In this section, we outline the theory behind feedback control, which is the basis for the design of the controls developed in [19, 43]. We then present a hierarchy of models for the dynamics of the interface of a thin film of water flowing down a plane in two dimensions (2D), which is the physical system of interest—this hierarchy is obtained using asymptotic expansions that lead to reduced-order models.
An ODE example
The theory of linear feedback control was first established for (systems of) ordinary differential equations (ODEs), see [51]. To introduce this methodology, we consider the simplest example of controlling a scalar ODE
$$\begin{aligned} {\dot{y}} = \lambda y, \quad y(0) = y_0, \end{aligned}$$
(1)
where the dot represents derivative with respect to time. It is well known that the solution of (1) is \(y(t) = y_0 e^{\lambda t}\) and that \(y(t)\rightarrow 0\) if \(\lambda < 0\) and \(y(t)\rightarrow \infty \) if \(\lambda \) is positive.
The main goal of linear control theory is to introduce a control to Eq. (1), i.e.
$$\begin{aligned} {\dot{y}} = \lambda y + f, \quad y(0) = y_0, \end{aligned}$$
(2)
and choose f in such a way that the solution is stabilised, i.e. driven to \(y = 0\) even for \(\lambda >0\). The simplest example of a proportional feedback control would be to choose \(f(t) = -\alpha y(t)\) for some positive constant \(\alpha \). It is clear that if \(\alpha \) is such that \(\lambda -\alpha < 0\), then \(y(t)\rightarrow 0\) as \(t\rightarrow \infty \) and we say that the control f(t) stabilises the solution to the ODE. The term feedback is used because the control uses information on the current state of the system; the control is called proportional since it is proportional to the current solution. A similar idea can be used in systems of ODEs
$$\begin{aligned} \dot{\mathbf {y}} = A \mathbf {y} + \mathbf {f}, \quad \mathbf {y}(0) = \mathbf {y_0}, \end{aligned}$$
(3)
where now \(\mathbf {y}, \, \mathbf {y_0}, \, \mathbf {f}\in {\mathbb {R}}^d\) and A is a \(d\times d\) matrix. Similarly, to the one-dimensional case, when \(\mathbf {f}=\mathbf {0}\), if the eigenvalues of A all have negative real part, the solution is asymptotically stable, i.e. \(\mathbf {y}(t)\rightarrow \mathbf {0}\) as \(t\rightarrow \infty \). The analogue of the previous control here is to use \(\mathbf {f}(t) = -\alpha \mathbf {y}(t) = -\alpha I \mathbf {y}(t)\), where I is the \(d\times d\) identity matrix. A simple calculation can be used to find the smallest \(\alpha \) necessary to stabilise the system, namely choose \(\alpha \) such that the eigenvalues of \(A-\alpha I\) all have negative real parts.
It is naturally more efficient to choose the controls using more information about the model. This can be done by designing the control as follows:
$$\begin{aligned} \dot{\mathbf {y}} = A\mathbf {y}+B\mathbf {f}, \quad \mathbf {y}(0) = \mathbf {y}_0. \end{aligned}$$
(4)
Here, B is a \(d\times M\) matrix that encodes some information about how one applies the controls (e.g. if the controls are localised in some part of the domain), and the controls are now \(\mathbf {f}\in {\mathbb {R}}^M\). Note that we can have \(M = d\) and \(B = I\), which is the case outlined above. It can be shown that under certain assumptions on the matrices A and B (namely the Kalman rank condition [51]), one can find a matrix K such that the eigenvalues of \(A+BK\) all have negative real part, and therefore, the controls \(\mathbf {f} = K\mathbf {y}\) stabilise the system. The matrix K can be computed using, e.g. a pole placement algorithm [24] or by solving a linear quadratic regulator problem [51].
In more realistic physical systems, the dynamics of the solution are modelled by a nonlinear system of ODEs,
$$\begin{aligned} \dot{\mathbf {y}} = F(y) + B\mathbf {f}, \quad \mathbf {y}(0) = \mathbf {y}_0, \end{aligned}$$
(5)
where F is some nonlinear function of y. In addition, physical continuum models provide (linear or nonlinear) partial differential equations (PDEs), e.g. a reaction–diffusion equation for the evolution of a population, tumour growth or other biological and chemical applications. Such PDEs take the general form
$$\begin{aligned} u_t = {\mathcal {L}} u + {\mathcal {N}}(u) + f, \end{aligned}$$
(6)
along with appropriate initial and boundary conditions. The subscript t denotes time derivative, and \({\mathcal {L}}, \, {\mathcal {N}}\) are linear and nonlinear spatial differential operators, respectively. By projecting this equation to an appropriate basis (e.g. taking Fourier transforms), one can write the PDE as an infinite-dimensional system of ODEs such as (5). Passing to this limit is not straightforward, even for linear PDEs [51]; however, in certain cases this is possible, even for nonlinear PDEs, as was shown for the Kuramoto–Sivashinsky equation in [4, 19]. In this case, in order to prove that the controls stabilise the full nonlinear system, one needs to use nonlinear stability analysis techniques, such as finding a Lyapunov function [25].
Reduced-order models for thin liquid films
This section is devoted to a brief description of the physical problem, its mathematical modelling and the asymptotic work that leads to the hierarchy of reduced-order models we are focussing on. Consider a thin film of water flowing down a 2D plane inclined at an angle \(\theta \) to the horizontal, as shown in Fig. 1. Throughout this discussion, we will be using the \((\cdot )^*\) notation to distinguish dimensional quantities from their undecorated dimensionless counterparts. The film thickness is denoted by \(h^*(x,t)\), and our main goal is to drive the system to its undisturbed flat interface, \(h^*(x,t) = h^*_0\). We will apply proportional feedback controls which are actuated by means of blowing or suction of water via slots in the wall, and the controls will be designed based on readings of the interfacial height.
The full mathematical model for the fluid motion is given by the Navier–Stokes equations for both liquid film and gas above it with a suitable nonlinear coupling at the interface. Due to the passive nature of the gas in our applications (in view of large density/viscosity ratios), one can restrict attention to the liquid film alone. The blowing/suction conditions at the wall \(y=0\) are
$$\begin{aligned} u^* = 0, \quad v^* = F^*(x^*,t^*), \end{aligned}$$
(7)
where \(u^*,v^*\) are the streamwise (parallel to the inclined plane) and transverse (perpendicular to the plane) velocities, respectively, as depicted in Fig. 1. The uncontrolled system admits a uniform flat film solution known as the Nusselt solution [21], given by \(h^*(x,t) = h_0^*\) and a semi-parabolic in \(y^*\) horizontal fluid velocity with surface value \(U^*_s = \frac{\rho ^* g^* \sin \theta (h_0^*)^2}{2\mu ^*}\), where \(g^*\), \(\rho ^*\) and \(\mu ^*\) are the acceleration of gravity and the fluid’s density and viscosity, respectively. An appropriate non-dimensionalisation of this problem allows us to define two important dimensionless parameters that characterise the system. The Reynolds number \({\text {Re}} = \frac{\rho ^* U^*_s h_0^*}{\mu ^*}\) and the capillary number \({\text {Ca}} = \frac{\mu ^* U^*_s}{\gamma ^*}\), which measure the relative importance between inertia and viscosity and between gravity and surface tension (represented by \(\gamma ^*\)), respectively.
As explained previously, full models such as the Navier–Stokes equations are computationally expensive to simulate. However, in the case of thin liquid films, the mean interface height \(h_0^*\) is much smaller than the length of the domain, \(L^* = nh^*_0\), so that we can define a long wave parameter \(\epsilon = \frac{h_0^*}{L^*} = n^{-1} \ll 1\). This disparity of scales facilitates a multiscale approach to derive from first principles hierarchies of reduced-order models. For the remainder of the modelling discussion, we use \(h_0^*\) and \(U^*_s\) as reference length and velocity scales alongside the defined groupings to transfer the system to its dimensionless counterpart (and drop the decorations accordingly). In this context, the requisite assumptions are:
- (A1):
-
(long-wave assumption) the geometrical aspect ratio \( \epsilon \) is small;
- (A2):
-
The Reynolds number Re is \({\mathcal {O}}(1)\);
- (A3):
-
Surface tension is sufficiently strong to appear at leading order, i.e. the capillary number is small, and \({\text {Ca}} = {\mathcal {O}}(\epsilon ^2)\) is the appropriate distinguished limit;
- (A4):
-
The controls F are small \(F = {\mathcal {O}}(\epsilon )\), implying weak blowing/suction.
Using assumptions (A1)–(A4) and asymptotic analysis techniques, Thompson et al. [43] derived two different long-wave models. Both models satisfy a mass conservation equation
$$\begin{aligned} h_t + q_x = F(x,t), \end{aligned}$$
(8)
and couple with an equation for the flux q(x, t). In the first model, the Benney equation, they obtain an explicit expression for q(x, t) and the model is a single PDE for the interfacial height h(x, t):
$$\begin{aligned} q(x,t)&= \frac{h^3}{3}\left( 2-2h_x\cot \theta +\frac{h_{xxx}}{{\text {Ca}}}\right) \nonumber \\&\quad +{\text {Re}}\left( \frac{8h^6h_x}{15}-\frac{2h^4 F}{3}\right) . \end{aligned}$$
(9)
The second model is the weighted residuals model, which describes the evolution of the interfacial height h(x, t) and the flux q(x, t):
$$\begin{aligned} \frac{2 {\text {Re}}}{5}h^2 q_t + q&= \frac{h^3}{3}\left( 2-2h_x\cot \theta +\frac{h_{xxx}}{{\text {Ca}}}\right) \nonumber \\&\quad +{\text {Re}} \left( \frac{18q^2h_x}{35} - \frac{34hqq_x}{35} + \frac{hqF}{5}\right) . \end{aligned}$$
(10)
We note that the controls appear as an inhomogeneous term F(x, t) in the mass conservation Eq. (8), and this structure plays a crucial role in the efficiency of these controls.
Due to the asymptotic reduction, the models do not directly provide the evolution of bulk quantities such as the streamwise and transverse velocities u and v, but these are known in terms of the interfacial height h(x, t) and the flux q(x, t) from the analysis. At leading order, and using the results from the weighted residuals model, for example, the fluid velocities are
$$\begin{aligned} u(x,y,t)&= \frac{3q}{h}\left( \frac{y}{h(x,t)}-\frac{y^2}{2h(x,t)^2}\right) , \end{aligned}$$
(11)
$$\begin{aligned} v(x,y,t)&= F(x,t) - \int _0^y u(x,y',t) \ \mathrm{d}y', \end{aligned}$$
(12)
thus providing a description of the flow field in the whole domain that can be compared with direct numerical simulations, for instance. Details on the numerical methodology behind solving these reduced-order models are provided in Appendix B. We underline, however, that we have used a unimodal perturbation of sufficiently small amplitude, typically of \({\mathcal {O}}(10^{-2})\), as initial interface shapes in all our numerical solutions, including higher up in the hierarchy at the DNS level to ensure consistency in the comparisons.
The above long-wave models are significantly more accessible computationally than the full Navier–Stokes equations, but they are still highly nonlinear and to date have not been tackled analytically to obtain rigorous results. Hence, further simplifications are necessary in order to make analytical progress. One can, for example, perform weakly nonlinear analysis to obtain a Kuramoto–Sivashinsky (KS) equation for small but nonlinear perturbations from a flat interface [21, 43]. The KS equation is a fourth-order nonlinear PDE having the same form as (6) with \({\mathcal {L}}u = u_{xxxx} + u_{xx}\) and \({\mathcal {N}}(u) = uu_x\). The KS equation appears in a plethora of applications and is widely studied since it is one of the simplest model PDEs which exhibit spatiotemporal chaotic behaviour. Over the last few decades, existence and uniqueness of solutions have been explored [41], different types of attractors have been characterised [14], and the route to chaos for solutions of the KS equation has been reported [31], thus exemplifying the range of interesting analytical and computational results that can be achieved even at this lowest member of the model hierarchy.
Some uncontrolled computations are presented next to showcase the underlying nonlinear dynamics. Figure 2 shows the comparison of the results for the evolution of the interfacial height using the KS equation (top-left panel), our most comprehensive long-wave model (the weighted residuals model, top-middle panel) and the virtual experiment solving the Navier–Stokes equations (top-right panel) for a film thickness of \(h^*_0 = 175\ \upmu \)m and an inclination angle of \(\theta = \pi /3\). These \(x-t\) plots are colour coded according to the film thickness with darker colours representing thinner regions. In all cases, after a short transient a nonlinear travelling wave emerges moving from left to right as seen from the straight lines in the \(x-t\) plane followed by wave troughs and crests. The bottom-left panel presents the evolution of the film thickness measured at a fixed station positioned at the centre of the \(L^*=64h^*_0\)-sized periodic domain used for the computations, highlighted by a black dashed line in the figure. The time-periodic signal (after an initial transient) once again verifies the presence of a nonlinear travelling wave of permanent form. Finally, the bottom-right panel presents a comparison between the saturated interfacial profiles of the resulting nonlinear wave, as obtained from both long-wave model and DNS predictions. Notably, from both the transient and the final state comparisons, there is excellent agreement between the long-wave model and the DNS—this is the case as long as the assumptions made in the derivation of the model are used for the DNS. This is in fact a stringent scenario for all the reduced-order models, as inertial effects and nonlinear features lead to specific forms of breakdown (which we will soon describe) in this region of the parameter space. However, a weighted-residuals approach is still reliable at this stage. By contrast, a setting weighed down by restrictive assumptions as for the KS equation leads to quantitatively different solutions unless the resulting dynamical behaviour is simple; this is illustrated by the convergence of the solution to the KS equation to a bimodal travelling wave, instead of the unimodal wave that both the weighted residuals and the DNS converge to, see bottom-right panel of Fig. 2.
Given these results, one could question the appropriateness of some of the reduced-order models in direct comparisons with DNS and experiments. (Of course, the dynamics supported are rich and the equations are of fundamental mathematical importance.) Notably, both the Benney and the KS equations are valid in very limited parameter regimes, making comparisons with experiments difficult. Even the weighted-residuals model solutions included in the comparisons with DNS in Fig. 2 are close to the boundary of their applicability, and numerical solution is already hindered by stiffness.
Simpler mathematical models, however, are key players in mathematical studies and help us to push conceptual boundaries to the point where the developed methodologies can be applied higher up in the model hierarchy. This is the approach taken here, and in particular, we subsequently use such methodologies in our virtual experiments with the aim of utilising them in real-world scenarios, e.g. physical experiments and applications. We should point out that small discrepancies still exist between any model and the full DNS. As illustrated, the long-wave model and DNS are in quantitative agreement (even during transient dynamics towards equilibrium coherent structures); hence, we are confident that comparisons and hybrid use of the two frameworks across a wide range of scenarios are appropriate.
More recently, there has been interest in the study of feedback control for the Kuramoto–Sivashinsky equation. Armaou and Christofides [4] explored the control of the zero solution (flat state) in small domains, while Gomes et al. [18, 19] generalised their results to long domains (where chaotic behaviour is observed) and to stabilising solutions with a chosen non-uniform interfacial shape. The results in [18, 19] show that any possible solution to the KS equation can be stabilised using a finite number of point actuated controls whose strength only depends on the difference between the observed and desired interfacial shapes. The number of control actuators depends only on the domain length, and the control rule can be computed using a standard pole placement algorithm [24]. Furthermore, the controls are robust to uncertainty in the problem parameters, as well as to small changes in the number of controls used. Motivated by the similar linear stability properties between the KS equation and the Benney equation (the simplest long-wave model), Thompson et al. [43] studied the control problem for two long-wave models: the Benney equation and the first-order weighted residual model, which acted as a test for the robustness of the controls across the full hierarchy of models. The authors start by showing that in the unrealistic scenario where one can observe the whole interface and actuate everywhere, the simplest proportional controls of the form
$$\begin{aligned} f(x,t) = -\alpha (h(x,t)-1), \end{aligned}$$
(13)
for some constant \(\alpha >0\) to be determined, efficiently drive the system towards the flat solution \(h(x,t) = 1\) (or indeed any desired solution H(x, t), by replacing 1 by H(x, t)). The critical value of \(\alpha \) can be computed from linear stability analysis of the Benney or KS equations, and it depends only on the Reynolds and capillary numbers. It is also shown that the critical \(\alpha \) for the Benney equation is sufficient to obtain linear stability of the weighted residuals model and indeed the full Navier–Stokes equations, by solving an Orr–Sommerfeld system. The critical value of \(\alpha \) for the Benney equation is calculated by solving a linear algebraic equation for the eigenvalues of the linearised system, while for the weighted residuals model the associated equation is quadratic. For the full model, the resulting Orr–Sommerfeld equation is a fourth-order differential equation for each eigenvalue, which needs to be solved numerically, as was done, e.g. in [43, Section III-C]. Even though the result for the Benney equation underestimates the boundaries of linear stability, leading to stronger controls (larger \(\alpha \)) than necessary, the relative simplicity and reduced computational cost offered by the Benney equation are a clear advantage of using reduced-order models. Nonlinear stability is confirmed by numerical simulations of the initial value problem.
In the more realistic case of a finite number of observations of the interface and a finite number of control actuators, Thompson et al. [43] use proportional feedback controls of the form
$$\begin{aligned} f(x,t) = -\alpha \sum _{j=1}^M \delta (x-x_j) (h(x_j-\phi ,t)-1), \end{aligned}$$
(14)
where \(\delta (\cdot )\) is the Dirac delta function, the control actuators are located at the positions \(x_j, \, j=1,\dots ,M,\) and observations of the interface are made at \(x=x_j-\phi \) for some displacement \(\phi \). Such a control protocol was shown to be efficient in stabilising the flat solution \(h(x,t) = 1\) for M sufficiently large. (In practice, \(M=5\) is usually sufficient for Reynolds and capillary numbers found in relevant flows—see Appendix A.) The authors explore other forms of controls, such as the case when the whole interface is observed (in which case one can use pole placement algorithms, similarly to the ones used for the KS equation), or when the number of observations and controls are different, in which case these can also be combined using dynamic observers [43, 51].
The latter control strategies are the most efficient in stabilising the flat solution for the Benney equation, but since their design is model dependent, their applicability across the hierarchy of models is unlikely to be accurate. For this reason, in this paper we chose to use DNS to study the applicability of full proportional controls (13) and point-actuated proportional controls (14) developed for the long-wave models. We will see that we cannot simply “translate” the controls designed for the long wave models (even the weighted residuals model) into the virtual experiment framework, since there are physical effects that appear at the DNS level which are not mitigated in the weighted residuals model. However, our study enables us to design a simple adaptation of the model-based control rules to attain desired control in the DNS, thus acting as a valuable guiding tool within the multi-dimensional parameter space and reducing computational time requirements by several orders of magnitude.
We point out that an alternative to proportional feedback controls would be to use an optimal control approach [48], where we would instead minimise a cost functional constrained by the PDEs which constitute our hierarchy of models. However, this approach is much more computationally expensive, as it involves a gradient descent algorithm that requires solving the highly nonlinear PDEs and corresponding adjoint equations multiple times [11], and not necessarily achievable theoretically, as functional analytical tools required to prove existence of an optimal control [48] are not available in these highly nonlinear settings. Furthermore, as we will discuss in Sect. 3, one of the strengths of the approach under consideration in the present context is that controls based on linear stability analysis of the reduced-order models will be sufficient in order to stabilise the full nonlinear system.
DNS solution of the Navier–Stokes equations
We have constructed a state-of-the-art computational framework in which we can conduct highly accurate in silico experiments of a real-world scenario, controlling a falling film down an inclined plane. This framework does not require any restrictive assumptions and is capable of resolving all the relevant scales and nonlinearities, thus enabling direct comparisons with real-world physical experiments, and indeed theoretical predictions that are devoid of experimental errors and challenges in imaging, measurement and data acquisition. As such, it constitutes a powerful environment to evaluate the mathematical model prior to refining the control methodology for specific applications. In addition, we can construct databases containing the entirety of the flow information without experimental restrictions and errors. Importantly, everything above holds in a general setting; however, we selected a classical fluid problem for which a range of well-known hierarchy of reduced models, numerical and experimental results are available. This allows us to focus on the most delicate aspect of our work, namely the efficiency and accuracy of controls constructed on reduced models as they are used in the control of progressively more complex systems, e.g. Navier–Stokes DNS. Furthermore, the fluid flow problem is of intrinsic importance and offers a rich landscape of solutions and interplay of physical effects and pertains to a wide range of industrial applications from coating technologies to cooling systems in microchips and multi-physics solutions for heat and mass transfer.
Details of the open-source computational platform we used, the
Flow Solver [32, 33], are provided in Appendix C. There we discuss the general direct numerical simulation methodology, details about the discretisation scheme, the volume-of-fluid method used to represent the fluid interface, as well as technical aspects related to the large scale solution effort, data gathering and post-processing. A rigorous validation procedure has also been implemented utilising both converged and transient model solutions, and also information regarding stability and regime boundaries in the target parameter space (discussed in detail in the following section).
The implementation is based upon monitoring a periodic domain of sufficient length compared to the film height (typically 64 times the film thickness) while being able to investigate all flow quantities of relevance in an unsimplified setting. This still enables us to import the full control toolkit presented at the end of Sect. 2.2, including either distributed (full-surface) or, more realistically, point-actuated controls based on discrete interfacial height measurements.
Distributed controls are approximated as piece-wise constant strips—numbering between 4 and 64 across the length of the periodic domain—attempting to mirror a realistic set-up with changeable parts and modular elements. We have noticed that beyond 16 elements the results no longer vary within the tested conditions, thus amounting to a sufficiently accurate representation of the full continuous set-up.
As in Thompson et al. [43], for the point-actuated scenario we require a special treatment of the localised control region. The Dirac \(\delta (x)\)-functions introduced in Eq. (14) are converted to smooth finite counterparts via \(s(x) = \exp [ (\cos (2\pi x) - 1)/w^2 ]\), where w denotes the smoothing window. As \(w \rightarrow 0\), we find \(s(x) \rightarrow \delta (x)\), but in practice we choose \(w^2\) to be of \({\mathcal {O}}(10^{-3} - 10^{-2})\) to preserve the nature of the intended effect while allowing for an efficient numerical solution of the resulting system given resolution and multiscale constraints.