The 'other' half of the continuity equation
I've been having a few thoughts about how we formulate the 'incompressible NavierStokes' system & would value a few insights.
Background: 1. We have 3 momentum equations  x,y,z cartesian directions 2. We begin with the transient form of the continuity eqn (all derivs are partials, r=density, N = nabla, v = velocity, D = substantial deriv): dr/dt + v.Nr + r*(N.v) = 0 or dr/dt + v.Nr + r*div(v) = 0 massage into form: div(v) = (1/r)*(dr/dt + v.Nr) = Dr/r For incompressible flow, we claim that Dr/r, & discard this to end up with: div(v) = 0 our wellknown 'friend'... 3. We use div(v)=0 to drop the velocity crossterm form the compressible momentum equations, with some manipulation. 4. We now form a system of 4 equations, reusing div(v)=0: 3 x fn(u,v,w,p) > 3 incompressible momentum eqns 1 x fn(u,v,w) > div(v)=0  A few provocative thoughts: 1. Is it a 'wise' idea to try use a closing equation deficient in p? Would it not be preferable to use an equation which contains p? We do not have 4 equations in 4 unknowns, but rather, 3 equations in 4 unknowns & one semiclosing equation (deficient in p). 2. In our restriction to obtain div(v)=0, without making specific conditions for all terms of Dr/r to be zero, have we not created a 'density wave' that is going to enter our solution space? 3. Further to point 2, if ' r=c*P ', as is the case for almost incompressible fluids, with 'c' being very, very small, we have then created a 'pressure wave' in our solution space. 4. With socalled 'steady solvers', are we not merely substituting 'time' with 'iteration number'? 5. It is interesting that the pressure 'blowup' is an often seen malady in cfd solutions.  A little food for thought. desA 
Re: The 'other' half of the continuity equation
> I've been having a few thoughts about how we formulate the
: 'incompressible NavierStokes' system & would value a few insights. The assumption of incompressibility does not necessarily mean density=constant. There are many incompressible flow combustion codes with strong density gradients. Incompressibility is normally invoked by splitting the pressure the into two components: a constant in space but possibly time varying background thermodynamic pressure and a small spatially varying component that is not involved in determining the fluid state but does enter in the momentum balance. Because the thermodynamic pressure is constant the fast moving acoustic waves have been eliminated which is usually the reason for invoking the assumption of incompressibiliy. If the flow is also assumed to be isentropic then the energy equation will have nothing useful to contribute and can be dropped. With a constant thermodynamic pressure and constant entropy then the density will also be constant. 
Re: The 'other' half of the continuity equation
Thanks Andy for your input  it is very much appreciated.
desA wrote: : I've been having a few thoughts about how we formulate the : 'incompressible NavierStokes' system & would value a few insights. Andy replied: :The assumption of incompressibility does not necessarily mean density=constant. There are many incompressible flow combustion codes with strong density gradients. Incompressibility is normally invoked by splitting the pressure the into two components: a constant in space but possibly time varying background thermodynamic pressure and a small spatially varying component that is not involved in determining the fluid state but does enter in the momentum balance. Because the thermodynamic pressure is constant the fast moving acoustic waves have been eliminated which is usually the reason for invoking the assumption of incompressibiliy. If the flow is also assumed to be isentropic then the energy equation will have nothing useful to contribute and can be dropped. With a constant thermodynamic pressure and constant entropy then the density will also be constant. desA's reply: :Now this is very interesting. Can I ask in which field you work  it would appear to be combustion judging form your early statement? I find the approach you're using to make complete sense. Now, I've seen a number of ideas around this incompressibility issue, including: density = const wrt x,y,z,t => div(v)=0 is direct constant fluid properties  no mention of spatial, or temporal. Also mention of a compressibility factor relating to density change. Less that 30% being practically considered 'incompressible'. Many of the engineering fluids & convection books are written this way. The mathemagicians seem to often simply use div(v)=0 as the guarantee of incompressibility. At very slow flows, would we still expect to see a pressurewave? I'm talking about as low at the onset of instability, say Re 40 odd over a cylinder? Thanks again Andy, for your input into the debate. desA 
Re: The 'other' half of the continuity equation
I just would like to add a simple clarification: Incompressibility means the effect of pressure on the density is small enough to be considered negligible (in Engineering Applications where the Mach number is less than 0.3 this has be proven experimentally), This does not exclude the effect of Temperature on density which Andy is referring to from his observations on combustion processes. Diaw, The incompressible Navier Stokes equations are an Engineering Assumption that is nice and good for design purposes only, but not for a research of the wave phenomnun you are talking about. Computer codes blow up because of rounding errors not because of the lack of pressure effects on the density.

Re: The 'other' half of the continuity equation
Welcome, Ahmed  great to have you aboard...
Ahmed wrote: I just would like to add a simple clarification: Incompressibility means the effect of pressure on the density is small enough to be considered negligible (in Engineering Applications where the Mach number is less than 0.3 this has be proven experimentally), This does not exclude the effect of Temperature on density which Andy is referring to from his observations on combustion processes. desA replies: I agree with this observation. So, how would engineers implement the 'continuity' closure equation, in a way that best expresses their observations? Ahmed wrote: Diaw, The incompressible Navier Stokes equations are an Engineering Assumption that is nice and good for design purposes only, but not for a research of the wave phenomnun you are talking about. Computer codes blow up because of rounding errors not because of the lack of pressure effects on the density. There is a lot of current research going into these 'blowup' solutions at present. Something is definitely there, in the physics, beyond the early numerical roundoff argument. Ironically with my current work, it is a very bad day should I come cross a numerical 'blowup'& I really torture the NS. I have managed to isolate what I call 'momentum waveforms' in the NS equations. I can show them via tensor & vector calculus  as well as in various simulations. In current work, I've been able to predict, via tensor calculus, where the solution bifurcations should occur. These waveforms are there alright  & in reasonably large dimension, especially at Reynolds numbers climb, with almost negligible compressibility & bulk modulus in the order of that for water. My current debate is what is the 'correct' continuity closure equation to use which can serve physical, mathematical & computing requirements. desA 
Re: The 'other' half of the continuity equation
Why don't you publish your findings?

Re: The 'other' half of the continuity equation
bystander wrote:
Why don't you publish your findings? desA replies: Valid question. All in good time. At the moment, it is somewhat of a rapid emergence of discovery & I'm fasttracking, with mostly handwritten notes as I go (over 30 kg of paper :) I have published some papers in my university seminars. You can read a little between the lines on this forum as to what I'm exploring. I'll follow up whether it would be acceptable to put these local papers up for review on a website. This stuff has been fun & I honestly wish I had a 48hour day. As things stand, I'm in the process of writing up another local paper & am in the process of possibly changing universities. I plan a series of papers during the next few months & am busy collecting suitable material. You would be most welcome, as would anyone else, to email me directly in the meantime & we can compare notes. desA 
Re: The 'other' half of the continuity equation
Diaw I am not going to write about the merits of Engineers in our modern life, you can see the end result around you by yourself. But the point is when Engineers make statements, these are generally correct statements because they are based on real life experiences. Now when you try to apply their statements it is your responsibility to make sure that you are having the right conditions to apply these statements, this is a very tricky point. In my previous posting I said when the flow Mach number is below 0.3, then the incompressibility assumption is valid, if you check all books written about Aerodynamics you will find this same assumption and a lot of experimental work has validated it. Now if you take this assumption outside the field of aerodynamics, it is your responsibility to make sure you meet the right conditions. Industrial accidents do happen everywhere no matter what the level of technology is (Just remember the Chernobeel reactor and who was conducting the tests at the time) I was a wittness of one of those accidents where life and property loss was enormous (to say the least), that accident was motivated by that same incompressibilty assumption, water, as you can check all technical references written by engineers and scientists, is treated as incompressible fluid, but the accident I am talking about happened because that assumption was taken blindly to its full extent, but neglecting the effect of temperature drop and how the whole physical process will take place no matter how you treat it mathematically. Excluding the compressibilty of matter because the Mach number is below 0.3 is valid only in Aerodynamic applications, surje waves do happen in Hydraulic applications at much lower speeds, and they develop because Water has compressibility Enjoy the reading of what is not written. My computer suffered from one of these accidents, and I had to install a new hard drive, I lost all my email system, I wish you could send me your email address How the preparations for the world cup are going on

Re: The 'other' half of the continuity equation
I would like you to log on the following web site: http://web.mit.edu/fluids/www/Shapiro/ncfmf.html Download the file entitled Pressure fields and acceleration If you are a carefull listener, you will notice how professor Shapiro referes to the water as "nearly incompressible" fluid. Enjoy your time

Re: The 'other' half of the continuity equation
Hi Ahmed,
Thanks for your comments. I am an Engineer by the way, on a Physics platform, & so I understand precisely where you are coming from. I've also been around the block a few times too many. :) I have also had the same approach to understanding incompressibility & that is why I'm working on gaining an understanding of the physical mechanisms at work, in order to bring the mathematicalmodeling techniques we rely on, closer to reality. This incompressible / nearlyincompressible issue is something not currently wellenough understood  it has been easier for folks to work at highspeed than at low speed. Thanks for the Shapiro link  I do have the set on file. I'll work through the link you've suggested  thanks so much for that. If you draw an analogy to how electromagnetic waves are generated, the physics books mention that these occur when a particle is accelerated  we should, thus, by inference 'expect' to see wave phenomena in fluid flows  we haven't really spent time looking. The point I'm currently trying to settle is the appropriate closure form of the 'Continuity Equation' to best represent the physics of the situation. I have shown concern for the traditional mathematical approach of a deficient in 'p' closure of form div(v)=0. Thanks for your input. My email is <momentumwaves@gmail.com> I look forward to your direct email. desA 
Re: The 'other' half of the continuity equation
Hi Ahmed,
I had another look at Shapiro this afternoon. A few things were pretty interesting, & these were not directly covered in his talk. They were the 'unsteady effects' in the following situatons: 1. Hydrogen bubbles coming onto the nose of an obstruction. Observe closely the bubble patterns, from some distance before impact. 2. Hydrogen bubbles moving into constricted flow region (1/2 converging nozzle). Observe the bubble packets as they accelerate towards the rights of the screen. 3. Hydrogen streamline continuous trace through same 1/2 nozzle. Observe bubble pattern to right of screen & tell me if it looks as smooth as the entry pattern. If not, what do you see? 4. The water test nozzle at startup. Two shots of this. Very interesting unsteady flowpatterns. Very interesting stuff. desA 
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