Reduction of Hydrodynamic Mixing Models on the Basis of the DMD Algorithm
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Journal of Engineering Physics and Thermophysics, Vol. 93, No. 6, November, 2020
HYDROGASDYNAMICS IN TECHNOLOGICAL PROCESSES REDUCTION OF HYDRODYNAMIC MIXING MODELS ON THE BASIS OF THE DMD ALGORITHM T. Yu. Sukharev† and D. L. Reviznikov
UDC 532.529
With the use of decomposition into dynamic modes, reduced models of hydrodynamic mixing have been constructed and rather exact space–time pictures of impurity distribution for various periodic regimes of mixing have been obtained with substantial reduction of computational expenditures. It is shown that after processing the results of solution of the problem on mixing in a rectangular cavern with mobile bottom and lid by the DMD method, the gain in the information storage amounts to more than 80%. The proposed approach can also be applied for processing experimental data. Keywords: dynamic modes, hydrodynamic mixing, Navier–Stokes equations, DMD algorithm, dispersed impurity, SVD truncation, finite-dimensional reduced models. Introduction. In experimental investigation and numerical simulation of hydrodynamic processes, an important role is played by the elucidation of the main characteristic features of flow. The information obtained in such a case can be used for both the analysis of the characteristic structures and formations in a medium [1–3] and for reducing complex systems with an infinite number of degrees of freedom to finite-dimensional dynamic systems. This creates the basis for constructing reduced mathematical models. There is a widely known procedure for identification of coherent structures, which is based on the method of principal components POD [4]. With the use of this procedure, a reduced model was constructed in work [5] for solving the inverse thermal conductivity problem. The solution of such kind of problems is of importance, for example, in identification of the thermal properties of materials [6]. However, the method of principal components has a number of drawbacks [7], two of which are most substantial: 1) the energy spectrum is not a correct measure for ranking coherent structures, 2) the information on the phase being the time characteristic of flow gets lost. Among the methods and approaches allowing one to overcome the indicated difficulties, we must single out the decomposition into dynamic modes (DMD method). This method was suggested in the theoretical work on investigation of the Koopman linear operator associated with a nonlinear dynamic system [8]. In [9], it was suggested to use the indicated method in the hydrodynamics for processing experimental and calculated data. In work [10], the DMD method was used in application to the analysis of schlieren pictures of a helium jet that were obtained in the course of measurements by the particle image velocimetry (PIV) method, which made it possible to identify and quantitatively describe the basic physical mechanisms in a flow. In recent years, the DMD method is being developed actively in both the context of theoretical analysis and construction of various modifications of the algorithm
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