Indeed, it is rather vertical turbulent diffusion that is often parameterized

by the friction term (in which roughness plays a role, among other things).

You can find an attempt at rigorous derivation of the friction term from vertical turbulent diffusion under standard asymptotic scaling assumptions for shallow water models in

Article Asymptotic derivation of the section-averaged Shallow Water ...

In the horizontal, turbulent diffusion models are employed in many cases, see e.g.

https://doi.org/10.1061/(ASCE)0733-9429(1998)124:5(493)

Many other references to similar work can be found in the literature. Clearly there is a problem when numerical diffusion from a low order method overlaps with that coming from the turbulence model, but this would happen in 3d as well.In some cases, I have argued myself that the magnitude of the numerical diffusion is APPROXIMATELY equivalent to what would be provided by a turbulence model, but this is very much resolution dependent and cannot be taken as a general rule. An interesting alternative to full 3D models are multilayer models, see

Article A Multilayer Method for the Hydrostatic Navier-Stokes Equati...

for a rigorous derivation and

DOI: 10.1016/j.jcp.2018.03.017

for an efficient discretization.

Ciao Luca,

many thanks for sharing your viewpoint with me. I must say this is all logical and make a lot of sense to me. I will read now the papers you have mentioned, many thanks for this!

Cheers

Aron

Just adding some comments to the Luca ones.

Using RANS modelling do not worry much about the low order scheme and numerical diffusion. The magnitude order of the RANS-based turbulence model is higher than the order of the local truncation error. The problem of the accuracy is relevant in LES formulation.

A "turbulent diffusion" mimicked by the action of the numerical scheme is an issue in ILES.

Hi Filippo,

we are talking about the 2d solutions of the SWE equation not about 3d RANS or ILES solvers or do you refer to 2d solutions using RANS or LES, if yes let me please know this publications. Otherwise 3d models are not of interest in this discussion. For the LES it would be interesting to see 2D LES for certain river engineering problems.

Otherwise I cannot reproduce your statement that the solution order for RANS models do not matter. If your momentum part is lower order your will just introduce significant numerical errors on the side of the well known part and by adding some turbulence closure will not make things better. Basically there is a huge range of applications such as storm surge modeling and propagation of nonlinear waves, which are strongly damped by 1st order schemes so basically I find your comments on that far to coarse. Here we did not talk at all about stratification effects and so on and so on. Therefore I must say that the order solution of the numerical schemes do matter a lot in real life applications.

Cheers

Aron

No matter about 2 or 3d assumption. Eddy viscosity models have greater impatto on the solution than local truncation error

Can u provide some real life examples for this?

Anyway as I said no matter what kind of approximation u have 1d, 2d or 3d the numerical error when solving using lower order methods just for the propagation part are very big especially when topographic effects are of concern ... let me see your examples ...

Thanks

Aron

Adding to Filippo's point .... RANS models typically approximate statistically stationary turbulence which is represented in the model by model steady state. The leading order diffusive error of the first order implicit method is Const. x timestep x |u^{n+1} - u^{n}|^2 where the superscript is the timestep. This obviously vanishes at steady state. Thus this error should not impact what is sought in RANS. Storm surge is a non-equilibrium phenomena- you'd use URANS or LES for it I'd imagine (as Filippo suggests). For time accuracy for those time adaptivity and higher formal order are useful . I think adaptivity is essential (and higher order methods needed for efficiency) for getting the right answer.

Hi Aron

According to the William's answer, the local truncation error should be differently considered for steady or transient flow problem. In a statistical approach you can adopt RANS for a problem having a statistically steady state or URANS (to be carefully considered for what means unsteadiness) for a problem that does not have a statistical steady state.

Assume for example the simple 1D Burgers equation as model test:

du/dt + d/dx(u^2/2) = d/dx( mu*du/dx)

Using the Reynolds time-averaging

d/dx (^2/2) = d/dx(mu*d/dx) + unresolved term

Now see the local truncation error for the LHS term, for example a first order upwind (O(h)) and compare it to the unresolved term after a turbulent viscosity-based model is introduced.

Hi William, Fillipo,

ti seems the problem is much more complicated than anticipated ... I need now to consolidate my thoughts and read a bit about all this and by the I was not very clear when talked about 2nd order schemes I was rather thinking of 2nd order of the spatial derivatives rather than the temporal ones ... I have some examples in mind. Sorry for not being precise but still this is very interesting as well.

Many thanks for your time and explanation.

Cheers

Aron