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Coherent structures in turbulent flows and their modeling by linear stability analysis

Transversal fluctuation of the precessing vortex core as a global coherent structure in a generic combustor setup, empirically extracted by proper orthogonal decomposition (left) and modeled by linear stability analysis (right). Flow direction is from bottom to top.

Turbulent shear flows, such as jets, wakes or mixing layers, are present in almost any technical flow. These flows feature a broad spectrum of vortices and eddies ranging from the large productive scales to the small dissipative scales. Coherent structures represent the eddies of the productive spectrum and they are typically the strongest in amplitude and the largest in space as well as they are conserved the longest in time. Many studies showed that most of these structures are driven by intrinsic flow instabilities. Key motivation of this research is to reveal these instabilities and to exploit them for effective flow control applications.

Linear stability analysis (LSA) can be used to quantitatively model the coherent fluctuations of such turbulent flow fields. Although originally developed for laminar flows only, linear stability theory may also be applied in turbulent environments. This approach has been successfully validated against empirically extracted coherent structures in several flow cases such as turbulent swirling jets [1]. However, some open questions remain, such as regarding closure of the turbulent-coherent interactions, asking for an appropriate turbulence model. This has been addressed by employing an eddy viscosity model that showed a significant improvement of the predictive capability [2]. Another open question regards the applicability of LSA to complex, technical flow setups. Here, it was demonstrated that the developed model still holds valid in flows including combustion [3] or in highly non-parallel flows of an industrial fuel injector [4]. Overall, LSA-based models have shown to provide valuable information about the physical cause of coherent structures that cannot be revealed by numerical simulations or experiments. 

Adjoint mode of the precessing vortex core in a generic combustor setup for axial (left), transverse (center) and out-of-plane (right) flow component. The adjoint mode characterizes the receptivity of the precessing vortex core with regard to flow actuation. Higher values at a location correspond to stronger effect of the actuator when positioned there.

In addition, LSA in its adjoint formulation may be used to develop very effective and tailored flow control solutions. The adjoint LSA allows to identify the source of the instability that drives the coherent structures and it reveals the regions of highest receptivity to flow modifications by passive or active flow control. Thereby, optimal and most effective actuator positions can be derived without exhaustive preliminary actuator tests [5].


[1] Oberleithner, K., Sieber, M., Nayeri, C., Paschereit, C., Petz, C., Hege, H.-C., Noack, B., and Wygnanski, I., “Three-dimensional coherent structures in a swirling jet undergoing vortex breakdown: stability analysis and empirical mode construction,” Journal of Fluid Mechanics, vol. 679, pp. 383–414, 2011.

[2] L. Rukes, C. O. Paschereit, and K. Oberleithner. “An assessment of turbulence models for linear hydrodynamic stability analysis of strongly swirling jets.” In: European Journal of Mechanics - B/Fluids 59 (Sept. 2016), pp. 205–218.

[3] P. Paredes, S. Terhaar, K. Oberleithner, V. Theofilis, and C. O. Paschereit. “Global and Local Hydrodynamic Stability Analysis as a Tool for Combustor Dynamics Modeling.” In: J. Eng. Gas Turbines Power 138.2 (Sept. 2015), p. 021504.

[4] Kaiser, T. L., Poinsot, T., and Oberleithner, K., “Stability and Sensitivity Analysis of Hydrodynamic Instabilities in Industrial Swirled Injection Systems,” Journal of Engineering for Gas Turbines and Power, vol. 140, no. 5, p. 051 506, 2018

[5] Müller, J. S., Lückoff, F. and Oberleithner, K., “Guiding actuator designs for active flow control of the precessing vortex core by adjoint linear stability analysis,” in ASME Turbo Expo 2018: Turbomachinery Technical Conference and Exposition, American Society of Mechanical Engineers, 2018.



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