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

[Ulf Schiller]

On 03/17/2016 10:51 AM, Wink, Markus wrote:
> Hello everybody,
> 
>> Your plot suggests that the inlet velocity is different from the outlet 
>> velocity. In that case it makes sense that the boundary effects have a 
>> different extent.
> 
> Sorry for the misleading picture. The first and last data point should of 
> cause be the velocity of the rhomboid. I set both rhomboids with the same 
> velocity (see tcl-script attached). So the boundary effects should have same 
> extent, which they clearly don't have. Could this also be related to a 
> slightly compressed fluid?
> 
>> In LB, as you say, the pressure is proportional to the density, so you could 
>> set the density at the outlet. This is in fact what the Zhou/He boundary 
>> condition effectively does.
> 
> Is it really that simple? I mean, for the outlet we have 5 unknown 
> distribution functions (the ones having a velocity vector pointing inwards to 
> the channel). If I set a certain density at the outlet, I only have one 
> equation for those 5 unknown distributions. I could also imply a normal 
> output boundary condition for the flow, so setting u_y and u_z to zero. This 
> gives me another two relations. As far as I know, Zou/He assume furthermore, 
> that for the non-equilibrium part of the distribution function there is a 
> bounce-back rule, giving the last two relations needed to solve the system of 
> equations. 
> I am just wondering, if it is promising to just set the density at the outlet 
> to a certain value (maybe in combination with normal output condition for the 
> velocity) to get rid of the output effects. Any idea/feeling about it?

Of course Zhou/He would be cleaner, but you'd need to implement that
(which is not difficult). Just setting the density will not satisfy the
normal and non-equilibrium conditions (which will affect the stress
tensor, but you have an artificial profile anyhow). I'm not sure what
will happen, but may be worth a quick'n'dirty try...

Cheers,
Ulf

> -----Ursprüngliche Nachricht-----
> Von: Ulf Schiller [mailto:address@hidden 
> Gesendet: Donnerstag, 17. März 2016 14:30
> An: Wink, Markus; address@hidden
> Betreff: Re: AW: [ESPResSo-users] No conservation of momentum/mass in LBM ??
> 
> Hi Markus,
> 
> On 03/17/2016 05:33 AM, Wink, Markus wrote:
>> 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. 
> 
> Well, in principle the density should be constant in an incompressible fluid. 
> Now, since you are adding momentum to the system through the inlet velocity, 
> I think the fluid is locally compressed at the inlet, so a (small) density 
> gradient may develop. So I think your observation makes sense. However, if 
> the density gradient becomes too large, it may lead to instabilities.
> 
>> 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?
> 
> I would say 1% is a pretty good agreement. You could check that you get the 
> expected convergence by reducing the grid size (see below).
> 
>> 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?
> 
> Your plot suggests that the inlet velocity is different from the outlet 
> velocity. In that case it makes sense that the boundary effects have a 
> different extent.
> 
>> 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).
> 
> In LB, as you say, the pressure is proportional to the density, so you could 
> set the density at the outlet. This is in fact what the Zhou/He boundary 
> condition effectively does.
> 
>> 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).
> 
> This could be related to the different velocities at the inlet and the 
> outlet. The mass flux should be constant far away from the boundaries though.
> 
>> 6) Have anyone ever checked the second order accuracy of the LBM in 
>> ESPResSo? 
> 
> I did that a long time ago, and would support the idea of adding a 
> corresponding test case. However, many of the LB features in ESPResSo, 
> including the integrator, are only first order accurate (either in time or 
> space).
> 
> Hope this helps.
> 
> Best wishes,
> Ulf
> 
>> -----Ursprüngliche Nachricht-----
>> Von: address@hidden 
>> [mailto:address@hidden
>> org] 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
>>
> 
> 
> --
> 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
> 


-- 
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