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Clausius2

Science Advisor

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During my undergraduate studies, I have been taught a lot about some important mathematical models for physics and engineering. The most important and heavy equations I have been shown have been:

1) Navier Stokes Equations for Fluid Flow (Reactive and NonReactive).

2) Structural Mechanics (Navier) Equations.

3) Maxwell Equations.

5) Analytical Mechanics (Lagrange & Hamilton) Equations.

All of them are partial differential equations and integral equations. I have felt the heaviness of these models and how difficult is to integrate them when one tries to reflect poorly how is the real event.

I think these equations are more or less "magic". They are not as simple mathematical models. In particular I am going to try to develop here an imaginary numerical experiment with Navier Stokes Equations (with which I am more familiar).

Imagine I want to simulate numerically the free turbulent jet in water (see the first picture attached). You know that there are an special set of equations for turbulent flow (the Reynolds Averaged Navier Stokes Equations RANS). I won't want to use this set, because they introduce an approach about Reynolds averaged values. I want to use the

But there is another aspect we have to pay attention. If I could compute this problem with DNS technique and with a "traditional" set of boundary conditions, I would not obtain a real turbulent jet, understanding that a complete correspondence with reality won't occur. In order to enhance the correct onset of turbulence, one should simulate the boundary conditions which make this flow to be turbulent: microscopic rugosity in walls and its interaction with boundary layer, unsteady effects, and flow self instability transported from the pipeline upstream.

I mean this: imagine I have the N-S equations for simulating this flow. I have also a set of [tex]N_b[/tex] boundary conditions, which reflects the most accurately possible the real boundary conditions of the flow (called [tex] N_{b(real)}[/tex], and also I have available a time [tex] t [/tex] of computation. The MAIN question of this thread is:

"If [tex] N_b\rightarrow N_{b(real)[/tex] allowing [tex] t\rightarrow[/tex] to be extraordinary larger (which means we must have a great power of computation), then would the numerical solution be the same than the real flow solution?? "

The answer to this question would have non-trivial consequences. I personally think the answer would be "yes it would be". But it would mean that N-S equations are not a model of reality, but they would tend asymptotically to reality.

What do you think?

1) Navier Stokes Equations for Fluid Flow (Reactive and NonReactive).

2) Structural Mechanics (Navier) Equations.

3) Maxwell Equations.

5) Analytical Mechanics (Lagrange & Hamilton) Equations.

All of them are partial differential equations and integral equations. I have felt the heaviness of these models and how difficult is to integrate them when one tries to reflect poorly how is the real event.

I think these equations are more or less "magic". They are not as simple mathematical models. In particular I am going to try to develop here an imaginary numerical experiment with Navier Stokes Equations (with which I am more familiar).

Imagine I want to simulate numerically the free turbulent jet in water (see the first picture attached). You know that there are an special set of equations for turbulent flow (the Reynolds Averaged Navier Stokes Equations RANS). I won't want to use this set, because they introduce an approach about Reynolds averaged values. I want to use the

**complete**Navier Stokes equations. This technique is the so-called Direct Numerical Simulation DNS. Well, in order to compute N-S equations I would need a GREAT number of computational points N, with N being of order [tex]Re_L^{9/4}[/tex] being [tex] Re\sim 10^7[/tex] the Reynolds Number for a typical turbulent flow. Well, the time of computation will be enormous too.But there is another aspect we have to pay attention. If I could compute this problem with DNS technique and with a "traditional" set of boundary conditions, I would not obtain a real turbulent jet, understanding that a complete correspondence with reality won't occur. In order to enhance the correct onset of turbulence, one should simulate the boundary conditions which make this flow to be turbulent: microscopic rugosity in walls and its interaction with boundary layer, unsteady effects, and flow self instability transported from the pipeline upstream.

I mean this: imagine I have the N-S equations for simulating this flow. I have also a set of [tex]N_b[/tex] boundary conditions, which reflects the most accurately possible the real boundary conditions of the flow (called [tex] N_{b(real)}[/tex], and also I have available a time [tex] t [/tex] of computation. The MAIN question of this thread is:

"If [tex] N_b\rightarrow N_{b(real)[/tex] allowing [tex] t\rightarrow[/tex] to be extraordinary larger (which means we must have a great power of computation), then would the numerical solution be the same than the real flow solution?? "

The answer to this question would have non-trivial consequences. I personally think the answer would be "yes it would be". But it would mean that N-S equations are not a model of reality, but they would tend asymptotically to reality.

What do you think?

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