Effect of bubble on the pressure spectra of oscillating grid turbulent flow at low Taylor-Reynolds number
Graphical abstract
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 at large Reynolds number and expressed as , where C is a universal constant. The scaling of the pressure spectrum in the inertial subrange region for low and high Taylor Reynolds number 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 , however till now, the experimental (Tsuji and Ishihara, 2003) results have been limited to moderate range 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 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 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 . Previously authors investigated flow behaviour in the oscillating grid system 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 .
- 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):where 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 () 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).
References (47)
- et al.
A continuum method for medeling surface tension
J. Computat. Phys.
(1992) - et al.
Some results on the Reynolds number scaling of pressure statistics in isotropic turbulence
Physica D
(2012) - et al.
Modulation of turbulent flow field in an oscillating grid system owing to single bubble rise
Chem. Eng. Sci.
(2018) - et al.
Experimental investigation on modulation of homogeneous and isotropic turbulence in the presence of single particle using time-resolved PIV
Chem. Eng. Sci.
(2016) - et al.
Analysis of turbulence energy spectrum by using particle image velocimetry
Procedia Eng.
(2014) - et al.
Comparison of specific energy dissipation rate calculation methodologies utilising 2D PIV velocity measurement
Chem. Eng. Sci.
(2015) - et al.
Force prediction by PIV imaging: a momentum-based approach
J. Fluids Struct.
(1997) Pressure fluctuation in isotropic turbulence’
Math. Proc. Cambridge Philos. Soc.
(1951)- et al.
PIV with high temporal resolution for the determination of local pressure reductions from coherent turbulent phenomena, 3rd International Workshop on Particle Image Velocimetry
(1999) - Cao, N., Chen, S., Doolen, G.D., 1999. Statistics and structures of pressure in isotropic turbulence. Physics of Fluids...
A multi-pulsed PTV technique for acceleration measurement, 3rd International Workshop on Particle Image Velocimetry
An algorithm to estimate unsteady and quasi-steady pressure fields from velocity field measurements
J. Exp. Biol.
Interaction of two parallel plane jets of different velocities
J. Visualization
Pressure spectrum in homogeneous turbulence
Phys. Rev. Lett.
Intermittency and scaling of pressure at small scales in forced isotropic turbulence
J. Fluid Mech.
Computation of pressure distribution using PIV velocity data, 3rd International Workshop on Particle Image Velocimetry
Turbulence
Evaluation of local energy dissipation rate using time resolved PIV
PIV measurement of pressure distributions about single bubbles
J. Nucl. Sci. Technol.
Particle Image Velocimetry for predictions of acceleration fields and forces within fluid flows
Meas. Sci. Technol.
Cited by (7)
Influence of bubble surface loading on particle-laden bubble rising dynamics in a fluid flow system
2023, Minerals EngineeringAnalysis of the flow pattern and periodicity of gas–liquid–liquid three-phase flow in a countercurrent mixer-settler
2022, Chinese Journal of Chemical EngineeringCitation Excerpt :The National Energy Technology Laboratory uses the fast Fourier transform (FFT) to quantify turbulent flow in gas–liquid bubble columns [23–26]. Hoque et al. [27] evaluated the pressure spectrum of a two-phase flow in an oscillating grid system using the FFT method. Janiga et al. [28] characterized the MI of the stirred tank using FFT analyses of the three-dimensional proper orthogonal decomposition temporal coefficients.
Numerical estimation of critical local energy dissipation rate for particle detachment from a bubble-particle aggregate captured within a confined vortex
2022, Minerals EngineeringCitation Excerpt :In several past studies, we have demonstrated the use of PIV to quantify the energy dissipation rate in presence of bubbles (Hoque et al., 2018c), single particle (Hoque et al., 2016) and multiple particles (Hoque et al., 2018a) in a near isotropic and homogeneous turbulent flow field. Also shown was direct estimation of pressure field from the PIV measured velocity field (Hoque et al., 2015a, Hoque et al., 2018b). More recently total kinetic energy (TKE) budgeting has been performed in the pseudo-turbulence generated by a rising bubble in a quiescent medium (Hoque et al., 2022).
Dynamics of a single bubble rising in a quiescent medium
2022, Experimental Thermal and Fluid ScienceCitation Excerpt :Due to these limitations, the TKE terms were evaluated based on the following two assumptions: (a) time-averaged statistical properties of the flow are symmetric along the x1 and x3 direction and (b) isotropy exists in the flow field [54]. The earlier studies of Dabiri et al. [55] and Hoque et al. [56] showed that this approach reasonably estimated the pressure field from velocity field produced by PIV measurement. Fig. 10 shows instantaneous pressure field around a rising bubble (Db = 3.53 mm) at two time instants overlayed with velocity field.
A critical analysis of turbulence modulation in particulate flow systems: A review of the experimental studies
2023, Reviews in Chemical Engineering