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On the mixed convection hydrodynamic force on a sphere
Aerosol Science 36 (2005) 1177 – 1181 www.elsevier.com/locate/jaerosci
Technical note
On the mixed convection hydrodynamic force on a sphere E. Mograbia , E. Bar-Ziva, b,∗ a Department of Mechanical Engineering, Ben-Gurion University of the Negev, P.O. Box 653, 84105 Beer-Sheva, Israel b Institute for Applied Research, Ben-Gurion University of the Negev, P.O. Box 653, 84105 Beer-Sheva, Israel
Received 14 October 2004; received in revised form 30 November 2004; accepted 2 December 2004
Abstract The hydrodynamic force on a sphere in mixed convection flow field for low Reynolds and Grashof numbers is considered. The note discusses the difficulties in obtaining analytical solutions and suggests an approximate expression for the combined force. The expression is compared with numerical results and with asymptotic derivations. Conditions for the validity of the expression and its advantages and drawbacks in comparison with previous results are discussed. 䉷 2004 Elsevier Ltd. All rights reserved. Keywords: Hydrodynamic force; Free convection; Mixed convection; Buoyant plume; Particle dynamics; Laminar flow
In a recent study, Mograbi and Bar-Ziv (2004) investigated the hydrodynamic force on a moving sphere when there is a temperature difference between the sphere and the surrounding fluid. They considered low Reynolds and Grashof numbers and proposed an empirical formula of the form FD = Ff (Re) + (P r)Gr 1/2 (),
(1)
in which F is force; subscripts D and f denote total force and isothermal forced-convection, respectively; Re is Reynolds number defined by Re=U d/, where U is particle velocity, d its diameter and is kinematic viscosity; Gr is Grashof number defined by Gr =g p d 3 /2 , where p =Tp −T∞ is temperature difference between the particle (Tp ) and the surrounding fluid (T∞ ), g is gravity acceleration and is the coefficient of thermal expansion; P r = /, is Prandtl number, where is thermal diffusivity; is mixed convection parameter defined by = Re2 /Gr. (P r) is an unknown function of Pr. ∗ Corresponding author. Tel.: +972 8 647 7109; fax: +972 8 647 2813.
E-mail address: [email protected], [email protected] (E. Bar-Ziv). 0021-8502/$ - see front matter 䉷 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.jaerosci.2004.12.002
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E. Mograbi, E. Bar-Ziv / Aerosol Science 36 (2005) 1177 – 1181
The Gr 1/2 law is a consequence of the similarity law of Zel’dovich (1937) and was suggested by Fendell (1968) and later by Hieber and Gebhart (1969, denoted as HG) as the scale of the buoyancyinduced velocity of the thermal plume far from the particle. Mograbi and Bar-Ziv implicitly connected this velocity with the free convection force in a similar procedure as for the force induced on a body due to a laminar wake downstream from a submerged body. Under this procedure the force can be shown to relate linearly with the averaged velocity in the cross section of an assumed thin wake region (Landau & Lifshitz, 1987, Section 21). The function () in Eq. (1) was introduced to account for the non-linear interaction of the underlying forced convection and the free convection regimes. Eq. (1) was an attempt to obtain an expression that is valid for the entire range of . It was found to be adequate for the case in which the imposed flow coincides with the rising buoyant plume (or particle motion in the direction of gravity). This configuration was denoted as co-flow by Mograbi and Bar-Ziv (2004). HG obtained results for large (i.e., a small buoyancy effect on a dominant forced convection). Indeed, since the work of HG there are ample studies on free and mixed convection, but the bulk of them are concerned with heat transfer (see discussion in Mograbi & Bar-Ziv, 2004). The validation of Eq. (1) was carried out by numerically simulating the steady-state Navier–Stokes equation subjected to Boussinesq condition (and for P r = 1), supported by experimental measurements. Also, it was shown that the function behaves according to the flow configuration; in opposed-flow it is valid asymptotically as → ∞. In the present note we examined more carefully the arguments and the approach to Eq. (1). The examination will help to understand better the physical significance of the linear and non-linear behavior of the force in connection with the two underlying mechanisms. Let us first recall briefly a known result, of the singular nature of the Stokes equation. The Stokes velocity field around a three-dimensional body (although consistent with regard to the convergence at infinite distance from the source) is valid only at distances shorter than the Oseen length /U (see e.g. Van Dyke, 1964); put differently, a linear zero-order inner term in a perturbation expansion is valid. If buoyancy is added into the Stokes equation the solution results in a diverging velocity field at infinity. This pathology can be explained by considering the solution to the Stokes–Boussinesq (SB) equations, momentum and energy, respectively, u − ∇p = g,
(2)
= 0,
(3)
in which is the Laplacian operator. Eqs. (2) and (3) are solved for axis-symmetric configurations using a procedure similar to the one used by Landau and Lifshitz (1987, Section 20) to yield a velocity field in spherical coordinates, u = U cos (f − f + 1)er + U sin (f − f /r − 1)e ,
(4)
in which f (r)(f = df/dr) satisfies the biharmonic inhomogeneous equation 2 f = a(/).
(5)
Constant a is defined by a = g/U . This result shows the pathology inherent in the SB equations, that is, if
falls off at infinity slower than O(r −2 ), the velocity diverges. This includes the problem of a sphere with uniform temperature for which ∼ O(r −1 ). This result was also obtained by Mahony (1957), Fendell
E. Mograbi, E. Bar-Ziv / Aerosol Science 36 (2005) 1177 – 1181
1179
(1968), and HG by solving the problem of an isothermal sphere using the solution of the stream function and employing the principle of minimum singularity (Van Dyke, 1964). The above-solution helps one to realize the physical mechanism responsible for the diverging field as opposed to the similar isothermal case. The radius of validity is defined by the inequality (u · ∇)u>g. Substituting the velocity field, Eq. (4), in the inertia term gives the order U 2 f 2 /r 5 . Considering Eq. (5) (f/r 4 ∼ g /U ) and substituting into the inequality yields r/d >Gr 1/n−3 ,
(6)
where n is the power of the temperature distribution, ∼ O(r −n ). From inequality (6) it is apparent that the radius of validity of the SB solution depends both on Gr (similar to Re in the isothermal case) and also on the manner in which the temperature decays at infinity. As n decreases, the radius of validity is shortened by orders of magnitude and, consequently, the field divergence is not surprising. Note also that for n 3 the result is everywhere valid and indeed the flow field would not diverge. It is interesting to recall the above pathology in connection with the derivation of Eq. (1). As noted by Mahony (1957), a consistent perturbation scheme for the free convection problem is not forthcoming because the equation in the far field is non-linear, and there is no natural way to simplify it in a manner similar to the Oseen linearization. In contrast, the isothermal Stokes zero-order term has been matched successfully with the Oseen solution (e.g. Proudman & Pearson, 1957) and it constitutes the leading term in the expansion. Hence it is apparent that there is no analytical solution for the coupled force at the entire range of . Nevertheless, aiming to preserve the rationale of Eq. (1) and yet to provide an explicit expression (though approximate) over the entire range of , we argue as follows: take the asymptotic nonlinear scale of the velocity field Ufc ∼ Gr 1/2 and superpose it with the linear Stokes solution without resorting to an hypothesized “universal” function (); the result is U ∼ U∞ + Ufc = (/d)Re(1 + Gr 1/2 /Re).
(7)
Hence FD / ∼ 3 Re(1 + Gr 1/2 /Re).
(8)
The factor √ 3 stems from the Stokes law (the expression in the parenthesis may be written more concisely as 1 + , where is the Richardson number Gr/Re2 = 1/). The questions that arise are: is Eq. (8) a valid approximation, and if so, under what conditions is it valid? To answer these questions we have compared numerical results with Eq. (8). Fig. 1 presents the force versus Re for Gr = 0.01 and 0.0001. For Gr = 0.01 large discrepancies at the middle range of Re exists, corresponding to ∼ O(1). The difference is not surprising because for Gr = 0.01 non-linear effects become dominant for r ∼ 10d (as obtained from inequality (6)). Also, in this range of the interaction of forced and free convection is most pronounced. For low Gr the discrepancies are less pronounced in absolute magnitude, again departing when ∼ O(1). By examining the relative errors obtained for the two Gr values it was found that the mean relative error is the largest for Gr = 0.01 with a value of ≈ 0.29; for the case Gr = 10−4 the mean relative error reduces to ≈ 0.13. The results suggest that the error tends to decrease with Gr (error analysis for Gr = 0.001 yielded a mean value of ≈ 0.22). It is concluded that Eq. (8) is not restricted to large but instead requires Gr → 0. This result also helps to gain insights into the behavior for strong non-linear coupling. When ∼ O(1) the non-linearity acts to diminish the contribution of the buoyant forces, as seen in Fig. 1 (this corresponds
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Fig. 1. Comparison between numerical results, asymptotic behavior, and the results of Hieber and Gebhart (1969) of FD as function of Re, for Gr = 0.01 and 0.0001.
to the behavior of the function (), in Eq. (1), discussed by Mograbi & Bar-Ziv, 2004). As Gr → 0 the non-linear interaction reduces and the two contributions are approximately superposed. Under this restriction the function () reduces to unity, i.e. zero interaction, and the contribution of free convection is considered constant (Gr 1/2 ) throughout the range of Re. On the other hand, for strong interaction restricted to > 1 the results of HG are much more accurate, as seen in Fig. 1 for Gr = 0.01. For reference, we present HG expression of the drag coefficient CB = 24/Re[0.5 + (0.6633/4) Re − (0.56/4) 2 + · · ·],
(9)
2 A, A = 0.25 d 2 ; this expression is the contribution of the buoyancy effect in which CB = FB /0.5 U∞ only, hence the subscript B. The drawback of Eq. (9) is that it does not reduce to the free convection force as Re → 0, rather diverging in the limit, as can be observed from Fig. 1. Instead of the correct Gr 1/2 Eq. (9) yields the first-order term Gr/Re which is unbounded at the origin. A comparison of Eq. (8) with Eq. (9) and results from numerical simulations for low Gr(10−4 ) is also presented in Fig. 1. The asymptotic solution diverges for ∼ O(1), whereas Eq. (8) approximates the behavior quite well throughout the entire range of Re. To conclude, for weak interaction between free and forced convection Eq. (8) was found to adequately represent the force acting on the particle. In the co-flow regime this explicit result is useful for the entire range of Re. In the opposed-flow it is restricted to large . For strong interaction a more precise result is offered by Mograbi and Bar-Ziv (2004).
E. Mograbi is grateful to the Kreitman Foundation for their financial and moral support as a Kreitman Fellow during the course of this study. This study was partially supported by US-Israel Binational Science Foundation (BSF) Grant number 2000163.
E. Mograbi, E. Bar-Ziv / Aerosol Science 36 (2005) 1177 – 1181
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References Fendell, F. E. (1968). Laminar natural convection about an isothermally heated sphere at small Grashof numbers. Journal of Fluid Mechanics, 34, 163–176. Hieber, C. A., & Gebhart, B. (1969). Mixed convection from a sphere at small Reynolds and Grashof numbers. Journal of Fluid Mechanics, 38, 137–159. Landau, L. D., & Lifshitz, E. M. (1987). Fluid mechanics. (2nd ed.), Oxford: Butterworth-Heinemann. Mahony, J. J. (1957). Heat transfer at small Grashof numbers. Proceedings of the Royal Society of London A, 238, 412–423. Mograbi, E., Bar-Ziv, E. 2004. Dynamics of a spherical particle in mixed convection flow field. Journal of Aerosol Science, accepted for publication, doi: 10.1016/j.jaerosci.2004.09.011. Proudman, I., & Pearson, J. R. A. (1957). Expansions at small Reynolds numbers for the flow past a sphere and a circular cylinder. Journal of Fluid Mechanics, 2, 237–262. Van Dyke, M. (1964). Perturbation methods in fluid mechanics. New York: Academic Press. Zel’dovich, Ya. B. (1937). Limiting laws of freely rising convective currents. Zhurnal eksperimental’noi i Teoreteicheskoi fiziki, 7, 1463(in Russian).
Technical note
On the mixed convection hydrodynamic force on a sphere E. Mograbia , E. Bar-Ziva, b,∗ a Department of Mechanical Engineering, Ben-Gurion University of the Negev, P.O. Box 653, 84105 Beer-Sheva, Israel b Institute for Applied Research, Ben-Gurion University of the Negev, P.O. Box 653, 84105 Beer-Sheva, Israel
Received 14 October 2004; received in revised form 30 November 2004; accepted 2 December 2004
Abstract The hydrodynamic force on a sphere in mixed convection flow field for low Reynolds and Grashof numbers is considered. The note discusses the difficulties in obtaining analytical solutions and suggests an approximate expression for the combined force. The expression is compared with numerical results and with asymptotic derivations. Conditions for the validity of the expression and its advantages and drawbacks in comparison with previous results are discussed. 䉷 2004 Elsevier Ltd. All rights reserved. Keywords: Hydrodynamic force; Free convection; Mixed convection; Buoyant plume; Particle dynamics; Laminar flow
In a recent study, Mograbi and Bar-Ziv (2004) investigated the hydrodynamic force on a moving sphere when there is a temperature difference between the sphere and the surrounding fluid. They considered low Reynolds and Grashof numbers and proposed an empirical formula of the form FD = Ff (Re) + (P r)Gr 1/2 (),
(1)
in which F is force; subscripts D and f denote total force and isothermal forced-convection, respectively; Re is Reynolds number defined by Re=U d/, where U is particle velocity, d its diameter and is kinematic viscosity; Gr is Grashof number defined by Gr =g p d 3 /2 , where p =Tp −T∞ is temperature difference between the particle (Tp ) and the surrounding fluid (T∞ ), g is gravity acceleration and is the coefficient of thermal expansion; P r = /, is Prandtl number, where is thermal diffusivity; is mixed convection parameter defined by = Re2 /Gr. (P r) is an unknown function of Pr. ∗ Corresponding author. Tel.: +972 8 647 7109; fax: +972 8 647 2813.
E-mail address: [email protected], [email protected] (E. Bar-Ziv). 0021-8502/$ - see front matter 䉷 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.jaerosci.2004.12.002
1178
E. Mograbi, E. Bar-Ziv / Aerosol Science 36 (2005) 1177 – 1181
The Gr 1/2 law is a consequence of the similarity law of Zel’dovich (1937) and was suggested by Fendell (1968) and later by Hieber and Gebhart (1969, denoted as HG) as the scale of the buoyancyinduced velocity of the thermal plume far from the particle. Mograbi and Bar-Ziv implicitly connected this velocity with the free convection force in a similar procedure as for the force induced on a body due to a laminar wake downstream from a submerged body. Under this procedure the force can be shown to relate linearly with the averaged velocity in the cross section of an assumed thin wake region (Landau & Lifshitz, 1987, Section 21). The function () in Eq. (1) was introduced to account for the non-linear interaction of the underlying forced convection and the free convection regimes. Eq. (1) was an attempt to obtain an expression that is valid for the entire range of . It was found to be adequate for the case in which the imposed flow coincides with the rising buoyant plume (or particle motion in the direction of gravity). This configuration was denoted as co-flow by Mograbi and Bar-Ziv (2004). HG obtained results for large (i.e., a small buoyancy effect on a dominant forced convection). Indeed, since the work of HG there are ample studies on free and mixed convection, but the bulk of them are concerned with heat transfer (see discussion in Mograbi & Bar-Ziv, 2004). The validation of Eq. (1) was carried out by numerically simulating the steady-state Navier–Stokes equation subjected to Boussinesq condition (and for P r = 1), supported by experimental measurements. Also, it was shown that the function behaves according to the flow configuration; in opposed-flow it is valid asymptotically as → ∞. In the present note we examined more carefully the arguments and the approach to Eq. (1). The examination will help to understand better the physical significance of the linear and non-linear behavior of the force in connection with the two underlying mechanisms. Let us first recall briefly a known result, of the singular nature of the Stokes equation. The Stokes velocity field around a three-dimensional body (although consistent with regard to the convergence at infinite distance from the source) is valid only at distances shorter than the Oseen length /U (see e.g. Van Dyke, 1964); put differently, a linear zero-order inner term in a perturbation expansion is valid. If buoyancy is added into the Stokes equation the solution results in a diverging velocity field at infinity. This pathology can be explained by considering the solution to the Stokes–Boussinesq (SB) equations, momentum and energy, respectively, u − ∇p = g,
(2)
= 0,
(3)
in which is the Laplacian operator. Eqs. (2) and (3) are solved for axis-symmetric configurations using a procedure similar to the one used by Landau and Lifshitz (1987, Section 20) to yield a velocity field in spherical coordinates, u = U cos (f − f + 1)er + U sin (f − f /r − 1)e ,
(4)
in which f (r)(f = df/dr) satisfies the biharmonic inhomogeneous equation 2 f = a(/).
(5)
Constant a is defined by a = g/U . This result shows the pathology inherent in the SB equations, that is, if
falls off at infinity slower than O(r −2 ), the velocity diverges. This includes the problem of a sphere with uniform temperature for which ∼ O(r −1 ). This result was also obtained by Mahony (1957), Fendell
E. Mograbi, E. Bar-Ziv / Aerosol Science 36 (2005) 1177 – 1181
1179
(1968), and HG by solving the problem of an isothermal sphere using the solution of the stream function and employing the principle of minimum singularity (Van Dyke, 1964). The above-solution helps one to realize the physical mechanism responsible for the diverging field as opposed to the similar isothermal case. The radius of validity is defined by the inequality (u · ∇)u>g. Substituting the velocity field, Eq. (4), in the inertia term gives the order U 2 f 2 /r 5 . Considering Eq. (5) (f/r 4 ∼ g /U ) and substituting into the inequality yields r/d >Gr 1/n−3 ,
(6)
where n is the power of the temperature distribution, ∼ O(r −n ). From inequality (6) it is apparent that the radius of validity of the SB solution depends both on Gr (similar to Re in the isothermal case) and also on the manner in which the temperature decays at infinity. As n decreases, the radius of validity is shortened by orders of magnitude and, consequently, the field divergence is not surprising. Note also that for n 3 the result is everywhere valid and indeed the flow field would not diverge. It is interesting to recall the above pathology in connection with the derivation of Eq. (1). As noted by Mahony (1957), a consistent perturbation scheme for the free convection problem is not forthcoming because the equation in the far field is non-linear, and there is no natural way to simplify it in a manner similar to the Oseen linearization. In contrast, the isothermal Stokes zero-order term has been matched successfully with the Oseen solution (e.g. Proudman & Pearson, 1957) and it constitutes the leading term in the expansion. Hence it is apparent that there is no analytical solution for the coupled force at the entire range of . Nevertheless, aiming to preserve the rationale of Eq. (1) and yet to provide an explicit expression (though approximate) over the entire range of , we argue as follows: take the asymptotic nonlinear scale of the velocity field Ufc ∼ Gr 1/2 and superpose it with the linear Stokes solution without resorting to an hypothesized “universal” function (); the result is U ∼ U∞ + Ufc = (/d)Re(1 + Gr 1/2 /Re).
(7)
Hence FD / ∼ 3 Re(1 + Gr 1/2 /Re).
(8)
The factor √ 3 stems from the Stokes law (the expression in the parenthesis may be written more concisely as 1 + , where is the Richardson number Gr/Re2 = 1/). The questions that arise are: is Eq. (8) a valid approximation, and if so, under what conditions is it valid? To answer these questions we have compared numerical results with Eq. (8). Fig. 1 presents the force versus Re for Gr = 0.01 and 0.0001. For Gr = 0.01 large discrepancies at the middle range of Re exists, corresponding to ∼ O(1). The difference is not surprising because for Gr = 0.01 non-linear effects become dominant for r ∼ 10d (as obtained from inequality (6)). Also, in this range of the interaction of forced and free convection is most pronounced. For low Gr the discrepancies are less pronounced in absolute magnitude, again departing when ∼ O(1). By examining the relative errors obtained for the two Gr values it was found that the mean relative error is the largest for Gr = 0.01 with a value of ≈ 0.29; for the case Gr = 10−4 the mean relative error reduces to ≈ 0.13. The results suggest that the error tends to decrease with Gr (error analysis for Gr = 0.001 yielded a mean value of ≈ 0.22). It is concluded that Eq. (8) is not restricted to large but instead requires Gr → 0. This result also helps to gain insights into the behavior for strong non-linear coupling. When ∼ O(1) the non-linearity acts to diminish the contribution of the buoyant forces, as seen in Fig. 1 (this corresponds
1180
E. Mograbi, E. Bar-Ziv / Aerosol Science 36 (2005) 1177 – 1181
Fig. 1. Comparison between numerical results, asymptotic behavior, and the results of Hieber and Gebhart (1969) of FD as function of Re, for Gr = 0.01 and 0.0001.
to the behavior of the function (), in Eq. (1), discussed by Mograbi & Bar-Ziv, 2004). As Gr → 0 the non-linear interaction reduces and the two contributions are approximately superposed. Under this restriction the function () reduces to unity, i.e. zero interaction, and the contribution of free convection is considered constant (Gr 1/2 ) throughout the range of Re. On the other hand, for strong interaction restricted to > 1 the results of HG are much more accurate, as seen in Fig. 1 for Gr = 0.01. For reference, we present HG expression of the drag coefficient CB = 24/Re[0.5 + (0.6633/4) Re − (0.56/4) 2 + · · ·],
(9)
2 A, A = 0.25 d 2 ; this expression is the contribution of the buoyancy effect in which CB = FB /0.5 U∞ only, hence the subscript B. The drawback of Eq. (9) is that it does not reduce to the free convection force as Re → 0, rather diverging in the limit, as can be observed from Fig. 1. Instead of the correct Gr 1/2 Eq. (9) yields the first-order term Gr/Re which is unbounded at the origin. A comparison of Eq. (8) with Eq. (9) and results from numerical simulations for low Gr(10−4 ) is also presented in Fig. 1. The asymptotic solution diverges for ∼ O(1), whereas Eq. (8) approximates the behavior quite well throughout the entire range of Re. To conclude, for weak interaction between free and forced convection Eq. (8) was found to adequately represent the force acting on the particle. In the co-flow regime this explicit result is useful for the entire range of Re. In the opposed-flow it is restricted to large . For strong interaction a more precise result is offered by Mograbi and Bar-Ziv (2004).
E. Mograbi is grateful to the Kreitman Foundation for their financial and moral support as a Kreitman Fellow during the course of this study. This study was partially supported by US-Israel Binational Science Foundation (BSF) Grant number 2000163.
E. Mograbi, E. Bar-Ziv / Aerosol Science 36 (2005) 1177 – 1181
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References Fendell, F. E. (1968). Laminar natural convection about an isothermally heated sphere at small Grashof numbers. Journal of Fluid Mechanics, 34, 163–176. Hieber, C. A., & Gebhart, B. (1969). Mixed convection from a sphere at small Reynolds and Grashof numbers. Journal of Fluid Mechanics, 38, 137–159. Landau, L. D., & Lifshitz, E. M. (1987). Fluid mechanics. (2nd ed.), Oxford: Butterworth-Heinemann. Mahony, J. J. (1957). Heat transfer at small Grashof numbers. Proceedings of the Royal Society of London A, 238, 412–423. Mograbi, E., Bar-Ziv, E. 2004. Dynamics of a spherical particle in mixed convection flow field. Journal of Aerosol Science, accepted for publication, doi: 10.1016/j.jaerosci.2004.09.011. Proudman, I., & Pearson, J. R. A. (1957). Expansions at small Reynolds numbers for the flow past a sphere and a circular cylinder. Journal of Fluid Mechanics, 2, 237–262. Van Dyke, M. (1964). Perturbation methods in fluid mechanics. New York: Academic Press. Zel’dovich, Ya. B. (1937). Limiting laws of freely rising convective currents. Zhurnal eksperimental’noi i Teoreteicheskoi fiziki, 7, 1463(in Russian).