We combine reduced order modelling and system identification to reconstruct the temporal evolution of large scale vortical structures behind the blades of a Rushton impeller. We performed Direct Numerical Simulations at Reynolds number 600 and employed proper orthogonal decomposition (POD) to extract the dominant modes and their temporal coefficients. We then applied the identification algorithm, N4SID, to construct an estimator that captures the relation between the velocity signals at sensor points (input) and the POD coefficients (output). We show that the first pair of modes can be very well reconstructed using the velocity time signal from even a single sensor point. A larger number of points improves accuracy and robustness, and also leads to better reconstruction for the second pair of POD modes. Application of the estimator derived at Re=600 to the flows at Re=500 and 700, shows that it is robust with respect to changes in operating conditions. 1This article has been accepted for publication and undergone full peer review but has not been through the copyediting, typesetting, pagination and proofreading process which may lead to differences between this version and the Version of Record. Please cite this article as

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A plane wake or jet supports sinuous and varicose instability modes. The nonlinear interaction between them following their linear development was described previously by Leib and Goldstein [“Nonlinear interaction between the sinuous and varicose instability modes in a plane wake,” Phys. Fluids A 1, 513–521 (1989)] using the strongly nonlinear non-equilibrium critical-layer approach in the case of the Bickley jet for which the frequencies of the sinuous and varicose modes have an integer ratio of 2. This paper develops the theory for general profiles where the frequencies of the sinuous and varicose modes are non-commensurable. The disturbance is quasi-periodic in time and space and must be expressed as a function of two phase variables. Using matched asymptotic expansions simultaneously with the multi-scale method, we derived a set of coupled evolution equations governing the development of the amplitudes and critical-layer vorticities of these modes. The evolution system is solved for the base-flow profiles mimicking those in experiments. The sinuous mode suppresses the varicose mode but also causes the latter to saturate in a highly oscillatory manner. The varicose mode inhibits the sinuous mode initially. However, in the later stage, it lends the sinuous mode a significantly higher saturating amplitude. For a wide range of initial modal compositions and Reynolds numbers, the ratio of the varicose mode amplitude to that of the sinuous mode eventually tends to an almost constant value in the range of 0.4–0.6, in line with the experimental measurement. Due to the self and mutual interactions, the vorticities roll up to form vortices, which are non-symmetric in the transverse direction and quasi-periodic in the streamwise direction as well as in time. With such an increased complexity, the vortices resemble those observed in experiments. The nonlinear interactions of the sinuous and varicose modes in the critical layer generate all harmonics in the main layer, as a result of which the perturbation is non-periodic and may even appear “random-like.”

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