© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. Education Community Fluid Flow: Overview of Fluid Flow Analysis

© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. Education Community Objectives  Become familiar with the underlying theory of fluid flow.  Understand fluid viscosity.  Differentiate between compressible and incompressible flow.  Examine the Navier-Stokes equation.  Understand how numerical methods apply.  Identify key design and simulation principles.  Learn from an example of Couette Flow and apply a what-if analysis. Section 5 – Fluid Flow Module 1: Overview Page 2

© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. Education Community Introduction to Fluid Flow  “Fluid” is a generic term used to describe both liquids and gases.  Fundamental laws such as conservation of mass, momentum and energy provide the equations that underlie these analyses.  In addition an Equation of State may also be used for finding unknown variables such as density and temperature.  Complex equations mostly require numerical solutions. Experimental Techniques /Regression Modelling Diagram (not to scale or proportion) approximating the relative applicability of CFD Section 5 – Fluid Flow Module 1: Overview Page 3

© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. Education Community Application of Computational Fluid Dynamics (CFD) The diversity of CFD has led to its extensive use in many applications:  Process and process equipment  Power generation, petroleum and environmental projects  Aerospace and turbomachinery  Automotive  Electronics / appliances /consumer products  HVAC / heat exchangers  Materials processing  Architectural design and fire research Today, CFD represents a major portion of fluid flow solutions (dimensions/proportions approximate). Section 5 – Fluid Flow Module 1: Overview Page 4

© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. Education Community Underlying Theory Conservative form of Navier-Stokes equation Continuity equation Energy equation Fluid Pressure and Velocity are the two main variables of interest in fluid flow analysis. Section 5 – Fluid Flow Module 1: Overview Page 5

© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. Education Community Understanding Viscosity  Viscosity is the measure of resistance to fluid flow.  Inviscid fluid is an ideal case in which viscous forces are absent.  Rarefied flow in the outer atmosphere can be approximated as a real life example of inviscid flow.  Equations such as the Euler and Bernoulli equations ignore effects of viscosity and thus are restricted to approximate analyses.  To analyze and predict flow behavior accurately, effects of viscosity cannot be ignored.  Viscous Fluids can be classified into: NewtonianDilatant BinghamPlastic Section 5 – Fluid Flow Module 1: Overview Page 6

© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. Education Community Understanding Viscosity Newtonian (Low Viscosity) e.g. Water Newtonian (high Viscosity) e.g. Honey Bingham-plastic e.g. Toothpaste Pseudo-plastic e.g. Styling Gel Dilatant e.g. Putty Strain rate (1/s)  (N/m 2 )  Fluid viscosity varies in behavior from simple Newtonian fluids to more complex Pseudo-plastic fluids.  Common engineering fluids are Newtonian (e.g. water, steam, air, oils). Section 5 – Fluid Flow Module 1: Overview Page 7

© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. Education Community Incompressible Flow  Incompressible flow is comparatively easy to solve.  As density is constant, fluid flow can be solved by continuity and momentum equations alone.  For all practical cases, air flow with Mach number below 0.3 can be treated as incompressible.  Similarly liquids, unless at extremely high pressure, can be treated as incompressible.  Although no liquid is truly incompressible, it is a very accurate approximation. Section 5 – Fluid Flow Module 1: Overview Page 8

© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. Education Community Compressible Flow  For compressible flow, as density is variable, the energy equation needs to be introduced, which relates density to temperature.  To solve for both these additional variables (density and temperature), a separate equation is also required.  The Boussinesq approximation or Equation of State can be used to relate density and temperature  The study of sound waves in air and choked flow in a converging diverging nozzle are common examples of compressible flow. The shock wave created by a supersonic jet aircraft is an example of compressible flow. Image courtesy of US Air Force and Wikipedia. Where: α is the coefficient of volume expansion. ρ o is the known value of density at temperature T o Section 5 – Fluid Flow Module 1: Overview Page 9

© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. Education Community Types of Flow and Navier-Stokes Equation  Compressible vs. Incompressible  Laminar vs. Turbulent  Steady vs. Unsteady  Navier-Stokes equations are the most generic equations able to apply to the different kinds of flow as mentioned above (in 3D or 2D).  e.g. blood flow, flow over aerofoil/hydrofoil, smoke/exhaust plume analysis  Navier-Stokes equations are fundamentally complex, but can take different forms and be simplified depending upon the nature of flow.  Some exact solutions to the Navier-Stokes equations exist for examples such as Poiseuillie flow, Couette flow and Stokes boundary layer. Turbulent flow vs Laminar flow Aerofoil flow Laminar flow Turbulent flow Section 5 – Fluid Flow Module 1: Overview Page 10

© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. Education Community The Navier-Stokes Equation A short representation of the Navier-Stokes equation is its vector form: For incompressible flow: This form can be converted into an algebraic equation by replacing derivative terms For the application of numerical methods, the above equation is discretized across a domain that is broken up into small regions (discussed in detail in later section). Section 5 – Fluid Flow Module 1: Overview Page 11

© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. Education Community How Numerical Methods Apply: Part I  Expanding the Navier-Stokes equation: Convective termsLocal accelerationPiezometric pressure gradient Viscous term The Cartesian form of the Navier-Stokes equation is given above. The spatial derivates are replaced with approximate algebraic equivalents. Section 5 – Fluid Flow Module 1: Overview Page 12

© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. Education Community How Numerical Methods Apply: Part II  The Navier -Stokes equation can be discretized into algebraic equations:  Algebraic equations can be solved by several available indirect (or iterative) numerical methods such as Gauss-Siedel or Jacobi iteration.  The Tridiagonal Matrix Algorithm (TDMA, or Thomas Algorithm) is a direct method and an alternate to Gaussian Elimination to solve the algebraic equations.  TDMA is easily programmable and a student can create code using TDMA as the algorithm of choice for solving equations.  Further details for discretization are provided in the next module. Section 5 – Fluid Flow Module 1: Overview Page 13

© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. Education Community Key Design and Simulation Principles  Convergence is analogous to a spiral, where the locus of the solution moves toward the center of the spiral and hence successive computations arrive closer to the exact answer.  The user has to stop the numerical solution based upon a pre-determined level of accuracy. Otherwise the solution would continue iterating ever closer toward the exact result without reaching it. Exact Solution Section 5 – Fluid Flow Module 1: Overview Page 14

© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. Education Community Performing Analysis  Convergence criteria:  Initial value  A good initial value for variables (speed and pressure) will result in fewer iterations.  Multiplier / under-relaxation factor  Controls the speed of progress toward a solution.  Iterations  The number of times the equations are processed.  Residual values  Indicator of differences of variables between two successive iterations. A fair idea of the above mentioned terms can be grasped by solving simultaneous algebraic equations through any iterative scheme (e.g., Gauss–Siedel, TDMA). Iterations Residual Section 5 – Fluid Flow Module 1: Overview Page 15

© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. Education Community Example: Couette Flow (Steady State)  Couette Flow  Assumptions  Model / geometric simplifications  Fluid properties (Constant vs variable parameters)  Boundary Conditions  Moving / stationary wall  Constant / variable pressure outlet / inlet u0u0 X Stationary Plate Moving Plate Y Newtonian viscosity Exact solution to Couette Flow is given by: Flow is steady Section 5 – Fluid Flow Module 1: Overview Page 16

© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. Education Community What-If Analysis  The following parameters can be changed and flow behavior can be investigated:  Upper plate velocity  Viscosity  Thickness between the plates  A video presentation for the steady flow module is available for setting up Couette Flow in Autodesk Simulation Multiphysics software.  By setting up the template for Couette flow as shown in the video, multiple what-if scenarios can be investigated. Section 5 – Fluid Flow Module 1: Overview Page 17

© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. Education Community Summary  This module covered the basics of fluid flow.  Fluid flow can be classified into compressible vs. incompressible, steady vs. unsteady and laminar vs. turbulent.  This identification has to be made by the user before any analysis.  Fluid viscosity is a major factor among the flow parameters.  The Navier-Stokes equation is a general equation that can apply to various kinds of fluid flow. Section 5 – Fluid Flow Module 1: Overview Page 18

© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. Education Community Summary  However, the Navier-Stokes equation consists of complex partial differential equations, and thus numerical methods are applied for practical solutions.  When numerical methods are applied, it is important to ensure that the solution converges.  If the solution does converge, the user must self-determine where to stop the calculation based on what accuracy is required.  Each successive computation brings the result closer to the actual value, but never to an exact answer. Section 5 – Fluid Flow Module 1: Overview Page 19