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
Dovetail stream, Derbyshire UK.
waves-video php
The Equations of Motion Differential Form (for a fixed volume element) The Continuity equation The Navier Stokes’ equations The Viscous Flow Energy Equation
Fluid Statics
Fluid Statics (V = 0) Continuity Momentum Energy Equation of Heat Conduction Equation of Hydrostatic Equilbrium Density is a constant (in time)
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?
Fluid Dynamics?
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
DNS example University of Groningen Color shows pressure
waves-video php
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
Kinematics Kinematics of Velocity
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
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
Francis turbine simulation ETH Zurich
Frame of Reference
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?
Francis turbine simulation ETH Zurich
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?
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
Example: 2D – Flow Over An Airfoil Take y x z Find consistent relations for the steamfuncitons (implicit or in terms of the velocity field).
Titan
Example: Spherical Flow Choose Flow takes place in spherical shells (no radial velocity). r erer ee ee r Find a set of streamfunctions.