3 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 dataresults have to be interpreted (corrected) for errors in simulation.
4 Background Limiting 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.
7 What is a CFD code?Converts chosen physics into discretized forms and solves over chosen physical domainGeometryDefinitionComputational Gridand DomainDefinitionBoundaryConditionsPreprocessingDiscretizationApproachSolutionApproachComputerUsageStrategyProcessingSolutionAssessmentSolutionDisplayPerformanceAnalysisPostprocessing
8 Problem Formulation Equations of Motion Conservation of mass (continuity) = particle identityConservation of linear momentum = Newton’s lawConservation of energy = 1st law of thermo (E)2nd law of thermo (S)Any others?????Most General Form: Navier-Stokes EquationsWritten 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).
9 Reduced Forms of Governing Equations Critical issue: modelingviscous and turbulentflow behavior
10 Complex Aircraft Analysis, Circa 1968 B747-100 with space shuttle Enterprise What is different with these aircraft from normal operation?
11 Reduced Forms of Governing Equations Euler EquationsCoupled system of 5 nonlinear first order PDE’sDescribes conservation mass, momentum, energyDescribes wave propagation (convective) phenomenaFull Potential EquationSingle nonlinear second order PDEDescribes conservation mass, energyConservation of momentum not fully satisfied in presenceof shocksP:otential Flow EquationSingle linear second order PDEDescribes incompressible flowNavier-Stokes EquationsCoupled system of 5 nonlinear second order PDE’sDescribes wave propagation phenomena dampedby viscosityMore Physics(More complex equations)Neglect viscosity &heat conductionIsentropic, irrotational flowsNeglect compressibilityMore Geometry(More complex grid generation)(More grid points)
13 Finite Volume Finite Element All based on discretization approaches Finite DifferenceFinite VolumeFinite ElementAll based ondiscretizationapproachesP.D.E.Lu=fDiscretizeSystem of LinearAlgebraic EqnsUp
14 Breakup Continuous Domain into a Finite Number of Locations Boundary ConditionB. C.B. C.Boundary Condition
15 Discretization & Order of Accuracy xfifi+1fi+2fi+3xi+1xi+2xi+3xixTaylor Series ExpansionPolynomial Function [Power Series]Accuracy Dependent on Mesh Size and Variable Gradients
16 Discretization Example Derivative approximation proportional to polynomial orderOrder of accuracy: mesh spacing, derivative magnitudeonly reasonable if product is small
17 Numerical Error Sources - I Truncation errorFinite polynomial effectDiffusion: acts like artificial viscosity & damps out disturbancesDispersion: introduces new frequencies to input disturbanceEffect is pronounced near shocksExact Diffusion Dispersion
18 Numerical Error Sources - II at t=400at t=0Traveling linear wave model problem
21 Time-Accurate vs. Time-Marching Time-marching: steady-state solution from unsteady equationsIntermediate solution has no meaningTime-accurate: time-dependent, valid at any time step
22 Numerical Properties of Method StabilityTendency of error in solution of algebraic equations to decayImplies numerical solution goes to exact solution of discretized equationsConvergenceSolution of approximate equations approaches exact set of algebraic eqns.Solutions of algebraic eqns. approaches exact solution of P.D.E.’s as x t 0GoverningP.D.E.’sL(U)DiscretizationSystem ofAlgebraic EquationsConsistencyExact SolutionUConvergenceas x t 0Approximate Solutionu
23 How good are the results? Assess the calculation forGrid independenceConvergence (mathematical): residuals as measure of how well the finite difference equation is satisfied.Look for location of maximum errorsLook for non-monotonicity
24 How good are the results? Convergence (physical): Check conserved properties: mass (for internal flows), atom balance (for chemistry), total enthalpy, e.g.
26 2-D Problem Setup Structured Grid / Data i,j+1i-1,ji,ji,j-1i+1,jX , iY, jUi,jStructured Grid / DataUnstructured Data / Structured Grid6135361137XYU3660101262
27 2-D Problem Setup Semi -Structured Grid / Unstructured Data 6135361137XYU3660101262Semi -Structured Grid / Unstructured DataUnstructured Data / Unstructured Grid6135361137XYU3660101262
28 Grid Generation Transformation to a new coordinate system Transformation to a stretched grid
29 Grid Generation - Generic Topologies More complicated grids can be constructed by combining the basic gridtopologies - cylinder in a ductBlock-structuredO + HOverset or ChimeraCartesian + PolarBoth take advantage of natural symmetries of the geometry
30 Grid Generation - Generic Topologies More complicated grids can be constructed taking advantage of simple elementsCartesian-stepwiseUnstructured-hybridDimension Unstructured Structured2D triangular quadrilateral3D tetrahedra hexahedra
33 Viscosity and Turbulence LaminarSteady UnsteadyTurbulentSteady Unsteady
34 Viscosity and Turbulence Properties Averaged Over Time Scale Much Smaller Than Global Unsteadiness
35 Viscosity and Turbulence Laminar viscosity modeled by algebraic law: SutherlandTurbulent viscosity modeled by 1 or 2 Eqn. ModelsRealizable k- model is most reliablek=turbulence kinetic energy = turbulence dissipationModel near wall behavior by:Wall integration; more mesh near wall, y+ 1-2Wall functions: less mesh, algebraic wall model, y+ 30-50
37 Finite Volume Cell centered Corner centered 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 centeredCorner centeredPiecewise constantinterpolationPiecewise linearInterpolation
38 2D Steady Flux Equation Finite-difference: centered in space scheme WENS Xi,j+1i-1,ji,j-1
39 Steady Governing Equations = transport coeff. / = diffusivityStart with generalized RANS equations
41 Fluent Solution Method Simple Scheme Solution algorithm:Staggered grid; convected on different grid from pressure.Avoids wavy velocity solutions
42 Fluent Solution Method Simple Scheme CV for u-eqn.Two sets of indices or one and one staggered at half-cell
43 Fluent Solution Method Simple Scheme CV for v-eqn.
44 Fluent Solution Method Simple Scheme CV for p-eqn.
45 Fluent Solution Method Simple Scheme 5-point computational molecules for linearized systemusing geographical not index notation
46 Fluent Solution Method Simple Scheme – Multidimensional Model 2-D and 3-D computational molecules using geographical not index notation
47 Fluent Operational Procedures Generate GeometryGenerate Computational GridSet Boundary ConditionsSet Flow Models: Equation of State, Laminar or Turbulent, etc.Set Convergence Criteria or Number of IterationsSet Solver Method and SolveCheck Solution Quality Parameters: Residuals, etc.Post-process: Line Plots, Contour PlotsExport Data for Further Post-processing
48 Suggested Fluent Development Path Read FlowLab FAQ notes [Barber Web site]Run FlowLab to familiarize yourself with GUI, solution process and post-processingRead Cornell University training notes [Handout]Develop a relevant validation-qualification process, i.e. compare with known analyses or dataDeveloping laminar flow in straight pipeDeveloping turbulent flow in a straight pipe [if appropriate]Convection processConvergent-divergent nozzle flow….