The concept of DOF in fluid mechanics and CFD
Hi,
Does anyone hear about the concept of degree of freedom (DOF) in the field of fluid mechanics and CFD? Can we apply it to our research? I have been thinking of this for sometime that every fluid atom or molecule can be considered as a particle and it should have three velocity components as three degrees of freedom, if the rotation movement is neglected. Therefore, for a small volume of gas, we will have billions of billions of DOF in the domain as a complicated dynamic system. For CFD, this idea will be a little simpler. For a 3D solid computational grid, every grid point is assigned three velocity components (3DOF) as well as some other variables (temperature, pressure...). The BC imposed on the domain can be seen as an elimination of DOF from this system. For example, a slip wall requires the normal velocity to be zero on the wall grid point; nonslip BC needs all velocity components to be zero; and a periodic BC make all variables of the grid points on a domain face identical to another face. They more or less decrease the DOF of grid points on the boundary. Base on how many DOFs are reduced, we can rate these BCs to predict their effects on the flow field: periodic > nonslip wall > slip wall. I did a computational aeroacoustic simulation for a selfoscillating flow sometime ago. A periodic sidewall generated lower sound pressure level (smaller oscillation) than a slip sidewall under other equal conditions. On the other hand, comparing CFD results with experiments, DOF reduction may or may not affect the simulation fidelity. The grid resolution is one thing, the more the closer to reality if we've good solver. For some artificial BC and assumptions (such as wall function), if these approximations agree well with reality, they work as a good simplification method. Because DOFs in real physics are limited in a close way as we assumed in these method. When they don't agree with reality, they will act as extra constraints to the system and produce negative effects to the results. Is my idea correct? Thank you for your opinions. 
Re: The concept of DOF in fluid mechanics and CFD
I don't quite see where you're going with this. Sure you can describe boundary conditions by enforcing constraints which essentially reduce the number of DOF at the boundary. That's what we all do in one way or another. Then what? All of the remaining 99.99% of the degrees of freedom of the system are not independent but connected by the conservation laws. We obtain them by solving the discretized governing equations.

Re: The concept of DOF in fluid mechanics and CFD
Hello,
DOF exist for decades in any field. In plain terms, DOF of system means the variables you want to solve. E.g. if you want to solve for only temperature distribution in a conducting material, then it has one DOF i.e. Temperature. Similary if you are solving for velocity (U V W) and pressure (P) then it has 4 DOF. Basically, DOF means the unknown parameters that needs to be determined. It is one of the minimum number of parameters needed to describe the state of a physical system. Your concept that providing the BCs reduces the DOF of the system is not correct. BC helps in solving the DOF, it does not reduces it. E.g. we provide initial conditions and BC for all the parameters to be solved in the 3D fluid system. Still you have to solve those variables. A simple example: In a conduction problem of a 1D bar, temperature distribution needs to determined. You provide temperature BC. Does it mean that it reduces the DOF? e.i. no need to solve for T?) OPS 
Re: The concept of DOF in fluid mechanics and CFD
Thanks for replying.
My idea is not how we solve the flow field using the concept of DOF, but how the change of DOF on boundary affects the final result, and based on this viewpoint, we can define different BC as strong BC or weak BC. For steady simulation, any variable on every grid point is finally solved resulting a fixed value. The DOF concept may not seem very meaningful. But for the timedependent flow simulation, especially the turbulent flow, the change of DOF will show its effects on the statistic results. For Mani's question, NS equation does not apply the values of the flow variables(Vx,Vy,Vz,T,P..) but their 1st, 2nd order spatial/temporal gradient. We can find infinite number of solutions to satisfy this gradient requirement without a BC. Therefore, NS equation doesn't reduce any DOF in the domain. Fundamentally, NS equation is nothing but a mathematical deduction from Newton's law and thermodynamics equations under certain assumptions. We can't say Newton's law and thermodynamics set constraints to DOFs. For OPS's question of 1D bar heat conduction, suppose we equally set 10 nodes on the bar, then we will have 10DOF for temperature. If we define the two ends of fixed values, then we have 8 DOFs left to solve. I don't see any problem with that. 
Re: The concept of DOF in fluid mechanics and CFD
That wasn't my question. My question is: Where are you going with this, and how is it different from current approaches.
Take compressible CFD for example. Boundary conditions are applied by recognizing that certain characteristics enter the flow field from outside, and others leave the flow field from inside, each carrying a certain DOF. We know those DOF variables (characteristic variables) exactly for the linearized inviscid equations (and approximately for the NS equations). Not just any set of DOF is suitable for boundary conditions (for example, the most suitable DOF are not given by flow variables like velocity or pressure, but by combinations, like entropy). As you said yourself, some boundary conditions will work and others won't. We already know that, and we know which ones will work. What do you have to offer to improve our understanding of boundary conditions? What exactly do you mean by the "concept of DOF" and how is it new? So far you haven't mentioned anything that we are not already doing. Maybe a concrete example will illuminate your idea. 
Re: The concept of DOF in fluid mechanics and CFD
My idea is not to design any new BC by using DOF setting, but to describe the existing BC in terms of DOF. As all BC we used in CFD are kind of approximation or simplification of the realistic boundary, using DOF can be more accurate and closer to the reality for such description. Actually the application of DOF is not limited to the boundary, it's about all the constraint we set up in the domain(such as wall function). And based on DOF analysis, we can evaluate the existing BC about their effect on the computational statistic result.
This idea is from my aeroacoustic simulation experience. We tried two different sidewall BC, periodic BC and slip BC, on our highly unsteady transonic flow simulation. The domain geometry is parallel and symmetrical, and other conditions are equal for the two cases. The slip BC case turned out to be the better one with higher sound pressure level(SPL) and closer to experiment. A direct mathematical explanation is that the periodic BC sometimes counteract the pressure gradient change across the BC and thus weakens the pressure oscillation. From the view of DOF, periodic BC set more constraints to the domain(even less DOF than a nonslip wall in experiment), so the oscillation must be weaker. If we use the DOF concept, we don't have to carry out the simulation to predict this difference. A BC works well because it's a correct approximation to reality. If not, because its approximation is wrong. We can interpret this as: the correct BC has the same or close DOF setup with reality, while the wrong BC has more constraint (DOF reduction) than the reality. A higher DOF reduction can only cause negative effect to the result such as a lower SPL. Isn't this an improved understanding of BC? 
Re: The concept of DOF in fluid mechanics and CFD
>We can interpret this as: the correct BC has the same or close DOF setup with reality, while the wrong BC has more constraint (DOF reduction) than the reality.
Correct, but that's only part of it. Not only do you need to get the *number* of constraints right, but you also need to constrain the correct *types* of degrees of freedom. Typically, these are not flow variables that we use to describe the solution, but nonlinear combinations of flow variables. Will your DOF analysis provide the correct combinations? >Isn't this an improved understanding of BC? I completely understand what you're saying, but it could only improve our understanding if we did not already know the answer.... right? In your aeroacoustics example, it sure is important that the boundary you are modeling is in accord with the reality of the experimental setup. It's not really a mathematical problem, supposed that you know the experimental setup it should be clear how to model the boundary. Furthermore, you might have counted the DOF of each BC beforehand and found out that the periodic one is more restrictive, does not model reflections, and therefore artificially dampens the oscillation. Would that have helped you? Even knowing in advance that one boundary condition will qualitatively yield higher oscillation amplitudes than the other, you still need to run both analyses to find out which one is quantitatively closer to the experimental result. I now kind of see where you are coming from, though. It seems that you were stabbing in the dark to find a suitable boundary condition, and finally go with the one that gives the best match with experiments. I am not sure if this is typical for aeroacoustics, but in many other applications you pretty much know what type of boundary condition to choose, by definition of the problem. 
Re: The concept of DOF in fluid mechanics and CFD
I don't know what kind of model you're using for your simulations but LES (Large Eddy Simulation) it's all about DOF reduction.
That is, the number of DOF of your simulation is the number of time and space grid points required to capture all the structures of the flow, what is needed for a DNS. The objective of the LES is the reduction of DOF. One of the biggest trouble in LES is the specification of boundary conditions because a great number of DOF have to be properly specified (respect to RANS), this is particularly true in regard of the statistics of the flow. This was just to giving you a picture. In respect of what you said, i don't think it's necessary to disturb DOF to understand that periodic BC's are an artificial costrainment not so real. In LES and DNS it is of great practical interest the distance about which the periodicity is applied in reference to the integral lenght scale. In your case an inviscid slip wall maybe is the best condition respect to the periodic one if the latter is applied on a dimension which is small in reference to the distance of correlation of pressure fluctuations. This should be investigated if not known. Regards 
Re: The concept of DOF in fluid mechanics and CFD
Dear bearcat,
Well, it is indeed a wellknown notion in fluid dynamics, but I am afraid you may find it rather heavily mathematical. Anyway, since you are interested, you may have a look at the paper "The dimension of attractors underlying periodic turbulent Poiseuille flow" by L.Keefe, P.Moin, and J.Kim published in J. Fluid Mechanics, 1992, v.242, pp. 129. Here is the abstract: ============= Using a coarse grained (16Ã33Ã8) numerical simulation, a lower bound on the Lyapunov dimension, D[λ], of the attractor underlying turbulent, periodic Poiseuille flow at a pressuregradient Reynolds number of 3200 has been calculated to be approximately 352. These results were obtained on a spatial domain with streamwise and spanwise periods of 1.6π, and correspond to a wallunit Reynolds number of 80 ============ Hope this helps. Regards Sergei 
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