Re: [ESPResSo-users] No conservation of momentum/mass in LBM ??

[Wink, Markus]

Hello everybody,

I checked the mass flux. It is constant over the length of the channel (some 
oscillations at the outlet, but I am not concerned about it). I revised the 
script and now I get the maximum velocity right (less than 1% deviation to the 
theoretical one). But I am still puzzling with some aspects. First of all I 
checked whether the density is constant. Within one plane perpendicular to the 
direction of flow, that is the case. Along the direction of transport, I notice 
a drop of the density. This makes sense to me, since the density is 
proportional to the pressure and I expect a linear pressure profile along the 
channel. 

Nevertheless, there are some questions left.

1) I have noticed, that in both cases, whether if I apply a body force to the 
fluid or an constant velocity inlet boundary condition, the maximum velocity of 
the profile is a bit lower than the expected one (although it is quite good 
with less than 1% deviation). I am just wondering, since I checked two 
"methods", whether this deviation lies in the nature of the LBM algorithm?
3) With a constant velocity inlet and constant velocity outlet I have inlet and 
outlet effects of cause a certain length (until the profile develops). I was 
expecting that the two lengths should be equal, since I have equivalent 
boundary conditions. In the appendix you will see, that this is not the case 
(it shows the velocity as a function of the x-position, while y- and z- are set 
to half the channel width/height). The entry effect seems to be much more 
pronounced, but I am not sure why. Does anyone have an idea?
4) As an outlet condition it would be neat to have a constant pressure boundary 
condition (with that, one would eliminate the outlet effect). I was thinking to 
put the outlet-nodes to a constant pressure via lbnode set. Is there a command 
for setting the pressure of a node to a given value (investigating the source 
code it seems, that there is only "lbnode x y z print pi" but not set).
5) What is the proper way to get the mean velocity out of the mass flux? If I 
sum up the velocities and divide by the cross section, I get a slight increase 
of v_mean along the channel (I have no idea why).
6) Have anyone ever checked the second order accuracy of the LBM in ESPResSo? 

Greetings and thanks a lot for your help! 

Markus




-----Ursprüngliche Nachricht-----
Von: address@hidden [mailto:address@hidden Im Auftrag von Ulf Schiller
Gesendet: Mittwoch, 16. März 2016 13:49
An: address@hidden
Betreff: Re: [ESPResSo-users] No conservation of momentum/mass in LBM ??

Hi Markus,

sorry for the confusion. In my earlier email I should have said mass flux of 
course, i.e., the Q in the formulas Kai provided. You can calculate that from 
the simulated velocity profile for each plane along the channel, and it should 
be constant.

Best,
Ulf

On 03/16/2016 08:21 AM, Georg Rempfer wrote:
> Assuming the fluid is not compressed (you could check that, but it's 
> likely true), the mass flux is proportional to the velocity. You claim 
> the average velocity in the channel direction is too low half way 
> between inlet and outlet. This implies that mass gets lost between the 
> inlet/outlet and the middle (or that the lb fluid is denser in the 
> middle). Can you check that?
> 
> On Wed, Mar 16, 2016 at 1:02 PM, Wink, Markus 
> <address@hidden <mailto:address@hidden>>
> wrote:
> 
>     Hello everybody,____
> 
>     __ __
> 
>     thank you for your answers. I did not get it. Which quantity is of
>     interest? Mass flux or momentum flux? I am not sure about it,
>     although to check whether mass conservation is fulfilled, both
>     should work, am I right?____
> 
>     __ __
> 
>     __ __
> 
>     Greetings____
> 
>     __ __
> 
>     Markus____
> 
>     ____
> 
>     __ __
> 
>     __ __
> 
>     *Von:address@hidden
>     <mailto:address@hidden>
>     [mailto:espressomd-users-bounces+markus.wink
>     <mailto:espressomd-users-bounces%2Bmarkus.wink>address@hidden
>     <mailto:address@hidden>] *Im Auftrag von *Georg Rempfer
>     *Gesendet:* Mittwoch, 16. März 2016 11:26
>     *An:* Kai Szuttor
>     *Cc:* address@hidden <mailto:address@hidden>
>     *Betreff:* Re: [ESPResSo-users] No conservation of momentum/mass in
>     LBM ??____
> 
>     __ __
> 
>     I agree with you argument, Markus. Mass conservation dictates that
>     the normal flow through every surface along the channel should be
>     the same (assuming the flow is incompressible). Together with the
>     fixed shape of the fully developed flow profile, this uniquely
>     determines the flow in regions far away from the inlet/outlet. So if
>     this does not come out correctly, mass conservation should be broken
>     somewhere. I don't think this is possible in the LB. Can you
>     calculate this flux through the surfaces along the channel and show
>     us where exactly it differs from the inlet/outlet?____
> 
>     __ __
> 
>     On Tue, Mar 15, 2016 at 5:09 PM, Kai Szuttor
>     <address@hidden <mailto:address@hidden>> 
> wrote:____
> 
>     Now with attachment :)
> 
>     Am 15/03/16 um 14:07 schrieb Ulf D Schiller:
>     > Did you check the flow rates directly, i.e., the momentum flux per
>     plane? Your argument seems correct, so I can only guess that there's
>     some
>     > flaw in the calculation of the mean velocity. I think there's an
>     expression for the flux in rectangular channels that one could use.
>     >
>     > Best,
>     > Ulf
>     >
>     > Sent from a mobile device.
>     >
>     >
>     > -------- Original message --------
>     > From: "Wink, Markus" <address@hidden
>     <mailto:address@hidden>>
>     > Date: 3/15/2016 8:47 AM (GMT-05:00)
>     > To: 'Ivan Cimrak' <address@hidden <mailto:address@hidden>>,
>     address@hidden <mailto:address@hidden>
>     > Subject: Re: [ESPResSo-users] No conservation of momentum/mass in
>     LBM ??
>     >
>     > Hi Ivan, Hi Florian,
>     >
>     >
>     >
>     >>/How did you compute the expected maximum velocity? As far as I
>     know, the poisseuille flow has an exact expression for the velocity
>     in the case
>     > of channel with circular cross section, and you have a rectangular
>     one.///
>     >
>     > / /
>     >
>     > I know the velocity of the rhomboid. Thus I know the mean velocity
>     of the fluid (assuming it is incompressible). I took that for
>     calculating the
>     > Reynoldsnumber, pressure gradient and theoretical velocity profile
>     (using the expression in the book  “Viscous Fluid Flow” of Frank M.
>     White).
>     >
>     >
>     >
>     > /> //The boundaries are momentum sinks. (Florian)/
>     >
>     > /> Now I read the comment of Florian -//does that mean that amount
>     of fluid is decreasing when no-slip is prescribed?/
>     >
>     > I still don’t get it. That the boundaries are momentum sinks, I
>     agree. Due to the present of the walls and the “friction” of the
>     fluid there, I
>     > achieve the poiseuille profile. But I still hold the opinion, that
>     the mean velocity of the fluid should be the same.
>     > Imagine the following physical experiment: you have a syringe pump
>     set up with a constant flow rate Q0 connected to a rectangular
>     channel having
>     > a cross section A=w*h. The fluid in the channel then has a mean
>     velocity of v_mean=Q/A. Assuming an incompressible medium, this
>     means the
>     > velocity should be the same at every slice normal the direction of
>     transport.
>     > In my simulation, the mean velocity should be velocity v0 of the
>     rhomboid.
>     >
>     > So I still don’t get the deviation to the theoretical value…
>     >
>     > Greetings Markus
>     >
>     >
>     >
>     >
>     >
>     >
>     >
>     >
>     *Von:address@hidden
>     <mailto:address@hidden>
>     > [mailto:espressomd-users-bounces+markus.wink
>     <mailto:espressomd-users-bounces%2Bmarkus.wink>address@hidden
>     <mailto:address@hidden>] *Im Auftrag von *Ivan Cimrak
>     > *Gesendet:* Dienstag, 15. März 2016 13:22
>     > *An:* address@hidden <mailto:address@hidden>
>     > *Betreff:* Re: [ESPResSo-users] No conservation of momentum/mass
>     in LBM ??____
> 
>     >
>     >
>     >
>     > Hi Markus,
>     >
>     >
>     >
>     >     Hello Everybody,
>     >
>     >
>     >
>     >     so far, in the LBM scheme only the body force is implemented
>     and no velocity/pressure boundary condition. So I was thinking on a
>     way of
>     >     mimicking a “velocity boundary” condition without changing the
>     source code. I am aware that one should, as a proper approach, using
>     Zou/He
>     >     boundary conditions and adjusting the distribution functions
>     according to the boundary conditions.
>     >
>     >
>     >
>     >     For that I have constructed a channel with rectangular cross
>     section and put the fluid inside. Furthermore, two rhomboids where
>     put inside,
>     >     one at the inlet of the channel, one at the outlet. The cross
>     section of the two rhomboids is equal to the cross section of the
>     channel,
>     >     furthermore they have a constant velocity v0.
>     >
>     >     My idea was, that, since the no-slip boundary condition is
>     implemented, I force the fluid nodes adjacent to the rhomboids to
>     have a constant
>     >     velocity, thus achieving constant velocity inlet/outlet condition.
>     >
>     >
>     >
>     >     As a result I achieve a poiseuille profile in the middle of
>     the channel (fully developed flow after inlet/outlet effects). The
>     qualitative
>     >     pressure gradient looks proper, too.
>     >
>     >     Nevertheless, the maximum velocity is not the same as I
>     expected (factor 3 to the expected one).
>     >
>     > How did you compute the expected maximum velocity? As far as I
>     know, the poisseuille flow has an exact expression for the velocity
>     in the case
>     > of channel with circular cross section, and you have a rectangular
>     one.
>     >
>     >
>     > I have checked the mean velocity. I would expect, that the mean
>     velocity of the fluid should be the velocity v0 of the rhomboid (due to
>     > mass/momentum conservation), I get less (10 %).
>     >
>     > This is strange. The amount of fluid at the inlet (integral of
>     velocity over the inlet surface, in this case is the velocity
>     constant over the
>     > inlet surface) should be the same as integral over the middle
>     cross section, as well as integral over the outlet surface....
>     Supposing you
>     > computed the average velocity as sum of velocities over the LB
>     nodes at middle cross section divided by number of these nodes, you
>     should have
>     > obtained the velocity at the inlet...
>     >
>     > Now I read the comment of Florian - does that mean that amount of
>     fluid is decreasing when no-slip is prescribed?
>     >
>     > Ivan
>     >
>     >
>     >
>     > What is wrong with my idea stated here? Obviously, something is
>     not correct, but I have no idea, what the reason for that is. Where
>     does the
>     > momentum vanish?
>     >
>     >
>     >
>     > Does anybody have an idea?
>     >
>     >
>     >
>     > Sincerely,
>     >
>     >
>     >
>     > Markus
>     >
>     >
>     >
>     >
>     >____
> 
>     __ __
> 
> 


--
Dr. Ulf D. Schiller
Assistant Professor
Department of Materials Science and Engineering Clemson University
161 Sirrine Hall
Clemson, SC 29634

Office: 299c Sirrine Hall
Phone: 1-864-656-2669
Fax: 1-864-656-5973

velocity_profile.png
Description: velocity_profile.png