Variational framework for flow optimization using seminorm constraints

We present a general variational framework designed to consider constrained optimization and sensitivity analysis of spatially and temporally evolving flows defined as solutions of partial differential equations. We particularly focus on seminorm constraints which naturally arise for instance when the quantity which we wish to optimize can have contributions from several terms in the PDE through different physical mechanisms in a specific physical system. We show that this case implicitly requires that constraints be placed on the magnitude of complementary (with respect to the first constraining seminorm) seminorms of initial perturbations such that the sum of these complementary seminorms defines a total “true” norm of the state vector. A simple (true) norm constraint naturally satisfies this property. Therefore, the use of this framework requires the introduction of new parameters which describe the relative magnitude of the initial perturbation state vector calculated using the various constrained complementary seminorms to the magnitude calculated using the true norm, even for linear problems. We demonstrate that any required optimization has to be carried out by prescribing these new parameters as initial conditions on the admissible perturbations; the influence and significance of each seminorm component, partitioning the initial total norm of the perturbation, can then be considered quantitatively. To demonstrate the utility of this framework, we consider an idealized problem, the (linear) nonmodal stability analysis of a mean flow given by a “Reynolds averaging” of the one-dimensional stochastically forced Burgers equation. We close the mean flow equation by introducing a turbulent viscosity to model the turbulent mixing, which we allow to evolve subject to a new transport equation. Since we are interested in optimizing the relative amplification of the perturbation kinetic energy (i.e., the perturbation's “gain”) this problem naturally requires the use of our new framework, as the kinetic energy is a seminorm of the full state velocity-viscosity vector, with a new adjustable parameter, describing the ratio of an appropriate viscosity seminorm to the sum of this viscosity seminorm and the kinetic energy seminorm. Using this framework, we demonstrate that the dynamics of the full system, allowing the turbulent viscosity to evolve subject to its transport equation, is qualitatively different from the behavior when the turbulent viscosity is “frozen” at a fixed, mean value, since a new mechanism of perturbation energy production appears, through the coupling of the evolving turbulent viscosity perturbation and the mean velocity field.

Figure 1

Schematic representation of the partition of the space $H^1\left(\Omega \right)$ through the choice of $n_c+1$ (here $n_c=5$) seminorm constraints. (a) Schematic representation of the generic case of final value optimization, where the objective seminorm is different from all the (initial) constraint seminorms. (b) Schematic representation of the special case of gain optimization, where the objective seminorm coincides with one of the constraint seminorms (and as a consequence, so do their kernel complementary spaces).

Figure 2

A schematic representation of the “Direct-Adjoint” loop process in order to find the optimal perturbation. We start with a guess ${\mathbf{q}}_0$, apply the initial condition constraint (4), then integrate the direct equation forward in time. This gives the direct state vector at time $T$ which allows us to define the “final” condition for the adjoint state vector using (28). We then integrate the adjoint equation backward in time from this final condition to obtain the “initial” adjoint state vector which allows us to compute the sensitivity with respect to the chosen initial condition on the (forward) state vector ${\mathbf{q}}_0$ using (29). We then use an appropriate optimization method in order to find the “best” initial condition achieving the maximum value of the objective functional defined in (6).

Figure 3

A schematic representation of the algorithm used to calculate the sensitivity of a general functional $\mathcal{I}$ of the optimal initial condition state vector ${\mathbf{q}}_0^{*}$ to constraint parameters $p_c$. We start from the optimal initial condition state vector ${\mathbf{q}}_0^{*}$ we obtained using the optimization framework and integrate the direct equation to obtain the “final” state vector ${\mathbf{q}}^{*}\left(T\right)$. (This step may not actually be required if the final state of the optimal direct state vector ${\mathbf{q}}^{*}$ has been saved in the last iteration of the optimization framework.) We then construct a new final adjoint state vector ${\mathbf{q}}^{*{†}_{\mathcal{I}}}\left(T\right)$, which construction depends on the particular choice of the functional $\mathcal{I}$. Finally a backward integration of the adjoint state vector leads to a new “initial” adjoint state vector ${\mathbf{q}}^{*{†}_{\mathcal{I}}}\left(0\right)$ which is needed in order to determine the required sensitivity.

Figure 4

(a) A schematic representation illustrating the sensitivity of the optimized functional $\mathcal{J}$ to a constraint parameter. (b) A schematic representation illustrating the sensitivity of the optimized functional $\mathcal{J}$ to an external parameter. Black lines are the level lines of the objective functional $\mathcal{J}$ [gray lines of part (b) of the figure correspond to the level lines of the functional for $p_e=p_e+\delta p_e$]. Thick black lines are the constraints (thick dashed line is the constraint for $p_c=p_c+\delta p_c$). Black circles represent the optimal locations in solution space for the state vector. In the case of the sensitivity with respect to an external parameter we can see that the terms $\frac{\delta \mathcal{J}}{\delta \mathbf{q}}$ and $\frac{\partial {\mathbf{q}}^{*}}{\partial p}$ are orthogonal whereas they are not in the case of a constraint parameter.

Figure 5

(a) Velocity $\overline{u}$ and (b) turbulent viscosity $\overline{{\nu }_t}$ solutions of the steady RAB equations (47) with boundary conditions $\overline{u}\left(0\right)=1$, $\overline{u}\left(1\right)=-1$ and $\overline{{\nu }_t}\left(0\right)=r\nu$, $\overline{{\nu }_t}\left(1\right)=r\nu$. The value of the laminar viscosity is set to $\nu =0.05$. Production and destruction coefficients are chosen to have the illustrative values $c_1=0.75$ and $c_2=2$. For $r=0$, no turbulent viscosity is created, and the flow is laminar. When $r\ne 0$, turbulent viscosity is generated in the zones of high gradient, which are, as a consequence, smoothed for $r=0.5$.

Figure 6

Optimal gain against time for the “LAM” case (plotted with a black line) and the “FROZ” case (plotted with a gray line). The amplification in the LAM case is due to the positive energy production term $P_{E_1}$ of Eq. (53), which can be seen as a focusing of the perturbation in the middle part of the domain. The gain then decreases very slowly due to the low value of the viscosity. In the FROZ case with $r=0.5$ [as defined in (45)] the transient growth remains but is reduced dramatically because of the larger total viscosity. For sufficiently large values of $r$, any transient growth can be completely suppressed due to strong viscous damping.

Figure 7

(a) LAM case; (b) FROZ case. Optimal initial conditions (in solid lines) and final state (in dashed lines) at global maximum gain time $T_{E_{\mathrm{opt}}}=1.15$, for $r=0.5$ as defined in (45). The perturbation is concentrated in the middle of the domain under the action of the base mean flow. For the FROZ case, the focusing is weaker (smaller base flow gradient), and the damping is larger.

Figure 8

(a) The variation of $G_N\left(T\right)$ [defined in (73)] with optimization time $T$ of the short time optimal (STO) perturbation (plotted with a black line) and long time optimal (LTO) perturbation (gray line) total norm for the FULL perturbation equations case, when the total norm ${\parallel ·\parallel }_N$ defined in (59) is optimized using an SVD analysis. (b) The variation of the ratio $C_0$ as defined in (64) (plotted with solid lines) and $C\left(T\right)$ as defined in (65) (plotted with dashed lines) with optimization time $T$ of the STO perturbation (black lines) and the LTO perturbation (gray lines) for both the optimal (black) and suboptimal (gray) mode. We notice that there is a competition between two modes: the short time optimal (STO) perturbation and the long time optimal (LTO) perturbation. The STO perturbation is typically associated with large values of $C_0$, meaning that the STO perturbation is initially “turbulent,” although it evolves to have $C\left(T\right)\ll 1$, meaning that ultimately the perturbation is almost exclusively composed of coherent perturbation velocity. On the other hand, the LTO perturbation is dominated at all times by the coherent perturbation velocity [$C\left(T\right)\ll C_0\ll 1$] and thus we refer to it as a “laminar” perturbation.

Figure 9

Variation of energy gain $G_E\left(T\right)$ as defined in (71) with the optimization time interval $T$ for the full linearized perturbation system of equations (49) for perturbations which optimize the total norm gain $G_N\left(T\right)$, as defined in (73). The STO perturbation (plotted with a black line) achieves very large values of $G_E\left(T\right)$, while the LTO perturbation (plotted with a gray line) is associated with substantially smaller values of $G_E\left(T\right)$, comparable to those obtained in the FROZ case. The STO perturbation has a larger energy gain than the LTO perturbation in the plotted time window but eventually has a larger decay rate and is thus less efficient in preserving a large energy gain.

Figure 10

Variation with $x$ of the initial perturbation (plotted with solid lines) and final perturbation (at $T=T_{E_{\mathrm{opt}}}$, plotted with a dashed line) of (a) coherent perturbation velocity $\tilde{u}$ and (b) perturbation turbulent viscosity $\tilde{\nu }$ for the STO perturbation. We notice a decay of the turbulent viscosity perturbation [through diffusion, destruction, and other mechanisms described by Eq. (55)] giving rise to large perturbation velocities in the final perturbation state vector. This perturbation is not present in either the LAM case or the FROZ case, and arises from the richer dynamics in the FULL case where the turbulent viscosity evolves in space and time.

Figure 11

Variation with $x$ of the initial perturbation (plotted with solid lines) and final perturbation (at $T=T_{E_{\mathrm{opt}}}$, plotted with a dashed line) of (a) coherent perturbation velocity $\tilde{u}$ and (b) perturbation turbulent viscosity $\tilde{\nu }$ for the LTO perturbation. The turbulent viscosity perturbation has a minor role in the dynamics. Indeed, the behavior of the coherent perturbation velocity $\tilde{u}$ is very similar to the behavior observed in the FROZ case [see Fig. 7b].

Figure 12

Variation of the energy seminorm gain $G_E\left(T\right)$ as defined in (71) with optimization time interval $T$ for values of $C_0=K_0/E_0$ ranging from ${10}^{-2}$ to ${10}^1$. Gray curves corresponds to $C_0={10}^{-2}$ and $C_0={10}^{-1}$, the black dashed curve corresponds to the balanced case $C_0=1$, and the black curve corresponds to $C_0=10$. We notice that when the contribution of the turbulent viscosity perturbation increases, higher gains are achieved. A whole family of black curves exists with gains evolving linearly with $C_0$ for larger values of $C_0$.

Figure 13

Variation with $C_0=K_0/E_0$ as defined in (64) of (a) optimal gain $G_{E_{\mathrm{opt}}}$ and (b) optimal times $T_{E_{\mathrm{opt}}}$ for the FULL case. We see that the optimal gain increases linearly with $C_0$, meaning that we can linearly extract as much energy as we want from the base flow and turbulent viscosity perturbation. The optimal time is also increasing with $C_0$, but not in such a neat fashion. The line types are the same as in Fig. 12.

Figure 14

Variation of coherent perturbation kinetic energy gain $G_E\left(T\right)$, as defined in (71) for the FROZ case (plotted with a gray line) and the FULL case in the limit $C_0\to 0$ (plotted with a black line). These results have been obtained with the same set of parameter values, and especially with $r=0.5$. Even in this limit, the results for the two cases differ quantitatively, both in optimal time and optimal gain predictions.

Figure 15

Evolution diagram with time of scaled perturbation turbulent viscosity $X_{\mathrm{opt}}\left(t\right)$ and perturbation velocity $Y_{\mathrm{opt}}\left(t\right)$, defined by (81) and (82), respectively, parameterized by time $t$ for an optimal perturbation with optimization interval $T=0.3$, for different values of the initial ratio $C_0$. We can clearly see the two different types of perturbation: laminar perturbations ($C_0\ll 1$) are plotted with gray lines, and turbulent perturbations ($C_0\gg 1$) are plotted with black lines which are the two limit cases. Initial conditions are represented with black circles, and lie on the curve $x+y=1$ (plotted with a short dashed line). The laminar perturbations do not have significant transient growth, while the turbulent perturbations in black have very large transient growth of the energy. An intermediate state (plotted with a dashed line) exists for $C_0=1$, corresponding to the point $(0.5,0.5)$ on this figure.

Figure 16

Evolution diagram with time of scaled perturbation turbulent viscosity $X_{\mathrm{opt}}\left(t\right)$ and perturbation velocity $Y_{\mathrm{opt}}\left(t\right)$, defined by (81) and (82), respectively, parameterized by time $t$ for a full-norm optimal perturbation (for $T=T_{E_{\mathrm{opt}}}$, $T=1.21$ for the STO perturbation and $T=0.12$ for the LTO perturbation) for the optimization of the total norm ${\parallel ·\parallel }_N$ defined in (59) using the SVD analysis as described in Sec. 4a3 a. The gray curve plots the time evolution of the LTO perturbation and the black curve plots the time evolution of the STO perturbation. By comparison with Fig. 15, there is apparently a close relationship between the STO perturbation and the turbulent perturbation from the FULL case, and the LTO perturbation and laminar perturbation from the FULL case.

Figure 17

Space-time plots of the sensitivity of the final optimized energy $E_{\mathrm{opt}}\left(T\right)$ for varying optimization time intervals $T$ with respect to (a) the mean flow velocity ${\nabla }_{\overline{u}}\mathcal{J}$ and (b) the mean flow turbulent viscosity ${\nabla }_{{\overline{\nu }}_t}\mathcal{J}$ for $C_0\gg 1$.

Figure 18

Variation of the sensitivity of the final optimal energy $E_{\mathrm{opt}}\left(T\right)$ with optimization time interval $T$ to (a) the viscosity ${\nabla }_{\nu }\mathcal{J}$, (b) the turbulent viscosity production coefficient ${\nabla }_{c_1}\mathcal{J}$, and (c) the turbulent viscosity destruction coefficient ${\nabla }_{c_2}\mathcal{J}$. Solid lines show the variation with optimization time interval of the total sensitivities, dashed lines show the variation of the mean flow sensitivities, and dashed-dotted lines show the variation of the perturbation sensitivities. Increasing $\nu$ decreases the value of the objective functional $\mathcal{J}$, and mean and perturbation contributions act in the same way, with a larger sensitivity due to the perturbation. A positive variation of $c_1$ also leads to a decrease of the objective functional, although the perturbation and mean flow sensitivity act oppositely, with the perturbation sensitivity being positively correlated with the total sensitivity. The change in $\mathcal{J}$ due to a variation in $c_2$ is opposite (in both mean and perturbation sensitivity) to the one observed with $c_1$, but is an order of magnitude smaller.