Sir, i saw in one paper...in that is mentioned we are indirectly implementing artificial viscocity in the equation....what does it mean

Some schemes like upwind schemes, when used for the convective terms introduce artificial viscosity in the system. This maybe the case.

I hope u are asking for the meaning of artificial viscosity. Here is the meaning. Now that u know the numerical methods we use do not give us accurate results. There is always error in our solutions. Hence we assume error is acceptable to a particlar level. Now most numerical schemes are unstable. They deflect and deviate so much. (numerical oscillations). so to reduce the deflections in solutions and to ensure a solid answer, we introduce some junk error term(these errors are always measured). These junk terms would decrease the oscillations and ensure stability. this is called artificial viscosity or numerical dissipation. The author must have mentioned that this viscosity is provided for inadvertantly. For a better and clear explanation, refer to Anderson Computational Fluid Dynamics. All the best

prapanj wrote:

I hope u are asking for the meaning of artificial viscosity. Here is the meaning. Now that u know *the numerical methods we use do not give us accurate results*. There is *always error in our solutions*. Hence we *assume error is acceptable to a particlar level*. Now most numerical schemes are unstable. They deflect and deviate so much. (numerical oscillations). so to reduce the deflections in solutions and to ensure a solid answer, *we introduce some junk error term* (these errors are always measured). These junk terms would decrease the oscillations and ensure stability. this is called artificial viscosity or numerical dissipation. The author must have mentioned that this viscosity is provided for inadvertantly. For a better and clear explanation, refer to Anderson Computational Fluid Dynamics. All the best

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diaw writes:

With all these *junk errors*, we trust CFD results implicitly? Oh boy... Interesting explanation, I must say...

desA

hello Mr. diaw i am just an undergraduate student. but i hope i am right when i said stability is obtained with some cpmpromise on errors. of course artificial viscosity induces some erorrs. doesn't it? I case there is a higher explanation, kindly edify me. thank u

jj would you like to tell me whats your qualification and on which thing your are working

prapanj wrote:

i am just an undergraduate student. but i hope i am right when i said stability is obtained with some cpmpromise on errors. of course artificial viscosity induces some erorrs. doesn't it? I case there is a higher explanation,

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diaw's reply:

I actually thought your overview was very interesting, not because I thought you were being flippant, but, because I think that you are rather observant.

In my research, I've observed that wave phenomena are essentially what many of the stabiliation techniques try to control. One trick is to increase viscosity to 'pull' the equations to the other side of the singularity line. This keeps the N-S response towards its 'dispersion nature', rather than its 'convection nature'.

A review of nonlinear mathematical reearch will show that wave phenomena can exist in the flows we routinely consider - this is the dual-nature of the N-S equations.

By introducing additional viscosity, the dispersion mechanism is biased in favour of stability - this will introduce errors.

desA

hi mr.diaw, could u please explain what is specific in your finding? i mean what is the advantage of pulling the NS to diffusion side. It is the sole purpose of art.viscosity right. Could u please elaborate the implication of ur concept.. its just a curiosity to know what big brains do..

prapanj wrote:

could u please explain what is specific in your finding? i mean what is the advantage of pulling the NS to diffusion side. It is the sole purpose of art.viscosity right. Could u please elaborate the implication of ur concept.. its just a curiosity to know what big brains do..

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diaw's reply:

Not at all sure about the 'big brains bit'... I'll try & offer a few thoughts on my research.

The incompressible N-S reach a critical steady limit when the convection & dispersion terms balance, forcing the unsteady term to activate. This is trivial in a 1D case, but more complex in a 2 & 3d case. It appears that the flow can undergo a form of resonance in the region of this crititical point. To the one side of this point, dispersion dominates while to the other side, convection dominates. The convection wave can, in its most basic form, swing to positive, or negative velocity values - thus introducing instability. The dispersion wave tends to be more sluggish in nature & damps out oscillations. This is why the addition of artificial viscosity assists in stability.

The N-S is a balancing act between two waveforms - convective & dispersive forms.

For a fluid like water, it has a tiny kinematic viscosity eg. 855e-9 m2/s - this is the multiplier for the dispersion terms. A very small viscosity addition will thus cause the dispersion term to increase rather rapidly & so stabilising the numerics quite nicely.

Remember that the typical numeric schemes which use method-of-weighted-resuiduals are, in themselves, averaging operations - think of an integral-averaging effect. This provides some intrinsic level of wave-damping for the physics & numerics, but sometimes it is not enough.

This is a rather simplistic explanation, but ot will give you something to think on.

desA

Could you mind to say more on the definition of convection & dispersion and their diffenece?

Ejaz tum nay tu degree bee galio may leee hay. Kuch sherem karo aisay batay iss forum paeer achee nay laghtee.

I will never open ur response and also suggest all members not to open his response because he is using abusive language and not responsing seriously................

Khan

Rocky wrote:

Could you mind to say more on the definition of convection & dispersion and their diffenece?

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diaw's reply:

In a very simple 1D form (for momentum equation):

u.du/dx => convection (partial deriv)

kvisc*d2u/dx2 => dispersion (partial deriv)

Convection basically convects (moves, shifts) a property across a flow field. This can be seen by substituting say density, or temperature for u as u.dT/dX eg.

dT/dt + u.dT/dx - alfa*d2T/dx2 = f(x,t) (all partials).

Dispersion tries to dissipate the property movement & often results in say amplitude decay & spreading effects. The pure heat equation applied to a 1D bar, for instance, shows a collapse of an initial temperature profile down to that of a mean line between terminal temperatures (with no thermal source).

The fun begins when u.dT/dx - alfa*d2T/dx2 = 0

I hope this helps a little.

diaw...

"For a fluid like water, it has a tiny kinematic viscosity eg. 855e-9 m2/s - this is the multiplier for the dispersion terms. A very small viscosity addition will thus cause the dispersion term to increase rather rapidly & so stabilising the numerics quite nicely."

Diaw, you are mixing up dispersion with diffusion - they are different things. Dispersion can, and does, happen when there is no viscosity in the system - basically it is a spreading of the initial disturbance due to the indivual wave compents making up a wave packet moving at different speeds. In this case the total energy, or more correctly the wave action, is conserved. In the case of diffusion the energy decays.

If you consider the simple equation u_t = a.d^n(u)/dx^n for n=1,2,3, etc (a is a constant) then you will see that disperion occurs when n is odd (n=1 is rather special since it corresponds to pure convection => constant group velocity => dispersionless) while diffusion (provided the sign of a is chosen correctly to give a wellposed problem) occurs when n is even. Solve the equations with a=1 and n=2 then n=3 for a Guassian intial condition to see the difference!

Tom.

diaw wrote originally:

"For a fluid like water, it has a tiny kinematic viscosity eg. 855e-9 m2/s - this is the multiplier for the dispersion terms. A very small viscosity addition will thus cause the dispersion term to increase rather rapidly & so stabilising the numerics quite nicely."

Tom's reply:

Diaw, you are mixing up dispersion with diffusion - they are different things. Dispersion can, and does, happen when there is no viscosity in the system - basically it is a spreading of the initial disturbance due to the indivual wave compents making up a wave packet moving at different speeds. In this case the total energy, or more correctly the wave action, is conserved. In the case of diffusion the energy decays.

diaw's reply:

Thanks Tom for your excellent insights. I've tended to look at dispersion in terms of its activity acting on one phase at a time - most surely as an energy-decay mechanism. I've not constrained the mechanism of phase energy-decay to exclude an energy-trade with other phases. In other words, down on one & up on the other.

If it is more correct to term this mechanism as diffusion, then I understand. As I'm understanding your explanation, you're looking at its effect on the wave group as a whole. Fair enough. I guess that the trick would then be to understand the phase energy interchange such the the group maintains its overall energy balance.

In your case mentioned, the group of wave-forms disperse, but group energy content is conserved - possibly either at individual phase energy conservation, or phase energy trading, but global energy conservation.

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Tom wrote:

If you consider the simple equation u_t = a.d^n(u)/dx^n for n=1,2,3, etc (a is a constant) then you will see that disperion occurs when n is odd (n=1 is rather special since it corresponds to pure convection => constant group velocity => dispersionless) while diffusion (provided the sign of a is chosen correctly to give a wellposed problem) occurs when n is even. Solve the equations with a=1 and n=2 then n=3 for a Guassian intial condition to see the difference!

diaw writes:

That's an interesting observation since it is clearly apparent in the KdV equation, with quasilinear convective & 3rd order 'dispersive' term. It 'throws' the dispersion term onto the imaginary axis for a simple wave-analysis.

So, I guess that from what you are saying, that the action of 'dispersion' can be either energy-dissipative (even), or energy-conservative (odd). If so... Eureka !!! This has triggered something about the mechanism of dispersion - time to write it down.

If the strict definition splits the actions into dispersion -> energy-conservative spreading mechanism, & diffusion being a pure energy-decay mechanism (no spreading) - so be it... I'll set this down in more detail this week.

Tom, as always, thanks for your excellent insights. It always gives me food-for-thought & I come away a little richer...

diaw...

Tom.

"If the strict definition splits the actions into dispersion -> energy-conservative spreading mechanism, & diffusion being a pure energy-decay mechanism (no spreading) - so be it... I'll set this down in more detail this week."

In this example the split in definition reflects the the fact that for n=2 it is a parabolic equation while for n=3 it is hyperbolic. In general it is safe to say that dispersion is of "hyperbolic" character while diffusion is "parabolic" in nature.

From a numerical analysis point of view the central differences usually give rise to a dispersive error (hence the oscillations) while the upwind scheme gives a diffusive error (i.e. the damping of oscillations).

Good evening, Tom. Thanks so much for replying (I was hoping you would)...

diaw wrote:

"If the strict definition splits the actions into dispersion -> energy-conservative spreading mechanism, & diffusion being a pure energy-decay mechanism (no spreading) - so be it... I'll set this down in more detail this week."

Tom replied:

In this example the split in definition reflects the the fact that for n=2 it is a parabolic equation while for n=3 it is hyperbolic. In general it is safe to say that dispersion is of "hyperbolic" character while diffusion is "parabolic" in nature.

diaw's reply:

That makes a lot of sense. This is giving me a more clear mental picture based on work I've done in the complex plane. I'll work it in with the hyperbolic & parabolic thoughts. Thanks very much.

Tom wrote:

From a numerical analysis point of view the central differences usually give rise to a dispersive error (hence the oscillations) while the upwind scheme gives a diffusive error (i.e. the damping of oscillations).

diaw's reply:

That is a very interesting review. Hadn't ever thought of it that way.

diaw writes further:

If I could now pull the two thoughts together in the form of the Momentum equations. Let's first work on a 1D transient case. We have both convection (dispersion) & diffusion at work. What interaction is there between the two phenomena? What happens when the convective term equals the diffusive term?

Further:

If we now expand this into a 3d system & re-collect into 'tensor form'. This leads to my question posted earlier regarding the 'tensor wave'. We have a 'dominant' direction - it seems - with a tensorlike waveform... I'd love to know if there is literature in this regard. I'm going into vector-bundles & so forth later this week in the hope of trying to assist me to get the 'tensor shapes' more clear in my head. I've already developed a compact form of the N-S in tensor form, which is very interesting.

I'd love to hear your wisdom on this one... Thanks so much for your input - it is always very much appreciated.

desA

"diaw's reply: That is a very interesting review. Hadn't ever thought of it that way."

This is pretty much the standard interpretation - you should be able to find a discussion of this in a text on numerical analysis (can't remember which book I read this in but see if you can find Randall Leveques lecture notes - they used to be on his webpage).

"diaw writes further: If I could now pull the two thoughts together in the form of the Momentum equations. Let's first work on a 1D transient case. We have both convection (dispersion) & diffusion at work. What interaction is there between the two phenomena? What happens when the convective term equals the diffusive term?"

In the continuous case nothing special happens unless either the viscosity is very small (or possibly very large as in the Oseen approximation). If the Reynolds number is large then a internal boundary layer forms (a bit like in a shock wave or mixing layer). In a finite difference (or any numerical approximation) you get a grid Peclet type condition - this can (and should in my opinion) be interpreted as a statement that you have at least 2-3 points resolving a steep gradient; i.e. the viscosity has to large enough (or the grid fine enough) that any steep gradients formed by convection fit within the resolved internal boundary layer.

"If we now expand this into a 3d system & re-collect into 'tensor form'. This leads to my question posted earlier regarding the 'tensor wave'. We have a 'dominant' direction - it seems - with a tensorlike waveform... I'd love to know if there is literature in this regard. I'm going into vector-bundles & so forth later this week in the hope of trying to assist me to get the 'tensor shapes' more clear in my head. I've already developed a compact form of the N-S in tensor form, which is very interesting."

A wave is statement about the type of solution you are looking for rather about what the solution is; A wave is a solution of the form u=q.exp( i(k.x - w.t) ) irrespective of whether u is a scalar (0th order tensor), vector (1st order tensor), matrix (2nd order tensor), etc.

For this reason I'm not sure what you mean by "tensor wave"

Diaw's reply:

Thanks for the pointer to "Randall Leveques lecture notes". I did manage to locate soem of his material on the website, I'll explore further, thanks very much.

"diaw writes further: If I could now pull the two thoughts together in the form of the Momentum equations. Let's first work on a 1D transient case. We have both convection (dispersion) & diffusion at work. What interaction is there between the two phenomena? What happens when the convective term equals the diffusive term?"

Tom's reply:

In the continuous case nothing special happens unless either the viscosity is very small (or possibly very large as in the Oseen approximation). If the Reynolds number is large then a internal boundary layer forms (a bit like in a shock wave or mixing layer). In a finite difference (or any numerical approximation) you get a grid Peclet type condition - this can (and should in my opinion) be interpreted as a statement that you have at least 2-3 points resolving a steep gradient; i.e. the viscosity has to large enough (or the grid fine enough) that any steep gradients formed by convection fit within the resolved internal boundary layer.

diaw's reply:

Very interesting findings. If I may ask, were these experiemental or numerical simulation? In terms of small viscosity - how small? Water, air, or some other fluid? Water is an odd one in that its kinematic viscosity is 855e-9 ms/s, some 18x lower than for air.

diaw writes further:

"If we now expand this into a 3d system & re-collect into 'tensor form'. This leads to my question posted earlier regarding the 'tensor wave'. We have a 'dominant' direction - it seems - with a tensorlike waveform... I'd love to know if there is literature in this regard. I'm going into vector-bundles & so forth later this week in the hope of trying to assist me to get the 'tensor shapes' more clear in my head. I've already developed a compact form of the N-S in tensor form, which is very interesting."

Tom's reply:

A wave is statement about the type of solution you are looking for rather about what the solution is; A wave is a solution of the form u=q.exp( i(k.x - w.t) ) irrespective of whether u is a scalar (0th order tensor), vector (1st order tensor), matrix (2nd order tensor), etc.

For this reason I'm not sure what you mean by "tensor wave"

diaw writes:

Let me paint some background by way of pointing towards the work of Volpert (AMS download available) regarding parabolic waveforms - then moving into the whole chaos, bifurcation, nonlinear analysis research. Volpert's work seemingly can be extrapolated up to systems of parabolic waveforms, which seem to include the momentum equations. This work claims the existance of wave solutions to this class of waveforms. I still have a mountain of reading to do here, so I'll not go in deeply.

I have then taken a 'birds-eye view' of the momentum equations from this vantage point & begun looking for geometric ways to best interpret the physical mechanisms at work in the tensor form of the momentum equations. From this viewpoint they 'look like' a 1D Burger's type equation, except that they represent the 3d fluid flow field. In form the 'look like' what I can only best describe as a 'tensor waveform'. I'm working completely with tensor & tensor-like structures... Lots more to read up & develop.

I've got the numerical evidence pretty well cleaned up on the existence of co-existant wave solutions within confined fluid domains. 'Waveprobes' have been developed which interogate the flow solutions on the trot. A lot of very interesting information is coming out of all this. I'm now at the stage of going back into the maths/physics to bed these findings further in theoretical terms.

At this juncture, it 'looks like' a wave-duality is present in the waveform - the trade-off between convection-diffusion. This is why I would like to know more about what folks observe at the equality point of convection=diffusion. I'm expecting some sort of resonance/vibration under some circumstances.

I'd be more than happy to send you a pre-print of an internal paper presented here last Friday. (Use an anonymous yahoo server, to keep the University hounds at bay - I'll send it there). It may not be too wise to put too much of my current research out in the open internet, before it gets into hardcopy...

desA

"diaw's reply: Very interesting findings. If I may ask, were these experiemental or numerical simulation? In terms of small viscosity - how small? Water, air, or some other fluid? Water is an odd one in that its kinematic viscosity is 855e-9 ms/s, some 18x lower than for air."

It's an observation based on solving the Navier-Stokes equations using matched asymptotic expansions in the limit of large Reynolds number. The formation of critical layers in unstable shear flows is a good example of this.

"diaw writes: Let me paint some background by way of pointing towards the work of Volpert..."

Volpert's book is about reaction diffusion equations not the Navier-Stokes equations. The NS equations have a fixed nonlinearity while in reaction diffusion problems you can pretty much "make up/change" the form of the nonlinearity to allow you to prove your required result. Another important difference, for incompressible flows, is that you have a constraint that the flow field be solenoidal - no such global constraints occur in reaction diffusion problems (except possibly on boundedness of the concentration which is enforced by suitable choice of the reaction terms). These "waves" are generally of the "wave of permenant form" type (e.g. solitary waves) rather than the usual oscillatory type. For these types of waves the reaction diffusion problem usually reduces to the solution of a nonlinear eigenvalue problem - they are realted to the existence of a Lie group symmetry (see Peter Olver's book on symmetries of odes and pdes).

Something you will need to watch out for when proven results using ideas/theorems from other papers is that rather innocuous changes to an equation can render the theoerem inapplicable - the devil's in the details as they say!

You'll probably need to read up (and solve the exercises since there is a great deal of subtelty in mathethatical proofs) on functional analysis if you follow this path -you'll probably need a pure(ish) maths lecturer to help as well. Peter Lax has written a nice book on functional analysis.

Still not sure about "tensor wave" - after all a vector is a tensor so the NS equations are already in tensor form! Do you really mean operator form? i.e. writing the equations as a single equation using linear and bilear differential operators.

Hope this doesn't discouraging, Tom.

Thanks Tom for your exceptional insights - always a lot of food for thought.

I had seen Volpert's work as having usefulness beyond the simple reaction-diffusion work, in that he seems to have extended the proofs to include the 'form of' the N-S. I'll go through it very carefully to clear my mind on that one. I'm focusing on the 'look of' & 'form' of the equations at the moment - a physicist's brain, I'm afraid.

I plan to work through Lie's work - it's on my reading list. Thanks for the pointer to Olver's work. I had a feeling that I needed to fully explore the symmetry aspects.

Tom wrote:

Still not sure about "tensor wave" - after all a vector is a tensor so the NS equations are already in tensor form! Do you really mean operator form? i.e. writing the equations as a single equation using linear and bilear differential operators.

diaw's reply:

Sure, the N-S are definitely in tensor (vector) form. What I'm exploring, I guess, are the inherent symmetries, properties of tensor operators & ?functional? groups. I've basically uncovered a 'Reynold's-type operator' & I'm working on. I'm doing this to look for the mechanisms (force-balances) at work. I'm treating the N-S terms as vector (tensor) operators in terms of force balances, but, go one step up in investigating the tensor operators from which the vector components arise. (Again, physicist's brain - think in pictures).

In terms of incompressible fluids, there is an 'unstable' vibration motion of the element that complies with isochoric motion at zero dilation. In 1D this is non-physical, but in 2D (to ~1D-dominant flows) & 3D this motion is possible physically. You could see this in the same way structural researchers use the unstable truss element - it can provide local instability - instead of the usual concept of a global instability (Champneys, Hunt, Thompson). The wave-forms you are looking for here are free & forced oscillation forms. I have the free-oscillation side pretty-well on show at the moment - as a result of the 'wave-probe' research - lots of very interesting pictures & numeric results - flowing fluids vibrate in very interesting ways. I'm planning to write that up soon - the only question is which Journal would find this of interest.

Tom wrote:

Hope this doesn't discouraging, Tom.

diaw's reply:

Never discouraged, just spurs me on to read more. Your reading list are wonderful, they have pointed me towards where I need to gain the necessary background. I'm thoroughly enjoying the process. Need a 48 hour day,though...

Thanks so much for your kindness, Tom, it is always appreciated. I haven't been writing much on CFD-online, because I've had my head deep in my research. I come up for air on the rare occasion.