Development of immersed boundary methods for isothermal and thermal flows 7

7.1 Forced convective heat and mass transfer around a stationary isolated sphere Knowledge concerning the flow around an isolated stationary sphere has been well accumulated by experime

The incompressible viscous thermal flow around spheres is a fundamental fluid dynamic and heat transfer problem with widespread scientific and engineering applications Bioreactors, industrial fluidized beds, combustion systems, and chemical processes, etc are among the well-known examples In spite of its simple and axisymmetric geometry, the sphere always induces fully three-dimensional flows which would admit complicated kinematics The forced convective heat transfer, as pointed out, is a passive scalar transport governed by the flow field such that the thermal field should experience corresponding variations The natural convective heat transfer, on the other

hand, is induced by buoyancy due to the temperature gradients so that the fluid and thermal fields interact intensively and are strongly coupled In the present study, the fluid behaviors as well as the heat transfer performances for both forced and natural convective heat transfer are numerically investigated Specifically, forced convective heat and mass transfer around a single stationary sphere, two tandem stationary spheres and a single streamwise rotating sphere, as well as natural convective heat transfer inside concentric and vertically eccentric spherical annuluses are simulated The thermal conditions on the sphere surface, in most cases, are set to be of Dirichlet type, for the convenience of direct comparison and analysis Heat flux condition is only considered in a few particular cases  

7.1 Forced convective heat and mass transfer around a stationary isolated sphere

Knowledge concerning the flow around an isolated stationary sphere has been well accumulated by experiments and numerical simulations during the past decades Just as its two-dimensional counterpart (flow around a circular cylinder), the problem of flow around a stationary sphere has been considered

as an excellent case for validating new numerical methodologies in three dimensions

The flow field around a single sphere is recognized to enjoy rich transition

modes, depending on the Reynolds number Re, which is frequently defined

as Re ρU D

μ∞

= based on the sphere diameter D and the uniform incoming

velocity U∞ At low Reynolds numbers, an axisymmetric vortex ring is formed behind the sphere and attached to its surface The flow is steady and topologically similar at various Re With an increase in Reynolds number, the vortex ring downstream of the sphere shifts off-axis and the flow no longer exhibits axial symmetry Although non-axisymmetric, the flow does, however, contain a plane of symmetry and remains steady When the Reynolds number increases further, a third transition mode occurs, at which stability of vortex ring is lost The flow is now unsteady but periodic The vortex is shed from the sphere surface with a single dominating frequency A continued increase in the Reynolds number would lead to very complex flow behaviors and is beyond our investigation These abundant phenomena and the critical Reynolds numbers at which the transition modes occur have been explored by many researchers Taneda (1956), using the flow visualization method, identified that the generation of an axisymmetric vortex ring occurred

at Re≈ 24 Magarvey & Bishop (1961), through the dye visualization, found that the wakes behind the liquid spheres exhibited the same vortex structure as that observed by Taneda (1956) Besides, he noticed that the stable and axisymmetric rings persisted up to Re=210 but developed into a non-axisymmetric pattern characterized by two parallel threads in the range

of 210<Re< 270 This double-thread, beyond Re=270, lost its stability and shed from the sphere Numerically, by employing a spectral element method, Tomboulides (1993) predicted the initial separation at Re=20 and a transition from axisymmetric to double-thread wakes at Re=212 The stability of the axisymmetric flow was also examined by Natarajan & Acrivos (1993) using a finite-element method, who suggested a regular bifurcation

at Re= 210 Johnson & Patel (1999) investigated the flow regimes at Reynolds numbers up to 300 both numerically and experimentally The lower and upper Reynolds number limits for steady axisymmetric regime was reported to be 20 and 210 while the onset of a periodic vortex shedding flow was recorded around Re= 270 There are still many other excellent efforts on the flow characteristics investigations, including the shedding frequency measurement (Achenbach 1974; Kim & Durbin 1988; Sakamoto & Haniu 1990), vortex structure visualization (Shirayama 1992), etc

The convective heat transfer from an isolated sphere has been the subject of extensive investigations, as summarized by Clift et al (1978) and Polyanin et

al (2002) Ahmed & Yovanovich (1994) proposed an approximate analytical solution in the range of Reynolds number 0≤Re≤ ×2 104 based on linearization of the energy equation Whitaker (1972), by examining his experimental data, provided a correlation for the average Nusselt number in a wide range of Reynolds numbers 1 Re 10≤ ≤ 5 Numerically, Dennis et al

(1973) calculated the heat transfer from an isothermal sphere at low values of Reynolds numbers up to 20 Dhole et al (2006) investigated the heat transfer characteristics in the steady symmetric flow regime for both the constant temperature and constant heat flux boundary conditions on the solid sphere surface

The convective heat and mass transfer from an isolated stationary sphere is governed by two characteristic parameters: the Reynolds number Re and the Prandtl number Pr In the current simulation, Pr is fixed at 0.71 and Re is taken as 100, 200, 250 and 300, which covers all the three flow regimes identified in the literature For the steady axisymmetric case (Re=100 and 200), both kinds of thermal boundary conditions have been considered on the sphere surface, due to the availability of published results for convenient comparison They are set as: the isothermal condition T B =1 and the isoflux

7.1.1 Steady axisymmetric flow regime

In a certain low Reynolds number range, a steady axisymmetric vortex ring is reported to form behind the sphere (Taneda 1956; Tomboulides 1993; Magnaudet et al 1995), which is often referred to as the “steady axisymmetric flow regime” Two Reynolds numbers of Re=100 and 200 are selected for the simulation and description of flow behaviors The three-dimensional vortex structures identified using the λ2-definition proposed by Jeong & Hussain (1995) are plotted in Fig 7.1(a)-(b), where the axisymmetry is clearly visualized, being toroidal and concave toward the sphere tail (without special illustration, the three-dimensional vortex structures in the following are all identified using the method of Jeong & Hussain (1995)) Streamlines in the ( , )x y -plane at both Reynolds numbers are presented in Fig 7.2 As can be

observed, the flow separates from the surface of the sphere and then rejoins, forming a closed separation bubble which attaches to the sphere surface It is also noted that the flows in both planes are symmetric about the centerline, and their topologies keep identical Variations exist only in the separation location, position of the vortex center and length of the recirculation region With Reynolds number increasing from Re=100 to 200, the separation point

on the sphere surface moves towards the front stagnation point while the vortex center extends downstream and the recirculation region becomes stretched

The steady-state thermal fields are plotted in terms of temperature contours (isotherms) in Figs 7.3 and 7.4, for isothermal condition and isoflux condition, respectively As expected, the isotherms are symmetric about the centerline for both types of thermal condition They cluster heavily around the front surface

of the sphere, indicating a large heat transfer rate there As the Reynolds number increases from 100 to 200, the clustering of isotherms on the front surface shows some enhancement Moreover, the isotherms around the back stagnation point which is thinly distributed for Re=100, are more densely spaced at Re=200 These observations can be further verified in Fig 7.5, where the local Nusselt number distribution along the sphere surface is presented (the angle θ is measured in the clockwise direction from the front stagnation point (θ = ) to the rear one (0° θ =180°)) Each curve manifests two peaks located around the front and rear stagnation points, with the former much higher than the latter The Nusselt number throughout the sphere surface

is enhanced as Re increases from 100 to 200 The profiles reported by Dhole

et al (2006) for both boundary conditions are included in Fig 7.5 as well, which show a good agreement with ours

The present results are also quantitatively validated in Table 7.1 by making a comparison of our calculated drag coefficient with the numerical measurements of Johnson & Patel (1999), Gilmanov et al (2003) and White (1974) Note that for all the problems in this chapter, the drag coefficient is

defined as

2 2

1

D D

F C

A further validation is implemented by examining the heat transfer rate on the sphere surface For the isothermal case, Table 7.2 lists the surface-averaged Nusselt number Nu obtained from the present results and those calculated from the published correlations in the literature Our numerical value is calculated by taking an area-integral of the local Nusselt number over the sphere surface and then making an average in the way

2

14

7.1.2 Steady planar-symmetric flow regime

The flow at Re=250 is taken as a representative to illustrate the flow features in the steady planar-symmetric flow regime Streamlines in both ( , )x y -plane and ( , ) x z -plane are presented in Fig 7.6, which clearly reveals

the loss of the axial-symmetry in the current flow regime Although non-axisymmetric, the flow remains steady and exhibits symmetric in the ( , )x z -plane It is also noted that the rear stagnation point in the ( , ) x y -plane

moves forward along the sphere surface and stays away from the centerline The three-dimensional vortex structure is plotted in Fig 7.1(c) where the vortex behind the sphere is different from the toroidal structure of axisymmetric case, and is developed into a double-threaded structure in the planar-symmetric flow regime

While the isotherms in the ( , )x z -plane still keep symmetric about the

centerline, the onset of non-axial symmetry, as expected, results in an asymmetric behavior of thermal field in the ( , )x y -plane (Fig 7.7) The local

Nusselt number along the sphere surface in the ( , )x z -plane is depicted in Fig

7.8 As compared to the axisymmetric cases (Re=100 and 200), the heat transfer is enhanced on the surface around the front stagnation point In the ( , )x y -plane (Fig 7.8(b)), the heat transfer rate on the front hemisphere

( 0 ≤ ≤θ 100 and 260 ≤ ≤θ 360 ) is coincident with that in the ( , )x z -plane, showing a symmetric behavior On the rear hemisphere,

anti-symmetry happens and the local peak no longer appears at θ =180 but moves forward to 170θ = Additionally, the peak value increases as compared to that in the ( , )x z -plane

The fluid and thermal behaviors at Re= 250, in forms of average drag coefficient and Nusselt number, are compared with those established ones in Tables 7.3 and 7.2 respectively, from which good agreements are observed

7.1.3 Unsteady periodic flow regime

As Re is increased above approximately 270, instability becomes so pronounced that flow unsteadiness is triggered In this periodic unsteady flow regime, flow at a Reynolds number of 300 was chosen as the case of interest The time evolution of drag coefficient and surface-averaged Nusselt number

on the sphere is well traced and plotted in Fig 7.9, from which a perfect periodic characteristic is observed, showing that the vortex is shedding periodically from the sphere The vortex shedding frequency is frequently described by the dimensionless Strouhal number, which, based on our calculation, is 0.133 and basically agrees well with the published value of 0.136 reported by Tomboulides (1993), and 0.137 provided by Johnson & Patel (1999)

The instantaneous in-plane streamlines in one vortex shedding cycle are plotted in Figs 7.10 and 7.11, corresponding to four equal-interval phases It

is clear from the pictures that the streamlines in the ( , )x z -plane (Fig 7.10)

remain symmetric throughout the cycle, indicating that the observed plane-symmetry for the steady flows is still present in the unsteady ones,

which agrees with the findings of Johnson & Patel (1999) and Kim & Choi (2002) The movement of the recirculation region as revealed in Johnson & Patel (1999), is also obviously observed in this work Streamlines in the symmetry plane, i.e ( , )x y -plane, as observed in Fig 7.11, show a sequence

of patterns similar to that of the steady case ( Re=250)

The three-dimensional wake structures in this flow regime become more complex As shown in Fig 7.1(d), the vortices are no longer attached to the sphere but are seen to shed regularly from the sphere surface and develop into

a pair of legs behind it The hairpin vortex structure, which has been reported

by Kim & Choi (2002) and Giacobello et al (2009), is nicely captured in the present work

Consistent with the streamlines in each phase, the isotherms in the ( , )x z -plane (Fig 7.12) are symmetric about the centerline throughout the

vortex shedding cycle and evolve following the motion of fluid flow Their distributions in the vicinity of the sphere surface, however, are almost independent to time Isotherms near the sphere surface in the ( , )x y -plane

(Fig 7.13) do not change with time either They always concentrate around the stagnation points, implying that the stagnation points do not change with time The above visualizations can be further clarified by the Nusselt number profiles plotted in Fig 7.14, from which it is observed that the Nusselt number

in the ( , )x z -plane (Fig 7.14(a)) are symmetric about θ =180° for all the

four phases They peak at the front and rear stagnation points of θ = 0°

and 180θ = °, with the former much larger than the latter Except for the

symmetric characteristic, the Nusselt number profiles in the ( , )x y -plane (Fig

7.14(b)) show similar behaviors However, the second peak on the sphere

surface no longer exhibits itself at θ =180° but moves forward to 160θ = °

approximately Now we can see that while the front stagnation point always

resides at θ = , the rear stagnation point in the ( , )0° x y -plane keeps moving

forward from θ =180° at Re=100 and 200, to θ ≈170° at Re=250

where T shed is the vortex shedding period, are presented in Table 7.4 and 7.2

respectively, and compared with the published results Once again, good

agreement is achieved

From the above simulation results, we can see that as the Reynolds number

increases, the flow instability increases correspondingly and the flow develops

from steady axisymmetric to unsteady plane symmetric While the drag force experienced by the sphere successively decreases, the overall heat transfer from the sphere surface is monotonically enhanced

7.2 Forced convective heat and mass transfer around a pair of tandem spheres

As demonstrated in the above subsection, complex flow behaviors can be produced from one isolated sphere in a free stream It is believed that flow phenomena would become more abundant and interesting when two identical spheres are located in tandem, due to the dynamic interaction between the wakes behind each sphere Consequently, flow past a pair of tandem spheres attracts quite a number of researchers

Pioneering studies, such as the ones conducted by Rowe & Henwood (1961), Lee (1979), Tsuji et al (1982), were mainly dedicated to the direct measurement of drag forces on interacting spheres Later, Tal et al (1984) investigated both the hydrodynamic and thermodynamic interaction of two tandem spheres in a steady uniform flow at Re=40 Zhu et al (1994) examined the effect of separation distance and Re on the drag forces for Reranging from 20 to 130 Prahl et al (2007) reported the variation of the drag and lift forces for Reynolds number of 50, 100 and 200 Yoon & Yang (2007),

by performing a parametric study, estimated flow-induced forces on two

arbitrarily positioned spheres at Re=300 All these studies found that the drag forces on both spheres were always less than that of an isolated one and that the reduction was much more significant for the downstream sphere Other than drag forces, more recent works have also concentrated on examining the wake structure and its effect on dynamic forces Zou et al (2005) paid their attentions to the effect of separation distance on the flow patterns at Re=250 and predicted three different flow regimes Following Zou’s work, Prahl et al (2009) investigated the force characteristics and shedding patterns under the influence of separation spacing at Re=300 The aforementioned studies clearly indicate that for two spheres in tandem

arrangement, the Reynolds number Re U D

ν∞

= (U∞ is velocity of the

uniform incoming flow) and their separation distance G ( G is the center-to-center distance between the spheres) are the two most important parameters The drag, lift forces and wake patterns strongly depend on them

In this subsection, forced convective heat and momentum transfer around a pair of tandem spheres are numerically studied at two Reynolds numbers: a low one of 40 and a moderate one of 300 Two separation distances of / 1.2

G D= and 2.5 are simulated for Re=40, and three separation distances

of G D/ =1.5 , 2 and 3 are considered for Re=300 Their flow patterns, drag and lift coefficients and heat transfer performances are well analyzed A

computational domain of size (25D G+ ) 20× D×20D is chosen such that the inlet is located 10D ahead of the upstream sphere and the outlet 15D

behind the downstream sphere, with the spheres located at (10D,10D,10D) and (10D G+ ,10D,10D) A non-uniform mesh is used for the domain discretization, with a locally fine resolution of h= Δ = Δ = Δ =x y z D/ 40covering the two spheres For the unsteady cases, a step size of Δ =t 0.001 is selected for the time integration

The flow field at this low Reynolds number is always axisymmetric regardless

of the separation distance, as seen from the three-dimensional vortex structures in Fig 7.15 The steady state streamlines plus isotherms in the ( , )x z -plane are depicted in Figs 7.16 and 7.17 for G D/ =1.2 and 2.5, respectively It is observed that the recirculation region behind the upstream sphere for G D/ =1.2(Fig 7.15) contacts the downstream sphere while it does not happen in the case of G D/ = 2.5, which, as a result, are frequently referred as the contacting case and non-contacting case, respectively In the contacting case, the downstream sphere is located so close to the upstream one that the stagnation-type flow in front of it is replaced with a recirculation zone, and the dense concentration of isotherms, which is supposed to be observed on the front surface of the sphere, does not show up either The existence of the recirculation zone corresponds to a low pressure region and results in a

considerable reduction of drag In the non-contacting case, the recirculation zone behind the upstream sphere is being stretched as compared to that of a single sphere Although the stagnation point and isotherms gathering reappear

at the front surface of the downstream sphere, its concentration, i.e., temperature gradient, is clearly weaker For both cases, flow separation occurs earlier on the upstream sphere and much later on the downstream one (for easy comparison, the streamlines for an isolated sphere is plotted in Fig 7.18) The recirculation region behind the downstream sphere, at the same time, significantly contracts itself These observations are in well agreement with those of Tal et al (1984) and Zhu et al (1994)

The local Nusselt number distribution along the sphere surface in its circumferential direction is given in Fig 7.19 for the contacting case

of G D/ =1.2 The result for the case of isolated sphere is also provided for convenience θ is measured in the same way as previously It is noted that the distribution of local Nusselt number Nu on the upstream sphere has not been changed as compared to the case of isolated sphere up to the location around 80θ = °, after which, large discrepancy is captured and Nu on the upstream sphere drops drastically This is due to the close contact of the downstream sphere such that there is no enough room for the entrainment of the cold free-stream fluid The local Nusselt number on the downstream sphere is significantly affected by the presence of upstream one A

considerable reduction is produced on its front surface and a maximum occurs

at around 100θ = ° where the isotherms are concentrated, which should approximately correspond to the separation location

The local Nusselt number distributions for the non-contacting case of / 2.5

G D= are plotted in Fig 7.20 On the upstream sphere, the large reduction in Nusselt number on its leeward side does not occur, as compared

to the contacting case It seems that the local Nusselt number almost recovers

to that for the isolated case On the downstream sphere, the local heat transfer rate is enhanced throughout the entire surface as compared to the contacting case, especially in the leeward side (approximately from 110θ = °

to 180θ = °)

A quantitative comparison of the mean drag coefficient and Nusselt number is made between our results and the experimental measurements of Rowe et al (1961) and numerical simulations of Tal et al (1984) and listed in Table 7.5 While the present drag coefficients are a little larger than the reference values, the Nusselt numbers agree quite well with those from Tal et al (1984) The results show that, the drag forces on both spheres are increased as the separation distance G D/ increases from 1.2 to 2.5 and the heat transfer rates are enhanced for both spheres

7.2.2 The case at Re=300

When the Reynolds number is increased to Re=300, the flow patterns will transit from one mode to another depending on the separation distance The present study considers three separation distances of G D/ =1.5, 2.0 and 3.0, corresponding to three different flow regimes: steady axisymmetric, steady plane symmetric and unsteady plane symmetric

Our calculations show that the flow at small separation distances of / 1.5

G D = and 2.0 would eventually reach a steady state While a toroidal-structured vortex (Fig 7.21(a)) is enveloping the spheres

at G D/ =1.5 , the vortex at G D/ =2.0 loses its axisymmetry and is characterized by two parallel threads behind the downstream sphere (Fig.7.21(b)) Recalling the flow pattern of an isolated sphere at the same Reynolds number, it seems that the placement of a second sphere in the wake can suppress the three-dimensional instabilities to a certain extent Fig 7.22 plots the steady-state streamlines and isotherms in the ( , )x y -plane

for G D/ =1.5, while the corresponding pictures for G D/ =2.0 are shown

in Fig 7.23 in both the ( , )x y - and ( , ) x z -plane Consistent with the

three-dimensional wake structures in Fig 7.21(b), a breakdown of the flow symmetry in the ( , )x y -plane has been captured as the separation distance

increases from G D/ =1.5 to 2.0 In both cases, it is observed that the recirculation region behind the upstream sphere is confined in the gap region

and closely contacts with the downstream sphere, and the stagnation-type flow

in front of the downstream sphere is therefore replaced with this recirculation zone

The isotherms in Fig 7.22 for G D/ =1.5 show weaker gathering around the rear stagnation point of 180θ = ° on the upstream sphere and disappearing

of dense concentration around θ = on the downstream sphere, as compared 0°

to the single isolated case (Fig 7.3) This is due to the close contact of the downstream sphere so that no room is available to allow the entrainment of cold free-stream fluid Nevertheless, the isotherms do exhibit relatively heavy cluster around θ =70° and 290θ = ° on the downstream sphere, resulting in peaks of local Nusselt number in Fig 7.24 Recalling the streamlines in Fig 7.22, these peaks approximately correspond to the reattachment points The thermal field for G D/ =2.0 is not axisymmetric, as seen from Fig 7.23 where the temperature contours are symmetric in the ( , )x z -plane and

asymmetric in the ( , )x y -plane In the ( , ) x z -plane, the isotherms around the

base (θ=180°) of the upstream sphere and the front ( θ = ) of the 0°downstream sphere are more concentrated than the former case of G D/ =1.5,

as clearly revealed from the Nusselt number distribution in Fig 7.24 Meanwhile, the heat transfer rate around the two reattachment points on the downstream sphere is enhanced as well Regarding the isotherms in the ( , )x y -plane, their concentration around the stagnation points is nicely

captured Although difference in the Nusselt number distribution does exist between the two enumerated planes, the locations and values of the maximum are coincident (Fig 7.25) with each other

At a separation distance of G D/ =3.0, the flow becomes unsteady but the vortices are shed off regularly from the spheres Zou et al (2005) detected an unsteady plane symmetric flow in their simulation, while Prahl et al (2009) reported that their flow at this separation distance presented a statically axisymmetric behavior To make a clear visualization of flow development at this unsteady case, the perspective and side view of the wake structure are presented in one cycle in Fig 7.26 A careful observation of these pictures shows that our results more closely resemble the findings of Zou et al (2005), presenting an unsteady plane-symmetric behavior The ( , )x y -plane continues

to present as a symmetry plane during the flow evolution process

The local Nusselt numbers along the sphere surfaces in the ( , )x z -plane and

( , )x y -plane corresponding to Fig 7.26 are shown in Fig 7.27 (the angle θ

is defined in the same way as before) The Nusselt number profiles on both spheres in the ( , )x z -plane are symmetric about θ =180°, revealing the in-plane symmetry of the isotherms This observation matches well with the temperature visualization in Fig 7.28 (only the first-quarter phase is presented

as a representative) Meanwhile, all of the profiles in Fig 7.27 maximize

around the front stagnation point of θ= and present a local peak around 0°the rear stagnation point of 180θ = ° On the downstream sphere, the Nusselt number exhibits a double-hump profile and maximizes at the two humps corresponding to 65 ~ 70θ = ° ° and θ =290 ~ 295° ° respectively This pair

of humps changes their locations and values slightly during one cycle Additionally, at θ= and0° θ =180°, two local peaks which are about one-third as large as the maximum value are also presented

The Nusselt number profiles in the ( , )x y -plane share many of thermal

behaviors as described above except for the symmetry On the upstream sphere, the maximum heat transfer rate exhibits at the front stagnation point and keeps the same throughout the cycle Meanwhile, a local peak is always captured around the rear stagnation point On the downstream sphere, the Nusselt number profiles show a double-hump pattern The two humps are not of equal height, with one higher than another The locations of the two humps are shifted periodically with time, due to the periodic formation and shedding of the vortices between the two spheres

The reliability of the present results is further confirmed in Table 7.6 by making a quantitative comparison of the obtained force coefficients with those reported by Yoon et al (2007) and Prahl et al (2009) It is observed that the drag coefficients on the upstream sphere in the present study are somewhat

larger than the reference ones, but basically our results agree well with reference data Different from the low Reynolds number case at Re=40, with an increase in separation distance, the drag force at Re=300 decreases

on the upstream sphere but increases on the downstream sphere Nevertheless, the drag forces on both spheres are reduced as compared to the isolated case The heat transfer rate, as listed in Table 7.7, however, increases monotonically

on both spheres as the separation distance increases from 1.5 to 3.0

7.3 Laminar flow past a streamwise rotating isothermal sphere

Flow over a rotating sphere is of fundamental importance in numerous engineering applications involving particle-particle or particle-wall collision Its flow characteristics have been recognized to rely significantly on the direction of rotation In the case of streamwise rotation where the sphere rotates in the same direction as the incoming flow, Schlichting (1979), based

on the work of Luthander & Rydberg (1935) and Hoskin (1955), summarized two specific features related to the variation of drag and separation line due to the rotation By means of numerical simulation, Kim & Choi (2002) shed light

on the modifications made by the streamwise rotation on the vortex structures behind the sphere Reynolds numbers of 100, 250 and 300, which cover the axisymmetric steady, non-axisymmetric steady and vortex shedding regimes for a stationary sphere were considered Their simulation results exhibited strong dependence of flow features on both the Reynolds number Re and the

rotational speed Ω In this section, the wake transitions identified in Kim & Choi (2002) are revisited in a forced convective laminar flow, in which a streamwise rotating sphere with uniform high temperature is placed in a cold incoming flow Results for some representative Reynolds numbers and rotational speeds, which are defined in the same way as in Kim & Choi (2002), are presented and compared with Kim & Choi (2002) Moreover, the effect of streamwise rotation on the heat transfer rate is examined A finite domain of size 25D×20D×20D is used as the computational domain, with the rotating sphere located at (10D,10D,10D) A non-uniform mesh is employed for its discretization, with a locally fine resolution of h= Δ = Δ = Δ =x y z D/ 40around the sphere Meanwhile, a step size of Δ =t 0.002 is used for the time integration

Fig 7.29 shows the three-dimensional vortex structures for 0.3Ω =and 1.0Ω = at Re=100 It is observed that they are both axisymmetric, which is consistent with the observations of Kim & Choi (2002) that the wake structures for all the rotational speeds 0≤Ω ≤1 at this Reynolds number remain steady and axisymmetric However, as compared to the toroidal vortex structure for the stationary case ( Ω =0) in Fig 7.1(a), the vortex for the rotating cases comprises of a shroud over the sphere and a threaded structure

in the near wake The streamlines in Figs 7.30 and 7.31 demonstrate that with Ω increasing from 0.3 to 1.0, the vortex becomes elongated and

stronger On the other hand, the contour lines of temperature, as the rotational speed is increased from 0.3 to 1.0, show an obvious tendency to gather towards the base region of the sphere This behavior is clearly reflected in Fig 7.32 Observing the Nusselt number on the sphere surface at different rotational speeds, it is found that while there is almost no difference between the stationary case and the low-speed rotating case of Ω =0.3, a tangible increase in Nu at the base region from θ =135° to 180° is noted in the high rotating speed case of 1.0Ω =

The instantaneous vortex structures for Re=250 at rotational speeds

of 0.1Ω = , 0.3 and 1.0 are presented in Fig 7.33 It has already been known that for a stationary sphere (Ω =0), the flow is steady and a pair of vortices with the same vorticity strength but opposite direction appears in the wake (Fig 7.1(c)) When the sphere is under streamwise rotation, the flow becomes unsteady At 0.1Ω = (Fig 7.33(a)), because of the introduction of positive streamwise vorticity from the sphere rotation, the aforementioned balance and symmetry have been broken and one vortex is strengthened while the other is weakened This effect is enhanced at Ω =0.3 (Fig 7.33(b)), where the weakened vortex disappears and the structure gives way to “single-threaded”

As the rotational speed increases to Ω ≥0.5 (Fig 7.33(c)), the

“double-threaded” structure is induced once again and the two vortices are twisted together forming a complex pattern

The time evolutions of the drag and lift coefficients are plotted in Fig 7.34 Apparently, the flows are unsteady at all the three rotational speeds It is seen that at small rotational speed of Ω =0.1 (Fig 7.34(a)) and 0.3 (Fig 7.34(b)), the two lift components C and Ly C Lz exhibit sinusoidal variations The magnitudes of the drag and lift, however, are constants and independent of time, indicating that the vortex structures created by the rotating sphere are in

a frozen state At the larger rotational speed of Ω =1.0 (Fig 7.34(c)), the magnitudes of the drag and lift vary regularly with time, illustrating that the flow is unsteady asymmetric and the strength of the vortices in the wake keeps varying in time

Due to the complexity of the flow structure as well as its evolution process, it

is difficult to provide a clear and full visualization of the thermal field However, the thermal field evolution, undoubtedly, strictly follows the development of the flow field In Fig 7.35, a preliminary view regarding the effect of sphere rotating on the heat transfer rate is provided by tracing the surface-averaged Nusselt number Nu Consistent with the drag coefficient, the surface-averaged Nusselt number remains constant for Ω =0.1 and 0.3 throughout the rotating process while it varies regularly at 1.0Ω = Furthermore, Nu increases with an increase in Ω , showing that the sphere rotation does produce some enhancement on the heat transfer rate

The modifications of vortex structures caused by the sphere rotation

at Re=300 are illustrated in Fig 7.36 for Ω =0.1, 0.6 and 1.0 Without rotation, the flow maintains planar-symmetry while the vortices are shedding from the sphere However, when the sphere experiences a rotation in the streamwise direction, the planar-symmetry is lost (Fig 7.36) The vortices in the wake are distorted and spiral around the wake centerline At 1.0Ω = , the vortex structures have already become very complicated (Fig 7.36(c)) Similar

to Re= 250, the flows at Re=300 are unsteady for all the rotational speeds considered, which, can also be classified as “frozen” and in an unsteady asymmetric state As observed from the time histories of drag and lift coefficients in Fig 7.37, the flow is unsteady asymmetric at low rotational speeds (Fig 7.37(a)), then develops into a “frozen” state at Ω =0.6, and becomes unsteady asymmetric once again at Ω =1.0 It is interesting to note that the rotational speed at which the flow becomes frozen for Re=300 is higher than that for Re= 250 We can see that the flow patterns reported by Kim & Choi (2002) have been successfully recovered in the present study

Despite the similarities, fundamental differences still exist in vortex structures between the frozen flows at Re=250 and Re=300 For the former case, a single strong vortex structure is present in the flow field (0.1 and 0.3) and it is elongated in the streamwise direction just as shown in Fig 7.33(a)-(b) For the latter case, the vortex structures in the frozen flow fields (Ω =0.6 in Fig

7.36(b)) have both positive and negative streamwise vorticity and spiral around the wake centerline

The time development of surface-averaged Nusselt number Nu is similarly traced for Re=300 at all the simulated rotating speeds in Fig 7.38, which follows the same variation characteristics as those described for the drag coefficient While the Nu increases monotonically with the increase of Ω , the heat transfer enhancement is very weak at Ω =0.1 and almost negligible

The overall performance of flow behavior and heat transfer from the rotating sphere is assessed in Fig 7.39 by displaying the variations of the time-mean C D and Nu with respect to the rotational speed The resultant

curves show that for a given Reynolds number, C D and Nu increase

with increasing rotating speed, whereas for a given rotating speed, they decrease with increasing Reynolds number It is only when the sphere rotates

at a sufficiently large speed and a higher Reynolds number, such as 0.1Ω =

at Re=300, that the heat transfer is significantly enhanced

7.4 Natural convective heat transfer between concentric and vertically eccentric spheres

The preceding subsections have extensively discussed the capability of the developed IBM solvers to solve forced convective heat transfer problems with

stationary and moving boundaries In this subsection, the applicability of the developed IBM solvers for natural convection where the velocity and temperature fields are strongly coupled is examined For this purpose, the problem of natural convective heat transfer between concentric and vertically eccentric spheres is considered Its geometric configuration is shown in Fig 7.40, where two spheres with radii R i and R o ( R i <R o) are located concentrically at O or eccentrically in the vertical direction at O′ , respectively Note that the eccentricity is deliberately assigned in the direction

parallel to gravity g such that the largest variations of heat transfer may

occur The eccentricity of the annulus is measured by the distance between the centers of the two spheres, and is defined as positive if the center of the outer sphere is located above the center of the inner sphere The angular position of the annulus, denoted by θ , is measured in a way shown in Fig 7.40 Assume that the annulus contains a viscous and incompressible Newtonian fluid, and the surfaces of the outer and inner spheres are respectively maintained at constant low and high temperatures T out and T in (T out >T in) Initially, the fluid is quiescent at a uniform temperature T out Due to the temperature gradients, the fluid near the hot surface of the inner sphere becomes less dense and rises upward while the fluid near the cold surface of the outer sphere is cooled and flows down, forming a convection current The fluid and thermal behaviors of the present problem are governed by four dimensionless

parameters: ratio of the outer and inner radius o

ratio

i

R R

on the effects of Rayleigh number R a and vertical eccentricity ε while keeping the other two parameters fixed at R ratio =2 and Pr=0.7 A uniform Eulerian mesh with resolution h=R i / 20 is used Without special illustration, all the results below are presented in a dimensionless form

Consideration is first given to the effect of Rayleigh number R a and we keep the vertical eccentricity at ε =0 Based on our calculations, the heat and fluid flows appear to be axisymmetric in the considered annulus Therefore, the flow visualization can be realized in any vertical axial plane In the present study, it is presented in a way that the streamlines are plotted on the left half while the isotherms are plotted on the right half in the selected plane Fig 7.41 shows the steady-state streamlines and isotherms at the Rayleigh numbers of

in Table 7.8 with those in Chiu & Chen (1996), from which a favorably

agreement is observed The isotherms, at the low Rayleigh number of

3

1 10

Ra= × , reveal a shape of concentric circles around the inner sphere,

presenting a dominant conduction behavior When the Rayleigh number

increases to 1 10× 4 and 1 10× 5, the isotherms develop into shapes of pretty

plume and are intensely clustered near the top surface of the outer sphere and

bottom surface of the inner sphere, implying that the convective flow has

begun to take control of the whole system Particularly, the strong uprising

plume impinges onto the top surface of the outer sphere at Ra= ×1 105,

leading to an intense concentration of isotherms The local Nusselt number

∂

= −

are depicted versus the angular position θ for the three Rayleigh numbers in

Fig 7.42 The same behavior is predicted for the three cases: the local heat

transfer rate always increases monotonically from the top to the bottom

surface, minimizing at θ = and maximizing at 0° θ =180° This observation

is exactly consistent with the distribution of isotherms shown in Fig 7.41

Benefited from the stronger convective motion due to the increase in Rayleigh

number, the local heat transfer rate is enhanced all along the way The

resultant surface-averaged Nusselt number Nu , calculated by

2

14