Wave propagation and group velocity

The main aim of this paper is to relate instability modes with modes obtained from proper orthogonal decomposition (POD) in the study of global spatio-temporal nonlinear instabilities for flow past a cylinder. This is a new development in studying nonlinear instabilities rather than spatial and/or temporal linearized analysis. We highlight the importance of multi-modal interactions among instability modes using dynamical system and bifurcation theory approaches. These have been made possible because of accurate numerical simulations. In validating computations with unexplained past experimental results, we noted that (i) the primary instability depends upon background disturbances and (ii) the equilibrium amplitude obtained after the nonlinear saturation of primary growth of disturbances does not exhibit parabolic variation with Reynolds number, as predicted by the classical Stuart-Landau equation. These are due to the receptivity of the flow to background disturbances for post-critical Reynolds numbers (Re) and multi-modal interactions, those produce variation in equilibrium amplitude for the disturbances that can be identified as multiple Hopf bifurcations. Here, we concentrate on Re = 60, which is close to the observed second bifurcation. It is also shown that the classical Stuart-Landau equation is not adequate, as it does not incorporate multi-modal interactions. To circumvent this, we have used the eigenfunction expansion approach due to Eckhaus and the resultant differential equations for the complex amplitudes of disturbance field have been called here the Landau-Stuart-Eckhaus (LSE) equations. This approach has not been attempted before and here it is made possible by POD of time-accurate numerical simulations. Here, various modes have been classified either as a regular mode or as anomalous modes of the first or the second kind. Here, the word anomalous connotes non-compliance with the Stuart-Landau equation, although the modes originate from the solution of the Navier-Stokes equation. One of the consequences of multi-modal interactions in the LSE equations is that the amplitudes of the instability modes are governed by stiff differential equations. This is not present in the traditional Stuart-Landau equation, as it retains only the nonlinear self-interaction. The stiffness problem of the LSE equations has been resolved using the compound matrix method.

Dynamical system approach to instability of flow past a circular cylinder

The response of the Blasius boundary layer to free-stream vortical disturbances of the convected gust type is studied The vorticity signature of the boundary layer is computed through the boundary-region equations, which are the rigorous asymptotic limit of the Navier-Stokes equations for low-frequency disturbances The method of matched asymptotic expansion is employed to obtain the initial and outer boundary conditions For the case of forcing by a two-dimensional gust, the effect of a wall-normal wavelength comparable with the boundary-layer thickness is taken into account The gust viscous dissipation and upward displacement due to the mean boundary layer produce significant changes on the fluctuations within the viscous region The same analysis also proves useful for computing to second-order accuracy the boundary-layer response induced by a three-dimensional gust with spanwise wavelength comparable with the boundary-layer thickness It also follows that the boundary-layer fluctuations of the streamwise velocity match the corresponding free-stream velocity component The velocity profiles are compared with experimental data, and good agreement is attained The generation of Tollmien-Schlichting waves by the nonlinear mixing between the two-dimensional unsteady vorticity fluctuations and the mean flow distortion induced by localized wall roughness and suction is also investigated Gusts with small wall-normal wavelengths generate significantly different amplitudes of the instability waves for a selected range of forcing frequencies This is primarily due to the disparity between the streamwise velocity fluctuations in the free stream and within the boundary layer

The pre-transitional Klebanoff modes and other boundary-layer disturbances induced by small-wavelength free-stream vorticity

Mixed convection flow past a vertical plate: Stability analysis and its direct simulation

The spatial stability properties of a mixed convection boundary layer developed over a heated horizontal flat plate is investigated under the linear and quasi-parallel assumptions by solving the Orr-Sommerfeld equations. The main aim of this work is to find a critical buoyancy parameter above which a qualitative change in the stability characteristics of the mixed convection boundary layer due to the influence of heat transfer is displayed. The equilibrium flow used here is that given by the similarity solution of [1] which implies the wall temperature to vary as the inverse square root of the distance from the leading edge of the plate. The Orr-Sommerfeld equation has been solved by using the compound matrix method (CMM)-- which locates all the modes in a chosen range of complex wave number, for spatial stability analysis. Cases with different buoyancy parameters have been investigated which were earlier reported incorrectly in [2]. Results indicate modifications of the hydrodynamic mode with certain buoyancy parameter for which continuous existence of instability as the flow develops downstream. The erroneous results in the above mentioned paper is associated with the wrong choice of compound variables used in CMM and the associated dispersion relation satisfied at the wall. Key Words: Mixed Convection Flow; Orr-Sommerfeld Equation; Spatial Stability Analysis; Compound Matrix Method (CMM)

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Linear Spatial Stability Analysis of Mixed Convection Boundary Layer over a Heated Plate

The spatial stability properties of a mixed convection boundary layer developing over a heated horizontal plate is studied here under linear and quasi-parallel flow assumption. The main aim of the present work is to find out if there is a critical buoyancy parameter that would indicate the importance of heat transfer in destabilizing mixed convection boundary layers, when the buoyancy effect is given by Boussinesq approximation. The undisturbed flow used here is that given by the similarity solution of [1] that implies the wall temperature to vary as the inverse square root of the distance from the leading edge of the plate. The stability of this flow has been investigated by using the compound matrix method (CMM)—that allows finding all the modes in the chosen range in the complex wave number plane for spatial stability analysis. Presented neutral curves for mixed convection boundary layer show the existence of two types of disturbances present simultaneously, for large buoyancy parameter. One notices very unstable high-frequency mode when the buoyancy parameter exceeds the above-mentioned critical value. This unstable thermal mode is in addition to the hydrodynamic mode of isothermal flow given by corresponding similarity profile. The calculated critical buoyancy parameter is shown to qualitatively match with experimental results.

K. Venkata Subbaiah

Papers

Spatial Stability for Mixed Convection Boundary Layer over a Heated Horizontal Plate