We apply direct transient growth analysis in complex geometries to investigate its role in the primary and secondary bifurcation/transition process of the flow past a circular cylinder. The methodology is based on the singular value decomposition of the Navier-Stokes evolution operator linearized about a two-dimensional steady or periodic state which leads to the optimal growth modes. Linearly stable and unstable steady flow at Re=45 and 50 is considered first, where the analysis demonstrates that strong two-dimensional transient growth is observed with energy amplifications of order of 10 3 at U X TID ~ 30. Transient growth at Re=50 promotes the linear instability which ultimately saturates into the well known von-Karman street. Subsequently we consider the transient growth upon the time-periodic base state corresponding to the von-Karman street at Re=200 and 300. Depending upon the spanwise wavenumber the flow at these Reynolds numbers are linearly unstable due to the so-called mode A and B instabilities. Once again energy amplifications of order of 10 3 are observed over a time interval of T/T=2, where T is the time period of the base flow shedding. In all cases the maximum energy of the optimal initial conditions are located within a diameter of the cylinder in contrast to the spatial distribution of the unstable eigenmodes which extend far into the downstream wake. It is therefore reasonable to consider the analysis as presenting an accelerator to the existing modal mechanism. The rapid amplification of the optimal growth modes highlights their importance in the transition process for flow past circular cylinder, particularly when comparing with experimental results where these types of convective instability mechanisms are likely to be activated. The spatial localization, close to the cylinder, of the optimal initial condition may be significant when considering strategies to promote or control shedding.

Three-dimensional linear BiGlobal instability of two-dimensional states over a periodic array of T-106/300 low-pressure turbine (LPT) blades is investigated for Reynolds numbers below 5000. The analyses are based on a high-order spectral//¡p element discretization using a hybrid mesh. Steady basic states are investigated by solution of the partial-derivative eigenvalue problem, while Floquet theory is used to analyse time-periodic flow set-up past the first bifurcation. The leading mode is associated with the wake and long-wavelength perturbations, while a second shortwavelength mode can be associated with the separation bubble at the trailing edge. The leading eigenvalues and Floquet multipliers of the LPT flow have been obtained in a range of spanwise wavenumbers. For the most general configuration all secondary modes were observed to be stable in the Reynolds number regime considered. When a single LPT blade with top to bottom periodicity is considered as a base flow, the imposed periodicity forces the wakes of adjacent blades to be synchronized. This enforced synchronization can produce a linear instability due to long-wavelength disturbances. However, relaxing the periodic restrictions is shown to remove this instability. A pseudo-spectrum analysis shows that the eigenvalues can become unstable due to the non-orthogonal properties of the eigenmodes. Three-dimensional direct numerical simulations confirm all perturbations identified herein. An optimum growth analysis based on singular-value decomposition identifies perturbations with energy growths O(10 5 ).

A direct transient growth analysis for two dimensional, three component perturbations to flow past a periodic array of T-106/300 low pressure turbine fan blades is presented. The methodology is based on a singular value decomposition of the flow evolution operator, linearised about a steady or periodic base flow. This analysis yields the optimal growth modes. Previous work on global mode stability analysis of this flow geometry showed the flow is asymptotically stable, indicating a non-modal explanation of transition may be more appropriate. The present work extends previous investigations into the transient growth around a steady base flow, to higher Reynolds numbers and periodic base flows. It is found that the notable transient growth of the optimal modes suggests a plausible route to transition in comparison to modal growth for this configuration. The spatial extent and localisation of the optimal modes is examined and possible physical triggering mechanisms are discussed. It is found that for longer times and longer spanwise wavelengths, a separation in the shear layer excites the wake mode. For shorter times and spanwise wavelengths, smaller growth associated with excitation of the near wake are observed.

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Public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing this collection of information. Send cornments regarding this burden estimate or any other aspect of this collection of information, including suggestions for redudng this burden to Department of Defense, Washington Headquarters Services, Directorate for Information SUPPLEMENTARY NOTES ABSTRACTThis final report covers the 35-month period from Ferbuary 1, 2003, the inception of the grant, to its end on December 31, 2005. Within the present effort BiGlobal primary and secondary (Floquet) instability analyses of Low Pressure Turbine (LPT) flows were performed for the first time. The T-106/300 LPT blade was selected and used for the analyses reported herein. In addition, two-and three-dimensional direct numerical simulations (DNS) were employed respectively in order to provide the nonparallel steady and unsteady two-dimensional basic flows to be analyzed and in order to cross-validate primary and secondary BiGlobal instability analysis results. Calculations were performed at chord Reynolds numbers of 900 and 5000 using both structured and unstructured grids. Finally, the first-ever transientgrowth analysis of a nonparallel flow field was performed in the context of the LPT flow investigated herein. In the entire range investigated, unstable three-dimensional modes were discovered; as the three-dimensional flow approaches the two-dimensional limit, the time-periodic basic state is recovered. FINAL REPORT Grant F49620-03-1-0295 (Theofilis) -"Global instability and control of low-pressure turbine flows"

Abstract. This paper presents a numerical algorithm for the linearized ow problem involving complex geometries where analytical solution is impossible. The method centres around calculation of an eigenvalue problem involving the linearised ow and its spatial adjoint, and yields the ow perturbations that grow the most in a prescribed time, the magnitude of that growth and the perturbations after the growth has occurred. Previous work has shown that classical stability analysis of ow past a low-pressure turbine blade gives only stable eigenvalues, which cannot explain transition to turbulence in this ow. The inital value problem for this fan blade is presented and demonstrates signicant perturbation growth, indicating that this growth may be the facilitator for transition in this case.

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