Optimization of Injectors Location in a Water Reservoirs Aeration Problem

The aeration process via injectors is used to combat the lack of oxygen in water reservoirs due to the eutrophication problem. The optimization of the injectors location allows to generate a better flow dynamic in order to remedy to this phenomena an preserve the water quality.We propose to adapt the topological gradient method to this problem.The idea consists of studing the topological sensitivity analysis of the consideredmodel based on the three-dimensional-dependent timeNavier-Stokes equationwith respect to the insertion of an injector in the flow domain. Finally the obtained results are used in a numerical algorithm and some numerical tests are presented to validate the used approach.


Introduction
During the warmer months of the year many lakes and water reservoirs experience some degree of thermal stratification.This phenomena inhibit vertical mixing between the surface layer and the bottom water.This can lead to anoxic conditions in the lower region.The lack of oxygen accelerates the eutrophication of the water which is characterized by a number of damaging effects on its quality.The dynamic aeration is one of used methods to treat the water eutrophication problem.This technique consists of inserting air by the means of injectors located at the bottom of the lake in order to generate a vertical motion mixing up the water of the bottom with that on the top, thus oxygenating the lower part by bringing it in contact with the surface air.
We focus in this work on the optimization of the injectors location in order to generate the best dynamic in the fluid with respect to the aeration purpose.The optimal injectors location is characterized as the solution to a topological optimization problem.The topological sensitivity method is used to solve this problem [1][2][3].To present this method, we consider the case in which Ω contains a single injector  , located at  ∈ R 3 and having a diameter .The topological sensitivity method provides an asymptotic expansion of an objective function  of the form  (Ω \  , ) =  (Ω) +  () +  () , ∀ ∈ Ω. ( This expression is called the topological asymptotic expansion and  is the topological sensitivity function.In order to minimize the cost function, the best location to insert the injector in Ω is where  is the most negative.In fact if () < 0, we have (Ω \  , ) < (Ω) for small .The function  can be used as a descent direction in the domain optimization process.This method is used in [4] to optimize the injector position.In this work a stationary model based on generalized Stokes equations is considered.This paper extend this approach to dependent time model based on three dimensional Navier-Stokes equations.In order to obtain a topological sensitivity analysis of the used model with respect to the insertion of an injector, the model is firstly discretized on time using the characteristics method.A generalized Stokes problem is obtained at each time step.The topological sensitivity analysis for this problem presented in [4] is used and then extended to the Navier-Stokes problem.
The paper is organized as follows.The used model is presented in Section 2. Section 3 is devoted to the optimization problem.A topological sensitivity analysis for the generalized Stokes equations is presented and extended to Navier-Stokes equations.Finally, we illustrate the efficiency of the proposed method by several numerical experiments in Section 4.

Physical Model
Let Ω be a three-dimensional flow domain representing the eutrophized lake.Different models can be used to describe the resulting two phase water air bubbles flow problem.A general framework is proposed in [5].In our case, such a problem leads to many difficulties mainly due to the complexity of the model itself and to the inherent high computing costs for such large-scale applications.For this reason, Legendre in [6] studied the case of one air bubble and its interactions with the fluid flow.This work has been extended to several air bubbles by Clement in [7] but in two-dimensional case.Using the fact that the water phase is dominant, a two-dimensional simplified model is proposed in [8].In this work, the water phase is governed by the three-dimensional dependent time Navier-Stokes equations in which we integrate the effect of momentum released by the injected bubbles by adding a local boundary condition for the velocity on the injector holes.In the presence of an injector  inj ⊂ Ω (see Figure 1), the velocity (, ) and the pressure (, ) solve the following system: where Ω  = Ω \  inj is the lake domain in the presence of the injector  inj ,  is the water viscosity, G is the gravitational force, Γ  is a free boundary (inlet/outlet), Γ  is the surface in contact with the atmosphere,  wind is the wind velocity, Γ  is the bottom lake boundary,  is the final time of simulation,  0 is the initial velocity field, and (, ) = (∇ + ∇  ) −  is stress tensor with  is unit outward normal vector.
For the sake of simplicity, we denote by Γ  = Γ  ∪Γ  ∪ inj and by

Optimization of Injectors Location
This section is dedicated to design an efficient method to optimize the injectors location in order to generate the best motion of the fluid.The topological optimization method is used for this purpose.

Topological Optimization Problem.
For the sake of simplicity, we will assume that the injectors are well separated and have the geometry form    , =   +   , 1 ≤  ≤ , where  is the shared diameter and   ⊂ R 3 are bounded and smooth domains containing the origin.The points   ∈ Ω, 1 ≤  ≤  determine the location of the injectors.The domains   describe the injectors geometries.
In the presence of injectors, the velocity   and the pressure   satisfy the following system: Note that for  = 0, ( 0 ,  0 ) is solution to For the optimization criteria, we assume that a "good" lake oxygenation can be described by a target velocity U  .Then, the cost function J  to be minimized is given by where Ω  ⊂ Ω is the measurement domain (the top layer, see Figure 2).We denote by  the design function defined by where   is the solution of ( 4).
The identification problem can be formulated as a topological optimization problem: find the optimal location of the injectors    , =   +   , 1 ≤  ≤ , inside the lake domain Ω minimizing the optimal design function : To solve this optimization problem (O  ) we will use the topological sensitivity analysis method [2,3,[9][10][11].It consists of studying the variation of the design function  with respect to a small topological perturbation of the domain Ω.

Time Discretization.
The Lagrangian representation of the flow is based on the function  : (, ; ) ∈ Ω × (0, ) 2 → (, ; ), where (, ; ) is the position at time  of the particle of fluid which is at point  at time  = .Thus  → (, ; ) is the parametric representation of the ajectory of the particle.Given the velocity , the trajectory may be determined from the initial value problem: Lemma 1.If | − | is sufficiently small, then the Jacobian of the homeomorphism  → (, ; ) is equal to 1.
Let NT be a positive integer, Δ = /NT and   = Δ.For  ∈ Ω,  =  +1 ,   ≤  ≤  +1 , 0 ≤  ≤ , (9) implies using Euler scheme that Then numerically the position (,  +1 ;   ) is determined by Using (10) it is shown in [12] that material derivative is as follows: This approximation is called characteristic method [13] and using Lemma 1 we show that ( 12) is  2 stable [14].Using (12), problem ( 4) is equivalent to solve the following generalized Stokes equation: Find  +1 and  +1 solutions of where  = 1/Δ, Note that for  = 0, at each time step ( 0 ,  0 ) is solution to Then, in order to obtain an asymptotic expansion of the cost function J  for the Navier-Stokes problem, we study first the topological sensitivity analysis for the generalized Stokes problem by giving an asymptotic expansion of the cost function   .

Topological Sensitivity Analysis for the Generalized Stokes
Problem.In this section we present the topological asymptotic expansion of the cost function   defined in (6) with respect to the insertion of a small injector  , = + inside the domain Ω.From the week formulation of ( 14) one can show that   is the solution to where the functional space V  , the bilinear form A  , and the linear form   are defined by Next, we denote by û the following extension of   in Ω: We begin first by presenting the topological asymptotic expansion of a cost function   defined on V  and verifying the following hypothesis.
Hypothesis 1. (i)  0 is differentiable with respect to ; its derivative is denoted by  0 ().
(ii) There exists a real number  such that ∀ ≥ 0 (iii) The adjoint problem has a unique solution  0 ∈ V 0 0 .
The asymptotic expansion described in the following theorem is valid for arbitrary shaped holes and all cost function verifying the Hypothesis 1.The case of spherical holes is given in (24) [11].Finally we present the sensitivity analysis corresponding to the cost function   defined in (6).

Theorem 2. If Hypothesis 1 holds, the function 𝐽 𝜀 has the following asymptotic expansion:
where  ∈  −1/2 () 3 is the solution to the boundary integral equation Proof.This proof is an adaptation of the proof presented in [11].
In the particular case where  = (0, 1), the density  is given explicitly by [10] In this case and under the hypothesis of Theorem 2, the asymptotic expansion becomes Proposition 3. The function   given by (6) satisfies the Hypothesis 1 with Then, The function   has the following expansion Proof.See [15].

Topological Sensitivity Analysis for the Navier-Stokes
Problem.In this section we consider the in stationary Navier-Stokes problem (4).We want to compute the variation (Ω \  , ) − (Ω) with respect to the insertion of a small injector in the fluid flow domain.Using ( 6), the variation of the design function ( 7) is given by where NT ∈ N * and   = Δ, with Δ = /NT is the time step.
By trapezoidal formula, we get Using the sensitivity analysis for the generalized Stokes equations (21) and Proposition 3, we deduce the following asymptotic expansion where () = ∑ NT =1 (  () +  −1 ()), with Here   0 and   are, respectively, the approximations at time   = Δ of the adjoint solution  0 (⋅) of problem (20) and  defined in (22).If  is the unit ball (0, 1), by (24) In this case, using (25).The design function  defined in (7) has the following expansion: where   0 and   0 are, respectively, solutions to the generalized Stokes problem (15) and its associated adjoint problem (20).

Numerical Results
This section is devoted to some numerical examples.Based on the previous theoretical results, we propose fast and accurate algorithms for the detection of optimal injectors location.
As described in the previous section, each injector Inj  , 1 ≤  ≤ , is modeled as a small hole    , =   +   around   ∈ Ω, having an injection velocity   inj .In order to use the explicit topological sensitivity expression we assume that   = (0, 1).
We recall that the optimal injectors location is characterized as the solution of the topological optimization problem min where Ω  is a measurement domain (see Figure 2), U  is an objective fluid flow, and   is the solution to the Navier-Stokes equations in Ω \ ∪  =1    , .The detection procedure is based on the following algorithm.(ii) For 1 ≤  ≤ 3, compute the sensitivity We note that the topological sensitivity function   depends on   0 and   0 solutions, respectively, to the direct and adjoint problems computed in the domain Ω (not in the perturbed one Ω \  , ) which is an interesting feature of the used approach.The mixed finite element method " 1 + bubble/ 1 " [16] is used for the space approximation of the problems (15) and (20).The obtained linear system is solved using Uzawa method [13].One can consult [12] for the existence, uniqueness of the solution and the analysis of the error estimates of the discrete solution ( ℎ ,  ℎ ).
In numerical tests, different values for NT and Δ are studied.The presented results in Figures 3-9 correspond to NT = 60 and Δ = 1.
Next, some numerical examples are presented.The first and second tests presented in Sections 4.1 and 4.2 concern, respectively, the study of the injector radius and the mesh step ℎ effects on the optimization precess.We consider Ω = [0, 1]× [0, 1]×[0, 1].The domain Ω is discretized by tetrahedral mesh with mesh step ℎ.
The third example concerns the identification of separated small injectors in a water reservoir.In the last example we discuss the detection of an injector line containing a sequence of small injectors representing the injection holes.For the two last tests presented in Sections 4.3 and 4.4, a 3D reservoir with length 8 m, width 4 m, and maximum height 3 m is considered.The domain is discretized using 16007 nodes and 85112 tetrahedral.

Sensitivity to the Injector's Radius.
The goal is to study the injector (considered as a sphere) radius effect .We consider the case of a unique injector  centered at  * and having radius .We introduce the following error function: where ‖ ⋅ ‖ is the usual norm in R 3 ,  * is the exact injector location, and  ℎ is the computed location using the mesh step ℎ.We presenet in Figure 3 the obtained errors for ℎ = 0.02 and different injector radius  varying from 0.02 to 0.1.We remark that error rises when  increases.

Sensitivity to the Mesh
Step.The mesh step effect is considered in this section.The error (35) is plotted in Figure 4 for a fixed radius  = 0.02 and different mesh step ℎ varying from 0.02 to 0.08.We remark that error rises when ℎ increases.

Detection of Separated Small
Injectors.This test concerns the detection of optimal location of separated small injectors.Each injector is characterized by a small hole.In the numerical computation, we have considered the case of three injectors   =   + (0, 1), 1 ≤  ≤ 3, where  < ℎ and ℎ is the mesh size equal in this case to 0.02.
The wanted velocity field U  is chosen as solution to the Navier-Stokes problem in Ω \ ∪ 3 =1   *  , where   *  =  *  + ℎ(0, 1), 1 ≤  ≤ 3 are the injectors to be detected.The exact locations  *  and the injection velocities   inj are described in Table 1.In Figure 5, we present the initial flow.The wanted flow is shown in Figure 6.
The isovalues of the topological sensitivity are plotted in Figure 7. Using Algorithm 4, the injectors locations are given by the local minima of the topological sensitivity .We show in Figure 7, respectively, on the left a vertical cut and on the right a horizontal cut of the topological sensitivity function  at each injector.One can observe that the local minima of the obtained topological sensitivity coincides with the exact location described in Table 1.
The obtained optimal locations are shown in Figure 8.The obtained velocity is shown in Figure 9. Since we have detected the exact location, the obtained velocity is identical to the wanted one.

Identification of an Injector Line.
The aim here is the detection of an injector line  (see Figure 10).It is approximated by a sequence of small injectors (holes).
In order to detect , we propose here an iterative process based on the following algorithm.
(c) determine the set   = { ∈ Ω  ,   () ≤   < 0}, where   is chosen in such a way that the cost function decreases,       detected injector.It is nearly identical to the wanted one (see Figure 11(a)).
In Figure 12 we plot the isovalues of   showing the detected zone   (where   is most negative) during the optimization process.
One can note that the proposed algorithm (Algorithm 5) permits to detect the injector line in only three iterations (Figure 13).Finally Figure 14 plots the objective function J  (  ).

Conclusion
In this paper a numerical study of the optimal injector location problem in lake aeration process has been studied.The topological sensitivity analysis method is used to solve the optimization problem.The proposed numerical algorithm is fast, is easily implemented, and only needs to solve the direct and the adjoint problems on a fixed grid.In practice we have obtained interesting numerical results.Like all numerical methods, the proposed approach has its own drawbacks: on one hand, in the case of the simultaneous creation of multiple holes.The topological sensitivity  provides information on where to create holes independently of its number.Then the obtained solution can be seen rather as a suggestion then as an optimal solution.On the other hand, the numerical realization via discretization may not take properly into account the correct influence of the shape of the hole assumed to be spherical of radius  → 0. To this aim we need a very fine grid which leads to many difficulties mainly due to the high computing costs necessary for such three-dimensional application.For this reason a parallel implementation of the numerical algorithm can be investigated in a forthcoming work.

Figure 6 :
Figure 6: The wanted velocity: a 2D cut on each injector.

8 − 2 .Figure 7 :
Figure 7: 2D cuts of the topological sensitivity  showing the exact location (black dots) and the local minima of  (red zones).

Figure 9 :
Figure 9: The obtained velocity: a 2D cut on each injector.

Figure 11 :Figure 12 :
Figure 11: The wanted and obtained fluid flow on the measurement domain Ω  .