CE 3305 Engineering FLUID MECHANICS Lecture 5: flow Visualization; Euler's equation

Outline Flow fields Euler’s Equation Application to Practical Cases

Flow patterns Flow patterns and visualization in real fluids Use markers such as: dye; smoke; heat are used to “see” how fluid moves The markers are “tracers,” and the tracer hypothesis is used to infer behavior of the flow field Timeline: a line formed by marking adjacent particles at some instant Pathline: the trajectory of a particular fluid particle Streakline: the trajectories of many different particles that pass through a common point in space Streamline: a line in a flow field that is tangent to velocity (at that point). Flow does not cross a streamline. (3D equivalent is called a streamtube) <visualization video>

Flow patterns Uniform flow: a flow field where velocity does not change along a streamline (velocity does not vary with position) Non-uniform flow: a flow field where velocity does vary with position.

Velocity field LaGrangian: consider an individual fluid particle Eulerian: consider a location in space Dimensionalit y (pg 121)

Velocity field Velocity of each particle expressed all at once is called the velocity field

Velocity field Usual convention is to use u,v,w as the velocity functions in a cartesian system

Velocity field Typically represented in: Cartesian coordinates Pathline coordinates.

Flow types Uniform flow: velocity constant along a pathline Non-uniform: velocity varies with position (Niagara Falls in a Canoe) Steady flow: velocity constant in time Unsteady flow: velocity varies with time

Flow types Laminar flow: velocity constant along a pathline Turbulent: velocity varies with position (Niagara Falls in a Canoe) Viscous: shear stresses impact flow Inviscid: shear stress small enough to ignore <turbulent and laminar video>

Flow types Boundary layer, wake, potential (inviscid) flow regions <flow around cylinder video>

Flow types Boundary layer, wake, potential (inviscid) flow regions

acceleration Recall from Mechanics: F = m a Acceleration is vital to relate forces to flow

acceleration Recall from Mechanics: F = m a Acceleration is vital to relate forces to flow

Particle kinematics (lagrangian) Individual particles

Fluid kinematics (eulerian) Pick a point in space; how does fluid behave at that point?

Eulerian reference system Pick a point in space

Eulerian reference system Pick a point in space

Eulerian reference system Pick a point in space

Euler’s equation We have just created a way to examine the acceleration vector in the context of a Eulerian reference frame

Example (application of definitions) Problem Statement

Example (application of definitions) Known: Geometry; Velocity field (functions); Linear variation along nozzle

Example (application of definitions) Unknown: Local acceleration at ½ distance along nozzle at time t=2 seconds. (either a numerical value, or a function)

Example (application of definitions) Governing Equations: Definition of position, velocity, and acceleration in Eulerian system

Example (application of definitions) Solution:

Example (application of definitions) Solution:

Example (application of definitions) Solution:

Example (application of definitions) Discussion: This problem is really an application of definitions and calculus The acceleration in this example is a function of time and position, but the local part is a constant at any location (not the same constant).

Example (application of euler’s equation) Problem:

Example (application of euler’s equation) Known: Constant acceleration Tank dimensions Water surface linear variation at free surface Euler’s equation applies (we will treat water as a rigid body) Pressure at free surface is 0 gage (this will be really important!)

Example (application of euler’s equation) Unknown: Value of constant acceleration in x-direction

Example (application of euler’s equation) Governing Equations: Euler’s equation Gravitational acceleration constant Water density (I assume the liquid is water; but it could be JP4)

Example (application of euler’s equation) Solution:

Example (application of euler’s equation) Solution:

Example (application of euler’s equation) Solution:

Example (application of euler’s equation) Solution:

Example (application of euler’s equation) Discussion: Liquid density did not matter because the term cancelled in Euler’s equation; So same result using water or mercury (energy required would be different) Expressed result in “gravity” reference (e.g. acceleration at 0.13G); then convert into numbers. What do you think the liquid would look like if accelerate at 1.0G?

Example (application of euler’s equation) Problem

Example (application of euler’s equation) Known: System at equilibrium Constant angular velocity Polar coordinates

Example (application of euler’s equation) Unknown: Equation of height change in two arms of the manometer

Example (application of euler’s equation) Governing equations: Euler’s equation

Example (application of euler’s equation) Solution

Example (application of euler’s equation) Solution

Example (application of euler’s equation) Solution

Example (application of euler’s equation) Solution

Example (application of euler’s equation) Solution

Example (application of euler’s equation) Discussion Manometer fluid density cancel Height change proportional to the radial component of acceleration Could express as multiples of gravity Centrifuges are used to generate huge accelerations (the angular velocity and radial arm length are squared) Centrifugal pumps work about the same way

Outline Bernoulli Equation Application to some practical cases

Bernoulli’s equation Sort of a derivation Start with Euler’s equation Textbook derives along a streamline (which saves a step). Start with Euler’s equation

Bernoulli’s equation Select a useful coordinate system Incompressible fluid

Bernoulli’s equation Write in differential form Only showing x- component to the right; similar structure for y and z. Z-component will have a weight term. Require irrotational flow (vorticity vanishes) <watch vorticity video>

Bernoulli’s equation Y and Z acceleration terms

Bernoulli’s equation Irrotational (zero vorticity) lets us refactor the cross- terms in the acceleration vector Euler’s equation after substitutions (still ugly calculus) Use chain rule

words

Bernoulli’s equation Rearrange the component equations Group terms within the partial differential operation Recall definition of the length of a vector in 3- space

Bernoulli’s equation Recall what a constant does when differentiated Three derivatives, all equal to each other and all equal to zero and all with respect to a different variable They must be the same function!

Bernoulli’s equation The textbook derives along a streamline (which by definition means flow is irrotational) Typically the equation is memorized as total head between two locations on the same streamline Bernoulli’s equation is a special case of Euler’s equation of motion It can be applied to compressible flow with minor modifications

Example using bernoulli’s equation Problem Statement

Example using bernoulli’s equation Known Total head in system Free surface and outlet pressure Water is working fluid

Example using bernoulli’s equation Unknown Velocity at outlet

Example using bernoulli’s equation Governing Equations Bernoulli’s equation

example Solution

example Discussion Water/oil same (specific weight cancels) Steady flow requires important assumption about relative “areas” Assumed pressure across “jet” is zero No frictional losses (yet – that’s coming soon!)

Example bernoulli’s equation Problem Statement

Example bernoulli’s equation Known: Working head Outlet velocity Working fluid (water)

Example bernoulli’s equation Unknown: Outlet pressure

Example bernoulli’s equation Governing equation: Bernoulli’s equation

example Solution

example Discussion Almost same as prior example; but to find pressure, we need to know the working fluid Outlet velocity specified, don’t know if it is a jet, so no assumption about pressure No frictional losses (yet!)

Bernoulli example Problem Statement

Bernoulli example Known: Speed of jet at fountain nozzle Vertical speed of water at apogee (high point) in fountain Working fluid (water)

Bernoulli example Unknown: Height of the jet (fountain)

Bernoulli example Governing Equations: Bernoulli’s equation

example Solution

example Solution

example Discussion Working fluid was irrelevant (it does matter when we introduce friction later on) Height controlled by exit speed Imagine the speed at the Bellagio Fountains in Las Vegas, Nevada (find a video)

Next Time Reynold’s Transport Theorem