Periodic flow: impose gradP such that Paverage!=0

Hi Luc,

On a general note regarding periodic boundaries, I do not think it is meaningful to consider that Paverage has a particular value in your cell. The cell is just a finite-sized window into an infinite problem, the latter showing a periodic pattern. The problem actually showing the exact same behaviours from one finite sized window to another.

Then, with an imposed pressure gradient in periodic boundaries, the window you're looking at may show a average pressure value P, but you could as well have chosen another window with an average pressure P + gradP * cellLength.

Obviously, I'm guessing here that your periodic simulation depends on (is affected by) the pressure gradient only, and not by the the average value.

But, as a matter of fact, I think that a problem that would depend both on pressure gradient and pressure average could not be simulated using periodic boundaries, since such a problem would not show a periodic pattern from one cell (or window) to another.

Jérôme

Thank you Jerome for your input. As I said, I struggle a little bit with the periodic boundaries concept.

Still, why would the current "pattern" (in the direction of the flow):

-gradP -- 0 -- +gradP / -gradP -- 0 -- +gradP / -gradP -- 0 -- +gradP

would be more "correct" (consistent?) than the following one:

0-- P -- 2gradP / 0-- P -- 2gradP / 0-- P -- 2gradP

The "windows" you mentioned would also experience the exact same behavior one relative to another, right? Furthermore, the effect of the average pressure would be taken into account, no?

Luc

Sorry, I meant:

-P -- 0 -- +P / -P -- 0 -- +P / -P -- 0 -- +P

and

0 -- P -- 2P / 0 -- P -- 2P / 0 -- P -- 2P

for gradP=P

From my understanding, I would more consider the pressure field along the infinite problem being something like

-infinity -- -infinity + gradP*L -- -infinity + gradP*2L / -infinity + gradP*2L -- -infinity + gradP*3L -- -infinity + gradP*4L / ..... / -gradP*L -- 0 -- +gradP*L / .... / infinity -gradP*2L -- infinity - gradP*L -- infinity

(from the "left" of the infinite problem, to the "right"...)

In your examples #2 or #3 I think you describe pressure discontinuity, with the different values across the cell edges (the "/"). Which is maybe not the way to go ?..

> "a problem that would depend both on pressure gradient and pressure average could not be simulated using periodic boundaries, since such a problem would not show a periodic pattern from one cell (or window) to another."

That's absolutely correct.

> "-P -- 0 -- +P / -P -- 0 -- +P / -P -- 0 -- +P"

The pressure drop per period (i.e. the macroscopic pressure gradient gradP) can be calculated by taking any two points separated by a distance k*L, with L the period length.
E.g. starting from the first point in your sequence and going forward by L:
> "-P -- 0 -- +P / -P"

gradP = -P - (-P) / L = 0

More generally a periodic flow problem necessarily leads to P(x)=a*x + p(x), where a is the macroscopic gradient and p(x) is a periodic fluctuation, so that grap(P)=dP/dx=a+dp/dx is a truly periodic function (and velocity as well since v=-k(grad(P)).

 Cheers
Bruno

Thank you guys. I guess I need some time to process this with respect to what I want to achieve.

Just a final comment:

If I can simulate separately:

1) an increase of pressure P in the cell (homogenized pressurization of the medium)

2) an increase of pressure gradient gradP along one direction of the cell (flow along a given direction)

and if, by combining the "commands" defining 1) and 2) I happen to obtain a result that seems to make sense in terms of overall response (i.e. cumulative effect of 1) and 2) in effective stress and associated deformation), would that be mechanically/physically correct of is it just wrong because "a problem that would depend both on pressure gradient and pressure average could not be simulated using periodic boundaries"?

Cheers

Luc

Hi,
You can indeed control the gradient and the mean pressure independently in a flow problem.
The problem is in the coupling. If crack aperture (or deformation) depends on local pressure and local pressure has a macroscopic gradient, then crack aperture is not periodic. This is incompatible with a periodic geometry.
My conclusion is that the gradient should be zero if what you want is the response to mean pressure.

Bruno