Advanced Fluid Mechanics

Class overview

Welcome to Advanced Fluids at Portland State University,! If you're here that means a couple of things. One is that you like fluids and decided to choose this as your elective. The other is that you are not afraid of math since you must have experienced by now that fluid mechanics is a field rich with some of the most advanced applied math as well as advanced experimental and numerical techniques. I think the best way to get the bug for fluid mechanics is to watch a bunch of videos of the stuff flowing. "Why does it look like that?" This is really the only question you need to inspire research in the field. The gallery of fluid motion is a great place to start.

One More Thing.

I like to refer to a "computer" as a "compooter"

This is a long developed habit of mine to remind myself anytime I'm using one for analysis that they are tools and are incredibly stupid. They should used very carefully, and results should verified and always compared to some theory. A lot can go wrong if one blindly believes the numbers scientific programs output are exact and infallible. More can go right (but slower) if the outputs are judged extremely skeptically.

Hopefully you will see a reoccurring theme in the class when we analyze these flow problems. Here is a pseudo algorithm that all the great flow solutions basically follow once the mountains of failed scribbles is rewritten in a clear form...

1. Come Up With A Problem Formulation: "Why Does it Look Like That?"

   1. What is the <quantity> during <flow situation>?
   2. Why does my data say <quantity> is proportional to <parameter>?

2. Draw A Really Good Picture Of Your Problem: "Mental Meshing!"

   1. Is it 2D or 3D
   2. Any symmetry?
   3. What is the dominant energy? Kinetic? Friction Work? Surface Energy? Rotational Kinetic?
   4. Solid Walls = No Slip Condition
   5. Free Surfaces = No Shear Condition

3. Reduce The Navier-Stokes Equations: "What Matters?"

   1. Physical Reasoning
   2. Ad-Hoc Mathematical Assumptions
   3. Scale The Terms Based On Physical Parameters and Dimensions

4. Make Your Problem Dimensionless: "Mental Pre-Conditioning"

   1. Use Scales From Step 3.
   2. If Domain Is Infinite, Semi-Infinite, or Moving Look For Similarity Solutions

5. Solve The Math Problem: "Characterize Velocity Field"

   1. Closed Form? (Calculus + Algebra)
   2. Approximately? (Asymptotics + Algebra)
   3. Numerically? (A Compooter)
   4. Return to Step 3 if 1-3 Fail.

6. Interpret Solution's Implications: "You're Smarter Than A Compooter"

   1. Look at dimensional form of solution
   2. Calculate Quantities of Interest
      1. Flowrate
      2. Pressure and Shear Functions
      3. Net Pressure and Net Shear (Integrated Functions)
   3. Note proportionality to all physical quantities
   4. Come up with rationale, pictures, analogies, and simpler stories to tell the steps 1-5.
   5. Come up with helpful ratios that tell implications of flow concisely.
   6. You own flavor to add to make you stand out!

I would say the hardest part is step 1. You can ask an infinite number of questions about fluids and flows, but only a small subset are well formulated. Well formulated means you can state your problem as a fully defined differential equations. Fully defined means you can posed a differential function and assign initial conditions (time = 0) and boundary conditions (space = edges). Luckily for the Navier-Stokes equations boundary conditions are always known due to the no-slip condition. This requirement on the differential equation is an analogy to the double-slit experiment for quantum mechanics. It is an experimental requirement we enforce to the general equation. If we didn't have to enforce no-slip the Navier-Stokes problem wouldn't be singular and life would be a little bit easier. However, nature is a cruel endlessly confusing muse. It is as if we academics are Picard and nature is Q. However, we can sometimes sidestep this most general conditions with three tricks!

• No-Shear Boundary Condition: If we have a liquid gas interface, like the surface of water waves, one simplification is to say the air is so less dense than the liquid is can't cause significant shear on the interface.

• Symmetry Boundary Condition: If the solid boundary has a geometric similarity and other technical assumptions hold we can solve for part of the liquid domain and infer the rest.

• Periodic Boundary Condition: The rigorous mathematical analysis of turbulence uses this as it is convenient for proofs, it also has application in homogenous turbulence

Once you have a well-posed problem it is a series of marks on paper or clicks on a keyboard that get you to a result. This very question of "well-posed" is still unanswered for the Navier Stokes equations. You win a million dollars if you can prove yes or no all questions we can ask are well posed given the governing equations. The proof of yes is very difficult, the proof of no requires a construction of a counter example or other trickery. Numerical solutions of the full 3D Navier-Stokes equations are notorious for blowing up in values. This is not a proof. We are not sure if this unbounded growth of velocity values in finite simulation time is due to the fact that a computer can't understand infinity or because the equation itself must diverge.

Steps 3 and 4 are sometimes interchangeable, in fact after enough time you will probably end up skipping step 3 altogether. However, never forget the rigor in step 3. Considerable leaps can be made by tuning assumptions in this step to yield more solutions.

Some of the greatest fluids scientist I've met in person can essentially just go from step 1 to 6 in their heads. They basically skip the middle steps because they have done them so many times. They just know how those parts will shake out. That is a Grand Master Jedi Level of fluid dynamics. I believe after this class you are set up to become one only if you wish to pursue the field further.

If you get a sense of scales you can immediately translate the Navier-Stokes equations into much simpler classical governing equations given some assumptions holding. These scales manifest themselves as non-dimensional numbers. One of them you may have heard of before. Its called....the Reynolds Number. Mathematically this number measures how non-linear the equation is given the problem. For an engineer or physicist, it ends up being a fantastic reference value to calculate many quantities of interest. After this class hopefully you should see this number is one of many that let you quickly arrive at governing equations.