GPE-EPIC
2nd International Congress on Green Process Engineering 2nd European Process Intensification Conference 14-17 June 2008 - Venice (Italy)
INTENSIFYING THE TURBULENT KINETIC ENERGY DISSIPATION RATE BY REDISTRIBUTING STREAMWISE VORTICITY
C. HABCHI(1,2), T. LEMENAND(1), D. DELLA VALLE(1), H. PEERHOSSAINI(1,*)
♦ (1) Thermofluids, Complex Flows and Energy Research Group
Laboratoire de Thermocinétique de Nantes (LTN), CNRS UMR 6607 École Polytechnique de l’Université de Nantes, rue Christian Pauc, B.P. 50609 44306 Nantes, France
♦ (2) Agence de l’Environnement et de la Maîtrise de l’Énergie (ADEME) 20 avenue du Grésillé, B.P. 90406
49004 Angers, France [email protected]
Abstract. Streamwise vortices are used in many applications for mixing processes, fast chemical reactions and heat transfer enhancement. In this work we study experimentally and numerically the effect of vorticity redistribution on the turbulent kinetic energy (TKE) dissipation rate in a modified-geometry high-efficiency vortex (HEV) mixer, where the vortex generator arrays are periodically rotated by 45° to better distribute the vorticity field. Attention focuses on the evolution and distribution of TKE dissipation rate since it describes quantitatively the mechanism of turbulent micromixing, which controls fast chemical reactions. It is found that redistribution of the vorticity field locally intensifies turbulent micromixing relative to the classical HEV mixer, leading to a global intensification of 20%. In addition, the alternating vortex generator arrays yield better homogenization of the turbulence field in the mixer.
Key-words: micromixing, turbulent energy dissipation, process intensification, vortex generators, static mixer-reactor, multifunctional heat exchanger.
INTRODUCTION Streamwise vorticity can be generated in two main ways:
1- Flow streamline curvature (Görtler, Dean roll-cells) where centrifugal force generates the streamwise vortices. Heat transfer intensification in these flows has been studied extensively by Peerhossaini1; mixing by Görtler instability, which is more appropriate for open flows, has been studied by Girgis and Liu2, among others.
2- In the domain of online mixers and multifunctional heat exchangers, longitudinal vortices can readily be generated by vortex generators such as inclined tabs in the present case. This is the case for instance in the high-efficiency vortex mixer (HEV) (Lemenand et al. 3 and Mokrani et al.4). Here several arrays of vortex generators produce a complex vortex system consisting of steady longitudinal counterrotating vortex pairs (CVP) and transient hairpin vortices close to the tabs5, which are modified, further downstream, by the subsequent arrays.
In this work we study experimentally and numerically the effect of vorticity redistribution on the turbulent kinetic energy dissipation rate in a modified HEV geometry compared to the classical HEV geometry. The classical HEV consists of a circular pipe in which aligned tab arrays are fixed. Each array is composed of four trapezoidal inclined tabs located at 90° in the tube cross section. In the modified- geometry HEV, each tab array is rotated by 45° with respect to its neighbor, leading to a periodic rotation.
Experiments were carried out by measuring instantaneous velocities using laser Doppler anemometry (LDA). Numerical simulations of the velocity and turbulence field in the flow are used to supplement the study of turbulent flow structures in this new geometry. This work focuses on the distribution of turbulent kinetic energy dissipation rate as a parameter quantifying turbulent mixing on the Kolmogorov scale.
EXPERIMENTAL STUDY
Apparatus and data acquisition
The test section used in this work is a straight circular pipe of 20 mm inner diameter along which seven rows of vortex generators are fixed. Each vortex generator array is rotated 45° with respect to its neighboring array. Each of seven arrays consists of four diametrically opposed trapezoidal mixing tabs5. Figure 1 shows schematically both aligned and alternating tab array geometries.
The tabs are inclined 30° with respect to the tube wall. Their role is to generate longitudinal vortices that enhance the turbulence and radial transfer over that in the empty duct. The test section is preceded by a preconditioner (2000 mm straight Plexiglas pipe) to produce a fully developed turbulent flow at the test section inlet, and is followed by a postconditioner (200 mm straight Plexiglas pipe). Care was taken that the connections between the different elements do not disturb the flow. A safety valve is added to the circuit as well as a pulsation absorber whose role is to limit the pressure fluctuations produced by the pump and thus ensure continuous and stable flow in the test section.
The temperature of the working fluid (water) is maintained constant at 298 K, so there is no significant effect on the turbulence of the physical properties of the working fluid. The experiments are carried out in a fully developed steady turbulent flow with Reynolds numbers 7500, 10000, 12500 and 15000. All measurements are taken on radial profiles 3 mm downstream of each array.
The measurements are performed using a Dantec LDA system (laser Doppler anemometry) equipped with a 10 W argon-ion laser source and two BSA-enhanced signal-processing units (57N20 BSA and 57N35 BSA enhanced models). A lens of 160 mm focal distance is used. A lightweight precision three-dimensional traversing mechanism controlled via PC is used to displace the measuring volume. The data-acquisition rate was 1-4 kHz and the sampling particle number was 30000.
Results and discussion of LDA measurements
To ensure the repeatability of LDA measurements, experiments were iterated four times for radial profiles at different positions for Reynolds number 15000. The relative standard deviation for the mean and RMS fluctuating velocities depends on the location of the measurement volume. It is maximal in the near-wall region, a low-velocity zone, and minimal in the flow core region. The global mean standard deviation is 6%
for the mean velocities and 5% for the fluctuating velocities.
Figure 2 shows profiles for mean and fluctuating axial velocities and a schematic description of the hairpin- like structures in the flow.
When the fluid flows through the tab array, the mean axial velocity accelerates due to the decrease in the flow cross-section. In the bulk region fory/R>0.45, the mean velocity is high and quasi-constant and the RMS velocity is low. In the “shear” zone located near the tab edge,0.3<y/R<0.45, an important velocity gradient and a peak in the fluctuating velocities can be observed. This peak coincides with the head of
Aligned arrays in HEV Alternating arrays in HEV Aligned arrays in HEV Alternating arrays in HEV Aligned arrays in HEV Alternating arrays in HEV
Figure 1. Aligned and alternating rows of vortex generators in HEV static mixer
-0.10.0 0.2 0.4 0.6 0.8 1.0 1.2 1.3
0.0 0.2 0.4 0.6 0.8 1.0
Reverse flow
Flow region
Reverse vortex
Radial position, y/R
Mean axial velocity, Um (m.s-1) Hairpin structure
head
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40
0.0 0.2 0.4 0.6 0.8 1.0
Reverse vortex Hairpin structure
head
Radial position, y/R
Fluctuating axial velocity, u (m.s-1)
Hairpin head
Reverse vortex
Hairpin neck Flow
Mixing tab
Wall Mixer centerline
(a) (b) (c)
Figure 2. (a) Mean and (b) fluctuating axial velocities 3 mm downstream of seventh tab array for Re= 15000, and (c) schematic of hairpin structures and reversed vortex
Figure 2). The second inflection point is a sign of a reversed vortex due to the stagnation flow behind the mixing tab in the wall region0< y/R<0.3. Mean and fluctuating axial velocities increase with Reynolds number, but have similar profiles. The flow seems to be established beyond the fourth array, so that the profiles shown at the seventh row correspond to the equilibrium situation.
Because of the circular geometry of the test section, it was not possible to measure the radial velocity components. However, radial velocities are conveniently described by the numerical simulation presented in the next section.
To calculate the turbulent energy dissipation, the determination of the convection velocity Uconv is required. In the present study, the convective velocity was estimated by using the method proposed by Van Doorn8 and successfully used by Lemenand et al.3, Mokrani et al.4 and Mohand Kaci et al.7 for the HEV, assuming the local isotropy of the turbulence in a one-dimensional mean flow:
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛ +
= 2 1 5 22 U U u
Uconv (1)
Figure 3 shows the longitudinal evolution of turbulence intensity defined by
Uconv
/ u
I = measured 3.5 mm from the tube wall (distance of tabs’ extremity from the wall) and at the centerline, in both aligned and alternating rows of vortex generators for Reynolds number 15000. It can be observed that the turbulence intensity in alternating arrays oscillates between that of aligned arrays.
Assuming a perfect tracer injected at the inlet: at line 1 the tracer trajectory passes periodically through high-turbulence zones, while at line 2 in aligned arrays it remains in the lower-turbulence zone between two tabs. This periodic oscillation is due to the periodic redistribution of the vorticity field in the mixer volume, and it enhances the homogeneity in the mixing process.
In the present study, the energy dissipation rate is calculated from the following expression, by using Batchelor’s9 model based on dimensional analysis:
( )
Λε 2
3
u2
=A (2)
whereΛ represents the scale of the energetic structures, which is the upper limit of the inertial domain in Kolmogorov energy cascades. The constantA,, determined by Mokrani et al.4 as 1.85, was used later by Lemenand et al.3 and Mohand Kaci et al.7 for turbulent flow in aligned rows of vortex generators. In homogenous turbulent flows, Λ is defined as:
( ) ( )
( )
x dru r x u x
∫
∞u +=
0
Λ 2 (3)
EvaluatingΛ using equation (3) requires the spatial autocorrelation (following Batchelor9). In the present study, the spatial macroscale was derived from the integral time scale given in equation (4):.
( ) ( ) ( )
t dT uT t u t
∫
u∞ +
=
0
ξ 2 (4)
Hence, the spatial macroscale reads:
Uconv
ξ
Λ= (5)
Line 2 Line 2 Line 2 Line 1 Line 1
(a)
Line 2 Line 2 Line 1 Line 1
(b)
0 1 2 3 4 5 6 7
0 5 10 15 20 25 30 35 40 45 50
Turbulent intensity, I (%)
Array number
Reactor Centerline Reactor Centerline (Mohand Kaci et al. [2006]) (a) Line 1 (b) Line 1 (Mohand Kaci et al. [2006]) (a) Line 2 (b) Line 2 (Mokrani et al. [2008])
Figure 3. Evolution of turbulence intensity at 3.5 mm from tube wall and at centerline in (a) aligned and (b) alternating arrays. Triangles indicate
locations of vortex generators; Re=15000
Figure 4 represents radial profiles of the turbulent energy dissipation
ε
calculated from equation (2), 3 mm downstream of the seventh tab array, in both aligned and alternated rows geometries. It can be observed that the presence of vortex generators strongly intensifies the turbulence energy dissipation in both geometries. In the alternating rows of mixing tabs,ε
is much greater than in classical aligned rows. The maximum energy dissipation is located near the tabs’ extremity, near the top of the hairpin-like structures visualized by Mokrani et al.4 in aligned rows of vortex generators (Figure 2). These turbulent structures destabilize the flow by adding high-frequency fluctuations.When passing the first tab array, a maximum value appears for the position near the tab extremity, with a large gradient.
The turbulent energy dissipation
ε
was also evaluated for the other Reynolds numbers (7500, 10000 and 12500); no qualitative effect of Reynolds number onε
was found in the range studied, butε
was found to increase globally with Re.NUMERICAL STUDY
The numerical simulations were carried out by using the finite volume CFD code Fluent®, which is double-precision, segregated, and implicit. Using the segregated approach, the governing equations are solved sequentially. To obtain second-order accuracy, quantities at the cell faces are computed using a multidimensional linear reconstruction approach. Pressure-velocity coupling is achieved by the SIMPLE algorithm. The mesh is a cell-centered finite volume discretization
Numerical procedure
When studying aligned rows of vortex generators, Mohand Kaci et al.7 showed that the more accurate turbulent model for the flow pattern in this geometry is the standard k−ε model, associated with a two- layer model to compute the wall region. The same model is used here to analyze fluid dynamics in alternating rows of vortex generators.
At the mixer inlet, the 1/7 power-law velocity profile is imposed. The turbulence kinetic energy and its dissipation rate at the entrance are based on 10% turbulence intensity.
Due to the symmetries, the studied geometry was reduced to 1/8 of the tube cross section. An unstructured three-dimensional mesh with hexahedral volumes is adopted and refined at all solid boundaries. The code was tested for several mesh densities and refined until no effect on the numerical results was detected. The appropriate meshing contains 724174 cells. Mesh quality was quantified by using equiangle skew parameter. Results show that 78.1% of the cells are of very good quality, 18.4% of good quality and 3.43% of mean quality. Series of simulations were carried out by varying the convergence stop-criterion values from 10-4 to 10-9. Beyond a convergence criterion of 10-7 there are no significant changes on the turbulence quantities; however, a convergence criterion value of 10-9 was used in numerical simulations. To validate the simulation accuracy, the global mean energy dissipation εglobal obtained from numerical simulations is compared to that obtained from the equation for pressure drop.
Results were in good agreement, with a relative standard deviation of 10-4. Results and discussion of numerical simulations
Figure 5 shows experimental and numerical results for different parameters and Reynolds number 15000.
Numerical simulations are in fair agreement with the LDA measurements of the dissipation rate. The main quantitative difference is located in the near-wall region due to the high noise in the LDA signal in this zone.
0.0 0.2 0.4 0.6 0.8 1.0
0 5 10 15 20 25 30 35 40 45 50 55
Radial position, y/R Turbulent dissipation rate, ε (m2.s-3)
Aligned arrays (Mohand Kaci et al. [2006]) Alternating arrays
Figure 4. Radial profiles of turbulent energy dissipation rate 3 mm downstream of the seventh tab array in aligned and
alternating rows for Re=15000
As generally observed on wing edges, a counterrotating vortex pair (CVP) is generated at each side of the tab. A common radial flow is induced, transporting low-momentum fluid from the near-wall region toward the high- momentum fluid at the mixer centerline. This mechanism greatly enhances the radial transfers.
While fluid passes through the tab wake zone, it is stretched and folded over due to the swirling motion induced by CVP. This mechanism occurs also over the tabs further downstream. A second type of periodic vortex develops from the upper tip of the mixing tabs, riding on the top of CVP and flowing downstream. These vortices, called hairpin-like structures, have been the subject of several studies (Dong and Meng6, Gretta and Smith5). Figure 6 shows the CVP and hairpin-like structures developed upon the tab, slightly downstream of the first array, which are obtained from numerical simulations, and compared with a LIF visualization performed by Mokrani et al.4. This figure shows that the common upflow induced by CVP ejects the near-wall fluid and forms the head (arch) of the hairpin-like structures, which further interact with CVP to intensify the mixing process. At the splitting point,(white arrow in Figure 6), the common upflow velocity induced by the CVP is divided into two opposite tangential velocities in the flow cross section.
It is clearly seen that the head of the hairpin vortices corresponds to positions of higher turbulent-energy dissipation, confirming the peaks of kinetic energy dissipation profiles obtained from LDA, which are located at the
tab height. Numerous studies5, 6 have shown that while flowing downstream of the tabs, CVP are transformed into hairpin-like structures that further on in the flow become the main contributors to the turbulent mixing process. Figure 7 shows contour maps for energy dissipation rate 3 mm downstream of
0.0 0.2 0.4 0.6 0.8 1.0
0 10 20 30 40 50 60
Numerical results Experiments
Line 1
Axial coordinate, z/L Turbulent dissipation rate, ε(m2.s-3)
(a)
0.0 0.2 0.4 0.6 0.8 1.0
0 10 20 30 40 50 60
Numerical results Experiments
Line 2
Axial coordinate, z/L Turbulent dissipation rate,ε(m2.s-3)
(b)
Figure 5. Experimental validation of longitudinal distribution of turbulence energy dissipation ε for line 1 (a) and line 2 (b) schematically represented, Re=15000
Counter-rotating Vortex pair Hairpin head
Splitting point
(a) Counter-rotating (b) Vortex pair Hairpin head
Splitting point
(a) Counter-rotating (b) Vortex pair Hairpin head
Splitting point
(a) (b)
Figure 6. Interaction of hairpin-like structures with counterrotating vortex pair (CVP), at a cross section 3 mm downstream of first array: (a) present numerical results for Re= 15000 and (b) LIF measurements by Mokrani et al.
(2008) for Re=1000
Figure 7. Evolution of turbulence energy dissipation 3 mm downstream of tab arrays (m2.s-3), Re= 15000
each row of vortex generators: maximum values of ε are at a distance from the wall equal to that of the tab tips, where the hairpin-like structures are generated.
CONCLUSIONS: EFFICIENCY OF VORTICITY REDISTRIBUTION Figure 8 compares the energy dissipation rate εmean profile
along the mixer, at the level of the tab tops, in the aligned and alternating configurations. When the flow encounters the first array, in both cases, εmean sharply increases and periodically attains the maximum value at each tab row.
Minimum values are located upstream of each tab. The advantage of the alternating geometry is clear in this figure: locally εmean increases more than twice at the second tab array, and by a factor 1.5 for the following tab arrays, compared to the aligned geometry. The global efficiencyζ , defined as the difference between the mean dissipation rates in the alternating geometry relative to the aligned tabs, is represented in Figure 9 for Reynolds
numbers from 7500 to 15000. It can be noted that the efficiency increases with the Reynolds number. For 7500
Re= the efficiency is still low
(
ζ =5%)
, but as the Reynolds number increases it reaches 5%=18.
ζ forRe=15000. This tendency of ζ suggests that it will increase for higher Re; however, the present study was limited to Re=15000 due to pumping system limitations. Global energy dissipation is strongly intensified by the redistribution of streamwise vortices caused by the alternating rows of vortex generators.
REFERENCES
1 Peerhossaini H., On the effect of streamwise vortices on wall heat transfer, Compact heat exchangers for process industries, ed. R. Shah, Begell House Publishers, New York, 1997.
2 Girgis I.G. and Liu J.T.C., Mixing enhancement via the release of strongly nonlinear longitudinal Görtler vortices and their secondary instabilities in the mixing region, J.
Fluid Mech. 468 (2002) 29-75.
3 Lemenand T., Dupont P., Della Valle D. and Peerhossaini H., Turbulent mixing of two immiscible fluids, ASME J. Fluids Eng. 127 (2005) 1132-1139.
4 Mokrani A., Castelain C. and Peerhossaini H., Experimental study of the influence of the rows of vortex generators on turbulence structure in a tube, Chem. Eng. Proc. (2008) in press.
5 Gretta W.J. and Smith C.R., The flow structure and statistics of a passive mixing tab, J. Fluids Eng. 115 (1993) 255-263.
6 Dong D. and Meng H., Flow past a trapezoidal tab, J. Fluid Mech. 510 (2004) 219-242.
7 Mohand Kaci H., Lemenand T., D. Della Valle and Peerhossaini H., Enhancement of turbulent mixing by
embedded longitudinal vorticity: a numerical study and experimental comparison, In Proc. FEDSM ASME, Miami (2006).
8 Van Doorn M., On Taylor’s hypothesis in turbulent shear flows, Internal note 811123, University of
Missouri-Rolla, 1981.
9 Batchelor G.K., The theory of homogeneous turbulence, Cambridge University Press, 1953.
0.0 0.2 0.4 0.6 0.8 1.0
0 1 2 3 4 5 6 7 8 9 10
Dimensionless axial coordinate, z/L Alternated arrays
Aligned arrays (Mohand Kaci et al. [2006])
Turbulent dissipation rate, εmean (m2.s-3)
Figure 8. Comparison of the performances of aligned and alternating rows of vortex generators, Re=15000
7500 10000 12500 15000
0 2 4 6 8 10 12 14 16 18 20 22
ζ (%)
Reynolds number, Re
Figure 9. Turbulence energy dissipation efficiency
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛ −
=
aligned aligned alternated
ε ε ζ 100ε
