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### Summary

The main focus of the second funding period of the project A02 are two research topics. On the one hand, the *shifted Proper Orthogonal Decomposition*, which has been developed during the first funding period, will be improved by investigating error estimators, parameter dependencies, and the preservation of the port-Hamiltonian structure. On the other hand, by means of these methods, reduced-order models will be developed for the combustion tubes and for their interaction with the downstream plenum.

### Summary

In this project a model reduction of reactive flows is developed. Model reduction aims to replace complex, high-dimensional models by models of much smaller dimension. Goal of this project is to improve the existing techniques for systems where transport phenomena are dominant. To this end an appropriate error estimator is developed and combined with a model reduction. The small model can then be adaptively improved by adding physically motivated ansatz-functions. By this approach a low order model of a pulsed combustion is derived. This is used for control and design of a pulsed detonation combustor.

The reduced order models shall not only describe the process of the combustion but also show the changes due to specific manipulation. The controllability in the context of mathematical fluid dynamics is determined via adjoint equations. For this the adjoint equations for reactive flows have to be differentiated and implemented.

The reduced models are then used to design the combustor and to control the combustion process.

Figure 1 shows an example of the solution of an adjoint equation, here the adjoint pressure. A desired performance of the system can be achieved if the adjoint equations are used as influence properties in a simulation. In this example the aim was to increase the pressure in the red rectangle. The higher pressure is shown with the lighter colour. The simulation was influenced in the area of the black rectangle. The solution is not trivial because the influence is highly changed due to the interaction with the flame in the middle of the figure.

As a first model reduction task, dynamics of a zero-dimensional, perfectly stirred, isobaric reactor has been considered. For this, the complex chemistry of combustion is modeled by the GRI3.0 scheme. Combustion is very sensitive with respect to the initial temperature and provides a challenging task for model reduction techniques. First, a simple interpolation approach has been used which bases on a training set with different initial temperatures. The performance of the reduced model is investigated by a simulation for an initial temperature which is not contained in the training set. The corresponding time course of the temperature is depicted in Fig. 2. The result reveals that this simple interpolation-based model reduction approach yields a reduced model which is not capable of describing the combustion for the considered initial temperature.

However, after adding physical information , i.e. the dependence of the ignition time on the initial temperature to this model reduction procedure, a satisfying reduced model has been obtained. The corresponding result (simulated temperature over time) is illustrated in Fig. 3 and shows that the behavior of the full-order system is well described by the reduced-order model. Moreover, the latter one has significantly smaller calculation times and storage requirements.

Currently, the applicability of the Discrete Empirical Interpolation Method (DEIM) is investigated. To this end, simulation tools developed by projects A03 and A04 are used to establish reduced order models for the shockless explosion combustion (SEC) and the pulsed detonation combustion (PDC), respectively.

### Publications

Lemke, M., A. Miedlar, J. Reiss, V. Mehrmann and J. Sesterhenn. Model reduction of reactive processes. In R. King, ed., Active Flow and Combustion Control 2014, vol 127 of NNFM, 234-362. Springer International Publishing, 2015.

Altmann, R. und P. Schulze: On the port-Hamiltonian structure of the Navier-Stokes equations for reactive flows. Preprint 25–2015, Institut für Mathematik, TU Berlin, 2015. http://www.math.tu-berlin.de/preprints/.

Beattie, C., V. Mehrmann und H. Xu: Port-Hamiltonian realizations of linear time invariant systems. Preprint 23–2015, Institut für Mathematik, TU Berlin, 2015. www.math.tu-berlin.de/preprints/.

Binder, A., V. Mehrmann, A. Miedlar und P. Schulze: A Matlab toolbox for the regularization of descriptor systems arising from generalized realization procedures. Preprint 24–2015, Institut für Mathematik, TU Berlin, 2015. www.math.tu-berlin.de/preprints/.

Reiss, J., P. Schulze und J. Sesterhenn: The shifted Proper Orthogonal Decomposition: A mode decomposition for multiple transport phenomena. Preprint 22–2015, Institut für Mathematik, TU Berlin, 2015. http://www.math.tu-berlin.de/preprints/.

Schulze, P. und B. Unger: Data-driven interpolation of dynamical systems with delay. Preprint 17–2015, Institut für Mathematik, TUBerlin, 2015. www.math.tu-berlin.de/preprints/.