Turbulence structure in a Taylor–Couette apparatus

Experimental Thermal and Fluid Science 32 (2007) 220–230 www.elsevier.com/locate/etfs

Turbulence structure in a Taylor–Couette apparatus Noah Fehrenbacher, Ralph C. Aldredge *, Joshua T. Morgan Department of Mechanical and Aeronautical Engineering, University of California, Davis, CA 95616, United States Received 11 May 2006; received in revised form 23 February 2007; accepted 24 March 2007

Abstract Turbulence measurements were made in a Taylor–Couette apparatus as a basis for future flame propagation studies. Results of the present study extend that of earlier work by more complete characterization of the featureless turbulence regime generated by the Taylor–Couette apparatus. Laser Doppler Velocimetry was used to measure Reynolds stresses, integral and micro time scales and power spectra over a wide range of turbulence intensities typically encountered by turbulent pre-mixed hydrocarbon–air flames. Measurements of radial velocity intensities are consistent with earlier axial and circumferential velocity measurements that indicated a linear relationship between turbulence intensity and the Reynolds number based on the average cylinder rotation speed and wall separation distance. Measured integral and micro time scales and approximated integral length scales were all found to decrease with the Reynolds number, possibly associated with a confinement of the largest scales (of the order of the cylinder wall separation distance). Regions of transverse isotropy were discovered in axial–radial cross correlations for average cylinder Reynolds numbers less than 6000 and are predicted to exist also for circumferential cross correlations at higher average Reynolds numbers, greater than 6000. Power spectra for the independent directions of velocity fluctuation exhibited 5/3 slopes, suggesting that the flow also has some additional isotropic characteristics and demonstrating the role of the Taylor–Couette apparatus as a novel means for generating turbulence for flame propagation studies. Ó 2007 Elsevier Inc. All rights reserved. Keywords: Taylor–Couette; Turbulence generation; Turbulence structure; Turbulent-flame propagation

1. Introduction Taylor–Couette (TC) flow has been the basis of a variety of viscous-flow investigations, from the viscosity measurements of Couette [1,2] to the study of medical filtration devices [3]. Ronney et al. [4] were the first to propose the TC apparatus for reaction-related research, studying aqueous isothermal reaction-front propagation. To date, several experiments have been carried out to extend Ronney et al.’s work to higher temperature turbulent-flame propagation realistic of typical hydrocarbon–air reactions, however flow field characterization has been limited to measured coldflow velocity profiles [5] used to determine the size of the created homogeneous region and flame speeds in terms of intensities [6,7].

*

Corresponding author. Tel.: +1 530 752 5016; fax: +1 530 752 4158. E-mail address: [email protected] (R.C. Aldredge).

0894-1777/$ - see front matter Ó 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.expthermflusci.2007.03.007

Andereck et al. [8] found 26 stable regimes existing in a TC apparatus depending on the rotation rates of each cylinder. Velocity measurements within many of these regimes, namely Taylor vortex and Couette flow, have been dedicated to studying existing bifurcations and sub and super critical phase transitions, however little has been done to quantify the featureless turbulence regime ideal for combustion related research. The featureless regime is the only truly homogeneous turbulent flow created in the TC apparatus completely devoid of any sustained largescale features [8]. If the outer cylinder is fixed, the onset of homogeneous turbulence will begin at an inner cylinder Reynolds number Rei > 50,000, defined by: Rei = xirid/m where xi is the positive inner cylinder angular velocity, ri is the inner cylinder radius, d is the annular gap, and m is the kinematic viscosity. The outer cylinder and average Reynolds numbers are defined to be Reo = xorod/m and Reave = 1/2(Rei + Reo). Luckily, much lower cylinder Reynolds numbers can reach the featureless turbulence

N. Fehrenbacher et al. / Experimental Thermal and Fluid Science 32 (2007) 220–230

221

Nomenclature U U V W u0 v0 w0 u0 v0 w0 QE RE(s) E(n) CH1 CH2

mean velocity in direction (cm/s) radial instantaneous velocity (cm/s) circumferential instantaneous velocity (cm/s) axial instantaneous velocity, cm/s radial fluctuation velocity (cm/s) circumferential fluctuation velocity (cm/s) axial fluctuation velocity (cm/s) radial RMS intensity (cm/s) circumferential RMS intensity (cm/s) axial RMS intensity (cm/s) Eulerian time cross-correlation function, dimensionless Eulerian autocorrelation function, dimensionless Eulerian frequency spectrum (cm2/s) signal from the single-component fiber optic probe signal from the two-component fiber optic probe

zone if both are counter rotated above a minimum value of Reave = 1000. In this case, it was found that Rei and Reo must be of the same order of magnitude, with the absolute value of the ratio Reo/Rei being slightly greater than unity. To be consistent with previous researchers [4–7], the ratio Reo/Rei was set equal to 1.4, placing the generated turbulence in the middle of the featureless regime. Many other experimental configurations have been employed for turbulent pre-mixed hydrocarbon flame studies that have ultimately failed to create an adequate homogeneous, steady, turbulent field. These configurations include fan-stirred media [9], rod-stabilized V-flames [10–12], and stagnation point burners [13,14]. Both stagnation point and V-flame burners suffer from having non-uniform turbulence properties along the flame brush area, creating a mean strain within the flow. Closed volume vessels such as fan-stirred bombs and the new cruciform burner [15] have a tendency of creating regions of inhomogeneity ahead of the flame making any theoretical assumptions based on a stationary flame outside a very small region somewhat questionable. Typically, high intensity turbulence can produce flame speeds as high as 400 cm/s [15]. These flames will remain in the homogeneous region of these closed vessels for only hundredths of a second due to their essentially zero mean flow. The TC apparatus on the other hand is able to sustain these stationary, uniform properties over many integral length and times scales while providing a nearly homogeneous region, 60 cm in length and more than 50% of the flame brush area, in the direction of flame propagation; a region close to ten times that of closed vessels. In addition, the TC apparatus is able to control the intensity of turbulence

Re d r i o

Reynolds number, dimensionless annulus width (cm) cylinder radius (cm) inner cylinder outer cylinder

Greek symbols s time, s TE Eulerian integral time scale (s) sE Eulerian micro time scale (s) KE Eulerian integral length scale (cm) h1 single-component fiber optic probe angle (rad) h2 two-component fiber optic probe angle (rad) x cylinder rotation rate (rad/s) m kinematic viscosity of the fluid at 298° and 1 atm (cm2/s)

independent of the mean flow while also minimizing the effects of heat loss and buoyancy by the appropriate selection of annulus width and cylinder Reynolds numbers. This attribute is particularly useful for potential stabilization of the flame for experimental studies of turbulentflame structure. Previous cold-flow turbulent measurements by Vaezi et al. [5] attempted to quantify the featureless turbulent regime in a TC apparatus by measuring the intensities in both the axial and circumferential directions. Laser Doppler Velocimetry (LDV) measurements found that the relative intensities were much higher in the circumferential direction than the axial, yet no quantitative analysis could be done at the time to determine the extent of anisotropy present. Due to the orientation of the apparatus and surrounding structures, further LDV measurements were incapable of capturing the radial velocity or any turbulent property related to the radial direction. The present experimental results extend those of Vaezi et al. [5] and previous researchers [4–6,16] to the radial direction using a newly developed two-component laser probe positioning system (LPPS). The objectives of this work are to quantify the featureless regime created in the TC apparatus for a large range of intensities typically seen in lean hydrocarbon–air flame reactions in terms of turbulence intensities, autocorrelation functions, normal stresses, frequency spectra, integral and micro time scales and estimations of the integral length scales in all three directions and to further identify and predict any isotropic features of the flow that may exist using axial–radial and axial–circumferential cross-correlation functions and Reynolds stresses.

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2. Experimental configuration The TC apparatus and experimental configuration is shown in Fig. 1. A pressurized supply of 99% pure medical air first enters a TSI 6-Jet Atomizer at 80 psig. Resulting axial mean flow velocities were controlled via an integrated pressure regulator to be between 3 cm/s and 7 cm/s for all experiments. The Atomizer seeds the flow with a mixture of silicon carbide (SiC) particles and pure olive oil. The mean particle diameters were 2 lm and 0.6 lm for silicon carbide and olive oil, respectively. The seeded air then enters the bottom of the apparatus through a 3/8-in. pipe and enters a chamber within the inner cylinder, where it stagnates against a flat plate and is distributed equally to the annulus through perforations of the inner cylinder. The inner cylinder is perforated with 900 2 mm diameter holes which are equally spaced in the bottom 6 cm of the inner cylinder. The seeded flow then travels up the annular region (1.1 cm) between the inner and outer cylinders that is open to the atmosphere at the top. The outer cylinder is composed of 3/8-in. thick Pyrex glass, which provides an optical window for LDV measurements. The inner cylinder is made of anodized black aluminum, which reduces the amount of reflections from the incident laser beams back into the probe. The radii ratio ri/ro = 0.877 between inner and outer cylinders was specifically defined to suppress

OUTFLOW PYREX OUTER CYLINDER

7.9 cm 1.1 cm

66 cm

FIBER-OPTIC LDV PROBES

both buoyancy and heat loss effects below 10% found by [16]. An aspect ratio, defining the ratio of the apparatus length to annulus width of 60, was chosen to be consistent with other investigators [4,14,17] to minimize end effects and to provide for a constant flame speed. The rotation of the two cylinders are independently controlled by a set of Danfross drivers and two 0.75-horsepower AC Baldor motors capable of counter rotation at rates up to 3450 revolutions per minute. The core of the LDV system is a water-cooled Lexel model 85 Argon Ion laser rated at 2.5 W and emits light of the wavelengths specific to Argon gas. From the aperture, the laser travels through a variable filter into a TSI model 9201 Colorburst multicolor separator. The Colorlink filters all the wavelengths except for 514.5 nm (green) and 488 nm (blue) and splits the laser into fiber optic lines, two for each color. One line of each color travels to the TSI model 9230 Colorlink multicolor receiver that shifts the frequency by 40 MHz. All signals are then routed to TSI 9800 series fiber optic probes. The output power of the probes is 61 mW/beam (514.5 nm) and 35 mW/beam (488 nm). This allows for two-component measurements of the measurement volume. The transmitting lens on each fiber optic probe has a focal length of 362.6 mm, having a fringe spacing within the measurement volume of 3.744 lm (green) and 3.547 lm (blue). The axial and radial velocities were measured simultaneously using a newly developed laser probe positioning system (LPPS) designed and built at UC Davis. For these measurements both the single and two-component probes were used, each supplying one set of beams of one color. One beam from each probe was frequency shifted by 1.5 MHz by downshifting the 40 MHz signal of the Colorlink by 38.5 MHz with the Flowsizer software. Both sets of beams were oriented in the vertical direction and focused onto a single point on the centerline of the inner cylinder for alignment. Using the LPPS, a 3D traversing system, a digital camera and a TV monitor each beam was focused onto the same point through the outer cylinder. The coincident measurement volumes were then moved back away from the inner cylinder until they were at the midpoint in the annular region (see Fig. 1). Due to the small degree of curvature of the Pyrex outer cylinder, there were no adverse beam deviation effects from this process. Twocomponent measurements for the axial and circumferential velocities were done using the two-component probe alone, where four beams exited the lens, two in the vertical plane and two in the horizontal plane. 3. Results and discussion

ANODIZED ALUMINUM INNER CYLINDER INFLOW Fig. 1. The Taylor–Couette apparatus and LDV configuration for radial and axial cold-flow turbulence measurements.

3.1. A region of transverse isotropy Turbulence intensities in the axial, radial and circumferential direction are compared to earlier measurements found by Vaezi et al. [5] in Fig. 2. These measurements are plotted versus the average cylinder Reynolds number

N. Fehrenbacher et al. / Experimental Thermal and Fluid Science 32 (2007) 220–230 90

QEuw ðsÞ ¼

Axial Radial Circumferential V.Vaezi et al. Axial V.Vaezi et al. Circumferential

80

50

QEvw ðsÞ ¼

40 30 20 10 0 2000

3000

ð2Þ

4000

5000

6000

7000

Reave

Fig. 2. Measured axial, radial and circumferential turbulence intensities versus Reave compared with measurements performed by previous researchers.

(Reave). Measurements were made approximately 6 cm above the perforated section of the inner cylinder in the center of the annular region. Each data point was created from the ensemble average of six separate experiments done at the same axial, radial, and circumferential position, each consisting of a minimum of 3000 data points. Because of the off axis arrangement of the laser probes, velocities calculated by the LDV processor had to be transformed to a orthogonal set of axis pertaining to the axial and radial directions shown in Fig. 1. The positive angle of each probe was measured relative to the horizontal to determine the coefficients of the transformation matrix shown below: 3 " # 2 1 1  cosðh2 Þðtanðh U cosðh1 Þðtanðh1 Þþtanðh2 ÞÞ 1 Þþtanðh2 ÞÞ 5 ¼4 tanðh2 Þ tanðh1 Þ W cosðh1 Þðtanðh1 Þþtanðh2 ÞÞ cosðh2 Þðtanðh1 Þþtanðh2 ÞÞ " # CH1  ð1Þ CH2

a

0.3 0.25

ð3Þ

Axial-Radial Reave = 1500 Axial-Radial Reave = 3000 Axial-Radial Reave = 4500

0.2

Axial-Radial Reave = 6000

0.15

Anisotropy begins Reave = 6000

0.1 0.05 0 -0.05 -0.1 0

0.01

0.02

0.03

0.04

0.05

t (s)

b Cross-Correlation Function

The error bars shown in the figure represent the standard deviation in data taken at each Reave. Both the axial and circumferential measured intensities correspond well with those previously found by Vaezi et al. [5] following a linear trend as a function of Reave. The radial intensities were also linear with a slightly larger slope, roughly equal to the axial intensity up to Reave = 4500 and coinciding with the previously measured circumferential intensities measured by Vaezi et al. [5] at Reave = 6000. The departure from the axial intensity is related to a shift from a transverse isotropic regime (isotropic under transverse rotations of the axial–radial plane) to an anisotropic regime found by examining the cross-correlation functions between the axial and radial directions. The Eulerian cross-correlation function for the axial– radial direction QEuw(s), is defined as:

v0 ðtÞw0 ðt þ sÞ v00 w00

In Fig. 3a the axial–radial cross-correlations versus s are plotted for all four values of Reave. Cross-correlation curves were calculated from the ensemble average of 6 experiments. Axial–radial cross-correlations are nearly isotropic below Reave = 6000, where ensemble averaged correlations are roughly zero. At higher Reave P 6000 however, velocity fluctuations show a distinct anisotropic exponential correlation at the same Reave intensities between the axial and radial directions. Axial–circumferential crosscorrelations shown in Fig. 3b were all anisotropic attaining distinct non-zero correlations at all Reave. Measured Reynolds stresses are plotted in Fig. 4 and reinforce previously mentioned cross-correlation predictions of an existing transverse isotropic regime in the axial–radial planes. Axial–radial Reynolds stresses remained zero within experimental uncertainties and nearly isotropic until Reave = 4500 after which they too became non-zero. Axial–circumferential Reynolds stresses were never zero

Cross-Correlation Function

Intensity (cm/s)

60

1000

u0 ðtÞw0 ðt þ sÞ u00 w00

where u 0 and w 0 are the radial and axial perturbation velocities and u 0 and w 0 are their respective root mean square intensities. The axial–circumferential cross-correlation is similarly defined as:

70

0

223

0.5 Axial-Circumferential Reave = 1500

0.4

Axial-Circumferential Reave = 3000

0.3

Axial-Circumferential Reave = 4500 Axial-Circumferential Reave = 6000

0.2

Anisotropic correlations at all Reave 0.1 0 -0.1 -0.2 0

0.05

0.1

0.15

0.2

τ (s)

Fig. 3. Ensemble-averaged Eulerian cross-correlation functions for the (a) axial–radial and (b) axial–circumferential directions for four different Reave.

224

N. Fehrenbacher et al. / Experimental Thermal and Fluid Science 32 (2007) 220–230 Axial Normal Stress Radial Normal Stress Circumferential Stress Axial-Radial Reynolds Stress Axial-Circumferential Reynolds Stess

7000

1

Autocorrelation Function

6000

Stress (cm2/s2)

5000 4000 3000 2000 1000 0

Direction of 0. 8

increasing Reave

0. 6 0. 4 0. 2 0

-1000 0

1000

2000

3000

4000

5000

6000

7000

Reave

-0 .2

0

Fig. 4. Measured axial, radial and circumferential normal stresses and Reynolds stresses in the axial–radial and axial–circumferential planes versus Reave.

increasing from 318 cm2/s2 to 2574 cm2/s2 over the measured range of Reynolds numbers. Normal stresses in each direction increased parabolically with Reave, as linear intensities would predict. The largest intensities and normal stresses created in the TC burner were 140.07 cm/s and 19,621 cm2/s2 respectively measured in the circumferential direction at Reave = 13,000. Normalized Reynolds stresses are displayed in Fig. 5 for both the axial–radial and axial–circumferential two-component measurements. Similar to Fig. 4, axial–radial normalized stresses were nearly zero and isotropic up until Reave = 4500 where they increased to 0.35u00 w00 at Reave = 6000. Axial–circumferential normalized Reynolds stresses remained fairly constant at 0.61 = v00 w00 suggesting that the anisotropic qualities present in the axial–circumferential planes in the featureless regime are likely to remain for all Reave. Error bars in Figs. 4 and 5 represent

Axial-Radial Normalized Reynolds Stress Axial-Circumferential Normalized Reynolds Stress

0.8 0.7 0.6

Normalized Stress

Circumferential Reave = 1500 Circumferential Reave = 3000 Circumferential Reave = 4500 Circumferential Reave = 6000 Axial Reave = 1500 Axial Reave = 3000 Axial Reave = 4500 Axial Reave = 6000 Radial Reave = 1500 Radial Reave = 3000 Radial Reave = 4500 Radial Reave = 6000

1. 2

0.5 0.4 0.3 0.2 0.1 0 -0.1 -0.2 0

1000

2000

3000

4000

5000

6000

7000

Reave

Fig. 5. Normalized Reynolds stresses in the axial–radial and axial– circumferential directions versus Reave.

0.02

0.04

0.06

0.08

0.1

τ (s)

Fig. 6. Autocorrelation functions in the of four different Reave. All curves are roughly exponential with time constants approximately equal to their respective integral time scales.

the standard deviation in respective measured intensities and resulting stresses. Fig. 6 shows the variations of the axial, radial, and circumferential ensemble Eulerian autocorrelation functions RE(s) versus time (s) at four different Reave. The Eulerian autocorrelation function is defined as: RE ðsÞ ¼

u0 ðtÞ  u0 ðt þ sÞ s  e TE u002

ð4Þ

where u 0 is the velocity fluctuation. All autocorrelation functions decayed exponentially with a time constant approximately equal to that of the integral time scale (TE). The arrow in this figure indicates the movement of correlation curves as Reave is increased. Fig. 7a clearly presents ensemble averaged autocorrelation trends in the axial direction versus Reave. Autocorrelation curves in each direction attained steeper exponential decays as Reave increased. More interesting were the differences between time correlations of different directions shown in Fig. 7b at Reave = 1500. Circumferential correlations were consistently broader in shape than those in the axial and radial directions. Since larger turbulence intensities narrow the shape of correlation curves, the broader distribution of correlations in the higher intensity laden circumferential direction must be geometry related. Turbulent scales in the TC apparatus are confined to the annular region, which are the largest in the circumferential (53 cm) and axial (66 cm) directions and smallest in the radial direction (1.1 cm). The ensemble averaged Eulerian integral time (TE) scales are plotted versus Reave in Fig. 8 on a log–log scale. Integral time scales were calculated by integrating each individual autocorrelation function, from s = 0 until the first time RE(s) = 0. Integral time scales fit an empirical formula of the form: T E ¼ aReb ave where a is in units of time

1.2 1 0.8 0.6 0.4 0.2 0 -0.2 0

0.05

0.1

0.15

0. 2

t (s)

Autocorrelation Function

b

1.2 Reave = 1500 Reave = 3000 Reave = 4500 Reave = 6000

1 0.8 0.6 0.4 0.2 0 -0.2 0

0.05

0.1

0.15

assuming that U =u0  1. Taylor’s Hypothesis assumes that the velocity fluctuations occurring at a fixed point are caused by the bulk motion of the passing turbulent flow with some mean velocity U [18]. Since TC turbulence is generated by shear in the circumferential, the relatively low axial mean velocity should have no impact or relation to any of the measured turbulent properties. Approximations made using the average cylinder speed instead were also found to be unacceptable, predicting radial length scales greater than the annulus width. Abdel-Gayed et al. [9] used an approximation for the mean velocity U of pffiffiffiffiffiffiffi ffi 8=pu00 by assuming that the flow was trivariant separable Gaussian, true of all isotropic flows. This form is more appropriate since turbulent properties are intensity related. However, since the equation is based on an isotropy and a mean generated turbulence, the constant of proportionality was not included and may be different depending on the spatial direction. Future two-point correlations should be studied to determine the correct constant. The Eulerian integral length scale is then approximated by the following equation: KE ¼ u00 T E

Fig. 7. (a) Autocorrelation functions demonstrating the broader distribution of time scales in the circumferential direction and (b) the decreasing nature of the largest time scales in the axial direction with Reave.

Axial Radial Circumferential Power (Circumferential)

0.1

ð5Þ

0.2

τ (s)

Power (Axial) Power (Radial)

Integral Time Scale (s)

225

KE ¼ U T E Axial Reave = 1500 Radial Reave = 1500 Circumferential Reave = 1500

-1.6605

ΤE = 9252.1Reave

0.01

ΤE = 366.64Reave

-1.3114

ΤE = 1900.6Reave

-1.5375

ð6Þ

The formula for the integral length scale KE, is an order of magnitude approximation, however any trends found are presumed to be representative of the flow. Calculated integral length scales are plotted on a log–log plot versus Reave for each direction in Fig. 9 and were fitted with the same form of empirical formula as the integral time scale where a now has units of (cm) and b is (non-dimensional). Interestingly, integral length scales in all three directions were found to decrease as Reave increased, the largest scales existing at the lowest Reave. One-dimensional energy spectra were calculated from the same data as in Fig. 3a. The Eulerian frequency spectrum is obtained from the cosine transform of the autocorrelation function where:

Axial Radial Circumferential

0.001 1000

10

10000

Reave Fig. 8. Integral time scales TE versus Reave in the axial, radial and circumferential directions.

and b is non-dimensional. Both constants are shown on each of their respective curves. All three directions showed a decrease in the integral time scale with increasing Reave. The integral length scales were estimated using a statistical approximation of Taylor’s Hypothesis, which relates the mean velocity and integral time scale to the integral length scale by the following simple product:

Power (Circumferential) Power (Radial)

Integral Length (cm)

a Autocorrelation Function

N. Fehrenbacher et al. / Experimental Thermal and Fluid Science 32 (2007) 220–230

Power (Axial)

ΛE = 1769.8Reave

ΛE = 14.835Reave

1

0.1 1000

-0.9781

ΛE = 62.808Reave

Reave

-0.4832

-0.6738

10000

Fig. 9. Integral length scales in the axial, radial and circumferential directions versus Reave.

226

N. Fehrenbacher et al. / Experimental Thermal and Fluid Science 32 (2007) 220–230 1.E-01

-5/3 slope

Axial Radial Circumferential Power (Circumferential) Power (Radial) Power (Axial)

0.025

Micro Time Scale (s)

E(n) (m^2/s )

1.E-02

1.E-03

1.E-04

0.02

τE = 305.72Re ave-1.3154 0.015

τE = 53.856Re ave-1.1203 0.01

0.005

1.E-05

τE = 23.231Re ave-1.0471 0 0

2000

4000

1.E-06 1

10

100

Fig. 10. A typical measured frequency spectrum for Reave = 4500 in the circumferential direction exhibiting a 5/3 slope for more than 1.5 decades.

EðnÞ ¼ 4u002

Z

1

dtRE ðtÞ cosð2pntÞ

ð7Þ

0

and where n is the frequency (in Hz). A typical frequency spectrum for the circumferential direction at Reave = 4500 is presented in Fig. 10. At high frequencies, the data exhibits the Kolmogoroff 5/3’s slope indicating that the circumferential direction acquires some additional isotropic properties. Trends for all power spectrum of increasing Reave show a shift of energy from the larger scales to the smaller scales as the inertial subrange is pushed out to higher frequencies with the value of E(n) as n ! 0 decreasing in proportion with TE. Hinze [18] found the value for this limit to be equal to 4TE, which agrees quite well with measured data. Fig. 11 compares frequency spectra measured in each of the three directions at Reave = 1500. At lowReave < 4500 energy in the circumferential direction was significantly larger than that of the axial and radial direc-

8000

Reave

1000

n (Hz)

6000

Fig. 12. Micro time scales in the axial, radial and circumferential directions versus Reave.

tions followed by the radial and axial direction respectively. At higher Reave P 4500, frequency spectra in different directions became more or less indistinguishable and energy contained at the smallest and largest scales between each direction were approximately equal. Fig. 12 presents the Taylor micro scales found from integrating the frequency spectrum E(n) where:  2Z 1 1=2 2p sE ¼ 002 dnEðnÞn2 ð8Þ u 0 Similar to trends for the integral time scale, Taylor micro scales were roughly fitted with an empirical power law of the form sE ¼ aReb ave where error bars represent the standard deviation in calculated data. The constants a and b have the same units as those for the integral time scales. Circumferential ensemble averaged micro scales were interestingly found to be the largest, followed by the axial and radial micro scales. Following Eqs. (7) and (8), the smallest scales created in the TC apparatus should be contained within the circumferential direction. Fig. 13 Axial Radial

0.014

Circumferential

Micro Time Scale (s)

^

0.012 0.01 0.008 0.006 0.004 0.002 0 0

1000

2000

3000

4000

5000

6000

7000

Reave

Fig. 11. Trends for frequency spectra in the axial, radial, and circumferential directions at Reave = 1500.

Fig. 13. Smallest recorded micro time scales created in the circumferential, radial, and axial directions versus Reave.

N. Fehrenbacher et al. / Experimental Thermal and Fluid Science 32 (2007) 220–230

presents the smallest micro time scales taken from one of the six individually sampled runs at each Reave. The smallest created micro scales are now shared between the circumferential and radial directions. One possible reason for smaller relative radial scales may be due the way in which the radial and circumferential energy is distributed. Circumferential energy was found to have a much broader range of scales seen in both frequency spectra and autocorrelations functions at the same Reave. The radial scales however were limited by the annulus width, in which a relatively narrower band of scales were permitted. Unless additional energy in the circumferential direction is forced to only create smaller scales, the broader distribution may make it possible for relatively larger micro scales to exist. 3.2. Oscillating phenomena

^

Oscillating phenomena were found in frequency spectra, and integral and micro time scales in which relative magnitudes of each periodically oscillated from one sequential experiment to the next. A series of six sequential measurements of 3000 data points each were made to capture at least one full period of this oscillatory behavior in each direction for frequency spectra and integral and micro time scale calculations. Fig. 14 attempts to illustrate this phenomenon in which half a frequency spectra oscillation in the circumferential direction at Reave = 1500 is shown. Arrows included in the figure represent the movement of the frequency spectra during half a period (maximum to minimum amplitude). Spectra in the circumferential and axial directions were found to show the greatest oscillatory motion at both low and high frequencies with a slight shift in the inertial subrange, suggesting that spectral oscillations

227

would have their largest impact on the integral and micro time scales. The total energy calculated from integrating each of the six consecutive spectra are included in the inset to show where each of the three chosen circumferential frequency spectra lye in the half period oscillation. Of particular interest were the phase differences of these spectral oscillations between each of the three directions. For Reave < 4500, axial and radial spectral oscillations were consistently 180° out of phase with the circumferential direction. Energy minima present in circumferential direction were accompanied by energy maxima in both the axial and radial directions at the same time suggesting that circumferential energy was periodically pumped to the other two directions. As Reave increased, spectra oscillations in the circumferential and radial direction became increasingly damped. At Reave = 6000, oscillations in all three directions were nearly zero as energy was more or less evenly distributed and axial and radial spectra oscillations were completely out of phase. The frequencies at which the spectra oscillations occurred were not found to be artificially created by the cylinder rotation rates or any multiple there of. Fig. 15 presents the circumferential micro scale oscillations. At the lowest Reave = 1500, micro scale oscillation amplitudes are their greatest approaching 12.7 ms, with the smallest micro scales of 6.3 ms and 7.7 ms during the second and forth experiments. As Reave increased, micro scale oscillations became increasingly damped, reducing amplitudes at Reave = 6000 to less than 0.3 ms from the original 12.7 ms in the circumferential direction. Micro scale oscillations were also found to occur in both the radial and axial directions. Radial micro scale oscillations were similar to the circumferential oscillations in that their amplitudes decreased with increasing Reave hinting that they to may redistribute some of their energy to either the axial or circumferential directions. Fig. 16 displays the micro scale oscillation amplitude of all three directions. Both circumferential and radial amplitudes decreased exponentially with Reave, Reave = 1500 Reave = 3000 Reave = 4500

0.035

Reave = 6000

Micro Time Scale (s)

^

0.03 0.025 0.02 0.015 0.01 0.005 0 0

Fig. 14. Circumferential Spectra oscillations at Reave = 1500. A half oscillation is shown in which larger integral and micro time scales are created as energy is lost to the axial and radial directions.

1

2

3

4

5

6

7

Experiment Number

Fig. 15. Micro scale oscillations in the circumferential direction versus experiment number.

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N. Fehrenbacher et al. / Experimental Thermal and Fluid Science 32 (2007) 220–230

a

Axial

Circumferential

0.1

0.012

Circumferential energy decrease, axial increase

0.09

Integral Time Scale (s)

Micro Time Scale Oscillation Amplitudes (s)

Circumferential Reave = 1500 Axial Reave = 1500 Circumferential Integral Time Mean Axial Integral TimeScale Mean

Radial

0.014

0.01

0.008

0.006

0.004

0.08 Circumferential energy increase, axial decrease

0.07 0.06 0.05 0.04 0.03 0.02 0.01 0

0.002

0

1

2

3

4

5

6

7

Re ave

0 2000

4000

6000

8000

Reave

Fig. 16. Micro scale oscillations amplitudes in the circumferential, radial, and axial directions versus average Reynolds number Reave.

indicating that they are initially gaining and losing more energy in their micro scales than that of the axial direction. Axial oscillations surpassed those in the radial direction at Reave = 4500 after which they too became damped. Micro scale oscillations between each of the three directions were roughly equal for Reave P 6000. Fig. 17a and b further depict the energy pumping process through two-component integral time scale measurements taken in the axial–circumferential and axial–radial experiments. Integral time scale oscillations in the axial and circumferential directions for Reave = 1500 were found to be consistently 180° out of phase and are shown in Fig. 17a. When circumferential integral time scales were at their maximum (energy minimum), axial scales were flatted to a local minimum region (energy maximum). As Figs. 7a, 8 and 14 remind us; energy addition creates smaller respective integral time scales. Trends for integral scales oscillations with increasing Reave found that axial and circumferential oscillations were in complete phase agreement below Reave = 6000, after which oscillations were randomly in and out of phase. Axial–radial integral time scale oscillations at all Reave are displayed in Fig. 17b. Oscillations between the axial and radial directions were consistently in phase with each other when the largest fluctuations in circumferential integral time scales occurred, and distributed energy between the three directions was most unequal. As Reave increased, radial and axial oscillations slowly became out of phase as circumferential oscillation amplitudes decreased and energy was more or less evenly distributed between each of the three directions. 3.3. Error analysis The scatter in the turbulence intensity, integral and micro time scale measurements represented by the error

b

Axial Reave = 1500 Radial Reave = 1500 Axial Reave = 3000 Radial Reave = 3000 Axial Reave = 4500 Radial Reave = 4500 Axial Reave = 6000 Radial Reave = 6000

0.04 0.035

Integral Time Scale (s)

0

Complete phase agreement

0.03 0.025 0.02 90% complete phase agreement

0.015 0.01

66% phase of the time

0.005 Completely out of phase

0

0

1

2

3

4

5

6

7

Experiment Number

Fig. 17. Simultaneously measured integral scale oscillations in the (a) axial and circumferential directions at Reave = 1500 and (b) the axial and radial directions at all Reave.

bars in Figs. 2, 4, 5, 8, 9, and 11 are attributed to several sources of uncertainty. First, the controller for the Baldor motors is accurate to within 17 rpm resulting in a Reave uncertainty of 208. This uncertainty in Reave manifests a 2.5 cm/s average uncertainty in the intensity u 0 . Second, the LDV signal processor is accurate to within 0.1% of the measured velocity, giving a maximum error of 0.074 cm/s at the highest intensities measured. Third, two-component axial–radial measurements required the use of the LPPS, whose probe angles are only accurate to within 1 degree of the angle measured resulting in a 0.8% accuracy in axial and radial velocities and an error of 0.82 cm/s. Forth, both the silicon carbide and olive oil cannot follow the flow perfectly due to their finite mass and inertia. A maximum uncertainty of 1% (the greater of the two) in measured intensities results, corresponding to an error of 0.74 cm/s at the highest intensity. Finally, a purely statistical uncertainty based on individual samples of 3000 data points resulting in an approximate uncertainty of 2% or 1.48 cm/s. The total uncertainty in measured intensities is then roughly 6.43 cm/s (about 4.5% of the highest inten-

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sities obtained). This value is obtained by summing the aforementioned contributions of intensity uncertainties and then multiplying the result by two to account for plus and minus deviations. The predicted level uncertainty of 6.43 cm/s in u 0 agrees well with the magnitudes of measured turbulence intensity scatter indicated by the error bars in Figs. 2, 4, 5, 8, 9, and 11. 4. Concluding remarks The present work further quantifies the featureless turbulent regime within the Taylor–Couette apparatus designed for future turbulent flame propagation studies [6,19]. Newly measured radial intensities were found to be linear, similar to trends found earlier in the axial and circumferential directions by Vaezi et al. [5]. Unlike the axial and circumferential intensity data whose lines are approximately parallel, the radial intensities were roughly equal to axial data at lower Reave and approached the circumferential data at higher Reave. It is predicted that a radial–circumferential transverse isotropic regime exists at Reave > 6000 where the radial and circumferential intensities are projected to merge. This assumption is supported by [5] whose previous circumferential measurements overlap with recently measured radial intensities recorded at Reave = 6000. Both cross-correlation functions and Reynolds stresses similarly showed a region of transverse isotropy in the axial–radial planes located between Reave = 1500 and Reave = 4500, after which Reynolds stresses became non-zero and cross-correlations showed an exponential increase near the s = 0 axis. Signs of anisotropy were found to exist in the axial and circumferential cross-correlations and Reynolds stresses at allReave, however calculated spectra for most individual samples exhibited a 5/3 slope suggesting that the flow still remains to have some isotropic properties. Sequentially measured frequency spectra also illuminated a very interesting phenomenon characterized as spectra oscillations. These oscillations manifested themselves in all physical turbulent properties related to the energy spectrum, namely time and length scales. The circumferentially generated shear was found to periodically supply energy from the circumferential direction through all scales to the axial and radial directions due to an initially unstable state. As energy was additionally supplied through larger Reave, oscillation amplitudes decreased and turbulent scales and frequency spectra in each direction became similar. The radial direction was also found to supply small amounts of energy to the axial direction as well at low Reave < 4500. At higher Reave P 6000, all three directions randomly distributed energy between each other. Integral and micro time scales were found to decrease with Reave independent of direction in which empirical relations were found. The largest time scales were found to be 93 ms, 35 ms, and 34 ms in the circumferential, axial, and radial directions respectively. Calculated ensemble aver-

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aged micro time scales provided interesting results in that circumferential scales were larger than those in the axial direction, resulting from spectra oscillations. Micro scale oscillations were found to be the largest in the circumferential direction where released energy was directly supplied to the axial and radial scales in phase. The smallest time scales created were divided between the circumferential (Reave 6 3000) and radial (Reave P 4500) directions. The overall smallest time scales created in the TC burner at a Reave = 6000 were 2.5 ms, 2.2 ms and 1.8 ms in the axial, circumferential, and radial directions respectively, roughly one and a half orders of magnitude smaller than their respective integral time scales. Integral length scale approximations were made using a form of Taylor’s Hypothesis, which relates the Eulerian integral length scale to the mean velocity feeding the turbulence along with the integral time scale. Abdel-Gayed et al. [9] took it a step further and replaced the mean velocity with the intensity, which makes more sense since the turbulence is shear and not mean generated. Though the AbdelGayed et al. [9] approximation presents a more meaningful relation to the size of the turbulent scales, his equation is still based off of a statistical mean quantity. It was the choice of the authors herein to disregard the proposed constant of proportionality in [9] and the resulting relation in Eq. (6) is made to be only an order of magnitude approximation. The resulting integral length scales were found to decrease with increasing Reave, similar to trends in integral and micro time scales, the radial direction having its largest scale structures 0.41 cm in length; on the order of half the annulus width (0.55 cm) predicted by previous researchers [4,6]. The largest scale structures created in the TC apparatus were found in the circumferential direction decreasing from a maximum value of 1.4 cm to 0.35 cm over the tested range of Reave. The results from this work go to quantify the turbulence structures generated in a Taylor–Couette (TC) apparatus over a large range of intensities typically seen in lean hydrocarbon–air mixtures. Experimental results in [5] show that the TC apparatus contains a homogeneous region that extends more than 60 cm in the axial direction and more than half the annulus width in the radial direction. Current results extend this work to show that in addition to creating an extremely large region of homogeneous flow, the TC apparatus also acquires a region of transverse isotropic flow in the axial and radial directions between 1500 6 Reave 6 4500. Measured intensities, integral and micro time scales and integral length scales all strongly suggest that an additional transverse isotropic regime may exist for Reave > 6000 in the radial and circumferential directions. Power spectra slopes in each direction following the 5/3 power law further define the TC generated turbulence in the featureless regime as attaining other desirable features characteristic of isotropic flow. Altogether, the TC apparatus has proved itself as a novel means of generating ideal turbulent characteristics suitable for flame propagation studies.

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