PIV-based fast pressure reconstruction and noise prediction of tandem cylinder configuration

Abstract

The present work proposes a fast and optimized experimental approach for pressure reconstruction and far-field noise prediction for flow past tandem cylinders based on time-resolved particle image velocimetry (PIV). The low-order reconstruction of the velocity fields based on proper orthogonal decomposition (POD) is applied, which effectively mitigates the incoherent measurement noise by selecting the low-order modes representing the dominant coherent structures. The preprocessing of velocity fields significantly improves the accuracy of both field and surface pressure fluctuations estimated by solving the Poisson equation. The time-marching enhancement algorithm uses the pressure field from the preceding snapshot as the initial guess in the iterative process, which accelerates convergence and reduces the computational cost for solving the Poisson equation of the PIV database with a large ensemble size. The estimated surface pressure fluctuations are used to predict the far-field noise through Curle’s analogy with the correction based on the spanwise correlation length. Comparisons are performed with reference signals, yielding good agreement on both pressure and noise spectra.

1 Introduction

Pressure fluctuations over the solid surface are key aerodynamic parameters and play a significant role in noise generation and structural vibration (Kim 1989). Conventional pressure fluctuation measurements typically rely on pointwise techniques like wall pressure transducers. Although providing high temporal resolution for dynamic analysis (Showkat Ali et al. 2018), the spatial resolution of measurement is limited by the geometry size of transducers, which fails to provide overall surface pressure distribution. The application of fast pressure sensitivity paint (PSP) enables the capture of the pressure fluctuations over the entire model surface. However, the pressure sensitivity and signal-to-noise ratio reduce significantly when being applied to low-speed flow (Peng and Liu 2019). Moreover, the pressure distribution related to coherent structures within the flow field cannot be acquired.

The recent developments in particle image velocimetry (PIV)-based pressure reconstruction provide access to extract pressure information from the instantaneous velocity database, enabling a non-intrusive and whole-field measurement of velocity and pressure simultaneously (van Oudheusden 2013). Particularly, the time-resolved PIV (TR-PIV) offers high temporal and spatial resolutions for the estimation of the instantaneous pressure fields, being applied to investigate the high-frequency dynamic behavior of turbulent boundary layer (Pröbsting et al. 2013; Ghaemi and Scarano 2013). Additionally, for aeroacoustic problems, the reconstructed pressure fluctuations from PIV can also be applied to predict the far-field noise with an acoustic analogy (Koschatzky et al. 2011; Pröbsting et al. 2015). As a result, pressure fluctuations and noise induced by the unsteady flow behavior can be estimated based on a single PIV measurement.

Generally, for PIV-based pressure reconstruction, the pressure gradient (∇p) is usually derived from the instantaneous velocity field based on the incompressible Navier–Stokes Eq. (1):

$$\nabla p = - \rho \frac{{{\text{D}}{\varvec{u}}}}{{{\text{D}}t}} + \mu \nabla^{2} {\varvec{u}}$$

(1)

where $\frac{D{\varvec{u}}}{Dt}$ is the material acceleration, ρ is the fluid density, u is the instantaneous velocity, and μ is the dynamic viscosity. Both Eulerian and Lagrangian approaches are used to calculate the material acceleration (van Oudheusden 2013). The former divides material acceleration into local and advective acceleration, being obtained from temporal and spatial derivatives of the instantaneous velocity, respectively. The latter needs a trajectory reconstruction of a ‘fluid parcel’ over time to calculate the temporal derivative of the velocity. The Eulerian approach performs better for rotational flow compared with the Lagrangian approach (de Kat and van Oudheusden 2011) and directly links to the PIV measurement grid, simplifying the process of extracting pressure gradients from PIV data (van Oudheusden 2013).

Subsequently, the pressure field is determined either by spatially integrating the pressure gradient directly or by solving the Poisson equation. Taking the divergence of Eq. (1) under the assumptions of inviscid flow yields the Poisson equation:

$$\Delta p = \nabla \cdot \nabla p = - \rho \nabla \cdot ({\varvec{u}} \cdot \nabla ){\varvec{u}}$$

(2)

where Δp is the source term of Poisson equation and can be directly derived from the velocity field. The pressure is then estimated by solving the equation through an iterative approach with known Δp and appropriate boundary conditions. Usually, direct spatial integration suffers from accumulating gradient errors along the integration paths. Solving the Poisson equation can inherently diffuse the errors globally (van Oudheusden 2013) due to the homogeneous spatial integration approach applied (Pirnia et al. 2020). Meanwhile, it has the inherent irrotational characteristic coming from the Laplacian operator Δ (Wang et al. 2016), alleviating the contaminating effects stemming from outliers within the velocity field. Therefore, solving the Poisson equation has been extensively employed in the pressure reconstruction and noise prediction for vortex-dominated flow and trailing edge noise (Ragni et al. 2022; Koschatzky et al. 2010; Pröbsting et al. 2015).

When determining the pressure fields, the boundaries of the pressure evaluation domain are significantly sensitive to velocity errors, as demonstrated by Auteri et al. (2015) through Monte Carlo analysis. For a Poisson solver, the error-sensitive boundary conditions are always mixed (Koschatzky et al. 2010; Pröbsting et al. 2013), for which pressure calculated from the Bernoulli equation is enforced on the relative steady and irrotational Dirichlet boundary and pressure gradient evaluated with Eq. (1) is given for the disturbed Neumann boundary with more intense velocity fluctuations (de Kat and van Oudheusden 2011). Velocity errors (ε_u) arise more frequently on the Neumann boundary due to the effects of out-of-domain motion (Hart 2000) and laser reflections close to the wall (Kähler et al. 2012), which further propagate into pressure gradient in the following form (van Oudheusden 2013):

$$\varepsilon_{{\nabla {\text{p}}}} \approx \rho \varepsilon_{{\varvec{u}}} \sqrt {\left( {\frac{1}{{2{\text{d}}t^{2} }} + \frac{{\left| {\varvec{u}} \right|^{2} }}{{2h^{2} }} + \left| {\nabla u} \right|^{2} } \right)}$$

(3)

where ε_∇p is the error of pressure gradient, h is the spacing between two grid nodes, dt is the time interval of the PIV measurement, and ∇u is the gradient of streamwise velocity. Besides, the source term of the Poisson equation is also affected by the spatially incoherent errors distributed in the velocity field, reducing the reconstructed pressure accuracy (Pan et al. 2016). Implementing advanced techniques such as irrotational correction on pressure gradient (Wang et al. 2016) and physics-informed neural network (PINN) incorporating the Navier–Stokes equations (Calicchia et al. 2023; Fan et al. 2023) can considerably enhance the accuracy of pressure reconstruction. However, the former accompanies an optimization problem, and the latter requires for the prior training process of neural network, resulting in expensive computational costs. Therefore, preprocessing to decrease the velocity errors is necessary for accurate pressure reconstruction.

As an effective data-driven modal decomposition technique, proper orthogonal decomposition (POD) permits the extraction of coherent spatial modes from a turbulent flow (Berkooz et al. 1993). The dominant structures containing high disturbance energy are represented by the low-order modes. The high-order modes are related to the random small-scale structures and incoherent errors, which can be removed during the denoising process based on POD low-order reconstruction (Taira et al. 2017). Zeng et al. (2022) found that POD outperforms the spectrum POD and discrete Fourier transformation (DFT) in principal component extractions, making it more suitable for data denoising for jet flow. Similarly, by using POD-based low-order reconstruction, the predominant coherent structures have been successfully extracted and highlighted for cylinder wake (Wang et al. 2015), turbulent boundary layer (Feng and Ye 2023), and vortex ring (Brindise and Vlachos 2017). When performing pressure reconstruction, the POD-based denoising approach also increases the overall accuracy for the artificial synthetic pulsatile flow (Zamir and Budwig 2002) and the Taylor vortex flow (Stuart 1986), demonstrating a better performance than the Lagrange minimization and low-pass filter proposed by Charonko et al. (2010). For the flow involving a submerged solid wall, the surface laser reflections (Theunissen et al. 2008), limited particle density near the wall, and high velocity gradient (Kähler et al. 2012) lead to an intensification of random errors in the wall vicinity, thereby contaminating the pressure field. Ye et al. (2018) have proven that POD-based low-order reconstruction removes the small-scale random turbulent structures close to the wall in the transitional boundary layer. Further analysis is needed to prove the validity of POD-based denoising for the estimation of surface pressure fluctuations.

Solving the Poisson equation also faces expensive computational costs due to the numerical iterative approach. To achieve result convergence, a large number of iterations is indispensable for updating the pressure field from an initial guess with a specific numerical iterative scheme. Fujisawa et al. (2005) applied the successive over-relaxation (SOR) scheme in Poisson solver to enhance computational efficiency. The iterative progress is accelerated by introducing a relaxation factor β (1 < β < 2) in SOR scheme that successively forms a weighted average between the prior and updated iterative value (Ford 2014). However, for time-resolved PIV measurements with high spatial resolution, where the sampling frequency is elevated up to 10 kHz and the ensemble sample size is more than ten thousand snapshots (Ghaemi et al. 2012; Ghaemi and Scarano 2013; Ragni et al. 2018), the computational cost is still non-trivial for SOR scheme. The efficiency of an iterative approach can be improved if the initial guess is very similar to the final solution (Wu 2005). For the high-sampling rate TR-PIV measurements, where the time interval between two consecutive snapshots is sufficiently small (Ghaemi et al. 2012; Pröbsting et al. 2015; Ragni et al. 2018), most of the turbulent structures still exist within the measurement domain after the limited convection (Taylor 1938). As a result, by deriving the initial guess of pressure from the previous instant following a time-marching estimation, the number of iterations could be reduced. This method has been successfully applied in shaking the box (STB) (Schanz et al. 2016) and sequential motion tracking enhancement multiplicative algebraic reconstruction technique (SMTE-MART) (Lynch and Scarano 2015), significantly improving the computational efficiency of the iterative approach. However, the time-marching estimation has not been tested to expedite the iterative procedure of solving the pressure Poisson equation, which requires further evaluation.

The flow-induced noise for a stationary and impermeable solid model in the flow primarily originates from the pressure fluctuations over the solid surface (Glegg and Devenport 2017). The corresponding noise can be calculated with the unsteady surface pressure fluctuations through Curle’s analogy (Curle 1955). However, with conventional pressure transducers, the limited spatial resolution of the measurement data fails to provide the detailed distribution of pressure required for noise prediction (Morris 2011). On the other hand, the surface pressure fluctuations derived from TR-PIV provide high temporal and spatial resolutions both in the flow field and on the solid surface, enabling the application of Curle’s analogy to predict the far-field noise (Lorenzoni et al. 2012; Zhang et al. 2018).

Flow past the tandem cylinders is ubiquitous in engineering applications, including the landing gear of airplanes and high-speed rail pantographs (Zhou and Mahbub Alam 2016). As the vortices shedding from the upstream cylinder periodically impinge on the downstream one, strong unsteady surface pressure fluctuations arise on the downstream cylinder, leading to prominent vortex-induced vibration (VIV) (Qin et al. 2017) and noise (Liu et al. 2015). Previous experimental and numerical research from NASA Langley Research Center (Lockard et al. 2007) and Exa Corporation (Brès et al. 2012) studied the noise generated by tandem cylinder. They found that the most intense near-field pressure radiation responsible for the far-field noise occurs near the upstream half of the downstream cylinder. Additionally, they observed that the noise radiation from the tandem cylinder configuration exhibits a dipole pattern, being centered slightly downstream of the second cylinder. However, there remains a lack of detailed experimental analysis on the relation between flow field, pressure fluctuations, and far-field noise.

The present study aims to estimate the unsteady pressure field and far-field noise for the tandem cylinder flow based on TR-PIV. The deterministic coherent structures are identified by the POD analysis and selected for low-order reconstruction to mitigate incoherent errors in the velocity field, thereby enhancing the accuracy of the following pressure reconstruction and far-field noise prediction. The instantaneous pressure field is reconstructed by solving the Poisson equation. The far-field noise is predicted based on reconstructed pressure through Curle’s analogy with spanwise correction. Time-marching enhancement algorithm is applied to improve the computational efficiency of pressure field reconstruction and noise prediction. Compared with the reference measurements, good agreements on surface pressure fluctuations and far-field noise are achieved.

2 Methodology

2.1 POD-based denoising

To mitigate errors that arise from PIV experiment and processing, POD low-order reconstruction is implemented in the present study. Sirovich (1987) improved the direct POD method into a snapshot method, which reduces the correlation matrix from the spatial to the temporal dimension and enables feasibility for PIV data. Generally, the POD analysis is employed for a scalar or vector field (Taira et al. 2017). The raw velocity fluctuation field u′(x,t) measured by PIV is represented by the optimal linear combination of orthogonal spatial modes (ϕ_n(x)):

$${\varvec{u}}^{\prime } (x,t) = \sum\limits_{n = 1}^{N} {a_{n} (t)\phi_{n} (x)}$$

(4)

where N is the total number of snapshots, a_n(t) is the temporal coefficient, and ϕ_n(x) is the corresponding orthogonal spatial modes. The optimal linear combination (4) is achieved by solving the eigenvalue problem of an autocovariance matrix C:

$$C = \frac{1}{2M}U_{f}^{T} U_{f}$$

(5)

where T denotes the transpose operator, M refers to the number of spatial positions in each snapshot, and U_f is the matrix that rearranges the streamwise and transverse velocity fluctuations (u′ and v′, respectively) of each snapshot into the columns, with a size of 2 M × N.

The eigenvalue (λ_n) obtained from the eigenvalue problem represents the contribution of the corresponding mode (ϕ_n(x)) to the total disturbance energy. Subsequently, the corresponding eigenvectors are rearranged with the decreasing order of λ_n into a matrix A. The normalized POD modes are calculated as:

$$\phi (x) = \frac{{U_{f} A}}{{\left\| {U_{f} A} \right\|}}$$

(6)

The nth rank mode ϕ_n(x) is extracted from the nth column of the matrix ϕ(x). The time coefficient a_n(t) is determined by projecting the velocity fluctuation matrix U_f onto the POD modes (ϕ(x)) (Wang et al. 2014).

To remove the random errors, the instantaneous velocity fields are reconstructed with low-order POD modes, highlighting the dominant unsteady flow features as:

$${\varvec{u}}_{{\varvec{s}}} (x,t) = \overline{{\varvec{u}}} + \sum\limits_{n = 1}^{{N_{tr} }} {a_{n} (t)\phi_{n} (x)}$$

(7)

where u_s(x, t) is the reconstructed instantaneous velocity field, which is subsequently used as the input to solve the Poisson equation. $\overline{{\varvec{u}}}$ is the time-averaged velocity, and N_tr denotes the number of POD modes used for low-order reconstruction. To preserve the dominant dynamic features and minimize incoherent small-scale errors in the reconstructed field u_s(x, t), the POD modes containing 70%, 80%, and 90% of the total disturbance energy are selected for low-order reconstruction in the following analysis.

2.2 Pressure reconstruction from time-resolved PIV data

For the flow field around and past the tandem cylinders, the contribution of out-of-plane components is relatively trivial to the pressure fluctuations compared with the in-plane ones (Violato et al. 2010). As a result, the flow field can be considered quasi-two-dimensional for pressure reconstruction. Under the assumption of two-dimensional, divergence-free, and inviscid conditions, the Poisson Eq. (2) in the Cartesian coordinate system is simplified to (van Oudheusden 2013):

$$\Delta p = - \rho \left\{ {\left( {\frac{\partial u}{{\partial x}}} \right)^{2} + 2\frac{\partial v}{{\partial x}}\frac{\partial u}{{\partial y}} + \left( {\frac{\partial v}{{\partial y}}} \right)^{2} } \right\}$$

(8)

where u and v are the streamwise and transverse components of u, respectively. Consequently, the source term of the Poisson equation Δp can be evaluated by five-points second-order spatial numerical differentiation of the velocity field.

The field pressure is determined by solving the Poisson equation through an iterative process, requiring appropriate boundary conditions, an initial guess, and a numerical iterative scheme. For the boundary conditions, it is impossible to prescribe the pressure for all the boundaries of the pressure evaluation domain and immersed wall. Therefore, the mixed boundaries are implemented based on turbulence intensity. The top and bottom edges of the pressure evaluation domain, which are parallel to the free-stream and have a low turbulence intensity level, are implemented with Dirichlet boundary condition through an extended version of the Bernoulli equation following de Kat and van Oudheusden (2011):

$$p_{{{\text{static}}}} + \frac{1}{2}\rho \left( {\overline{{\varvec{u}}} \cdot \overline{{\varvec{u}}} + {\varvec{u}}^{\boldsymbol{^{\prime}}} \cdot {\varvec{u}}^{\boldsymbol{^{\prime}}} } \right) = p_{\infty } + \frac{1}{2}\rho u_{\infty }^{2}$$

(9)

where p_static is the pressure prescribed on the Dirichlet boundaries, u′ is the velocity fluctuations on the Dirichlet boundaries of each TR-PIV snapshot, and p_∞ and u_∞ are the free-stream static pressure and velocity, respectively. The remaining boundaries, including the inlet and outlet of the pressure evaluation domain, and the immersed solid surface of the cylinder, are set as Neumann boundaries due to their higher turbulence intensity, where the pressure gradient is prescribed by using Eq. (1). To increase the accuracy of pressure reconstruction, the velocity fields after POD low-order reconstruction are used to calculate Poisson source term, Neumann, and Dirichlet boundary conditions.

After determining the field pressure, the surface pressure of the cylinder is estimated by averaging the data integrated from the field pressure within a 3 × 3 kernel centered at the field pressure grid point, which is closest to the related surface location (x_s), as outlined by Jux et al. (2020):

$$p_{{\text{s}}} = \frac{1}{{N_{{{\text{point}}}} }}\sum\limits_{f = 1}^{f = 9} {(\nabla p_{{{\text{f}} \to {\text{s}}}} {\text{d}}x_{{{\text{f}} \to {\text{s}}}} + p_{{\text{f}}} )}$$

(10)

where p_s is the surface pressure on the discrete points over the cylinder surface, ∇p_f→s is the corresponding pressure gradient from the field pressure grid nodes (x_f) to the discrete surface point (x_s) evaluated through Eq. (1), dx_f→s = x_s − x_f is the distance between the surface point and surrounding nodes, and p_f is the field pressure on the field grid nodes. The details of the surface pressure evaluation are shown in Fig. 1a. The field pressure on the field grid nodes (black square) outside the cylinder surface is used for integration. In the present work, the surface pressure fluctuations at 72 discrete points over the downstream cylinder surface with a separation of 5° are estimated, providing the overall surface pressure distribution. The complete flowchart for both field and surface pressure estimation in the present work is shown in Fig. 1b.

Fig. 1

The approach of pressure reconstruction in the present work, a details of the surface pressure evaluation, and b the complete flowchart for both field and surface pressure reconstruction

2.3 Time-marching enhancement algorithm

In the time-marching enhancement algorithm, the SOR scheme is applied to accelerate the iteration progress. The updated value of pressure after kth iterations on the (i, j) node is in the form of:

$$p_{i,j}^{k} = \frac{\beta }{4}\left( {p_{i + 1,j}^{k - 1} + p_{i - 1,j}^{k} + p_{i,j + 1}^{k - 1} + p_{i,j - 1}^{k} - h^{2} \Delta p_{i,j} } \right) + (1 - \beta )p_{i,j}^{k - 1}$$

(11)

where (i, j) is the node index of the velocity field grid, β (1 < β < 2) is the relaxing factor determining the weighted proportion of updated value, k is the current iteration step, and h is the nodal spacing (vector pitch). For the first iteration step, the pressure (p) of the right term of Eq. (11) with superscript (k − 1) is given by an initial guess, which the iterative update of pressure in all the following steps is based on.

Taylor (1938) proposed the hypothesis that the turbulent eddies can be considered as ‘frozen’ and convect downstream with maintaining the coherence of the structures. Considering the small time interval between adjacent snapshots of TR-PIV measurement, the pressure distributions at two consecutive sampling instants exhibit spatial similarity in flow topology, with only a limited shift in the convective direction. Therefore, the initial guess is set as the solution of the Poisson equation from the preceding snapshot, which reduces the discrepancy between the initial value and the final solution for the SOR scheme, thereby decreasing the iteration number required for convergence. Meanwhile, the iterative progress also becomes stable. Therefore, a larger relaxation coefficient (β closer to 2) in the SOR scheme can be selected, further accelerating convergence. For the first snapshot of the PIV database where no preceding snapshot exists, the free-stream pressure is applied as the initial guess. The iterative process of the time-marching enhancement algorithm applied in Poisson solver is shown in Fig. 2.

Fig. 2

The iterative process of the time-marching enhancement algorithm (BC and IG represent the boundary condition and initial guess, respectively)

2.4 Far-field noise prediction

Curle (1955) extended Lighthill’s theory (Lighthill 1952) to incorporate the influence of solid boundaries upon far-field noise. The far-field acoustic pressure (p_a) of a stationary, rigid solid surface immersed in a fluid field comprises a dipole source related to solid surface loading and a quadrupole source stemming from the free-field turbulence. In low Mach number airflow, the contribution of the quadrupole noise source and viscosity effects are negligible. As a result, the acoustics pressure is calculated following Glegg and Devenport (2017):

$$p_{{\text{a}}} ({\varvec{x}},\tau ) \approx \frac{{x_{i} }}{{4\pi |{\varvec{x}}|^{2} c_{\infty } }}\int_{S} {\left[ {\frac{{\partial \left( {p^{\prime } \delta_{ij} n_{j} } \right)}}{\partial \tau }} \right]_{{\tau^{ * } }} {\text{d}}S({\varvec{y}})}$$

(12)

where x_i is the ith component of the listener’s position vector x, c_∞ is the speed of sound, p′ is the pressure fluctuations on the solid surface, δ_ij is the Kronecker delta, n_j is the normal unit vector of the surface, τ denotes the time, τ^* indicates the pressure fluctuations of sound source occurred at the retarded time τ^* = τ − |x|/c_∞, dS is the surface integral unit, and y denotes to the coordinate system of sound source.

In Curle’s analogy, the two-dimensional flow assumption is applied, contributing to an over-estimation of the far-field noise (Lorenzoni et al. 2012; Zhang et al. 2018). The frequency-dependent spanwise coherence based on velocity fields measured in the streamwise–spanwise (x–z) plane is considered to correct the predicted noise. The spanwise coherence is extracted from the spanwise evolution of streamwise velocity fluctuations as:

$$\gamma (f,z) = \sqrt {\frac{{\left| {\Phi_{uu} \left( {f,z_{0} ,z} \right)} \right|^{2} }}{{\left| {\Phi_{uu} (f,z,z)} \right|\left| {\Phi_{uu} (f,z_{0} ,z_{0} )} \right|}}}$$

(13)

where γ(f, z) is the spanwise coherence at location z. z₀ is the reference spanwise location. Φ_uu(f, z₀, z) denotes the cross-power spectral density of streamwise velocity fluctuations between two spanwise locations, and Φ_uu(f, z, z) and Φ_uu(f, z₀, z₀) are the power spectral density (PSD) of streamwise velocity fluctuations at the spanwise location z and the reference location z₀, respectively. For each frequency, the spanwise coherence is fitted with a Gaussian curve concerning the spanwise location (z), as:

$$\sqrt {\gamma^{2} (f,z)} = e^{ - z/l(f)}$$

(14)

The spanwise correlation length is obtained following Roger et al. (2006), as:

$$Lz(f) = \sqrt {\pi /2} l(f)$$

(15)

Finally, the far-field noise predicted by Curle’s analogy is corrected by subtracting 10lgL/Lz from the corresponding sound pressure level (SPL) at each frequency.

3 Experimental setup

3.1 Flow facility and model geometry

The effects of POD denoising and time-marching enhancement algorithm on pressure reconstruction and far-field noise prediction are tested based on TR-PIV measurement of the flow over tandem cylinders. The experiments were conducted in the anechoic wind tunnel of the Institute of Fluids Engineering at Zhejiang University. The test section of the wind tunnel has a cross-sectional area of 0.4 × 0.5 m² with a turbulence intensity of less than 0.5%. The maximum operation speed is 60 m/s. A pair of circular cylinders arranged in tandem were installed in the symmetry plane of the test section, extending over the entire span. The diameter of the cylinders (D) is 15 mm. The streamwise spacing between two cylinders is 4.2D. The origin of the coordinate system is located at the center of the downstream cylinder. The x, y, and z axes correspond to the streamwise, transverse, and spanwise directions, respectively. The installation of the model is shown in Fig. 3a. The experiments were performed at the free-stream velocities of u_∞ = 8 and 10 m/s. The corresponding Reynolds numbers based on cylinder diameter (Re_D = u_∞D/ν) equal to 0.81 and 1.01 × 10⁴, respectively.

Fig. 3

Conceptual sketch of the model installation and PIV setup, a perspective view, b FOV in x–y plane, and c FOV in x–z plane

3.2 Time-resolved PIV

Time-resolved PIV is used to capture the instantaneous velocity fields. The flow was seeded with water–glycol droplets of approximately 1 μm diameter. Illumination is provided by a high-speed laser (Nd: YLF, 527nm Beamtech Vlite-Hi-527-50 K, 2 × 50 mJ per pulse at 1 kHz). To avoid obstruction of illumination due to the presence of the models, the laser was separated into two equal beams by a 50% splitter. Two overlapped laser sheets of 1.5 mm thickness were provided through a series of cylindrical lenses, enabling the illumination of all the azimuth angle of the downstream cylinder, as shown in Fig. 3a. A Photron FASTCAM Mini AX100 (1024 × 1024 pixel, 20 μm/pixel) CMOS camera equipped with a Micro-Nikkor 105mm prime objective was used to capture the particle images. The aperture of the prime objective was set at f_# = 4. To increase the temporal resolution, the camera sensor was cropped into 896 pixels × 608 pixels, resulting in an acquisition frequency (f_s) of 4 kHz. The pulse separation Δt was set to 40 and 25 μs for u_∞ = 8 and 10 m/s, respectively. The field of view (FOV) of the measurement is 80.9 mm × 54.9 mm (x × y), starting from the streamwise location of x = − 2.4D. The digital image resolution is 11.1 pixels/mm. As the dominant sound source of tandem cylinder flow (Geyer 2022), the downstream cylinder is located at the center of FOV, as shown in Fig. 3b. The data acquisition was taken over 1 s, corresponding to an ensemble data size of 4000 image pairs. LaVision Davis 10 software was used for PIV system synchronization, data acquisition, and processing. All particle images acquired were cross-correlated with multiple iterations to calculate the velocity fields. The final integration window size of 16 × 16 pixels with 75% overlap was applied. The resultant vector pitch is 0.36 mm, corresponding to 42 vectors per cylinder diameter.

To account for the effect of spanwise coherence on the far-field noise, planar PIV measurement was also taken over the x–z plane, as shown in Fig. 3c. The FOV is 57 mm × 70 mm, starting from x = − 3.84D. With a cropped sensor size of 640 pixels × 748 pixels, the digital image resolution is 11.2 pixels/mm. The sampling frequency and time remain the same, capturing a total number of 4000 image pairs. The spanwise velocity fields were also evaluated by cross-correlation method, employing the same final interrogation window size and overlap as the planar PIV in the x–y plane, yielding a vector pitch of 0.36 mm.

3.3 Surface pressure fluctuations

Surface pressure fluctuation signals were simultaneously sampled by a miniature Knowles FG-23329-P07 transducer. The miniature transducer is 2.5 mm in diameter and has a circular sensing area of 0.8 mm. The frequency response is ± 3 dB within in the range of 100 Hz–10 kHz. Prior to the experiments, the transducer was calibrated in situ inside the acoustic chamber against a GRAS 40PH-10 free-field microphone with known magnitude and phase in the frequency domain. The calibration result demonstrates a relatively consistent sensitivity of 25.2 mV/Pa from 100 to 2000 Hz. It was flash-mounted in the downstream cylinder at the symmetry plane, coinciding with the x–y plane of the PIV measurement. The data were acquired using a National Instrument PXle-4499, with a sampling frequency of 51.2 kHz and an acquisition time of 10 s. The direction of the miniature transducer was adjusted by rotating the cylinder to α = 45°, 90°, 120°, and 150° with respect to the negative streamwise direction, as shown in Fig. 3b by the red points.

3.4 Acoustic measurement

After completing the PIV measurements and removing all the PIV-related equipment from the acoustic chamber, the far-field noise measurement was taken by three GARS 40PH-10 CCP microphones (1/4 inch; frequency response: ± 1 dB; frequency range: 10 Hz–20 kHz; max. output: 135 dB ref. 20 μPa), with sensitivities of 59.2, 54.1, and 52.9 mv/Pa, respectively. The data were acquired by a National Instruments PXIe-4499 at a sampling frequency of 40 kHz for a duration of 30 s. The microphones are arranged in an arc shape, located 1 m away from the center of the downstream cylinder. The orientations of the microphones were aligned at 75°, 90°, and 120° with respect to the negative streamwise direction, respectively. The selection of these angles is similar to those used in the benchmark (Lockard 2009), preliminarily revealing the dipole properties of the tandem cylinder noise.

4 Assessment results

4.1 Flow characteristics

The cross-plane contours of non-dimensional turbulence intensity for u_∞ = 8 and 10 m/s are depicted in Fig. 4. The vectors of time-averaged velocity are superimposed, by subtracting 0.4u_∞. Generally, depending on the Re_D and spacing between cylinders, the flow fields around tandem cylinders feature with reattachment and co-shedding states. For the reattachment state, the shear layers from the upstream cylinder attach onto the downstream one, with the formation of a weak recirculation zone within the gap between the two cylinders. On the other hand, these shear layers roll up and develop into alternating shedding vortices within the gap for the co-shedding state (Elhimer et al. 2016). These vortices eventually impinge on the downstream cylinder, resulting in significant tonal and broadband noise (Geyer 2022).

Fig. 4

Contour of turbulence intensity at x–y plane for different flow conditions, a reattachment state for u_∞ = 8 m/s, b co-shedding state for u_∞ = 8 m/s, and c co-shedding state for u_∞ = 10 m/s. The vectors of time-averaged velocity field are superimposed by subtracting 0.4u_∞. The magenta and green solid lines denote the Dirichlet and Neumann boundaries, respectively

At u_∞ = 8 m/s, a bi-stable state (Igarashi 1981) is observed, where the flow field between the upstream and downstream cylinders alternates intermittently between reattachment (Fig. 4a) and co-shedding states (Fig. 4b). The turbulence intensity distributions for reattachment and co-shedding states are consistent with those of tandem square cylinder reported by More et al. (2015). At u_∞ = 10 m/s, only the co-shedding state emerges (Fig. 4c), with overall higher level of turbulence intensity.

As shown in Fig. 4, the top and bottom boundaries of the pressure evaluation domain (magenta solid lines) exhibit low turbulence intensity (about 0.1), reaching a relatively steady condition closer to the free-stream. Consequently, the Dirichlet boundary condition is applied here, enforcing static pressure (p_static) through Eq. (9). Conversely, high turbulence intensity is observed at the inlet and outlet of the pressure evaluation domain, as well as the solid surface of the downstream cylinder (green solid lines) due to the crossing and impingement of Karman vortices. The Neumann boundary condition is given here with the instantaneous pressure gradient obtained at each time instant through Eq. (1) (Jux et al. 2020).

4.2 Effect of POD low-order reconstruction on the velocity field

POD low-order reconstruction is applied as a data denoising technique to reduce the random errors introduced during PIV measurement and processing. The small-scale and random structures corresponding to the high-frequency incoherent noise in the high-order modes are filtered out (Brindise and Vlachos 2017; Taira et al. 2017), increasing the signal-to-noise ratio of the instantaneous velocity fields. For two different flow conditions (u_∞ = 8 and 10 m/s), an ensemble size of 4000 snapshots is applied for POD analysis, following the procedure described in Sect. 2.1.

The contribution of the first 100 modes to the total disturbance energy is shown in Fig. 5. Different from the mode energy distribution for the flow past an isolated cylinder (Wang. et al. 2014; Van Oudheusden et al. 2005), in which the first two modes cover more than 70% of the total disturbance energy, the large energy variation among POD modes diminishes in the tandem cylinder flow. This uniform energy distribution is attributed to the generation of multi-scale structures (Ye et al. 2018), arising from the impingement of spanwise vortices on the downstream cylinder. In the present study, the first and second modes represent 25.6% and 21.0%, and 20.9% and 17.2% of total disturbance energy for u_∞ = 8 and 10 m/s, respectively.

Fig. 5

Energy contribution of the first 100 POD modes; the symbols and solid lines represent the energy content of each mode and the cumulative energy of the first nth modes, respectively

To perform a quantitative assessment of the denoising effect, low-order modes containing 70%, 80%, and 90% of the total disturbance energy are selected to reconstruct the instantaneous velocity fields, as shown in Fig. 6. All three energy contents selected here have covered the first two dominant modes, corresponding to the vortex shedding. The raw velocity fields are also presented for comparison. Random outliers arise at the inlet and outlet of the measurement domain, around the cylinder, and in the vortex-dominated areas within the gap between the two cylinders and wake behind the downstream cylinder, as shown by the red cycles in Fig. 6a and b. These outliers are attributed to out-of-domain particle motion (Hart 2000) and the out-of-plane motions induced by the three-dimensionality Kelvin–Helmholtz instability after the rolling up of the shear layer from both cylinders (Aasland et al. 2022). These errors subsequently propagate into boundary conditions and Poisson source term, thus reducing the accuracy of the estimated pressure (Pan et al. 2016). With the implementation of POD low-order reconstruction, the spurious vectors are effectively suppressed, while the dominant structures are accentuated for all the energy contents (see Fig. 6c–h), indicating a notable outlier removing effect.

Fig. 6

Instantaneous velocity fields of raw and POD low-order reconstructed velocity. a, c, e, and g: u_∞ = 8 m/s and b, d, f, and h: u_∞ = 10 m/s. Velocity fields reconstructed using low-order POD modes contain (c) and (d) 90%, (e) and (f) 80%, and (g) and (h) 70% of total energy. The vectors are superimposed by subtracting 0.4u_∞ to better elucidate the vortical structures

Further quantitative estimation of the random errors of the instantaneous velocity field is performed. Following to Ghaemi et al. (2012), the random errors of streamwise and transverse velocity (ε_u and ε_v, respectively) are determined through the premultiplied PSD of corresponding velocity fluctuations (fΦ_uu and fΦ_vv, respectively), equal to the root square of the noise level of the premultiplied PSD. The premultiplied PSD of velocity fluctuations near the head of the downstream cylinder (primary sound source, x/D = − 0.74, y/D = 0) for raw and various POD low-order reconstructed velocity input of u_∞ = 10 m/s are evaluated, as presented in Fig. 7. The noise level is given by the magnitude of the high-frequency plateau of the premultiplied PSD shown by the black dash lines. A notable decrease on the noise level is achieved when employing POD low-order reconstruction. The reduction of noise levels from 0.55 and 0.50 m²/s² to a minimum of 0.035 and 0.018 m²/s² is observed for the streamwise and transverse components of 70% POD low-order reconstructed velocity, respectively. This corresponds to the random errors of velocity decrease from 0.74 and 0.71 m/s to 0.19 and 0.13 m/s for ε_u and ε_v, respectively. Figure 7 also indicates that the cut-off frequency of velocity increases when the POD low-order reconstruction contains 80% and 90% energy. However, a reduction in the cut-off frequency is observed for the 70% POD low-order reconstruction, accompanied by a distinct magnitude loss of the premultiplied PSD in the low- to medium-frequency range. This suggests that 70% POD low-order reconstruction leads to an over-smoothing effect on large-scale structures within the velocity field.

Fig. 7

Premultiplied PSD of streamwise and transverse velocity fluctuation near the head of downstream cylinder at x/D = − 0.74, y/D = 0, a fΦ_uu, b fΦ_vv

The related random error of total displacement is calculated as:

$$\varepsilon_{{{\text{ds}}}} = \sqrt {\varepsilon_{u}^{2} + \varepsilon_{v}^{2} }\Delta t$$

(16)

where Δt is the pulse separation of the measurement frames. Following the approach above, the random errors of the streamwise, transverse velocity, and total displacement (ε_u, ε_v, and ε_ds, respectively) at the top and inlet of the measurement domain, and near the head of downstream cylinder are calculated and summarized in Table 1. For the top of the measurement domain, the displacement error decreases to 0.01 pix when the reconstructed velocity fields occupy less than 80% energy, which is much lower than 0.1 pix detected in the free-stream reported by de Kat and van Oudheusden (2011). Considerable reductions are also achieved both at the inlet edges of the measurement domain and near the surface of the cylinder, providing velocity input with higher accuracy for further pressure field reconstruction and far-field noise prediction.

Table 1 Estimation of the random error based on premultiplied PSD

4.3 Pressure reconstruction

To avoid the over-smoothing effect, the raw and low-order reconstructed velocity fields with 80% and 90% of the total energy contents are used as input of the Poisson solver to estimate the pressure field. Comparisons are made with the reference pressure signals simultaneously measured with the surface pressure transducer at the azimuthal location of α = 45° (see Fig. 8) for u_∞ = 8 and 10 m/s to assess the accuracy of pressure reconstruction. To compare with the surface pressure reconstructed from PIV data, the pressure signal acquired by the miniature transducer is downsampled to 4 kHz.

Fig. 8

Time series of estimated and measured pressure signals (p′/p′_rms) at α = 45°, a and c: u_∞ = 8 m/s and b and d: u_∞ = 10 m/s

The time series of normalized pressure fluctuations p′/p′_rms (where p′_rms denotes the root mean square of pressure fluctuations) is shown in Fig. 8 for a duration of 50 ms. All the signals are band-pass filtered in the range of f = [30, 1500] Hz, with the upper corresponding to the cut-off frequency of the premultiplied PSD observed in Fig. 8. For all the cases, the periodicity of the pressure fluctuation signal due to vortex shedding and impingement is well captured. The estimated pressure based on the raw velocity field exhibits densely distributed high-frequency noises superposed over the signal with low-frequency periodic variation (see Fig. 8a and b). With the application of POD low-order reconstruction, the high-frequency noise is significantly suppressed, providing a smoother periodic pressure signal with better agreement with the reference measurement signal. Moreover, Fig. 8c and d suggests that the pressure signals estimated from 80 to 90% POD low-order reconstructed velocity also preserve the secondary peaks related to the superharmonics of the vortex shedding and small-scale turbulent structures, ensuring higher accuracy in broadband.

To further quantify the improvement in accuracy of estimated pressure after POD low-order reconstruction, the cross-correlation coefficient R_pp between pressure signals estimated from PIV and measured by surface pressure transducer is calculated as:

$$R_{{{\text{pp}}}} = \frac{{\left\langle {p^{\prime }_{{{\text{PIV}}}} p^{\prime }_{{{\text{mic}}}} } \right\rangle }}{{\sigma_{{{\text{PIV}}}} \sigma_{{{\text{mic}}}} }}$$

(17)

where p′_PIV and p′_mic denote the pressure fluctuation signals estimated from PIV and measured by the surface pressure transducer, ⟨⟩ is the ensemble average operator, and σ is the standard deviation of pressure fluctuations. Results of R_pp for u_∞ = 8 and 10 m/s at α = 45° of the downstream cylinder are plotted in Fig. 9. The R_pp for both free-stream velocities denotes that pressure estimated based on POD low-order reconstruction velocity agrees better with the reference measurement signals than that from the raw velocity field. The peak values of R_pp increase as the high-order modes containing incoherent small-scale structures are removed, achieving a maximum of 0.82 and 0.67 when using 80% POD low-order reconstruction for u_∞ = 8 and 10 m/s, respectively. The periodic vortical structures are highlighted by POD low-order reconstruction, thereby better resolving the dominant periodic pressure signal. Time interval Δτ between two negative peaks is 11 ms and 8.5 ms for u_∞ = 8 m/s and 10 m/s, respectively. The corresponding Strouhal number (St = D/(Δτu_∞)) is 0.17 and 0.18, respectively. The results agree with that reported by Ljungkrona et al. (1991) for the case of Re_D = 2 × 10⁴ with a separation between cylinders of 4D.

Fig. 9

Cross-correlation coefficient of pressure fluctuations R_pp between estimated signals based on PIV data and reference signals measured by pressure transducers at α = 45° for a and b: u_∞ = 8 m/s and c and d: u_∞ = 10 m/s. (b) and (d) are the enlarged plots around the peak of R_pp

The peak value of cross-correlation coefficient for pressure signals obtained from reference measurement and PIV-based reconstructions derived from various velocity inputs for different azimuthal α and u_∞ is summarized in Table 2. For all the cases, the peak values exceed 0.5 when using the POD low-order reconstruction as the velocity inputs, with the enhancement from 0.1 to 0.3 in comparison with using the raw velocity field as input.

Table 2 Peak value of R_pp for various velocity inputs at α = 45°, 90°, 120°, and 150°

To further assess the improvement by POD low-order reconstruction in the frequency domain, the PSD of p′/p′_rms at α = 45° and 150° of the downstream cylinder is estimated and compared with the reference measurement signals for both free-stream velocities, as shown in Fig. 10. For all the cases, the pressure fluctuation spectra estimated from the velocity fields after POD low-order reconstruction exhibit good agreements with the reference signal at both peak frequencies and broadband in the medium- to high-frequency range, while for the spectra estimated based on the raw velocity field (green line), only the peak frequency is captured, performing a relatively larger disparity with the reference PSD magnitude. In the medium- to high-frequency range, the spectra exhibit like white noise, becoming flattened due to the incoherent small-scale errors in the raw velocity field.

Fig. 10

PSD of p′/p′_rms, a and c correspond to u_∞ = 8 m/s at α = 45° and 150°, respectively. b and d correspond to u_∞ = 10 m/s at α = 45° and 150°, respectively

Moreover, the accuracy of the pressure estimation in the high-frequency range depends on the energy contents employed for POD low-order reconstruction. At α = 45°, the energy content of the velocity and pressure fluctuations in the low-frequency range is predominant, which is attributed to the periodic impingement of the large-scale Karman vortices shed from the upstream cylinder to the downstream one, while the high-frequency-related random error has a limited energy content. Removing high-order modes by only 10% energy contents sufficiently reduces the incoherent small-scale noise, yielding better agreement with the reference signal compared with that containing 80% energy, as shown in Fig. 10a and b. Conversely, for α = 150° near the recirculation zone (Fig. 10c and d), abundant small-scale structures arise and undulate in the spanwise direction due to the shear layers separating (Lynch and Wagner 2018). These induce high-content high-frequency random errors in the velocity field, which finally propagate into the estimated pressure (de Kat and van Oudheusden 2011). Therefore, employing 80% POD low-order reconstruction (with a greater removal of high-order energy) effectively suppresses the higher-content high-frequency noise for cases at α = 150°, enhancing the accuracy of pressure reconstruction in the medium- to high-frequency range.

The instantaneous pressure fields reconstructed with various velocity inputs (raw and POD low-order reconstructed) for u_∞ = 8 and 10 m/s are depicted in Fig. 11 by the pressure coefficient (C_p = (p − p_∞)/(0.5ρu_∞²)). The same time instant is selected as the instantaneous velocity fields in Fig. 6. With the implementation of POD low-order reconstruction on the velocity fields, the denoising effect significantly reduces the small-scale random errors in the pressure fields, making the dominant coherent structures and their corresponding pressure distribution more evident. As shown in Fig. 11, the convective vortices within the gap between two cylinders and the wake behind the downstream cylinder produce local pressure minima. And local maxima pressure is observed at about α = 45° of the downstream cylinder, indicating the impingement of vortical structures on the solid surface. A movie of the instantaneous pressure field for u_∞ = 10 m/s with 80% energy after POD reconstruction is provided (supplementary material: Movie 1).

Fig. 11

Instantaneous pressure fields. a, c, and d represent for u_∞ = 8 m/s and b, e, and f represent for u_∞ = 10 m/s. Pressure fields are reconstructed from (a) and (b) raw velocity, (c) and (d) 90% POD low-order reconstruction, and (e) and (f) 80% POD low-order reconstruction. The vectors are superimposed by subtracting 0.4u_∞ to better elucidate the vortical structures

4.4 Effect of time-marching enhancement algorithm

To assess the potential of the time-marching enhancement algorithm in reducing the computational cost, the pressure fields iteratively estimated through the SOR approach are analyzed to characterize the convergence speed. The initial guess in the SOR approach is set with the pressure of the preceding snapshot (time-marching enhancement algorithm) and free-stream static pressure for comparison. The cross-plane contour of the estimated instantaneous pressure field derived from 90% POD low-order reconstructed velocity of u_∞ = 10 m/s is shown in Fig. 12. To compare the convergence of the two methods, three iterative steps are presented, namely 100, 1000, and 3000. When implementing the time-marching enhancing algorithm (Fig. 12a), 100 iterations are enough for the detection of the negative pressure zones corresponding to the spanwise vortices (labeled ‘A’ and ‘C’) and the positive pressure zone in the area of impingement (labeled ‘B’). Increasing the iteration numbers further removes the over-estimation of the positive pressure zone, yielding similar pressure distribution for the case of 1000 and 3000 steps. However, for setting the initial guess with free-stream static pressure, the identification of the negative pressure zones is challenging when the iteration number is small (≤ 1000), with an over-estimation of pressure throughout the entire field (Fig. 12d and e). The data convergence is significantly delayed to more than 3000 steps. The results prove that the initial guess provided by the time-marching enhancement algorithm closely approximates the pressure distribution of the final solution, resulting in faster iteration convergence.

Fig. 12

Cross-plane contour of instantaneous pressure field derived from 90% POD low-order reconstructed velocity of u_∞ = 10 m/s with various iterative steps (100, 1000, and 3000 iterations, respectively). a–c: Time-marching enhancement algorithm and d–f: setting the initial guess as free-stream static pressure in the SOR approach. The vector fields are shown by subtracting 0.4u_∞

The iterative processes of surface pressure evaluation with two types of initial guess in the SOR approach are also monitored. Figure 13 illustrates the updates of normalized surface pressure deviation (p^k− p)/(0.5ρu_∞²) of 72 dispersed points on the solid surface of the downstream cylinder (as mentioned in Sect. 2.2). Here, p^k denotes the pressure updated after kth iterations, and p is the final converged pressure.

Fig. 13

Updates of pressure deviation ((p^k− p)/0.5ρu_∞²)) during the iterative process. Solid and dash lines represent the cases utilizing time-marching enhancing algorithm and setting free-stream pressure as the initial guess in the SOR approach, respectively

When employing free-stream static pressure as the initial guess in the SOR approach, significant pressure deviation and non-monotonic trend are observed for all dispersed points before reaching 1000 iterations (dash lines in Fig. 13), indicating a chaotic pressure distribution and data oscillation during iterative updating. On the contrary, the pressure deviation is noticeably smaller for the time-marching enhancing algorithm (solid lines in Fig. 13), decreasing rapidly and stably toward convergence for all the monitored points. Data convergence is obtained when the pressure deviation is lower than 0.005 (corresponding to 0.26 Pa). As a result, it is found that 1951 iterations are sufficient for convergence using the time-marching enhancing algorithm, which is 32.6% faster than the 2897 required when setting the free-stream static pressure as the initial guess.

4.5 Far-field noise prediction

Far-field noise prediction through Curle’s analogy based on the pressure fluctuations estimated from PIV measurement directly relates the flow behavior to the generation of aerodynamic noise (Koschatzky et al. 2010). In this section, the far-field noise is predicted and compared with the results of the reference far-field microphone measurements under u_∞ = 10 m/s, corresponding to the co-shedding state.

Since the flow past tandem cylinders is not perfectly two-dimensional, the spanwise correction based on an additional planar PIV measurement in the x–z plane is applied, performed close to the head of the downstream cylinder (x/D = − 1.5) where high-level pressure fluctuations and far-field noise are produced. The spanwise coherence and correlation length at different frequencies are illustrated in Fig. 14, calculated following Eqs. (13), (14), and (15). The result indicates the high spanwise coherence in the low-frequency range, particularly at the shedding frequency (f = 120 Hz) and its first harmonics (f = 230 Hz). The resultant Lz reaches maximum values of 24.7 mm and 15.5 mm for the shedding frequency and its first harmonics, respectively.

Fig. 14

Spanwise coherence analysis for u_∞ = 10 m/s, a spanwise coherence (γ). b Spanwise correlation length (Lz)

The predicted sound pressure level (SPL) through Curle’s analogy with spanwise correction is shown in Fig. 15. Comparison is made with the reference microphone measurements positioned 1 m away from the downstream cylinder oriented at 75°, 90°, and 120° with respective to the negative streamwise direction. The fundamental and first harmonics peaks attributed to the vortex shedding are accurately identified, reaching a good agreement with the reference microphone measurement for all three types of velocity inputs. When the raw velocity fields are applied as the input, the predicted SPL exhibits a plateau in the medium- to high-frequency range for three different orientations, similar to the high-frequency noise observed in the premultiplied PSD of velocity fluctuations (Fig. 7) and the PSD of surface pressure fluctuations (Fig. 9). This is mainly attributed to the small-scale errors in the velocity fields, which are spatial randomly and temporal incoherently distributed. Consequently, these errors cause the high-frequency noise in the velocity spectra and propagate into the same frequency range in the spectra of pressure and far-field noise. When using the POD low-order reconstructed velocity fields as input, the plateau disappears, and the magnitude of the SPL spectrum gradually attenuates with a similar slope to the reference far-field noise measurement in the medium- to high-frequency range (400 Hz < f < 1200 Hz). In this attenuating frequency band, the averaged deviation between the prediction and reference measurement decreases from 21 dB for the raw velocity to 12 and 7 dB for 90% and 80% POD low-order reconstruction, respectively. In addition, the third harmonic peak also emerges from the white noise-like broadband noise at f = 330 Hz. Results suggest that POD low-order reconstruction effectively suppresses the high-frequency noise of the predicted SPL, thereby enhancing the accuracy of noise predictions. For the low-frequency range (f < 100 Hz), the disagreement is attributed to reflections of acoustic waves of long wavelengths, which are not adequately dampened in the anechoic wind tunnel and received by microphones (Brès et al. 2012).

Fig. 15

Sound pressure level (SPL) of far-field noise predicted based on PIV and measured by the far-field microphone, with the listeners located 1 m away from the downstream cylinder, a 75°, b 90°, and c 120°

5 Conclusion

In the present work, we propose a fast and accurate approach for TR-PIV-based pressure reconstruction and far-field noise prediction. The POD low-order reconstruction and time-marching enhancing algorithm are applied to improve the estimation accuracy and elevate the computational efficiency, respectively. The analysis is performed for flow field past tandem cylinders at Re_D = 0.81 and 1.01 × 10⁴. An acquisition frequency of 4 kHz was used for TR-PIV to assure a full coverage of the interested frequency band of velocity fields. The measurements of surface pressure and far-field noise were also taken to provide reference signals. Through pressure reconstruction and noise prediction, the instantaneous velocity fields, pressure fields, and far-field noise for tandem cylinder flow are acquired simultaneously through one single experimental approach.

The POD low-order reconstruction is applied to remove the random errors and small-scale turbulence within the instantaneous velocity fields. A notable reduction of over 65% for the displacement error is obtained, highlighting the predominant vortical structures due to vortex shedding.

The pressure is obtained by solving the Poisson equation through an iterative process. The POD-based denoising process greatly mitigates the error in the velocity field, particularly on the boundaries of the pressure evaluation domain. This results in better agreement of surface pressure in comparison with the reference pressure signal, specifically for the medium- and high-frequency range. All resultant peak cross-correlation coefficient values exceed 0.5.

The time-marching enhancement algorithm is applied by using the reconstructed pressure field from the proceeded time instant as the initial guess of current instant in the SOR scheme. The iteration steps to obtain data convergence of surface pressure are reduced by 32.6% compared with that using free-stream pressure as the initial guess. The data oscillation during the iterative process is also suppressed.

The final prediction on far-field noise is performed through Curle’s analogy. Spanwise correlation correction derived from the x–z PIV data is conducted to account for the three-dimensionality of the tandem cylinder flow. The POD-based denoising improves the accuracy for identifying the peaks of the total noise and broadband noise compared with the reference measurement. The assessments based on pressure reconstruction and far-field noise prediction both reveal the validity and efficiency of the present approach.