3CFD 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.
4Background 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.
7What is a CFD code?Converts chosen physics into discretized forms and solves over chosen physical domainGeometryDefinitionComputational Gridand DomainDefinitionBoundaryConditionsPreprocessingDiscretizationApproachSolutionApproachComputerUsageStrategyProcessingSolutionAssessmentSolutionDisplayPerformanceAnalysisPostprocessing
8Problem 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).
9Reduced Forms of Governing Equations Critical issue: modelingviscous and turbulentflow behavior
10Complex Aircraft Analysis, Circa 1968 B747-100 with space shuttle Enterprise What is different with these aircraft from normal operation?
11Reduced 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)
13Finite Volume Finite Element All based on discretization approaches Finite DifferenceFinite VolumeFinite ElementAll based ondiscretizationapproachesP.D.E.Lu=fDiscretizeSystem of LinearAlgebraic EqnsUp
14Breakup Continuous Domain into a Finite Number of Locations Boundary ConditionB. C.B. C.Boundary Condition
15Discretization & Order of Accuracy xfifi+1fi+2fi+3xi+1xi+2xi+3xixTaylor Series ExpansionPolynomial Function [Power Series]Accuracy Dependent on Mesh Size and Variable Gradients
16Discretization Example Derivative approximation proportional to polynomial orderOrder of accuracy: mesh spacing, derivative magnitudeonly reasonable if product is small
17Numerical 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
18Numerical Error Sources - II at t=400at t=0Traveling linear wave model problem
21Time-Accurate vs. Time-Marching Time-marching: steady-state solution from unsteady equationsIntermediate solution has no meaningTime-accurate: time-dependent, valid at any time step
22Numerical 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
23How 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
24How good are the results? Convergence (physical): Check conserved properties: mass (for internal flows), atom balance (for chemistry), total enthalpy, e.g.
262-D Problem Setup Structured Grid / Data i,j+1i-1,ji,ji,j-1i+1,jX , iY, jUi,jStructured Grid / DataUnstructured Data / Structured Grid6135361137XYU3660101262
272-D Problem Setup Semi -Structured Grid / Unstructured Data 6135361137XYU3660101262Semi -Structured Grid / Unstructured DataUnstructured Data / Unstructured Grid6135361137XYU3660101262
28Grid Generation Transformation to a new coordinate system Transformation to a stretched grid
29Grid 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
30Grid Generation - Generic Topologies More complicated grids can be constructed taking advantage of simple elementsCartesian-stepwiseUnstructured-hybridDimension Unstructured Structured2D triangular quadrilateral3D tetrahedra hexahedra
33Viscosity and Turbulence LaminarSteady UnsteadyTurbulentSteady Unsteady
34Viscosity and Turbulence Properties Averaged Over Time Scale Much Smaller Than Global Unsteadiness
35Viscosity 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
37Finite 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
382D Steady Flux Equation Finite-difference: centered in space scheme WENS Xi,j+1i-1,ji,j-1
39Steady Governing Equations = transport coeff. / = diffusivityStart with generalized RANS equations
41Fluent Solution Method Simple Scheme Solution algorithm:Staggered grid; convected on different grid from pressure.Avoids wavy velocity solutions
42Fluent Solution Method Simple Scheme CV for u-eqn.Two sets of indices or one and one staggered at half-cell
43Fluent Solution Method Simple Scheme CV for v-eqn.
44Fluent Solution Method Simple Scheme CV for p-eqn.
45Fluent Solution Method Simple Scheme 5-point computational molecules for linearized systemusing geographical not index notation
46Fluent Solution Method Simple Scheme – Multidimensional Model 2-D and 3-D computational molecules using geographical not index notation
47Fluent 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
48Suggested 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….