Características del software de modelación Bentley WaterGEMS

In accordance with the selected flow of this study (a parallel boundary layer), corresponding to a parallel flow over a flat plate, the development and evolution of small-scale perturbations are studied in a semi- infinite Cartesian domain over an adequately-long flat plate, as depicted in figure2.1. Accordingly, four types of boundaries are required to be prescribed. The conditions in the upper boundary far from the wall are covered by the assumption of infinitesimal or vanishingly small amplitudes as z → ∞, as described above

in equation (2.22). The upstream and downstream boundaries of the domain are considered, essentially, as inflow and outflow regions, respectively. The adopted treatment of the boundary on the limiting wall ensures the no-slip condition. Likewise, a methodology has been introduced to numerically deal with the semi-infinite domain. Furthermore, given the periodicity of the near-wall structures, the spanwise direction is treated using a Fourier expansion approach. In the following sections a further examination of each of the last four boundaries is presented.

In order to bring clarity to this discussion and for illustration purposes figure2.2shows a schematic represen- tation of the domainΩand boundaries which will be referenced later on. Specifically, boundariesΓ1 andΓ2 are the inlet and outlet of the domainΩ, for non-streamwise-periodic simulations, respectively. Γ3represents the location of the wall bounding the flow at z = 0, and Γ4 represents the upper boundary located at an asymptotic locationz → ∞. The spanwise direction has been intentionally omitted as it will be treated as

periodic and, as discussed in §2.2.3, expanded in independent Fourier modes.

Figure 2.2: Two-dimensional side visualisation of the physical domainΩ.

Boundary conditions on the streamwise direction

For the numerical simulations, the streamwise domain was kept long enough to roughly cover two to three times the anticipated streamwise length of the near-wall streaks. Letting the total domain length in the streamwise direction be represented byLx, every streamwise location is bounded byx∈[0, Lx]. The limits in

this range define the inlet or upstream inflow boundary atx= 0(boundary Γ1 in figure2.2), and the outlet or downstream outflow boundary atx=Lx (boundaryΓ2 in figure2.2).

The numerical code employed in this work focused on two different type of numerical experiments: short-time evolutions (non-periodic simulations) and long-time evolutions (periodic simulations). Depending upon such a criteria different type of boundary conditions were considered for these inlet and outlet regions.

Non-periodic simulations For those cases where the interest was set upon relatively short simulation times, non-periodic simulations were setup. In this case the inlet or inflow boundary has been set as a zero value boundary condition for primary variables, so

with the upstream influence of the perturbations considered to be negligible, as long as the downstream location of such perturbations remains reasonably far from the inlet. In cases where intentionally a source of perturbation was located nearby the inlet, a simple non-reflective boundary condition was used, ensuring first derivatives equal to zero,

∂φ(x= 0, y, z, t)

∂x = 0 (2.31)

and so establishing a non-reflective nature for the inlet region.

For the farthest downstream boundary, the outlet region in the non-periodic simulations, a non-reflecting boundary condition has been defined for all cases as

∂2_{φ(x=Lx, y, z, t)}

∂x2 = 0 (2.32)

This condition has proved to be extremely satisfactory, especially for transient growth-decay type phenomena. Equally, it contributes to subdue the presence of any spurious oscillation and limits reflections of numerical information back into the domain.

Periodic simulations A slight change in the code allowing periodicity in the streamwise direction has been performed when long periods of simulation have been required, so avoiding excessively long computational domains. Thus, a periodic boundary condition has been set to the upstream boundary as

φ(x= 0, y, z, t) =φ(x=Lx, y, z, t) (2.33)

Wall-normal direction and boundary conditions at the wall

A detailed discussion on aspects associated with boundary conditions at the wall for the methodology hereby adopted are thoroughly discussed in Davies and Carpenter [71]. Nevertheless, in such work the main focus was on compliant walls and boundary conditions for moving surfaces. In this thesis, the study these boundary conditions for wall-normal direction are simpler because all applications explored in here correspond to flows over rigid walls. The interest is focused on the stability of different parallel unperturbed mean base flow profiles to small disturbances with no moving walls. In this case the boundary conditions for perturbations

are given by u e =u(x, y, z= 0, t) = 0 (2.34) v e =v(x, y, z= 0, t) = 0 (2.35) w e (x, y, z= 0, t) =η(x, y, t) (2.36)

where (2.34) and (2.35) account for no-slip condition and (2.36) allows us to prescribe any transpiration condition, if it is required. In cases where no transpiration velocity is present,η= 0satisfies incompressibility

at the wall. In the formulation adopted here, boundary conditions for perturbations of wall-normal velocity, as prescribed by (2.36), can be applied directly to the Poisson equation (2.28). In this way any prescription ofwcan be used directly as a Dirichlet condition on this equation.

On the other hand, the prescription of boundary conditions for vorticity is usually a nontrivial issue in this kind of formulation. In the formulation here adopted, however, such imposition may be carried out easily using the definition of the secondary variables. Such definitions would also allow to naturally associate boundary conditions u

e

and v

e

with the vorticity transport equations, if needed. In any case, for the formulation in this thesis the vorticity boundary conditions are replaced by integral constraints obtained directly from the definitions of the secondary variables

Z ∞ z=0 ωy dz=−u e − Z ∞ z=0 ∂w ∂xdz (2.37) Z ∞ z=0 ωxdz=v e + Z ∞ z=0 ∂w ∂ydz (2.38)

These expressions evidently act as constraints on the vorticity transport equations, ensuring at the same time the no-slip condition and incompressibility.

Dealing with the wall-normal semi-infinite domain

Since the wall-normal extension of the physical domain is semi-infinite, with the solid boundary surface located at z = 0 (Γ3 in figure 2.2), it is necessary to adopt a numerical strategy that can deal with such domain. For numerical purposes different approaches to mapping or truncating the semi-finite domain have been presented in [78–81]. Asaithambi [79] highlights problems related to stability and convergence when attempting to directly solve the equation for the entire mapped semi-infinite domain. In order to deal with such a semi-infinite domain, the present work follows the method presented in [38,82,83], where a coordinate

transformation defined as

ζ= L∞

z+L∞ (2.39)

is employed to transform the physical regionz∈[0,∞)on to the bounded computational domainζ∈(0,1]

with the parameter L_∞ being a stretching factor. In general, the stretching factor can be any value such that the minimum mapped coordinateζcorresponds to a sufficiently large coordinatezenough to be greater than the boundary layer thickness. At this location it is safe to assume that the mean velocity profile asymptotically approaches the free stream limit and that, as it was previously defined, primary and secondary variables asymptotically approach zero and effectively vanish. An additional effect of the mapping defined by (2.39) is that every derivative in the wall-normal direction must be modified to account for such re-mapping. As such,z-derivatives must be expressed now as

∂φ ∂z =− ζ2 L∞ ∂φ ∂ζ (2.40) ∂2_φ ∂z2 = ζ4 L2 ∞ ∂2_φ ∂ζ2 + 2ζ3 L2 ∞ ∂φ ∂ζ (2.41)

for every field variableφ. It can be appreciated from (2.40) and (2.41) that this mapping also ensures that if derivatives inζ remain bounded asζ→0, then derivatives inzvanish asz→ ∞. In other words, if∂φ/∂ζ is bounded then∂φ/∂zvanishes approaching the upper boundary for any fieldφ.

Spanwise direction. Fourier modes expansion.

In the spanwise direction, in the present formulation, one of the main features of the near-wall structures it is exploited: their coherent periodicity and relative spanwise uniformity. Owing to the well-known characteristic of the structures under study, a spectral Fourier representation of the perturbations in the spanwise direction has been incorporated and defined in the model as:

φ(x, y, z, t) = 1 2π Z ∞ −∞ ˆ φβ(x, z, t)eiβydβ (2.42)

for every field variableφ, withβ denoting a spanwise wavenumber defined also by a spanwise wavelengthλy as

β= 2π

λy (2.43)

An attractive compensation of this assumption in the model is that, once the governing equations have been linearized, each spanwise or Fourier mode is independent of all others. This allows to study stability of any

chosen laminar or turbulent profile to small perturbations at individual spanwise modes in an independent manner. In this case no boundary conditions are required for the spanwise direction. Two- and three- dimensional simulations can be implemented straightforwardly using the same mathematical model with a proper selection and superposition ofβ modes; though in both cases it is required to use a parallel, spanwise uniform mean flow. Finally, advantage of modern parallel computing techniques, as discussed later in §3.1.3, can be taken with relative ease in the numerical implementation.