1) What is the difference between Reynolds 1st & 2nd transport equation?

2) What is the task of turbulence model & why it is needed in the calculation of a turbulent flow?

3) How can I classify turbulence model based on time averaged Reynolds equation & what is the example of each model?

4) Galerkin's FEM for elliptic, Kv=F. What is v?

5) What is staggered grid? What is the importance of the grid in calculation of the flow field?

THANX.

1) I'm sorry but i don't know such a difference between them.

4) Idem. I presume it is the unknowns vector, the solution you are looking for when you solve an elliptic problem.

2) The task of a turbulence model comes from a simple fact, if you want to simulate a flow with CFD you should use such a big number of nodes (in space and time...and growing with Re number) that it would be impossible to simulate high Re flows.

So, what happen if we use just the number of nodes we can use for? A completely different dynamic behaviour of your flow and completely different (and wrong) prediction of global parameters as force coefficients and so on.

Why this happens? Because the nonlinear dynamic of the Navier-Stokes Equations is such that (roughly speaking) phenomena happening on different space scales (in example, bigger and smaller than your grid spacing) are not independent of each other so if you eliminate a part of these scales(that is what happens when you use a grid not fine enough) the remaining part of the flow (which is represented on your grid) behaves in a very different and wrong way. So, to simulate high Re number flows, the turbulent ones, we should, in principle, use a grid such that every single structure of the flow is well resolved up to the dissipation scale, the scale at which the viscous dissipation is strong enough to damp every turbulent fluctuation. Unfortunately, this scale is very small and becomes smaller with increasing Re number, so it is impratical, if not impossible at all, to properly solve such kind of flows without some tricks. It is needed to point out that this is entirely due to the nonlinear term of the N.S. equations (which links different space/time scales). If we are working, for example, with heat diffusion in solids, no such problems exists and the grid spacing required is entirely drived by the accuracy required (which is still problem dependent, due to the truncation error behaviour, but in a very different and linear fashion).

The idea to circumvent this problem is to do some kind of operations on the Navier-Stokes equations that will make them solvable with an affordable number of grid points. Two popular approach are RANS and LES.

In RANS, to cut the smallest scales of the flow, those we can't represent on our grid, the N.S. equations are time averaged. When time averaging the N.S. equations, we split the velocity field in a time averaged part, say U, and a fluctuating part, u'. Obviously, because we are now trying to solve just for U, we need an evolution equation for it which, as said before, comes from time averaging the original N.S.

Unfortunately, again, this operation give rise to some unknown-unclosed terms, the Reynolds Stress, dependent on the turbulent fluctuations u' which are not represented on our grid. This is where the turbulence modeling comes in. After the time averaging procedure we have an equation of the form:

NS ( U ) = f ( u' )

to make it solvable, that is to eliminate the unknown term f ( u' ), a turbulent model is a relation of the form:

f ( u' ) = M ( U )

so, in RANS, we effectively solve an equation like the following:

NS ( U ) = M ( U )

Different model exists, that is different functional relations of the form M ( U ).

The very important issue with this approach is that the model has to represents from the smallest, almost damped, fluctuations (which are almost universal) up to the biggest unsteady turbulent structures which are very problem dependent. As a consequence, there is no universal model and every model has at least a tuning part, that is different numeric constants to set empirically to fit the solution with the biggest possible part of experimental data. Another drawback comes from the fact that these models are derived for turbulent flows so the laminar part of the flow, where fluctuations are not present like in the viscous sublayer, has to be explicitly accounted for. The same is true for transition which has to be explicitly modeled.

Moreover, the RANS approach give rise to a steady equation and it's simple unsteady extension, namely URANS, give rise to non-obvious results.

Just to give you a taste of it, in LES the approach is very different, instead of a time averaging a local space averaging is employed, which is a low pass space filtering. This, in some way, give rise to similar closure problems and some modeling is still required.

3) RANS models are usually classified in first and second order models. First order models are usually based on the Boussinesque Hypotesis and reduce the modeling of the unknown term f ( u' ) to one scalar term, usually known as turbulent viscosity. The second order models are based on a formal derivation of the evolution equation for the unclosed terms. These equations, in turn, are still unclosed for some of their terms so modeling enters at this second stage.

Moreover, first order models have their own classification based on the way the turbulent viscosity (t.v.) is effectively computed. In fact, following a dimensional reasoning and the several connections with kinetic theory of gas, it is usually assumed to be the product of a velocity scale and a lenght scale. In turn we have:

a) Algebraic models: the t.v. is calculated with an algebraic relation based on local flow parameters. In example the mixing lenght model with the lenght scale based on the local distance from the wall

b) One-Half equation Models: the t.v. is based on a relation in which one of the parameters is solution of an ordinary differential equation (ODE). In example the mixing lenght model in boundary layers computations with the lenght being the solution of an ODE solved in streamwise direction

c) One Equation Models: the t.v. is based on a relation in which one of the parameters is solution of a partial differential equation (PDE). In example, one of the first modification of the mixing lenght model in which the velocity scale was assumed to be the square root of the turbulent kinetic energy k. An evolution equation is derived for k, modeling is introduced for its unclosed terms and the local solution of this equation is used to derive the local value of the t.v. A modern example is the Spalart-Allmaras turbulence model in which a single evolution equation for the t.v. itself is solved.

d) One and One-Half equations models: As before but one of the parameters is the solution of an ODE the other of a PDE. For example using the k equation (PDE) for the velocity scale and an ODE for the lenght scale.

e) Two equation models. Two PDE for two parameters are solved and the t.v. is based on the product of the two parameters each raised at some power, based on dimensional reasoning. k-e, k-omega and all their modifications are some examples.

Some models employ more than two equations, like the v2-f model. Other models are based on a more complex parametrization of the t.v., up to second order terms, based on the strain-rate tensor Sij. The latter are called Algebraic (Reynolds) Stress Models or ASM.

This is just a taste, probably not very well explained, but this is a very wide field and is difficult to go deeper without writing a book.

5) The difference between staggered and colocated grid is where the velocity and pressure are located on your grid. When all the variables are stored at the same locations the grid is named colocated; when the velocity is stored at some of the faces of a control volume and the pressure at the centroids of such volumes the grid is named staggered.

Several issues are connected with the choice of the grid, mostly due to the resulting discretization of the equations and the properties of such discretization.

However there are several difference between RANS and LES approach. In RANS, because of the nature of the equations, several tricks are available that, without serious drawbacks, make the cell-centered finite volume approach with second order upwind discretization a feasible choice. Moreover, the grid spacing required in RANS is not drived by some spectral requirement and is nearly Re independent except for some near wall requirements.

Because of the nature of the LES equations these are not feasible choices for LES; the same second order discretization is a questionable choice (because the turbulent models also are second order). Several other issues are connected with the discrete conservation of the kinetic energy; moreover the grid spacing has a mild Re dependency only for shear-free flows while near-wall flows have a full Re dependency.

However this is again just a taste and a more complete view can be gained from the several papers available online. The Stanford CTR page is a very useful source with a lot of material on several area of the LES approach.

Hope this helps.