Elsevier

Journal of Fluids and Structures

Volume 75, November 2017, Pages 174-192
Journal of Fluids and Structures

On lift and drag decomposition coefficients in a model reduction framework using pressure-mode decomposition (PMD) analysis

https://doi.org/10.1016/j.jfluidstructs.2017.09.003Get rights and content

Abstract

Proper orthogonal decomposition (POD) has been extensively used for developing reduced-order models (ROM) in fluid mechanics. In most of the research, velocity POD modes are employed for analysis while there is less focus on using pressure POD modes. In fact, pressure POD modes can be beneficial to gain physical insight to the aerodynamic forces acting on a structure. In this study, we simulate the flow past a circular cylinder and compute the velocity and pressure POD modes from the data ensemble. We then perform the novel process of pressure mode decomposition (PMD). We consider the localized pressure POD modes on the cylinder surface, integrate each mode on the cylinder surface, and decompose them into normal and streamwise components, namely lift (LDC) and drag (DDC) decomposition coefficients, respectively. The LDC and DDC are scalar quantities and are independent of spatial coordinates. These coefficients provide insight to the contribution of each mode in the development of aerodynamic forces. The lift and drag coefficients are expanded in a Galerkin fashion using the decomposition coefficients. The temporal coefficients are computed through a mapping function based on a quadratic stochastic estimator. The main contribution and strength of our research is the PMD analysis and to develop an efficient ROM for aerodynamic forces which shows promising results for circular and elliptic cylinders.

Introduction

Flow past a bluff body is often studied to understand complex mechanisms in the production of unsteady aerodynamic forces. These forces may cause damage or even failure owing to fatigue in the structure. Many researchers have investigated the phenomenon of vortex shedding in bluff bodies through experimental techniques Roshko (1955), Bearman (1969), Gerrard (1966), Tritton (1959b) and numerical methods Tsaia and Chena (1998), Meneghini and Bearman (1995), Bearman (1967), Kravchenko and Moin (2000). Large amount of flow field data is generally recorded that can be analyzed through different post-processing techniques such as conditional and non-conditional techniques (Antonia, 1981). These techniques can be used for extraction of coherent vortical structures in the flow that helps us to understand the flow physics and related instabilities. In a conditional technique, sampling of flow is done only during those intervals that satisfy some predetermined criteria. On the other hand, a non-conditional technique has no constraint of predetermined conditions so it can easily be applied for investigation of coherent structures in flows. Proper orthogonal decomposition (POD) is one example of the non-conditional technique employed by Bakewell and Lumley (1967) in turbulent flows. POD is also known as the Karhunen–Loeve expansion in statistics and principal component analysis in image processing or empirical orthogonal functions in meteorology. It is based on a two-point correlation tensor and an effective statistical technique for data reduction and feature extraction. It provides a tool to formulate an optimal basis or minimum degrees of freedom (or modes) that are required to model a dynamical system.

POD can also serve as a building block to develop reduced-order models (ROMs) for complex flows (Noack et al., 2008), flow control design Graham et al. (1999a), Graham et al. (1999b), Akhtar and Nayfeh (2010), Kunisch and Volkwein (1999), Akhtar et al. (2015), image processing (Sirovich and Kirby, 1987), uncertainty quantification and optimization (Yue, 2012). In general, a POD-ROM can be constructed in two steps: (a) Computation of the POD modes from the ensembled flow field data and (b) Galerkin projection of the governing equations onto a space spanned by a small number of the POD modes. Most of the research in developing the POD-ROM focuses on the velocity field. Flow past a circular cylinder is a canonical problem, which is often employed to validate the accuracy and stability of the ROM Noack and Eckelmann (1994), Ma and Karniadakis (2002), Noack et al. (2003), Akhtar and Nayfeh (2010). These models are limited to low Reynolds numbers, however, recent work on closure modeling has extended the application of the POD-ROM to turbulent flows as well Wang et al. (2011), Noack et al. (2008), Wang (2012), Östh et al. (2014).

The POD-ROM can also be used for design and optimization of engineering systems involving fluid–structure interaction where accurate prediction of aerodynamic forces is critical. Although the aerodynamic forces are mainly dependent on the fluid pressure acting on the structure, limited research is available on the development of POD-ROMs for the pressure field Noack et al. (2005), Akhtar et al. (2009b), Ghommem et al. (2013), Rehman et al. (2013), Akhtar et al. (2008). Noack et al. (2005) showed that the accuracy of POD-ROM can be significantly improved by introducing a pressure term representation for incompressible shear flows. They demonstrated that lack of pressure term causes amplitude errors which cannot be corrected by incorporating more number of modes. Akhtar et al. (2009b) developed the pressure POD-ROM from the Galerkin projection of the pressure Poisson equation onto the pressure POD modes. They also computed the aerodynamic forces using the pressure modeled through POD-ROM.

In structural engineering, mode shapes and types have already been used for different industrial applications, such as damage detection. Chen and Kareem (2008) focused on structural vibrations where they identified critical structural modes and provided a guideline for selection of the most critical structural modes while performing a coupled flutter analysis. The study helps in capturing the underlying physics of coupled flutter by identifying the critical structural modes. Fayyadh and Razak (2011) showed that in case of damage, the mode shapes deviate towards the damaged location unlike the undamaged case. Lee et al. (2012) presented pressure and vorticity force analysis on an impulsively started finite plate. They related CL and CD to the various sources of vorticity on or near the wing plate that provide better perspective for flow control. However, this study did not consider the effect of modes on lift and drag forces. Ghommem et al. (2013) and Rehman et al. (2013) introduced the concept of force decomposition coefficients in pressure POD modes. These numbers are scalar quantities that can be used, in a Galerkin expansion setting, for regeneration of coefficients of lift (CL) and drag (CD) rather than the pressure POD modes.

This paper investigates the pressure mode decomposition (PMD) on the aerodynamic forces acting on circular and elliptic cylinders. We focus on the local pressure POD modes on the cylinder surface which is mainly responsible for the generation of aerodynamic forces. In PMD, each segment of the surface pressure mode is computed providing an insight to its contribution towards lift and drag. We integrate each mode on the cylinder surface and decompose it into normal and streamwise components, namely lift (LDC) and drag (DDC) decomposition coefficients, respectively. The LDC and DDC are scalar quantities and are independent of spatial coordinates. Using these coefficients as in the POD modes, the lift (CL) and drag (CD) coefficients can be expanded in a Galerkin fashion.

A stochastic estimator technique is employed to compute the temporal coefficients in the Galerkin expansion. Adrian (1977) and Cole et al. (1992) proposed and employed the Linear Stochastic Estimation (LSE) technique realizing that the statistical information contained within the two-point correlation tensor developed for POD calculation could be combined with instantaneous information for estimating the flow field. Bonnet et al. (1994) proposed a complementary technique which combines the POD and LSE to obtain the time dependent POD expansion coefficients from instantaneous velocity data on coarse hot wire grids. Taylor and Glauser (2004) modified the technique and approximated the instantaneous velocity field from the surface pressure data and termed it the modified complementary technique (mLSE). In another numerical study, Cohen et al. (2004) investigated feedback flow control of the wake of a “D” shaped cylinder using the POD approach. They obtained the steady-state streamwise velocity u and pressure p data from 100 equally-spaced snapshots over approximately 15 vortex-shedding cycles at Re =300. For their closed-loop control design, the qi(t) could not be measured directly and therefore they designed an estimator to compute the states of the system using the Linear Stochastic Estimator (LSE) approach. The q13 are mapped onto the extracted sensor signals from the pressure signals ps as qm(t)=s=1nCsmps(t), where n is the number of sensors and the Csm represent the coefficients of the linear mapping. This approach of measuring the pressure field is sensitive to the number of sensors and their configurations, and depends on the accuracy of the estimator.

In this paper, we develop the ROM of the pressure field by using the quadratic stochastic estimator (QSE) to relate temporal coefficients of the pressure to the temporal coefficients of the velocity field through a mapping function. On the basis of temporal and decomposition coefficients, we identify the critical modes in lift and drag forces. We also investigate the shape effects and compute the decomposition coefficients for the flow past elliptic cylinders with varying eccentricities. This study will provide a new dimension towards understanding the contribution of an individual pressure POD mode and its distribution on the structure by using the PMD analysis. It is an important step for designing flow control techniques based on the ROM. Depending on the nature of an engineering system, sensors or actuators placement can be used to control individual modes (Akhtar et al., 2015). We emphasize that the main contribution and strength of our research is the PMD analysis and to develop an efficient ROM for aerodynamic forces which shows promising results for circular and elliptic cylinders.

Section snippets

Numerical methodology

We use a parallel CFD code for simulating a two-dimensional incompressible flow past elliptic cylinders Akhtar et al. (2009b), Akhtar (2008). The governing equations include the continuity and the Navier–Stokes equations that can be represented in a strong conservative form as follows: u=0, N[u]tu+(uu)+p1ReΔu=!0,where u = (u,v) represents the velocity field and p denotes the pressure. The Reynolds number is defined as Re = ULy/νs, where U is the freestream and νs shows

Low-dimensional representation of flow quantities

A schematic of the strategy to develop a ROM for the lift and drag forces is shown in Fig. 3. The ROM is constructed in two steps: (a) Computation of the LDC and DDC and (b) Computation of the pressure temporal coefficients. In the first step, we generate snapshots of the velocity and pressure fields from the flow simulations. We then compute the velocity and pressure POD modes using the method of snapshots (Sirovich, 1989). After that, we perform the PMD analysis and compute the LDC and DDC.

Numerical results and discussion

We record 100 snapshots of the flow field past a circular cylinder over a shedding cycle and ensemble them into the velocity and pressure snapshot matrices. We compute the mean velocity and pressure fields from the snapshot data. We then compute the Φi and ψi for first ten POD modes (M=10) using velocity and pressure snapshot matrices, respectively. To see the contours of the first 10 POD modes for velocity, the readers are referred to Akhtar (2008). Fig. 4 plots the normalized eigenvalues (λi

Extension of the model to elliptic cylinders

We analyze the shape effects of elliptic cylinders on the decomposition coefficients by simulating the flow with different eccentricities. We consider two cases of elliptic cylinders with varying thickness ratio, i.e. τ=0.5 and 0.7 and compare the results obtained for the circular cylinder (τ=1). Using steady-state time histories of lift and drag coefficients, we perform the fast Fourier transform and compute the dominant frequencies. Fig. 17, Fig. 18 show the time histories and power spectra

Conclusion

The study proposed a novel PMD technique for the ROM of aerodynamic forces on a structure. The study mainly focused on the decomposition coefficients which are generated by integrating the pressure POD modes along the surface of cylinder. We computed the pressure and velocity POD modes using the snapshot data of pressure and velocity fields obtained through numerical simulation. We implemented the QSE for computation of temporal coefficients. Lift and drag coefficients are modeled in a Galerkin

Acknowledgment

IA would like to thank COMSTECH-TWAS for their support under contract Research Grant Agreement (RGA) No. 11-208 RG/ENG/AS-C-UNESCO FR:3240262645.

References (57)

  • WangZ. et al.

    Proper orthogonal decomposition closure models for turbulent flows: A numerical comparison

    Comput. Methods Appl. Mech. Engrg.

    (2012)
  • ZangY. et al.

    A non-staggered grid, fractional step method for time dependent incompressible Navier-Stokes equations in curvilinear coordinates

    J. Comput. Phys.

    (1994)
  • AdrianR.J.

    On the role of conditional averages in turbulence theory

    Turbul. Liq.

    (1977)
  • AkhtarI.

    Parallel Simulations, Reduced-Order Modeling, and Feedback Control of Vortex Shedding Using Fluidic Actuators, Ph.D. Thesis

    (2008)
  • AkhtarI. et al.

    Using functional gains for effective sensor location in flow control: A reduced-order modelling approach

    J. Fluid Mech.

    (2015)
  • AkhtarI. et al.

    A van der pol–duffing oscillator model of hydrodynamic forces on canonical structures

    J. Comput. Nonlinear Dyn.

    (2009)
  • Akhtar, I., Nayfeh, A., Ribbens, C., A Galerkin model of the pressure field in incompressible flows, in: 46th AIAA...
  • AkhtarI. et al.

    Model based control of laminar wake using fluidic actuation

    J. Comput. Nonlinear Dyn.

    (2010)
  • AkhtarI. et al.

    On the stability and extension of reduced-order Galerkin models in incompressible flows

    Theor. Comput. Fluid Dyn.

    (2009)
  • AkhtarI. et al.

    A new closure strategy for proper orthogonal decomposition reduced-order models

    J. Comput. Nonlinear Dyn.

    (2012)
  • AntoniaR.

    Conditional sampling in turbulence measurement

    Annu. Rev. Fluid Mech.

    (1981)
  • AusseurJ.M. et al.

    Experimental development of a reduced-order model for flow separation control

    AIAA Paper

    (2006)
  • BakewellH.P. et al.

    Viscous sublayer and adjacent wall region in turbulent pipe flow

    Phys. Fluids

    (1967)
  • BearmanP.

    On vortex shedding from a circular cylinder in the critical Reynolds number regime

    J. Fluid Mech.

    (1969)
  • BearmanP.W.

    The effect of base bleed on the flow behind a two-dimensional model with a blunt trailing edge

    Aero Quarterly

    (1967)
  • BonnetJ. et al.

    Stochastic estimation and proper orthogonal decomposition: Complementary techniques for identifying structure

    Exp. Fluids

    (1994)
  • BrazaM. et al.

    Numerical study and physical analysis of the pressure and velocity fields in the near wake of a circular cylinder

    J. Fluid Mech.

    (1986)
  • Cohen, K., Siegel, S., Luchtenburg, M., McLaughlin, T., Seifert, A., Sensor Placement for Closed-Loop Flow Control of a...
  • Cited by (0)

    View full text