CE 3305 Engineering FLUID MECHANICS Lecture 6: bernoulli’s equation for a fluid

Outline Bernoulli Equation Application to some practical cases

Bernoulli’s equation Sort of a derivation 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

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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