Partial Differential Equations - Modelling and ... - ResearchGate
Partial Differential Equations - Modelling and ... - ResearchGate Partial Differential Equations - Modelling and ... - ResearchGate
246 J.-M. Brun and B. Mohammadi contribution here is the generalization of these solutions in a non-symmetric travel-time based metric to account for non-uniform winds. We add constraint such that solutions built with this approach to be solution of the direct model (i.e. flow equations and transport model for a passive scalar). In particular, the divergence free condition for the generated winds, conservation, positivity and linearity of the solution of transport equations are requested. Numerical examples show a comparison of this approach with a PDE based simulation. Examples also show multi-source configurations as well as sensitivity analysis of detected pollution. This is useful for both source identification and risk analysis. 2 Reduced-Order Modelling One aims to model very large multi-scale phenomenon present in agricultural phyto treatment of cultures. The different entities to account for range from rows of plants to water attraction basins and one should also consider local topography and atmospheric conditions. It is, therefore, obvious that modelling phenomenon falling in length scales below a few meters becomes inevitable. Consider the calculation of a state variable V (p), function of independent variables p. Our aim is to define a suitable search space for the solution V (p) instead of considering a general function space. This former approach is what one does in finite element methods, for instance, where the solution is expressed in some subspace S({W N }) described by the functional basis chosen {W N }, with the quality of the solution being monitored either through the mesh quality and/or increasing the order of the finite element [Cia78]. In all cases, the size of the problem is large 1 ≪ N < ∞ and if the approach is consistent, the projected solution tends to the exact solution when N →∞. In a low-complexity approach, one replaces the calculation of V (p) bya projection over a subspace S({w n }) generated, for instance, by {w n }, a family of solutions (‘snapshots’) of the initial full model (p → V (p)). In particular, one aims n ≪ N [VP05]. In our approach, we aim to remove the calculation of these snapshots as this is not always an easy task. We take advantage of what we know on the physic of the problem and replace the direct model p → V (p) byan approximate model p → v(p) easier to evaluate. This is a very natural way to proceed, as often one does not need all the details on a given state. Also it is sufficient for the low-complexity model to have a local validation domain: one does not necessarily use the same low-complexity model over the whole range of the parameters. We have used this approach in the incomplete sensitivity concept where the linearization is performed not for the direct model but for an approximate state equation [MP01].
Reduced-Order Modelling of Dispersion 247 2.1 Near-Field Solution The first step is to model the solution at the outlet of the injection device used to expand the phyto treatment in between rows. One important hypothesis is to assume two different time scales based on the injection velocity and the velocity at which the injection source moves. The injection velocity being much higher, one assumes the local concentration at the outlet of the injection device to be established instantaneously. This instantaneous local flow field is devoted to vanish immediately and not to affect the overall atmospheric circulation. This injection velocity is only designed to determine the part of the pollutant leaving near-ground area and being candidate for transport over large distances (see Section 2.2). These are strong hypotheses which seriously reduce the search space for the solution. One considers a cylindrical local reference frame where z indicates the motion direction for the vehicle in the field. One looks for local injection solutions of the form: u l ∼ f 1 (r)g 1 (θ)(zh 1 (z)+(1− h 1 (z))r) and c l ∼ f 2 (r)g 2 (θ)h 2 (z), (1) where the subscript l reads for local. r is a unit vector having its origin at the injection point and visiting the unit circle around this point in the plan perpendicular to z. This defines an instantaneous flow field around the injection point. c l denotes the local distribution of a passive scalar. f i (r), i =1, 2, are solutions of a control problem for the assimilation of experimental data by a PDE based model obtained by dimension reduction of the Navier–Stokes and transport equations [Fin00, RT81, Sum71, Bru06]. These experimental data show that after injection both the flow velocity and phyto products concentration drop to nearly zero after three rows of vegetation. g i (θ), i =1, 2, are Gauss distributions describing the characteristics of the injection device and are provided by the manufacturer. h i (z), i =1, 2, include the characteristics of the vegetation by assimilation of experimental data and inform on how the density of the vegetation deviates the flow horizontally. h 1 (z) ∈ [−1, 1] is an erf function, odd and monotonic increasing, and h 2 (z) ∈ [0, 1] is a Gauss distribution. At this level, one includes compatibility conditions coming from the governing equations. In particular, one aims for the conservation condition to hold for the concentration of the passive scalar, the flow field to be divergence free and both variables to verify an advection equation: ∫ ∇·u l = u l ·∇c l =0 c l dv =given. (2) R 3 To summarize, the coefficients in functions f i , g i , h i , i =1, 2, are a solution of an assimilation problem for experimental data under the constraint (2) [Bru06]. From now, one expresses the variables in a global Cartesian reference frame where z denotes the vertical axis.
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246 J.-M. Brun <strong>and</strong> B. Mohammadi<br />
contribution here is the generalization of these solutions in a non-symmetric<br />
travel-time based metric to account for non-uniform winds. We add constraint<br />
such that solutions built with this approach to be solution of the direct model<br />
(i.e. flow equations <strong>and</strong> transport model for a passive scalar). In particular,<br />
the divergence free condition for the generated winds, conservation, positivity<br />
<strong>and</strong> linearity of the solution of transport equations are requested.<br />
Numerical examples show a comparison of this approach with a PDE based<br />
simulation. Examples also show multi-source configurations as well as sensitivity<br />
analysis of detected pollution. This is useful for both source identification<br />
<strong>and</strong> risk analysis.<br />
2 Reduced-Order <strong>Modelling</strong><br />
One aims to model very large multi-scale phenomenon present in agricultural<br />
phyto treatment of cultures. The different entities to account for range<br />
from rows of plants to water attraction basins <strong>and</strong> one should also consider<br />
local topography <strong>and</strong> atmospheric conditions. It is, therefore, obvious that<br />
modelling phenomenon falling in length scales below a few meters becomes<br />
inevitable.<br />
Consider the calculation of a state variable V (p), function of independent<br />
variables p. Our aim is to define a suitable search space for the solution V (p)<br />
instead of considering a general function space. This former approach is what<br />
one does in finite element methods, for instance, where the solution is expressed<br />
in some subspace S({W N }) described by the functional basis chosen<br />
{W N }, with the quality of the solution being monitored either through the<br />
mesh quality <strong>and</strong>/or increasing the order of the finite element [Cia78]. In all<br />
cases, the size of the problem is large 1 ≪ N < ∞ <strong>and</strong> if the approach is<br />
consistent, the projected solution tends to the exact solution when N →∞.<br />
In a low-complexity approach, one replaces the calculation of V (p) bya<br />
projection over a subspace S({w n }) generated, for instance, by {w n }, a family<br />
of solutions (‘snapshots’) of the initial full model (p → V (p)). In particular,<br />
one aims n ≪ N [VP05].<br />
In our approach, we aim to remove the calculation of these snapshots<br />
as this is not always an easy task. We take advantage of what we know on<br />
the physic of the problem <strong>and</strong> replace the direct model p → V (p) byan<br />
approximate model p → v(p) easier to evaluate. This is a very natural way to<br />
proceed, as often one does not need all the details on a given state. Also it is<br />
sufficient for the low-complexity model to have a local validation domain: one<br />
does not necessarily use the same low-complexity model over the whole range<br />
of the parameters. We have used this approach in the incomplete sensitivity<br />
concept where the linearization is performed not for the direct model but for<br />
an approximate state equation [MP01].
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