1. © 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. www.autodesk.com/edcommunity Education Community Fluid Flow: Overview of Fluid Flow Analysis

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. www.autodesk.com/edcommunity 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

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. www.autodesk.com/edcommunity 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

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. www.autodesk.com/edcommunity 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

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. www.autodesk.com/edcommunity 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

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. www.autodesk.com/edcommunity 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

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. www.autodesk.com/edcommunity 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

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. www.autodesk.com/edcommunity 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

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. www.autodesk.com/edcommunity 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

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. www.autodesk.com/edcommunity 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

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. www.autodesk.com/edcommunity 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

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. www.autodesk.com/edcommunity 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

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. www.autodesk.com/edcommunity 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

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. www.autodesk.com/edcommunity 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

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. www.autodesk.com/edcommunity 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

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. www.autodesk.com/edcommunity 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

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. www.autodesk.com/edcommunity 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

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. www.autodesk.com/edcommunity 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

19. © 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. www.autodesk.com/edcommunity 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