Instability, coherent structures and turbulent mixing in cross sheared flow in stratified environments

Abstract

In nature and engineer, substantial transitional scenarios to turbulent flow commence from a perturbed shear flow without curved streamline, which having been theoretically assumed as a parallel shear flow. In Cartesian coordinates, the parallel shear flow satisfy V ≪U, where U and V represent the basic laminar flow velocity component in x and y directions respectively. In addition, the principal direction of the imposed perturbations is also assumed to be aligned with the basic parallel shear flow. However, for flows with a large horizontal extent such as geographical flows, V is possibly comparable to U and therefore the basic laminar flow misaligned with the principal wavenumber direction of the external perturbations. The flow in which V ∼U is called cross shear flow, and the ratio ξ =V/U is called cross shear ratio. This thesis introduces a cross basic velocity component V into the linearized perturbation equations, namely, the Orr-Somerfield equation for viscous shear flow and the Taylor- Goldstein equations for inviscid shear stratified flow, which takes into account of the influences of V as well as the wavenumber in the y direction. By solving the perturbation equation for cross shear (stratified) flow with the QZ algorithm, the stability features of several cross shear flows, such as cross Poiseuille-Couttee flow, cross linear-hyperbolic flow, and cross shear stratified flow are obtained and compared to the classic linear stability analysis results for different parallel shear (stratified) flows, such as plane Poiseuille flow, plane Couttee flow, and plane mixing layer with same hyperbolic function. It is found based on linear stability analysis that the introduced V will either suppress or prompt the original laminar to turbulence in parallel shear flow, depending on whether V itself is a stable or unstable parallel shear (stratified) flow. A special but general case, in which a cross shear stratified flow with U and V having the same hyperbolic function profiles, is further examined with direction numerical simulation (DNS) under guidance of linear stability analysis. It is observed that in cross shear stratified (CSS) flow the orthogonal instability modes in the V direction coexist with the forced KH instability in the U direction, during the entire primary instability stage. The cross shear ratio ξ =V/U, which is the governing parameter for CSS flow, further distinguish three different initial modes between the forced mode and the orthogonal mode. The coherent structures, energy transfer and mixing are further investigated for these three initial CSS instability modes at different Reynolds number, Richardson number and cross shear ratio. Correlations between mixing/energy properties and ξ as well as Richardson number are obtained. The transition to turbulence of the shear convective boundary layer (SCBL) flow, where parallel shear flow coexists with dissipative thermal convection flow, is also preliminarily investigated. The linear stability results on SCBL flow, which is obtained by solving the modified Taylor-Goldstein equations with an unstratified factor defined in this thesis, show the Rayleigh-Benard mode coexists with Kelvin-Helmholtz mode.