Complexity

Complexity / 2021 / Article

Research Article | Open Access

Volume 2021 |Article ID 3912014 | https://doi.org/10.1155/2021/3912014

Nejib Smaoui, Alaa El-Kadri, Mohamed Zribi, "On the Control of the 2D Navier–Stokes Equations with Kolmogorov Forcing", Complexity, vol. 2021, Article ID 3912014, 18 pages, 2021. https://doi.org/10.1155/2021/3912014

On the Control of the 2D Navier–Stokes Equations with Kolmogorov Forcing

Academic Editor: Chongyang Liu
Received02 Jun 2020
Revised12 Dec 2020
Accepted12 Apr 2021
Published30 May 2021

Abstract

This paper is devoted to the control problem of a nonlinear dynamical system obtained by a truncation of the two-dimensional (2D) Navier–Stokes (N-S) equations with periodic boundary conditions and with a sinusoidal external force along the x-direction. This special case of the 2D N-S equations is known as the 2D Kolmogorov flow. Firstly, the dynamics of the 2D Kolmogorov flow which is represented by a nonlinear dynamical system of seven ordinary differential equations (ODEs) of a laminar steady state flow regime and a periodic flow regime are analyzed; numerical simulations are given to illustrate the analysis. Secondly, an adaptive controller is designed for the system of seven ODEs representing the approximation of the dynamics of the 2D Kolmogorov flow to control its dynamics either to a steady-state regime or to a periodic regime; the value of the Reynolds number is determined using an update law. Then, a static sliding mode controller and a dynamic sliding mode controller are designed for the system of seven ODEs representing the approximation of the dynamics of the 2D Kolmogorov flow to control its dynamics either to a steady-state regime or to a periodic regime. Numerical simulations are presented to show the effectiveness of the proposed three control schemes. The simulation results clearly show that the proposed controllers work well.

1. Introduction

In this work, we study the dynamics as well as the adaptive and the sliding mode control problem of seven-mode truncation system of the 2D Navier–Stokes equations with periodic boundary conditions and a sinusoidal external force along the x-direction. This type of forcing is known as the Kolmogorov forcing and the resulting flow is known as the 2D Kolmogorov flow.

In 1958, Kolmogorov [1] introduced the 2D Kolmogorov flow as an example to study transition to turbulence. The model has been successfully used to study 2D turbulent flows in atmospheric, oceanic, and astrophysical flows due to the weak dependence of the velocity field on the third dimension [2, 3]. In the field of magnetohydrodynamics, Kolmogorov flow has been extensively used and was easily reproduced by suitably placed electrical and magnetic fields. Bondarenko et al. [4] observed this flow inside a specific electrically conducting fluid driven by an electromagnetic field [5]. In the literature, linear and nonlinear stability analysis of this flow was investigated for different domain sizes and forcing wave numbers [69]. Numerical simulations and investigations helped in the advancement of the understanding of Kolmogorov flow. In particular, it has been shown that the Kolmogorov flow dynamics exhibits complex structures transforming periodic states to chaotic attractors through a sequence of bifurcations including period doubling, period tripling and gluing bifurcations [1015]. Some of these structures were realized in experimental laboratories [4, 16].

In the last four decades, several reduced order models that approximate the dynamics of the 2D Kolmogorov flow were constructed using the Fourier Galerkin approach [10, 11, 1725]. Franceschini and Tebaldi [21] constructed a five-mode truncation ODE system of the 2D Kolmogorov flow when the external force acts on the mode (2, −1). In [21], a number of steady states and Hopf bifurcations were observed up to a Reynolds number equal to 50. Later on, in 1987, She [12] investigated the metastability and vortex pairing of Kolmogorov flow when the external force along the x-direction acts on the mode (0, 8). Also, Nicolaenko and She [14] studied the dynamics of the coherent structures, homoclinic cycles, and vorticity expolosion in Kolmogorov flow. In 1997, using the Karhunen–Loéve decomposition and symmetry, Smaoui and Armbruster [11] used a computationally effective method to construct a reduced order system of nonlinear ODEs that approximates the dynamics of Kolmogorov flow when the external force acts on the mode (0, 2). One decade later, Chen [23] and Chen et al. [24, 25] obtained a reduced order system of ODEs when the force acts on the mode (0, 4).

Recently, the control problem of nonlinear PDEs with periodic boundary conditions has been the subject of many research studies [2636]. Because of the infinite dimensional nature of these PDEs, the practical implementation of such controllers is a very difficult task. As a consequence, attempts were made to approximate these PDEs based on ODE approximations. The idea of inertial manifold to obtain such reduced systems of ODEs was introduced by Foias et al. [37]. Other efforts to construct systems of ODEs that capture the dynamics of the original PDEs were made by other researchers [2629, 3842]. Smaoui and Zribi [2628] constructed reduced order ODE systems that approximate the dynamics of the 2D Navier–Stokes equations using the truncated Fourier expansion method when the external force along the x-direction acts on the mode (0, k). Moreover, Smaoui [29] derived a seven-mode truncation system of ODEs and proposed controllers for its dynamics using Lyapunov based controllers. Extensive numerical simulations were presented to show the different behavior of Kolmogorov flow for Reynolds number range , and Lyapunov-based controllers were designed to control the dynamics of the system of ODES to different attractors. Although the control problem of parabolic PDEs has been investigated, the control problem of the different finite dimensional approximations of the 2D Kolmogorov flow is not completely investigated.

The main contribution of this paper is the design of an adaptive controller as well as a static and a dynamic sliding mode controllers to control the dynamics of the seven-mode truncation ODEs system of the 2D Navier–Stokes equations. The seven-mode truncation ODEs’ system was completely derived by Smaoui [29]. This ODEs’ system is the lowest dimensional system obtained so far that captures the dynamics of the 2D Navier–Stokes equations with sinusoidal external force , where . We should emphasize here that that the design of such controllers for this well-known partial differential equation has not been treated elsewhere in the literature. First, the dynamics of this 2D Navier–Stokes equation described by a laminar steady-state regime and a periodic flow regime is briefly analyzed. Then, an adaptive control law and a static and a dynamic sliding mode control laws are designed and applied to the system of ODEs to control its dynamics either to a steady state or to a periodic state. It should be noted that other types of controllers for nonlinear systems such as observed-based finite-time tracking sliding mode control, output feedback active suspension control, robust sliding mode control, and adaptive dynamic programming-based decentralized sliding mode control were explored by different investigators from various disciplines [4351].

The paper is organized as follows. In Section 2, the 2D Kolmogorov flow equations and their Fourier Galerkin approximation described by a seventh-order nonlinear ODE system are presented. The dynamics of the reduced order ODE system described by a laminar steady-state regime and a periodic flow regime is described in Section 3. Section 4 presents the design of an adaptive control law which is used to regulate the states of the reduced order ODE system to a desired fixed state or to a periodic state without the knowledge of the Reynolds number. Section 5 presents sliding mode control laws to control the dynamics of the steady state and periodic regimes. The theoretical developments are verified by numerical simulations. Specifically, a static as well as a dynamic sliding mode controllers are proposed for the system of ODEs. Finally, some concluding remarks are given in Section 6.

2. A Seven-Mode Truncation O.D.E System of the 2D Kolmogorov Flow

The “basic 2D Kolmogorov flow” was introduced by Kolmogorov [1] as an example to study transition to turbulence. This flow is the solution of the 2D Navier–Stokes equations:with force and with periodic boundary conditions in two directions . The kinematic viscosity is , where is the Reynolds number and the pressure is .

A system of ODEs can be derived from the Navier–Stokes equations (1) and (2) for by expanding , and using the following Fourier expansion forms:where is a wave vector with integer components, , and the reality condition must hold.

The equation for iswhere and is the component of with respect to .

In [29], the following system of seven ODEs was constructed by considering the set of vectors, , , , , , , and , and their negatives in equation (4):

Remark 1. It can be easily checked that system (5) is invariant under the following symmetries:where , and are reflection symmetries across the x-axis, the y-axis, and the origin, respectively. Therefore, it can be concluded that with the identity transformation form an Abelian group: .

3. The Laminar Regime of the Kolmogorov Flow

In this section, we analyze the dynamics of the seven-mode truncation ODE system given by (5) when the Reynolds number and corresponding to a steady-state laminar regime and a periodic regime, respectively.

At , numerical simulations of system (5) shows that the system has seven fixed points. These points are classified as four asymptotically stable points and three unstable points. Figure 1 presents the phase plane of the system for four different initial conditions, and Figure 2 depicts the vorticity of the corresponding four asymptotically stable fixed points. It can be verified that these fixed points can also be generated by applying the symmetries described in Section 2.

At , the numerical simulations show the existence of four stable periodic orbits arising from a Hopf bifurcation at that remain stable up to (see Figure 3). Figure 4 presents the vorticity corresponding to one of the four periodic regimes at different times. It can be checked that the other stable periodic orbits can also be generated by applying the symmetries described in Section 2.

4. An Adaptive Controller of the 2D Kolmogorov Flow

In this section, we design an adaptive controller to control different dynamics of the seven-mode truncation ODE system given by (5).

4.1. Dynamic Model of the Error System

The model of the first seventh-order ODE system, the master system, is

The model of the second seventh-order ODEs system, the slave system, is

Note the addition of controllers in the dynamic model of the slave system. These controllers will be designed to force the states of the slave system to follow the states of the master system.

Define the errors as the subtraction of the states of the master system from the states of the slave system, which can be written as follows:

Using equations (7)–(9), the dynamical model of the error system can be written as follows:where .

4.2. An Adaptive Controller for the 2D Kolmogorov Flow

In this section, an adaptive-based controller is designed to drive the states of the system in (8) to asymptotically converge to the states of the system in (7) without knowledge of the value of the Reynolds number.

Let the gains be positive scalars. Also, let the control gains be positive scalars such that

Also, define the estimate of the parameter as .

The control scheme is given by the following theorem.

Theorem 1. The application of the adaptive controller,withto the error model given by the set of ODEs (10) guarantees the convergence of the errors to zero as tends to infinity.

Proof. Define the parameter error as ; then, .
Now, consider the Lyapunov function candidate such thatUsing the model of the error system given by (10), the control law given by (11), (12), and the update law of the parameter given by (13), the derivative of with respect to time is such thatorSince the design parameters are positive scalars, then it is concluded that the Lyapunov function defined by equation (14) is positive definite and its derivative is negative semidefinite. Also, since and are bounded, then invoking Barbalat’s lemma, the error functions in (9) asymptotically converge to zero as tends to infinity.

Therefore, it can be concluded that the states of system (8) asymptotically converge to the states of system (7) as tends to infinity.

Numerical simulations were carried out for the proposed adaptive controller. The values of the control gains are taken to be , , , and . The values of the control gains are , , , , , , , and . The simulation results for the master system with corresponds to an initial condition and for the slave system with corresponds to an initial condition . At the beginning of the simulations, the controllers , , , and are set to zero for the first 30 seconds. Then, the control law given by (12) is switched on.

The simulation results of system (8) with the proposed control scheme given by (12) and (13) when the Reynolds number and are presented in Figures 5 and 6, respectively. Note that the dynamics at corresponds to a steady state regime and for corresponds to a periodic regime.

Figures 5 and 6 depict the simulation results for the dynamics of the slave system before and after the control is switched on. Figure 5 shows how the control drags the dynamics of a periodic flow regime of the slave system to a steady state fixed point flow regime of the master system. On the contrary, Figure 6 shows how the dynamics of a steady state fixed point flow regime of the slave system is dragged into the dynamics of a periodic flow regime of the master system.

Figure 7 presents the case when both the master system and the slave system are simulated with the same Reynolds number but with different initial conditions (i.e., for the master system, and for the slave system). The values of the control gains used in this case are the same used in the previous two cases. Figure 7 shows how a periodic regime can be dragged into another symmetric periodic regime.

Therefore, it can be concluded that the numerical simulations clearly show that the proposed adaptive controller is able to force the states of the slave system to converge to the states of the master system even though the exact value of is not known.

5. Sliding Mode Controllers for the 2D Kolmogorov Flow

In this section, sliding mode controllers are proposed to control the dynamics of the seven-mode truncation ODE system presented in (5). The choice of this type of controllers is motivated by the fact that sliding mode controllers are known for their robustness and their insensitivity to modelling errors [4351].

Recall from the previous section that the model of the error system is as follows:where . We will design a static as well as a dynamic sliding mode controllers to force the states of the slave system given by (8) to the states of the master system given by (7).

5.1. A Static Sliding Mode Controller

Define the sliding surfaces such asand define the signum function as follows:

Let the control gains and be positive scalars. The control scheme is introduced by the following theorem.

Theorem 2. The sliding mode control lawwhen applied to the error system (17) guarantees the convergence of the errors to zero.

Proof. Taking the time derivatives of along the trajectories of the errors given by (17) and applying the controllers given by (20), we obtainLet the Lyapunov function candidate be such thatUsing the control law given by (20) and the error model given in (17), the derivative of with respect to time is such thatTherefore, for , for . Hence, the trajectories associated with the discontinuous dynamics given by (21) converge to zero from any initial condition in a finite time given that and () are chosen to be sufficiently large and positive scalars. It should be noted that sincewhere are positive scalars; then, the above inequality is sufficient to ensure the finite time attractiveness of the sliding surfaces .
The reaching time is upper bounded by a function of (0) (i = 1, ..., 4) [52].
Therefore, it can be concluded from (18) that the errors , and converge to zero in finite time.
After such a finite time, the errors’ equations of , and can be written asDefine the vector of reduced errors such that . The system of ODEs given by (25) can be written asorwhereThe characteristic polynomial is . One of the roots of is . The other two roots are located in the left-half of the complex plane since the coefficients of are always positive. Therefore, it can be concluded that the characteristic polynomial is Hurwitz and the matrix is a stable matrix. Therefore, we can conclude that .
Hence, it can be concluded that the errors , and converge to zero in finite time, while the errors , and converge to zero asymptotically.

Therefore, the proposed static sliding mode controller forces the states of the slave system given by (8) to converge to the states of the master system given by (7).

The performance of the proposed static sliding mode controller is simulated using the MATLAB software. Numerical simulations were carried out for the proposed sliding mode controller for three cases. The values of the control gains are taken to be , , , and . The values of the control gains are chosen to be , , , and . Three cases are simulated: for the first case, we choose , and the initial conditions for the master system in (7), and and the initial condition for the slave system in (8). The first state in this case corresponds to a periodic orbit, while the second state corresponds to an asymptotically stable orbit. For the second case, we choose and the initial conditions for the master system in (7), and and the initial conditions for the slave system in (8) system. The two states in this case correspond to two symmetric asymptotically stable orbits. For the third case, we choose and the initial condition x0 = [0.67, −0.32, 0.59, 2.55, 0.165, 4.31, −1.27] for the master system in (7) and and the initial condition for the slave system in (8). Note that the two states in this case correspond to two symmetric stable periodic orbits. Moreover, at the beginning of the simulations in each case, the controllers , , , and are set to zero for the first 10 seconds. Then, the control law given by (20) is switched on to force system (8) to synchronize with system (7).

Figure 8 presents the simulation results for Case 1. Figure 8(a) depicts the norm of the error versus time. Also, the states and versus time and and versus time are plotted in Figures 8(b) and 8(c), respectively. Figure 8(d) plots the state and versus and ; the figure shows the efficacy of the static sliding mode controller to drive the dynamics from one attractor to a different attractor. Figure 9 shows the simulation results for Case 2, and Figure 10 shows the results for Case 3. In the three cases, it is shown how the error converges to zero. Hence, it can be concluded that the designed static sliding mode control law in (20) is able to synchronize the ODE systems in (7) and (8) when these systems have the same or different Reynolds numbers, but they start from two different initial conditions. Clearly, the simulation results indicate that the proposed static sliding mode controller works well.

Remark 2. Sliding mode controllers were implemented on many systems. However, most of the work on the control of the Navier–Stokes equations has been theoretical; numerical algorithms were developed to verify the theoretical results. The implementation of controllers on the Navier–Stokes equations is generally an open research area. It should be mentioned that a few works discussed some implementations issues related to the control of the Navier–Stokes equations. For example, Yan et al. [53] briefly described a practical control algorithm for these equation; Vazquez and Krstic [54] discussed some implementation issues related to a closed-form feedback controller for stabilization of the linearized 2D Navier–Stokes Poiseuille System.
In addition, it should be mentioned that constrained sliding-mode control is an active research area, and we are not considering it in the proposed work. However, for completeness, we refer the reader to [35, 36, 55, 56].

5.2. A Dynamic Sliding Mode Controller

In this section, we design a dynamic sliding mode control to a system of seven ODEs derived from the 2D Navier–Stokes equation.

Let be positive scalars and and be sufficiently large positive scalars. Define the sliding surfaces such as

Theorem 3. The dynamic sliding mode control law,when applied to the error system (17) guarantees the convergence of the errors to zero in a finite time. Hence, the states of the slave system given by (8) converge to the states of the master system given by (7).

Proof. Taking the time derivatives of the sliding surfaces along the trajectories of the errors given by equation (17), we obtainLet the Lyapunov function candidate be such thatTaking the time derivative of with respect to the trajectories given by (31), we obtainSince and for , hence converges to zero in finite time.
The trajectories associated with the unforced discontinuous dynamics equation (32) exhibit a finite-time reach ability to zero from any initial conditions provided the constants are sufficiently large and positive.
Since are driven to zero in finite time, the errors are governed after such a finite time by the following first-order dynamics:Since are positive scalars, then the errors will then converge asymptotically to zero. Hence, the dynamic sliding mode controller guarantees the asymptotic convergence of the errors to zero. Using the same argument as in Theorem 1 of Section 5, one can show that also converge asymptotically to zero.

The performance of the proposed dynamic sliding mode controller is simulated using the MATLAB software. Numerical simulations were carried out for the proposed dynamic mode controller for three cases discussed in Section 5.1. The values of the control gains are taken to be , , , and . The values of the control gains are chosen to be , , , and ; the values of are such that , , , and . Figure 11 presents the simulation results for Case 1. Figure 11(a) depicts the norm of the error versus time. Also, the states and versus time and and versus time are plotted in Figures 11(b) and 11(c), respectively. Figure 11(d) plots the state and versus and ; the figure shows the efficacy of the dynamic sliding mode controller to drive the dynamics from one attractor to a different attractor. Figure 12 shows the simulation results for Case 2, and Figure 13 shows the results for Case 3. In the three cases, it is shown how the error converges to zero. Hence, it can be concluded that the designed dynamic sliding mode control law in (30) is able to synchronize the ODE systems in (7) and (8) when these systems have the same or different Reynolds numbers, but they start from two different initial conditions. Clearly, the simulation results indicate that the proposed dynamic sliding mode controller works well.