We consider the frequency domain form of proper orthogonal decomposition (POD), called spectral proper orthogonal decomposition (SPOD). Spectral POD is derived from a space–time POD problem for statistically stationary flows and leads to modes that each oscillate at a single frequency. This form of POD goes back to the original work of Lumley (Stochastic Tools in Turbulence, Academic Press, 1970), but has been overshadowed by a space-only form of POD since the 1990s. We clarify the relationship between these two forms of POD and show that SPOD modes represent structures that evolve coherently in space and time, while space-only POD modes in general do not. We also establish a relationship between SPOD and dynamic mode decomposition (DMD); we show that SPOD modes are in fact optimally averaged DMD modes obtained from an ensemble DMD problem for stationary flows. Accordingly, SPOD modes represent structures that are dynamic in the same sense as DMD modes but also optimally account for the statistical variability of turbulent flows. Finally, we establish a connection between SPOD and resolvent analysis. The key observation is that the resolvent-mode expansion coefficients must be regarded as statistical quantities to ensure convergent approximations of the flow statistics. When the expansion coefficients are uncorrelated, we show that SPOD and resolvent modes are identical. Our theoretical results and the overall utility of SPOD are demonstrated using two example problems: the complex Ginzburg–Landau equation and a turbulent jet.

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Spectral proper orthogonal decomposition and its relationship to dynamic mode decomposition and resolvent analysis

Hydrodynamic and hydromagnetic stability

Corrugated conductive partition effects on MHD free convection of CNT-water nanofluid in a cavity

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Physical Fluid Dynamics

Steady mixed convection in a differentially heated square enclosure with an active rotating circular cylinder

The proper orthogonal decomposition (POD) has become a very useful tool in the analysis and low-dimensional modelling of flows. It provides an objective way of identifying the ‘coherent’ structures in a turbulent flow. The application of POD to the case of a thermally driven two-dimensional flow of air in a horizontal rotating cylinder is presented. The data for the POD analysis are obtained by numerical integrations of the governing equations of mass, momentum and energy. The decomposition based on POD modes or eigenfunctions is shown to converge to within 5% deviation of the computational data for a maximum of 15 modes for the different cases. The presence of degenerate eigenvalues is an indicator of travelling waves in the flow, and this is confirmed by symmetry in both space and time for the corresponding eigenfunctions. Wave speeds are also determined for these travelling waves. Furthermore, low-dimensional models are constructed employing a Galerkin procedure. The low-dimensional models yield accurate qualitative as well as quantitative behaviour of the system. Not more than 20 modes are required in the low-dimensional models to accurately model the system dynamics. The ability of low-dimensional models to accurately predict the system behaviour for the set of parameters different from the one they were constructed from is also examined. Copyright © 2010 Curtin University of Technology and John Wiley & Sons, Ltd.

Proper orthogonal decomposition and its applications

A proper orthogonal decomposition (POD) analysis and low-dimensional modelling of thermally driven two-dimensional flow of air in a horizontal rotating cylinder, subject to the Boussinesq approximation, is considered. The problem is unsteady due to the harmonic nature of the gravitational buoyancy force with respect to the rotating observer and is characterized by four dimensionless numbers: gravitational Rayleigh number (Rag), the rotational Rayleigh number (RaΩ), the Taylor number (Ta) and Prandtl number (Pr). The data for the POD analysis are obtained by numerical integration of the governing equations of mass, momentum and energy. The POD is applied to the computational data for RaΩ varying in the range 102–106 while Rag and Pr are fixed at 105 and 0.71 respectively. The ratio of Ta to RaΩ is fixed at 100 so that the results apply to physically realistic situations. A new criterion, in the form of appropriately defined error norms, for assessing the truncation error of the POD expansion is proposed. It is shown that these error norms reflect the accuracy of the POD-based reconstructions of a given data ensemble better than the widely employed average energy criterion. The translational symmetry in both space and time of the pair of modes having degenerate (equal) eigenvalues confirms the presence of travelling waves in the flow for several different RaΩ values. The shifts in space and time of the structure of the degenerate modes are utilized to estimate the wave speeds in a given direction. The governing equations for the fluctuations are derived and low-dimensional models are constructed by employing a Galerkin procedure. For each of the five values of RaΩ, the low-dimensional models yield accurate qualitative as well as quantitative behaviour of the system. Sufficient modes are included in the low-dimensional models so that the modelling of the unresolved scales of motion is not needed to stabilize their solution. Not more than 20 modes are required in the low-dimensional models to accurately model the system dynamics. The ability of low-dimensional models to accurately predict the system behaviour for a set of parameters different from those from which they were constructed is also examined.

Proper orthogonal decomposition and low-dimensional modelling of thermally driven two-dimensional flow in a horizontal rotating cylinder

On the outflow boundary condition for external incompressible flows: A new approach

In the present work, numerical simulations are performed to investigate the vortex shedding suppression phenomenon for mixed convective flows past a square cylinder in the large-scale heating regime. A full compressible flow model with variable transport and thermo-physical properties is employed to capture large-scale heating effects. The Reynolds number, Prandtl number, and Mach number are kept constant at Re = 100, Pr = 0.71, and M = 0.1, while the cylinder inclination (ϕ), free-stream inclination (α), and over-heat ratio (ϵ) are varied in the range [0, 45°], [0, 90°], and [0, 2], respectively. The governing equations are solved numerically using the particle velocity upwind (PVU-M+) scheme. The buoyancy parameter which governs the vortex shedding suppression process in the non-Boussinesq model is identified as RiNB=ϵFr222+ϵ, where Fr is the Froude number. Using the Stuart-Landau model, the neutral curves separating the steady and unsteady flow regimes are generated in the ϵ–ϕ and ϵ–α parametric spaces. The neutral curves show qualitatively similar characteristics as observed for Boussinesq models. The relative contribution of various large-scale heating effects in suppression of vortex shedding is also highlighted. This reveals that buoyancy effects followed by variations in transport properties play a major role in suppression of vortex shedding. The findings are also applicable to a range of low Re (O(100)) as supported by data obtained at Re = 130 for ϕ = 40°. The mechanism of vortex shedding suppression has been analyzed and extended for the large-scale heating scenarios.

Vortex shedding suppression in mixed convective flow past a square cylinder subjected to large-scale heating using a non-Boussinesq model

The present study involves a numerical investigation of buoyancy induced two-dimensional fluid motion in a horizontal, circular, steadily rotating cylinder whose wall is subjected to a periodic distribution of temperature. The axis of rotation is perpendicular to gravity. The governing equations of mass, momentum and energy, for a frame rotating with the enclosure, subject to Boussinesq approximation, have been solved using the Finite Difference Method on a Cartesian colocated grid utilizing a semi-implicit pressure correction approach. The problem is characterized by four dimensionless parameters: (1) Gravitational Rayleigh number Ra g ; (2) Rotational Rayleigh number Ra Ω ; (3) Taylor number Ta; and (4) Prandtl number Pr. The investigations have been carried out for a fixed Pr=0.71 and a fixed Ra g =10 5 while Ra Ω is varied from 10 2 to 10 7

Nadeem Hasan

Papers

Proper orthogonal decomposition and its applications

Proper orthogonal decomposition and low-dimensional modelling of thermally driven two-dimensional flow in a horizontal rotating cylinder

On the outflow boundary condition for external incompressible flows: A new approach

Vortex shedding suppression in mixed convective flow past a square cylinder subjected to large-scale heating using a non-Boussinesq model

The Dynamics of Two-Dimensional Buoyancy Driven Convection in a Horizontal Rotating Cylinder