Elsevier

Chemical Engineering Science

Volume 190, 23 November 2018, Pages 28-39
Chemical Engineering Science

Effect of bubble on the pressure spectra of oscillating grid turbulent flow at low Taylor-Reynolds number

https://doi.org/10.1016/j.ces.2018.05.048Get rights and content

Highlights

  • Pressure field estimated from N-S equation using PIV velocity data.

  • Pressure spectrum determined with and without bubble for Reλ = 12 to 60.

  • With bubble, pressure spectra exhibited a slope less steep than −7/3.

  • With bubble, length scale ratios Lp/L and λp deviated from single phase DNS data.

  • ε (m2/s3) from pressure spectrum deviated by ∼32% max compared to velocity spectrum.

Abstract

For many engineering applications, measurements of velocity and pressure distributions in the system are of fundamental importance to provide insights into the flow characteristics. In the present study, the experiments were carried out in an oscillating grid system in absence and presence of two different bubble diameters (Db  =  2.70 and 3.52 mm) rise using the non-intrusive two-dimensional (2D) particle image velocimetry (PIV) at the low Taylor-Reynolds number (Reλ) ranging from 12 to 60. Using the measured PIV velocity data, the instantaneous pressure fluctuations were estimated by integrating the full viscous form of the Navier-Stokes (N-S) equation. The obtained pressure field was compared with three dimensional (3D) computational fluid dynamics (CFD) simulation which was found to be in good agreement. The pressure spectra of single and two phase flow cases were evaluated by taking Fast Fourier transformation (FFT) of the computed pressure fluctuations. A spectral slope of −7/3 was found in the inertial subrange of the single-phase pressure spectrum. In contrast, the two-phase pressure spectrum exhibited a slope less steep than −7/3 in the inertial subrange because of the extra production of turbulence in the presence of bubble. For single-phase flow, the ratio of pressure integral length scale to the velocity integral length scale (Lp/L) was found to be ∼0.67, and the pressure Taylor microscale (λp) was approximately 0.79 ± 0.03 of the velocity Taylor microscale (λ) within the Taylor-Reynolds number range studied. The scaling ratios based on the single-phase experimental results were compared with existing theory and DNS results and found to accord well; however, these ratios deviate from the theoretical values for two-phase flow. Also, the energy dissipation rate was evaluated based on the pressure spectrum and found to be over-predicted (∼32%) compared those calculated from the velocity spectrum.

Introduction

In recent years, considerable interest has emerged in the study of pressure spectra related to turbulence research. Understanding the behaviour of pressure spectra and related statistics in turbulent flow can enlighten many fields of fundamental research interest such as gas-fluid or fluid-solid interactions including drag and lift forces, flow separation, eddy shedding, etc., which are affected by the variation of velocity and pressure. Also, in a turbulent flow field, pressure fluctuations lead to the production of sound and cavitation which are of significant practical interests (Pumir, 1994). Many researchers have experimentally and numerically studied the statistics of pressure (Dong et al., 2001, Donzis et al., 2012, Pearson and Antonia, 2001, Tsuji and Ishihara, 2003), however, studies on pressure spectrum in the literature are scarce due to the difficulty involved in measuring pressure in the laboratory experiments. Conventional pressure measurement relies on the use of probes which due to their significant physical presence distort the flow field and affects the reliability of the results. The practical limitation involved in the manufacturing of miniature probe comparable to the size of small length scales of the flow field often obscures relevant information.

The limitation resulting from inaccuracy of estimating the pressure in an intrusive manner is often circumvented by determining pressure from the measured particle image velocimetry (PIV) velocity information. There are now several studies considering the evaluation of the pressure field from the application of the Navier-Stokes (N-S) equations to the measured velocity field by PIV (Baur and Köngeter, 1999, Unal et al., 1997). The pressure distribution can be obtained directly from the momentum equation by performing the spatial integration over the flow domain along different paths depending on the reference particle location. However, the obtained pressure may be different if the integration is achieved along the different paths. A proper integration path is needed to minimise the errors of integration and decrease the effect of uncertainties related with velocity measurement (Fujisawa et al., 2004). The other method involves the use of Poisson equation which can be utilised to predict the pressure field (Fujisawa et al., 2004, Gurka et al., 1999, Hosokawa et al., 2003) by taking the gradient of Navier-Stokes equation where the source term yields two parts of the noticeably distinctive character. The so-called rapid part represents the direct interaction between the gradient of turbulent velocity fluctuation and the gradient of the mean velocity while the slow part represents the turbulence-turbulence interaction (Tsuji et al., 2007). It is noted that Neumann type boundary condition is needed for the pressure fluctuations, which complicates the application of this method (Jaw et al., 2009).

From the aforesaid studies (Baur and Köngeter, 1999, Fujisawa et al., 2004, Gurka et al., 1999, Hosokawa et al., 2003, Jaw et al., 2009, Tsuji et al., 2007, Unal et al., 1997), it is apparent that the pressure field is attainable from PIV velocity data from the direct solution of the Navier-Stokes equation. These studies however did not emphasise on the accurate estimation of the material derivative term which is a critical requirement for the appropriate estimations of pressure field. There are few studies (Chang et al., 1999, Jakobsen et al., 1997, Jensen and Pedersen, 2004, Jensen et al., 2001, Liu and Katz, 2006) available in the literature where the material derivative term has been shown to be estimated robustly. Chang et al. (1999) employed a single-camera system with double exposure in the first image and single exposure in the second image to calculate this acceleration term. Liu and Katz (2006) estimated the material acceleration term and derived the pressure field with high precision using a four-exposure PIV system invloving two cameras. Nonetheless, these multi-exposure PIV systems involves cumbersome calibration procedure requiring cross-correlation to precisely orient the two cameras to the same field of view (FoV). Liu and Katz (2006) reported that a bias error in velocity estimate in the order of 0.1 would lead to significant errors in the pressure field. Recently, Dabiri et al. (2014) presented an appropriate method for pressure field estimation from the PIV velocity data using median polling of several integration paths. Their technique is relatively simple yet effective in estimating the pressure gradient term with higher accuracy. Such research endeavours in recent time have opened up the possibility to explore the pressure-based flow statistics from a direct solution of Navier-Stokes equation utilizing the velocity field data.

Similar to velocity spectrum, it has been hypothesised that the Kolmogorov’s similarity hypothesis (Kolmogorov, 1941) may also hold true for pressure spectrum. According to Kolmogorov’s similarity hypothesis (Kolmogorov, 1941), the spectral slopes of velocity and pressure spectra in the inertial sub-region are −5/3 and −7/3 over the wavenumber space, respectively (Monin and Yaglom, 1975, Sreenivasan and Antonia, 1997). The pressure spectra exhibits an inertial subrange (κη<<1) at large Reynolds number and expressed as Ep(κ)=Cρ2ε4/3κ-7/3, where C is a universal constant. The scaling of the pressure spectrum in the inertial subrange region for low and high Taylor Reynolds number (Reλ=urmsλ/ν) is one of the most-debated topics in turbulence (Meldi and Sagaut, 2013). Direct numerical simulation (DNS) has made a significant contribution revealing the pressure spectrum characteristics in turbulence (Cao et al., 1999, Gotoh and Fukayama, 2001, Gotoh and Rogallo, 1999, Kim and Antonia, 1993, Meldi and Sagaut, 2013, Pumir, 1994, Vedula and Yeung, 1999). A brief overview of the pressure spectrum slope and inertial range decade is reported in Table 1. It can be noted that in the case of the pressure spectrum, the spectral slope was found to be in the range −7/3 to −5/3 in several DNS results (Cao et al., 1999, Gotoh and Rogallo, 1999, Vedula and Yeung, 1999) (see Table 1). These findings cast doubts on Kolmogorov’s scaling for the pressure spectrum while there is a substantial experimental and numerical evidence of the accuracy of Kolmogorov scaling for the velocity spectrum. It is noted that the DNS (Cao et al., 1999, Gotoh and Fukayama, 2001, Gotoh and Rogallo, 1999, Kim and Antonia, 1993, Lesieur et al., 1999, Meldi and Sagaut, 2013, Pumir, 1994, Vedula and Yeung, 1999) results have been obtained over a wide range of Taylor Reynolds number (5Reλ5000), however till now, the experimental (Tsuji and Ishihara, 2003) results have been limited to moderate Reλ range (200Reλ1200) only.

Despite all the DNS works on the pressure fluctuation and spectrum of homogeneous turbulence, only a few experimental works were found in the literature. Although such a comparison is difficult (Tsuji et al., 2007), but Pearson and Antonia, 2001, Tsuji and Ishihara, 2003 were able to compare their results with DNS data. Pearson and Antonia (2001) experimentally studied the Reynolds number dependency of turbulent velocity and pressure increments. Their comparison indicated that the magnitude and Reλ dependency of the mean square pressure gradient based on the joint Gaussian approximation were incorrect. Tsuji and Ishihara (2003) measured pressure fluctuations in a turbulent jet using a condenser microphone and piezo-resistive transducer. The power law exponent and the proportionality constant of normalised pressure spectrum were discussed from the standpoint of Kolmogorov universal scaling. The power law with scaling exponent ∼ −7/3 was confirmed at Reλ600 which is a significantly higher Taylor Reynolds number than needed for inertial sub-range scaling of velocity statistics. Later, Tsuji et al. (2007) experimentally investigated the detailed pressure statistics in high Reynolds number (5870Reθ18300) based on the momentum thickness (θ) of the turbulent boundary layers and found a spectral slope of −1 slope in the inertial region while the Kolmogorov slope (−7/3) was not observed.

The foregoing discussion clearly illustrates the increasing use of the non-intrusive technique of pressure estimation from PIV velocity data and highlights the continued discrepancy on the scaling of the pressure spectrum which demands more systematic experimental investigation especially at the low Taylor-Reynolds number. Additionally, no studies are reported yet up to the best of the author’s knowledge on the scaling of the two-phase turbulence pressure spectrum in the inertial region. Therefore, it was thought desirable to conduct a systematic experiment in such a system which can produce nearly-isotropic turbulent flow field at low Reλ. Previously authors investigated flow behaviour in the oscillating grid system 12Reλ60 for single-phase flow and quantified turbulence length scales and energy dissipation rate (Hoque et al., 2015), turbulence modulation in the presence of particle (Hoque et al., 2016) and bubbles (Hoque et al., 2018) using velocity spectrum approach. A specific interest of this study was therefore to determine the pressure spectrum slope for oscillating grid turbulent (OGT) system in the absence and presence of single bubble (henceforth, the velocity field at different grid Reynolds number inside the oscillating grid system in absence bubble is referred as “single–phase” flow while in the presence of bubble is referred as “two-phase” flow) and analysis the statistics of pressure fluctuations and corresponding energy dissipation rate. The specific aims of the present study were to:

  • 1.

    Determine the instantaneous pressure field from the PIV velocity data by integrating the full viscous form of the Navier-Stokes (N-S) equation and compare these results with CFD predictions using a finite volume method.

  • 2.

    Determine the characteristics of pressure spectra in terms of length scales and inertial subrange slope.

  • 3.

    Critically analyse the scaling the ratio of pressure to velocity integral length scale and compare these results with the available DNS results.

  • 4.

    Examine the well-known Batchelor’s prediction (Batchelor, 1951) of pressure and velocity Taylor microscale ratio for single-phase and two-phase flow at 12Reλ60 .

  • 5.

    Quantify the specific energy dissipation rate from the velocity and pressure spectra and provide a comparison of these two methods.

In line with the stated aims, rest of the paper is organized in the following order: in Section 2, details pertaining to the velocity measurement methodologies are described; analysis of the velocity and pressure data is then presented in Section 3; results and discussions are presented in Section 4 followed by conclusions of the present study in Section 5.

Section snippets

Velocity field measurement

In the present study, an oscillating grid system (Hoque et al., 2013, Hoque et al., 2014) was used as a source of single-phase homogeneous and isotropic flow. The turbulent flow was generated by using two vertical grids (150 × 150 mm2) inside a rectangular tank (300 mm length × 200 mm width × 180 mm depth). A schematic representation of the grid-generated turbulence is shown in Fig. 1.

The gates were made of an aluminium plate with a thickness of 6 mm, a mesh size (M) of 30 mm and an overall

Pressure estimation

The general expressions involved in determining the instantaneous pressure field from the time-resolved velocity field are outlined in this section. Under the incompressible flow condition for given fluid density (ρ) and viscosity (µ), the Navier-Stokes equation can be used to estimate the pressure gradient from the measured velocity field as follows (Van Oudheusden, 2013):p=-ρDuDt+μ2uwhere Du/Dt is the material acceleration term.

The instantaneous material acceleration (Du/Dt) in Eq. (1) can

Estimation of pressure field from the PIV data

The instantaneous pressure fields of single-phase and two-phase flow obtained by integrating complete Navier-Stokes expression given in Eq. (1) based on the measured velocity data at Reλ = 12 are shown in Fig. 2a and b. Although periodic, the flow field in OGT is indeed complex due to active grid motion and merging of flow structures of different length scales. At an arbitrary time instant, both positive pressure and negative pressure zones can be seen randomly distributed in the field of view

Conclusions

Using PIV velocity data for single and two-phase flow, the instantaneous pressure field was estimated by integrating the full viscous form of the Navier-Stokes (N-S) equation at low Taylor Reynolds numbers (12Reλ60) to study the flow field pressure spectrum and related scaling properties. The present study focused on three questions: (a) does the pressure spectrum follow Kolmogorov’s hypothesis for both single and two-phase flow case at low Reλ? (b) does the theoretical ratio (0.81) of

Acknowledgement

One of the authors, Mohammad Mainul Hoque, would like to acknowledge the scholarships (UNIPRS and UNRSC 50:50) provided by the University of Newcastle, Australia, to pursue the research work. Also, all authors gratefully acknowledge the financial support provided by the Australian Research Council (Grant number LP160101181).

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