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Could someone help me devise a short series of steps for solving an arbitrary fluid flow problem? Often the most difficult part of these problems is just figuring out what path to take in solving them, and I am trying to coalesce in my mind the best general procedure in preparation for my final exam.

To give you an objective example of what I mean: to analogize with classical physics, the general procedure is to

  1. Identify the forces (body forces, contact forces, normal forces)
  2. Break the system into dimensional components
  3. Solve the resulting system of equations. This works for "most" systems.

I'm looking for a similar fluid flow procedure. This is my best attempt:

  1. Write down the Navier Stokes equations for either cartesian, cylinder, or spherical coordinates. This gives you a momentum balance.
  2. Cross out all terms based on assumptions (e.g. cross out viscosity if flow is inviscid. Cross out angular terms if the flow is symmetric within a cylinder, etc.)
  3. Write down the continuity equation. This gives you a mass balance, so now you have mass and momentum taken care of.
  4. Write down a constitutive equation relating any further unknowns to the properties of the material.

Any suggestions for modification or elaboration? I haven't included anything in there about boundary layers or thermal boundary layers or turbulent/laminar flow, etc. What would be the best spot to put these considerations?

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2 Answers

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I belive you have it pretty much settled already. If I was to change anything, I would shrink instead of adding more items:

  1. Identify the relevant quantities of your system: Energy, Momentum, entropy, electric charge, mass ...

    Which may or may not be conserved. If you have boundary conditions, most probably you don't have energy and/or momentum conservation on the system alone. Either way, you need to write continuity equations for the quantities of your system, which may have source terms that are due to this non-conservations.

    So, if think on a non-relativistic, free of boundaries fluid, with no additional conserved charge.

    The relevant variables would be the (in principle well defined) velocity $\vec v$, the mass density $\rho$, the stress tensor $\sigma$, which gives the interaction inside the fluid, and also the external force density that acts on the fluid, $\vec f$.

  2. Write general equations for your system based on this general laws. Navier Stokes would come from the conservation of momentum, with a particular choice for the stress tensor.

    So, an example of equations in this case is to put an non-relativistic fluid without boundaries, so the general equations would (mostly) respect Galilei symmetries and you would end up with:

    $\frac{\partial \rho}{\partial t}+\nabla \cdot \left(\rho \vec v\right)=0$

    $\frac{\partial}{\partial t}\left(\rho \vec v\right)+\nabla \cdot \left(\rho \vec v\otimes \vec v + \sigma\right)=\vec f$

    So you end up with 2 continuity equations, the first without source, which describes the conservation of mass of the fluid. The second, which describes the conservation of momentum of the fluid, in princible have a source term, that is the external force density (think on gravity).

    You would still need an equation for the conservation of energy and entropy.

    Here is also where you would need to include boundary conditions and the sort. You can try inserting this the $\vec f$ (at least numerically, like in particle methods), or writing constraining equations, which can be, for example, zero relative velocity near a wall with a viscous fluid, or finite energy and momentum system with no free boundaries.

  3. Write down the Constitutive Equations. I believe this is the trickiest part, since you can't deduce the general form of them from symmetry principles alone. This is where you need to include the Equation of State (EoS) for the system, and also where you need to include memory effects if applicable.

    Here, if you are studying newtonian fluids, you have the following form of the stress tensor:

    $\sigma_{ij} = p\delta_{ij}- \eta\left(\frac{\partial v_i}{\partial x_j} + \frac{\partial v_j}{\partial x_i}\right) - \delta_{ij} (\zeta-\frac{2}{3}\eta) \nabla \cdot \mathbf{v}$

    Where you now need an equation of state for $\eta$ and $\zeta$ and $p$, but in principle, they are functions of $\rho$ and $T$(temperature), but that's it. Once you arrive here, you should have a 'complete' hydrodynamical description of your system.

    For the 'working physicist/engineer', most probably you will need to, at least on some point, work out a computational solution to your problems, since most of the time hydrodynamics, even in it's simplest forms, are very hard to attack only with pencil and paper. Thus, I would add another item:

  4. Formulate your problem numerically: Discretize your problem, write down what would be the approximate equations, either using mesh-full (Finite Differences, Finite Elements...) or mesh-free (Smoothed Particle Hydrodynamics...) methods, and implement on the computer using your favorite programming language.

    Much of the advances on the practical point of view are done this way, instead of trying to solve 'by hand'. This is even more important with non-newtonian fluids.

    Still it's important not to forget alternative approaches to specific cases. The first example that comes to my mind is the Statistical Theory of Turbulence (K41 theory and the ones that followed it). Even though you don't get all the information that is there to get, you can attack the problem phenomenologically, which in many cases is more than you get if you attack the problem directly.

Lastly:

I haven't included anything in there about boundary layers or thermal boundary layers or turbulent/laminar flow, etc. What would be the best spot to put these considerations?

I believe they would fit on section 2, since, in the case of turbulence, you usually treat different from laminar flow, thus requiring a different set of equations. Consequently you would also need to adapt 1 and 3, but, in my view, the crux is at 2.

There are groups that try to solve turbulent flows directly, but it's necessarily via numerical methods, and it's incredibly costly to do.

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Discretize the equations for conservation of matter, momentum, energy, and any other relevant physical quantity, and solve them on a computer. Increase the resolution of the simulation until the results converge.

(Then think really hard about the results, supplement the detailed calculations with analytic treatments of simplified analogous cases, etc. But you asked for the most general procedure, and this is it).

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