Reduced-order description of fluid flow with moving boundaries by proper orthogonal decomposition

Abstract

The approach of proper orthogonal decomposition (POD) has been extensively adopted for fluid dynamics in fixed geometries. This technique is examined here for fluid flow with moving boundaries; in the context of cavitating and phase change flows, and fluid–membrane interaction. The purpose is to assess the capability of POD in extracting the salient features and offering a compact representation to the CFD solutions associated with boundary movement. The cavitating flow simulations are investigated to distill the effect of turbulence modeling, between the Launder–Spalding and a filter-based turbulence models. The lower-order eigenmodes of the flow field, for both turbulence models, show different flow structures and global parameters between higher and lower cavitation numbers. The effect of multi-timescales produced by the filter-based turbulence model is discerned by POD analysis. For 3-D, membrane wing flows, very few POD modes seem sufficient for accurate representation of the velocity field. However, reduced-order analysis of the aerodynamic performance, which is strongly dictated by pressure, may be coarsened by moving membrane dynamics. The flow with fusion is further considered for its solid–liquid phase front propagation. While few modes can sufficiently construct the flow field for the later interval of the flow, a larger number of POD modes are required to provide the flow scales for the initial part of the phase change process.

Introduction

The art of scientific computations has lately received impetus due to the improvement in computer hardware resources. With the advent of faster processors, efficient memory modules, and parallel computing techniques, simulation of complex phenomena in nature is becoming increasingly feasible. However, the challenges in performing a computation and analyzing the available data, increase manifold with the level of complexity in the governing equations. Despite the fact, efforts in encountering these difficulties are rewarding. Numerical simulations yield comprehensive data, which may be impractical as an experimental task. These data not only provide deep insight into the physics of the problem, but also concurrently assist in the appraisal and enhancement of various models that are deployed for the computations.

Effective utilization of the colossal information yielded by simulations certainly merits development of efficient post-processing strategies. Any tool that offers a succinct expression to the available data is obviously a great asset to the computational endeavor. In the present study, we focus on the capabilities of the statistical technique of proper orthogonal decomposition (POD), in context of probing flows with moving boundaries. POD essentially captures intrinsic and salient flow structures of a time-dependent field by a statistical approach. It yields a set of orthogonal basis functions, termed as ‘POD modes’ or ‘eigenmodes’, which capture the principle flow features. The extraction of eigenmodes is further complimented by a rapid convergence of the energy associated with each mode. Consequently, a reduced-order investigation of the data is facilitated through a fewer number of POD modes.

POD was first introduced by Lumley (1967) to investigate coherency of turbulent flow structures. Since this seminal contribution, POD has been a popular tool to extract systematically hidden, but deterministic, structures in turbulent flows and can be extensively found in literature. Gradual progress in this area has established the application of this technique to laminar flows, as well as to flows under incompressible and compressible conditions. The latest studies and relevant issues of POD implementation are reviewed hereafter. Aubry et al. (1988) built a five-mode, reduced-order model for the wall region of a fully turbulent channel. Their effort was extended by Podvin (2001), who provided numerical validation of a ten-dimensional model for the wall layer. The ability of the ten-dimensional model to produce intermittent features, which are reminiscent of bursting process in a wall layer, was also demonstrated. Comprehensive POD studies on turbulent mixing layers are available (Delville et al., 1999, Ukeiley et al., 2001), which not only identify the large-scaled structures in the layer, but also model the dynamic behavior of the lower-order modes. Prabhu et al. (2001) explored the effect of various flow control mechanisms in a turbulent channel, on the flow structure in POD modes. The POD modes of various flow control mechanisms showed significant differences close to the wall but were similar in the channel core. Similarly, while Liberzon et al. (2001) employed POD to study vorticity characterization in a turbulent boundary layer, Kostas et al. (2002) adopted the approach to probe PIV data for the backward-facing step flow. Picard and Delville (2000) investigated the effect of longitudinal pressure distribution on velocity fluctuations in the turbulent shear layer of a subsonic round jet, using POD. Annaswamy et al. (2002) examined ‘edge-tones’ of an aircraft nozzle by analyzing the POD modes of azimuthal pressure distribution of a circular jet. Cizmas and Palacios (2003) gained insight on the turbine rotor–stator interaction with a lower-order POD investigation. They effectively utilized the time history and phase-plane plots of the POD coefficients to unravel the key dynamics in the flow behavior. POD-based investigations on pulsed jet flow field (Bera et al., 2001), temperature field in flow over heated waves (Gunther and Rohr, 2002), and many such flow cases provide further evidence on the wide applicability of the technique. As mentioned earlier, POD has been successfully extended to laminar flow cases for extracting the principle features from time-dependent flow data (Ahlman et al., 2002).

Despite the ongoing progress in POD, several issues of its applicability to variable density and compressible flow are lately being examined. As indicated before, POD essentially yields a series, which rapidly converges towards the norm of a variable q(r, t). Several past studies intuitively adopted scalar-valued norms for the convergence criterion. For example, each flow variable namely pressure, density, or any velocity component was separately decomposed into POD modes. However, Lumley and Poje (1997) observed that for variable density flows, such as buoyancy-driven flows, an independent POD analysis may decouple the physical relationship between the flow variables. They suggested that the norm selection should incorporate the density variations into the velocity field, to achieve the convergence of a physically relevant quantity––mass rather than mere velocities––through the POD procedure. However, simultaneous use of two flow variables also poses an important issue of deciding the significance of each variable in convergence process. Lumley and Poje (1997) suggested a vector form to q(r, t) as q(r, t) = [C₁u, C₁v, C₁w, C₂ρ′]. In addition, they provided a mathematical analysis to optimize the values of the weighing factors for expediting the convergence. Colonius et al. (2002) extended the above argument by examining the impact of the norm selection on POD analysis of compressible flow over a cavity. POD modes independently obtained by scalar-valued norms of pressure and velocity were compared to those derived by vector-based norms. The choice of the vector, $q(r\text{,}t)=[u\text{,}v\text{,}w\text{,}\sqrt{\frac{2}{\gamma -1}}c]$, was chosen with ingenuity so as to yield a norm as shown below:

$$\Vert q(r\text{,}t)\Vert ={\left[\underset{\Omega }{\int }\left(u^2+v^2+w^2+\frac{2}{\gamma -1}{c}^2\right)\mathrm{d}{r}^3\right]}^{\frac{1}{2}}$$

The above norm effectively yielded a linear series that converged towards the stagnation enthalpy instead of mere kinetic energy. Colonius et al. (2002) further reported that scalar-valued POD modes were unable to capture key processes such as acoustic radiation, which heavily rely on coupling mechanisms between the variables. In comparison, the vector-valued POD modes were in cohort with the compressible flow dynamics. Ukeiley et al. (2002) employed $q(r\text{,}t)=[\frac{\rho }{{\rho }_{\infty }}\text{,}\frac{u}{U_{\infty }}\text{,}\frac{v}{U_{\infty }}\text{,}\frac{w}{U_{\infty }}\text{,}\frac{T-T_{\infty }}{T_0-T_{\infty }}]$ to perform POD on numerical data of compressible mixing layer. The variables, as shown above, were normalized by their freestream values to ensure a rational weighing of their fluctuations. A slow convergence towards the multi-variable norm was reported. POD implementation was also shown to face serious issues in case of supersonic flows due to presence of shock fronts. Lucia et al. (2002) noticed that though the bulk flow can be modeled by few eigenmodes, a larger number of eigenmodes are required to accurately capture the discontinuity in the flow. They circumvented the issue by employing domain decomposition in their POD implementation. Though reasonable success in the reduced-order representation of the shock was reported, issues of extending the decomposition technique to moving shock fronts are still unresolved.

In this study, we extend the POD methodology to investigate three distinct cases of time-dependent simulations. While two of the chosen cases are two-phase flow problems, the third case comprises fluid–structural interaction. To the best of our knowledge, the efficacy of POD representation in context of either two-phase flows or aeroelasticity problems has been insufficiently probed. The existence of a moving boundary, and in turn its numerical modeling strategy register a significant impact on the dynamic behavior of the solution and the flow scales. As a result, there is a need to assess the fidelity of POD representation for such computations to instill confidence into the approach. Furthermore, some of the computations investigated in this study are fairly exploratory, with the use of newly developed models and insufficient experimental support. The eigenmodes and the unsteady behavior of their respective scalar coefficients for these computations are employed to distill the impact of the numerical models on the solution. Thus, a twofold objective of appraisal and examination of reduced-order flow description for three distinct and interesting simulations motivates the present endeavor.