Perturbation enstrophy decay in Poiseuille and Couette flows according to Synge's method

In this work we derive the conditions for no enstrophy growth for bidimensional perturbations in the plane Couette and Poiseuille flows. We follow the method of vorticity proposed by Synge in 1938 (see the Semi-Centennial Puplication of the Amer. Math. Soc., equation 12.13, and the more detailed version in the Proc. of the Fifth Inter. Congress of Applied Mechanics, pages 326-332), which is actually based on the analysis of the spatially averaged enstrophy. We find that the limit curve in the perturbation wavenumber-Reynolds number map differs from the limit for no energy growth (see e.g. Reddy 1993). In particular, the absolute stability region for the enstrophy is wider than that of the kinetic energy, and the maximum Reynolds number giving the monotonic enstrophy decay, at all wavenumbers, is 155 and 80 for the Poiseuille and Couette flows, respectively. It should be noted that in past literature the energy-based analysis was preferred to Synge's enstrophy analysis. This, possibly, for two reasons: the low diffusivity of the 1938 Vth ICAM proceedings and the objectively very complicated analytical treatment required. Nevertheless, the potentiality of this method seems high and therefore it is interesting nowadays to exploit it by means of the symbolic calculus

A Lower Bound for Transient Enstrophy Growth of Two-Dimensional Internal Traveling Waves

This study provides the temporal monotonic decay region of the wavenumber-Reynolds number stability map, for the enstrophy of any two-dimensional perturbations traveling in the incompressible and viscous plane Poiseuille and plane Couette flows. Mathematical difficulty related to this problem was due to the unknown boundary conditions on the perturbation vorticity, which left the problem open since the first historical studies conducted by J. L. Synge 234 (1930s). By extending Synge’s work to the non-modal approach, we provide the smallest Reynolds number, Re Ω , allowing transient growth of perturbations’ integral-enstrophy. As a noticeable result, the enstrophy monotonic decay region inside the parameters space is wider than the kinetic energy one. The shape, evolution and wall vorticity of optimal-enstrophy streamfunctions will also be discussed. Concurrently, this study considers the dispersive nature of wavy perturbations. Building on our previous study, we show how the coexistence of dispersion andnondispersion at fixed value of the flow control parameter can affect the morphology and evolution of wave packets in the plane Poiseuille flow. Short waves experience mild growth but travel nondispersively and generate compact structures. Dispersive wave components show the largest enstrophy growth and are responsible for the morphology of the spot core. Both components are relevant in the dynamics of transitional structures

Linear waves in sheared flows. Lower bound of the vorticity growth and propagation discontinuities in the parameters space

This study provides sufficient conditions for the temporal monotonic decay of enstrophy for two-dimensional perturbations traveling in the incompressible, viscous, plane Poiseuille and Couette flows. Extension of J. L. Synge's procedure (1938) to the initial-value problem allowed us to find the region of the wavenumber-Reynolds number map where the enstrophy of any initial disturbance cannot grow. This region is wider than the kinetic energy's one. We also show that the parameters space is split in two regions with clearly distinct propagation and dispersion properties

Fabrication and characterisation of novel functionally-graded lattice materials using additive manufacturing

Porous and cellular materials are frequently found in nature as well as applications in biomedical, automotive, and aeronautical engineering. Their unusual properties are a direct consequence of their micro-structure. These materials have proven to be difficult to mimic synthetically. Additive manufacturing is a promising approach to fabricate this class of materials, as it provides great control over the lattice micro-architecture. A fabrication technique is developed using fused deposition modelling with affordable 3D printers, to produce extrusion with continuously varying cross-sectional size. The novel approach proposed here achieves this by manipulating process parameters. Significant intended variation in the axial and bending stiffness of extrusion is achieved. An excellent agreement between the target diameter of the extruded filament and the diameter of the fabricated filaments confirms a reasonably linear response of the machine following simple incompressible flow of molten material. The adaptability of a fixed bore nozzle to produce variable diameter extrusion was characterised via three quantities α, αS, αB that respectively represent adaptable range in geometry, axial stiffness and bending stiffness of the extruded filaments. The ovality of the extruded filaments thus produced was quantified and was seen to have a significant impact on their bending stiffness. Following successful fabrication of variable diameter extrusions, rectangular bi-layer lattice strips, with spatially varying bending stiffness, were fabricated. Their bending response is asymmetric about their length-wise centre. This asymmetric response was found to be consistent with a simple one-dimensional theory of post-buckled mode shape arising from a functionally graded beam. The response shows high curvature in parts of the structure with relatively softer struts compared to the stiffer regions, which is consistent with expectations. Finally, bi-layer square lattice films with spatially varying stiffness were fabricated. The bent surface of the planar structure shows strong spatial variations in bending response. This asymmetric response is well captured by the linear buckling mode shapes obtained from finite element analysis. Encouraged by the success in fabrication and analysis, a host of mathematical problems including response of woodpile lattices when properties vary spatially were solved