# Control Volumes?

1. Mar 21, 2005

### Q_Goest

I'm looking for some terminology. Numerical methods of analysis seem to use different terminology for the same things, such as an "element" or "finite element" used for stress analysis, versus a "control volume" in thermodynamics. I've also seen both terms used in CFD. Is there a proper term for the elements something is broken up into? And how would you define a "node"?

The boundary around this element or volume is often refered to as a control surface or even a control boundary. But is that what a finite element analysis program will use as terminology or is there some other term?

What do you call the physical interactions between volumes or elements? For a stress analysis we have stresses or forces acting on the elements. In CFD analysis, we may have mass moving through the boundary, heat flux, or other influences on the volume or element. Is there a name for these influences?

2. Mar 22, 2005

### Clausius2

As far as I see you know how is the stuff in structural mechanics, but you're little confused as Fluid Mech. is involved, aren't you?.

The fact is nowadays there is people who employ Finite Element Methods to solve CFD problems. Former CFD expertises rejected (it gave good results in structural mechanics but not in Fluid Mech.) to employ this technique with Fluid Mechanics. Because of that, Finite Differences and Finite Volumes have become the most spread methods to solve CFD problems.

Of course a Finite Volume is not the same as a Finite Element. While a Finite Volume is a control volume where Integral Conservation Laws must be satisfied, a Finite Element is a region of the space composed by some computational nodes in its boundary and where the solution is virtually computed and solved imposing a some condition of minimum residual. Also the solution on the nodes is pondered by nodal functions which interpolate the solution across the finite element.

Also there are interaction between elements whatever they are, because by definition there are global constraints such as continuity and momentum transferring that every element must yield simultaneously.

By the way I'm not an expert on FEM at all. I only used it with a commercial code ANSYS, I have never programed it at all. My knowledge is more related to Finite Differences. Therefore whatever the stupidity I have just said, let it me know.

3. Mar 22, 2005

### Q_Goest

Clausius, thanks for the response. I had a look for the terms Finite Volume Method and Finite Element Method and found some decent definitions at Mathworld

Finite Volume Method:
The finite volume method is a numerical method for solving partial differential equations that calculates the values of the conserved variables averaged across the volume…
Ref: http://mathworld.wolfram.com/FiniteVolumeMethod.html

Finite Element Method:
A method for solving an equation by approximating continuous quantities as a set of quantities at discrete points, often regularly spaced into a so-called grid or mesh.
Ref: http://mathworld.wolfram.com/FiniteElementMethod.html

It seems these are numerical methods used when breaking something up with a mesh. They regard how the calculations are performed. If we break something up into small bits, I'm guessing that these are methods used to calculate the physical interactions between the bits. Is that correct? What I'm more interested in is a general concept. I'll try to explain starting with a control volume:

Control Volume:
A control volume is a fixed region in space chosen for the thermodynamic study of mass and energy balances for a flowing system. …
Ref: http://www.engineersedge.com/thermodynamics/control_volume.htm

The control volume concept isn't a numerical method. It's a more general concept. But I don't believe people use the term when describing the volumes into which something is broken up in order to perform a numerical analysis on it. We don't generally talk about a structure that we're doing a finite element analysis on as being broken up into control volumes (or do you?). And we don't generally talk about a fluid that we're doing a CFD analysis on as being broken up into control volumes (or do you?).

So getting back to and refining the original questions,
1) Is there a word for the small chunks of 'stuff' that something is broken up into when performing some type of numerical analysis? Granted, a control volume and a control mass is slightly different, but is there a general term?

2) Is the correct term for the boundary that surrounds that small chunk a "control surface" or is there another term?

3) And is there any terminology which is defined as the physical interactions which cross that boundary? (By "physical interactions" I mean a term which all the stresses, forces, mass flows, electric flux, heat flux, etc… can be lumped into such as 'boundary effect', which is some physical interaction that occurs at the boundary. Note that I just made up the term "boundary affect" to give you an idea of what I'm looking for.)

Thanks.

4. Mar 23, 2005

### Clausius2

Of course I do. When you're computing a problem, the physical domain is divided into small cells usually called "control volumes".

Take a look at my figure attached. It is a mesh of the Soyuz/ST rocket fairing, computed by means of elliptic method. I made it some time ago for a project of a CFD course. It is prepared to compute on it a discretized form of N-S equations with a Finite Difference Method. But anyway, it could be employed with a Finite Volume Method (FVM). Each curvilinear cell you see is a control volume, which is rounded by a control surface.

Why is it called "control volume". If you are familiar with Integral Form of the N-S equations, those equations handle finite regions of space, called "control volumes" in the general case. You integrate those equations over such domain. The aim of FVM is to do the same, but the main difference with a "macroscopic" integral method is that it allows non-uniform profiles of the fluid variables. Each cell must yield an integral conservation law (Continuity, Momentum and Energy). Due to the interaction of each cell (some fluxes which go out of some cell will come into another), all the fluxes are coupled. Pay attention to the word "flux", because I mean the surface integral of some fluid variable (momentum flux, mass flux...). That says to you there is a border of the control volume called "control surface".

Coupling all the integral conservations laws for each control volume you obtain the entire flow field. Each control volume is also controlled and located, so it forms a definited entity.

Commercial codes such as Fluent works that way. FVM gives a major elasticity at computing, without serious problems of meshing a complex geometry nor numerical instabilities.

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5. Mar 23, 2005

### Q_Goest

Clausius, thanks for the feedback again. I'm working on a presentation here and want to make sure the terminology I'm using is correct. I started using "control volume" and "control surface" to describe the elements but wasn't sure if those were the correct terms. Sounds like you're saying they are, at least for CFD. Do you know if that holds true for an FEA stress analysis too? Do you call the elements "control volumes" or do you call them elements or nodes? FEA and CFD analysis weren't very common when I was in college, so although the concept seems fairly basic, I need to ensure the terminology I'm using is valid.

Also, any idea what the best terminology is for question #3 above? In your case, you have mass flowing into and out of your CV's, as well as perhaps some shear stresses or pressures which operate on each control surface. Is there a general term to describe these inputs as they cross over between two CV's?

6. Mar 23, 2005

### Clausius2

You're welcome.

In FEA you talk about "nodes" and "elements". Never talk about "control volumes". The fact is that the information you found about it seems to me to be accurate.

The interaction between one cell and another in steady state is given by an integral extended over some surface $$S_j$$ which is the frontier of the CV:

$$\int_S \overline{F}\cdot \overline{dS}=0$$

$$\overline{F}$$ is the so-called set of principal fluxes. These fluxes are composed of mass flux $$\rho \overline{v}$$, momentum flux $$\rho \overline{v}\overline{v}$$ and energy flux $$\rho (e+v^2/2)\overline v$$. It could be substracted from momentum flux the stress-pressure forces $$PI-\tau'$$, where I is the identity tensor.

That equation represents the conjunction of all the integral conservation laws represented in vectorial form. "Fluxes of fluid variables" is the expression you're looking for.