Hello Everyone, this is the problem. Lets assume the interface of two viscous incompresible liquids, where the upper one has a density much lower than the other, and lets consider initialy static conditions. Put rigid conditions below the system and let the upper liquid surface be free, and take two dimentions. When a perturbation is applied to the surface of the upper fluid, for example a periodic one, then for a realistic simulation at the interface : how is it going to respond ?. and, what kind of left and right boundary conditions are adecuate to represent this motion ?.

I appreciate any comments

Manuel Rodriguez

The respond of an interface essential depends on the Froude number -- whether it is sub- or supercritical. In any case, the length of computational domain must be of order not less then 10 characteristic wave length. There are several possible types of outer boundary conditions -- Sommerfeld´s, Orlansky or damping zone (or any combination). There are a lot of articles about these conditions for potential flows, see, e. g. the Proceedings of International Conference on Water Waves and Floating Bodies. For viskous fluids I know the applications of only the last method (see Schumann, ~PRADS´97)

With best regards

Hi.

I am afraid, this answer is not exactly accurate. Your asnwer is more suitable for the case of a steady wave, like in the case of a flow past a ship.

1. If the initial conditions are periodic then the solution is periodic, too. The boundary conditions are then periodicity conditions, and it is natural to take the computation domain length to be equal to the period, hence one wavelength, if you wish.

If the initial conditions are not periodic, say, just a single hump on the surface, then waves are formed. When these waves reach the borders of the coputation domain, they are reflected, and this is not physical. Hence, you have either limit the time until the reflected waves return to the region of interest, and then the required size of the domain depends on the wave speed, not length, or construct non-reflecting boundary conditions, which is difficult, or put the boundaries so far that the waves are weak enough there due to the action of viscosity. Then the distance to them depends on the viscosity and cannot be determined in the terms of the wavelength only.

2. Froude number is the proper parameter in the case when there is a velocity scale. Again, like a ship moving or a river running through rapids. Here, you have the wave speed but you do not have another velocity, so it is hard to define sub- or supercritical flow conditions.

Note, that here we will have at least two wave speeds because there are two surfaces, lower and upper fluid layers. Since it is given that density of the bottom layer is much bigger, these velocities will be very different, too. So, one can expect serious difficulties in calculating such a flow. It can be recommended to apply some asymptotic approach first, to simplify the problem. On the other hand, may be, somebody here around has experience in calculating systems with two very different wave speeds?

Best wishes, Sergei

Hi, thank You for your response.There are some topics to explain: 1. These approaches suit for a general type of flow, not only for steady. Moreover, for steady problems there are several special tricks depending on the method of solution (accomodation of collocation points for potential problems, for instance).

2. The presence of periodicity would be very nice and removes naturally all the problems with side boundaryes, but a viscous flow cannot be periodical in space without any source of energy (e. g. wind)

3. To remove side boundaries so far that 'the waves are weak enough there due to the action of viscosity' seems me impossible. I do not know such works.

4. It seems me, that for any waves, a wave covers one wave length for one period. Thus for obtaining steady solution (as a rule, ~10 periods) one can obtain a necessary condition for the length of computational domain of order 10 wavelength

5. If the perturbance moves, it gives a reference speed for determining a Froude number. For a immovable perturbance, the Froude number can be determined through characteristic period of perturbation. The Froude number determines here whether the flow is sub- or supercritical, as mentioned abobe, and thus, whether there are periodical-type or soliton-type solutions. This is of essential importance for determining the parameters of computations.

6. I would assume that there are three actual wave speeds (one for free surface and two for interface) for a sharp density discontinuity and a continuous spectrum for smooth one.

7. Constructing non-reflecting boundary conditions is indeed very difficult and demands not so theoretical background as numerical trials. If one has a good enough computer, I would recommend numerical damping as the most simple and robust. But it demands about two-wavelength additional computational zone with non-utilized solition.

Mit best regards, Vladimir

>but a viscous flow cannot be periodical in space without any source of energy (e. g. wind)

Unsteady viscous flow can. Again, if the initial condition is periodic in space, then the solution is periodic in space for all t. Since in the problem posed for us there is no source of energy, the eventual state, the steady state, will be static, without any motion at all. Note, that this static solution is periodic, too. (Because a constant satisfies the definition of a periodic function, namely there exist a>0 such that for any x f(x+a)=f(x). Let us be precise and explicit.)

>If the perturbance moves, it gives a reference speed for determining a Froude number.

Yes.

In the problem posed (this is important. NOT in the probems you imagine. In that specific one which is posed. This is the key point) we cannot prescribe the speed of perturbance. Hence, this Froude number is a not a parameter. It has to be found from the solution. Well, etc.

Well, let us stop it. Not interesting. You are right, as far as the problems you have in mind are concerned. Do you know that small problem used to check the ability for abstract reasoning?

Two travellers came to the river. There was a boat which could carry only one person at a time. Travellers crossed the river using the boat and went further. How did they do it?

If you do not imagine what is not there in the formulation, the solution comes within a second, usually .

Best wishes. Yours Sergei

I do not know. And how? With best regards, Vladimir

They were, initially, on the opposite sides of the river.

(1).This way of thinking is quite dangerous because of the ill-defined initial conditions. If the early warning system can not detect whether initially the missile on the radar screen is moving away from or comming toward you, a nuclear ICBM could be launched in 10 minuites by accident. Creative thinking should be on the solution to the real problem, rather than on the action to the ill-defined initial condition problem. One should not attempt to solve a problem when it is not properly defined.(2).Back to the original question about the interface wave motion. I think the amplitude of the free surface motion is one important parameter, and the depth of the upper layer is also important. The force acting on the interface will depend upon the dynamic force acting on it and the velocity at the interface. If low density also means low viscosity, then the interface is not likely to respond to the free surface motion directly because of the much higher viscosity of the bottom layer fluid. It will probably do so at certain frequencies only. I think, the situation is similar to that of a car shock absorber,where the tire motion can be related to the free surface motion, and the car motion can be related to the interface motion. For a bad shock absorber (poor damping), the road motion will be transmitted directly to the car. The only difference is that when the depth of the upper layer fluid is relatively large(relative to the free surface amplitude), the actual dynamic pressure and the velocity at the interface will be greatly reduced.

Hi, John.

>This way of thinking is quite dangerous

Which way? Abstract?

Incompetent reasoning can be dangerous. Abstract reasoning (=mathematics) is just a powerful tool, no more, no less.

>because of the ill-defined initial conditions.

Is it about Manuel's problem or river crossing?

Neither is ill-defined. With river, if one adds to the formulation the assumption that travellers are on the same side, it becomes ill-posed, since has no solution, but this is the fault of the man. The fact that they crossed the river is important information from which it follows that they were on the same side. In its original form, the problem has a unique solution.

And after all, it is just a children test. When Manuel posted his question, he probably wanted an answer to it, not to something else.

>If the early warning system can not detect whether initially the missile on the radar screen is moving away from or comming toward you, a nuclear ICBM could be launched in 10 minuites by accident.

Excellent illustration in my favor. If the system adds its own assumption about the direction of motion, it can. If not, it cannot. If it does add it, there is an error in the code governing its behaviour, or a decision was made in advance about the proper reaction in such a case.

Back to business.

>I think the amplitude of the free surface motion is one important parameter, and the depth of the upper layer is also important.

Sure.

Sergei

Thank you very much for your opinion, I guess I must have watched too much T.V. news recently. Anyway, there is really no direct link between your example and my story. Just the random noise. That's all.