Response of a plate in turbulent channel flow: Analysis of fluid–solid coupling

Abstract

The paper performs simulation of a rectangular plate excited by turbulent channel flow at friction Reynolds numbers of 180 and 400. The fluid–structure interaction is assumed to be one-way coupled, i.e, the fluid affects the solid and not vice versa. We solve the incompressible Navier–Stokes equations using finite volume direct numerical simulation in the fluid domain. In the solid domain, we solve the dynamic linear elasticity equations using a time-domain finite element method. The obtained plate averaged displacement spectra collapse in the low frequency region in outer scaling. However, the high frequency spectral levels do not collapse in inner units. This spectral behavior is reasoned using theoretical arguments. The resonant vibration is stronger at the third natural frequency than at the first natural frequency. We explain this behavior by comparing the fluid and solid length scales. We further study the sources of plate excitation using a novel formulation. This formulation expresses the average displacement spectrum of the plate as an integrated contribution from the fluid sources within the channel. Analysis of the sources reveals that at the plate natural frequencies, the contribution of the fluid sources to the plate excitation peaks in the buffer layer. The corresponding wall-normal width is found to be $\approx 0.75\delta$. The integrated contribution of the overlap and outer regions together to the plate response is comparable to that from the buffer region for $R{e}_{\tau }=180$ and exceeds the buffer region contribution for $R{e}_{\tau }=400$. We analyze the decorrelated features of the sources using spectral Proper Orthogonal Decomposition (POD) of the net displacement source. We enforce the orthogonality of the modes in an inner product with a symmetric positive definite kernel. The dominant spectral POD mode contributes to the entire plate excitation. The contribution of the remaining modes from the different wall-normal regions undergo destructive interference resulting in zero net contribution. The envelope of the dominant mode further shows that the intensity of the sources peaks in the buffer region and the wall-normal width of the sources extend well into the outer region of the channel.

Introduction

The coupling between a turbulent flow and the resulting structural excitation is a problem of interest in marine, civil and aerospace engineering. In this paper, we investigate this coupling in a canonical setting — linear one-way coupled (fluid affects solid, but not vice versa) response of an elastic plate in turbulent channel flow (Pope, 2001) due to wall-pressure fluctuations alone. Specifically, we address the question — how much do the fluid sources at different wall-normal locations contribute to the plate excitation for different frequencies, and what are the salient features of these fluid sources? We answer this question with a novel formulation that combines Direct Numerical Simulation (DNS) data, Green’s function formulation and spectral Proper Orthogonal Decomposition (POD). For brevity, we will sometimes refer to wall-pressure fluctuations as just wall-pressure.

The one-way coupling between the fluid sources and plate excitation can be broken into two parts: (i) fluid source–wall-pressure fluctuation coupling, and (ii) wall-pressure fluctuation–plate excitation coupling. Note that we neglect the wall-shear stress contribution to the plate forcing. We further classify the techniques to investigate the fluid source–wall-pressure fluctuation coupling into — scaling variables-based, Green’s function-based and conditional averaging-based techniques. We discuss some features of the wall-pressure fluctuation sources identified by each of these techniques.

Identification of the scaling variables for the power-spectral density (PSD)/wavenumber spectrum of wall-pressure fluctuation yields qualitative information of the wall-normal region of the fluid sources. The wall-pressure PSD in the low ($\omega \delta ∕u_{\tau }<5$), mid ($5<\omega \delta ∕u_{\tau }<100$) and high frequency ranges ($\omega \delta ∕u_{\tau }>0.3R{e}_{\tau }$) scale with the potential flow variables (${\rho }_f,{\delta }^{\ast },U_o$), outer flow variables (${\rho }_f,\delta ,{\tau }_w$), and inner flow variables (${\rho }_f,{\nu }_f,{\tau }_w$), respectively (Farabee and Casarella, 1991), where $\omega$ is the angular frequency, ${\rho }_f$ is the fluid density, $\delta$ is the boundary layer thickness, ${\delta }^{\ast }$ is the displacement thickness of the boundary layer, $U_o$ is the centerline velocity, ${\tau }_w$ is the wall-shear stress, $u_{\tau }=\sqrt{{\tau }_w∕{\rho }_f}$ is the friction velocity, ${\nu }_f$ is the kinematic viscosity of the fluid, and the friction Reynolds number $R{e}_{\tau }$ is defined as $u_{\tau }\delta ∕{\nu }_f$. Thus, the sources responsible for the low, mid and high frequency wall-pressure fluctuations are predominantly in the potential, outer and inner region of the turbulent boundary layer, respectively.

The Green’s function-based techniques (Chang III et al., 1999, Anantharamu and Mahesh, 2020) yield quantitative information of the sources of wall-pressure fluctuation. The premultiplied streamwise wavenumber spectrum and the PSD of the wall-pressure fluctuations in a turbulent channel show peaks at ${\lambda }_x^{+}=300$ (Panton et al., 2017) and ${\omega }^{+}\approx 0.35$ (Hu et al., 2006) for $R{e}_{\tau }=180-5000$, respectively, where ${\lambda }_x$ is the streamwise wavelength, and ＋ indicates normalization with viscous units (${\nu }_f$ and $u_{\tau }$). The dominant contributors to this inner peak are in the buffer region of the channel (Anantharamu and Mahesh, 2020). The approach of Anantharamu and Mahesh (2020) that identified this dominant contribution (i) combines DNS data with the Green’s function formulation to express the wall-pressure PSD (${\varphi }_{pp}\left(\omega \right)$) as integrated contribution ($\Gamma (r,s,\omega )$) from all wall-parallel plane pairs, ${\varphi }_{pp}\left(\omega \right)={\iint }_{-\delta }^{+\delta }\Gamma (r,s,\omega )\mathrm{dr}\mathrm{ds}$, (ii) accounts for the relative phase difference between the contributions from different wall-parallel planes neglected in the previous Green’s function approach of Chang III et al. (1999), and (iii) yields a distribution of sources in the wall-normal direction instead of a wall-normal region as indicated by the scaling variables. Further, the methodology identified decorrelated features of wall-pressure fluctuation sources using spectral Proper Orthogonal Decomposition (POD). The identified dominant wall-pressure source at the linear and premultiplied wall-pressure PSD peak frequency resembled tall and inclined patterns, respectively.

The conditional averaging-based technique (Ghaemi and Scarano, 2013) yields patterns of the flow structure that are correlated to a particular wall-pressure fluctuation event. The time history of the wall-pressure fluctuation signal at a point on the wall shows occasional positive and negative high amplitude wall-pressure peaks. The conditionally averaged flow fields show coupling between a hairpin vortex and the high amplitude peaks (Ghaemi and Scarano, 2013). The flow structure responsible for the positive and negative high amplitude wall-pressure peak at a point are the sweep and ejection event occurring above it, respectively. The ejection event responsible for the negative peak occurs upstream of the hairpin head in between the quasi-streamwise vortices. The sweep event that leads to the positive peak occurs downstream of the hairpin head.

The dynamic linear elasticity equations describe the wall-pressure fluctuation–plate excitation coupling. This one-way coupled FSI approach is valid for small linear deformation ($d{u}_{\tau }∕{\nu }_f<1$) of the plate, where $d$ is the wall-normal displacement. The approach generally uses (i) plate theories (e.g. Poisson–Kirchhoff) to describe the deformation, and modal superposition to obtain the response, (ii) frequency domain since steady state response is usually the quantity of interest, and (iii) a model wavenumber-frequency spectrum (Corcos, 1964, Chase, 1980, Hwang, 1998) for the spatially homogeneous wall-pressure fluctuations as input. Note that the model wavenumber-frequency spectrum usually requires a model PSD (Bull, 1967, Smol’Iakov and Tkachenko, 1991, Goody, 2004). The mode shapes and natural frequencies of the plate required to perform modal superposition can be obtained analytically for simple boundary conditions and geometry. For complicated boundary conditions and geometry, Finite Element Method (FEM) is used to compute the modal decomposition.

The wall-pressure fluctuation–plate excitation coupling has been previously investigated in wavenumber space (Hwang and Maidanik, 1990, Blake, 2017). The modal force PSD of the plate can be expressed as the wavenumber integral (Blake, 2017)

$${\varphi }_{f_j{f}_j}\left(\omega \right)={\iint }_{-\infty }^{+\infty }{\phi }_{pp}(k_1,k_3,\omega ){\left|S_j(k_1,k_3)\right|}^2{\mathrm{dk}}_1{\mathrm{dk}}_3,$$$$S_j(k_1,k_3)={\int }_a^{a+L_x}{\int }_b^{b+L_z}{S}_j(x,z)e^{i(k_{1}x+k_{3}z)}\mathrm{dx}\mathrm{dz},$$

where $a$ and $b$ are the origins of the plate in the streamwise and spanwise directions, $L_x$ and $L_z$ are the lengths of the plate in the streamwise and spanwise directions, ${\varphi }_{f_j{f}_j}\left(\omega \right)$ is the modal force PSD of the $j$th mode shape, ${\phi }_{pp}(k_1,k_3,\omega )$ is the wall-pressure wavenumber-frequency spectrum and ${\left|S_j(k_1,k_3)\right|}^2$ is the modal shape function. From the above equation, we observe that the modal shape function couples the wall-pressure wavenumber-frequency spectrum to the modal force. The relative contribution of different wavenumber regions to the modal force spectra depends on the mode order ($j$), boundary conditions, and the ratio of the streamwise modal wavenumber ($k_{m,j}$) to the convective wavenumber at the natural frequency of the mode (Hwang and Maidanik, 1990). The high streamwise wavenumber ($k_1∕k_{m,j}\gg 1$) portion of ${\left|S_j(k_1,k_3)\right|}^2$ decays as $k_1^{-6}$, $k_1^{-4}$ and $k_1^{-2}$ for clamped, simply supported and free boundary conditions on all edges (Blake, 2017). Thus, plates with free boundary conditions accept more of the high streamwise wavenumber component of the wall-pressure fluctuations. Further, special wall-pressure fluctuation models that separately approximate the high and low wavenumber portion of the wall-pressure fluctuation wavenumber-frequency spectrum can be derived and used to obtain the response of plates (Hambric et al., 2004). Hambric et al. (2004) showed good agreement between FEM response of a plate excited by the modified Corcos model of Hwang (1998) and an equivalent edge forcing model which only models the convective component in the modified Corcos model for a plate with three edges clamped and one edge free. This shows the importance of the convective region of wall-pressure fluctuation spectrum for plates with free boundary conditions. For a plate with all four edges clamped, FEM response from a low wavenumber excitation model showed good agreement with the modified Corcos model, thus highlighting the dominance of low wavenumber contribution for clamped boundary condition.

Experiments by Zhang et al. (2017) have shown coupling between flow structures and the response of a compliant wall in a turbulent channel flow. The large positive and negative deformation of the compliant wall is coupled to the ejection and sweep events, respectively, occurring above it (Zhang et al., 2017). Conditionally averaged flow fields show that these events are related to the high amplitude pressure peaks and hairpin vortices that surround the local deformation of the compliant wall. For large deformation of the compliant wall, the plate deflection affects the near-wall turbulence. The compliant wall deflection into the buffer layer breaks the near-wall streaks and the associated quasi-streamwise vortices, and induces more spanwise coherence (Rosti and Brandt, 2017).

In this paper, we develop a formulation to obtain the wall-normal distribution of intensity and relative phase of the fluid sources responsible for the plate excitation. Previous research works do not yield such quantitative information of the fluid sources. The main idea is to express the plate averaged displacement PSD as a double wall-normal integral of the ‘net displacement source’ cross-spectral density (CSD) ${\Gamma }^a(r,s,\omega )$ across the height of the channel. The analysis framework combines the volumetric DNS data, Green’s function solution of the pressure fluctuation and modal superposition, and builds on the previous work of Anantharamu and Mahesh (2020). We then apply the framework to explain the one-way coupled FSI simulation results of an elastic plate in turbulent channel flow at $R{e}_{\tau }=180$ and $400$. The fluid and solid simulations make use of finite volume DNS and time-domain FEM, respectively. Further, the decorrelated fluid sources that contribute the most to plate response are obtained using spectral POD of the net displacement source CSD.

The organization of the paper is as follows: In Section 2, we describe the computational domain, mesh resolution, and the FSI simulation details. Section 3 discusses the novel one-way coupling analysis framework. In Section 4.1, we discuss the obtained one-way coupled FSI results. We compare the fluid and solid length scales for the first three modes in Section 4.2. Section 4.3 discusses the spectral features of the net displacement source CSD and in Section 4.4, we identify the decorrelated features of the fluid source using spectral POD. Finally, we summarize the results in Section 5.

Note that $x$, $y$ and $z$ denote the streamwise, wall-normal and spanwise coordinates, respectively. Superscripts/subscripts $f$ and $s$ denote fluid and solid quantities, respectively.