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The Dean instability in power-law and Bingham fluids in a curved rectangular duct
J. Non-Newtonian Fluid Mech. 165 (2010) 163–173
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The Dean instability in power-law and Bingham fluids in a curved rectangular duct H. Fellouah a,∗ , C. Castelain b , A. Ould-El-Moctar b , H. Peerhossaini b a
Department of Mechanical and Materials Engineering, Queen’s University, Kingston, ON K7L3N6, Canada Thermofluids, Complex Flows and Energy Research Group, Laboratoire de Thermocinétique, UMR CNRS 6607, Ecole Polytechnique de l’Université de Nantes, Rue Christian Pauc, B.P. 90604, F-44306 Nantes Cedex 3, France b
a r t i c l e
i n f o
Article history: Received 19 May 2009 Received in revised form 23 October 2009 Accepted 23 October 2009
Keywords: Yield-stress fluid Power-law fluid Dean instability Curved channel Secondary flow
a b s t r a c t The laminar flow of power-law and yield-stress fluids in 180◦ curved channels of rectangular cross section was studied experimentally and numerically in order to understand the effect of rheological fluid behavior on the Dean instability that appears beyond a critical condition in the flow. This leads to the apparition of Dean vortices that differ from the two corner vortices created by the channel wall curvature. Flow visualizations showed that the Dean vortices develop first in the near-wall zone on the concave (outer) wall, where the shear rate is higher and the viscosity weaker; then they penetrate into the centre of the channel cross section where power-law fluids have high viscosity and Bingham fluids are unyielded in laminar flow. Based on the complete formation on the concave wall of the new pairs of counter-rotating vortices (Dean vortices), the critical value of the Dean number decreases as the power-law index increases for the power-law fluids, and the Bingham number decreases for the Bingham fluids. For power-law fluids, a diagram of critical Dean numbers, based on the number of Dean vortices formed, was established for different axial positions. For the same flow conditions, the critical Dean number obtained using the axial velocity gradient criterion was smaller then that obtained with the visualization technique. © 2009 Elsevier B.V. All rights reserved.
1. Introduction Curved channel flows are used in many industrial applications to provide compactness and high heat and mass-transfer rates so as to enhance mixing in laminar flow regimes. The flow in curved channels is characterized by a secondary flow in the form of a pair of counter-rotating symmetrical vortices in the duct cross section (Fig. 1a and c), these corner vortices are called end cells or Eckman vortices by Finlay and Nandakumar [1] and they are a consequence of the channel wall curvature. Their existence was shown analytically for the first time by Dean [2] in tubes of circular cross section. The commonly used control parameter for a flow in a curved duct is the Dean number, De, defined as Um Dh De =
Dh Rc
(1)
where Um is the axial mean velocity, is the kinematic viscosity, Dh is the hydraulic diameter and Rc is the mean radius of curvature. The Dean number is therefore the product of the Reynolds num-
∗ Corresponding author at: Department of Mechanical and Materials Engineering, Queen’s University, 130 Stuart Street, Kingston, ON K7L3N6, Canada. Tel.: +1 613 533 6000x77252; fax: +1 613 533 6489. E-mail address: [email protected] (H. Fellouah). 0377-0257/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.jnnfm.2009.10.009
ber (based on axial mean velocity Um through a pipe of hydraulic diameter Dh ) and the square root of the curvature ratio. Note that the hydraulic diameter is defined as Dh =
4(cross section area) . wetted perimeter
Beyond a critical value of Dean number, new pairs of counterrotating vortices appear on the concave (outer) wall of the duct. They are due to the Dean instability and are called here Dean vortices. The number of Dean vortices depends on the cross sectional shape of the curved channel: there is one pair in circular and rectangular cross sections of small aspect ratio (Fig. 1b), and two or more pairs in rectangular cross sections of high aspect ratio (Fig. 1d). Although non-Newtonian fluids (power-law and with yield stress) have great importance in the food, pharmaceutical and chemical industries, there have been few studies of the effect of non-Newtonian properties on the Dean instability. These properties have a strong impact on flow behavior and lead to flow instabilities and dynamic nonlinearity. Cho and Hartnett [3] reviewed works on the flow of power-law fluids flow in curved channels. Ranade and Ulbrecht [4], who studied experimentally fully developed flow in a circular toroidal channel, showed that the power-law character of the fluid affects the velocity profile and that an increase in the power-law index flattens the veloc-
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Fig. 1. Schematic of the various vortices in curved channels. (a and c) Flow without instability; (b and d) flow with Dean instability.
ity profile. The secondary flow was more pronounced in the wall zones because the viscosity there was lower than at the cross section centre. The secondary flow intensity decreases with increasing power-law index. In curved channels of rectangular cross section, Shanthini and Nadakumar [5] showed numerically that the development of the flow was the same in Newtonian and non-Newtonian fluids. An increase in the critical Dean number (to be defined precisely for the non-Newtonian fluids below), and decrease in Fanning friction factor were observed with a decrease in the power-law index. In literature, the criterion commonly used to detect the onset of the Dean instability is qualitative and based on the visual appearance of the Dean vortices. For Newtonian fluids in a curved channel of square cross section, Bara et al. [6], Ghia and Sokhey [7], and Hille et al. [8] found critical Dean numbers of 137, 143, and 150, respectively. These discrepancies seem to arise from the use of a qualitative criterion in all these studies. Fellouah et al. [9] showed that, in both numerical and experimental visualizations, the critical Dean number tends to be systematically overestimated. In numerical studies visualization techniques depend strongly on the number of iso-value contour lines selected. In other words, for the same flow visualization, increasing the number of iso-value contour lines used may show two different flow situations; flow without an instability (for a small number of iso-value lines), and with an instability (large number of iso-value lines). Experimentally, an instability was signaled by the appearance of Dean vortices, but these vortices were visible only when highly amplified. Fellouah et al. [9] developed a new method to detect the critical Dean number for the appearance of a new vortex pair. Their accurate criterion for this instability threshold, based on the radial gradient of the axial velocity, relies on the shear stress and is highly sensitive to modification in the secondary flow caused by the formation of Dean vortices. In the present work, the criterion proposed by Fellouah et al. [9] and developed later in this text is used to detect the critical Dean number. The aim of the present work was to understand the effect of the non-Newtonian rheological parameters on the Dean instability, in particular the effect of the power-law index and the yield stress. The paper is organized as follows: Section 2 describes the formulation of non-Newtonian fluid behavior. Sections 3 and 4 respectively present the numerical methodology and the experimental apparatus and methods. Section 5 discusses the numerical and experimental results for the instability threshold obtained with the visualization technique and the axial velocity gradient criterion.
2. Non-Newtonian fluid behavior and formulation To study numerically the effect of the power-law index and the yield stress in non-Newtonian fluids on the Dean instability, two categories of fluids were used. First, power-law fluids, with constitutive relation displayed in Eq. (2), where (Pa) is the shear stress, ˙ (s−1 ) is the shear rate, n is the power-law index, and k (Pa s−1 ) is the fluid consistency. Secondly, Bingham fluids with constitutive relation displayed in Eq. (3) where w (Pa s) is the fluid viscosity at the wall and 0 is the yield stress. = k|| ˙ n
= 0 + w ˙ for > 0 ˙ = 0 for ≤ 0
(2) (3)
These yield-stress fluids are characterized by the nondimensional Bingham number which is the ratio of the yield and viscous stresses and defined as Bm =
0 Dh w Um
(4)
The experimental fluids used in this study were chosen for their rheological behavior and for some practical considerations as well: they must be transparent to allow the use of optical techniques for flow visualization, stable to thermomechanical effects, and non-toxic. Water was used as the reference Newtonian fluid. The power-law non-Newtonian fluid chosen for this study was an aqueous solution of carboxymethylcellulose (CMC) manufactured by Hercules Inc. (grade 7H4C), obtained by mixing a CMC powder in water. Fig. 2a shows the variation of the shear stress with shear rate for different concentrations of CMC solutions with the shear rate varying from 3 to 1000 s−1 . A Carrimed (model Weissenberg) rheometer of cone-and-plate geometry and constant shear rate was used to measure the rheological properties of the fluids. The cone has a diameter of 50 mm and a cone angle of 2◦ , a configuration recommended in the user’s manual for shear rates between 2 × 10−5 and 6 × 103 s−1 . A posterior verification of the measured velocity profiles showed that the shear rates encountered in the channel were of order 10–100 (closer to 10) s−1 , showing that the rheological measurements were carried out in the appropriate domain. The fitting of the experimental data to the power-law fluid model (Eq. (2)) was satisfactory and allows calculation of n and k for each concentration (Table 1). Fig. 2b shows the variation of viscosity with shear rate for the same CMC concentrations.
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Fig. 3. Rheological measurements of the Carbopol solution for different concentrations. Table 2 Rheological parameters of the Carbopol solution at different concentrations obtained by extrapolation of the Herschel–Bulkley fit. Carbopol (%)
0 (Pa)
n
k (Pa s−n )
0.1 0.15 0.2 0.3
7 10 17 25
0.4 0.36 0.34 0.33
13.23 32.12 45.51 62.3
its rheological properties.
Fig. 2. Rheological measurements of the CMC solution for different concentrations. (a) Shear stress vs. shear rate; (b) viscosity vs. shear rate.
To the best of our knowledge, no real Bingham fluid satisfies the above experimental requirements (transparency, stability, and the yield-stress relation). An aqueous gel of an acrylic acid polymer (Carbopol 940) of different concentrations was chosen for this fluid, the rheology of which can be closely represented by the Herschel–Bulkley model (also called the nonlinear viscoplastic model), displayed in Eq. (5). It was prepared by dissolving the polymer powder (hygroscopic white powder) in demineralized water and then raising the PH value to neutral by adding a NaOH solution. A small amount (0.05% of the quantity of NaOH added) of ethylenediaminetetraacetic acid (EDTA) was added in order to increase the resistance to destruction of this aqueous gel; it has no effect on the fluid PH. The quantity of the EDTA and the fluorescein (for flow visualization) added to the working fluid had negligible effect on Table 1 Rheological parameters of the CMC solution at different concentrations obtained by extrapolation of the power-law relation fit. CMC (%)
n
k (Pa s−n )
0.25 0.3 0.4 0.5 0.6 0.7
0.75 0.73 0.68 0.6 0.48 0.49
0.097 0.15 0.3 0.67 2.28 2.75
= 0 + k|| ˙ n ˙ = 0
for > 0 for ≤ 0
(5)
For the Carbopol solution, two different rheometers of coneand-plate geometry were used: the Carrimed (model Weissenberg), which is a constant shear rate rheometer, and the Carrimed-CS, which is a constant-torque rheometer. The yield-stress measurements could be done only with a rheometer that allows measurements with controlled stress. The principle of this technique is to apply successively increasing stresses to the working fluid; as long as the stress does not exceed the yield stress of the fluid, the response is that of a solid. Fig. 3 shows the variation of shear stress with shear rate for different concentrations of Carbopol solutions with shear rate varying from 0.1 to 1000 s−1 . The parameters in Table 2 are obtained by using the Herschel–Bulkley model (Eq. (5)) to fit the data in Fig. 3. The yield stresses in Table 3 are the real yield stresses of the Carbopol solution obtained by using the constant-torque rheometer (rather than that obtained by extrapolation of the Herschel-Bulkley fit) and are smaller than those given in Table 2. We now consider the definition of the Dean number in rectangular cross section ducts for the non-Newtonian fluids used in this study. As expressed before, the Dean number is directly related to the Reynolds number (Eq. (1)). Expressions for the Reynolds numTable 3 Real yield stress of the Carbopol solution at different concentrations. Note these value are obtained by constant stress rheometry, rather than that obtained by extrapolation of the Herschel–Bulkley fit. Carbopol (%)
0 (Pa)
0.15 0.2 0.3
0 1 5
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Table 4 Dimensionless geometrical parameter proposed by Delplace and Leuliet [11]. b/a
1 7.113
1.33 7.238
4 7.774
5 9.116
8 9.787
∞ 12
ber for Newtonian fluids flowing in tubes of circular cross section and between two parallel plates have been established by Metzner and Reed [10]. Using the Metzner and Reed [10] relation, Delplace and Leuliet [11] expressed the Reynolds number for power-law fluids in ducts of square and rectangular cross section for different aspect ratios as Re =
2−n n Dh Um
(6)
n
k{(24n + )/(24 + )n} n−1
Fig. 4. Coordinates system.
where is the fluid density. This expression is based on a dimensionless geometrical parameter known for many rectangular cross sections (Table 4). In the rectangular cross section, the expression of the Reynolds number for the Herschel–Bulkley (H–B) fluids was obtained by multiplying the expression of the Reynolds number for the powerlaw fluids in the same geometry by a factor Rec related to the yield stress. The expression of the Reynolds number for the Herschel–Bulkley (H–B) fluids in the rectangular cross section is displayed in Eq. (7) where Re|Power-Law fluids is the Reynolds number for power-law fluids in ducts of rectangular cross section (Eq. (6)) and Rec is the contribution of the yield stress in the flow displayed in Eq. (7). ReH–B = Re|Power-Law
fluids
(7)
Rec
Rec is defined in Devienne et al. [12] and displayed in Eq. (8) where 0 = 0 /w , is the dimensionless unsheared plug radius and w is the wall shear stress.
Rec
= (1 − 0 ) 1 −
n 02 0 − n+1 n+1
n
(8)
In the present work, w is the mean wall shear stress related to the pressure gradient and knowing the value of 0 from the rheological study, a value of 0 is determined. Note that all non-Newtonian fluids used in this paper can be regarded as special cases of the Herschel–Bulkley model. If n = 1 and 0 = 0 the model reduces to the Newtonian model. If n = / 1 and 0 = 0 the model reduces to the power-law model, and if n = 1 and 0 = / 0, this model reduces to the Bingham model. 3. Numerical methodology The continuity and time-independent three-dimensional Navier–Stokes equations governing incompressible laminar flow in a curved channel of rectangular cross section were solved using the well established code Fluent. This code has been used extensively in the calculation of complex flows (see Refs. [13–15] for recent examples). The numerical methodology for Newtonian fluids was developed in Fellouah et al. [9]. Here, we extend the earlier work (Ref. [9]) to include the methodology for non-Newtonian fluids. The apparent viscosity of the fluid ap = /, ˙ needed for the expression of the shear stress tensor, is displayed in Eq. (9) for the Herschel–Bulkley fluids. ap =
o + k˙ n−1 ˙
(9)
This apparent viscosity presents a singularity when the shear rate vanishes. To overcome this singularity, a user defined function (UDF) based on a model valid for both sheared and unsheared areas was developed and added to the CFD code. Three models have
been developed in the literature: the Papanastasiou model [16], the Bercovier and Engelman model [17], and the O’Donovan and Tanner [18] model (bi-viscosity model). These three models have the same physical meaning: they transform the unsheared zone of the yield-stress fluid into a high-viscosity region. In the present study, the Papanastasiou model, displayed in Eq. (10), is used with a stress exponent of m = 105 higher than the minimum m = 103 recommended for use of the Papanastasiou model with confidence. ap =
0 (1 − e−m˙ ) + k˙ n−1 ˙
(10)
where m is the stress growth exponent. Fig. 4 shows the system coordinates used to study the curved rectangular channel flow. Experiments were performed in a channel with rectangular cross section of width b = 160 mm and a height a = 20 mm; b/a = 8 and Rc /Dh = 10 are the aspect and the curvature ratios, respectively. Various aspect ratios and curvature ratios were studied in the numerical work. Velocity components in the x, y and z directions are u, v and w, respectively. The flat mid-planes are the planes that pass by the centre of the channel cross section (x = 0) and are parallel to the flat walls; the curved mid-planes are the planes that pass by y = 0 and are parallel to the curved walls. The dimensions of the curved ducts studied were chosen such that the height a = 20 mm was constant. A symmetry condition about the plane x = 0 was used to reduce the computational burden for large geometries (b/a ≥ 8). From Fig. 4, the hydraulic diameter is Dh =
2ab . a+b
The segregated solver is used to iterate the nonlinear partial differential equations since this solver is better adapted to the large grids used in our work. To calculate the pressure at the nodes the body-force-weighted scheme is used, which works well when inertial forces are not negligible. The SIMPLEC algorithm is used for pressure–velocity coupling in order to obtain the pressure in the continuity equation. For the momentum equation, the second order quick scheme is used because it is recommended in the presence of rotating flows. The no-slip boundary condition at the wall is used. Gambit was used to generate the grid or mesh for the Fluent CFD solver. Calculations start at a straight channel upstream of the curved duct. At the entry of the straight channel, a uniform velocity profile equal to the mean velocity is imposed. At the channel exit, the flow is hydrodynamically developed and enters the curved duct. Convergence is controlled by following the evolution of the residuals for the velocity components and the continuity equation. In the convergence, the evolution residuals must converge toward small (in all cases less than 10−6 ) and stable quantities. The time necessary to reach convergence varied; it depended on the curvature and aspect ratios and on the fluid considered. To optimize the mesh grid, several grids were tested for a curved duct of square
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167
Fig. 5. Experimental apparatus.
cross section, ranging from 20 to 34 nodes on the flat and curved walls and from 5 to 10 mm between two successive grid surfaces (see Ref. [9] for details). For this work, it was reasonable to use 24 nodes on the flat and curved walls, corresponding to square meshes of 0.83 mm × 0.83 mm on the surface, and 10 mm between two adjacent grid surfaces. For yield-stress fluids, the Dean number depends on the dimensionless radius of the plug-flow region 0 = 0 /w . For this reason, 0 was initialized to start simulations for a given Dean number. At the convergence of these simulations, the pressure loss ( p) was obtained on the straight channel. Given the relation between the pressure loss and the average wall shear stress w (w = (Dh /4)( p/L), where L is the axial length of the straight channel), the average wall shear stress w was determined. Since the yield shear stress 0 is known for the fluid used, the exact value of 0 was calculated and used to correct the Dean number.
Fig. 6. Flow visualization observed at different cross sections, b/a = 8, Rc /Dh = 10, power-law fluid (n = 0.75).
4. Experimental setup The experimental setup used was the same as that described in Fellouah et al. [9] and consists of a water channel facility (Fig. 5). A volumetric pump generates flow from the tank to the test section, which was preceded by a 2.6 m straight PVC duct for the development of Poiseuille flow and also to ensure stable and clean inlet conditions. The test section was composed of a 100 mm straight channel, a curved channel with curvature angle 180◦ and curvature ratio 10, and finally another 100 mm straight channel. The test section was made of Plexiglas to facilitate flow visualization and had a rectangular cross section a × b = 20 mm × 160 mm, giving an aspect ratio b/a = 8 large enough to minimize the influence of the side walls. A numerically controlled milling machine was used to fabricate the 180◦ bend and the straight length to precise dimensions (±0.02 mm). The bend and straight length were finely polished at the ends. A 200 mm PVC straight channel at the test section exit eliminates any effect of the adapter (rectangular–circular) on the flow there. The flow was visualized by the laser-induced fluorescence (LIF) technique. Fluorescein solutions of mass concentration 1 mg/l were used so that the density variation in the working fluid due to the tracer addition was negligible. Dye was injected with a syringe upstream of the settling chamber and was distributed throughout the channel section to allow overall visualization. 5. Results and discussions In this work, the critical Dean number was obtained by two methods: first by using the secondary flow visualization technique, and second by using the axial velocity gradient.
5.1. Secondary flow visualizations Unless otherwise indicated, in all flow visualization photographs, the concave and convex walls were at the top and the bottom respectively and the main flow was perpendicular and away from the page. The numerical visualizations were given by the helicity contour lines. The helicity is defined as the dot product of vorticity and the velocity vector displayed in the following equation: H = (∇ × V )V
(11)
It provides insight into the vorticity component aligned with the fluid stream. Vorticity is a measure of the rotation of fluid elements as they move in the flow field. 5.1.1. Power-law fluids Fig. 6 shows numerical and experimental flow visualization images for a power-law fluid of n = 0.75, at a Dean number close to De = 260 for different observation locations (from 60◦ to 180◦ ). This figure shows two symmetrical corner vortices near the flat end-walls, at the 60◦ position. In the central part of the test section no Dean vortices are visible. A first pair of Dean vortices, due to the Dean instability, appears at the 80◦ position in the central region. The number of visible Dean vortices pairs increases downstream in the curved channel: two pairs at the 90◦ position, three pairs at 120◦ , and four pairs at 150◦ and at the curved channel exit (180◦ ). Each pair of Dean vortices appears as a mushroom-shaped structure. The mushroom ‘stems’ are associated with the upwash regions, where fluid is transported from the concave wall to the cross section centre. The mushroom ‘caps’ correspond to the vor-
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tex core regions. The bright regions between each mushroom pair correspond to the downwash region where the fluid is transported from the cross section centre to the concave wall. Numerically, at the 60◦ position, the helicity contour lines confirm the configuration of the corner vortices seen in the visualization, but their elongation to the centre of the channel cross section shows the presence of a secondary flow in this region too. This secondary flow was not observed experimentally since it was weak in this region; only at the 120◦ position did several Dean vortices begin to form. The number of Dean vortices pairs at 150◦ is four and the vortex strength is intensified at 180◦ . Both numerical and experimental visualizations show that the first pair of Dean vortices appears at the centre of the channel cross section. This is due to the symmetrical circulation of the corner vortices and the extension of their influence to the centre of the cross section. It can be noted that the Dean vortices are less developed in the numerical results: they remain more confined close to the concave walls, whereas in experiments they rapidly occupy the full channel height. Fig. 7 presents the effect of the Dean number on the flow of the power-law fluid in the exit section of the curved rectangular channel (180◦ ). The experimental flow visualizations show a flow
Fig. 7. (Continued).
Fig. 7. Flow visualization observed for different Dean numbers at = 180◦ , b/a = 8, Rc /Dh = 10, power-law fluid (n = 0.75).
with two symmetrical corner vortices up to De = 110. At De = 122, the tracer shows perturbations that are confined in the vicinity of the concave wall; these perturbations increase in intensity with increasing Dean number to form a mushroom stem at De = 128. The stem continues to develop and at De = 139 gives rise to a mushroom form; at De = 150 a pair of fully developed Dean vortices is visible. In this flow development, the primary difference from the Newtonian case was due to the power-law index of the fluid; characterized by weak viscosity in regions of high shear stress near the wall and high viscosity in the regions of weak shear stress. Thus, Dean vortices develop easily in the wall region, but their difficulty in penetrating the high-viscosity region and entraining the fluid particles delays the achievement of their final form, the mushroom-shaped structure. The pair of Dean vortices observed at De = 150 intensifies at De = 163, a second pair of vortices is observed at De = 170, a third pair between De = 185 and De = 248, and a fourth pair appears at De = 263. This was the largest number of Dean vortices seen up to De = 342. Numerically, the first perturbation was observed at De = 150. Three pairs of Dean vortices were identifiable at De = 180; they continue their development at De = 200 and the fourth pair appears at De = 220. The vortex intensity increases with increasing Dean number and they become irregularly spaced at De = 300. Two more Dean vortices pairs appear at De = 320 and 340. Comparison of the experimental and numerical visualizations revealed a difference in the dynamic development of the Dean vortices. In fact, the evolution was clearer in the experimental study,
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Fig. 8. Flow visualization obtained at = 180◦ for the Newtonian fluid (water n = 1) and the power-law fluid (n = 0.75), b/a = 8, Rc /Dh = 10.
where the Dean instability starts in the wall region on the concave side of the channel and then develops into Dean vortices. The development of the flow and the maximum number of Dean vortices obtained numerically and experimentally differ; probably because the numerical study does not take into account the physical constraints and perturbations inherent in the experiments. To give a visual feeling for the effect of the power-law index on the Dean instability, Fig. 8 shows the experimental flow evolution at the exit section (180◦ ) of the rectangular curved channel for several Dean numbers for a Newtonian (n = 1) and a power-law (n = 0.75) fluid. In these visualizations, the Dean vortices appear rapidly in the Newtonian fluid and occupy the entire channel cross section. However, in the power-law fluid they appear gradually, first in the wall region of the concave side and then developing to occupy the entire cross section channel width. This is due to the variation in viscosity in the power-law fluid: it is weak near the wall and high in the cross section centre. If the critical Dean number is considered to correspond to the complete formation of the first Dean vortices (the first mushroom-shaped structure), these visualizations showed a shift in critical Dean number in the power-law fluid from 137 for the Newtonian fluid to 150 for the power-law fluid (n = 0.75). If, on the other hand, the critical Dean number is considered to correspond to the beginning of tracer perturbation on the concave wall, this shift in the critical Dean number was less pronounced. This result is plausible since, if the shear stress is large enough to increase the non-Newtonian fluid viscosity to that of the Newtonian fluid, then the balance between viscous and centrifugal force must remain unchanged and must result in the same critical Dean number. For high Dean numbers (around 250), the flow structure was similar for the two types of fluids. In order to show the influence of the power-law index, a power-law fluid with a weaker power-law index was considered numerically. Fig. 9 presents the secondary flow patterns at the exit of a 180◦ curved rectangular channel for different Dean numbers for a power-law fluid of power-law index 0.5. It can be seen that at low Dean numbers the vortices are confined near the concave wall, similar to the experimental observations for the power-law fluid (n = 0.75). Fig. 10 shows the development of the flow as a function of the critical Dean number and the position in a curved channel. The lower area corresponds to the flow without the Dean instability.
Fig. 9. Flow visualization observed for different Dean numbers at = 180◦ , b/a = 8, Rc /Dh = 10, power-law fluid (n = 0.5).
Fig. 10. Critical Dean numbers, obtained from the experimental visualizations, at different angular positions, b/a = 8 and Rc /Dh = 10, power-law fluid (n = 0.75). Area 1: flow without Dean instability; Area 2: flow with 1 pair of Dean vortices; Area 3–5: flow with 2–4 pairs of Dean vortices, respectively.
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Table 5 Critical Dean number obtained from the experimental flow visualization for the Newtonian fluid (Ref. [9]) and the power-law fluid (n = 0.75), b/a = 8, Rc /Dh = 10. Angular position (◦ )
Number of vortices pairs
Water (n = 1)
CMC 0.25% (n = 0.75)
180
1 2 3 4
130 135 140 230
150 170 185 263
150
1 2 3 4
150 160 185 270
162 187 239 285
120
1 2 3 4
160 180 200 285
181 215 258 329
90
1 2 3
190 220 260
222 243 314
80
1 2 3
220 260 320
272 300
The other areas correspond to the flow with the Dean instability. This figure shows that the Dean instability starts at an angular position of 80◦ and that the critical Dean number decreases in the axial direction. This is due to the axial development of the flow along the curved channel, which increases the secondary flow intensity. The evolution of the critical Dean number in Fig. 10 is analogous to that in the Newtonian fluid. At a given angular position, the secondary flow intensity increases with increasing Dean number and hence extra Dean vortices appear. Table 5 summarizes the critical Dean numbers for the power-law fluid (n = 0.75) at different positions and compares them with those for the Newtonian fluid. 5.1.2. Yield-stress fluids The qualitative study described above of the effect of the power-law index showed that the onset of the Dean instability, as characterized by the appearance of the Dean vortices as a mushroom-shaped structure, was delayed as this index decreases. Another aim of the present work was to study the Dean instability in a yield-stress fluid. For this purpose, as a first step, the flow evolution of a Bingham fluid along a curved channel (b/a = 8, Rc /Dh = 10) for a Dean number of 405 (chosen to be supercritical to show Dean vortex development) was studied numerically. Results are presented as helicity contour lines in Fig. 11. From positions 90◦ to 150◦ , the helicity contour lines show no variation in the flow pattern. These contour lines revealed information related to the shape of the recirculation zones that exist only in the sheared zone near the flat walls. These zones differ in form from those obtained for the Newtonian fluids, which extend further into the channel cross section centre. In fact, the helicity contour lines are bent around the unsheared region of the Bingham fluid. A similar but less pro-
Fig. 11. Numerical flow visualization observed at different cross sections, De = 405, b/a = 8, Rc /Dh = 10, Bingham fluid (Bm = 10).
nounced deformation was also observed in the power-law fluid. At an angular position of 180◦ , two more closed helicity contour lines appear near the concave wall in the centre of the channel cross section; they are confined in the sheared region. In addition to the disadvantages related to the qualitative detection of the onset of Dean vortex based on the observation of helicity contours there is another fact to be considered in the case of yield-stress fluids. In this case, the helicity contour lines presented in Fig. 11 should be considered carefully, they revealed the complicated nature of this yield-stress fluids flow with features not encountered in Newtonian flows. The determination of the exact shape of the yielded and unyielded surfaces in general Herschel–Bulkley fluid flows remains difficult due to the singular nature of the constitutive relation [19]. The Papanastasiou model used here is an approximation to the singular model and, as explained above, when ˙ → 0 this model behaves like viscous fluids. By definition then, the predictions of such model are only viscous approximations to the ideal Herschel–Bulkley flow. Hammad et al. [20] and Burgos and Alexandrou [19] found that during the flow of the Herschel–Bulkley fluids, dead zones (or stagnant flow) are formed in the corners of the channel. This can lead to predict vortices in cases where the flow is stagnant. The stagnant zones exist due to the low stress levels and, therefore, the lack of deformation of the fluid elements. As we will see below, by no way our results concerning the detection of the onset of Dean instability are affected by such presentation of contour lines. 5.2. Detecting the onset of Dean instability As shown in the flow visualizations, the instability always starts in the centre of the midplane (y = 0) near the concave wall. For this reason, the critical Dean number was detected only at this part of the plane. The first step consists in measurement for different Dean numbers, both numerically and experimentally, the axial velocity along segments [AA ] and [BB ] (Fig. 12). In this experiment, the length of these segments was 40 mm, which was sufficient to detect the formation of the first pair of Dean vortices. Then, for each x-coordinate position, the axial velocity gradient (dw/dy) was obtained. In evaluating the experimental instability criterion, the data were contaminated with measurement noise and an inverse method was applied for filtering. An easy and fast way to detect these inflection points was to calculate the slope of the axial velocity gradient profile along the x-coordinate for different Dean numbers.
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Fig. 12. Position of the measurements in the cross section.
Figs. 13 and 14 show the slope of the axial velocity gradient respectively for a Newtonian fluid and a power-law (n = 0.75) fluid obtained numerically and experimentally at the 180◦ position (channel exit). They show that the formation of the Dean vortices modifies the profile of the axial velocity gradient in the centre of the mid-plane, as demonstrated by the formation of one or more inflection points in the profile. These inflection points are due to the formation of the Dean vortices, which modify the wall shear stress. These slopes are uniform for low Dean numbers and contain perturbations (undulations) due to the normal velocity gradient caused by the formation of the Dean vortices for high Dean numbers. The difference in shape of the slopes obtained numerically (sinusoidal) and experimentally (asymmetric) can be attributed to measurement errors, primarily caused by vibration of the curved channel at large Dean numbers because of high pressure.
Fig. 14. Slope of the axial velocity gradient profiles in the centre of the mid-plane for different Dean numbers, power-law fluid (n = 0.75), b/a = 8 and Rc /Dh = 10, = 180◦ . (a) Numerical result; (b) experimental result.
Fig. 13. Slope of the axial velocity gradients profile in the centre of the mid-plane for different Dean numbers, Newtonian fluid, b/a = 8 and Rc /Dh = 10, = 180◦ . (a) Numerical result; (b) experimental result.
The variation in the slope form, from uniform to perturbed, is considered a signature of the Dean instability threshold. The critical Dean numbers can readily be determined, both numerically and experimentally, from Figs. 13 and 14. These results show that the numerical or experimental critical Dean numbers obtained by measuring the axial velocity gradient are below those obtained by numerical and experimental visualizations. This result confirms the precision of the criterion based on the axial velocity gradient adopted here to determine the onset of the Dean vortices. Henceforth, unless noted, critical Dean numbers were obtained by using the axial velocity gradient criterion. Fig. 15 shows the influence of the two rheological parameters, the power-law index and the Bingham number, on the onset of Dean instability. The critical Dean number decreases with increasing power-law index n due to the local change in apparent viscosity: the viscosity is weaker near the wall and stronger at the centre of the cross section, and thus it is difficult for unstable Dean vortices to form in the centre. All of these factors lead to a delay in the onset of the Dean instability with increasing power-law index. Increasing the Bingham number delays the onset of the Dean instability. In addition, when the Bingham number increases, the yield-stress effect on the flow also increases. To overcome the effect of the yield stress, the inertial forces and hence the Dean number must increase as well. Previous observations by Alexandrou et al. [21] showed that
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Fig. 17. Effect of aspect ratio on Dean instability onset, numerical result. Rc /Dh = 10.
Fig. 15. Effect of rheological parameters on the onset of Dean instability, numerical result.
Fig. 16. Effect of curvature ratio on the Dean instability onset, numerical result.
the size of the plug region increases with the Bingham number, as confirmed in the present work. The effect of the curvature ratio (Rc /Dh ) on the critical Dean number for two aspect ratios b/a = 1 and 8 is shown in Fig. 16. In particular, no curvature ratio effect on the critical Dean number for aspect ratio 1 was observed, whereas for aspect ratio 8, the critical Dean number decreases rapidly when the curvature ratio increases
to 10 and then becomes constant. This evolution is enhanced by the decrease in length of the curved channel; the hydraulic diameter was constant and therefore a reduction in curvature ratio implies a decrease in length of the curved channel. For a square cross section, this reduction is weak. The shape of the curve in this figure shows that the variation in the critical Dean number with curvature ratio is independent of the rheological parameters, which induce only a shift in the critical Dean number values. This result is confirmed in Fig. 17, showing the evolution of the critical Dean number with aspect ratio for different types of fluids (the curvature ratio was held constant at 10). The variation in critical Dean number with duct aspect ratio is noticeably less regular than with Rc /Dh . The instability, like the main flow, develops along the curved channel. Fig. 18 shows the onset of the Dean instability at different angular positions for Rc /Dh = 10 and b/a = 8. The axial position of the onset of instability increases with decreasing power-law index and Bingham number. This position is further downstream in the curved channel for the Bingham fluid than for the power-law fluid. In fact, it is more difficult for the Dean vortices to penetrate the yield region of the Bingham fluid. The numerical results in Fig. 18 show that the critical Dean number reaches its constant value more quickly when the power-law index increases and the Bingham number decrease, demonstrating that the development of the Dean instability along the curved channel is faster in Newtonian than in non-Newtonian fluids. For the non-Newtonian fluids the variation in viscosity, which is higher in the centre of the channel cross section, is responsible for this slow development of Dean instability.
Fig. 18. Evolution of the critical Dean number with the angular position, Rc /Dh = 10, b/a = 8.
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6. Conclusions The effects of the power-law index and yield stress on the Dean instability in curved channels of rectangular cross section were studied numerically and experimentally. The experimental study was carried out in a curved channel of aspect ratio 8 and curvature ratio 10, and was extended numerically to various other aspect and curvature ratios. The results showed that in one hand, in the case of the complete formation of Dean vortices, the critical value of the Dean number decreases with increasing power-law index and decreasing Bingham number. On the other hand, the onset of the Dean vortices in the wall zone of the concave wall was similar in all fluids: varying the rheological parameters merely shifted the position of the critical Dean number. Comparison of the numerical and experimental results using the radial gradient of axial velocity gradient criterion was satisfactory in predicting the critical Dean number. The flow visualizations showed that the effect of the yield stress on the shape of the flow appears to be the dominant parameter. Acknowledgments This research was supported by the French Ministry of Research and Technology. The authors are grateful for their financial support. References [1] W.H. Finlay, K. Nandakumar, Onset of two-dimensional cellular flow in finite curved channels of large aspect ratio, Phys. Fluids A 2 (7) (1990) 1163–1174. [2] W.R. Dean, Note on the motion of fluid in a curved pipe, Philos. Mag. 4 (1927) 208–223. [3] Y.I. Cho, J.P. Hartnett, Non-Newtonian fluids in circular pipe flow, Adv. Heat Transfer 15 (1982) 59–141. [4] V.R. Ranade, J.J. Ulbrecht, Velocity profiles of Newtonian and non-Newtonian toroidal flows measured by a LDA technique, Chem. Eng. Commun. 20 (1982) 253–272.
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[5] W. Shanthini, K. Nandakumar, Bifurcation phenomena of generalized Newtonian fluids in curved rectangular ducts, J. Non-Newtonian Fluid Mech. 22 (1986) 35–60. [6] B. Bara, K. Nandakumar, J.H. Masliyah, An experimental and numerical study of the Dean problem: flow development towards two-dimensional multiple solutions, J. Fluid Mech. 244 (1992) 339–376. [7] K.N. Ghia, J.S. Sokhey, Laminar incompressible viscous flow in curved ducts of rectangular cross-section, Trans. ASME I: J. Fluids Eng. 99 (1977) 640–648. [8] P. Hille, R. Vehrenkamp, E.O. Schulz-Dubois, The development and structure of primary and secondary flow in a curved square duct, J. Fluid Mech. 151 (1985) 219–241. [9] H. Fellouah, C. Castelain, A. Ould El Moctar, H. Peerhossaini, A criterion for detection of the onset of Dean instability in Newtonian flows, Eur. J. Mech. B: Fluids 25 (2006) 505–531. [10] A.B. Metzner, J.C. Reed, Flow of non-Newtonian fluids: correlation of the laminar, transition, and turbulent-flow regions, AIChE J. 1 (1955) 434–440. [11] F. Delplace, J.C. Leuliet, Generalized Reynolds number for the flow of power law fluids in cylindrical ducts of arbitrary cross-section, Chem. Eng. J. Biochem. Eng. J. 56 (2) (1995) 33–37. [12] R. Devienne, J.C. Leuliet, P. Sourlier, J.F. Maingonnat, M. Lalande, M. Lebouche, Incidence du seuil d’écoulement sur les performances thermodynamiques des échangeurs, in: Proceedings 2nd Colloque TIFAN, La Baule, 1989, pp. 79– 95. [13] R.J. Poole, B.S. Ridley, Development-length requirements for fully developed laminar pipe flow of inelastic non-Newtonian liquids, J. Fluids Eng. 129 (2007) 1281–1287. [14] Z. Huang, J.A. Olsen, R.J. Kerekes, S.I. Green, Numerical simulation of the flow around rows of cylinders, Comput. Fluids 35 (2006) 485–491. [15] L.Y. Hu, L.X. Zhou, J. Zhang, M.X. Shi, Studies on strongly swirling flows in the full space of a volute cyclone separator, AIChE J. 51 (3) (2005) 740–749. [16] T.C. Papanastasiou, Flow of materials with yield, J. Rheol. 31 (5) (1987) 385–404. [17] M. Bercovier, M. Engelman, A finite element method for incompressible nonNewtonian flows, J. Comput. Phys. 36 (1980) 313–326. [18] E.J. O’Donovan, R.I. Tanner, Numerical study of the Bingham squeeze film problem, J. Non-Newtonian Fluid Mech. 15 (1984) 75–83. [19] G.R. Burgos, A.N. Alexandrou, Flow development of Herschel–Bulkley fluids in a sudden three-dimensional square expansion, J. Rheol. 43 (3) (1999) 485– 498. [20] K.J. Hammad, G.C. Vradis, M.V. Ötügen, Laminar flow of a Herschel–Bulkley fluid over an axisymmetric sudden expansion, Trans. ASME: J. Fluids Eng. 123 (2001) 588–594. [21] A.N. Alexandrou, T.M. McGilvreay, G. Burgos, Steady Herschel–Bulkley fluid flow in three-dimensional expansions, J. Non-Newtonian Fluid Mech. 100 (2001) 77–96.
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Journal of Non-Newtonian Fluid Mechanics journal homepage: www.elsevier.com/locate/jnnfm
The Dean instability in power-law and Bingham fluids in a curved rectangular duct H. Fellouah a,∗ , C. Castelain b , A. Ould-El-Moctar b , H. Peerhossaini b a
Department of Mechanical and Materials Engineering, Queen’s University, Kingston, ON K7L3N6, Canada Thermofluids, Complex Flows and Energy Research Group, Laboratoire de Thermocinétique, UMR CNRS 6607, Ecole Polytechnique de l’Université de Nantes, Rue Christian Pauc, B.P. 90604, F-44306 Nantes Cedex 3, France b
a r t i c l e
i n f o
Article history: Received 19 May 2009 Received in revised form 23 October 2009 Accepted 23 October 2009
Keywords: Yield-stress fluid Power-law fluid Dean instability Curved channel Secondary flow
a b s t r a c t The laminar flow of power-law and yield-stress fluids in 180◦ curved channels of rectangular cross section was studied experimentally and numerically in order to understand the effect of rheological fluid behavior on the Dean instability that appears beyond a critical condition in the flow. This leads to the apparition of Dean vortices that differ from the two corner vortices created by the channel wall curvature. Flow visualizations showed that the Dean vortices develop first in the near-wall zone on the concave (outer) wall, where the shear rate is higher and the viscosity weaker; then they penetrate into the centre of the channel cross section where power-law fluids have high viscosity and Bingham fluids are unyielded in laminar flow. Based on the complete formation on the concave wall of the new pairs of counter-rotating vortices (Dean vortices), the critical value of the Dean number decreases as the power-law index increases for the power-law fluids, and the Bingham number decreases for the Bingham fluids. For power-law fluids, a diagram of critical Dean numbers, based on the number of Dean vortices formed, was established for different axial positions. For the same flow conditions, the critical Dean number obtained using the axial velocity gradient criterion was smaller then that obtained with the visualization technique. © 2009 Elsevier B.V. All rights reserved.
1. Introduction Curved channel flows are used in many industrial applications to provide compactness and high heat and mass-transfer rates so as to enhance mixing in laminar flow regimes. The flow in curved channels is characterized by a secondary flow in the form of a pair of counter-rotating symmetrical vortices in the duct cross section (Fig. 1a and c), these corner vortices are called end cells or Eckman vortices by Finlay and Nandakumar [1] and they are a consequence of the channel wall curvature. Their existence was shown analytically for the first time by Dean [2] in tubes of circular cross section. The commonly used control parameter for a flow in a curved duct is the Dean number, De, defined as Um Dh De =
Dh Rc
(1)
where Um is the axial mean velocity, is the kinematic viscosity, Dh is the hydraulic diameter and Rc is the mean radius of curvature. The Dean number is therefore the product of the Reynolds num-
∗ Corresponding author at: Department of Mechanical and Materials Engineering, Queen’s University, 130 Stuart Street, Kingston, ON K7L3N6, Canada. Tel.: +1 613 533 6000x77252; fax: +1 613 533 6489. E-mail address: [email protected] (H. Fellouah). 0377-0257/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.jnnfm.2009.10.009
ber (based on axial mean velocity Um through a pipe of hydraulic diameter Dh ) and the square root of the curvature ratio. Note that the hydraulic diameter is defined as Dh =
4(cross section area) . wetted perimeter
Beyond a critical value of Dean number, new pairs of counterrotating vortices appear on the concave (outer) wall of the duct. They are due to the Dean instability and are called here Dean vortices. The number of Dean vortices depends on the cross sectional shape of the curved channel: there is one pair in circular and rectangular cross sections of small aspect ratio (Fig. 1b), and two or more pairs in rectangular cross sections of high aspect ratio (Fig. 1d). Although non-Newtonian fluids (power-law and with yield stress) have great importance in the food, pharmaceutical and chemical industries, there have been few studies of the effect of non-Newtonian properties on the Dean instability. These properties have a strong impact on flow behavior and lead to flow instabilities and dynamic nonlinearity. Cho and Hartnett [3] reviewed works on the flow of power-law fluids flow in curved channels. Ranade and Ulbrecht [4], who studied experimentally fully developed flow in a circular toroidal channel, showed that the power-law character of the fluid affects the velocity profile and that an increase in the power-law index flattens the veloc-
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Fig. 1. Schematic of the various vortices in curved channels. (a and c) Flow without instability; (b and d) flow with Dean instability.
ity profile. The secondary flow was more pronounced in the wall zones because the viscosity there was lower than at the cross section centre. The secondary flow intensity decreases with increasing power-law index. In curved channels of rectangular cross section, Shanthini and Nadakumar [5] showed numerically that the development of the flow was the same in Newtonian and non-Newtonian fluids. An increase in the critical Dean number (to be defined precisely for the non-Newtonian fluids below), and decrease in Fanning friction factor were observed with a decrease in the power-law index. In literature, the criterion commonly used to detect the onset of the Dean instability is qualitative and based on the visual appearance of the Dean vortices. For Newtonian fluids in a curved channel of square cross section, Bara et al. [6], Ghia and Sokhey [7], and Hille et al. [8] found critical Dean numbers of 137, 143, and 150, respectively. These discrepancies seem to arise from the use of a qualitative criterion in all these studies. Fellouah et al. [9] showed that, in both numerical and experimental visualizations, the critical Dean number tends to be systematically overestimated. In numerical studies visualization techniques depend strongly on the number of iso-value contour lines selected. In other words, for the same flow visualization, increasing the number of iso-value contour lines used may show two different flow situations; flow without an instability (for a small number of iso-value lines), and with an instability (large number of iso-value lines). Experimentally, an instability was signaled by the appearance of Dean vortices, but these vortices were visible only when highly amplified. Fellouah et al. [9] developed a new method to detect the critical Dean number for the appearance of a new vortex pair. Their accurate criterion for this instability threshold, based on the radial gradient of the axial velocity, relies on the shear stress and is highly sensitive to modification in the secondary flow caused by the formation of Dean vortices. In the present work, the criterion proposed by Fellouah et al. [9] and developed later in this text is used to detect the critical Dean number. The aim of the present work was to understand the effect of the non-Newtonian rheological parameters on the Dean instability, in particular the effect of the power-law index and the yield stress. The paper is organized as follows: Section 2 describes the formulation of non-Newtonian fluid behavior. Sections 3 and 4 respectively present the numerical methodology and the experimental apparatus and methods. Section 5 discusses the numerical and experimental results for the instability threshold obtained with the visualization technique and the axial velocity gradient criterion.
2. Non-Newtonian fluid behavior and formulation To study numerically the effect of the power-law index and the yield stress in non-Newtonian fluids on the Dean instability, two categories of fluids were used. First, power-law fluids, with constitutive relation displayed in Eq. (2), where (Pa) is the shear stress, ˙ (s−1 ) is the shear rate, n is the power-law index, and k (Pa s−1 ) is the fluid consistency. Secondly, Bingham fluids with constitutive relation displayed in Eq. (3) where w (Pa s) is the fluid viscosity at the wall and 0 is the yield stress. = k|| ˙ n
= 0 + w ˙ for > 0 ˙ = 0 for ≤ 0
(2) (3)
These yield-stress fluids are characterized by the nondimensional Bingham number which is the ratio of the yield and viscous stresses and defined as Bm =
0 Dh w Um
(4)
The experimental fluids used in this study were chosen for their rheological behavior and for some practical considerations as well: they must be transparent to allow the use of optical techniques for flow visualization, stable to thermomechanical effects, and non-toxic. Water was used as the reference Newtonian fluid. The power-law non-Newtonian fluid chosen for this study was an aqueous solution of carboxymethylcellulose (CMC) manufactured by Hercules Inc. (grade 7H4C), obtained by mixing a CMC powder in water. Fig. 2a shows the variation of the shear stress with shear rate for different concentrations of CMC solutions with the shear rate varying from 3 to 1000 s−1 . A Carrimed (model Weissenberg) rheometer of cone-and-plate geometry and constant shear rate was used to measure the rheological properties of the fluids. The cone has a diameter of 50 mm and a cone angle of 2◦ , a configuration recommended in the user’s manual for shear rates between 2 × 10−5 and 6 × 103 s−1 . A posterior verification of the measured velocity profiles showed that the shear rates encountered in the channel were of order 10–100 (closer to 10) s−1 , showing that the rheological measurements were carried out in the appropriate domain. The fitting of the experimental data to the power-law fluid model (Eq. (2)) was satisfactory and allows calculation of n and k for each concentration (Table 1). Fig. 2b shows the variation of viscosity with shear rate for the same CMC concentrations.
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Fig. 3. Rheological measurements of the Carbopol solution for different concentrations. Table 2 Rheological parameters of the Carbopol solution at different concentrations obtained by extrapolation of the Herschel–Bulkley fit. Carbopol (%)
0 (Pa)
n
k (Pa s−n )
0.1 0.15 0.2 0.3
7 10 17 25
0.4 0.36 0.34 0.33
13.23 32.12 45.51 62.3
its rheological properties.
Fig. 2. Rheological measurements of the CMC solution for different concentrations. (a) Shear stress vs. shear rate; (b) viscosity vs. shear rate.
To the best of our knowledge, no real Bingham fluid satisfies the above experimental requirements (transparency, stability, and the yield-stress relation). An aqueous gel of an acrylic acid polymer (Carbopol 940) of different concentrations was chosen for this fluid, the rheology of which can be closely represented by the Herschel–Bulkley model (also called the nonlinear viscoplastic model), displayed in Eq. (5). It was prepared by dissolving the polymer powder (hygroscopic white powder) in demineralized water and then raising the PH value to neutral by adding a NaOH solution. A small amount (0.05% of the quantity of NaOH added) of ethylenediaminetetraacetic acid (EDTA) was added in order to increase the resistance to destruction of this aqueous gel; it has no effect on the fluid PH. The quantity of the EDTA and the fluorescein (for flow visualization) added to the working fluid had negligible effect on Table 1 Rheological parameters of the CMC solution at different concentrations obtained by extrapolation of the power-law relation fit. CMC (%)
n
k (Pa s−n )
0.25 0.3 0.4 0.5 0.6 0.7
0.75 0.73 0.68 0.6 0.48 0.49
0.097 0.15 0.3 0.67 2.28 2.75
= 0 + k|| ˙ n ˙ = 0
for > 0 for ≤ 0
(5)
For the Carbopol solution, two different rheometers of coneand-plate geometry were used: the Carrimed (model Weissenberg), which is a constant shear rate rheometer, and the Carrimed-CS, which is a constant-torque rheometer. The yield-stress measurements could be done only with a rheometer that allows measurements with controlled stress. The principle of this technique is to apply successively increasing stresses to the working fluid; as long as the stress does not exceed the yield stress of the fluid, the response is that of a solid. Fig. 3 shows the variation of shear stress with shear rate for different concentrations of Carbopol solutions with shear rate varying from 0.1 to 1000 s−1 . The parameters in Table 2 are obtained by using the Herschel–Bulkley model (Eq. (5)) to fit the data in Fig. 3. The yield stresses in Table 3 are the real yield stresses of the Carbopol solution obtained by using the constant-torque rheometer (rather than that obtained by extrapolation of the Herschel-Bulkley fit) and are smaller than those given in Table 2. We now consider the definition of the Dean number in rectangular cross section ducts for the non-Newtonian fluids used in this study. As expressed before, the Dean number is directly related to the Reynolds number (Eq. (1)). Expressions for the Reynolds numTable 3 Real yield stress of the Carbopol solution at different concentrations. Note these value are obtained by constant stress rheometry, rather than that obtained by extrapolation of the Herschel–Bulkley fit. Carbopol (%)
0 (Pa)
0.15 0.2 0.3
0 1 5
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Table 4 Dimensionless geometrical parameter proposed by Delplace and Leuliet [11]. b/a
1 7.113
1.33 7.238
4 7.774
5 9.116
8 9.787
∞ 12
ber for Newtonian fluids flowing in tubes of circular cross section and between two parallel plates have been established by Metzner and Reed [10]. Using the Metzner and Reed [10] relation, Delplace and Leuliet [11] expressed the Reynolds number for power-law fluids in ducts of square and rectangular cross section for different aspect ratios as Re =
2−n n Dh Um
(6)
n
k{(24n + )/(24 + )n} n−1
Fig. 4. Coordinates system.
where is the fluid density. This expression is based on a dimensionless geometrical parameter known for many rectangular cross sections (Table 4). In the rectangular cross section, the expression of the Reynolds number for the Herschel–Bulkley (H–B) fluids was obtained by multiplying the expression of the Reynolds number for the powerlaw fluids in the same geometry by a factor Rec related to the yield stress. The expression of the Reynolds number for the Herschel–Bulkley (H–B) fluids in the rectangular cross section is displayed in Eq. (7) where Re|Power-Law fluids is the Reynolds number for power-law fluids in ducts of rectangular cross section (Eq. (6)) and Rec is the contribution of the yield stress in the flow displayed in Eq. (7). ReH–B = Re|Power-Law
fluids
(7)
Rec
Rec is defined in Devienne et al. [12] and displayed in Eq. (8) where 0 = 0 /w , is the dimensionless unsheared plug radius and w is the wall shear stress.
Rec
= (1 − 0 ) 1 −
n 02 0 − n+1 n+1
n
(8)
In the present work, w is the mean wall shear stress related to the pressure gradient and knowing the value of 0 from the rheological study, a value of 0 is determined. Note that all non-Newtonian fluids used in this paper can be regarded as special cases of the Herschel–Bulkley model. If n = 1 and 0 = 0 the model reduces to the Newtonian model. If n = / 1 and 0 = 0 the model reduces to the power-law model, and if n = 1 and 0 = / 0, this model reduces to the Bingham model. 3. Numerical methodology The continuity and time-independent three-dimensional Navier–Stokes equations governing incompressible laminar flow in a curved channel of rectangular cross section were solved using the well established code Fluent. This code has been used extensively in the calculation of complex flows (see Refs. [13–15] for recent examples). The numerical methodology for Newtonian fluids was developed in Fellouah et al. [9]. Here, we extend the earlier work (Ref. [9]) to include the methodology for non-Newtonian fluids. The apparent viscosity of the fluid ap = /, ˙ needed for the expression of the shear stress tensor, is displayed in Eq. (9) for the Herschel–Bulkley fluids. ap =
o + k˙ n−1 ˙
(9)
This apparent viscosity presents a singularity when the shear rate vanishes. To overcome this singularity, a user defined function (UDF) based on a model valid for both sheared and unsheared areas was developed and added to the CFD code. Three models have
been developed in the literature: the Papanastasiou model [16], the Bercovier and Engelman model [17], and the O’Donovan and Tanner [18] model (bi-viscosity model). These three models have the same physical meaning: they transform the unsheared zone of the yield-stress fluid into a high-viscosity region. In the present study, the Papanastasiou model, displayed in Eq. (10), is used with a stress exponent of m = 105 higher than the minimum m = 103 recommended for use of the Papanastasiou model with confidence. ap =
0 (1 − e−m˙ ) + k˙ n−1 ˙
(10)
where m is the stress growth exponent. Fig. 4 shows the system coordinates used to study the curved rectangular channel flow. Experiments were performed in a channel with rectangular cross section of width b = 160 mm and a height a = 20 mm; b/a = 8 and Rc /Dh = 10 are the aspect and the curvature ratios, respectively. Various aspect ratios and curvature ratios were studied in the numerical work. Velocity components in the x, y and z directions are u, v and w, respectively. The flat mid-planes are the planes that pass by the centre of the channel cross section (x = 0) and are parallel to the flat walls; the curved mid-planes are the planes that pass by y = 0 and are parallel to the curved walls. The dimensions of the curved ducts studied were chosen such that the height a = 20 mm was constant. A symmetry condition about the plane x = 0 was used to reduce the computational burden for large geometries (b/a ≥ 8). From Fig. 4, the hydraulic diameter is Dh =
2ab . a+b
The segregated solver is used to iterate the nonlinear partial differential equations since this solver is better adapted to the large grids used in our work. To calculate the pressure at the nodes the body-force-weighted scheme is used, which works well when inertial forces are not negligible. The SIMPLEC algorithm is used for pressure–velocity coupling in order to obtain the pressure in the continuity equation. For the momentum equation, the second order quick scheme is used because it is recommended in the presence of rotating flows. The no-slip boundary condition at the wall is used. Gambit was used to generate the grid or mesh for the Fluent CFD solver. Calculations start at a straight channel upstream of the curved duct. At the entry of the straight channel, a uniform velocity profile equal to the mean velocity is imposed. At the channel exit, the flow is hydrodynamically developed and enters the curved duct. Convergence is controlled by following the evolution of the residuals for the velocity components and the continuity equation. In the convergence, the evolution residuals must converge toward small (in all cases less than 10−6 ) and stable quantities. The time necessary to reach convergence varied; it depended on the curvature and aspect ratios and on the fluid considered. To optimize the mesh grid, several grids were tested for a curved duct of square
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Fig. 5. Experimental apparatus.
cross section, ranging from 20 to 34 nodes on the flat and curved walls and from 5 to 10 mm between two successive grid surfaces (see Ref. [9] for details). For this work, it was reasonable to use 24 nodes on the flat and curved walls, corresponding to square meshes of 0.83 mm × 0.83 mm on the surface, and 10 mm between two adjacent grid surfaces. For yield-stress fluids, the Dean number depends on the dimensionless radius of the plug-flow region 0 = 0 /w . For this reason, 0 was initialized to start simulations for a given Dean number. At the convergence of these simulations, the pressure loss ( p) was obtained on the straight channel. Given the relation between the pressure loss and the average wall shear stress w (w = (Dh /4)( p/L), where L is the axial length of the straight channel), the average wall shear stress w was determined. Since the yield shear stress 0 is known for the fluid used, the exact value of 0 was calculated and used to correct the Dean number.
Fig. 6. Flow visualization observed at different cross sections, b/a = 8, Rc /Dh = 10, power-law fluid (n = 0.75).
4. Experimental setup The experimental setup used was the same as that described in Fellouah et al. [9] and consists of a water channel facility (Fig. 5). A volumetric pump generates flow from the tank to the test section, which was preceded by a 2.6 m straight PVC duct for the development of Poiseuille flow and also to ensure stable and clean inlet conditions. The test section was composed of a 100 mm straight channel, a curved channel with curvature angle 180◦ and curvature ratio 10, and finally another 100 mm straight channel. The test section was made of Plexiglas to facilitate flow visualization and had a rectangular cross section a × b = 20 mm × 160 mm, giving an aspect ratio b/a = 8 large enough to minimize the influence of the side walls. A numerically controlled milling machine was used to fabricate the 180◦ bend and the straight length to precise dimensions (±0.02 mm). The bend and straight length were finely polished at the ends. A 200 mm PVC straight channel at the test section exit eliminates any effect of the adapter (rectangular–circular) on the flow there. The flow was visualized by the laser-induced fluorescence (LIF) technique. Fluorescein solutions of mass concentration 1 mg/l were used so that the density variation in the working fluid due to the tracer addition was negligible. Dye was injected with a syringe upstream of the settling chamber and was distributed throughout the channel section to allow overall visualization. 5. Results and discussions In this work, the critical Dean number was obtained by two methods: first by using the secondary flow visualization technique, and second by using the axial velocity gradient.
5.1. Secondary flow visualizations Unless otherwise indicated, in all flow visualization photographs, the concave and convex walls were at the top and the bottom respectively and the main flow was perpendicular and away from the page. The numerical visualizations were given by the helicity contour lines. The helicity is defined as the dot product of vorticity and the velocity vector displayed in the following equation: H = (∇ × V )V
(11)
It provides insight into the vorticity component aligned with the fluid stream. Vorticity is a measure of the rotation of fluid elements as they move in the flow field. 5.1.1. Power-law fluids Fig. 6 shows numerical and experimental flow visualization images for a power-law fluid of n = 0.75, at a Dean number close to De = 260 for different observation locations (from 60◦ to 180◦ ). This figure shows two symmetrical corner vortices near the flat end-walls, at the 60◦ position. In the central part of the test section no Dean vortices are visible. A first pair of Dean vortices, due to the Dean instability, appears at the 80◦ position in the central region. The number of visible Dean vortices pairs increases downstream in the curved channel: two pairs at the 90◦ position, three pairs at 120◦ , and four pairs at 150◦ and at the curved channel exit (180◦ ). Each pair of Dean vortices appears as a mushroom-shaped structure. The mushroom ‘stems’ are associated with the upwash regions, where fluid is transported from the concave wall to the cross section centre. The mushroom ‘caps’ correspond to the vor-
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tex core regions. The bright regions between each mushroom pair correspond to the downwash region where the fluid is transported from the cross section centre to the concave wall. Numerically, at the 60◦ position, the helicity contour lines confirm the configuration of the corner vortices seen in the visualization, but their elongation to the centre of the channel cross section shows the presence of a secondary flow in this region too. This secondary flow was not observed experimentally since it was weak in this region; only at the 120◦ position did several Dean vortices begin to form. The number of Dean vortices pairs at 150◦ is four and the vortex strength is intensified at 180◦ . Both numerical and experimental visualizations show that the first pair of Dean vortices appears at the centre of the channel cross section. This is due to the symmetrical circulation of the corner vortices and the extension of their influence to the centre of the cross section. It can be noted that the Dean vortices are less developed in the numerical results: they remain more confined close to the concave walls, whereas in experiments they rapidly occupy the full channel height. Fig. 7 presents the effect of the Dean number on the flow of the power-law fluid in the exit section of the curved rectangular channel (180◦ ). The experimental flow visualizations show a flow
Fig. 7. (Continued).
Fig. 7. Flow visualization observed for different Dean numbers at = 180◦ , b/a = 8, Rc /Dh = 10, power-law fluid (n = 0.75).
with two symmetrical corner vortices up to De = 110. At De = 122, the tracer shows perturbations that are confined in the vicinity of the concave wall; these perturbations increase in intensity with increasing Dean number to form a mushroom stem at De = 128. The stem continues to develop and at De = 139 gives rise to a mushroom form; at De = 150 a pair of fully developed Dean vortices is visible. In this flow development, the primary difference from the Newtonian case was due to the power-law index of the fluid; characterized by weak viscosity in regions of high shear stress near the wall and high viscosity in the regions of weak shear stress. Thus, Dean vortices develop easily in the wall region, but their difficulty in penetrating the high-viscosity region and entraining the fluid particles delays the achievement of their final form, the mushroom-shaped structure. The pair of Dean vortices observed at De = 150 intensifies at De = 163, a second pair of vortices is observed at De = 170, a third pair between De = 185 and De = 248, and a fourth pair appears at De = 263. This was the largest number of Dean vortices seen up to De = 342. Numerically, the first perturbation was observed at De = 150. Three pairs of Dean vortices were identifiable at De = 180; they continue their development at De = 200 and the fourth pair appears at De = 220. The vortex intensity increases with increasing Dean number and they become irregularly spaced at De = 300. Two more Dean vortices pairs appear at De = 320 and 340. Comparison of the experimental and numerical visualizations revealed a difference in the dynamic development of the Dean vortices. In fact, the evolution was clearer in the experimental study,
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Fig. 8. Flow visualization obtained at = 180◦ for the Newtonian fluid (water n = 1) and the power-law fluid (n = 0.75), b/a = 8, Rc /Dh = 10.
where the Dean instability starts in the wall region on the concave side of the channel and then develops into Dean vortices. The development of the flow and the maximum number of Dean vortices obtained numerically and experimentally differ; probably because the numerical study does not take into account the physical constraints and perturbations inherent in the experiments. To give a visual feeling for the effect of the power-law index on the Dean instability, Fig. 8 shows the experimental flow evolution at the exit section (180◦ ) of the rectangular curved channel for several Dean numbers for a Newtonian (n = 1) and a power-law (n = 0.75) fluid. In these visualizations, the Dean vortices appear rapidly in the Newtonian fluid and occupy the entire channel cross section. However, in the power-law fluid they appear gradually, first in the wall region of the concave side and then developing to occupy the entire cross section channel width. This is due to the variation in viscosity in the power-law fluid: it is weak near the wall and high in the cross section centre. If the critical Dean number is considered to correspond to the complete formation of the first Dean vortices (the first mushroom-shaped structure), these visualizations showed a shift in critical Dean number in the power-law fluid from 137 for the Newtonian fluid to 150 for the power-law fluid (n = 0.75). If, on the other hand, the critical Dean number is considered to correspond to the beginning of tracer perturbation on the concave wall, this shift in the critical Dean number was less pronounced. This result is plausible since, if the shear stress is large enough to increase the non-Newtonian fluid viscosity to that of the Newtonian fluid, then the balance between viscous and centrifugal force must remain unchanged and must result in the same critical Dean number. For high Dean numbers (around 250), the flow structure was similar for the two types of fluids. In order to show the influence of the power-law index, a power-law fluid with a weaker power-law index was considered numerically. Fig. 9 presents the secondary flow patterns at the exit of a 180◦ curved rectangular channel for different Dean numbers for a power-law fluid of power-law index 0.5. It can be seen that at low Dean numbers the vortices are confined near the concave wall, similar to the experimental observations for the power-law fluid (n = 0.75). Fig. 10 shows the development of the flow as a function of the critical Dean number and the position in a curved channel. The lower area corresponds to the flow without the Dean instability.
Fig. 9. Flow visualization observed for different Dean numbers at = 180◦ , b/a = 8, Rc /Dh = 10, power-law fluid (n = 0.5).
Fig. 10. Critical Dean numbers, obtained from the experimental visualizations, at different angular positions, b/a = 8 and Rc /Dh = 10, power-law fluid (n = 0.75). Area 1: flow without Dean instability; Area 2: flow with 1 pair of Dean vortices; Area 3–5: flow with 2–4 pairs of Dean vortices, respectively.
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Table 5 Critical Dean number obtained from the experimental flow visualization for the Newtonian fluid (Ref. [9]) and the power-law fluid (n = 0.75), b/a = 8, Rc /Dh = 10. Angular position (◦ )
Number of vortices pairs
Water (n = 1)
CMC 0.25% (n = 0.75)
180
1 2 3 4
130 135 140 230
150 170 185 263
150
1 2 3 4
150 160 185 270
162 187 239 285
120
1 2 3 4
160 180 200 285
181 215 258 329
90
1 2 3
190 220 260
222 243 314
80
1 2 3
220 260 320
272 300
The other areas correspond to the flow with the Dean instability. This figure shows that the Dean instability starts at an angular position of 80◦ and that the critical Dean number decreases in the axial direction. This is due to the axial development of the flow along the curved channel, which increases the secondary flow intensity. The evolution of the critical Dean number in Fig. 10 is analogous to that in the Newtonian fluid. At a given angular position, the secondary flow intensity increases with increasing Dean number and hence extra Dean vortices appear. Table 5 summarizes the critical Dean numbers for the power-law fluid (n = 0.75) at different positions and compares them with those for the Newtonian fluid. 5.1.2. Yield-stress fluids The qualitative study described above of the effect of the power-law index showed that the onset of the Dean instability, as characterized by the appearance of the Dean vortices as a mushroom-shaped structure, was delayed as this index decreases. Another aim of the present work was to study the Dean instability in a yield-stress fluid. For this purpose, as a first step, the flow evolution of a Bingham fluid along a curved channel (b/a = 8, Rc /Dh = 10) for a Dean number of 405 (chosen to be supercritical to show Dean vortex development) was studied numerically. Results are presented as helicity contour lines in Fig. 11. From positions 90◦ to 150◦ , the helicity contour lines show no variation in the flow pattern. These contour lines revealed information related to the shape of the recirculation zones that exist only in the sheared zone near the flat walls. These zones differ in form from those obtained for the Newtonian fluids, which extend further into the channel cross section centre. In fact, the helicity contour lines are bent around the unsheared region of the Bingham fluid. A similar but less pro-
Fig. 11. Numerical flow visualization observed at different cross sections, De = 405, b/a = 8, Rc /Dh = 10, Bingham fluid (Bm = 10).
nounced deformation was also observed in the power-law fluid. At an angular position of 180◦ , two more closed helicity contour lines appear near the concave wall in the centre of the channel cross section; they are confined in the sheared region. In addition to the disadvantages related to the qualitative detection of the onset of Dean vortex based on the observation of helicity contours there is another fact to be considered in the case of yield-stress fluids. In this case, the helicity contour lines presented in Fig. 11 should be considered carefully, they revealed the complicated nature of this yield-stress fluids flow with features not encountered in Newtonian flows. The determination of the exact shape of the yielded and unyielded surfaces in general Herschel–Bulkley fluid flows remains difficult due to the singular nature of the constitutive relation [19]. The Papanastasiou model used here is an approximation to the singular model and, as explained above, when ˙ → 0 this model behaves like viscous fluids. By definition then, the predictions of such model are only viscous approximations to the ideal Herschel–Bulkley flow. Hammad et al. [20] and Burgos and Alexandrou [19] found that during the flow of the Herschel–Bulkley fluids, dead zones (or stagnant flow) are formed in the corners of the channel. This can lead to predict vortices in cases where the flow is stagnant. The stagnant zones exist due to the low stress levels and, therefore, the lack of deformation of the fluid elements. As we will see below, by no way our results concerning the detection of the onset of Dean instability are affected by such presentation of contour lines. 5.2. Detecting the onset of Dean instability As shown in the flow visualizations, the instability always starts in the centre of the midplane (y = 0) near the concave wall. For this reason, the critical Dean number was detected only at this part of the plane. The first step consists in measurement for different Dean numbers, both numerically and experimentally, the axial velocity along segments [AA ] and [BB ] (Fig. 12). In this experiment, the length of these segments was 40 mm, which was sufficient to detect the formation of the first pair of Dean vortices. Then, for each x-coordinate position, the axial velocity gradient (dw/dy) was obtained. In evaluating the experimental instability criterion, the data were contaminated with measurement noise and an inverse method was applied for filtering. An easy and fast way to detect these inflection points was to calculate the slope of the axial velocity gradient profile along the x-coordinate for different Dean numbers.
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Fig. 12. Position of the measurements in the cross section.
Figs. 13 and 14 show the slope of the axial velocity gradient respectively for a Newtonian fluid and a power-law (n = 0.75) fluid obtained numerically and experimentally at the 180◦ position (channel exit). They show that the formation of the Dean vortices modifies the profile of the axial velocity gradient in the centre of the mid-plane, as demonstrated by the formation of one or more inflection points in the profile. These inflection points are due to the formation of the Dean vortices, which modify the wall shear stress. These slopes are uniform for low Dean numbers and contain perturbations (undulations) due to the normal velocity gradient caused by the formation of the Dean vortices for high Dean numbers. The difference in shape of the slopes obtained numerically (sinusoidal) and experimentally (asymmetric) can be attributed to measurement errors, primarily caused by vibration of the curved channel at large Dean numbers because of high pressure.
Fig. 14. Slope of the axial velocity gradient profiles in the centre of the mid-plane for different Dean numbers, power-law fluid (n = 0.75), b/a = 8 and Rc /Dh = 10, = 180◦ . (a) Numerical result; (b) experimental result.
Fig. 13. Slope of the axial velocity gradients profile in the centre of the mid-plane for different Dean numbers, Newtonian fluid, b/a = 8 and Rc /Dh = 10, = 180◦ . (a) Numerical result; (b) experimental result.
The variation in the slope form, from uniform to perturbed, is considered a signature of the Dean instability threshold. The critical Dean numbers can readily be determined, both numerically and experimentally, from Figs. 13 and 14. These results show that the numerical or experimental critical Dean numbers obtained by measuring the axial velocity gradient are below those obtained by numerical and experimental visualizations. This result confirms the precision of the criterion based on the axial velocity gradient adopted here to determine the onset of the Dean vortices. Henceforth, unless noted, critical Dean numbers were obtained by using the axial velocity gradient criterion. Fig. 15 shows the influence of the two rheological parameters, the power-law index and the Bingham number, on the onset of Dean instability. The critical Dean number decreases with increasing power-law index n due to the local change in apparent viscosity: the viscosity is weaker near the wall and stronger at the centre of the cross section, and thus it is difficult for unstable Dean vortices to form in the centre. All of these factors lead to a delay in the onset of the Dean instability with increasing power-law index. Increasing the Bingham number delays the onset of the Dean instability. In addition, when the Bingham number increases, the yield-stress effect on the flow also increases. To overcome the effect of the yield stress, the inertial forces and hence the Dean number must increase as well. Previous observations by Alexandrou et al. [21] showed that
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Fig. 17. Effect of aspect ratio on Dean instability onset, numerical result. Rc /Dh = 10.
Fig. 15. Effect of rheological parameters on the onset of Dean instability, numerical result.
Fig. 16. Effect of curvature ratio on the Dean instability onset, numerical result.
the size of the plug region increases with the Bingham number, as confirmed in the present work. The effect of the curvature ratio (Rc /Dh ) on the critical Dean number for two aspect ratios b/a = 1 and 8 is shown in Fig. 16. In particular, no curvature ratio effect on the critical Dean number for aspect ratio 1 was observed, whereas for aspect ratio 8, the critical Dean number decreases rapidly when the curvature ratio increases
to 10 and then becomes constant. This evolution is enhanced by the decrease in length of the curved channel; the hydraulic diameter was constant and therefore a reduction in curvature ratio implies a decrease in length of the curved channel. For a square cross section, this reduction is weak. The shape of the curve in this figure shows that the variation in the critical Dean number with curvature ratio is independent of the rheological parameters, which induce only a shift in the critical Dean number values. This result is confirmed in Fig. 17, showing the evolution of the critical Dean number with aspect ratio for different types of fluids (the curvature ratio was held constant at 10). The variation in critical Dean number with duct aspect ratio is noticeably less regular than with Rc /Dh . The instability, like the main flow, develops along the curved channel. Fig. 18 shows the onset of the Dean instability at different angular positions for Rc /Dh = 10 and b/a = 8. The axial position of the onset of instability increases with decreasing power-law index and Bingham number. This position is further downstream in the curved channel for the Bingham fluid than for the power-law fluid. In fact, it is more difficult for the Dean vortices to penetrate the yield region of the Bingham fluid. The numerical results in Fig. 18 show that the critical Dean number reaches its constant value more quickly when the power-law index increases and the Bingham number decrease, demonstrating that the development of the Dean instability along the curved channel is faster in Newtonian than in non-Newtonian fluids. For the non-Newtonian fluids the variation in viscosity, which is higher in the centre of the channel cross section, is responsible for this slow development of Dean instability.
Fig. 18. Evolution of the critical Dean number with the angular position, Rc /Dh = 10, b/a = 8.
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6. Conclusions The effects of the power-law index and yield stress on the Dean instability in curved channels of rectangular cross section were studied numerically and experimentally. The experimental study was carried out in a curved channel of aspect ratio 8 and curvature ratio 10, and was extended numerically to various other aspect and curvature ratios. The results showed that in one hand, in the case of the complete formation of Dean vortices, the critical value of the Dean number decreases with increasing power-law index and decreasing Bingham number. On the other hand, the onset of the Dean vortices in the wall zone of the concave wall was similar in all fluids: varying the rheological parameters merely shifted the position of the critical Dean number. Comparison of the numerical and experimental results using the radial gradient of axial velocity gradient criterion was satisfactory in predicting the critical Dean number. The flow visualizations showed that the effect of the yield stress on the shape of the flow appears to be the dominant parameter. Acknowledgments This research was supported by the French Ministry of Research and Technology. The authors are grateful for their financial support. References [1] W.H. Finlay, K. Nandakumar, Onset of two-dimensional cellular flow in finite curved channels of large aspect ratio, Phys. Fluids A 2 (7) (1990) 1163–1174. [2] W.R. Dean, Note on the motion of fluid in a curved pipe, Philos. Mag. 4 (1927) 208–223. [3] Y.I. Cho, J.P. Hartnett, Non-Newtonian fluids in circular pipe flow, Adv. Heat Transfer 15 (1982) 59–141. [4] V.R. Ranade, J.J. Ulbrecht, Velocity profiles of Newtonian and non-Newtonian toroidal flows measured by a LDA technique, Chem. Eng. Commun. 20 (1982) 253–272.
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