What's the problem with turbulence models near the wall region?
What causes than most turbulence models need of wall functions? Why do we need wall functions and not simply use the turbulence model to represent the whole fluid?
Do the law of the wall apply for 2d flows as well? 
The turbulence modes assume that the energy may flow "freely" form larger to smaller vortices. In the near of walls this is not correct anymore: The fluid gets more viscous near walls. To take this into account the wall functions are used.

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You can find details in the books of Wilcox, of Pope, etc. 
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Now, what a wall function is? It is a model for the variation of flow quantities near a wall. There are laws of the wall for the velocity (component parallel to wall) as well as for other flow and turbulent quantities. These are all derived using, more or less, the same assumptions. A good overview can be find here: https://www.researchgate.net/profile...functions.pdf So, why would you need a wall function? It turns out that there are two main reasons, according to the variable for which you employ the wall function. The primary reason is for variables like the velocity itself. Basically, as the Re number increases, the velocity profile near a wall becomes steeper. So, while nothing in the equations you are solving prohibits you to invest more mesh points near the wall and eventually fully resolve the velocity profile on your grid, this is largely impractical in most cases. The wall function can, in this case, be interpeted as an advanced interpolation method (in contrast to the linearpolynomial ones used without wall function), one with some embedded knowledge of the underlying function, so that you can compute the wall stress from the first near wall cell velocity even for very coarse meshes. The downside is that, for the particular physics at play, the actual velocity profile might not actually match the assumed underlying function. Still, in these cases, the practical "better than nothing" approach is used. So, in this first case, in practice, the wall function is needed if you don't want to use a lot of mesh points. This is true both in (U)RANS and LES, the latter being just more demanding in terms of required mesh resolution and physics to be represented in the wall model. The second reason for which you might need a wall function is for some turbulence models (e.g., basic versions of keps and reynolds stress models), whose equations are not valid in the buffer and viscous layers near the wall. In this case, you are actually forced to avoid grids fine enough to resolve the wall (unless special tricks are used). It's just that the model underlying assumptions are not valid in the near wall region. Obviously, once some turbulence variable forces you to use a wall function (more precisely, a grid requiring it), all the other variables require it too. Note that the other turbulence models not having this problem, basically kw family and SpalartAllmaras, still need a wall function for their variables in the first case, when the grid is not fine enough (still, by design, for the SA model, such wall function is identical to the wall resolved case). 
But in 2D flows you don't model the wall region
I see, but my concern is because when you have a 2D model, you only get planar dimension results and the wall effects are included as a boundary condition. Do I still need to include some kind of "law of the wall" as a boundary condition if I already included a friction law f.g Chezy or Manning?

> all effects are included as a boundary condition. Do I still need to include some kind of "law of the wall"
The boundary conditions describe what happens straight at the wall. B.c. are no model in closer sense, but rather a physical fact. The don't 'model' what happens in the near of the boundary, they are spaceless (they don't incorporate a volume but only the boundary area). Wall functions describe the volume near the boundary, simplified with a model. 
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Basically, in such cases, you have all the information mixed in the model, from 3D effects to turbulence. 
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