Evaluation of Local Energy Dissipation Rate using Time Resolved PIV

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1 Evaluation of Local Energy Dissipation Rate using Time Resolved PIV Mohammad Mainul Hoque 1, Mayur J. Sathe 2, Jyeshtharaj B. Joshi 3,4, Geoffrey M. Evans 5,* 1,2,5 Department of Chemical Engineering 1,2,5 The University of Newcastle, Callaghan, NSW 2308, Australia 1 mohammadmainul.hoque@uon.edu.au, 2 Mayur.Sathe@newcastle.edu.au, 5,* Geoffrey.Evans@newcastle.edu.au 3,4 Department of Chemical Engineering, Institute of Chemical Technology, Mumbai , India 3,4 Homi Bhabha National Institute, Anushaktinagar, Mumbai , India 3,4 jbjoshi@gmail.com Abstract Numerous industrial approaches input large quantities of power to achieve a desired outcome. The question is whether that power is most efficiently transformed into process to achieve the specific purpose of heat, mass transfer, mixing etc. To optimize heat/mass transfer, small scale eddies should have maximum energy and they should be close to the interface. In contrast, optimum mixing performance is achieved by large eddies distributed uniformly over entire volume of contactor. In summary, the Energy budget ; i.e. how the supplied energy is distributed between eddies of different sizes is very important. Such a distribution is represented by the turbulent energy spectrum. In the current work, we utilize high speed particle image velocimetry (PIV) to record the variation of liquid velocity over a plane in time-resolved fashion. The local energy spectrum is obtained by taking FFT of velocity time series at each point in planar PIV data. A model energy spectrum by Pope ( Turbulent Flows Cambridge University Press, 2000) was fitted to experimental spectrum giving the energy flux (equivalent to the turbulent energy dissipation rate) at each point. The salient feature of this method is the ability to calculate local energy dissipation rate without using the estimates of velocity gradients. The results obtained in current work are less susceptible to the noise in PIV velocity data. The local energy spectrum and energy dissipation rate measurements will enable us to put a close check on the energy budget in the process equipment. It will allow tailoring the turbulence to the specific application, significantly enhancing the operating efficiency. Keywords-Energy spectrum; PIV; Transport Phenomena I. INTRODUCTION The reality is that no turbulent flow is homogeneous and isotropic. In addition, there are many types of turbulence based on the boundary conditions, the forces of the body, and other auxiliary parameters: incompressible, compressible, sheared homogeneous, heterogeneous and stratified, magnetohydrodynamics, superfluid turbulence, and so on. They are similar in some respects (for example, they are very dissipative), but also different in some respects (e.g., the topology of the large structure is different). This situation is somewhat similar to that of chemistry while all compounds have the same basic elements; they are as different from each other. In another turning point, a considerable simplification of the general problem of dynamic turbulence was conducted by Taylor (1935) with the introduction of the concept of homogeneous and isotropic turbulence; i.e. turbulence is statistically translation invariant, reflection and rotation of the coordinate axes. Experimentally, the nearly homogeneous and isotropic turbulence was developed in the late 1930 using uniform grids of bars in a wind tunnel (e.g. Comte-Bellot and Corrsin 1966). The use of tensor isotropic turbulence was introduced by von Karman (1937) who also studied the dynamic effects of the isotropic (Von Karman and Howarth, 1938). Taylor (1938) derived an equation for turbulent vortex and almost simultaneously, has pioneered the use of the Fourier transform and spectral representation. Since that time, isotropic turbulence has been the testing ground for most analytical theories of turbulence. Enter Kolmogorov (1941b) and his revolutionary postulate that small scales of turbulence are isotropic statistically, no matter how the turbulence is produced. This postulate, coupled with Kolmogorov's other hypothesis, for short-known in the jargon as K41, has allowed several detailed predictions to be made with regard to the scaling properties of ''small-scale'' turbulence. The spirit of K41, a major fore-runner for which are Richardson's (1922) qualitative ideas of self-similar distribution of turbulent eddies, is to assume that the ''small'' scales of turbulence are universal, even though the ''large'' scales are definite to a given flow-or class of flows with the same boundary conditions. While a full understanding of a turbulent flow requires attention to small and large scales (including the mixture varies from stream to flow), K41 assumes that the small scales can be understood independently of the characteristics that determine large scales. In particular, towards the upper end of the small-scale range (so-called inertial subrange), K41 shows

2 that spectral energy density,, varies with the following wavenumber, k, according to: =. (1) Here is the rate at which energy is dissipated by the lowend small-scale, and C is a constant unknown but universal. K41 is based on the principle that the large scales at which energy is injected transfer to small scales where it is dissipated through a series of steps, each of which is less dissipation and only involves the interaction of neighboring eddies. The use of regular grids in a uniform flow generate relatively simple turbulence may have started with the work of Simmons & Salter (1934) to calculate the dissipation rate. Subsequent measurements were made by Dryden, Schubauer, Mock & Skramstad (1937) and Dryden (1943), see also Taylor (1953, part 11), Corrsin (1942), Batchelor and Townsend (1947, 1948, Tsuji and Hama (1953), Grant & Nisbet (1957), Wyatt (1955) and others. It is remarkable that, despite the highly oriented and homogeneous character generator, the turbulent motion may be 40 or 50 mesh lengths downstream is statistically homogeneous (in planes parallel to the grid) almost to the accuracy of measurements and is approximately isotropic. As there is no residual mean shear, there is no continuous source of turbulent energy, and turbulence decays with distance downstream. The rate of decay of energy is roughly equal to the rate of viscous dissipation, and was one of the persistent objects of theoretical study since the pioneering paper of isotropic turbulence by Taylor (1935 b). Working with grid generated turbulence was performed by Schreck and Kleis (1993), Geiss et al. (2004) and Hussainov et al. (2000). The first used a vertical water channel, while the latter two studied an air/solid system. All three found that, even for a relatively small volume load, there is an overall decrease in the turbulent kinetic energy. In addition, the flow seems to become more anisotropic downstream of the grid (Geiss et al. 2004). Experimental work associated noteworthy are studies of the effective speed of sedimentation of particles in a homogeneous and isotropic turbulent flow, for example Aliseda et al. (2002) and Yang & Shy (2005). These studies are closely related, but their focus is on the various aspects of two-way coupling (i.e. the change in the sedimentation rate of a particle due to turbulence). Similarly, studies have been conducted to investigate mainly the so-called group of particles due to the interaction of fluid and particles (Eaton & Fessler 1994). In the case of PIV, Westerweel et al. (1997) improved the spatial resolution by nearly a factor of 10, through the use of a window offset. Only very recently have there been attempts to quantify the effect of the limited spatial resolution associated with the finite size of the interrogation window in PIV (Scarano 2003; Foucaut et al. 2004; Saikrishnan et al. 2006; Poelma et al ). Particle Image Velocimetry (PIV) is now a widely utilized measurement technique in the investigation of turbulent engineering flows. Its proven ability to capture time series of instantaneous, two- or three-component velocity data over a planar field of interest allows the fluid dynamist to analyses structures forming, passing through, and evolving through the measurement domain. PIV allows us to uncover fluid dynamic phenomena previously extremely difficult to detect with point measurement techniques, whilst also providing a very efficient means of measuring single- and multi-point statistical data such as time average velocities, Reynolds stresses and spatial correlation functions. Often there is significant turbulent energy in the flow field being measured that is contained in length scales similar to, or smaller than, the interrogation cells into which the field of interest has been divided for PIV analysis. Because of this, reported second order statistics such as Reynolds stresses and turbulent kinetic energy will be contaminated by a spatial filtering effect. Especially, time resolved PIV can give instantaneous particle position with local liquid velocity fluctuations. Therefore, in this paper an attempt has been made to estimate the rate of local energy dissipation using PIV. The geometry of the device-generated grid is a very important factor influencing the extent of turbulence. The Energy spectrum obtained for the first Fourier transform (FFT) was taken at each point of the PIV planar data. Also, the model of the energy spectrum given by Stephen B. Pope (2000) has been applied with the experimental spectrum in the inertial subrange where the -5/3 Kolmogorov law applies. II. EXPERIMENTAL APPARATUS In this study, a turbulence oscillating generator similar to Yang and Shy (2003) was used to produce stationary near isotropic turbulent flow fields. In such devices, the oscillating grids create a system of jets and wakes merging with each other to produce a sustained with zero mean flow turbulence. The turbulence intensity was varied simply by changing the frequency of vibration, strokes size and/or the separation distance between the grids. A relationship between these operational parameters and the characteristics of the resulting turbulent field has been established (Shy et al., 1997). A schematic representation of the grid generated turbulence used here is shown in Fig. 1. The device comprises a rectangular Perspex tank with a width of 300 mm including a pair of vertically oriented grids of size 150 mm 150 mm. The gates were made of aluminum foil with a thickness of 6mm with a mesh size of 30 mm (the distance between the centers of two successive openings was 30 mm) and an overall opening of 64%. Horizontally and vertically aligned square openings were made using a technique of laser cutting precision. The design specifications mentioned above were set according to the literature to prevent the formation of unstable flow structures (Yang et al., 2003). Rods supported by linear bearings were then used to connect the grids stepper motors through eccentric cams. To minimize the risk of transmission of engine vibration to the flow field, the tank was been installed on a separate (Zellouf et al., 2005) supporting bench. In general, the system was designed to operate at different frequencies and strokes. A design consideration for adjusting the separation distance between the networks has also been made. The motors were installed so that grids displaced inand-out, rather than side-by-side. The flow turbulence characteristics in these conditions for specific frequencies have already been determined by Yang and Shy (2003) for a

3 Synchronizer Grid Tank Laser Eccentric Cam Camera Supporting Bench Linear Guide Camera Stepper Motor Fig. 1: A schematic representation of grid generated turbulence similar system. Since the same design principles as those of Yang and Shy (2003) are used to make our turbulent generator, it is reasonable to assume that the same flow fields are formed in the same operational conditions. A systematic approach was used in the experimental work. Experimental studies were conducted using seeding consisting of glass beads particles. First, the operating parameters, including frequency of vibration, the size of the stroke and the distance between the grids were created. A turbulent flow field was then established by the reciprocating out-of-phase grids. The movement of particles was captured using a high speed camera. Phantom5 software was used to analyze the data obtained in the region of isotropic turbulence, which is centrally located between the grids. In the present study, the particle slip velocity is equal to its settling velocity since the liquid mean velocity relative to the wall is zero. III. THEORETICAL MODEL The literature describes various methods where energy spectra were used to estimate the value of the energy dissipation, ε. Few of these methods have been developed for data obtained from a single probe hot film anemometry (HFA) points. The methods are mainly based on Taylor series approximations and have been studied mostly for the case of homogeneous and isotropic turbulence. These methods have linked energy spectra and structure function exhibitors by an algebraic relationship. In another approach, the methodology of the structure function using multiple HFA probes is illustrated in the literature by Lindborg (1999), Pope (2000) and Kang et al. (2003) for calculating ε for the conditions in which the spatial distance between the two points is maintained such that they are within the inertial range. Particle image velocimetry (PIV) can be used to obtain many data points simultaneously. Consequently, the linked energy spectra and structure function becomes generalized and requires simultaneously solving the set of nonlinear equations. The devolvement of second and third order structure functions, and, energy spectrum,, as described by Pope (2000) is given below: The function of energy spectrum can be written as:, (2) where and are specified non-dimensional functions. The specification of is equal to:! " # $ %, (3) where & ' is taken to be 2, and ( is a positive constant. The specification of is equal to: )*&+,-./ 0 1( ,( 45, (4) where - also an positive constant. The function determines shape of the interval containing the energy and tends to unity for large. Similarly, determines the shape of the dissipation range, and tends to unity for small. Under the inertial range and are essentially unity, so that the Kolmogorov (1941) -5/3 spectrum with a constant C is recovered.

4 is the total energy spectrum. It can be separated into longitudinal 33 3 and transverse 3 spectrums. Lindborg (1999), Pope (2000) and Kang et al. (2003) have suggested a procedure on the basis of structure function. Such theories predict velocity structure functions of second and third order, and can be written D 3 *+) D 3 *,B, D 3 *+) D 3 *,B. (6) The relationship and can be obtained as follows: F 0 H F " =6L HM HF ', (7) where the first term of left hand side including the time derivative was omitted by Kolmogorov. In an inhomogeneous flow, which still can qualify as locally homogeneous to a good approximation, varies with position. In the laboratory reference frame the first term of left hand side in equation (7) is zero, but in the system moving with the mean flow it is not zero, since the turbulence decays along the mean flow direction. The first term of the right-hand side of equation (7) shows the effect of diffusivity. To calculate the time dependent term, N O,B NB, then Kolmogorov s second similarity hypothesis N NB and the model for the decay of the dissipation N NB= Q R were used. Thus equation (7) becomes: M FS = 0 T4VF W Y Z 3[ V \ ]^_`a WVF W # b, (8) where c =15LD Ffg 3 is Taylor microscale. Also, according to literature the values of and S have been varied in the range and , respectively. Further, one dimensional energy spectrum related to the 3D energy spectrum as given by the following equations: i 33 3 =G h V1 WJ, (9) 3 = 3 kh j l. (10) k The one dimensional energy spectra were estimated using eqs. (9)-(10) with, written in eq. (2). Results are compared to the measured longitudinal spectra, and the procedure was repeated until a good agreement between the fitted and original one dimensional spectrum was achieved. IV. RESULTS AND DISCUSSION The isotropic turbulence has been studied most, theoretically as well as experimentally. The geometrical and kinematic relation involved in turbulence has been studied through correlations and spectrum functions, and the dynamic behavior of these correlations and spectrum functions (decay of turbulence). To date it has been possible to obtain a satisfactory and nearly complete picture of the formal geometrical relations only when these are based upon conditions of isotropy. It has not yet been possible to obtain general, complete solutions of the differential equations involved. No complete solutions are known for the various correlations and spectrum functions, or for their changes with time, except for one single limiting case. However, by making some additional assumption or postulates that applies more or less approximately, it has been possible to obtain solutions of some restricted validity. In this paper, 33 3 and 3 components has been computed for a given. In this paper, a model spectrum (Fig. 2) which provides a reasonably accurate representation of measured turbulence spectra has been plotted. Fig.1 shows the various spectra, 33 3 and 3 given by this model for isotropic turbulence. 1.00E E E E E E E E E E-08 Longitudinal Spectrum E11(k1) Enegy Spectrum E(k) Transverse Spectrum E22(k1) 1.00E E E E E E+02, 3 t 3 Fig. 2: Comparison of spectra in isotropic turbulence The three dimensional energy spectra were plotted for arbitrary values of energy dissipation rate, integral and Kolmogorov length scales. The values of parameters used to plot Fig. 2 are =1.5, =50,( =6.78,& ' =2; =0.5,- =5.2,( =0.4 and - = These values produce good agreement with the measured longitudinal spectra across the radial locations, in the inertial range as well as dissipation range. All three spectra exhibit power law behaviour & = 5/3 in the centre of the wavenumber range. At high wavenumber, the spectra decay more swiftly than a power of, consistent with the underlying velocity field being infinitely differentiable. At low wavenumber, tends to zero. In contrast, the one dimensional spectra are maximum at zero wavenumber. This again illustrates the fact the onedimensional spectra contain contributions from wavenumber greater than 3. According to the Kolmogorov hypothesis, in any turbulent flow at sufficiently high Reynolds number, the highwavenumber portion of the velocity spectra adopts particular universal forms. And the forms of the Kolmogorov Spectra can be obtained via two different routes. The implication of the Kolmogorov hypotheses for the second and third order structure function is given in eqs. (5) and (6). The first route is to obtain the Kolmogorov spectra as the appropriate Fourier transform of the structure functions. However, the second route, which is simpler though less rigorous, is to apply the Kolmogorov hypothesis directly to the spectra.

5 1.00E E E vwj E E-03 E11_k_u E22_k_u E11_k_v E22_k_v 1.00E E E E E E+02 3 t 3 Fig. 3: Comparison of the longitudinal and transverse energy spectrum at frequency 10 khz for grid generated turbulence system. The comparison between longitudinal and transverse energy spectrum at frequency 10 khz has been plotted in Fig. 3 for different directions. To plot the spectra, the Fourier transform is used to evaluate the energy of eddies as a function of frequency. Here, the wavenumber has been calculated by using 2u/c. It can be observed that in both directions the longitudinal and transverse energy spectrum follows similar trend in the inertial sub range. All four spectra are overlapping within experimental accuracy; highlighting the fact the velocity field in the current experimental system represents nearly homogeneous and isotropic turbulence. In Fig. 4 the longitudinal energy spectra (experimental and fitted theoretical) have been plotted for different grid oscillation frequency. Figure 5 shows transverse energy spectra for the same grid oscillation frequency and curve fitting parameters. Here, the frequency of the grid varies from 5-12 khz. The theoretical model (Pope, 2000) has been fitted to each experimental spectrum by varying the energy dissipation rate, integral and Kolmogorov length scales. It has been observed that the energy dissipation rate increased with the increase of grid turbulence frequency. From Fig. 4 it can be observed that there is a general trend of a slope close to -5/3 in the inertial subrange. The spectral slope of -5/3 is synonymous with Kolmogorov s theory. It is based on the fact that the time required for the energy transfer from larger eddy to smaller eddy by vortex stretching is much smaller than the time required for the dissipation eddy energy by viscous dissipation. The model spectra for the transverse direction shown in Fig. 5 have been plotted using the same fitting parameters as the longitudinal spectra. It can be observed that the model spectrum does not fit the experimental transverse spectra very well. A principle reason for the poor fit is in the nature of the expression for transverse model spectrum, which has been derived for geometries in which there is flow in one principal direction. In our experimental geometry, flow is homogeneous and isotropic, with no net flow in any single direction; as proven by the overlapping spectra in Fig. 3. However, both the experimental and model transverse energy spectrum follow the Kolmogorov -5/3 slope in the inertial sub range. CONCLUSIONS Measurement of turbulence in a system generated grid was carried out. 2D velocity was measured for the flow generated by oscillating grid using time resolved PIV technique. The experimental data were analyzed to get the longitudinal and transverse energy spectrum using Fast Fourier Transform (FFT). The energy dissipation rate in the grid generated turbulence calculated using the current methodology by fitting a model spectrum was found to agree with the available literature. The spectral slope of -5/3 was found to be present in the energy spectrum in the inertial sub range. The experimental longitudinal and transverse spectra for both velocity components overlap, indicating homogeneous and isotropic turbulence. Using the current technique, energy dissipation rate can be calculated over entire plane. The profiles of ε provide the deepest insights into the nature of the flow inside, and provide guidelines for the intensification of the operation.

6 1.E+03 1.E+02 1.E+01 1.E+00 1.E-01 1.E-02 1.E E-04 Model Longitudinal Energy Spectrum eq. (7); epsilon= E-05 Experimental Longitudinal Energy Spectrum att 5 khz Model Longitudinal Energy Spectrum eq. (7); epsilon= E-06 Experimental Longitudinal Energy Spectrum at 8 khz 1.E-07 Model Longitudinal Energy Spectrum eq. (7); epsilon=0.05 Experimental Longitudinal Energy Spectrum at 10 khz Model Longitudinal Energy Spectrum eq. (7); epsilon= E-08 Experimental Longitudinal Energy Spectrum at 12 khz 1.E E E E E E+02 1.E+01 3 t 3 Fig. 4: Comparison of longitudinal energy spectra (solid lines) with the experimental data at different frequency 1.E+00 1.E E-02 1.E-03 Transverse Energy Spectrum eq.(8); epsilon=0.003 Experimental Transverse Energy Spectrum at 5 khz Transverse Energy Spectrum eq.(8); epsilon=0.02" Experimental Transverse Energy Spectrum at 8 khz 1.E-04 Transverse Energy Spectrum eq.(8); epsilon=0.05 Experimental Transverse Energy Spectrum at 10 khz Transverse Energy Spectrum eq.(8); epsilon=0.08 Experimental Transverse Energy Spectrum at 12 khz 1.E-05 1.E-02 1.E-01 1.E+00 1.E+01 1.E+02 3 t 3 Fig. 5: Comparison of the transverse energy spectra (solid lines) with the experimental data

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