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10:45   Instability, Transition and Control of Turbulent Flows 6 10:45 15 mins #74 Transfer functions for flow predictions in wall-bounded turbulence Kenzo Sasaki, Ricardo Vinuesa, André Cavalieri, Philipp Schlatter, Dan Henningson Abstract: The proposal of this work is to obtain time-domain predictions for turbulent flows applying system identification methods in order to develop single- and multiple-input linear and nonlinear transfer functions. Such transfer functions, which are built in the frequency/transverse wavenumber domains, are taken to the physical domain by means of a double inverse Fourier transform, being then treated as convolution kernels. Once available, such kernels allow the prediction of the unsteady behaviour of the velocity fluctuations at a given wall-normal position (output) from measurements at other wall-normal positions (inputs). These models are built directly from temporal and spanwise data, which permits a considerable flexibility in terms of the definition of inputs and outputs. For the current application, this data was taken from a large-eddy simulation of a zero-pressure gradient turbulent boundary layer of Reynolds number based on the momentum thickness up to 8200. Compelling performance is obtained depending on the relative position between inputs and outputs, where a correlation between the predicted and LES fields at the near-wall peak, predicted from wall-shear stress, was found to be of 0.84, depending on the method of choice. The results also indicate that 45% of the energy in the near-wall peak is linearly correlated with the outer layer structures. 11:00 15 mins #146 Identification of the pattern of breakdown based on binary sequence statistics and cellular-automaton simulations Wen Zhang, Hao Guo, Peiqing Liu, Minping Wan, Jianchun Wang, Shiyi Chen Abstract: The laminar-turbulent transition of the boundary layer flow is not only an intriguing problem in fluid dynamics, but also of great importance in aerospace engineering, since it could cause a significant increase in the skin friction and enhance the efficiency of the heat transfer. From a spatio-temporal aspect [1], the pattern of breakdown (i.e. the generation, propagation and mergence of turbulent spots) is important to reveal transition mechanism especially when the measurement of the flow prior to the breakdown is difficult to be carried out. In a recent experimental study about the boundary layer transition induced by a two-dimensional low-profile step (a typical surface imperfection on aircraft wings), as shown in Fig. 1, the intermittency factor agrees well with the conventional theory, but distinct difference can be observed in the results of the burst rate ($B/B_{max}$). Then a systematic investigation of the pattern of breakdown is motivated by this phenomenon. The theoretical analysis from a probabilistic view indicates that a distributed breakdown process is most likely to be responsible for the deviation in the turbulent burst statistics, which is in contrast with the concentrated breakdown hypothesis in the conventional theory. Numerical simulations for various patterns of breakdown are carried out with a cellular-automaton model [2] to validate the theoretical analysis, and provide additional statistics of the breakdown process to make comparison with the experimental data. 11:15 15 mins #193 An input-output approach to evaluating flow response to spatially varying actuator geometries Igal Gluzman, Dennice Gayme Abstract: Recent transfer function based approaches, such as input-output [1] and resolvent analysis [2], have provided valuable insights into the structure of turbulent and transitional wall-bounded flows. In this work, we build upon these approaches by extending the applicable class of inputs. In particular, we take advantage of the linearity of the input-output framework to represent spatially varying actuator arrays as a superposition of point source inputs at different densities to create a pattern that represents the desired actuator configuration. Here, we consider flow in laminar and turbulent boundary layers and apply step, impulse and impulse train inputs in spatial patterns that represent two actuators that are commonly used in experimental studies: a dielectric-barrier-discharge vortex generator (DBD-VG) [3] and a plasma actuator electrode with a saw-tooth geometry [4]. Figure 1 shows the resulting steady state response for the two actuators applying a step input. The results demonstrate that this model reproduces the vortical structures with features reminiscent of those seen in experimental studies. In particular the vorticity plots in panels (c) and (d) suggest the existence of counter-rotating streamwise vortices between the high- and low-speed streaks. The correspondence between the model results and studies suggest that this approach shows promise as a tool for identifying the types of structures produced under various actuation geometries, and may therefore be useful in design of flow control schemes. 11:30 15 mins #378 Adjoint sensitivity of turbulence using unstable invariant solutions Davide Lasagna Abstract: Fundamental understanding of the relation between output functionals of interest and input parameters or other external forcings can often be gained using adjoint sensitivity methods. For turbulent flows, however, with unstable dynamics, the adjoint equation possesses unstable linear modes producing exponentially growing solutions when integrated backwards in time, the symptom of the high sensitivity of chaotic systems to small perturbations. Hence, when the output functional is a long-time turbulent average it becomes difficult to extract physically relevant mechanisms that control the long-term dynamics because sensitivity is tightly coupled to instability \cite{bib:luchini-quadrio}. In this work, we show that specialising adjoint sensitivity methods to temporally recurrent unstable invariant solutions of the Navier-Stokes equations, i.e. (relative) periodic orbits, can circumvent the issue. Following our previous work \cite{bib:lasagna}, we show that on a periodic orbit, the adjoint equation must be complemented by periodic boundary conditions in time. This produces bounded solutions that reveals the true influence of external forcing and parameter perturbations. Using Floquet theory, we show that sensitivity and instability decouple and that sensitivity is now largest along the most neutral Floquet modes, suggesting that structural bifurcations, marginality and loss of hyperbolicity can play a fundamental role. Here, we apply the approach to invariant solutions embedded in Kolmogorov flow, i.e.~two dimensional turbulence with energy injected mono-chromatically at large scales at $Re=40$ \cite{bib:chandler}. We consider energy dissipation rate as the functional of interest driving the adjoint vorticity equation. In figure 1-(a), we visualise a long relative periodic orbit lying in a frequently visited region of state space, projected onto the normalised energy dissipation rate/power input plane. Panel (b) shows a vorticity snapshot at $t=0$, with contours representing the stream function. Panel (d) shows the time evolution of the norm of the adjoint vorticity field obtained using the classical integration (IVP) and as a solution of a periodic boundary value problem (BVP). In both cases, we use the orbit of panel (a) to drive the adjoint operator. Using the classical integration approach, the norm of the adjoint vorticity $\eta(\mathbf{x}, t)$ field grows asymptotically along the most unstable adjoint Floquet mode at an exponential rate approximately equal to $0.12$. By contrast, when formulated as a boundary value problem, the adjoint solution remains bounded. Physically relevant mechanism can be highlighted. This is visualised in panel (c), showing a snapshot of the norm of the gradient of the adjoint vorticity field $\|\nabla\eta(\mathbf{x}, t)\|$ at $t=0$, where the field displays strong localisation features at the intersection of the large scale coherent structures occupying the domain. 11:45 15 mins #291 ACTIVE FLOW CONTROL OF THE LOGARITHMIC LAYER Anna Guseva, Miguel P. Encinar, Javier Jiménez Abstract: Active flow control for years has been a vivid topic of fluid dynamics research. It is of especial importance for wall-bounded turbulent flows, where intense dissipation at the wall can produce undesirable effects. One successful control approach is to apply at the boundary a velocity field opposite to the observed in the buffer layer \cite{bib:choi1994}. Physically it is equivalent to having a "virtual wall" above the real one. As a consequence, small quasi-streamwise vortices are expelled away and wall friction is reduced \cite{bib:jimenez1994}. On the contrary, applications like mixing or prevention of boundary layer separation require enhancement of turbulence and local drag. Unlike in experiments or numerical simulations, in the real world the information about the flow above the wall is hard to obtain. Recently there has been a lot of interest in flow reconstruction based only on the wall measurements \cite{bib:oehler2018,bib:encinar2018}. The main problem of such flow reconstructions is the lack of fidelity far from the wall. The wall is impacted mostly by attached eddies and their size grows with the vertical distance. The farther from the wall, the more information about small-scale flow structure is not accessible \cite{bib:encinar2018}. Nevertheless, practical considerations suggest that only relatively large and slow eddies can be controlled due to the resolution restrictions of measuring sensors. This draws our attention away from the buffer layer (with passing time of the eddies $\sim$ ms) to the logarithmic layer control (time $\sim$ s). A theoretical framework for modifying large flow scales was proposed in \cite{bib:schoppa1998}, but its implementation is challenging. The focus of this work is on creating a control strategy that can be reproduced in experimental facilities. By acting on the flow from the wall, we aim to affect the eddies of relatively large wavelengths $(\lambda/h > 0.1)$ and alter friction created by their presence. We reconstruct the wall-normal velocity in the log-layer with the linear stochastic estimation method of \cite{bib:encinar2018}, using pressure and wall-parallel shear stress fluctuations at the wall (Fig.\ref{fig:1}). Preliminary implementation of opposition control on the large scales results in substantial drag increase, indicating that we are able to significantly affect those scales and have plenty of control authority (Fig.\ref{fig:2}). In the conference we present further development of this strategy. We compare the control efficiency of applying different wavelength bands at the wall and check the impact of imposed boundary conditions on the statistics of velocity fluctuations. Finally, we assess the general applicability of our control to the existing measurement techniques. 12:00 15 mins #556 Mean DNS adjoint solutions of turbulent Navier-Stokes flows Sophie Knechtel, Joern Sesterhenn Abstract: Applying the adjoint optimization method to turbulent DNS solutions of the Navier-Stokes equations poses severe challenges due to the linear nature of this approach. According to linear stability theory, the Navier-Stokes operator has unstable fixed points for turbulent flows and thus the linear adjoint solution, calculated backward in time, diverges (see e.g. [1]). In optimization and control problems the objective functional is often a time averaged quantity and the linear adjoint approach is bound to fail. In order to avoid this sequence of first linearizing and then averaging, which does not work for reasons stated above, the approach proposed here is to calculate the adjoint operator directly from the averaged Navier-Stokes equations. Using a statistically stationary DNS solution, the Navier-Stokes operator can be averaged over time. This results in the RANS equations, for which the higher averaging moments (e.g. the Reynolds stresses) are known. Deriving the adjoint equation from here, however, needs the partial derivation with respect to the time averaged state variables, which means that explicit closure relations are necessary nonetheless. Following results in mean flow analysis, these closure relations are especially important for wall-bounded flows, where viscid instabilities dominate. On the other hand, for free shear flows and wakes, the derivations of the higher averaging moments are neglectable for linear mean flow analysis (see for example [2] or [3]). In this talk, the stability properties of a 2D turbulent channel flow and a 2D turbulent shear flow are analyzed and compared. Following the adjoint approach discribed above, a simple test case is created, in which the gradient of an objective functional with respect to one parameter is calculated. For validation purposes, the same gradient is calculated via a finite difference from two Navier-Stokes solutions. It can be shown that the derivations of the higher averaging moments with respect to the mean state variables can be neglected for the 2D shear layer case, in accordance to the results in mean flow analysis (see figure 1). In order to obtain a valid adjoint solution for the channel flow, several closure ansatzes (e.g. the Boussinesque hypothesis) are compared and their ability to calculate the correct gradient of the objective functional is evaluated. 12:15 15 mins #601 Reinforcement learning versus linear control of Rayleigh–Bénard convection Gerben Beintema, Luca Biferale, Alessandro Corbetta, Pinaki Kumar, Federico Toschi Abstract: Thermally driven turbulent flows are common in many natural as well as in many industrial applications. The presence of a (turbulent) flow can greatly enhance the heat transfer with respect to its conductive value. It is therefore extremely important from both fundamental and applied points of view to understand if and how it is possible to control the heat transfer in thermally driven flows. In this work, we aim at maintaining a Rayleigh–Bénard convection (RBC) cell in its conductive state beyond the critical Rayleigh number where convection sets in for the uncontrolled set-up. Stabilization methods based on locally changing the boundary temperature, depending on the current flow state and using linear control \cite{howle1997active}, have seen limited success for Rayleigh number Ra 2.5*10^4 mostly due to the strong non-linear effects present at high Ra. In this work, we take advantage of recent developments in Reinforcement Learning (RL) to find control algorithms for RBC and we compare with linear control. In general, RL defines protocols of how an agent can interact with its environment to learn a policy (i.e. control algorithm) in order to accomplish a specific goal and is thus not constrained to linear methods. We train RL agents with controllable, parallelized and GPU-based 2D lattice Boltzmann simulations of RBC in order to be able to run many simulations concurrently and, therefore, to be able to quickly gather training data. Preliminary results show that RL-based control can perform on par or better than linear control methods with only minimal guidance.