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Modal and Non-Modal Stability of the Heated Flat-Plate Boundary Layer with Temperature-Dependent Viscosity

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This paper presents a modal and non-modal stability analysis of the boundary layer developed on a hot plate. A liquid-type temperature-dependent viscosity model has been considered to account for the viscosity variation in the boundary layer region. The base flow is uniform and parallel to the surface at the leading edge. The base flow solution is obtained using an open-source finite volume source code. The Reynolds number (Re) is defined based on the displacement thickness (δ*) at the inlet of the computation domain. The spectral collocation method is used for spatial discretization of governing stability equations. The formulated generalized eigenvalue problem (EVP) is solved using Arnoldi’s iterative algorithm with the shift and invert strategy. The global temporal eigenmodes are calculated for the sensitivity parameter β from 1 to 7, Re = 135, 270, and 405, and the span wise wave-number N from 0 to 1. The modal and non-modal stability analysis have been performed to study the least stable eigenmodes and the optimal initial conditions and perturbations (using mode superposition), respectively. The global temporal eigenmodes are found more stable for β > 0 at a given value of N. Thus, heating the boundary layer within the considered range of β (0 < β ≤ 7) leads to the stabilization of flow. The optimal energy growth increases with the β due to reducing the perturbation energy loss. Tilted elongated structures of the optimal perturbations are found near the outflow boundary. However, the length scale of the elongated cellular mode structure reduces with increase in β. The same qualitative structure of the optimal perturbations has been found at a given value of N.

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ACKNOWLEDGMENTS

The authors would like to thank the supercomputing facility of Marwadi University, established by the Gujarat Council of Science and Technology (GUJCOST), for availing it for base flow computations and the Science and Engineering Research Board (SERB) for availing computing facility established under the research grant CRG/2020/005698 for stability computations.

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Authors and Affiliations

  1. Department of Mechanical Engineering, Marwadi University, Rajkot, India

    M. Thummar & R. Bhoraniya

  2. Department of Mechanical Engineering, Indian Institute of Technology, Gandhinagar, India

    V. Narayanan

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  1. M. Thummar
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Thummar, M., Bhoraniya, R. & Narayanan, V. Modal and Non-Modal Stability of the Heated Flat-Plate Boundary Layer with Temperature-Dependent Viscosity. Fluid Dyn 58, 450–475 (2023). https://doi.org/10.1134/S0015462822601632

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