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Marangoni convection in model of floating zone under microgravity
Journal of Crystal Growth 229 (2001) 601–604
Marangoni convection in model of floating zone under microgravity Z. Zenga,*, H. Mizusekia, K. Shimamuraa, K. Higashinob, T. Fukudaa, Y. Kawazoea a
Institute for Materials Research, Tohoku University, Kawazoe Laboratory, 2-1-1 Katahira, Aoba-ku, Sendai 980-8577, Japan b Space Experiment System Development Department, IHI, Tokyo 190-1297, Japan
Abstract Marangoni convection in a half-zone model, a simplification of the floating zone model, is steady and axisymmetric for Marangoni number Ma5Mac, but the azimuthal flow after instability (Ma>Mac) breaks the axial-symmetry, and m-fold symmetry of the flow pattern is observed. The symmetry number m (azimuthal wave number) depends strongly on the aspect ratio, As (As=height/radius), of the half-zone. Based on the results of numerical simulation, the nature of the correlation between m and As is investigated # 2001 Elsevier Science B.V. All rights reserved. PACS: 47.20.Dr Keywords: A1. Convection; A1. Fluid flows; A1. Heat transfer; A2. Floating zone technique
1. Introduction Benefitting from the reduction of buoyancy convection and hydrostatic pressure under microgravity, the floating zone becomes a promising containerless technique to realize large and highquality crystal. Marangoni convection in floating zone is important in controlling mass and heat transfer, and therefore final crystal quality. Since the importance of Marangoni convection in floating zone was pointed out [1], a large number of literature has contributed to the study of Marangoni convection in floating zone and *Corresponding author. Tel. +81-22-215-2057; fax: +81-22215-2052. E-mail address: [email protected] (Z. Zeng).
especially in a floating zone model: half-zone. The half-zone model consists of a liquid bridge held by surface tension and suspended between two rigid discs with different temperatures, as in Fig. 1. The main flow features in floating-zone can be captured in half-zone, therefore the half-zone model is widely adopted in the study of Marangoni convection, and it is also adopted in the present study. The mechanism leading to the instability is thermal (hydrothermal waves) for high-Pr flow [2–4] while it is mechanical (inertial) in low-Pr melts [4,5]. The instability is accompanied by a change of flow pattern: from axisymmetry to mfold (approximate) symmetry. The symmetry number m, also called the azimuthal wave number, has been found to depend strongly on As. The relationships between As and m: As*m=2 and
0022-0248/01/$ - see front matter # 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 0 2 4 8 ( 0 1 ) 0 1 2 3 6 - 2
602
Z. Zeng et al. / Journal of Crystal Growth 229 (2001) 601–604
As*m 2.2 based on linear stability analysis [4] and experiment [6–8], respectively, were reported. A fundamental question arises from the above reports: what is the nature of the correlation for As*m 2.2 or 2? An investigation on the correlation between m and As is conducted in the present paper.
Fig. 1. Geometry and coordinate system of the present model.
2. Correlation between Azimuthal wave number m and As The study on the correlation between m and As is based on the results of the numerical simulations for three-dimensional unsteady Marangoni convection in half-zone. The details on physical and mathematical models and numerical technique can be found in Ref. [9]. In half-zone model, unbalance surface tension drives the surface flow from hot to cold discs, and this flow penetrates into bulk of liquid in some degree due to viscosity. The backflow is brought about by continuity and an axisymmetric convection pattern is formed. The axisymmetry is broken after instability. Here, we only concentrate on the investigation on the spatial structure characteristics of flow after the first instability. Fig. 2 shows the flow structure in z*=0.5 for high-Pr flow (Pr=16.08) with As=1. A structure characteristic of two-fold symmetry in z-cut is observed. This flow is in the periodic oscillatory regime, which is confirmed by checking the frequency spectrum obtained by fast Fourier
Fig. 2. (a) Dimensionless temperature contour line (increment 0.025), and (b) projected velocity vector at z*=0.5 within one period for Pr=16.08, Ma=2.64 104 and As=1.
Z. Zeng et al. / Journal of Crystal Growth 229 (2001) 601–604
Fig. 3. Two vortex-pairs model.
transfer of temperature–time curve. The two vortex-pairs model as shown in Fig. 3 can be used to describe this flow structure. For low-Pr flow (Pr=0.01) with As=1, two vortex-pairs and 2-fold symmetrical temperature structure are also obtained as same in Ref. [5]. This stationary threedimensional flow after the first instability is different from periodic oscillatory flow for highPr flow. Beside the temporal feature, the azimuthal surface flow directing from cold to hot spots for low-Pr flow is contrary with high-Pr flow. These different spatio-temporal features imply the difference of instability mechanism between low-Pr and high-Pr flow. Here, there is no intention to explain the mechanism of instability, and interested readers are referred to [2–5]. In fact, the small disturbance always exists not only in real experiment but also in numerical simulation, because the numerical error cannot be avoided completely. For the small driving force (Ma5Mac), all small disturbances are always damped, and a steady axisymmetric flow structure is observed. With the increment of the driving force (Ma), instability occurs while small disturbance is amplified for Ma>Mac. The vortex structure in r–f plane is formed as shown in Fig. 4. Due to the cyclic boundary condition along azimuthal direction, the disturbance vortex in z-cut should always appear in pairs and cover the whole circumference region, and the number of vortex pairs (symmetry number m) is decided by the size of disturbance vortex in zcut. As reported by Preisser et al. [6], the radial extension of roll equaled the zone length H when As41, and it means the characteristic vortex size (LB) of basic flow in r–z plane is about H (refer to
603
Fig. 4. The sketch for evaluation of characteristic size of basic vortex in r–z plane and disturbance vortex in r–F plane.
Fig. 4). It seems plausible to assume that the characteristic size of disturbance vortex (LD) approximately equals the vortex size of basic flow (LB). In other words, LD H for As41. For m vortex pairs with characteristic size H, the covered circumference size is about 2mH, and which should equal roughly the whole circumference size, 2p(RH/2), then As*m p(10.5*As) 1.573.14 for As41 is derived. This conclusion can explain why the azimuthal wave number m strongly depends on As. A good evidence in literature to support the above analysis can been found in Ref. [10], in which 1.64As*m43.2 was reported for Pr=0.02 and As=0.4–1.8 with linear stability analysis. We should bear in mind that As can vary continuously, and the number of vortex pairs m can only be an integer, therefore the above explanation can predict approximately m2 [INT(A), INT(A)+1] with A=p(10.5*As) while INT(A) is taken as an integer with A15INT(A)4A. Here, the expected m by m 2 [INT(A), INT(A)+1] is derived based on a rough evaluation on vortex size. Therefore, a small deviation between expected m and observed m is possible, but it does not deny the above explanation on the nature of correlation between m and As. A comparison between resultant m by numerical simulation and expected m by above method is listed in Table 1, and agreement is good. Comparing with resultant m by numerical simulation as in Table 1 for As=0.2 and 0.3, although there is a deviation of 1 with our expected m, this deviation is smaller than expected m based on correlation of As*m 2.2 [6] and As*m=2 [4].
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Z. Zeng et al. / Journal of Crystal Growth 229 (2001) 601–604
Table 1 Comparison with resultant m and expected m by [INT(A), INT(A)+1] for As41 with Pr=16.08 and Ma=2.64 104 a Aspect ratio (As)
Resultant m
Expected m
1.0 0.9 0.7 0.6 0.55 0.5 0.4 0.3 0.2
2 2 3 3(4) 4 4(5) 6 10 16
[1, 2] [1, 2] [2, 3] [3, 4] [4, 5] [4, 5] [6, 7] [8, 9] [14, 15]
a
The azimuthal wave number in the beginning of the oscillation is given in parenthesis.
model is proposed to describe observed flow structure in z-cut. By evaluating approximately the characteristic size of disturbance vortex in r–F as the characteristic vortex size of basic flow in r–z plane for As41, the correlation between m and As is derived as As*m p(10.5*As) 1.573.14 for As41. This conclusion can explain why the selection of flow pattern (azimuthal wave number m) strongly depends on the As, and it is insensitive with Pr and Ma. The expected m based on derived correlation between m and As has a good agreement with resultant m by numerical simulation.
References As shown in Table 1, the translation of flow pattern from m=4 in the beginning of instability to m=3 for As=0.6 and from m=5 to m=4 for As=0.5 is observed. This implies that the vortex size is slightly adjustable in the evolution of convection for a fixed As. Due to slight adjustable vortex size, it is easy to understand a minor effect of Ma and Pr on the m: m=2 or 3 depends on Pr [10] and Ma [11] for As=1. Compared to the As, the effects of Ma and Pr on m are small. In fact, based on the above explanation, we can derive that the m is mainly decided by As, and is insensitive with Ma (Re) and Pr. This derived conclusion is supported by numerous results in the literature.
3. Conclusions Based on the numerical simulation of Marangoni convection in half-zone, the vortex-pair
[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
C.E. Chang, W.R. Wilocx, J. Crystal Growth 28 (1975) 8. M.K. Smith, S.H. Davis, J. Fluid Mech. 132 (1985) 119. H.C. Kuhlmann, H.J. Rath, J. Fluid Mech. 247 (1993) 247. M. Wanschura, V.M. Shetsova, H.C. Kuhmann, H.J. Rath, Phys. Fluids 7 (1995) 912. M. Levenstam, G. Amberg, J. Fluid Mech. 297 (1995) 357. F. Preisser, D. Schwabe, A. Scharmann, J. Fluid Mech. 126 (1983) 545. R. Velten, D. Schwabe, A. Scharmann, Phys. Fluids A3 (1991) 267. S. Frank, D. Schwabe, Exp. Fluids 23 (1997) 234. Z. Zeng, H. Mizuseki, K. Higashino, Y. Kawazoe, J. Crystal Growth 204 (1999) 395. G. Chen, A. Lizee, B. Roux, J. Crystal Growth 180 (1997) 638. S. Yasuhiro, T. Sato, Y. Akiyama, N. Imaishi, Numerical simulation of three-dimensional oscillatory Marangoni flow in an adiabatic cylindrical half-zone of Pr=1.02 fluid with an aspect ratio of unity, to appear.
Marangoni convection in model of floating zone under microgravity Z. Zenga,*, H. Mizusekia, K. Shimamuraa, K. Higashinob, T. Fukudaa, Y. Kawazoea a
Institute for Materials Research, Tohoku University, Kawazoe Laboratory, 2-1-1 Katahira, Aoba-ku, Sendai 980-8577, Japan b Space Experiment System Development Department, IHI, Tokyo 190-1297, Japan
Abstract Marangoni convection in a half-zone model, a simplification of the floating zone model, is steady and axisymmetric for Marangoni number Ma5Mac, but the azimuthal flow after instability (Ma>Mac) breaks the axial-symmetry, and m-fold symmetry of the flow pattern is observed. The symmetry number m (azimuthal wave number) depends strongly on the aspect ratio, As (As=height/radius), of the half-zone. Based on the results of numerical simulation, the nature of the correlation between m and As is investigated # 2001 Elsevier Science B.V. All rights reserved. PACS: 47.20.Dr Keywords: A1. Convection; A1. Fluid flows; A1. Heat transfer; A2. Floating zone technique
1. Introduction Benefitting from the reduction of buoyancy convection and hydrostatic pressure under microgravity, the floating zone becomes a promising containerless technique to realize large and highquality crystal. Marangoni convection in floating zone is important in controlling mass and heat transfer, and therefore final crystal quality. Since the importance of Marangoni convection in floating zone was pointed out [1], a large number of literature has contributed to the study of Marangoni convection in floating zone and *Corresponding author. Tel. +81-22-215-2057; fax: +81-22215-2052. E-mail address: [email protected] (Z. Zeng).
especially in a floating zone model: half-zone. The half-zone model consists of a liquid bridge held by surface tension and suspended between two rigid discs with different temperatures, as in Fig. 1. The main flow features in floating-zone can be captured in half-zone, therefore the half-zone model is widely adopted in the study of Marangoni convection, and it is also adopted in the present study. The mechanism leading to the instability is thermal (hydrothermal waves) for high-Pr flow [2–4] while it is mechanical (inertial) in low-Pr melts [4,5]. The instability is accompanied by a change of flow pattern: from axisymmetry to mfold (approximate) symmetry. The symmetry number m, also called the azimuthal wave number, has been found to depend strongly on As. The relationships between As and m: As*m=2 and
0022-0248/01/$ - see front matter # 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 0 2 4 8 ( 0 1 ) 0 1 2 3 6 - 2
602
Z. Zeng et al. / Journal of Crystal Growth 229 (2001) 601–604
As*m 2.2 based on linear stability analysis [4] and experiment [6–8], respectively, were reported. A fundamental question arises from the above reports: what is the nature of the correlation for As*m 2.2 or 2? An investigation on the correlation between m and As is conducted in the present paper.
Fig. 1. Geometry and coordinate system of the present model.
2. Correlation between Azimuthal wave number m and As The study on the correlation between m and As is based on the results of the numerical simulations for three-dimensional unsteady Marangoni convection in half-zone. The details on physical and mathematical models and numerical technique can be found in Ref. [9]. In half-zone model, unbalance surface tension drives the surface flow from hot to cold discs, and this flow penetrates into bulk of liquid in some degree due to viscosity. The backflow is brought about by continuity and an axisymmetric convection pattern is formed. The axisymmetry is broken after instability. Here, we only concentrate on the investigation on the spatial structure characteristics of flow after the first instability. Fig. 2 shows the flow structure in z*=0.5 for high-Pr flow (Pr=16.08) with As=1. A structure characteristic of two-fold symmetry in z-cut is observed. This flow is in the periodic oscillatory regime, which is confirmed by checking the frequency spectrum obtained by fast Fourier
Fig. 2. (a) Dimensionless temperature contour line (increment 0.025), and (b) projected velocity vector at z*=0.5 within one period for Pr=16.08, Ma=2.64 104 and As=1.
Z. Zeng et al. / Journal of Crystal Growth 229 (2001) 601–604
Fig. 3. Two vortex-pairs model.
transfer of temperature–time curve. The two vortex-pairs model as shown in Fig. 3 can be used to describe this flow structure. For low-Pr flow (Pr=0.01) with As=1, two vortex-pairs and 2-fold symmetrical temperature structure are also obtained as same in Ref. [5]. This stationary threedimensional flow after the first instability is different from periodic oscillatory flow for highPr flow. Beside the temporal feature, the azimuthal surface flow directing from cold to hot spots for low-Pr flow is contrary with high-Pr flow. These different spatio-temporal features imply the difference of instability mechanism between low-Pr and high-Pr flow. Here, there is no intention to explain the mechanism of instability, and interested readers are referred to [2–5]. In fact, the small disturbance always exists not only in real experiment but also in numerical simulation, because the numerical error cannot be avoided completely. For the small driving force (Ma5Mac), all small disturbances are always damped, and a steady axisymmetric flow structure is observed. With the increment of the driving force (Ma), instability occurs while small disturbance is amplified for Ma>Mac. The vortex structure in r–f plane is formed as shown in Fig. 4. Due to the cyclic boundary condition along azimuthal direction, the disturbance vortex in z-cut should always appear in pairs and cover the whole circumference region, and the number of vortex pairs (symmetry number m) is decided by the size of disturbance vortex in zcut. As reported by Preisser et al. [6], the radial extension of roll equaled the zone length H when As41, and it means the characteristic vortex size (LB) of basic flow in r–z plane is about H (refer to
603
Fig. 4. The sketch for evaluation of characteristic size of basic vortex in r–z plane and disturbance vortex in r–F plane.
Fig. 4). It seems plausible to assume that the characteristic size of disturbance vortex (LD) approximately equals the vortex size of basic flow (LB). In other words, LD H for As41. For m vortex pairs with characteristic size H, the covered circumference size is about 2mH, and which should equal roughly the whole circumference size, 2p(RH/2), then As*m p(10.5*As) 1.573.14 for As41 is derived. This conclusion can explain why the azimuthal wave number m strongly depends on As. A good evidence in literature to support the above analysis can been found in Ref. [10], in which 1.64As*m43.2 was reported for Pr=0.02 and As=0.4–1.8 with linear stability analysis. We should bear in mind that As can vary continuously, and the number of vortex pairs m can only be an integer, therefore the above explanation can predict approximately m2 [INT(A), INT(A)+1] with A=p(10.5*As) while INT(A) is taken as an integer with A15INT(A)4A. Here, the expected m by m 2 [INT(A), INT(A)+1] is derived based on a rough evaluation on vortex size. Therefore, a small deviation between expected m and observed m is possible, but it does not deny the above explanation on the nature of correlation between m and As. A comparison between resultant m by numerical simulation and expected m by above method is listed in Table 1, and agreement is good. Comparing with resultant m by numerical simulation as in Table 1 for As=0.2 and 0.3, although there is a deviation of 1 with our expected m, this deviation is smaller than expected m based on correlation of As*m 2.2 [6] and As*m=2 [4].
604
Z. Zeng et al. / Journal of Crystal Growth 229 (2001) 601–604
Table 1 Comparison with resultant m and expected m by [INT(A), INT(A)+1] for As41 with Pr=16.08 and Ma=2.64 104 a Aspect ratio (As)
Resultant m
Expected m
1.0 0.9 0.7 0.6 0.55 0.5 0.4 0.3 0.2
2 2 3 3(4) 4 4(5) 6 10 16
[1, 2] [1, 2] [2, 3] [3, 4] [4, 5] [4, 5] [6, 7] [8, 9] [14, 15]
a
The azimuthal wave number in the beginning of the oscillation is given in parenthesis.
model is proposed to describe observed flow structure in z-cut. By evaluating approximately the characteristic size of disturbance vortex in r–F as the characteristic vortex size of basic flow in r–z plane for As41, the correlation between m and As is derived as As*m p(10.5*As) 1.573.14 for As41. This conclusion can explain why the selection of flow pattern (azimuthal wave number m) strongly depends on the As, and it is insensitive with Pr and Ma. The expected m based on derived correlation between m and As has a good agreement with resultant m by numerical simulation.
References As shown in Table 1, the translation of flow pattern from m=4 in the beginning of instability to m=3 for As=0.6 and from m=5 to m=4 for As=0.5 is observed. This implies that the vortex size is slightly adjustable in the evolution of convection for a fixed As. Due to slight adjustable vortex size, it is easy to understand a minor effect of Ma and Pr on the m: m=2 or 3 depends on Pr [10] and Ma [11] for As=1. Compared to the As, the effects of Ma and Pr on m are small. In fact, based on the above explanation, we can derive that the m is mainly decided by As, and is insensitive with Ma (Re) and Pr. This derived conclusion is supported by numerous results in the literature.
3. Conclusions Based on the numerical simulation of Marangoni convection in half-zone, the vortex-pair
[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
C.E. Chang, W.R. Wilocx, J. Crystal Growth 28 (1975) 8. M.K. Smith, S.H. Davis, J. Fluid Mech. 132 (1985) 119. H.C. Kuhlmann, H.J. Rath, J. Fluid Mech. 247 (1993) 247. M. Wanschura, V.M. Shetsova, H.C. Kuhmann, H.J. Rath, Phys. Fluids 7 (1995) 912. M. Levenstam, G. Amberg, J. Fluid Mech. 297 (1995) 357. F. Preisser, D. Schwabe, A. Scharmann, J. Fluid Mech. 126 (1983) 545. R. Velten, D. Schwabe, A. Scharmann, Phys. Fluids A3 (1991) 267. S. Frank, D. Schwabe, Exp. Fluids 23 (1997) 234. Z. Zeng, H. Mizuseki, K. Higashino, Y. Kawazoe, J. Crystal Growth 204 (1999) 395. G. Chen, A. Lizee, B. Roux, J. Crystal Growth 180 (1997) 638. S. Yasuhiro, T. Sato, Y. Akiyama, N. Imaishi, Numerical simulation of three-dimensional oscillatory Marangoni flow in an adiabatic cylindrical half-zone of Pr=1.02 fluid with an aspect ratio of unity, to appear.