2D-flow in a tube: Continuity and Navier-Stokes

Mike Jey · August 9, 2017, 12:55

Hi people,

I want to do some basic stuff with Python 2.7 an the finite differences method. Now I have the following case:

- 2D-flow in a tube, steady state, not compressible
- at the inlet there ist a constant velocity profile so ( $\frac{du}{dy}=0$ )
- the flow is developing into a convex profile

I want to find out the length "L" of the tube, where the velocity profile isn't developing any more.

The basic equation for continuity and Navier-Stokes are:

$\frac{du}{dx} + \frac{dv}{dy}= 0$

$u \frac{du}{dx} + v \frac{du}{dy}= -\frac{1}{\rho} \frac{dp}{dx} +\nu (\frac{d^2u}{dx^2} + \frac{d^2u}{dy^2})$

An assumption is, that I say the velocity "v" is everywhere zero. So every term with "v" in the Navier-Stokes and Continuity equation is zero and I have to solve only the u-Momentum (as shown). In addition to this I say that $\frac{dp}{dx}$ is also zero.

After this assumptions I have these equations left (Continuity and navier-stokes):

$\frac{du}{dx}= 0$

$u \frac{du}{dx} = \nu (\frac{d^2u}{dx^2} + \frac{d^2u}{dy^2})$

If I look to my continuity equation and insert it into the navier-stokes, there is:

$0 = (\frac{d^2u}{dx^2} + \frac{d^2u}{dy^2})$

So in my eyes it doesn't make sense ? It means: The velocity profile is not dependent on the fluid. So it doesn't matter, if there is water or air ? I don't think so.(?)

I tried the case without inserting $\frac{du}{dx}=0$ and I solved the equation $u \frac{du}{dx} = \nu (\frac{d^2u}{dx^2} + \frac{d^2u}{dy^2})$ in Python. It works and there are meaningful results.

But I don't understand. Actually I have to insert the continuity into the Navier-stokes, isn't it ?

agd · August 10, 2017, 15:31

For steady continuity a non-zero gradient du/dx in the flow will have to be accompanied by a non-zero dv/dy, hence v cannot be zero everywhere. I think you have made too many assumptions and the apparent conflict in your results is indicative of this. Once you reach fully developed flow, du/dx is zero and the pressure gradient is balanced by the viscous stress. Then v is zero. But in the development region you cannot make this assumption.

FMDenaro · August 10, 2017, 16:03

What you are trying to do is the study of the developing boundary layer along a flat plate (top and bottom walls), so you have to use the Prandtl equations.
Just consider that the flow evolves until the thickness of the BL is equal to H/2 (H the distance between the two walls). You can use the law delta(x) to compute this lenght that depends on the viscosity of the fluid (or Re number).

A FD solver can be used to check the theory.

LuckyTran · August 10, 2017, 16:47

You've made way too many assumptions. Your solution is u=constant everywhere, which doesn't even make sense.

du/dy=0 is only at the inlet, not everywhere. This should be obvious in a pipe flow because u=something at the inlet but u=0 on walls because of the no-slip condition.
From where do you get dp/dx=0 ? Even for a flat plate dp/dx is not zero and for a pipe it must eventually be a non-zero constant when the flow in the pipe is fully developed.
From where do you get v = 0?

This is the advection-diffusion equation for a parallel flow (flows very small gradients in the y direction). If you solve this equation, you will indeed get a result that looks like a flow because it is an advection equation. But it has limitations. For learning finite differences this is actually a really good equation to try and solve because it contains non-linear advection on top of diffusion. Usually diffusion problems are 1st problem introduced.

ztdep · August 11, 2017, 09:42

I think you should not make any assumption since the velocity is developing.

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August 10, 2017, 15:31		#2
agd Senior Member Join Date: Jul 2009 Posts: 357 Rep Power: 18	For steady continuity a non-zero gradient du/dx in the flow will have to be accompanied by a non-zero dv/dy, hence v cannot be zero everywhere. I think you have made too many assumptions and the apparent conflict in your results is indicative of this. Once you reach fully developed flow, du/dx is zero and the pressure gradient is balanced by the viscous stress. Then v is zero. But in the development region you cannot make this assumption.

August 10, 2017, 16:03		#3
FMDenaro Senior Member Filippo Maria Denaro Join Date: Jul 2010 Posts: 6,773 Rep Power: 71	What you are trying to do is the study of the developing boundary layer along a flat plate (top and bottom walls), so you have to use the Prandtl equations. Just consider that the flow evolves until the thickness of the BL is equal to H/2 (H the distance between the two walls). You can use the law delta(x) to compute this lenght that depends on the viscosity of the fluid (or Re number). A FD solver can be used to check the theory.

August 10, 2017, 16:47		#4
LuckyTran Senior Member Lucky Join Date: Apr 2011 Location: Orlando, FL USA Posts: 5,675 Rep Power: 66	You've made way too many assumptions. Your solution is u=constant everywhere, which doesn't even make sense. du/dy=0 is only at the inlet, not everywhere. This should be obvious in a pipe flow because u=something at the inlet but u=0 on walls because of the no-slip condition. From where do you get dp/dx=0 ? Even for a flat plate dp/dx is not zero and for a pipe it must eventually be a non-zero constant when the flow in the pipe is fully developed. From where do you get v = 0? This is the advection-diffusion equation for a parallel flow (flows very small gradients in the y direction). If you solve this equation, you will indeed get a result that looks like a flow because it is an advection equation. But it has limitations. For learning finite differences this is actually a really good equation to try and solve because it contains non-linear advection on top of diffusion. Usually diffusion problems are 1st problem introduced.

August 11, 2017, 09:42		#5
ztdep Senior Member p ding Join Date: Mar 2009 Posts: 427 Rep Power: 19	I think you should not make any assumption since the velocity is developing.