/~barbertj

Outline • Background Issues • Codes, Flow Modeling, and Reduced Equation Forms • Numerical Methods: Discretize, Griding, Accuracy, Error • Data Structure, Grids • Turbulence • Fluent

CFD Connection to Other Solution Approaches • CFD (numerical) approach is most closely related to experimental approach, i.e. • can arbitrarily select physical parameters (tunnel conditions) • output is in form of discrete or point data • results have to be interpreted (corrected) for errors in simulation.

BackgroundLimiting Factors - I • Computer size: • Moore’s law: First postulated by Intel CEO George Moore. Observation that logic density of silicon integrated circuits has closely followed curve: Bits per sq. in.(and MIPS) doubles power of computing (speed and reduced size), thereby quadrupling computing power every 24 months. Calculations per second per year for $1000.

What is a CFD code? Converts chosen physics into discretized forms and solves over chosen physical domain Geometry Definition Computational Grid and Domain Definition Boundary Conditions Preprocessing Discretization Approach Solution Approach Computer Usage Strategy Processing Solution Assessment Solution Display Performance Analysis Postprocessing

Problem FormulationEquations of Motion Conservation of mass (continuity) = particle identity Conservation of linear momentum = Newton’s law Conservation of energy = 1st law of thermo (E) 2nd law of thermo (S) Any others????? Most General Form: Navier-Stokes Equations • Written in differential or integral (control volume) form. • Dependent variables typically averaged over some time scale, shorter than the mean flow unsteadiness (Reynolds-averaged Navier-Stokes - RANS equations).

Reduced Forms of Governing Equations Critical issue: modeling viscous and turbulent flow behavior

Complex Aircraft Analysis, Circa 1968B747-100 with space shuttle Enterprise What is different with these aircraft from normal operation?

Navier-Stokes Equations • Coupled system of 5 nonlinear second order PDE’s • Describes conservation mass, momentum, energy • Describes wave propagation phenomena damped • by viscosity Neglect viscosity & heat conduction Isentropic, irrotational flows Neglect compressibility • Euler Equations • Coupled system of 5 nonlinear first order PDE’s • Describes conservation mass, momentum, energy • Describes wave propagation (convective) phenomena • Full Potential Equation • Single nonlinear second order PDE • Describes conservation mass, energy • Conservation of momentum not fully satisfied in presence • of shocks • P:otential Flow Equation • Single linear second order PDE • Describes conservation mass, energy • Describes incompressible flow • Conservation of momentum not fully satisfied in presence • of shocks Reduced Forms of Governing Equations More Physics (More complex equations) More Geometry (More complex grid generation) (More grid points)

Finite Difference • Finite Volume • Finite Element All based on discretization approaches P.D.E. Lu=f Discretize System of Linear Algebraic Eqns Up

Breakup Continuous Domain into a Finite Number of Locations Boundary Condition B. C. B. C. Boundary Condition

f fi fi+1 fi+2 fi+3 x xi+2 xi xi+1 xi+3 x Discretization & Order of Accuracy • Taylor Series Expansion • Polynomial Function [Power Series] • Accuracy Dependent on Mesh Size and Variable Gradients

Discretization Example • Derivative approximation proportional to polynomial order • Order of accuracy: mesh spacing, derivative magnitude • only reasonable if product is small

Numerical Error Sources - I • Truncation error • Finite polynomial effect • Diffusion: acts like artificial viscosity & damps out disturbances • Dispersion: introduces new frequencies to input disturbance • Effect is pronounced near shocks Exact Diffusion Dispersion

Numerical Error Sources - II at t=400 at t=0 Traveling linear wave model problem

Numerical Error Sources - III at t=400

Numerical Error Sources - IV at t=400

Time-Accurate vs. Time-Marching • Time-marching: steady-state solution from unsteady equations • Intermediate solution has no meaning • Time-accurate: time-dependent, valid at any time step

Numerical Properties of Method • Stability • Tendency of error in solution of algebraic equations to decay • Implies numerical solution goes to exact solution of discretized equations • Convergence • Solution of approximate equations approaches exact set of algebraic eqns. • Solutions of algebraic eqns. approaches exact solution of P.D.E.’s as x  t  0 Governing P.D.E.’s L(U) System of Algebraic Equations Discretization Consistency Exact Solution U Approximate Solution u Convergence as x  t  0

How good are the results? • Assess the calculation for • Grid independence • Convergence (mathematical): residuals as measure of how well the finite difference equation is satisfied. • Look for location of maximum errors • Look for non-monotonicity

How good are the results? • Convergence (physical): Check conserved properties: mass (for internal flows), atom balance (for chemistry), total enthalpy, e.g.

61 i,j+1 60 62 U36 Ui,j Y Y, j i-1,j 36 i,j 37 i+1,j 35 i,j-1 10 11 12 X , i X 2-D Problem Setup • Structured Grid / Data • Unstructured Data / Structured Grid

61 61 60 62 60 62 U36 U36 Y Y 35 36 37 36 37 35 10 11 12 10 11 12 X X 2-D Problem Setup • Semi -Structured Grid / Unstructured Data • Unstructured Data / Unstructured Grid

Grid Generation Transformation to a stretched grid Transformation to a new coordinate system

Grid Generation - Generic Topologies • More complicated grids can be constructed by combining the basic grid • topologies - cylinder in a duct Block-structured O + H Overset or Chimera Cartesian + Polar Both take advantage of natural symmetries of the geometry

Grid Generation - Generic Topologies • More complicated grids can be constructed taking advantage of simple elements Cartesian-stepwise Unstructured-hybrid Dimension Unstructured Structured 2D triangular quadrilateral 3D tetrahedra hexahedra

Viscosity and Turbulence

Viscosity and Turbulence Laminar Steady Unsteady Turbulent Steady Unsteady

Viscosity and TurbulenceProperties Averaged Over Time Scale Much Smaller Than Global Unsteadiness

Viscosity and Turbulence • Laminar viscosity modeled by algebraic law: Sutherland • Turbulent viscosity modeled by 1 or 2 Eqn. Models • Realizable k- model is most reliable • k=turbulence kinetic energy •  = turbulence dissipation • Model near wall behavior by: • Wall integration; more mesh near wall, y+  1-2 • Wall functions: less mesh, algebraic wall model, y+  30-50

Finite Volume • Basic conservation laws of fluid dynamics are expressed in terms of mass, momentum and energy in control volume form. • F.V. method: on each cell, conservation laws are applied at a discrete point of the cell [node]. • Cell centered • Corner centered Piecewise constant interpolation Piecewise linear Interpolation

 X N W E S 2D Steady Flux Equation Finite-difference: centered in space scheme i,j+1 i-1,j i,j-1

Steady Governing Equations  = transport coeff.  /  = diffusivity Start with generalized RANS equations

Fluent Solution MethodSimple Scheme SIMPLE: Semi-Implicit Method for Pressure Linked Equations

Fluent Solution MethodSimple Scheme • Solution algorithm: • Staggered grid; convected on different grid from pressure. • Avoids wavy velocity solutions

Fluent Solution MethodSimple Scheme CV for u-eqn. Two sets of indices or one and one staggered at half-cell

Fluent Solution MethodSimple Scheme CV for v-eqn.

Fluent Solution MethodSimple Scheme CV for p-eqn.

Fluent Solution MethodSimple Scheme 5-point computational molecules for linearized system using geographical not index notation

Fluent Solution MethodSimple Scheme – Multidimensional Model 2-D and 3-D computational molecules using geographical not index notation

Fluent Operational Procedures • Generate Geometry • Generate Computational Grid • Set Boundary Conditions • Set Flow Models: Equation of State, Laminar or Turbulent, etc. • Set Convergence Criteria or Number of Iterations • Set Solver Method and Solve • Check Solution Quality Parameters: Residuals, etc. • Post-process: Line Plots, Contour Plots • Export Data for Further Post-processing

Suggested Fluent Development Path • Read FlowLab FAQ notes [Barber Web site] • Run FlowLab to familiarize yourself with GUI, solution process and post-processing • Read Cornell University training notes [Handout] • Develop a relevant validation-qualification process, i.e. compare with known analyses or data • Developing laminar flow in straight pipe • Developing turbulent flow in a straight pipe [if appropriate] • Convection process • Convergent-divergent nozzle flow • ….

Fluent Lecture Dr. Thomas J. Barber engr.uconn/~barbertj