1. AOE 5104 Class 8 Online presentations for next class: –Kinematics 2 and 3 Homework 3 (thank you) Homework 4 (6 questions, 2 graded, 2 recitations, worth double, due in 2 weeks) Class next Tuesday will be given by Dr. Aurelien Borgoltz No office hours next week

3. Dovetail stream, Derbyshire UK. http://www.mikejs.com/photos/2002_der.htmlhttp://www.mikejs.com/photos/2002_der.html

4. http://gizmodo.com/gadgets/top/earthrace-boat-carbon-trimaran-stabs-through- waves-video-195323.php

5. The Equations of Motion Differential Form (for a fixed volume element) The Continuity equation The Navier Stokes’ equations The Viscous Flow Energy Equation

6. Fluid Statics

7. Fluid Statics (V = 0) Continuity Momentum Energy Equation of Heat Conduction Equation of Hydrostatic Equilbrium Density is a constant (in time)

8. Example: Liquid at Rest Under Gravity z, k Water resevoir g Body force per unit mass Momentum equation (density constant) Expand and compare terms From this we see that pressure is constant with x and y. Then Integrate and get that pressure is proportional to depth, or Variation: Fluid compressible (like air)? Variation: Water in a rotating tank?

9. Fluid Dynamics?

10. The Equations of Motion Differential Form (for a fixed volume element) The Continuity equation The Navier Stokes’ equations The Viscous Flow Energy Equation These form a closed set when two thermodynamic relations are specified

11. DNS example University of Groningen Color shows pressure

12. http://gizmodo.com/gadgets/top/earthrace-boat-carbon-trimaran-stabs-through- waves-video-195323.php

14. The Equations of Motion Differential Form (for a fixed volume element) The Continuity equation The Navier Stokes’ equations The Viscous Flow Energy Equation These form a closed set when two thermodynamic relations are specified

15. Kinematics Kinematics of Velocity

16. Kinematic Concepts - Velocity 1.Fluid Line. Any continuous string of fluid particles. Moves with flow. Cannot be broken. Fluid loop – closed fluid line. 2.Particle Path. Locus traced out by an individual fluid particle. 1 234

17. Kinematic Concepts - Velocity 3.Streamline. A line everywhere tangent to the velocity vector. Never cross, except at a stagnation point. No flow across a streamline. 4.Streamsurface. Surface everywhere tangent to the velocity vector. Surface made by all the streamlines passing through a fixed curve in space. No flow through a stream surface. 5.Streamtube. Streamsurface rolled so as to form a tube. No flow through tube wall. Flow

18. Francis turbine simulation ETH Zurich http://www.cg.inf.ethz.ch/~bauer/turbo/research_gallery.html

19. Frame of Reference

20. Mathematical Description Flow dsds V Streamline 1. Streamlines 2. Streamsurfaces Make up a function  ( x,y,z,t ) so that surfaces  = const. are streamsurfaces.  is called a ‘streamfunction’.  1 = const.  2 = const. 3. Relationship between 1 and 2 Consider a streamline that sits at the intersection of two streamsurfaces. The two streamsurfaces must be described by two different streamfunctions, say  1 and  2 At any point on the streamline the perpendicular to each streamsurface, and the velocity must all be normal to each other So, what about that mathematical relationship?

21. Francis turbine simulation ETH Zurich http://www.cg.inf.ethz.ch/~bauer/turbo/research_gallery.html

22. Mathematical Description Flow dsds V Streamline 1. Streamlines 2. Streamsurfaces Make up a function  ( x,y,z,t ) so that surfaces  = const. are streamsurfaces.  is called a ‘streamfunction’.  1 = const.  2 = const. 3. Relationship between 1 and 2 Consider a streamline that sits at the intersection of two streamsurfaces. The two streamsurfaces must be described by two different streamfunctions, say  1 and  2 At any point on the streamline the perpendicular to each streamsurface, and the velocity must all be normal to each other So, what about that mathematical relationship?

23. Flow  1 = const.  2 = const. Mathematical Description where  =  (x,y,z,t) and scalar To find  we take So, Steady flow:  = ,Incompressible flow:  = 1, Unsteady flow:  meaningless

24. Example: 2D – Flow Over An Airfoil Take y x z Find consistent relations for the steamfuncitons (implicit or in terms of the velocity field).

25. Titan

26. Example: Spherical Flow Choose Flow takes place in spherical shells (no radial velocity). r erer ee ee   r Find a set of streamfunctions.