Generalized Serre duality

We introduce a notion of generalized Serre duality on a Hom-finite Krull-Schmidt triangulated category $\mathcal{T}$. This duality induces the generalized Serre functor on $\mathcal{T}$, which is a linear triangle equivalence between two thick triangulated subcategories of $\mathcal{T}$. Moreover, the domain of the generalized Serre functor is the smallest additive subcategory of $\mathcal{T}$ containing all the indecomposable objects which appear as the third term of an Auslander-Reiten triangle in $\mathcal{T}$; dually, the range of the generalized Serre functor is the smallest additive subcategory of $\mathcal{T}$ containing all the indecomposable objects which appear as the first term of an Auslander-Reiten triangle in $\mathcal{T}$. We compute explicitly the generalized Serre duality on the bounded derived categories of artin algebras and of certain noncommutative projective schemes in the sense of Artin and Zhang. We obtain a characterization of Gorenstein algebras: an artin algebra $A$ is Gorenstein if and only if the bounded homotopy category of finitely generated projective $A$-modules has Serre duality in the sense of Bondal and Kapranov

A numerical investigation of self-sustained cavity flow oscillations

Two-dimensional (2-D) cavity flow physics and oscillation control were investigated through solutions of the Reynolds-Averaged Navier-Stokes equations coupled with a two-equation k-? turbulence model. Effects of the leading edge modifications including compression ramps, expansion surfaces and mass injection on supersonic cavity flow oscillation were investigated. Different flow mechanisms were observed at Mach 1.5 and 2.5. The study proposed a different explanation of the flow control mechanism when mass injection was used. An optimal mass injection rate was identified. A further improvement on the 2-D model was made by considering the effect of the turbulent viscous sub-layer using the Integration-to-the-wall boundary condition. The results confirmed that the shear layer instability reduced gradually from Mach 1.5 to 3.5. An improvement in SPL prediction was achieved. Dominant modes were also correctly predicted.The capability of the model was extended for the investigation of the 3-D compressible unsteady turbulent flow physics. It was validated against a hypersonic symmetric corner flow. The turbulent effect was modelled by a two-equation k-? turbulence model. A laminar cavity flow oscillation at Mach 1.5 was predicted. The result showed a self-sustained pressure oscillation. The predicted pressure oscillation was dominated by the second mode and its frequency was 5702H z which was close to the measured value of 5900H z. The SPL discrepancies with the measurements were within 2.3dB. A secondary symmetric flow pattern inside the cavity displayed a 3-D effect and showed the effect of the side wall within a spanwise distance of 2D. Further simulation of the turbulent cavity flow at Mach 1.5 showed a weak pressure oscillation, which indicated the RANS is sensitive to the choice of turbulence model. However, the surface flow pattern and surface pressure distribution were consistent with the measurements. The strongest surface pressure oscillation was observed near the rear corner on the centre plane. A secondary symmetric flow pattern also existed. A spanwise wave was found in the cavity. At the trailing plate, a flow separation was formed in the laminar boundary layer but was not observed in the turbulent boundary layer