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: Unsteady Flow

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  Understand unsteady flow.  Examine the unsteady form of the Navier–Stokes Equation.  Study the Courant Number for unsteady flow.  Learn from an example: Unsteady flow past a cylinder Section 5 – Fluid Flow Module 5: Unsteady Flow 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 Unsteady Flow: Part I  Flow conditions such as pressure, velocity or even domain boundaries change with time.  Relatively complex and time consuming to solve.  It is important to choose a reasonable “time step” size.  The rate of progress in time depends upon the steepness of the time gradient.  The smaller the time step, the higher the stability and accuracy.  However, more computational resources are required with smaller steps. AccuracyStability Computational Resources Section 5 – Fluid Flow Module 5: Unsteady Flow 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 Unsteady Flow: Part II  Time steps can be as large as hours, days or months when studying effect of anthropogenic greenhouse gases in Earth’s atmosphere.  Time steps can be microseconds when studying shock wave phenomena.  In order to work out an ideal time step, the Courant Number is generally kept under one (for implicit scheme).  All turbulent flows are essentially unsteady, laminar flows on the other hand can be steady as well as unsteady.  A good example of unsteady flow is seen in reciprocating devices such as piston engines and compressors. Section 5 – Fluid Flow Module 5: Unsteady Flow 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 Unsteady Flow: Part III  Consider the example of a four stroke internal combustion engine, focusing on the exhaust cycle.  As the piston rises and the valve opens, the flow velocity at the outlet valve increases, approaches a maximum and then decreases.  This is one of the most common cases of unsteady flow that is solved by CFD.  It gives an insight into flow that is difficult to get through prototype testing.  This information is used to design the exhaust manifold.  Setting up the right time-step is crucial. Section 5 – Fluid Flow Module 5: Unsteady Flow 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 Navier–Stokes Equation  Time derivative takes prominence and the strength of the time derivative will dictate the size of the time step used for analysis.  The time step refers to the size of the leap in temporal domain when progressing in time.  The higher the time derivative, the smaller the time step for numerical analysis. Section 5 – Fluid Flow Module 5: Unsteady Flow 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 Courant Number  Also termed as the Courant–Friedrichs–Lewy condition (CFL condition), the Courant Number is a useful tool to evaluate the time step size for unsteady flow cases.  The Courant Number for a 1 dimensional case can be given as:  The Courant Number for a 2 dimensional case can be given as: For a solution to be stable, the Courant number should be less than or equal to 1 Where: u is the flow speed, Δt is the time step Δx is the grid size Section 5 – Fluid Flow Module 5: Unsteady Flow 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 Example: Flow Across a Cylinder: Part I  Flow shows different behavior at different flow speeds.  A Von Karman Vortex Street (a fascinating phenomenon for fluid flow enthusiasts) can be observed at 40 < Re < 200,000.  Re = Reynolds Number  A two-part video for this module on unsteady flow covers setting up, solving and viewing results for a Von Karman Vortex Street. y x Section 5 – Fluid Flow Module 5: Unsteady Flow 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 Example: Flow Across a Cylinder: Part II  Various patterns of flow across a cylinder are seen at different Reynolds numbers.  Creeping Flow Re<10  Attached Vortices 10<Re<40  Von Karman vortex trail 40<Re<200,000  Fully turbulent wake Re>200,000 Section 5 – Fluid Flow Module 5: Unsteady Flow 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 Summary  Unsteady fluid flows occur in a variety of real life situations.  From starting and stopping phases of engines and wind turbines to vehicles for transport, unsteady flow patterns are frequently encountered.  When analyzing flow variation with time, the analysis is conducted for a series of time steps, which progress in time.  The size of these time steps depends upon the time gradient.  The higher the time gradient, the lower the time step should be. Section 5 – Fluid Flow Module 5: Unsteady Flow 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 Summary  The Courant Number helps estimate the proper time step.  If the duration of the time step is very low, the amount of computation required becomes extremely high.  Thus a compromise must be sought which depends upon the available time and computer resources as well as required accuracy. Section 5 – Fluid Flow Module 5: Unsteady Flow Page 11