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

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**Outline Background Issues**

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

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**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.

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**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.

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**Outline Background Issues**

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

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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

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**Problem Formulation Equations 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).

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**Reduced Forms of Governing Equations**

Critical issue: modeling viscous and turbulent flow behavior

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**Complex Aircraft Analysis, Circa 1968 B747-100 with space shuttle Enterprise**

What is different with these aircraft from normal operation?

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**Reduced Forms of Governing Equations**

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 incompressible flow Navier-Stokes Equations Coupled system of 5 nonlinear second order PDE’s Describes wave propagation phenomena damped by viscosity More Physics (More complex equations) Neglect viscosity & heat conduction Isentropic, irrotational flows Neglect compressibility More Geometry (More complex grid generation) (More grid points)

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**Outline Background Issues**

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

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**Finite Volume Finite Element All based on discretization approaches**

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

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**Breakup Continuous Domain into a Finite Number of Locations**

Boundary Condition B. C. B. C. Boundary Condition

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**Discretization & Order of Accuracy**

x fi fi+1 fi+2 fi+3 xi+1 xi+2 xi+3 xi x Taylor Series Expansion Polynomial Function [Power Series] Accuracy Dependent on Mesh Size and Variable Gradients

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**Discretization Example**

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

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**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

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**Numerical Error Sources - II**

at t=400 at t=0 Traveling linear wave model problem

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**Numerical Error Sources - III**

at t=400

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**Numerical Error Sources - IV**

at t=400

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**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

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**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) Discretization System of Algebraic Equations Consistency Exact Solution U Convergence as x t 0 Approximate Solution u

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**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

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**How good are the results?**

Convergence (physical): Check conserved properties: mass (for internal flows), atom balance (for chemistry), total enthalpy, e.g.

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**Outline Background Issues**

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

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**2-D Problem Setup Structured Grid / Data**

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

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**2-D Problem Setup Semi -Structured Grid / Unstructured Data**

61 35 36 11 37 X Y U36 60 10 12 62 Semi -Structured Grid / Unstructured Data Unstructured Data / Unstructured Grid 61 35 36 11 37 X Y U36 60 10 12 62

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**Grid Generation Transformation to a new coordinate system**

Transformation to a stretched grid

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**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

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**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

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**Outline Background Issues**

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

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**Viscosity and Turbulence**

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**Viscosity and Turbulence**

Laminar Steady Unsteady Turbulent Steady Unsteady

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**Viscosity and Turbulence Properties Averaged Over Time Scale Much Smaller Than Global Unsteadiness**

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**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

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**Outline Background Issues**

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

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**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 centered Corner centered Piecewise constant interpolation Piecewise linear Interpolation

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**2D Steady Flux Equation Finite-difference: centered in space scheme**

W E N S X i,j+1 i-1,j i,j-1

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**Steady Governing Equations**

= transport coeff. / = diffusivity Start with generalized RANS equations

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**Fluent Solution Method Simple Scheme**

SIMPLE: Semi-Implicit Method for Pressure Linked Equations

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**Fluent Solution Method Simple Scheme**

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

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**Fluent Solution Method Simple Scheme**

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

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**Fluent Solution Method Simple Scheme**

CV for v-eqn.

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**Fluent Solution Method Simple Scheme**

CV for p-eqn.

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**Fluent Solution Method Simple Scheme**

5-point computational molecules for linearized system using geographical not index notation

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**Fluent Solution Method Simple Scheme – Multidimensional Model**

2-D and 3-D computational molecules using geographical not index notation

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**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

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**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 ….

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CHAPTER 8 APPROXIMATE SOLUTIONS THE INTEGRAL METHOD

CHAPTER 8 APPROXIMATE SOLUTIONS THE INTEGRAL METHOD

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