# Incompressible fluid flow.

• stserkan
In summary,The Navier-Stokes equation in cylindrical coordinates can be used to describe steady, two-dimensional flow. The derivative of velocity in the radial direction is zero, and terms can be cancelled out on the basis of incompressible flow. When this equation is used to solve a problem, it is important to verify the code is correct.

#### stserkan

a perfect and incompressible fluid flowing in circular path about a center. the flow is two-dimensional and steady. I need to find differential relation between the pressure and the tangential velocity and the radial distance.?? I couldn't find which eqn i can use for this problem. I first thought that continuity eqn is ok with this problem but i couldn't figure how to relate it with pressure. If you help i would be persuaded.

The Navier-Stokes equation in cylindrical coordinates.

Yea, just either write the N-S in cylindrical, or convert them to cylindrical and then cancel out terms that are zero.

For example, since there can be no flow tangent to the wall, either the inside, or the outside, you can say for incompressible flow that velocity in the radial direction is 0. Since that velocity is zero everywhere and stays everywhere, then the derivative of velocity in the radial direction is also zero. You can also cancel out several terms in the momentum equation on the basis of incompressible flow.

I love it when we just cancel out terms in the N-S equations & stroll off to an answer - so easy on paper. We are all taught this method (maths, engineering), but, I wonder how many folks try validating these assumptions afterwards?

When this stuff is done using numerical simulation, the answer often turns out to give some surprises - precisely, in fact, because there it is not possible to cancel the terms, & the N-S are allowed to express themselves through their full connectivity. Cancellation of terms often turns out to be a problem in some flow-fields because we have changed the equation of motion into something else.

It would be useful for the OP to try & run up a simulation using say Freefem++, if he is really interested in understanding the problem, & see if the visualisation agrees with his approximate N-S answer. Try using properties for water, or air - you may find out something interesting, depending on your velocity. Welcome to the real world of engineering.

desA

Thank you both. I solved it.

Perhaps the problem is with the code then. As my advisor says, "The longer you are in the field of CFD, the less you trust the answers that the codes give you."

For an analytic problem, with perfect flow, you should be able to cancel out terms. We can verify flat plate boundary layers on computer codes, even though we canceled many terms out of the equations, and then iterated an 'analytic' solution.

I'm glad you solved the problem stserkan. In real life as desA is saying, crazy stuff does happens in numerical solutions though, some physical, some non-physical.

But desA, the new "thing" in codes is verifying the codes. You verify the codes using MMS and other things like that. I can't imagine verifying an analytic solution with an unverified code, which 99% of all codes are.

minger said:
But desA, the new "thing" in codes is verifying the codes. You verify the codes using MMS and other things like that. I can't imagine verifying an analytic solution with an unverified code, which 99% of all codes are.

The problem with most current CFD codes for so-called incompressible flow, is that the designers seem to have missed some of the physics. When this missing physics then tries to force its way back into the solution, is where the things begin to go horribly wrong.

So, how do the code developers sort it out? Well, they apply so-called stabilisation schemes, which in turn often end up solving the wrong equation of motion anyway. :rofl:

The funny thing about the Navier-Stokes equation is that they really don't like having terms taken out of them. In fact, the set of equations often used for solving incompressible N-S is deficient in the continuity equation due to a mathemagical assumption that div(v)=0, where, in practice, this is impossible. This assumption, in turn, leads to an over-sensitivity to wave phenomena - a catch-22 situation all round. The cfd folks then add in smoothers & all kinds of weird & wonderful stuff, only to end up trying to compensate for the physics they left out.

A truly strange world.

Actually, in real flow fields, the eqns of motion express themselves differently in different parts of the domain, subject to local boundary-conditions. In this way, the equations are left free to determine their own destiny, without our undue interference. Sometimes we play too much.

desA

If I have steady flow of water through a pipe please tell me why del . V =0 is a bad assumption?
Assumptions of cancelling out terms in the NS equation is not done arbitrarily. Ussually after non-dimensionalizing and order of magnitude analysis.