Optimization of Injectors Location in a Water Reservoirs Aeration Problem

Abstract

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 considered model based on the three-dimensional-dependent time Navier-Stokes equation with 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.

1. 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 [1–3]. To present this method, we consider the case in which Ω contains a single injector 𝜔_𝑧,𝜀 located at 𝑧∈ℝ³ and having a diameter 𝜀. The topological sensitivity method provides an asymptotic expansion of an objective function 𝑗 of the form 𝑗Ω⧵𝜔_𝑧,𝜀=𝑗(Ω)+𝜀𝛿𝑗(𝑧)+𝑜(𝜀),∀𝑧∈Ω. (1) This expression is called the topological asymptotic expansion and 𝛿𝑗 is the topological sensitivity function. In order to minimize the cost function, the best location to insert the injector in Ω is where 𝛿𝑗 is the most negative. In fact if 𝛿𝑗(𝑧)<0, we have 𝑗(Ω⧵𝜔_𝑧,𝜀)<𝑗(Ω) for small 𝜀. The function 𝛿𝑗 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.

2. 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 𝜔_inj⊂Ω (see Figure 1), the velocity 𝑢(𝑥,𝑡) and the pressure 𝑝(𝑥,𝑡) solve the following system: 𝜕𝑢𝜕𝑡+𝑢⋅∇𝑢−𝜈Δ𝑢+∇𝑝=𝒢inΩ_𝑖×]0,𝑇],∇⋅𝑢=0inΩ_𝑖×]],𝑢()0,𝑇⋅,0=𝑢⁰inΩ_𝑖,𝜎𝑢,𝑝⋅𝑛=0onΓ_𝑛×]0,𝑇],𝑢=𝑢_windonΓ_𝑠×]0,𝑇],𝑢=0onΓ_𝑤×]0,𝑇],𝑢=𝑢_injon𝜕𝜔_inj×]0,𝑇], (2) where Ω_𝑖=Ω⧵𝜔_inj is the lake domain in the presence of the injector 𝜔_inj, 𝜈 is the water viscosity, 𝒢 is the gravitational force, Γ_𝑛 is a free boundary (inlet/outlet), Γ_𝑠 is the surface in contact with the atmosphere, 𝑢_wind is the wind velocity, Γ_𝑤 is the bottom lake boundary, 𝑇 is the final time of simulation, 𝑢⁰ is the initial velocity field, and 𝜎(𝑢,𝑝)=𝜈(∇𝑢+∇𝑢^𝑇)−𝑝𝐼 is stress tensor with 𝑛 is unit outward normal vector.

Figure 1 

Domain Ω.

For the sake of simplicity, we denote by Γ_𝑖=Γ_𝑠∪Γ_𝑤∪𝜕𝜔_inj and by 𝑢_𝑑=𝑢_windonΓ_𝑠,0onΓ_𝑤,𝑢_injon𝜕𝜔_inj. (3)

3. 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.

3.1. Topological Optimization Problem

For the sake of simplicity, we will assume that the injectors are well separated and have the geometry form 𝜔_{𝑧𝑘,𝜀}=𝑧_𝑘+𝜀𝜔^𝑘,1≤𝑘≤𝑚, where 𝜀 is the shared diameter and 𝜔^𝑘⊂ℝ³ are bounded and smooth domains containing the origin. The points 𝑧_𝑘∈Ω,1≤𝑘≤𝑚 determine the location of the injectors. The domains 𝜔^𝑘 describe the injectors geometries.

In the presence of injectors, the velocity 𝑢_𝜀 and the pressure 𝑝_𝜀 satisfy the following system: 𝜕𝑢_𝜀+𝑢𝜕𝑡^𝜀⋅∇𝑢_𝜀−𝜈Δ𝑢_𝜀+∇𝑝_𝜀=𝒢inΩ⧵^𝑚_𝑘=1𝜔_{𝑧𝑘,𝜀}×]0,𝑇],∇⋅𝑢_𝜀=0inΩ⧵^𝑚_𝑘=1𝜔_{𝑧𝑘,𝜀}×]],𝑢0,𝑇_𝜀=𝑢_𝑑]],𝜎𝑢onΓ×0,𝑇_𝜀,𝑝_𝜀⋅𝑛=0onΓ_𝑛×]0,𝑇]. (4) Note that for 𝜀=0, (𝑢₀,𝑝₀) is solution to 𝜕𝑢₀+𝑢𝜕𝑡⁰⋅∇𝑢₀−𝜈Δ𝑢₀+∇𝑝₀=𝒢inΩ×]0,𝑇],∇⋅𝑢₀]],𝑢=0inΩ×0,𝑇₀=𝑢_𝑑]],𝜎𝑢onΓ×0,𝑇₀,𝑝₀⋅𝑛=0onΓ_𝑛×]0,𝑇]. (5) For the optimization criteria, we assume that a “good” lake oxygenation can be described by a target velocity 𝒰_𝑔. Then, the cost function 𝒥_𝜀 to be minimized is given by 𝒥_𝜀𝑢_𝜀=^𝑇₀𝐽_𝜀𝑢_𝜀𝑑𝑡with𝐽_𝜀𝑢_𝜀=_Ω𝑚𝑢_𝜀−𝒰_𝑔²𝑑𝑥, (6) where Ω_𝑚⊂Ω is the measurement domain (the top layer, see Figure 2).

Figure 2 

The top layer Ω𝑚.

We denote by 𝑗 the design function defined by 𝑗Ω⧵^𝑚_𝑘=1𝜔_{𝑧𝑘,𝜀}=𝒥_𝜀𝑢_𝜀, (7) where 𝑢_𝜀 is the solution of (4).

The identification problem can be formulated as a topological optimization problem: find the optimal location of the injectors 𝜔_{𝑧𝑘,𝜀}=𝑧_𝑘+𝜀𝜔^𝑘,1≤𝑘≤𝑚, inside the lake domain Ω minimizing the optimal design function 𝑗: 𝒪_𝜀Find𝑧^∗_𝑘𝑗∈Ω,1≤𝑘≤𝑚,suchthatΩ⧵⋃^𝑚_𝑘=1𝜔_𝑧^∗_𝑘,𝜀∪^𝑚_𝑘=1=min_{𝜔𝑧𝑘,𝜀⊂Ω}𝑗Ω⧵⋃^𝑚_𝑘=1𝜔_{𝑧𝑘,𝜀}. (8)

To solve this optimization problem (𝒪_𝜀) we will use the topological sensitivity analysis method [2, 3, 9–11]. It consists of studying the variation of the design function 𝑗 with respect to a small topological perturbation of the domain Ω.

3.2. Time Discretization

The Lagrangian representation of the flow is based on the function 𝑋∶(𝑥,𝑠;𝑡)∈Ω×(0,𝑇)²→𝑋(𝑥,𝑠;𝑡), where 𝑋(𝑥,𝑠;𝑡) is the position at time 𝑡 of the particle of fluid which is at point 𝑥 at time 𝑡=𝑠. Thus 𝑡→𝑋(𝑥,𝑠;𝑡) is the parametric representation of the ajectory of the particle. Given the velocity 𝑢, the trajectory may be determined from the initial value problem: 𝑑𝑋()(𝑋()),𝑋()𝑑𝑡𝑥,𝑠;𝑡=𝑢𝑥,𝑠;𝑡,𝑡𝑥,𝑠;𝑠=𝑥. (9)

Lemma 1. If |𝑠−𝑡| is sufficiently small, then the Jacobian of the homeomorphism 𝑥→𝑋(𝑥,𝑠;𝑡) is equal to 1.

Proof. See [12].

Let NT be a positive integer, Δ𝑡=𝑇/NT and 𝑡^𝑛=𝑛Δ𝑡. For 𝑥∈Ω, 𝑠=𝑡^𝑛+1, 𝑡^𝑛≤𝑡≤𝑡^𝑛+1, 0≤𝑛≤𝑁, (9) implies using Euler scheme that 𝑥−𝑋𝑥,𝑡^𝑛+1;𝑡^𝑛=^𝑡𝑛+1_𝑡𝑛𝑢𝑋𝑥,𝑡^𝑛+1𝑋;𝑡,𝑡𝑑𝑡≈Δ𝑡𝑢𝑥,𝑡^𝑛+1;𝑡^𝑛+1,𝑡^𝑛+1=Δ𝑡𝑢𝑥,𝑡^𝑛+1. (10) Then numerically the position 𝑋(𝑥,𝑡^𝑛+1;𝑡^𝑛) is determined by 𝑋𝑥,𝑡^𝑛+1;𝑡^𝑛=𝑥−Δ𝑡𝑢𝑥,𝑡^𝑛+1. (11) Using (10) it is shown in [12] that material derivative is as follows: 𝑑𝑢=𝑑𝑡𝜕𝑢+()𝑢𝜕𝑡𝑢⋅∇𝑢=𝑥,𝑡^𝑛+1−𝑢𝑋𝑥,𝑡^𝑛+1;𝑡^𝑛,𝑡^𝑛.Δ𝑡 (12) This approximation is called characteristic method [13] and using Lemma 1 we show that (12) is 𝐿² stable [14].

Using (12), problem (4) is equivalent to solve the following generalized Stokes equation: Find𝑢^𝑛+1and𝑝^𝑛+1solutionsof𝛼𝑢_𝜀^𝑛+1−𝜈Δ𝑢_𝜀^𝑛+1+∇𝑝_𝜀^𝑛+1=𝐹_𝜀^𝑛+1inΩ⧵^𝑚_𝑘=1𝜔_{𝑧𝑘,𝜀},∇⋅𝑢_𝜀^𝑛+1=0inΩ⧵^𝑚_𝑘=1𝜔_{𝑧𝑘,𝜀},𝑢_𝜀^𝑛+1=𝑢_𝑑𝜎𝑢onΓ,_𝜀𝑛+1,𝑝_𝜀^𝑛+1⋅𝑛=0onΓ_𝑛, (13) where 𝛼=1/Δ𝑡, 𝐹_𝜀^𝑛+1=(1/Δ𝑡)𝑢^𝑛_𝜀(𝑋(𝑥,𝑡^𝑛+1;𝑡^𝑛))+𝒢,  𝑢_𝜀^𝑛+1, and 𝑝_𝜀^𝑛+1 are the approximations of 𝑢_𝜀 and 𝑝_𝜀 on time 𝑡^𝑛+1. Then, at each time step, we have to solve a steady state generalized Stokes problem having the following generic form: 𝛼𝑢_𝜀−𝜈Δ𝑢_𝜀+∇𝑝_𝜀=𝐹_𝜀inΩ⧵^𝑚_𝑘=1𝜔_{𝑧𝑘,𝜀},∇⋅𝑢_𝜀=0inΩ⧵^𝑚_𝑘=1𝜔_{𝑧𝑘,𝜀},𝑢_𝜀=𝑢_𝑑𝜎𝑢onΓ,_𝜀,𝑝_𝜀⋅𝑛=0onΓ_𝑛. (14) Note that for 𝜀=0, at each time step (𝑢₀,𝑝₀) is solution to 𝛼𝑢₀−𝜈Δ𝑢₀+∇𝑝₀=𝐹₀inΩ,∇⋅𝑢₀𝑢=0inΩ,₀=𝑢_𝑑𝜎𝑢onΓ,₀,𝑝₀⋅𝑛=0onΓ_𝑛. (15) Then, in order to obtain an asymptotic expansion of the cost function 𝒥_𝜀 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 𝐽_𝜀.

3.3. Topological Sensitivity Analysis for the Generalized Stokes Problem

In this section we present the topological asymptotic expansion of the cost function 𝐽_𝜀 defined in (6) with respect to the insertion of a small injector 𝜔_𝑧,𝜀=𝑧+𝜀𝜔 inside the domain Ω. From the week formulation of (14) one can show that 𝑢_𝜀 is the solution to 𝑢_𝜀∈𝒱_𝜀,𝒜_𝜀𝑢_𝜀,𝑤=𝑏_𝜀(𝑤),∀𝑤∈𝒱⁰_𝜀,𝑢_𝜀∣Γ=𝑢_𝑑,𝑢_{𝜀∣𝜕𝜔𝑧,𝜀}=𝑢_inj, (16) where the functional space 𝒱_𝜀, the bilinear form 𝒜_𝜀, and the linear form 𝑏_𝜀 are defined by 𝒱_𝜀=𝜃∈𝐻¹Ω⧵𝜔_𝑧,𝜀³,div𝜃=0inΩ⧵𝜔_𝑧,𝜀,𝒱⁰_𝜀=𝒱_𝜀∩𝐻¹₀Ω⧵𝜔_𝑧,𝜀³,𝒜_𝜀(𝑢,𝑤)=𝛼_{Ω⧵𝜔𝑧,𝜀}𝑢𝑤𝑑𝑥+𝜈_{Ω⧵𝜔𝑧,𝜀}∇𝑢∇𝑤𝑑𝑥∀𝑢,𝑤∈𝒱_𝜀,𝑏_𝜀(𝑤)=_{Ω⧵𝜔𝑧,𝜀}𝐹_𝜀𝑤𝑑𝑥∀𝑤∈𝒱_𝜀. (17) Next, we denote by 𝑢_𝜀 the following extension of 𝑢_𝜀 in Ω: 𝑢_𝜀=𝑢_𝜀𝑢inΩ,_injin𝜔_𝑧,𝜀. (18) We begin first by presenting the topological asymptotic expansion of a cost function 𝐽_𝜀 defined on 𝒱_𝜀 and verifying the following hypothesis.

Hypothesis 1. (i) 𝐽₀ is differentiable with respect to 𝑢; its derivative is denoted by 𝐷𝐽₀(𝑢).
(ii) There exists a real number 𝛿𝐽 such that ∀𝜀≥0𝐽_𝜀𝑢_𝜀−𝐽₀𝑢₀=𝐷𝐽₀𝑢₀𝑢_𝜀−𝑢₀+𝜀𝛿𝐽+𝑜(𝜀). (19)
(iii) The adjoint problem 𝒜₀𝑣₀,𝑤=−𝐷𝐽₀𝑢₀𝑤,∀𝑤∈𝒱⁰₀, (20) has a unique solution 𝑣₀∈𝒱⁰₀.

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 𝐽_𝜀 defined in (6).

Theorem 2. If Hypothesis 1 holds, the function 𝐽_𝜀 has the following asymptotic expansion: 𝐽_𝜀𝑢_𝜀−𝐽₀𝑢₀=𝜀−_𝜕𝜔𝜂𝑦𝑑𝑠𝑦⋅𝑣₀(𝑧)+𝛿𝐽+Δ𝑡𝑜(𝜀), (21) where 𝜂∈𝐻^−1/2(𝜕𝜔)³ is the solution to the boundary integral equation _𝜕𝜔𝐸𝑦−𝑥𝜂(𝑥)𝑑𝑠(𝑥)=𝑢_inj−𝑢₀(𝑧),∀𝑦∈𝜕𝜔, (22) with (𝐸, Π) is the fundamental solution of the Stokes equations: 𝐸(𝑦)=(1/8𝜋𝜈𝑟)(𝐼+𝑒_𝑟𝑒^𝑇_𝑟), Π(𝑦)=𝑦/4𝜋𝑟³ with 𝑟=||𝑦||, 𝑒_𝑟=𝑦/𝑟 and 𝑒^𝑇_𝑟 is the transposed vector of 𝑒_𝑟.

Proof. This proof is an adaptation of the proof presented in [11].

In the particular case where 𝜔=𝐵(0,1), the density 𝜂 is given explicitly by [10] 𝜂𝑦=−3𝜈2𝑢₀(𝑧)−𝑢_inj,∀𝑦∈𝜕𝜔. (23) In this case and under the hypothesis of Theorem 2, the asymptotic expansion becomes 𝐽_𝜀𝑢_𝜀−𝐽₀𝑢₀=𝜀6𝜋𝜈𝑢₀(𝑧)−𝑢_inj⋅𝑣₀(𝑧)+𝛿𝐽+Δ𝑡𝑜(𝜀). (24)

Proposition 3. The function 𝐽_𝜀 given by (6) satisfies the Hypothesis 1 with 𝐷𝐽₀𝑢₀(𝑤)=2_Ω𝑚𝑢₀−𝒰_𝑔𝑤𝑑𝑥,∀𝑤∈𝒱₀,𝛿𝐽=0. (25) Then, The function 𝐽_𝜀 has the following expansion 𝐽_𝜀𝑢_𝜀−𝐽₀𝑢₀=2_Ω𝑚𝑢₀−𝒰_𝑔𝑢_𝜀−𝑢₀𝑑𝑥+Δ𝑡𝑜(𝜀). (26)

Proof. See [15].

3.4. 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 𝑗(Ω⧵𝜔_𝑧,𝜀)−𝑗(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 𝑗Ω⧵𝜔_𝑧,𝜀−𝑗(Ω)=𝒥_𝜀𝑢_𝜀−𝒥₀𝑢₀=^𝑇₀𝐽_𝜀𝑢_𝜀𝑑𝑡−^𝑇₀𝐽₀𝑢₀=𝑑𝑡_{NT𝑛=1𝑡𝑛𝑡𝑛−1}𝐽_𝜀𝑢_𝜀−𝐽₀𝑢₀𝑑𝑡, (27) where NT∈ℕ^∗ and 𝑡^𝑛=𝑛Δ𝑡, with Δ𝑡=𝑇/NT is the time step.

By trapezoidal formula, we get ^𝑡𝑛_𝑡𝑛−1𝐽_𝜀𝑢_𝜀(⋅,𝑡)−𝐽₀𝑢₀()=⋅,𝑡𝑑𝑡Δ𝑡2𝐽_𝜀𝑢^𝑛_𝜀−𝐽₀𝑢^𝑛₀+𝐽_𝜀𝑢_𝜀^𝑛−1−𝐽₀𝑢₀^𝑛−1. (28) Using the sensitivity analysis for the generalized Stokes equations (21) and Proposition 3, we deduce the following asymptotic expansion 𝑗Ω⧵𝜔_𝑧,𝜀(Ω)=1−𝑗2𝜀Δ𝑡𝑔(𝑧)+𝑜(𝜀), (29) where 𝑔(𝑧)=∑^NT_𝑛=1(𝛿𝑗^𝑛(𝑧)+𝛿𝑗^𝑛−1(𝑧)), with 𝛿𝑗^𝑛(𝑧)=−_𝜕𝒪𝜂^𝑛𝑦𝑑𝑠𝑦⋅𝑣^𝑛₀(𝑧). (30) Here 𝑣^𝑛₀ and 𝜂^𝑛 are, respectively, the approximations at time 𝑡^𝑛=𝑛Δ𝑡 of the adjoint solution 𝑣₀(⋅𝑡) of problem (20) and 𝜂 defined in (22). If 𝜔 is the unit ball 𝐵(0,1), by (24) 𝛿𝑗^𝑛(𝑧)=6𝜋𝜈𝑢^𝑛₀(𝑧)−𝑢_inj⋅𝑣^𝑛₀(𝑧). (31) In this case, using (25). The design function 𝑗 defined in (7) has the following expansion: 𝑗Ω⧵𝜔_𝑧,𝜀(Ω)×−𝑗=3𝜋𝜈𝜀Δ𝑡_NT𝑛=1𝑢^𝑛₀(𝑧)−𝑢_inj⋅𝑣^𝑛₀(𝑧)+𝑢₀^𝑛−1(𝑧)−𝑢_inj⋅𝑣₀^𝑛−1(𝑧)+𝑜(𝜀), (32) where 𝑢^𝑛₀ and 𝑣^𝑛₀ are, respectively, solutions to the generalized Stokes problem (15) and its associated adjoint problem (20).

4. 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_𝑘,1≤𝑘≤𝑚, is modeled as a small hole 𝜔_{𝑧𝑘,𝜀}=𝑧_𝑘+𝜀𝜔^𝑘 around 𝑧_𝑘∈Ω, having an injection velocity 𝑢^𝑘_inj. In order to use the explicit topological sensitivity expression we assume that 𝜔^𝑘=𝐵(0,1).

We recall that the optimal injectors location is characterized as the solution of the topological optimization problem min_{𝜔𝑧𝑘,𝜀⊂Ω}^𝑇_0Ω𝑚𝑢_𝜀−𝒰_𝑔²𝑑𝑥𝑑𝑡, (33) where Ω_𝑚 is a measurement domain (see Figure 2), 𝒰_𝑔 is an objective fluid flow, and 𝑢_𝜀 is the solution to the Navier-Stokes equations in Ω⧵∪^𝑚_𝑘=1𝜔_{𝑧𝑘,𝜀}.

The detection procedure is based on the following algorithm.

Algorithm 4. (i) For 𝑛=1,NT(a)compute 𝑢^𝑛₀, a solution to the generalized Stokes equations (15), (b)compute 𝑣^𝑛₀, a solution to the associated adjoint problem (20).
(ii) For 1≤𝑘≤3, compute the sensitivity 𝛿𝑗^𝑘(𝑧)=^NT_𝑛=1𝑢^𝑛₀(𝑧)−𝑢^𝑘_inj⋅𝑣₀^𝑛(𝑧)+𝑢₀^𝑛−1(𝑧)−𝑢^𝑘_inj⋅𝑣₀^𝑛−1(𝑧),∀𝑧∈Ω. (34)

We note that the topological sensitivity function 𝛿𝑗^𝑘 depends on 𝑢^𝑛₀ and 𝑣^𝑛₀ solutions, respectively, to the direct and adjoint problems computed in the domain Ω (not in the perturbed one Ω⧵𝜔_𝑧,𝜀) which is an interesting feature of the used approach. The mixed finite element method “𝑃¹+bubble/𝑃¹” [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 (𝑢_ℎ,𝑝_ℎ).

In numerical tests, different values for NT and Δ𝑡 are studied. The presented results in Figures 3–9 correspond to NT=60 and Δ𝑡=1𝑠.

Figure 3 

Injector radius effect.

Figure 4 

Mesh step effect.

Figure 5 

The initial flow.

Figure 6 

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

(a)

(b)

Figure 7 

2D cuts of the topological sensitivity 𝛿𝑗 showing the exact location (black dots) and the local minima of 𝛿𝑗 (red zones).

Figure 8 

Obtained injectors location.

Figure 9 

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

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.

4.1. Sensitivity to the Injector’s Radius

The goal is to study the injector (considered as a sphere) radius effect 𝑟. We consider the case of a unique injector 𝜔 centered at 𝑧^∗ and having radius 𝑟. We introduce the following error function: ()=𝑧erℎ,𝑟^ℎ−𝑧^∗,2𝑟 (35) where ‖⋅‖ is the usual norm in ℝ³, 𝑧^∗ is the exact injector location, and 𝑧_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 𝑟 varying from 0.02 to 0.1. We remark that error rises when 𝑟 increases.

4.2. 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 𝑟=0.02 and different mesh step ℎ varying from 0.02 to 0.08. We remark that error rises when ℎ increases.

4.3. 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 𝜔_𝑘=𝑧_𝑘+𝑟𝐵(0,1),1≤𝑘≤3, where 𝑟<ℎ and ℎ is the mesh size equal in this case to 0.02.

The wanted velocity field 𝒰_𝑔 is chosen as solution to the Navier-Stokes problem in Ω⧵∪³_𝑘=1𝜔_𝑧^∗_𝑘, where 𝜔_𝑧^∗_𝑘=𝑧^∗_𝑘+ℎ𝐵(0,1),1≤𝑘≤3 are the injectors to be detected. The exact locations 𝑧^∗_𝑘 and the injection velocities 𝑢^𝑘_inj are described in Table 1.

Table 1 

The exact locations 𝑧^∗_𝑘,1≤𝑘≤3 and their associated injection velocities.

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 𝛿𝑗. We show in Figure 7, respectively, on the left a vertical cut and on the right a horizontal cut of the topological sensitivity function 𝛿𝑗 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.

4.4. Identification of an Injector Line

The aim here is the detection of an injector line 𝜎 (see Figure 10). It is approximated by a sequence of small injectors (holes).

Figure 10 

Injector.

In order to detect 𝜎, we propose here an iterative process based on the following algorithm.

Algorithm 5. (i) Initialization: choose Ω₀=Ω, and set 𝑘=0.
(ii) Repeat until the target is reached:

The injector 𝜎 is detected iteratively, 𝜎_𝑘+1=𝜎_𝑘∪𝐻_𝑘, with 𝜎₀=∅. The constant 𝑐_𝑘 depends on the most negative value of 𝛿𝑗_𝑘. In the presented test, 𝑐_𝑘=0.8𝛿_𝑘 with 𝛿_𝑘=min_{𝑧∈Ω𝑘}𝛿𝑗_𝑘(𝑧).

The obtained results are shown in Figures 11–14. Figure 11(b) illustrates the obtained fluid flow created by the detected injector. It is nearly identical to the wanted one (see Figure 11(a)).

(a)  𝒰𝑔

(b) 𝑢𝜀 after 3 iterations.

Figure 11 

The wanted and obtained fluid flow on the measurement domain Ω𝑚.

(a)

(b)

Figure 12 

Isovalues of 𝛿𝑗𝑘 showing the detected zone 𝐻𝑘 (red zone) during the optimization process: a vertical cut (a), a horizontal cut (b).

Figure 13 

Injector position with respect to 𝛿𝑗3.

Figure 14 

Objective function 𝒥𝜀(𝑢𝜀).

In Figure 12 we plot the isovalues of 𝛿𝑗_𝑘 showing the detected zone 𝐻_𝑘 (where 𝛿𝑗_𝑘 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_𝜀(𝑢_𝜀).

5. 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 𝛿𝑗 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 𝜀→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.

Acknowledgments

This project was supported by King Saud University, Deanship of Scientific Research, College of Science Research Center.