Heat Mass Transfer (2007) 43:923–933 DOI 10.1007/s00231-006-0174-y

ORIGINAL

Numerical study of transient laminar natural convection heat transfer over a sphere subjected to a constant heat flux Kanichi Saito Æ Vasudevan Raghavan Æ George Gogos

Received: 14 December 2005 / Accepted: 14 July 2006 / Published online: 23 August 2006
Springer-Verlag 2006

Abstract Transient laminar natural convection over a sphere which is subjected to a constant heat flux has been studied numerically for high Grashof numbers (105 £ Gr £ 109) and a wide range of Prandtl numbers (Pr = 0.02, 0.7, 7, and 100). A plume with a mushroom-shaped cap forms above the sphere and drifts upward continuously with time. The size and the level of temperature of the transient cap and plume stem decrease with increasing Gr and Pr. Flow separation and an associated vortex may appear in the wake of the sphere depending on the magnitude of Gr and Pr. A recirculation vortex which appears and grows until ‘‘steady state’’ is attained was found only for the very high Grashof numbers (105 £ Gr £ 109) and the lowest Prandtl number considered (Pr = 0.02). The appearance and subsequent disappearance of a vortex was observed for Gr = 109 and Pr = 0.7. Over the lower hemisphere, the thickness of both the hydrodynamic (dH) and the thermal (dT) boundary layers remain nearly constant and the sphere surface is nearly isothermal. The surface temperature presents a local maximum in the wake of the sphere whenever a vortex is established in the wake of the sphere. The surface pressure recovery in the wake of the sphere increases with decreasing Pr and with increasing Gr. For very small Pr, unlike forced convection, the ratio dT/dH remains close to unity. The results are in good agreement

K. Saito Æ V. Raghavan Æ G. Gogos (&) Department of Mechanical Engineering, N104 Walter Scott Engineering Center, University of Nebraska-Lincoln, Lincoln, NE 68588-0656, USA

with experimental data and in excellent agreement with numerical results available in the literature. A correlation has also been presented for the overall Nusselt number as a function of Gr and Pr.

List of CD CD,p CD,l c¢p ~ F0 F¢D g¢ Gr h¢h h
0 k¢ Nu Nuh p¢d Pr q¢s r R¢0 r¥, z¥ t T¢ ~ V Vr Vh z

symbols total drag coefficient pressure drag coefficient viscous drag coefficient specific heat capacity at constant pressure body force in momentum equation drag force acceleration due to gravity Grashof number for a prescribed surface heat flux Local heat transfer coefficient average heat transfer coefficient thermal conductivity overall Nusselt number local Nusselt number motion pressure Prandtl number surface heat flux spherical radial coordinate sphere radius outer computational boundary time dimensional temperature velocity vector radial velocity polar velocity radial transformed coordinate

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Greek symbols b¢ Gz, Gh dsource dH dT h l¢ t¢ n q¢ r w x

coefficient of thermal volumetric expansion diffusion coefficients in general governing differential equation source term in general governing differential equation hydrodynamic boundary layer thickness thermal boundary layer thickness spherical polar coordinate absolute viscosity kinematic viscosity non-dimensional temperature density dependent variable in general differential equation non-dimensional stream function non-dimensional vorticity function

Subscripts ¥ s P, N, S, E, W n, s, e, w

ambient condition surface conditions main grid points staggered grid points

Superscripts ¢ dimensional value

1 Introduction Natural convection from a sphere provides interesting challenges as well as practical applications. There are fundamental unanswered questions concerning the nature of flow separation due to natural convection from spheres. The curvature effect inherent to a sphere surface contributes to rapid growth of the boundary layer followed by a flow separation as the momentum of the fluid becomes sufficiently high to overcome the increase in pressure. Consequently, the boundary layer assumptions are invalid in such a region. The full Navier–Stokes equations need to be employed to investigate the plume development and the recirculation vortex that may be present in the wake of the sphere. The spherical geometry is relevant to many industrial processes such as combustion and vaporization of fuel droplets, spray drying and heat transfer from packed beds of spherical bodies.

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Although numerous analytical, computational and experimental investigations have been conducted on laminar natural convection adjacent to a vertical flat plate, only limited literature is available for natural convection over spheres. Chiang et al. [1] provided numerical solutions of the boundary layer equations employing a semi-analytical series truncation method. Temperature and velocity profiles at various angular positions along the sphere surface and local heat transfer results for Pr = 0.7 were presented. Lien et al. [2] presented local heat transfer results for a nonNewtonian fluid with blowing and suction. Huang et al. [3] studied the influence of Prandtl number (Pr = 0.7, 7) and surface mass transfer with blowing and suction on the Nusselt number and the wall shear stress. Hatem et al. [4] studied numerically the steady state laminar mixed convection around a rotating sphere. The sphere surface was heated by a non-uniform heat flux, however, results were also presented for pure natural convection with a uniform surface heat flux for Pr = 0.7 and 7. All the above studies employed the laminar boundary layer equations. Kurdyumov and Linan [5] have solved the Boussinesq equations to investigate the steady free convective flow due to a point source of heat and heated spheres. They have considered small and moderate Grashof numbers and Prandtl numbers of 0.72 and 7 and proposed a correlation expression for laminar flow Nusselt number. With regards to experimental data, Jaluria et al. [6] presented results for natural convection over a hemisphere in water (Gr = 3.65 · 108 and Pr = 7) for a constant surface heat flux. A steady temperature increase along a hemisphere surface in the direction toward downstream was reported. The purpose of the present study is to solve the full transient Navier–Stokes equations for a solid sphere subjected to a constant heat flux. The SIMPLEC [7] numerical method was used to obtain solutions over a wide range of high Grashof numbers (105 £ Gr £ 109) and a wide range of Prandtl numbers (Pr = 0.02, 0.7, 7 and 100). Our numerical simulations reveal flow separation at the top of the sphere for very large Gr and very small Pr. The presence of flow separation and associated recirculation vortex in the wake of the sphere is discussed as a function of Gr, Pr and time. Results for both overall Nusselt number and drag coefficients are presented.

2 Theoretical model A sphere of fixed radius R¢0 having uniform and constant surface heat flux q¢s is placed in a cold and

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stagnant fluid environment maintained at temperature and pressure of T¢¥ and p¢¥. Gradually, the buoyancy effect induces an upward axisymmetric flow field around the sphere. Near the top of the sphere, hot fluid rises and forms into an evolving plume. The simulations have been carried out using the following assumptions: (a) the sphere is kept stationary at all times, (b) the fluid is Newtonian, of infinite extent, and contains a single, inert chemical compound, (c) the flow is buoyancy-driven, laminar and axisymmetric, and (d) the bulk viscosity coefficient, radiation effects, viscous dissipation and compressibility effects are negligible. The assumption that the flow is axisymmetric may not be completely valid for high speed flows, especially for large Grashof and Prandtl numbers. However, the predictions regarding the distribution of surface quantities agree well with experimental results available in the literature for a Grashof number around 4 · 108 and Prandtl number of 7 (see Sect. 4). Note that throughout the present study, the notation of prime is used to indicate dimensional values. The dimensional governing equations for an incompressible fluid with constant properties can be written in vector form ~0 ¼ 0 r0
V

ð1Þ

~0 0 @V 2 ~0 ~ 0
r 0 ÞV ~ 0 ¼ r0 p0 þ ~ q0 0 þ q0 ðV F þ l 0 r0 V @t

ð2Þ

q0 c0p




 @T 0 ~ 0
r0 ÞT 0 ¼ k0 r0 2 T 0 þ ð V @t0

ð3Þ 0

~ 0 is the velocity vector, ~ where V F is the body force, q¢ is the density, t¢ is the time, p¢ is the pressure, T¢ is the temperature, and l¢, c¢p and k¢ are viscosity, heat capacity at constant pressure and thermal conductivity, respectively. Employing the Boussinesq approximations, the terms ðr0 p0 þ ~ F 0 Þ can be replaced by – g¢q¢b¢(T¢ –T¢¥) – ¢p¢d, where g¢ is acceleration due  to  gravity, p¢d is the motion pressure @p0 and b0 ¼  q10 @T is the coefficient of thermal 0 0 p

volumetric expansion. Employing spherical coordinates (r¢, h), the physical geometry and the surrounding fluid flow are illustrated in Fig. 1. The above equations along with appropriate boundary conditions can be cast in dimensionless form using r¼

r0 ¼ ez ; R00

t¼

Vr ¼

R00 0 V; t0 r

Gr ¼

0 0 0 R04 0 g qS b ; k 0 t0 2

t0 0 t; R02 0

pd ¼

R02 0 p0 ; q 0 t0 2 d

R00 0 T0  T0 V h ; n ¼ 0 0 10 ; 0 t qS R0 =k 0 0 0 cp q t Pr ¼ k0

Vh ¼

where V¢r and V¢h are the radial and polar fluid velocity components, m¢ is the kinematic viscosity, n is the nondimensional temperature, Pr is the Prandtl number and Gr is the Grashof number based on the sphere radius R¢0 and the surface heat flux q¢s. An exponential function for the non-dimensional radius (r = ez) is used to produce finer grids near the surface of the sphere, where pressure, velocity and temperature gradients are the largest. Using the exponential function, Eqs. 1, 2 and 3 may be rearranged and cast into a single generalized dimensionless differential equation of the form  @   @   @  3z re sin h þ Vr re2z sin h þ Vh re2z sin h @t @z @h     @ @r @ @r z z Cz e sin h Ch e sin h ¼ þ þ dsource @z @z @h @h ð4Þ where r is the dependent variable of interest and Gz and Gh are the radial and polar diffusion coefficients, respectively. The values of r, Gz, Gh and dsource for the conservation equations are given below. (a)

Mass continuity equation

r ¼ 1;

Cz ¼ Ch ¼ 0;

dsource ¼ 0

ð5Þ

Fig. 1 Buoyancy driven flow over a sphere

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(b)

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Radial momentum CD;l ¼ 4 Cz ¼ Ch ¼ 1;

r ¼ Vr ;

0

dsource ¼ Vh2 e2z sin h  e2z sin h 2 (c)

@pd  Grne3z cos h sin h @z

@ ðVh ez sin hÞ  2Vr ez sin h @h

ð6Þ

2z

CD ¼ CD;p þ CD;l

2z

ð11Þ

h0h R00 1 ¼ 0 nS k

ð12Þ

and

Nu ¼

h
0 R00 2 ¼ Rp k0 n sin h dh 0 S

ð13Þ

Energy equation Cz ¼ Ch ¼ 1=Pr;

dsource ¼ 0

ð8Þ 3 Numerical method

At time t¢ = 0 the fluid surrounding the sphere is at rest. For t¢ > 0, a constant heat flux is imposed and the no slip boundary is applied at the sphere surface. At the outer boundary, the fluid temperature is taken as steady and uniform for the inflow region, whereas for the outflow region the temperature gradient is considered negligible. The initial and boundary conditions are summarized below. Initial conditions

For z  0 and all h : (b)

The total drag coefficient is given by

Nuh ¼

@pd þ Grne3z sin2 h ¼ Vr Vh e sin h  e sin h @h Vh ez @Vr þ 2ez sin h  sin h @h ð7Þ

r ¼ n;

(a)

ð10Þ

as

r ¼ Vh ; Cz ¼ Ch ¼ 1;

(d)

@Vh sin2 h dh @z S

The local and overall Nusselt numbers are expressed

Polar momentum

dsource

Zp

pd ¼ 0; Vr ¼ 0; Vh ¼ 0;

n¼0

Boundary conditions

For z ¼ 0 and all h : V r ¼ 0; @n=@z ¼ 1

V h ¼ 0;

For h ¼ 0; p and all z :

Vh ¼ 0;

@pd =@h ¼ 0; @V r =@h ¼ 0; @n=@h ¼ 0 As z ! 1 and all h : pd ¼ 0; V h ¼ 0; @ðe2z Vr Þ=@z ¼ 0 @n=@z ¼ 0;

n ¼ 0;

if V r < 0 ðinflow regionÞ;

if V r > 0 ðoutflow regionÞ

The pressure and viscous drag coefficients are given as CD;p ¼ 2

Zp 0

and

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pd jS sin 2h dh

ð9Þ

The governing equations are discretized using the finite control volume formulation and the hybrid scheme [8]. Referring to Fig. 2, the motion pressure and the temperature are evaluated at the main grid points P, E, W, N and S, and the velocities are calculated at the staggered locations e, w, n and s. The sphere surface corresponds to z = 0 and z = z¥ indicates the ‘‘computational infinity’’. This is an approximation which causes negligible error in heat transfer and stress distribution near the sphere surface when z¥ is adequately large. The velocity components, temperature and motion pressure from the previous time step have been used to evaluate the coefficients and source term in the algebraic equations. The SIMPLEC algorithm developed by Van Doormaal and Raithby [7] has been employed to evaluate the motion pressure field. The equations are solved by the ADI method. The modified TDMA solver with a relaxation parameter of 1.85 [7] was used along each of the two alternating directions and the solution is obtained through iterations within a time step. For an indication of convergence in the iteration process, an average Euclidean norm has been calculated and the convergence is determined by checking whether the norm after certain number of iterations is less than or equal to the product of the average norm of the first iteration and a residual reduction factor. The residual reduction factor is taken as 0.1 for velocities and temperature, and 0.2 for pressure correction as recommended by Van Doormaal

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Fig. 2 Grid network

and Raithby [7]. Extensive simulations have been done to select the appropriate numerical parameters. The parameters listed in Table 1 ensure that the final solutions are reasonably independent of grid size (Dz and Dh), time increment (Dt), and ‘‘computational infinity’’ ðr1 ¼ ez1 Þ: The number of time steps required for ‘‘steady state’’ is denoted as nt. The ‘‘steady state’’ is discussed in Sect. 4.2.

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effect of the Grashof number on the temperature field for Pr = 0.7 is shown in Fig. 3. The higher temperature of the sphere surface creates an upward directed flow (opposite direction of gravity vector). At a very early time (not shown), conductive heat transfer is predominant when compared to convective effects, resulting in almost spherically symmetric isotherms. The fluid flows parallel to the sphere surface and gradually realigns itself along the axis of symmetry at the top of the sphere. As time proceeds, the convective strength increases and stronger buoyancy induced motion dominates both momentum and heat transfer processes. As a consequence, the temperature contours are stretched along the flow direction. The entrained fluid rises and then rolls downward forming a vortex ring velocity pattern, which drifts upwards and evolves into a transient plume near the top of the sphere. Progressively, the stem of the plume becomes thinner as the plume cap advances upward, creating the mushroomshaped plume observed in the temperature contours. The cap keeps growing and eventually moves out of the computational domain. Figure 3 shows that the polar extent of the transient cap and plume stem decrease with increasing Gr. Note here that the non-dimensional time is multiplied by Gr1/2 to accommodate the different rates of plume growth; the dimensional speed of the plume cap increases with Gr. The effect of the Prandtl number on the plume is shown in Fig. 4 for Gr = 108. The size and the level of temperature of the transient cap and plume

4 Results and discussion 4.1 Validation The present numerical code has been validated by comparing results with existing data available in the literature. Comparisons are only briefly presented in this section and will be discussed extensively in later sections. The temperature and velocity profiles, and the sphere surface temperature distribution for Pr = 0.7 have been compared with the solutions obtained by Chiang et al. [1] (see Fig. 11). The surface temperature distribution for Pr = 7 was also checked against the numerical results by Huang et al. [3] and the experimental results by Jaluria et al. [6] (see Fig. 10). Comparison with the experimental data is satisfactory, whereas comparison with the numerical results by Huang et al. is excellent (within 2% for Gr = 109).

Table 1 Numerical parameters used in the present study Gr

105

107

108

109

4.2 Temperature and flow field Extensive simulations were carried out for the range 105 £ Gr £ 109 and Pr = 0.02, 0.7, 7, and 100. The

Parameter

Dz Dh Dt r¥ nt Dz Dh Dt R¥ nt Dz Dh Dt R¥ nt Dz Dh Dt R¥ nt

Pr 0.02

0.7

7

100

2.0 · 10–2 2 1.6 · 10–4 54.6 1,480 1.0 · 10–2 1 1.6 · 10–5 20.1 1,480 7.5 · 10–3 1 5.0 · 10–6 9.49 1,500 5.0 · 10–3 1 2.0 · 10–6 4.48 1,190

1.0 · 10–2 1 1.6 · 10–4 20.1 1,480 7.5 · 10–3 1 1.6 · 10–5 9.49 1,480 5.0 · 10–3 1 5.0 · 10–6 4.48 1,500 2.5 · 10–3 0.5 2.0 · 10–6 4.48 1,190

7.5 · 10–3 1 1.6 · 10–4 9.49 2,960 7.5 · 10–3 0.5 1.6 · 10–5 9.49 2,960 5.0 · 10–3 0.5 5.0 · 10–6 4.48 3,000 2.5 · 10–3 0.25 2.0 · 10–6 4.48 2,380

7.5 · 10–3 1 1.6 · 10–4 9.49 4,440 5.0 · 10–3 0.5 1.6 · 10–5 4.48 4,440 2.5 · 10–3 0.5 5.0 · 10–6 4.48 7,500 1.5 · 10–3 0.25 2.0 · 10–6 4.48 9,520

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Fig. 3 Temperature contours for Pr = 0.7 and different Grashof numbers at different times (Dn = 0.01)

Fig. 4 Temperature contours for Gr = 108 and different Prandtl numbers at different times

stem decrease with increasing Pr. In fact the plume becomes so thin that for Pr = 100, it no longer presents a mushroom-shape. It is also observed that the speed of the plume development becomes much slower for higher Pr. As the plume cap rises continuously from the top of the sphere, the flow and temperature fields near the surface of the sphere change barely and the velocity and temperature fields could be considered to have reached ‘‘steady state’’. The ‘‘steady state’’ temperature and velocity profiles as a function of the radial distance are shown in Figs. 5, 6 and 7 for Gr = 108 and Pr = 0.7. The temperature at the surface of the sphere is the maximum and decreases gradually with the radial distance. Up to h 120, the temperature profiles differ slightly as the temperature decays to zero within a small distance from the sphere. This means that the thickness of the thermal boundary layer remains nearly constant and the sphere surface is almost isothermal for approximately h < 120. For h > 120, the surface temperature increases rapidly with h. Figure 6 shows

that the polar velocity increases steeply to its local maximum and decreases moderately with increasing radial distance. The locations where Vh has a peak value shift away from the sphere surface with increasing h. The shifting is only within 3% of the sphere radius for h £ 120 and significantly larger for higher angles. The thickness of the hydrodynamic boundary layer remains nearly constant over a narrower range (approximately h < 90). The maximum in Vh steadily increases up to about 150 and decreases rapidly for higher h as the plume region is approached. For approximately h > 170, Vh first increases and then levels off with radial distance. As shown in Fig. 7, the magnitude of the radial velocity is negative for approximately h < 120 which indicates the inflow region where the fluid flows towards the sphere. For the outflow region, the maximum in Vr continues to increase and sustains its strength to larger radial distances from the sphere surface with increasing h. In addition, the radial velocity is much smaller in magnitude than the polar velocity for approximately

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929 Table 2 Thickness of thermal and hydrodynamic boundary layer at h = 90 for Pr = 0.7 Gr

dT

dH

dT/dH

105 107 108 109

0.576 0.220 0.136 0.0819

1.915 0.325 0.182 0.109

0.299 0.677 0.748 0.749

Table 3 Thickness of thermal and hydrodynamic boundary layer at h = 90 for Gr = 108

Fig. 5 Temperature profiles as a function of radial distance for Gr = 108 and Pr = 0.7

Fig. 6 Polar velocity profiles as a function of radial distance for Gr = 108 and Pr = 0.7

Fig. 7 Radial velocity profiles as a function of radial distance for Gr = 108 and Pr = 0.7

Pr

dT

dH

dT/dH

0.02 0.7 7 100

0.610 0.136 0.0592 0.0317

1.06 0.182 0.231 0.398

0.574 0.748 0.257 0.0794

h < 150. This relation between the velocity components (Vr << Vh) constitutes one of the boundary layer assumptions and beyond this point, where the radial velocity becomes more dominant, the boundary layer analysis will no longer be valid. The ‘‘steady state’’ thermal and hydrodynamic boundary layer thicknesses for various Grashof numbers and Prandtl numbers at h = 90 are shown in Tables 2 and 3, respectively. The outer edges of the boundary layers are defined such that the temperature or polar velocity is within 1% of their maximum values. The effect of the Grashof number on the boundary layer thicknesses is relatively simple. Both the thermal and the hydrodynamic boundary layer thickness decrease with increasing Gr, however, the hydrodynamic boundary layer thickness is more sensitive to changes in the Grashof number. The effect of the Prandtl number, on the other hand, is relatively more complex. In forced convection, the thermal and hydrodynamic boundary layers develop independently of each other so that dH << dT for Pr << 1 and dH >> dT for Pr >> 1. For natural convection, however, the thermal boundary layer is always smaller than the hydrodynamic boundary layer regardless of the magnitude of the Prandtl number since the driving force of fluid flow is caused by the temperature gradients. Similarly as for forced convection, the ratio dT/dH is very small for large Pr. For very small Pr, however, the ratio dT/dH remains closer to unity. The boundary layer growth due to the curvature effect for various Grashof numbers and Pr = 0.7 is shown in Fig. 8. For 105 £ Gr £ 109, the thermal boundary layer thickness remains relatively constant up to h  120 and increases steeply for approximately h > 150. The hydrodynamic boundary layer starts

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growing faster at an earlier angular position when compared to the thermal boundary layer. 4.3 Wake of the sphere Recirculation vortex was found only at the very high Grashof numbers (107 £ Gr £ 109) and very low Prandtl number (Pr = 0.02). The polar extent of the vortex decreases and its strength increases with Gr while the radial extent decreases slightly. The evolution of vortex formation is depicted in Fig. 9 for Gr = 109 and Pr = 0.02. The symbols w and x represent the nondimensional stream function and the non-dimensional vorticity, respectively, which are defined as Vr ¼ 

1 @w e2z sin h @h

ð14Þ

During early times when all the isotherms are nearly spherically symmetric, there is no vortex present. At approximately tGr1/2 = 11.25, as the mushroom-shaped plume cap starts forming, a thin vortex having the angular extent of approximately p/6 appears. As the stem of the plume becomes thinner at approximately tGr1/2 = 15, the vortex thickens in height and matures in size. During the formation, the temperature distribution and the strength of the vortex change drastically (see isothermal and vorticity contours), although its size and shape change weakly. In fact, the polar extent of the vortex changes slightly during its formation. The appearance and disappearance of a vortex has been predicted in the case of Gr = 109 and Pr = 0.7. The vortex matures at approximately tGr1/2 = 37.5. It starts shrinking afterwards and eventually disappears at approximately tGr1/2 = 67.5.

and 4.4 Surface temperature and surface pressure ~ x¼rV

ð15Þ

Fig. 8 Influence of Grashof number on a thermal and b hydrodynamic boundary layers as a function of h for Pr = 0.7

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Figure 10 shows the ‘‘steady state’’ surface temperature and surface pressure as a function of h for different Grashof numbers and Pr = 7. For approximately h > 150, the surface temperature increases steeply (or the local Nusselt number decreases steeply) with h, indicating the large growth of the thermal boundary layer. The present numerical results for Pr = 7 are compared with the numerical results by Huang et al.

Fig. 9 Evolution of vortex formation for Gr = 109 and Pr = 0.02

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[3], and the experimental data for a heated invertedhemisphere obtained by Jaluria et al. [6] for Gr = 3.65 · 108. Huang et al. [3] solved numerically the boundary layer equations. In view of this, they reported results only up to h = 90. The present study shows an excellent agreement with Huang et al. [3], especially at the higher Grashof numbers. On the other hand, the results from the present study differ substantially from the experimental results for higher h. The reason for the disagreement could be the different geometry in the experimental study. Since the experiment was conducted using a hemisphere (flat surface on the top), the curvature effects close to h = 90 (edge of the hemisphere) are much different from a whole sphere. Figure 10b shows the weak pressure recovery at higher Gr for approximately h > 150; however, the adverse pressure gradient is not strong enough to cause a recirculation vortex. Figure 11 shows the ‘‘steady state’’ surface temperature and surface pressure for different Grashof numbers for Pr = 0.7. Except for the magnitude, the surface

temperatures assume very similar profiles as in the case of Pr = 7. The surface pressures, however, indicate larger pressure recovery for Pr = 0.7. The location where the pressure starts to recover shifts to earlier angular position as Gr decreases. In addition, the surface temperatures are compared with the solution obtained by Chiang et al. [1] using thin laminar boundary layer assumptions. The present analysis is in excellent agreement with their work for approximately h < 120 and for very high Grashof numbers since the thin boundary layer assumptions are valid under these conditions. As expected, the boundary layer theory completely breaks down for large angles, as can be seen in Fig. 11a. Figure 12 shows the effect of the Prandtl number on the ‘‘steady state’’ surface temperature and pressure. For Pr = 0.02, the surface temperature reaches a maximum and then decreases with h. This local behavior is due to the presence of the vortex. Table 4 presents the ‘‘steady state’’ values of the Nusselt number scaled with the sphere radius, and

Fig. 10 Influence of Grashof number on a surface temperature and b surface pressure as a function of h for Pr = 7

Fig. 11 Influence of Grashof number on a surface temperature and b surface pressure as a function of h for Pr = 0.7

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viscous, pressure and total drag coefficients for all the simulations performed in the present study. The Nusselt number increases with Gr and Pr. The total drag coefficient decreases with increasing Pr and decreasing Gr. A correlation for the ‘‘steady state’’ overall Nusselt number based on the sphere diameter (NuD), as a function of Grashof number based on the sphere diameter (GrD) and Prandtl number (Pr) has been obtained for the complete range of parameters presented in Table 4, and is given in Eq. 16. The values of the Grashof number in Table 4, which is based on the sphere radius, have been multiplied by 16 and the values of Nusselt number in Table 4 have been multiplied by 2, in order to scale the quantities by the sphere diameter. 0:25 NuD ¼ 2 þ 0:531 Gr0:2 D Pr

obtained from the above correlation. This correlation can be employed to calculate the overall Nusselt number for natural convection over a sphere subjected to constant heat flux quite accurately for a wide range of high Grashof numbers (105 £ Gr £ 109) and a wide range of Prandtl numbers (Pr = 0.02, 0.7, 7 and 100). The Nusselt number in the present study (constant heat flux condition at the sphere surface) varies as Gr0.2 D . It can be recalled from any fundamental heat transfer textbook that the Nusselt number for an isothermal sphere varies as Gr0.25 D . This illustrates the fundamental difference between the constant flux and isothermal boundary conditions at the sphere surface. However, the dependence on Prandtl number is the same in both cases (Pr0.25). Also, the limiting value of the Nusselt number at vanishing Prandtl number is the same for both cases (NuD = 2, when Pr fi 0).

ð16Þ

In Fig. 13, symbols indicate the data given in Table 4, scaled by the sphere diameter. The solid line is

5 Conclusions The full transient Navier–Stokes equations for natural convection over a solid sphere subjected to a constant heat flux were solved. The Boussinesq approximation and the SIMPLEC numerical method were employed to compute the transient and ‘‘steady state’’ results for a wide range of high Grashof numbers (105 £ Gr £ 109) and a wide range of Prandtl numbers (Pr = 0.02, 0.7, 7 and 100). Table 4 Overall Nusselt number (Nu), pressure (CD,p), viscous (CD,l) and total (CD) drag coefficients for 105 £ Gr £ 109 and Pr = 0.02, 0.7, 7 and 100

Fig. 12 Influence of Prandtl number on a surface temperature and b surface pressure as a function of h for Gr = 108

123

CD,p/Gr3/5

CD,l/Gr3/5

CD/Gr3/5

Nu

Gr = 105 Pr = 0.02 Pr = 0.7 Pr = 7 Pr = 100

38.73 8.029 1.843 1.475

31.30 9.523 2.795 2.081

70.03 17.55 4.638 3.555

2.137 5.254 7.969 17.37

Gr = 107 Pr = 0.02 Pr = 0.7 Pr = 7 Pr = 100

67.49 9.504 3.094 0.941

31.48 9.254 3.926 1.365

98.97 18.76 7.020 2.306

4.234 12.08 21.35 38.98

Gr = 108 Pr = 0.02 Pr = 0.7 Pr = 7 Pr = 100

82.60 10.18 3.206 0.991

31.41 9.178 3.897 1.394

114.01 19.36 7.103 2.385

6.250 18.69 33.36 59.70

Gr = 109 Pr = 0.02 Pr = 0.7 Pr = 7 Pr = 100

97.57 10.77 3.305 1.012

31.38 9.122 3.883 1.400

129.0 19.89 7.187 2.412

9.449 29.13 52.35 93.41

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4.

5. Fig. 13 Correlation for ‘‘steady-state’’ overall Nusselt number (NuD) as a function of GrD and Pr. Symbols show the data presented in Table 4, scaled by sphere diameter. Solid line corresponds to Eq. 16

At an early time, conductive heat transfer is predominant compared to convective effects. As time proceeds, the convective strength increases and stronger buoyancy induced motion dominates both momentum and heat transfer processes. An axisymmetric plume with a mushroom-shaped cap forms at the top of the sphere. ‘‘Steady state’’ is defined only for the surface conditions or quantities in the vicinity of the sphere. The following are the main results discussed in the present study 1.

2.

3.

The size and the level of temperature of the transient cap and plume stem decrease with increasing Gr and Pr. The plume development is slower for lower Gr and higher Pr. The thickness of both the hydrodynamic and the thermal boundary layers remain nearly constant and the sphere surface is nearly isothermal over the lower hemisphere. The effect of the Prandtl number on the relative thickness of the thermal and hydrodynamic boundary layers has been clearly delineated. In forced convection, the thermal and hydrodynamic boundary layers develop independently of each other so that dH << dT for Pr << 1 and dH >> dT

for Pr >> 1. For natural convection, however, the thermal boundary layer is always smaller than the hydrodynamic boundary layer regardless of the magnitude of the Prandtl number since the driving force of fluid flow is caused by the temperature gradients. Similarly as for forced convection, the ratio dT/dH is very small for large Pr. For very small Pr, however, the ratio dT/dH remains close to unity. The surface temperature presents a local maximum in the wake of the sphere whenever a vortex is established in the wake of the sphere. The surface pressure recovery in the wake of the sphere increases with decreasing Pr and increasing Gr. Very fine grid is required in the wake region of the sphere. A recirculation vortex which appears and grows until ‘‘steady state’’ is attained was found only for the very high Grashof numbers (107 £ Gr £ 109) and very low Prandtl number (Pr = 0.02). The appearance and subsequent disappearance of a vortex was observed for Gr = 109 and Pr = 0.7.

References 1. Chiang T, Ossin A, Tien CL (1964) Laminar free convection from a sphere. J Heat Transfer 86:537–542 2. Lien F, Chen C, Cleaver JW (1986) Analysis of natural convection flow of micro-polar fluid about a sphere with blowing and suction. J Heat Transfer 108:967–970 3. Huang M, Chen C (1987) Laminar free convection from a sphere with blowing and suction. J Heat Transfer 109:529–532 4. Hatem N, Philippe Ch, Mbow C, Kabdi Z, Najoua S, Daguenet M (1996) Numerical study of mixed convection around a sphere rotating about its vertical axis in a Newtonian fluid at rest and subject to a heat flux. Num Heat Transfer A 29:397– 415 5. Kurdyumov VN, Linan A (1999) Free convection from a point source of heat, and heat transfer from spheres at small Grashof numbers. Int J Heat Mass Transfer 42:3849–3860 6. Jaluria Y, Gebhart B (1975) On the buoyancy-induced flow arising from a heated hemisphere. Int J Heat Mass Transfer 18:415–431 7. Van Doormaal J, Raithby GD (1984) Enhancements of the simple method for predicting incompressible fluid flows. Num Heat Transfer 7:147–163 8. Patankar SV (1980) Numerical heat transfer and fluid flow. Hemisphere Publishing Corporation, New York

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