Numerical on laminar flow
How to Solve for Laminar Flow in a Frictionless Cylindrical Pipe
Setting Up the Problem
Write down your knowns and unknowns about your system.The first thing you want to do is take a moment to organize all of the information provided to you in the problem. If a known has a value associated with it, right down the value next to the variable name.
- While most of the common values have variables associated with them, like m=mass, feel free to pick any variable, as long as you can keep track of it.
Figure out what you are solving for. Sometimes this is obvious, but other times it may be a little more complicated figuring out exactly what you need to know. Many times there are multiple things that you are trying to solve for, take some time to write down all of these, and figure out what would be the best order to do this in.
- At this point in time, the hardest part for you will be figuring what order to solve things in. As you get better, this will become more obvious.
Draw and label your diagram. This is the most important steps in setting up your problem, and arguably one of the most important steps in solving it. Make sure to be thorough and accurate, as you will be referencing it a lot as you solve your problem. A mislabeled diagram can alter how you solve the rest of the problem, leaving you with the wrong answer.
- Make sure to include as many of your knowns and unknowns as possible in your diagram, this will make it a lot easier to visualize your system.
Determine your velocity profile. This step requires, first, answering a couple of basic questions. The first thing to determine, is what direction your flow is in. Since we have laminar flow, it should only be in one direction. Secondly, what direction is your flow changing in, this may be obvious, but can be a little confusing sometimes. Your velocity will rarely change in the same direction as it flows. Next plot an approximation of your velocity profile in your diagram.
- Plotting will be helpful later on, and should force you to really think about what assumptions you made, and if they are realistic.
Align your units properly. This step is simple but crucial. You want to make sure all of your units are in the same system of measure. Often time’s different values are calculated using different devices, and the units don’t match other devices. This will help you to avoid making errors later on that will be difficult to catch if you go back through.
- Always use the SI system of units, no matter what you are given.
- Customary and American Engineering units are often confusing, and in some cases, have multiple units to measure the same thing. This will get challenging to keep straight as you work through the problem, and often results in unnecessary errors.
Solving the Problem
Determine your solving method. There are many ways to go about solving a fluid dynamic process, each valid for different situations. Choosing the correct solving method will save you a lot of headache down the line. The example problem uses the Navier-Stokes equation, as this is the most generalized of all of the ways to solve fluid dynamics problems.
- The Navier-Stokes equation is not always the best way to solve these kinds of problems, and as you learn more about the field, you will learn better ways to handle certain problems, but for now, the Navier-Stokes equation should handle any problems you need to tackle.
Simplify the equation of continuity for the system. This is a straightforward and simple equation to simplify, as you should have all of the necessary information by this point. This equation is only valid for steady-state flow processes, meaning you have no mass accumulation in your pipe.
- Usually you just end up with zero on each side, but occasionally you learn that one of your partial velocities is zero, making the Navier-Stokes equation easier to simplify.
Write out the Navier-Stokes equations for a cylindrical coordinate plane.The Navier-Stokes equation is a generalized equation used for solving all type of fluid problems. It looks very complicated at first, but once you get used to it, you should have no trouble using it.
- There are different equations for each coordinate plane, so make sure that you use the ones for a cylindrical system
Remove any terms that equal zero. Due to the laminar nature of your flow, many things should instantly drop out of your equation. Pay close attention to this, as this will drastically simplify your problem and help down the line. Now rewrite your equations without the zero terms, this should give you three very basic formulas.
- Don't go overboard when canceling out terms, it is easy to remove a term you may need, which will completely change your final answer.
Define boundary conditions and plug in values. Your boundary conditions are what will help you solve for the constants of integration while going through your problem. Boundary conditions change depending on the system, and sometimes are difficult to figure out, so pay very close attention to how your velocity profile is constructed. Next refer to step one, and plug in any of your knowns. This will once again help to simplify your equation.
- Classic examples of boundary conditions are, shear stress is zero in the center of the pipe, velocity is zero at the wall if they are stationary, and anytime a fluid touches a moving wall, the fluid has the same velocity as the wall.
Solve for your shear stress.Shear stress is the force perpendicular to your fluid flow, it is the force slowing your fluids velocity in a frictionless pipe. Shear stress is represented by the greek letter tau, and can be seen on the right side of the Navier-Stokes equation. Shear stress is a function of velocity and viscosity, so once you have the shear stress, you will be able to get your velocity.
- The reason you solve for shear stress first is that, almost always, the velocity terms in the Navier-Stokes equation will go to zero, but shear stress will stay.
Solve for your velocity profile equation. At this point, solving for velocity should be as simple as manipulating a few equation. You final equation will be a function of your knowns from your first step.
- Note that these equations are helpful for solving pressure drop across a system, shear stress, drag force, and many other useful things
Confirm that your flow was in fact laminar. Throughout the process of solving this equation, you assumed that the flow was laminar.
Video: Laminar flow through circular pipe,part-3,unit-6,Fm
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Date: 11.12.2018, 06:13 / Views: 55231