Jaman Hossain Gorla IJFDRes
See discussions, stats, and author profiles for this publication at: https://https://www.sodocs.net/doc/1d1948182.html,/publication/259772627 Fluctuating Free Convection Flow Along Heated Horizontal Circular Cylinders
ARTICLE in INTERNATIONAL JOURNAL OF FLUID MECHANICS RESEARCH · JANUARY 2009
DOI: 10.1615/InterJFluidMechRes.v36.i3.20
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International Journal of Fluid Mechanics Research, Vol. 36, No. 3, 2009
Fluctuating Free Convection Flow
Along Heated Horizontal Circular Cylinders?
M.A.Hossain1,Md.Kamrujjaman1,and Rama Subba Reddy Gorla2
1Department of Mathematics,University of Dhaka
Dhaka-1000,Bangladesh
E-mail:anwar@https://www.sodocs.net/doc/1d1948182.html,
2Department of Mechanical Engineering,Cleveland State University
Cleveland,OH44115,USA
Unsteady free convection?ow of a viscous incompressible?uid along an in?nite horizontal cylinder with a small-amplitude oscillation of the surface tem-
perature about a mean temperature has been https://www.sodocs.net/doc/1d1948182.html,ing appropriate trans-
formations,the governing equations reduced to local nonsimilarity equations for
both the steady and unsteady parts of the solutions,which are obtained numeri-
cally employing the implicit?nite difference method together with the Keller box
elimination scheme.The effects of pertinent parameters,such as the Strouhal
number,St,and the Prandtl number,Pr,on the local?uctuating skin-friction and
heat transfer coef?cients as well as on their amplitude and phase are shown graph-
ically on the entire surface of the cylinder0≤x≤π.
***
Nomenclature
a radius of the cylinder;
C f speci?c heat at constant pressure;
f dimensionless stream function;
g acceleration due to gravity;
Gr Grashof number;
Nu Nusselt number;
Pr Prandtl number;
St Strouhal number;
T temperature of the?uid;
T w temperature of the heated surface;
T∞temperature of the ambient?uid;
?Received18.12.2007
207
ISSN1064-2277
c 2009Begell House,Inc.
T0reference temperature;
u velocity in the x-direction;
v velocity in the y-direction;
x coordinates measuring distance along the cylinder;
y coordinates measuring distance normal to the cylinder;
βcoef?cient of volume expansion;
αcoef?cient of thermal diffusivity;
νkinematic viscosity;
?angle made by the outward normal from the cylinder;
φtemperature function;
θdimensionless temperature function;
ρdensity of the?uid.
Introduction
In the study of the laminar boundary layer in two-dimensional?ows,one aspect that has re-ceived much attention in recent years is that concerned with boundary layer responses to imposed oscillations.Lighthill[1]was the?rst to have studied the unsteady forced?ow of a viscous in-compressible?uid past a?at plate and a circular cylinder with a small-amplitude oscillation in the free stream.Subsequent authors have extended Lighthill’s solution,but in general have retained the small-α1approximations(0≤α1≤1is the amplitude parameter).Gibellato[2]and Gosh[3] restricted themselves to the case of the semi-in?nite plate(n=0),and independently extended the small-ε[ωx/U0(x)is the frequency parameter]expansion to several terms.Ghosh also extended the large-εexpansion,but considered only the viscous,not the thermal boundary layer.Rott and Rosenzweig[4]did the same for the general Falkner–Skan boundary layer?ow[taking the free stream velocity U0(x)=kx n,for any n],paying particular attention to the stagnation point case (n=1).Lam and Rott[5]gave an exhaustive mathematical treatment of the linear?rst-order(in α1)problem,including numerical calculation of the?rst15terms of the small-εexpansion.Finally, Gerston[6]repeated the small-and large-εexpansion to the second order inα1,in order to examine the interaction between the oscillatory boundary layer and the mean?ow.The small-amplitude ap-proximation has been shown to be unnecessary in both the small-εcase and the large-εcase,Lin[7], Gibson[8],but this work has not been followed up except in a recent paper by Ishigaki[9],who considered the case of stagnation point?ow(n=1).Oscillations for the laminar boundary layer over the semi-in?nite rigid plane are investigated by Pedly[10]for the free stream velocity.He considers four quantities as possible candidates for the criterion:the magnitude of the maximum skin friction,the phase of this maximum,the overall amplitude of the skin-friction variation,and the mean skin friction.He calculates the skin friction and heat transfer for small-and large-frequency parameterε.
The problem of unsteady free convection?ow along a vertical plate with oscillating surface temperature was studied by Nanda and Sharma[11]and Eshghy et al.[12].In consideration of the class of problems,Muhuri and Maiti[13]and Verma[14]have analyzed the effect of?uc-tuation in the surface temperature on the unsteady free convection from a horizontal plate.All the above investigations are based on the assumption that the surface temperature oscillates with a small amplitude about the uniform mean temperature employing the Karman–Pohlhausen approximate integral method.Based on the linearized theory,Kelleher and Yang[15]have studied the heat trans-fer responses of a laminar free convection boundary layer along a vertical heated plate to surface temperature oscillations,when the mean surface temperatureθw(x)is proportional to x n,where x
208
is the distance measured from the leading edge of the plate.The results were presented in terms of skin friction and heat transfer for smallε,which measures the distribution of the frequency of oscillation in the surface temperature and buoyancy force.Hossain et al.[16]consider electrically conducting?uid along a vertical plate in the presence of the variable transverse magnetic?eld when the surface temperature of the plate oscillates with small amplitude about the mean surface temper-ature.The mean temperature is assumed to vary as a power of x that measures the distance from the leading edge.
Heat transfer from heated cylinders is encountered in various applications,such as heat ex-changers,cooling systems,and electronic equipment.The literature on a free convection boundary layer over a cylinder is not as extensive as for a?at plate.Merkin[17]considered the case of a free convection boundary layer on an isothermal horizontal circular cylinder in a viscous?uid.It appears that Merkin was the?rst to present a complete solution to this problem for Newtonian?uid using Blasius and G¨o rtler series expansion methods along with an integral method and a?nite difference scheme.Hossain et al.[18]have studied the effects of radiation on mixed convection boundary layer?ow along a vertical cylinder.
Merkin[19]?rst considered the free convection?ow of a viscous incompressible?uid along an in?nite horizontal cylinder when the temperature of the cylinder is oscillating harmonically.Very recently,Gorla[20]has investigated the unsteady combined convection from a horizontal circular cylinder to a transverse?ow.In this study,the free stream time-dependent velocity was assumed to be sinusoidal and the boundary layer response due to both low as well as high frequencies of oscillation has been discussed.
In this paper,the free convection?ow of a viscous incompressible?uid along an in?nite hor-izontal cylinder when the temperature of the cylinder is oscillating sinusoidically about a mean temperature T0with small amplitude has been considered.The motion is caused by the action of oscillating buoyant body forces on the?uid near the https://www.sodocs.net/doc/1d1948182.html,ing appropriate transformations, the governing equations reduced to local nonsimilarity equations for both the steady mean?ow and for the?uctuating?ow,solutions of which are obtained numerically employing implicit?nite difference method together with the Keller box elimination method[21].Effects of pertinent pa-rameters such as the Strouhal number St and the Prandtl number Pr on the local skin friction and local heat transfer coef?cients for the?uctuating?ows are shown graphically on the entire surface of the cylinder0≤x≤π.The numerical values of the local skin friction and also of the local heat transfer for the steady mean?ow are also presented in tabular form and compared with the solutions obtained by Merkin[17],Nazar et al.[22],and Hossain et al.[23]for the?uid when Pr=1.0. The?ow patterns in terms of streamline and isotherm are also shown graphically with the effects of different values of Pr and St.
1.Basic Equations
Here,we consider a two-dimensional unsteady free convection boundary layer?ow of a viscous incompressible?uid over a horizontal circular cylinder of radius a,the surface temperature of which is assumed to be a sinusoidal oscillation with small amplitude about a constant mean temperature. The mean surface temperature of the cylinder is maintained at T0and the ambient temperature of the?uid is assumed to be T∞(such that T0>T∞).The coordinates are chosen such that x-axis is measured along the surface of the cylinder,the lowest point being the origin x=0,and the y-axis is de?ned to be the distance measured normally outward from the cylinder.The angle x/a is taken to be the angle made by the outward normal with the downward vertical.The?ow con?guration and the coordinate system are shown in Fig.1.
209
y
g
T
Fig.1.Physical model and coordinate system.
We further assume that the ?uid properties are constant,except that the density variations within the ?uid are allowed to contribute to the buoyancy forces.Under the usual boundary layer approxi-mation,the equations for conservation of mass,momentum,and energy are given as
?u ?x +?v
?y
=0,(1)
?u ?t +u ?u ?x +v ?u ?y =gβφsin x a +ν?2u
?y
,(2)
?φ?t +u ?φ?x +v ?φ?y =α?2φ
?y
2,(3)
where (ˉu ,ˉv )are the velocity components along the (ˉx ,ˉy )axes,νis the kinematic viscosity,αis the
thermal diffusivity,φ=T ?T ∞is the temperature of the ?uid,g is the gravitational acceleration,ρis the density of the ?uid,μis the dynamic viscosity,and βis the coef?cient of thermal expansion.The boundary conditions to be satis?ed are
u =v =0,φ=T 0[1+εsin ωt ]at y =0;
ˉu →0,
θ→0
as
ˉy →∞.
(4)
where ωis the frequency of oscillation,ε 1is a positive real number,and t is the time.
2.Transformation of Equations
The boundary condition Eq.(4)to the surface temperature suggests having the solutions in the
following form:
u =u s +εexp (iωt )u 1,
v =v s +εexp (iωt )v 1,θ=θs +εexp (iωt )θ1,
(5)
210
and the required solutions are the imaginary parts of the functions given in Eq.(5).It can be seen that the functions u s,v s,andθs represent the set of solution functions for the steady-state?ow and that should satisfy the following equations:
?u s ?x +
?v s
?y
=0,(6)
u s ?u s
?x
+v s
?u s
?y
=θs sin x+ν
?2u s
?y2
,(7) u s
?θs
?x
+v s
?θs
?y
=α
?2θs
?y2
,(8)
Corresponding boundary conditions for the steady-state situation are
u s=v s=0,θs=1at y=0;
u s→0,θs→0as y→∞.
(9)
In Eqs.(10)–(12),u1,v1,andθ1are the unsteady part of the solutions,equations for which are obtained as
?u1?x +
?v1
?y
=0,(10)
u s ?u1
?x
+v s
?u1
?y
+u1
?u s
?x
+v1
?u s
?y
+iωu1=θ1sin x+ν
?2u1
?y2
,(11) u s
?θ1
?x
+v s
?θ1
?y
+u1
?θs
?x
+v1
?θs
?y
+iωθ1=α
?2θ1
?y
,(12)
with the boundary conditions
u1=v1=0,θ1=1at y=0;
u1→0,θ1→0as y→∞.
(13)
To transform the set of equations to convenient form for integration,we de?ne the following one-parameter group of transformation for the dependent and the independent variables:
ξ=x
a
,η=Gr1/4
y
a
,[u s,u1]=
a
ν
Gr1/2x[f s(ξ,η),f (ξ,η)], [v s,v1]=νGr1/4ξ[f s(ξ,η),f(ξ,η)],φ=T0(θs,θ),
(14)
where Gr=gβT0a3/ν2is the Grashof number,and
f s+f s f s?f 2s+sinξ
ξ
θs=ξ
f s
?f s
?ξ
?f s
?f s
?ξ
,(15)
1 Pr θ s+f sθ s=ξ
f s
?θs
?ξ
?θ s
?f s
?ξ
.(16) 211
Here,Pr is de?ned to be the ratio of the kinematic viscosity to the thermal diffusivity of the?uid and is known as Prandtl number.
The appropriate boundary conditions to be satis?ed by Eqs.(15)and(16)are
f s(ξ,0)=f s(ξ,0),θs(ξ,0)=1,
f s(ξ,∞)=0,θs(ξ,∞)=0.
(17)
The local nonsimilarity Eqs.(15)–(17)have been investigated by Merkin[17]for the case of steady free convection boundary layer?ow of a viscous incompressible?uid along an isothermal horizontal circular cylinder.It appears that Merkin was the?rst to present a complete solution to this problem for Newtonian?uid using Blasius and G¨o rtler series expansion methods along with an integral method and a?nite difference scheme.
In the present analysis,solutions of the set of local nonsimilarity Eqs.(15)–(17)for the steady-state situation are obtained numerically employing the implicit?nite difference method discussed in the following section.
In practical applications,the physical quantities of principle interest are the skin friction and the Nusselt number,which are de?ned as
C f=Gr?3/4a2
μν
τw,Nu=Gr?1/4
a
k(T w?T∞)
τw,
where
τw=μ
?u
?y
y=0
,q w=?k
?T
?y
y=0
.
For the steady-state?ow,we have the following expression for the skin friction and the Nusselt number:
C fs
ξ
=f (ξ,0),Nu s=?θ s(ξ,0),(18) respectively.
Typical numerical values of the local skin friction C fs and the local rate of heat transfer Nu s obtained from the above expressions are obtained for Pr=1against the curvature parameterξare found in Tables1and2,and compared and found to be in excellent agreement with the correspond-ing solutions obtained by Merkin[17],Nazar et al.[22],and Hossain et al.[23].
Now,we are at the position to get the?uctuating parts of the momentum and the energy equa-tions from Eqs.(11)and(12)together with the transformations given in Eq.(14).The equations are obtained as follows:
f +f s f +ff s?2f f s?i St f +sinξ
ξ
θ=ξ
f s
?f
?ξ
+f
?f s
?ξ
?f s
?f
?ξ
?f
?f s
?ξ
,(19)
1 Pr θ +f sθ ?i Stθ+fθ s=ξ
f s
?θ
?ξ
+f
?θs
?ξ
?θ
?f s
?ξ
?θ s
?f
?ξ
,(20)
where St=ωa2/(νGr1/2)is Strauhal number,which depends on the frequency of oscillation of the?uctuating?ow.
212
Table1.
Comparison of the local skin friction coef?cient C fs(steady part)
for different values of curvature parameterξwhile Pr=1
ξPresent Hossain et al.[23]Nazar et al.[22]Merkin[17]
study(2005)(2002)(1976)
π/60.41510.41510.41480.4151
π/30.75390.75390.75420.7558
π/20.95410.95410.95450.9549
2π/30.96960.96960.96980.9756
5π/60.77390.77390.77400.7822
π0.32640.32640.32650.3391
Table2.
Comparison of the rate of heat transfer Nu s(steady part)
for different values of curvature parameterξwhile Pr=1
ξPresent Hossain et al.[23]Nazar et al.[22]Merkin[17]
study(2005)(2002)(1976)
π/60.41610.41610.41610.4161
π/30.40050.40050.40050.4007
π/20.37400.37400.37410.3745
2π/30.33550.33550.33550.3364
5π/60.28120.28120.28110.2825
π0.19170.19170.19160.1945 The corresponding boundary conditions Eq.(13)become
f(ξ,0)=f (ξ,0)=0,θ(ξ,0)=1;
(21)
f (ξ,∞)=0,θ(ξ,∞)=0.
One can see the set of Eqs.(15),(16),(19)and(20)together with the boundary conditions(17) and(21)are coupled and hence we need to get the solutions of these equations all at one time to get better results.
3.Solution Methodology
We integrate the locally nonsimilar partial differential equations(15),(16),(19)and(20)by using a well-established implicit?nite difference method.To begin with,the partial differential equations(21),(22),(27)and(28)are?rst converted into a system of?rst-order equations in y. The resulting equations are expressed in?nite difference forms by approximating the functions and their derivatives in terms of central differences.On denoting the mesh points in the(ξ,η)plane by ξi andηj,where i=0,1,...,M and j=1,2,...,N,central difference approximations are made, such that those equations involvingξexplicitly are centered at(ξi?1/2,y j?1/2)and the remainder are approximated at(ξi,ηj?1/2),whereηj?1/2=(ηj+ηj?1).This procedure results in a set of
213
nonlinear difference equations for the unknowns atξi in terms of their values atξi?1.To solve the resulting equations,Newton’s iteration technique,known in this context as the Keller box method, is used(Cebeci and Bradshaw[24]).Recently,this method was discussed in more detail and used ef?ciently by Hossain et al.[16,18,23,25]in studying the effect of oscillating surface temperature on the natural convection?ow from a vertical?at plate.To initiate the process atξ=0,we ?rst prescribe the pro?les for the functions f s,f s,f s,θs,θ s and which are obtained from the solutions of Eqs.(21),(22),(27)and(28).These pro?les f,f ,f ,θ,θ are then employed in the Keller box scheme,which has second-order accuracy to march stepwise along the boundary layer.For any given value ofξ,the iterative procedure is stopped to obtain the?nal velocity and temperature distributions when the difference in computing the velocity,the temperature,and the species concentration in the latest iteration is less than10?6,i.e.,δf≤10?6,where the superscript ηi denotes the number of iterations.Throughout the computations,a nonuniform grid has been used by setting y=sinh(j/a).Such a grid makes ef?cient use of computational time and computer memory.In the computations,the maximumηmax ranged from10.05to12.00as the value of Pr decreases to0.025,and for the curvature parameterξof the cylinder ranging from0.0toπ.Details of the algorithm of the above method for the present problem are given in the Appendix.
To initiate the solutions,we here considered as the initial pro?les from the exact solutions of the following sets of equations that is obtained for x=0:
f s+f s f s?f 2s+θs=0,(22)
1
Pr
θ s+f sθ s=0,(23)
f +f s f +ff s?2f f s?i St f +θ=0,(24)
1
Pr
θ +f sθ ?i Stθ+fθ s=0,(25) together with the boundary conditions
f s(0)=f s(0)=0,θs(0)=1,f(0)=f (0)=0,θ(0)=1;
f s(∞)=0,θs(∞)=0,f (∞)=0,θ(∞)=0.
(26)
Solutions of the above set of equations are obtained employing the sixth-order implicit Runge–Kutta method together with the Nachtsheim and Swigert iteration technique.
Once we know the solutions of the set of Eqs.(15)–(17)and Eqs.(19)–(21),we readily get the values of the physical quantities,namely,the shear stress and the rate of heat transfer at the surface of the cylinder,which are important from the experimental point of view.
We may now?nd the expressions for the amplitudes|A1|and|A2|and the phase anglesα1and α2for the corresponding local skin friction and heat transfer coef?cient for the?uctuating?ow and temperature?eld from the following relations:
|A1|=(f 2r+f 2i)1/2,|A2|=(θ 2r+θ 2i)1/2(27)
and
α1=arctan f i
f r
,α2=arctan
θ i
θr
,(28)
where,f r,f i,θr andθ i are relatively the real and imaginary parts of the coef?cients f (x,0)and θ (x,0).
214
4.Results and Discussion
Natural convection?ows driven by oscillating surface temperature of a horizontal cylinder are very important in many applications.The relative physical extent(η)of the two effects in the convection region is governed by the magnitudes of the Prandtl number and by their relative values.
For the steady state,other authors such as Merkin[17]have discussed situations with the effects of varying the parameter Pr.Here,we have revisited the solutions obtained by Merkin[17]for Pr=1.0.The numerical solutions,entered to Tables1and2,start at the lower stagnation point of the cylinder,ξ=0,with initial pro?les given by Eq.(16)along with the boundary conditions(17) and proceed round the cylinder up to the upper stagnation point,xi=π.Representative results for the local skin-friction coef?cient and the rate of heat transfer have been obtained for different positionsξwith Pr=1.0.The present results are compared with that of Nazar et al.[22]and Hossain et al.[23]and are found to be in excellent agreement.
From Eqs.(19)and(20),one can see that the?uctuating parts of the?ow and the temperature ?elds are not only depended on the curvature but also on the physical quantities,such as the Prandtl number Pr and the frequency of oscillation on the surface temperature,ω,the corresponding param-eter depending on the Strauhal number St.The solutions obtained by the aforementioned methods are expressed in terms of amplitude and phase of the rates of heat transfer.Here,we restrict our discussion to the aiding or favorable case only,for?uids with Prandtl number Pr=0.7,0.05,and 0.025,which represents air at20?C at1atm,and for values of the Strouhal number St=0.0,0.1, 0.5,and1.0.
The numerical values of the amplitude A1and phaseα1local skin friction for the?uctuating ?ow for?uids having values of Prandtl number Pr=0.1,0.72,and1.0while St=0.5are depicted in Figs.2a and b,respectively.It can be seen from these?gures that the amplitude of the skin friction increases initially with increase of the curvatureξand reaches its maximum value nearξ=π/2on the surface of the cylinder and then it leads to decrease.The behavior of this change of curvature is similar to that appearing for steady-state?ow.On the other hand,the phase of oscillation in the?uctuating skin friction decreases owing to the increasing value of the curvature.This trend on increase is slower in the region nearξ=π/2and decreases faster in the downstream region near the upper stagnation point of the cylinder.We further see that both the amplitude and the phase of oscillations in the skin friction increase owing to the decrease in the value of the Prandtl
b)
Fig.2.Local skin friction for S t=0.5and different values of Pr:
a)amplitude A1,b)phase angleα1.
215
b)
Fig.3.Local skin friction for different values of S t while Pr =0.1:
a)amplitude A 1,b)phase angle α1.
1
23
00.1
0.2
0.3
0.4
0.5
0.6[
A 2
Pr=1.0Pr=0.72Pr=0.1
a)
b)
Fig.4.Local rate of heat transfer for S t =0.5and
different values of Pr :
a)amplitude A 2,b)phase angle α2.
b)
Fig.5.Local rate of heat transfer for Pr =0.1and different values of S t :
a)amplitude A 2,b)phase angle α2.
216
number.This is expected,since owing to the decrease in the values of the Prandtl number,both the momentum and thermal boundary layers increase,which will lead to a rise in amplitude of the disturbances in the?ow?eld.
Figs.3a and b represent the numerical values of the amplitude A1and the phaseα1,respec-tively,obtained for the?uctuating surface skin friction and heat transfer coef?cients for values of the Strauhal number St=0.1,0.5,and1.0while?uid’s Prandtl number Pr=0.1.From this?gure, it can be seen that values of both the amplitude and phase of the skin friction increases owing to the increase in the value of St,i.e.,due to the increase in the frequency of oscillation of the sur-face temperature.We further notice that the relative maximum value of the amplitude moves away from the central point of the cylinder surface toward the upper stagnation point.On the other hand, relative minimum values of the phaseα1of the?uctuating skin-friction coef?cient increases owing to the increase in values of the Strauhal number St.It is also interesting to notice that the rate of decrease in the values ofα1is less with the increase of the curvatureξof the cylindrical surface from its central position(ξ=π/2)while the value of St increases.
Now,we try to observe the effect of the pertinent physical quantities Pr and St on the amplitude A2and the phaseα1of the?uctuating surface rate of heat transfer coef?cients along the surface of the cylinder.The numerical values of amplitude A2and phaseα1of the?uctuating surface rate of heat transfer coef?cients are shown in Figs.4a and b,respectively,for the?ow of?uid having the Prandtl number Pr=0.1,0.72,and1.0,while the frequency parameter St=0.5.It can be seen from these?gures that both the amplitude and phase of the?uctuating rate of heat transfer decrease with the decreasing value of the Prandtl number.We further observe that there is a decrease in the values of A2and increase inα2owing to the increase of the distanceξmeasured along the surface from its lower stagnation point at given values of the Prandtl number.
Finally,in Figs.5a and b,we have displayed the numerical values of A2andα2showing the effect of the increasing values of the Strauhal number St=0.1,0.5,and1.0)while Prandtl number Pr=0.1.From these?gures,we may see that there is an increase in the values of both the amplitude A2and phaseα2of oscillation in the?uctuating rate of heat transfer when there is an increase in the value of the Strauhal number St.We further notice that the rate of increase in the value of the phase angleα2is greater near the upper stagnation point(ξ=π)for higher values of St.
Now,we are at the position to discuss the effects of the different physical parameters on the transient skin friction and surface heat?ux.The?uctuating skin friction and Nusselt number for small-amplitude oscillating?ow may be obtained from the following relations:
C f
=f s(ξ,0)+ε[f i(ξ,0)cosωt+f r(ξ,0)sinωt],(29)ξ
and
Nu=θ s(ξ,0)+ε[θ i(ξ,0)cosωt+θ r(ξ,0)sinωt],(30) In Figs.6and7,numerical values of the?uctuating skin-friction coef?cient against the dimen-sionless time with amplitudeε=0.1have been depicted for three values of the Prandtl number (i.e.,for Pr=0.1,0.72,and1.0),while St=0.5.Solutions are also presented for different values of the curvatureξ=π/4,π/2,3π/4,andπ.From these?gures,it is observed that at every station of Pr,owing to increase in the value of xi,there is a decrease in the amplitude of oscillation in the ?uctuating skin-friction coef?cient;which is expected,since in the region near the upper stagnation point the?ow is dominated by the mean buoyancy force rather than the surface temperature oscil-lation.On the other hand,when the value of Pr is increased,the magnitude of the skin friction gets
217
c)
a)Pr=0.1,b)Pr=0.72,c)Pr=1.0.
c)
Fig.7.Transient skin-friction coef?cient againstτ=ωt for different values of S t when Pr=0.1andε=0.1:
a)S t=0.1,b)S t=0.5,c)S t=1.0.
decreased at everyτstation,and this effect can be observed in Fig.6.From Fig.7,one can see that due to increase in the value of the Strauhal number,the amplitude of the?uctuating skin-friction coef?cient increases and this is because the increase of the Strauhal number leads to the increase in frequency in oscillation of the?uid molecules near the surface,which ultimately contributes an increase of the skin-friction coef?cient.
Now we can discuss the effect of the Strauhal number St and Prandtl number Pr on the?uctu-ating surface heat?ux.In Fig.8,the numerical values of the?uctuating surface heat?ux,obtained for values of Pr=0.1,0.72,and1.0whileξ=π/4,π/2,3π/4,andπ,are depicted againstτin the range of0to2π.Here also,we see that the amplitude of oscillation of the surface heat?ux de-creases owing to the increase of the curvature of the cylinder for all kinds of?uids having different Prandtl number.Through comparison between the three?gures for Pr=0.1,0.72,and1.0,it can be seen that increase in the values of Pr leads to a decrease in the magnitude of the surface heat?ux, and this also reduces to the amplitude of oscillations in the surface heat?ux,and this is because an increase in the value of the Prandtl number reduces the thermal boundary layer thickness.
Finally,effects of the frequency of oscillation of the surface temperature of the cylinder surface, that is of the Strauhal number,as well as of the curvature on the?uctuating surface heat?ux are shown in Fig.9against the dimensionless time variableτ,while Pr=0.1.One can see from this?gure that the amplitude of oscillation of the?uctuating surface heat?ux increases due to the increase in the frequency of oscillation of the surface temperature,which is because the increase in the frequency of oscillation of the surface temperature will enhance the surface heat?ux.
218
c)
Fig.8.Heat transfer coef?cient against τ=ωt for different values of Pr when S t =0.5and ε=0.1:
a)Pr =0.1,b)Pr =0.72,c)Pr =1.0
.
c)
Fig.9.Heat transfer coef?cient against τ=ωt for different values of S t when Pr =0.1and ε=0.1:
a)S t =0.1,b)S t =0.5,c)S t =1.0.
Conclusions
In the present study,we have investigated the ?uctuating natural convection ?ows driven by an oscillating surface temperature from a horizontal https://www.sodocs.net/doc/1d1948182.html,ing the appropriate transformation,the boundary layer momentum and energy equations are reduced to coupled local nonsimilarity equations,which are integrated numerically employing the Keller box method.The steady-state problem that was investigated by Merkin [17]has been revisited by the aforementioned method and found to be in excellent agreement with the present results.The results for the ?uctuating ?ow and temperature distributions have been discussed in terms of amplitude and phase of local shear stress and surface heat ?ux coef?cients with the effect of the physical parameters Pr and St .Effects of the same parameters are also shown on the ?uctuating shear stress and surface heat ?ux as well as on the ?uctuating streamlines and isotherms.The following conclusions may be drawn from the above studies.
1.The amplitude of the skin friction increases initially with an increase of the curvature ξand reaches its maximum value near ξ=π/2on the surface of the cylinder and then it leads to a decrease.And the phase of oscillation in the ?uctuating skin friction decreases owing to an increasing value of the curvature.This trend of increase is slower in the region near ξ=π/2,which then decreases rapidly in the downstream region near the upper stagnation point of the cylinder.
2.Both the amplitude and the phase of oscillations in the skin friction increase owing to a de-crease in the value of the Prandtl number.
219
3.Both of the amplitude and phase of the skin friction increases owing to an increase in the
value of the Strauhal number.The relative maxima of the amplitude move away from the central point of the cylinder surface toward the upper stagnation point.And the relative min-imum values of the phaseα1of the?uctuating skin-friction coef?cient increases owing to an increase in value of St.The rate of decrease in the value ofα1is less with the increase of the curvatureξof the cylindrical surface from its central point(ξ=π/2)while the value of St increases.
4.The amplitude A2and phaseα2of the?uctuating rate of heat transfer decrease with the
decrease of Prandtl number.There is a decrease in the values of A2and increase inα2owing to an increase of the distanceξfor a given Pr.
REFERENCES
1.Lighthill,M.J.,The Response of Laminar Skin Friction and Heat Transfer to Fluctuations in
the Stream Velocity,Proc.Roy.Soc.Lond.,1954,A224,pp.1–23.
2.Gibellato,S.,Strato limito attorno ad una lastra piana investita da un?uido incompressibile
dotato di una velocita’che e’soma di una parte constante e di una parte alternata,Atti Accad.
Torino,1955,89,pp.180and90,pp.1.
3.Gosh,A.,Contribution a l’etude de la couche limite laminaire instationnaire,Publ.Sci,de la
Ministe’re de l’Air,No.381,1961.
4.Rott,N.and Rosenzweig,M.L.,On the Response of the Laminar Boundary Layers,J.Mech.
Phys.,1960,36,pp.173.
https://www.sodocs.net/doc/1d1948182.html,m,S.H.and Rott,N.,Theory of Linearized Time-Dependent Boundary Layers,Cornell Univ.
GSAE Rep.AFOSR TN-60-1100,1960.
6.Gerston,K.,Heat Transfer in Laminar Boundary Layers with Oscillating Outer Flow,
AGARDograph,1965,97,pp.423.
7.Lin,C.C.,Motion in the Boundary Layer with a Rapidly Oscillating External Flow,In:Proc.
IX Int.Congr.Appl.Mech.,V ol.4,Brussels,1956,pp.155–167.
8.Gibson,W.E.,Unsteady Boundary Layers,Ph.D.Diss.,MIT,Cambridge,MA,1957.
9.Ishigaki,H.,Periodic Boundary Layer Near a Two-Dimensional Stagnation Point,J.Fluid
Mech.,1970,43,pp.477–486.
10.Pedley,T.J.,Two-Dimensional Boundary Layers in a Free Stream which Oscillates without
Reversing,J.Fluid Mech.,1972,55,pp.359–383.
11.Nanda,R.S.and Sharma,V.P.,Free Convection Laminar Boundary Layers in Oscillatory Flow,
J.Fluid Mech.,1963,15,pp.419–428.
12.Eshghy,S.,Arpaci,V.S.,and Clark,J.A.,The Effect of Longitudinal Oscillation on Free
Convection from Vertical Surface,ASME J.Appl.Mech.,1965,32,pp.183–191.
13.Muhuri,P.K and Maiti,M.K.,Free Convection Oscillatory Flow from a Horizontal Plate,
Int.J.Heat Mass Transfer,1967,10,pp.717–732.
14.Verma,R.L.,Free Convection Fluctuating Boundary Layer on a Horizontal Plate,J.Appl.
Math.Mech.,1982,63,pp.483–487.
15.Kelleher,M.D.and Yang,K.T.,Heat Transfer Response of Laminar Free Convection Boundary
Layers Along a Vertical Heated Plate to Surface-Temperature Oscillations,J.Appl.Math.Phys., 1968,19,pp.31–44.
16.Hossain,M.A.and Alim,M.A.,Effect of Thermal Radiation on Natural Convection Over
Cylinders of Elliptic Cross Section,Acta Mech.,1998,129,pp.177–186.
220
17.Merkin,J.H.,Free Convection Boundary Layer on an Isothermal Horizontal Cylinder,In:
ASME/AICHE Heat Transfer Conf.,St.Louis,MO,1976.
18.Hossain,M.A,Das,S.K,and Pop,I.,Heat Transfer Response of MHD Free Convection Flow
along a Vertical Plate to Surface Temperature Oscillations,Int.J.Non-Lin.Mech.,1998,23,pp.541–553.
19.Merkin,J.H.,Oscillatory Free Convection from an In?nite Horizontal Cylinder,J.Fluid Mech.,
1967,30,pp.561–576.
20.Gorla,R.S.R.,Unsteady Mixed Convection Due to Time-Dependent Free Stream Velocity,
Heat Mass Transfer ,2003,39,pp.639–644.
21.Keller,H.B.,Numerical Methods in Boundary Layer Theory,Ann.Rev.Fluid Mech.,1978,10,
pp.417–433.
22.Nazar,R.,Amin,N.,and Pop,I.,Free Convection Boundary Layer Flow on a Horizontal Cir-cular Cylinder with Constant Heat Flux in a Micropolar Fluid,Int.J.Appl.Mech.Eng.,2002,7,pp.409–431.
23.Hossain,M.A.,Molla,M.M.,and Gorla,R.S.R.,Natural Convection Flow Form an Isother-mal Horizontal Circular Cylinder with Temperature Dependent Viscosity,Heat Mass Transfer ,2005,41,pp.594–598.
24.Cebeci,T.and Bradshaw,P.,Physical and Computational Aspects of Convective Heat Transfer ,
Springer-Verlag,New York,1984.
25.Hossain,M.A.,Kutubuddin,M.,and Pop,I.,Radiation-Conduction Interaction on Mixed Con-vection from a Horizontal Circular Cylinder,Heat Mass Transfer ,1999,35,pp.307–314.
Appendix A.
In the present work,an ef?cient and accurate implicit ?nite difference method has been em-ployed to solve the system of Eqs.(15)–(17)and Eqs.(19)–(21)in the (ξ,η)coordinates with parameters Pr and St .To begin with,the above-mentioned equations are written in terms of a system of ?rst-order equations as follows:
?f s
?η
=U s ,?U s
?η
=V s ,?θs
?η
=W s ,?V s ?η+p 1(f s V s ?U 2
s )+p 3G s =ξ U s ?U s ?ξ?V s ?f s ?ξ
,1Pr ?W s ?η+p 1f s W s =ξ U s ?G s ?ξ?W s ?f s
?ξ
,
(A1)
?f r
?η
=U r ,?U r
?η
=V r ,?θr
?η
=W r ,?V r
?η+p 1(f s V r +f r V s )?p 2U r U s +p 4U i +p 3G r =ξ
U s
?U r
?ξ+U r ?U s ?ξ?V s ?f r ?ξ?V r ?f s ?ξ
,1Pr ?W r ?η+p 1(f s W r +f r W s )?p 4G i =ξ U s ?G r ?ξ+U r ?G s ?ξ?W r ?f s ?ξ?W s ?f r
?ξ
,
(A2)?f i
?η
=U i ,?U i
?η=V i ,?θi
?η
=W i ,221
?V i
?η+p 1(f s V i +f i V s )?p 2U i U s ?p 4U i +p 3G i =ξ
U s ?U i ?ξ+U i ?U s ?ξ?V s ?f i ?ξ?V i ?f s
?ξ
,
1Pr ?W i ?η+p 1(f s W i +f i W r )?p 4G r =ξ U s ?G i ?ξ+U i ?G s ?ξ?W i ?f s ?ξ?W s ?f i
?ξ
,
(A3)
where p 1=1,p 2=2,p 3=sin ξ/ξ,and p 4=St .Then,the corresponding boundary conditions become
f s (ξ,0)=U s (ξ,0)=0,G s (ξ,0)=1,
U s (ξ,∞)=0,
G s (ξ,∞)=0,f r (ξ,0)=U r (ξ,0)=0,
G r (ξ,0)=1,U r (ξ,∞)=0,
G r (ξ,∞)=0,f i (ξ,0)=U i (ξ,0)=0,
G i (ξ,0)=0,U i (ξ,∞)=0,
G i (ξ,∞)=0.
Now we consider a net placed on the (ξ,η)plane,which is de?ned by
ξ0=0:ξn =ξn ?1+k n ,n =1,2,3,...,M ;η0=0:
ηj =ξj ?1+h j ,
j =1,2,3,...,N.
Now we approximate the quantities
(f s,j ,U s,j ,V s,j ,f r,j ,U r,j ,V r,j ,f i,j ,U i,j ,V i,j ,G s,j ,W s,j ,G r,j ,W r,j ,G i,j ,W i,j )at points (ξn ,ηj )of the net
(f n s,j ,U n s,j ,V n s,j ,f n r,j ,U n r,j ,V n r,j ,f n i,j ,U n i,j ,V n i,j ,G n s,j ,W n s,j ,G n r,j ,W n r,j ,G n i,j ,W n
i,j ),which we will call the net functions.We also employ the notation g n j for points and quantities midway between the net points and for any net function as follows:
ξn ?1/2≡12
(ξn
+ξn ?1),ηj ?1/2≡
1
2
(ηj ?ηj ?1),g n ?1/2
j
≡
12
(g n j +g n ?1j ),g n j ?1/2≡
12
(g n j ?g n j ?1).If g n
j denotes the value of any variable at (ξn ,ηj ),then the variables and their derivative at the point (ξn ?1/2,ηj ?1/2)are replaced by
g n ?1/2
j ?1/2=14
(g n j +g n j ?1+g n ?1j +g n ?1j ?1), ?g ?ξ n ?1/2
j ?1/2
=
14k n
(g n j +g n
j ?1?g n ?1j ?g n ?1j ?1),
?g ?η n ?1/2
j ?1/2
=
14h n
(g n j +g n j ?1?g n ?1j ?g n ?1j ?1).222
Now we write the difference equations that are to approximate equations(A1)–(A3).We start writing the?nite difference approximations for the points(ξn,ηj?1/2)using the central difference derivatives as
f n s,j?f n s,j?1
h j =U n
s,j?1/2
,
U n s,j?U n s,j?1
h j
=V n
s,j?1/2
,
G n s,j?G n s,j?1
h j
=W n
s,j?1/2
,
f n r,j?f n r,j?1
h j =U n
r,j?1/2
,
U n r,j?U n r,j?1
h j
=V n
r,j?1/2
,
G n r,j?G n r,j?1
h j
=W n
r,j?1/2
,
f n i,j?f n i,j?1
h j =U n
i,j?1/2
,
U n i,j?U n i,j?1
h j
=V n
i,j?1/2
,
G n i,j?G n i,j?1
h j
=W n
i,j?1/2
.
So,the equations(A1)–(A3)are then expressed in?nite difference form by approximating the functions and their derivatives,using the center difference about the midpoints(ξn?1/2,ηj?1/2). We get the following nonlinear difference equations:
1 2(L n+L n?1)=ξn?1/2
U n?1/2
s
U n s?U n?1
s
k n
?V n?1/2
s
f n s?f n?1
s
k n
,
1 2(M n+M n?1)=ξn?1/2
U n?1/2
s
G n s?G n?1
s
k n
?W n?1/2
s
f n s?f n?1
s
k n
,
1 2(N n+N n?1)=ξn?1/2
U n?1/2
s
U n r?U n?1
r
k n
+U n?1/2
r
U n s?U n?1
s
k n
?V n?1/2
s
f n r?f n?1
r
k n
?V n?1/2
r
f n s?f n?1
s
k n
,
1 2(O n+O n?1)=ξn?1/2
U n?1/2
s
G n r?G n?1
r
k n
+U n?1/2
r
G n s?G n?1
s
k n
?W n?1/2
r
f n s?f n?1
s
k n
?W n?1/2
s
f n r?f n?1
r
k n
,
1 2(P n+P n?1)=ξn?1/2
U n?1/2
s
U n i?U n?1
i
k n
+U n?1/2
i
U n s?U n?1
s
k n
?V n?1/2
s
f n i?f n?1
i
k n
?V n?1/2
i
f n s?f n?1
s
k n
,
1 2(Q n+Q n?1)=ξn?1/2
U n?1/2
s
G n i?G n?1
i
k n
+U n?1/2
i
G n s?G n?1
s
k n
?W n?1/2
i
f n s?f n?1
s
k n
?W n?1/2
s
f n i?f n?1
i
k n
.
223
Rearranging these equations,we can write
(V s)n+α1(f s V s)n?α2(U2s)n+p3(G n s)+α
V n?1
s
f n s?V n s f n?1
s
=R n?1,
1 Pr (W s)+α1(f s W s)?α
U s G n s?U n s G n?1
s
+W n s f n?1
s
?W n?1
s
f n s
=T n?1,
(V r)n+α1
(f s V r)n+(f r V s)n
?α2(U s U r)n+p3(G n r)+p4(U n i)
?α
U n?1
s
U n r+U n r U n?1
s
?V n r f n?1
s
?V n?1
s
f n r
=R1n?1,
1 Pr (W r)n+α1
(f s W r)n+(f r W s)n
?p4(G i)n
?α
U n?1
s
G n r+U n r G n?1
s
?W n r f n?1
s
?W n?1
s
f n r
=T1n?1,
(V i)n+α1
(f s V i)n+(f i V s)n
?α2(U s U i)n+p3(G n i)?p4(U n r)
?α
U n?1
s
U n i+U n i U n?1
s
?V n i f n?1
s
?V n?1
s
f n i
=R2n?1,
1 Pr (W i)n+α1
(f s W i)n+(f i W s)n
+p4(G r)n
?α
U n?1
s
G n i+U n i G n?1
s
?W n i f n?1
s
?W n?1
s
f n i
=T2n?1,
(A4)
where we have used the abbreviations
α=ξn?1
k n
,α1=p1+α,α2=p2+2α,
R n?1=?L n?1?α
(U2s)n?1?(V s f s)n?1
?p3(G s)n?1,
L n?1=
V s+p1(f s V s?U2s)+p3G s
n?1
,
T n?1=?M n?1+α
(f s W s)n?1?(U s G s)n?1
,
M n?1=
1
Pr
W s+p1f s W s
n?1
, 224
R1n?1=?L1n?1?α
?2(U s U r)n?1+(V s f r)n?1+(V r f s)n?1
,
L1n?1=[V r+p1(f s V r+f r V s)?p2U r U s+p4U i+p3G r]n?1,
T1n?1=?M1n?1+α
?(U s G r)n?1?(U r G s)n?1+(f s W r)n?1+(W s f r)n?1
,
M1n?1=
1
Pr
W r+p1(f s W r+f r W s)?p4G i
n?1
,
R2n?1=?L2n?1?α
?2(U s U i)n?1+(V s f i)n?1+(V i f s)n?1
,
L2n?1=[V i+p1(f s V i+f i V s)?p2U i U s?p4U r+p3G i]n?1,
T2n?1=?M2n?1+α
?(U s G i)n?1?(U i G s)n?1+(f s W i)n?1+(W s f i)n?1
,
M2n?1=
1
Pr
W i+p1(f s W i+f i W s)+p4G r
n?1
.
If we assume
f n?1 s,j ,U n?1
s,j
,V n?1
s,j
,f n?1
r,j
,U n?1
r,j
,V n?1
r,j
,f n?1
i,j
,U n?1
i,j
,V n?1
i,j
,
G n?1
s,j
,W n?1
s,j
,G n?1
r,j
,W n?1
r,j
,G n?1
i,j
,W n?1
i,j
to be known for0≤j≤J,Eq.(A4)is represented by system of15J+15equations for the solutions of15J+15unknowns
f n s,j,U n s,j,V n s,j,f n r,j,U n r,j,V n r,j,f n i,j,U n i,j,V n i,j,
G n s,j,W n s,j,G n r,j,W n r,j,G n i,j,W n i,j,
where j=0,1,2,...,J.These nonlinear systems of algebraic equations are then linearized by Newton’s quasi-linearization method,for which we de?ne the iterations for all functions as
F(i+1) ,j =F(i)
,j
+δF(i)
,j
.(A5)
We then insert the right-hand side of the expression Eq.(A5)in place of
f s,j,U s,j,V s,j,f r,j,U r,j,V r,j,f i,j,U i,j,V i,j,
G s,j,W s,j,G r,j,W r,j,G i,j,W i,j,
in Eq.(A4),and dropping the terms that are quadratic in
δf(i)
s,j ,δU(i)
s,j
,δV(i)
s,j
,δf(i)
r,j
,δU(i)
r,j
,δV(i)
r,j
,δf(i)
i,j
,δU(i)
i,j
,δV(i)
i,j
,
δG(i)
s,j
,δW(i)
s,j
,δG(i)
r,j
,δW(i)
r,j
,δG(i)
i,j
,δW(i)
i,j
,
225