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Influence of double slot suction (injection) into water boundary layer flows over sphere
International Communications in Heat and Mass Transfer 36 (2009) 646–650
Contents lists available at ScienceDirect
International Communications in Heat and Mass Transfer j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / i c h m t
Influence of double slot suction (injection) into water boundary layer flows over sphere☆ S. Roy a,⁎, P. Saikrishnan b, Bishun D. Pandey c a b c
Department of Mathematics, Indian Institute of Technology Madras, Chennai-600 036, Tamilnadu, India Department of Mathematics, National Institute of Technology Tiruchirapalli, Tiruchirappali-620 015, Tamilnadu, India Department of Mathematics, The Ohio State University, Marion, OH 43302, USA
a r t i c l e
i n f o
Available online 22 May 2009 Keywords: Slot suction (injection) Sphere
a b s t r a c t The objective of this work is to study the effect of non-uniform double slot suction (injection) into steady laminar water boundary layers over a sphere with variable viscosity and Prandtl number. The implicit finite difference scheme with quasi-linearization technique has been applied to find the non-similar solutions and also to overcome the difficulties arising at the starting point of the stream-wise coordinate, at the edges of the slot and at the point of separation. Results indicate that the separation can be delayed by non-uniform double slot suction and also by moving the slots downstream direction but the effect of non-uniform double slot injection is just the opposite. © 2009 Elsevier Ltd. All rights reserved.
1. Introduction A review of the non-similarity solution methods along with the citations of some relevant publications is given in an earlier study by Dewey and Gross [1]. Since then, there have been several studies pertaining to non-similar flows by finite difference method [2,3] and an implicit finite difference method in combination with quasi-linearization technique [4,5]. Fluid viscosity and thermal conductivity are the main governing fluid properties in the laminar water boundary layer forced flow and hence their variations can be expected to affect separation. Further, mass transfer through a slot strongly influences the development of a boundary layer along a surface and in particular can prevent or at least delay separation of the viscous region. Different studies [6–10] show the effect of single slot suction (injection) into steady compressible and water boundary layer flows over two-dimensional and axisymmetric bodies. Moreover, Roy [11] and Subhashini et al. [12] have investigated the influence of non-uniform double slot suction (injection) on compressible boundary layer flows over cylinder and yawed cylinder, respectively. Also, in more recent studies, Roy et al. [13], and Roy and Saikrishnan [14] have reported the influence of non-uniform double slot suction (injection) on an incompressible boundary layer flow over a slender cylinder and within a diverging channel, respectively. In the present investigation, the non-similar solutions have been obtained starting from the origin of the stream-wise coordinate to the point of separation (zero skin friction in the stream-wise direction) using quasi-linearization technique with an implicit finite difference scheme. The present analysis may be useful in understanding many boundary layer flow problems of practical importance, for example, in ☆ Communicated by A.R. Balakrishnan and S. Jayanti. ⁎ Corresponding author. E-mail address: [email protected] (S. Roy). 0735-1933/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.icheatmasstransfer.2009.04.007
suppressing recirculating bubbles and controlling transition and/or delaying the boundary layer separation over control surfaces. 2. Mathematical formulation We consider the steady laminar boundary layer forced convection flow of water with temperature dependent viscosity and Prandtl number over a sphere placed in a uniform stream. An orthogonal curvilinear coordinate system has been chosen in which coordinate x measures the distance from the forward stagnation point along a meridian and y is the distance normal to the body surface. The radius of a section normal to the axis of the sphere at a distance x along the meridian from the pole is written as r(x) and it is assumed that r(x) is large compared with the boundary layer thickness. The fluid is assumed to flow with moderate velocities, and the temperature difference between the wall and the free stream is small (b40 °C). In the range of temperature considered (i.e., 0 °C–40 °C), the variation of both density and specific heat (Cp), of water, with temperature is less than 1% and hence they are taken as constants (Table 1). However, since the variations of thermal conductivity (k) and viscosity (µ) [and hence Prandtl number (Pr)] with temperature are quite significant, the viscosity and Prandtl number are assumed to vary as an inverse linear function of temperature (T) [6,15]: μ=
1 1 and Pr = b1 + b2 T c1 + c2 T
ð1Þ
where b1 = 53:41; b2 = 2:43; c1 = 0:068 and c2 = 0:004:
ð2Þ
The numerical data, used for these correlations, are taken from [16]. Eqs. (1) and (2) are reasonably good approximations for liquids such as
S. Roy et al. / International Communications in Heat and Mass Transfer 36 (2009) 646–650 Table 1 Values of thermo-physical properties of water at different temperatures [16].
Nomenclature A Cf Cp Ec F G k N Nu Pr r(x) R Re T u, v x, y _ x _ _ x0, x1
dimensionless mass transfer parameter skin friction coefficient in the x-direction specific heat at constant pressure dissipation parameter dimensionless velocity component in the x-direction dimensionless temperature thermal conductivity (μμ ), viscosity ratio ∞ Nusselt number Prandtl number radius of the section normal to the axis of the sphere radius of the sphere Reynolds number temperature dimensional velocity components in x-, and y-directions, respectively dimensional meridional and normal distances, respectively dimensionless meridional distance slot location parameters
Temperature (T)
Density (ρ)
Specific heat (Cp)
Thermal conductivity (k)
Viscosity (µ)
(°C)
(g/cm3)
(J 107/kg K)
(erg 105/cm s K)
(g 10− 2/cm s)
Prandtl number, Pr
0 10 20 30 40 50
1.00228 0.99970 0.99821 0.99565 0.99222 0.98803
4.2176 4.1921 4.1818 4.1784 4.1785 4.1806
0.5610 0.5800 0.5984 0.6154 0.6305 0.6435
1.7930 1.3070 1.0060 0.7977 0.6532 0.5470
13.48 9.45 7.03 5.12 4.32 3.55
The following transformations Z x
ue r 2 x d ; u∞ R R 0 1 = 2 2n f ðn; ηÞ; ψðx; zÞ = u∞ R Re n=
Greek symbols β pressure gradient _ _ ▵η, ▵x step sizes in η- and x-directions respectively η, ξ transformed coordinates µ dynamic viscosity ρ density Ψ dimensional stream function ω⁎ slot length parameter
ð3Þ
y
−1
μ Pr
Ty
y
+
μ 2 u : ρCp y
T − Tw ; T∞ − Tw
2 NFη + f Fη + βðnÞ 1 − F = 2n FFn − fn Fη ;
ð6Þ
2 u −1 2 NPr Gη + f Gη + NEc e Fη = 2n FGn − fn Gη ; η u∞
ð7Þ
ð4Þ
μ μ∞
where N =
=
b1 + b2 T∞ b1 + b2 T
=
Pr =
1 1 = ; c1 + c2 T a3 + a4 G
a1 =
b1 + b2 Tw ; b1 + b2 T∞
a3 = c1 + c2 Tw ; βðnÞ =
Ec =
2n due ; ue dn
u2∞ ; Cp ðT∞ − Tw Þ
1 a1 + a2 G ;
a2 =
b2 ðT∞ − Tw Þ ; b1 + b2 T∞
a4 = c2 ðT∞ − Tw Þ; α 1 ðnÞ =
4n dr ; r dn
4Tw = ðTw − T∞ Þ;
u = ue fη = ue F; v= −
rue α ðnÞ f + 2nfn + βðnÞ + 1 − 1 ηF : 1=2 2 Rð2nReÞ
The transformed boundary conditions are ð5Þ
F ðn; 0Þ = 0; F ðn; ∞Þ = 1;
The boundary conditions are: where f = uðx; 0Þ = 0; vðx; 0Þ = vw ðxÞ; T ðx; 0Þ = Tw = constant; uðx; ∞Þ = ue ðxÞ; T ðx; ∞Þ = T∞ = constant:
u∞ Rρ ; μ∞
η
water, particularly for small wall and ambient temperature differences. As the fluid is incompressible, the contribution of heating due to compression is very small and it has been neglected. The effect of viscous dissipation is included in the analysis. The fluid at the edge of the boundary layer is maintained at a constant temperature T∞ and the body has a uniform temperature Tw. Under the foregoing assumptions, the boundary layer equations governing the flow are given by [6,15]:
−1 uux + vuy = ue ðue Þx + ρ μuy ;
Re =
reduce the system of partial differential Eqs. (3)–(5) into a nondimensional form given by
Subscripts ∞ conditions in the free stream e, w conditions at the edge of the boundary layer and on the surface, respectively _ x, η, ξ partial derivatives with respect to these variables
ðruÞx + ðrvÞy = 0;
Re 1 = 2 ue r y ; u∞ R R 2n R Aψ u= ; r Ay
η=
R Aψ ; r Ax
v= − G=
uTx + vTy = ρ
647
fw = − n
Gðn; 0Þ = 0; Gðn; ∞Þ = 1;
Rη 0
Fdη + fw and fw is given by
−1=2
1 = 2 Z x Re r 1 v ðxÞdx: 2 R u∞ w 0
ð8Þ
648
S. Roy et al. / International Communications in Heat and Mass Transfer 36 (2009) 646–650
The free stream velocity distribution for the case of axi-symmetric flow over a sphere and the distance from the axis of the body are given by ue 3 = sin x; u∞ 2
r = sin x; R
x=
x : R
ð9Þ
Consequently, the expressions for ξ, β(ξ), α1(ξ) and fw can be expressed as
Substituting Eq. (10) in Eqs. (6) and (7), we obtain 2 NFη + fFη + βðxÞ 1 − F = 2BðxÞ FFx − fx Fη ;
ð11Þ
2 u −1 2 NPr Gη + fGη + NEc e Fη = 2BðxÞ FGx − fx Gη ; η u∞
ð12Þ
η
where N =
1 a1 + a2 G,
and Pr =
1 a3 + a4 G.
The boundary conditions become 1 2 n = P1 P3 ; 2
2 −2 β = P3 P2 cos x; 3
α 1 = 2β;
8 0 > > > > > > −1 −1 = 2 > > C ðx; x0 Þ > AP1 P3 > > > > > < −1 −1 = 2 AP1 P3 C ðx0; x0 Þ fw = > > > −1 −1 = 2 > > AP1 P3 fC ðx0; x0 Þ + C ðx; x1 Þg > > > > > > > > AP −1 P −1 = 2 fC ðx; x Þ + C ðx; x Þg > : 1 0 0 1 1 3
F ðx; 0Þ = 0; Gðx; 0Þ = 0; F ðx; ∞Þ = 1; Gðx; ∞Þ = 1; ;
x b x0
;
x0 V x V x0
;
x0 V x V x1
;
x1 V x V x1
;
x N x1
Rη where f = 0 Fdη + fw . The skin friction coefficient at the wall can be expressed in the form: 1=2
Cf ðReÞ
9 −1 = 2 Nw Fη ; sin xP2 P3 w 2
=
NuðReÞ
sinfðω − 1Þx − ωx0 g + sinx0 ðω + 1Þ
;
x b x0
;
x0 V xV x
;
x0V xV x1
;
x1 V x V x1
;
x N x1
0
_ _ where ω, x0 and x1 are the free parameters which determine the slot _ length and slot locations, respectively. The function v w(x ) is _ continuous for all values of x and it has non-zero values only in the _ _ _ _ intervals [x0, x0⁎] and [x1, x1⁎]. The reason for taking such a function is that it allows the mass transfer to change slowly in the neighbourhood of leading and trailing edges of the slots. It may also be noted that more practical suction/injection velocity profiles for specific case study can be represented by interpolating polynomial as continuous function and used as in the present investigation to study the effect on the location of point of separation. The parameter A N 0 or A b 0 according to whether there is a suction or an injection. It is convenient _ to express Eqs. (6) and (7) in terms of x instead of ξ. Eq. (9) gives the _ relation between ξ and x as A A = BðxÞ ; An Ax
where BðxÞ = 3 − 1 tan 2x P3 P2−1 .
3 −1 = 2 Gη ; P2 P3 w 2
RðAT Ay Þw ðT∞ − Tw Þ.
ð15Þ
From Eqs. (14) to (15), it is clear that (Fη)w and
3. Results and discussion
Here vw is taken as
n
=
(Gη)w are the crucial parameters which characterize skin frictions and heat transfer of the fluid flow.
P1 = 1 − cosx; P2 = 1 + cosx and P3 = 2 + cosx:
8 > > 0 > > > > > −1 1 > > > Re 2 2 > > −u 2 A sinfω ðx − x0 Þg > ∞ > 2 > > > > > > < vw = 0 > > > > > > −1 1 > > Re 2 2 > > > −u∞ 2 A sinfω ðx − x1 Þg > > 2 > > > > > > > : 0
−1=2
where Nu =
sinfðω + 1Þx − ωx0 g − sinx0 − ; ðω + 1Þ
ð14Þ
½μ ðAu AyÞw and Nw = a1 +1a2 Gw = a11 = constant. where Cf = 2 ρu 2 ∞ Similarly, the heat transfer co-efficient in terms of Nusselt number can be written as
where C ðx; x0 Þ =
ð13Þ
ð10Þ
Eqs. (11) and (12) with boundary conditions (13) have been solved numerically using an implicit finite difference scheme in combination with the quasi-linearization technique [8,17]. The non-linear coupled partial differential Eqs. (11) and (12) were first linearized using quasilinearization method [8,17] and then, the linearized equations were expressed in difference form using central difference scheme in the _ η-direction and a backward difference scheme in the x-direction. Finally, the equations reduced to a system of linear algebraic equations with a block tri-diagonal structure which is solved using Varga's algorithm [18]. The step size in the η-direction has been chosen ▵η = 0.01 throughout the computation, as it has been found that further decrease in ▵η does not change the results up to the _ _ fourth decimal place. In the x-direction, ▵x = 0.01 has been used _ _ for small values of x ≤ 0.5 then it has been decreased to ▵x = 0.005. _ _ This value of ▵x has been used for x ≤ 1.25, thereafter the step size _ has been reduced further, ultimately choosing a value ▵x = 0.0001 in the neighbourhood of the point of zero skin friction. This has been done because the convergence becomes slower when the point of vanishing skin friction is approached. Computations were carried out for various values A(− 0.6 ≤ A ≤ 2.0) and Ec(−0.1 ≤ Ec ≤ 0). In order to validate our numerical results, comparison of the skin friction results [Cf(Re)1/2] has been made with the steady results of Eswara and Nath [4] who studied the problem of an unsteady non-similar two-dimensional and axisymmetric water boundary layer with variable viscosity and Prandtl number. The results are found to be in excellent agreement with the present results and the comparisons are not shown here to brief the manuscript. The effect of non-uniform suction of a single slot located at _ _ x0 = 0.75 or x0 = 1.30 on velocity gradient is compared with that over _ a non-uniform suction of double slots positioned at x0 = 0.75 and _ x1 = 1.30 in Fig. 1. Further, it may be noted that the total mass flow rate _ _ in double slots located at x0 = 0.75 and x1 = 1.30 is applied into the
S. Roy et al. / International Communications in Heat and Mass Transfer 36 (2009) 646–650
649
_ _ Fig. 1. Effect of single and double slot suctions on Fη(x , 0) and Gη(x , 0) when T∞ = 18.7, _ Tw = 28.7, Ec = −0.05 and ω⁎ = 2π. (———) double slot locations at x 0 = 0.75 and _ x̄1 = 1.3; (– – –) single slot location at x 0 = 0.75 or 1.3; (…………) no slot.
_ _ single slot at x0 = 0.75 or x0 = 1.30. It is noticed in Fig. 1 that the total mass flow rate of double slots is less effective in delaying separation than the single slot having the same total mass flow rate when the single slot is positioned at the downstream location i.e., at the same location of second slot in the double slot combinations. Thus, results presented in Fig. 1 indicate that mass flow rate and position of the slot significantly influence in delaying the point of separation. But it may be remarked that the use of double slots helps to maintain the maximum values of the velocity and temperature gradients or equivalently skin friction and heat transfer coefficients as lower order of magnitudes compared to those quantities in single slot case (see Fig. 1). For example, in Fig. 1, the maximum values of velocity and temperature gradients are approximately 2.5 and 12, respectively, for double slots whereas the maximum values of those corresponding quantities are approximately 4.5 and 25, respectively. In both the double and single slot suction cases, the velocity gradients gradually
Fig. 2. Effect of suction parameter (A N 0) on velocity and temperature gradients _ _ ω⁎ = 2π, T∞ = 18.7, Tw = 28.7 and Ec = −0.05. Slot locations at x 0 = 0.5, x1 = 1.4.
Fig. 3. Effect of injection parameter (A b 0) on velocity and temperature gradients when _ _ ω⁎ = 2π, T∞ = 18.7, Tw = 28.7 and Ec = − 0.05. Slot locations at x0 = 0.25, x 1 = 1.0.
increase from the leading edge of the slots, attain a maximum and then start decreasing at the rear end of the slots. _ The effect of suction parameter (A N 0) on velocity gradient Fη(x, 0) _ and temperature gradient, Gη(x, 0) in the case of non-uniform double slot suction is presented in Fig. 2. The velocity and temperature gradients increase with the increase of mass transfer rates. Also, the point of separation moves further downstream with the increase of the mass transfer rates. The graph of velocity and temperature gradients in
_ _ Fig. 4. Effect of slot locations (x0, x1) on velocity and temperature gradients when ω⁎ = 2π, _ _ A = 1.25, T∞ = 18.7, Tw = 28.7 and Ec= −0.05. (———) slot locations at x0 = 0.25, x1 = 1.0; _ _ (– – – – – – –) slot locations at x0 = 0.5, x1 = 1.4; (…………) no slot.
650
S. Roy et al. / International Communications in Heat and Mass Transfer 36 (2009) 646–650
the case of non-uniform injection in double slots is presented in Fig. 3. It is observed that the point of separation moves further upstream by increasing the magnitude of the injection parameter (A b 0). It is noticed from Fig. 4 that the point of separation moves downstream when positions of the slots are moved further downstream. Thus, the point of separation can be delayed by non-uniform double slot suction (A N 0) and also by positioning the slots further downstream.
4. Conclusions The present study effectively compares the significance of nonuniform single and double slot suction/injection of laminar water boundary layer flows over a sphere. The numerical investigation shows that the point of separation can be delayed using non-uniform double slot suction by increasing the mass transfer rate and also by positioning the slots further downstream. The results also indicate that the effect of non-uniform double slot injection is just the opposite. The investigation also reveals the effective comparisons on total mass flow rate and locations of the single slot and double slots in delaying/controlling the steady flow separation.
Acknowledgement Authors are thankful to the anonymous reviewer for his critical comments in modifying the manuscript. One of the authors S. Roy is grateful to the Council of Scientific and Industrial Research (CSIR), New Delhi for the financial assistance to carry out this work under the sponsored research project scheme.
References [1] C.F. Dewey, J.F. Gross, Exact Solutions of the Laminar Boundary Layer Equations, Advances in heat transfer, vol. IV, Academic Press, New York, 1967, p. 317. [2] T. Davis, G. Walker, On solution of the compressible laminar boundary layer equations and their behaviour near separation, J. Fluid Mech. 80 (1977) 279–292. [3] B.J. Venkatachala, G. Nath, Non-similar laminar incompressible boundary layers with vectored mass transfer, Proc. Indian Acad. Sci. (Eng. Sci) 3 (1980) 129–142. [4] A.T. Eswara, G. Nath, Unsteady two-dimensional and axi-symmetric water boundary layers with variable viscosity and Prandtl number, Int. J. Eng. Sci. 32 (1994) 267–279. [5] S. Roy, Non-uniform mass transfer or wall enthalphy into a compressible flow over yawed cylinder, Int. J. Heat Mass Transfer 44 (2001) 3017–3024. [6] P. Saikrishnan, S. Roy, Non-uniform slot injection (suction) into water boundary layers over (i) a cylinder and (ii) a sphere, Int. J. Eng. Sci. 41 (2003) 1351–1365. [7] S. Roy, H.S. Thakar, Compressible boundary layer flow with non-uniform slot injection (or suction) over i) a cylinder and ii) a sphere, Heat Mass Transf. 39 (2003) 139–146. [8] S. Roy, P. Saikrishnan, Non-uniform slot injection (suction) into steady laminar water boundary layer flow over a rotating sphere, Int. J. Heat Mass Transfer 46 (2003) 3389–3396. [9] S. Roy, P. Saikrishnan, Non-uniform slot injection (suction) into water boundary layer flow past yawed cylinder, Int. J. Eng. Sci. 42 (2004) 2147–2157. [10] S.V. Subhashini, H.S. Takhar, G. Nath, Non-uniform mass transfer or wall enthalpy into a compressible flow over a rotating sphere, Heat Mass Transf. 43 (2007) 1133–1141. [11] S. Roy, Non-uniform multiple slot injection (suction) or wall enthalpy into a steady compressible laminar boundary layer, Acta Mech. 143 (2000) 113–128. [12] S.V. Subhashini, H.S. Takhar, G. Nath, Non-uniform multiple slot injection (suction) or wall enthalpy into a compressible flow over a yawed circular cylinder, Int. J. Therm. Sci. 42 (2003) 749–757. [13] S. Roy, Prabal Datta, R. Ravindran, E. Momoniat, Non-uniform double slot injection (suction) on a forced flow over a slender cylinder, Int. J. Heat Mass Transfer 50 (2007) 3190–3194. [14] S. Roy, P. Saikrishnan, Multiple slot suction/injection into an exponentially decreasing free stream flow, Int. Com. Heat Mass Transfer 35 (2008) 163–168. [15] H. Schlichting, Boundary Layer Theory, Springer, New York, 2000. [16] D.R. Lide (Ed.), CRC Handbook of Chemistry and Physics, 71st ed, CRC Press, Boca Raton, Florida, 1990. [17] K. Inouye, A. Tate, Finite difference version of quasilinearization applied to boundary layer equations, AIAA J. 12 (1974) 558–560. [18] R.S. Varga, Matrix Iterative Analysis, Springer, New York, 2000, p. 220.
Contents lists available at ScienceDirect
International Communications in Heat and Mass Transfer j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / i c h m t
Influence of double slot suction (injection) into water boundary layer flows over sphere☆ S. Roy a,⁎, P. Saikrishnan b, Bishun D. Pandey c a b c
Department of Mathematics, Indian Institute of Technology Madras, Chennai-600 036, Tamilnadu, India Department of Mathematics, National Institute of Technology Tiruchirapalli, Tiruchirappali-620 015, Tamilnadu, India Department of Mathematics, The Ohio State University, Marion, OH 43302, USA
a r t i c l e
i n f o
Available online 22 May 2009 Keywords: Slot suction (injection) Sphere
a b s t r a c t The objective of this work is to study the effect of non-uniform double slot suction (injection) into steady laminar water boundary layers over a sphere with variable viscosity and Prandtl number. The implicit finite difference scheme with quasi-linearization technique has been applied to find the non-similar solutions and also to overcome the difficulties arising at the starting point of the stream-wise coordinate, at the edges of the slot and at the point of separation. Results indicate that the separation can be delayed by non-uniform double slot suction and also by moving the slots downstream direction but the effect of non-uniform double slot injection is just the opposite. © 2009 Elsevier Ltd. All rights reserved.
1. Introduction A review of the non-similarity solution methods along with the citations of some relevant publications is given in an earlier study by Dewey and Gross [1]. Since then, there have been several studies pertaining to non-similar flows by finite difference method [2,3] and an implicit finite difference method in combination with quasi-linearization technique [4,5]. Fluid viscosity and thermal conductivity are the main governing fluid properties in the laminar water boundary layer forced flow and hence their variations can be expected to affect separation. Further, mass transfer through a slot strongly influences the development of a boundary layer along a surface and in particular can prevent or at least delay separation of the viscous region. Different studies [6–10] show the effect of single slot suction (injection) into steady compressible and water boundary layer flows over two-dimensional and axisymmetric bodies. Moreover, Roy [11] and Subhashini et al. [12] have investigated the influence of non-uniform double slot suction (injection) on compressible boundary layer flows over cylinder and yawed cylinder, respectively. Also, in more recent studies, Roy et al. [13], and Roy and Saikrishnan [14] have reported the influence of non-uniform double slot suction (injection) on an incompressible boundary layer flow over a slender cylinder and within a diverging channel, respectively. In the present investigation, the non-similar solutions have been obtained starting from the origin of the stream-wise coordinate to the point of separation (zero skin friction in the stream-wise direction) using quasi-linearization technique with an implicit finite difference scheme. The present analysis may be useful in understanding many boundary layer flow problems of practical importance, for example, in ☆ Communicated by A.R. Balakrishnan and S. Jayanti. ⁎ Corresponding author. E-mail address: [email protected] (S. Roy). 0735-1933/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.icheatmasstransfer.2009.04.007
suppressing recirculating bubbles and controlling transition and/or delaying the boundary layer separation over control surfaces. 2. Mathematical formulation We consider the steady laminar boundary layer forced convection flow of water with temperature dependent viscosity and Prandtl number over a sphere placed in a uniform stream. An orthogonal curvilinear coordinate system has been chosen in which coordinate x measures the distance from the forward stagnation point along a meridian and y is the distance normal to the body surface. The radius of a section normal to the axis of the sphere at a distance x along the meridian from the pole is written as r(x) and it is assumed that r(x) is large compared with the boundary layer thickness. The fluid is assumed to flow with moderate velocities, and the temperature difference between the wall and the free stream is small (b40 °C). In the range of temperature considered (i.e., 0 °C–40 °C), the variation of both density and specific heat (Cp), of water, with temperature is less than 1% and hence they are taken as constants (Table 1). However, since the variations of thermal conductivity (k) and viscosity (µ) [and hence Prandtl number (Pr)] with temperature are quite significant, the viscosity and Prandtl number are assumed to vary as an inverse linear function of temperature (T) [6,15]: μ=
1 1 and Pr = b1 + b2 T c1 + c2 T
ð1Þ
where b1 = 53:41; b2 = 2:43; c1 = 0:068 and c2 = 0:004:
ð2Þ
The numerical data, used for these correlations, are taken from [16]. Eqs. (1) and (2) are reasonably good approximations for liquids such as
S. Roy et al. / International Communications in Heat and Mass Transfer 36 (2009) 646–650 Table 1 Values of thermo-physical properties of water at different temperatures [16].
Nomenclature A Cf Cp Ec F G k N Nu Pr r(x) R Re T u, v x, y _ x _ _ x0, x1
dimensionless mass transfer parameter skin friction coefficient in the x-direction specific heat at constant pressure dissipation parameter dimensionless velocity component in the x-direction dimensionless temperature thermal conductivity (μμ ), viscosity ratio ∞ Nusselt number Prandtl number radius of the section normal to the axis of the sphere radius of the sphere Reynolds number temperature dimensional velocity components in x-, and y-directions, respectively dimensional meridional and normal distances, respectively dimensionless meridional distance slot location parameters
Temperature (T)
Density (ρ)
Specific heat (Cp)
Thermal conductivity (k)
Viscosity (µ)
(°C)
(g/cm3)
(J 107/kg K)
(erg 105/cm s K)
(g 10− 2/cm s)
Prandtl number, Pr
0 10 20 30 40 50
1.00228 0.99970 0.99821 0.99565 0.99222 0.98803
4.2176 4.1921 4.1818 4.1784 4.1785 4.1806
0.5610 0.5800 0.5984 0.6154 0.6305 0.6435
1.7930 1.3070 1.0060 0.7977 0.6532 0.5470
13.48 9.45 7.03 5.12 4.32 3.55
The following transformations Z x
ue r 2 x d ; u∞ R R 0 1 = 2 2n f ðn; ηÞ; ψðx; zÞ = u∞ R Re n=
Greek symbols β pressure gradient _ _ ▵η, ▵x step sizes in η- and x-directions respectively η, ξ transformed coordinates µ dynamic viscosity ρ density Ψ dimensional stream function ω⁎ slot length parameter
ð3Þ
y
−1
μ Pr
Ty
y
+
μ 2 u : ρCp y
T − Tw ; T∞ − Tw
2 NFη + f Fη + βðnÞ 1 − F = 2n FFn − fn Fη ;
ð6Þ
2 u −1 2 NPr Gη + f Gη + NEc e Fη = 2n FGn − fn Gη ; η u∞
ð7Þ
ð4Þ
μ μ∞
where N =
=
b1 + b2 T∞ b1 + b2 T
=
Pr =
1 1 = ; c1 + c2 T a3 + a4 G
a1 =
b1 + b2 Tw ; b1 + b2 T∞
a3 = c1 + c2 Tw ; βðnÞ =
Ec =
2n due ; ue dn
u2∞ ; Cp ðT∞ − Tw Þ
1 a1 + a2 G ;
a2 =
b2 ðT∞ − Tw Þ ; b1 + b2 T∞
a4 = c2 ðT∞ − Tw Þ; α 1 ðnÞ =
4n dr ; r dn
4Tw = ðTw − T∞ Þ;
u = ue fη = ue F; v= −
rue α ðnÞ f + 2nfn + βðnÞ + 1 − 1 ηF : 1=2 2 Rð2nReÞ
The transformed boundary conditions are ð5Þ
F ðn; 0Þ = 0; F ðn; ∞Þ = 1;
The boundary conditions are: where f = uðx; 0Þ = 0; vðx; 0Þ = vw ðxÞ; T ðx; 0Þ = Tw = constant; uðx; ∞Þ = ue ðxÞ; T ðx; ∞Þ = T∞ = constant:
u∞ Rρ ; μ∞
η
water, particularly for small wall and ambient temperature differences. As the fluid is incompressible, the contribution of heating due to compression is very small and it has been neglected. The effect of viscous dissipation is included in the analysis. The fluid at the edge of the boundary layer is maintained at a constant temperature T∞ and the body has a uniform temperature Tw. Under the foregoing assumptions, the boundary layer equations governing the flow are given by [6,15]:
−1 uux + vuy = ue ðue Þx + ρ μuy ;
Re =
reduce the system of partial differential Eqs. (3)–(5) into a nondimensional form given by
Subscripts ∞ conditions in the free stream e, w conditions at the edge of the boundary layer and on the surface, respectively _ x, η, ξ partial derivatives with respect to these variables
ðruÞx + ðrvÞy = 0;
Re 1 = 2 ue r y ; u∞ R R 2n R Aψ u= ; r Ay
η=
R Aψ ; r Ax
v= − G=
uTx + vTy = ρ
647
fw = − n
Gðn; 0Þ = 0; Gðn; ∞Þ = 1;
Rη 0
Fdη + fw and fw is given by
−1=2
1 = 2 Z x Re r 1 v ðxÞdx: 2 R u∞ w 0
ð8Þ
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S. Roy et al. / International Communications in Heat and Mass Transfer 36 (2009) 646–650
The free stream velocity distribution for the case of axi-symmetric flow over a sphere and the distance from the axis of the body are given by ue 3 = sin x; u∞ 2
r = sin x; R
x=
x : R
ð9Þ
Consequently, the expressions for ξ, β(ξ), α1(ξ) and fw can be expressed as
Substituting Eq. (10) in Eqs. (6) and (7), we obtain 2 NFη + fFη + βðxÞ 1 − F = 2BðxÞ FFx − fx Fη ;
ð11Þ
2 u −1 2 NPr Gη + fGη + NEc e Fη = 2BðxÞ FGx − fx Gη ; η u∞
ð12Þ
η
where N =
1 a1 + a2 G,
and Pr =
1 a3 + a4 G.
The boundary conditions become 1 2 n = P1 P3 ; 2
2 −2 β = P3 P2 cos x; 3
α 1 = 2β;
8 0 > > > > > > −1 −1 = 2 > > C ðx; x0 Þ > AP1 P3 > > > > > < −1 −1 = 2 AP1 P3 C ðx0; x0 Þ fw = > > > −1 −1 = 2 > > AP1 P3 fC ðx0; x0 Þ + C ðx; x1 Þg > > > > > > > > AP −1 P −1 = 2 fC ðx; x Þ + C ðx; x Þg > : 1 0 0 1 1 3
F ðx; 0Þ = 0; Gðx; 0Þ = 0; F ðx; ∞Þ = 1; Gðx; ∞Þ = 1; ;
x b x0
;
x0 V x V x0
;
x0 V x V x1
;
x1 V x V x1
;
x N x1
Rη where f = 0 Fdη + fw . The skin friction coefficient at the wall can be expressed in the form: 1=2
Cf ðReÞ
9 −1 = 2 Nw Fη ; sin xP2 P3 w 2
=
NuðReÞ
sinfðω − 1Þx − ωx0 g + sinx0 ðω + 1Þ
;
x b x0
;
x0 V xV x
;
x0V xV x1
;
x1 V x V x1
;
x N x1
0
_ _ where ω, x0 and x1 are the free parameters which determine the slot _ length and slot locations, respectively. The function v w(x ) is _ continuous for all values of x and it has non-zero values only in the _ _ _ _ intervals [x0, x0⁎] and [x1, x1⁎]. The reason for taking such a function is that it allows the mass transfer to change slowly in the neighbourhood of leading and trailing edges of the slots. It may also be noted that more practical suction/injection velocity profiles for specific case study can be represented by interpolating polynomial as continuous function and used as in the present investigation to study the effect on the location of point of separation. The parameter A N 0 or A b 0 according to whether there is a suction or an injection. It is convenient _ to express Eqs. (6) and (7) in terms of x instead of ξ. Eq. (9) gives the _ relation between ξ and x as A A = BðxÞ ; An Ax
where BðxÞ = 3 − 1 tan 2x P3 P2−1 .
3 −1 = 2 Gη ; P2 P3 w 2
RðAT Ay Þw ðT∞ − Tw Þ.
ð15Þ
From Eqs. (14) to (15), it is clear that (Fη)w and
3. Results and discussion
Here vw is taken as
n
=
(Gη)w are the crucial parameters which characterize skin frictions and heat transfer of the fluid flow.
P1 = 1 − cosx; P2 = 1 + cosx and P3 = 2 + cosx:
8 > > 0 > > > > > −1 1 > > > Re 2 2 > > −u 2 A sinfω ðx − x0 Þg > ∞ > 2 > > > > > > < vw = 0 > > > > > > −1 1 > > Re 2 2 > > > −u∞ 2 A sinfω ðx − x1 Þg > > 2 > > > > > > > : 0
−1=2
where Nu =
sinfðω + 1Þx − ωx0 g − sinx0 − ; ðω + 1Þ
ð14Þ
½μ ðAu AyÞw and Nw = a1 +1a2 Gw = a11 = constant. where Cf = 2 ρu 2 ∞ Similarly, the heat transfer co-efficient in terms of Nusselt number can be written as
where C ðx; x0 Þ =
ð13Þ
ð10Þ
Eqs. (11) and (12) with boundary conditions (13) have been solved numerically using an implicit finite difference scheme in combination with the quasi-linearization technique [8,17]. The non-linear coupled partial differential Eqs. (11) and (12) were first linearized using quasilinearization method [8,17] and then, the linearized equations were expressed in difference form using central difference scheme in the _ η-direction and a backward difference scheme in the x-direction. Finally, the equations reduced to a system of linear algebraic equations with a block tri-diagonal structure which is solved using Varga's algorithm [18]. The step size in the η-direction has been chosen ▵η = 0.01 throughout the computation, as it has been found that further decrease in ▵η does not change the results up to the _ _ fourth decimal place. In the x-direction, ▵x = 0.01 has been used _ _ for small values of x ≤ 0.5 then it has been decreased to ▵x = 0.005. _ _ This value of ▵x has been used for x ≤ 1.25, thereafter the step size _ has been reduced further, ultimately choosing a value ▵x = 0.0001 in the neighbourhood of the point of zero skin friction. This has been done because the convergence becomes slower when the point of vanishing skin friction is approached. Computations were carried out for various values A(− 0.6 ≤ A ≤ 2.0) and Ec(−0.1 ≤ Ec ≤ 0). In order to validate our numerical results, comparison of the skin friction results [Cf(Re)1/2] has been made with the steady results of Eswara and Nath [4] who studied the problem of an unsteady non-similar two-dimensional and axisymmetric water boundary layer with variable viscosity and Prandtl number. The results are found to be in excellent agreement with the present results and the comparisons are not shown here to brief the manuscript. The effect of non-uniform suction of a single slot located at _ _ x0 = 0.75 or x0 = 1.30 on velocity gradient is compared with that over _ a non-uniform suction of double slots positioned at x0 = 0.75 and _ x1 = 1.30 in Fig. 1. Further, it may be noted that the total mass flow rate _ _ in double slots located at x0 = 0.75 and x1 = 1.30 is applied into the
S. Roy et al. / International Communications in Heat and Mass Transfer 36 (2009) 646–650
649
_ _ Fig. 1. Effect of single and double slot suctions on Fη(x , 0) and Gη(x , 0) when T∞ = 18.7, _ Tw = 28.7, Ec = −0.05 and ω⁎ = 2π. (———) double slot locations at x 0 = 0.75 and _ x̄1 = 1.3; (– – –) single slot location at x 0 = 0.75 or 1.3; (…………) no slot.
_ _ single slot at x0 = 0.75 or x0 = 1.30. It is noticed in Fig. 1 that the total mass flow rate of double slots is less effective in delaying separation than the single slot having the same total mass flow rate when the single slot is positioned at the downstream location i.e., at the same location of second slot in the double slot combinations. Thus, results presented in Fig. 1 indicate that mass flow rate and position of the slot significantly influence in delaying the point of separation. But it may be remarked that the use of double slots helps to maintain the maximum values of the velocity and temperature gradients or equivalently skin friction and heat transfer coefficients as lower order of magnitudes compared to those quantities in single slot case (see Fig. 1). For example, in Fig. 1, the maximum values of velocity and temperature gradients are approximately 2.5 and 12, respectively, for double slots whereas the maximum values of those corresponding quantities are approximately 4.5 and 25, respectively. In both the double and single slot suction cases, the velocity gradients gradually
Fig. 2. Effect of suction parameter (A N 0) on velocity and temperature gradients _ _ ω⁎ = 2π, T∞ = 18.7, Tw = 28.7 and Ec = −0.05. Slot locations at x 0 = 0.5, x1 = 1.4.
Fig. 3. Effect of injection parameter (A b 0) on velocity and temperature gradients when _ _ ω⁎ = 2π, T∞ = 18.7, Tw = 28.7 and Ec = − 0.05. Slot locations at x0 = 0.25, x 1 = 1.0.
increase from the leading edge of the slots, attain a maximum and then start decreasing at the rear end of the slots. _ The effect of suction parameter (A N 0) on velocity gradient Fη(x, 0) _ and temperature gradient, Gη(x, 0) in the case of non-uniform double slot suction is presented in Fig. 2. The velocity and temperature gradients increase with the increase of mass transfer rates. Also, the point of separation moves further downstream with the increase of the mass transfer rates. The graph of velocity and temperature gradients in
_ _ Fig. 4. Effect of slot locations (x0, x1) on velocity and temperature gradients when ω⁎ = 2π, _ _ A = 1.25, T∞ = 18.7, Tw = 28.7 and Ec= −0.05. (———) slot locations at x0 = 0.25, x1 = 1.0; _ _ (– – – – – – –) slot locations at x0 = 0.5, x1 = 1.4; (…………) no slot.
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S. Roy et al. / International Communications in Heat and Mass Transfer 36 (2009) 646–650
the case of non-uniform injection in double slots is presented in Fig. 3. It is observed that the point of separation moves further upstream by increasing the magnitude of the injection parameter (A b 0). It is noticed from Fig. 4 that the point of separation moves downstream when positions of the slots are moved further downstream. Thus, the point of separation can be delayed by non-uniform double slot suction (A N 0) and also by positioning the slots further downstream.
4. Conclusions The present study effectively compares the significance of nonuniform single and double slot suction/injection of laminar water boundary layer flows over a sphere. The numerical investigation shows that the point of separation can be delayed using non-uniform double slot suction by increasing the mass transfer rate and also by positioning the slots further downstream. The results also indicate that the effect of non-uniform double slot injection is just the opposite. The investigation also reveals the effective comparisons on total mass flow rate and locations of the single slot and double slots in delaying/controlling the steady flow separation.
Acknowledgement Authors are thankful to the anonymous reviewer for his critical comments in modifying the manuscript. One of the authors S. Roy is grateful to the Council of Scientific and Industrial Research (CSIR), New Delhi for the financial assistance to carry out this work under the sponsored research project scheme.
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