Manipulation of turbulent flow for drag reduction and heat transfer enhancement 4

The symmetric spherical dimple with smooth rounded edgesee Figures 4.1 and 4.3 considered in the present study is described bythe following depth functions: x − x ci2+ z − z ci2 is the h

Chapter 4

Heat transfer over asymmetric

In this chapter, medium deep (h/D ≥ 5%) asymmetric dimples with

dif-ferent depth ratio and skewness are investigated systematically to evaluatethe effects of these factors This study is deemed somewhat unique becausethe asymmetric dimple studied here is created by skewing the deepest pointthus keeping the shape of dimple’s rim as circular, so that the coveragearea of dimpled surface remains unchanged Additionally, multiple dimplesinstead of single dimple (Isaev et al., 2000b, 2003; Kornev et al., 2010)are arranged in the channel to investigate the interactions between them.The systematic and quantitative thermo-aerodynamic performance factors

measured in terms of friction factor, Nusselt number, area and volume

goodness factors (Shah and London, 1978) are presented to indicate the

trends and optimal configuration of the asymmetric dimple Furthermore,

the flow and thermal field structures (especially the secondary flow induced

by vortex) are investigated to illustrate the possible mechanisms leading toenhanced thermo-aerodynamic performance of the asymmetric dimple vis-a-vis symmetric dimple

The investigation of flow and heat transfer over a single hemisphericalcavity (Terekhov et al., 1995, 1997) showed that sharpness of dimple candouble the friction penalty compared to dimple with rounded edge Thus,dimple with smooth rounded edge are selected in this study for heat transferenhancement The symmetric spherical dimple with smooth rounded edge(see Figures 4.1 and 4.3) considered in the present study is described bythe following depth functions:

(x − x ci)2+ (z − z ci)2 is the horizontal distance

between a dimple’s surface point to the center axis of dimple, where (x ci , z ci)

is the center of the i th dimple at the plane of the channel floor R, h, d and

r are respectively the dimple’s radius of curvature in inner region, depth,

nominal diameter and rounded edge’s radius Other parameters are given

by the following equations:

D=2XEd2XIr

h

Figure 4.1: Sectional drawing of a single dimple

For the cases in which there are N dimples on the channel floor, the

composite depth function is given by the summation of the individual depthfunctions:

where Y = 0 indicates the center plane of the channel, and Y = −H

indicates the flat surface around the dimple (dimple horizon) where thedimples reside

The asymmetric dimple discussed in this study is established byskewing the symmetric spherical dimple in a systematic manner like simpleshear deformation (details are shown in Appendix E) Thus the outerboundary of the asymmetric dimple on the channel floor remains circular(i.e the print diameter is kept unchanged) instead of being changed intoother shapes like teardrop or oval (Chyu et al., 1997; Isaev et al., 2000b,2003) This produces higher coverage ratio of dimpled region than othermodified dimples investigated previously (Chyu et al., 1997; Isaev et al.,2000b, 2003), leading reasonably to higher augmentation of heat transfer.Parameter skewness is defined to quantify how much the center of dimple

has been shifted in the x and z directions:

where Δx and Δz are respectively the displacements of dimple’s center in

x and z direction (see Figure 4.2) More details can be found in Appendix

E

In the following, a channel with length L = 10 √

3, width W = 10 and half channel height H = 1 is taken as the main/working computational

domain For all the dimpled cases discussed here, only the lower wallconsists of dimples, while the upper wall is always flat In this study,eight dimples are placed in a staggered pattern on the lower wall of thechannel (see Figures 4.3 and 4.4) The flow is driven by the prescribed

mean pressure gradient β = −1 in x-direction with frictional Reynolds

dp(-h) displacement of the center of dimple

Figure 4.2: Dimple’s surface along the streamwise centerline of dimple, the

displacement of the deepest point is ΔX = dp(−h), the fluid flows from

left to right

number Re τ = 180 The grid resolution of current study is 160× 128 × 96,

so the mesh size is fairly similar to that of mesh 3 in Table 2.3 albeit betterresolution for the dimple features

To explore the asymmetric dimple’s capability to enhance heat transferefficiency, three different cases are investigated: Case 1 (flat plate), Case

2 (depth ratio h/D = 10%) and Case 3 (depth ratio h/D = 15%) Tables

4.1 and 4.2 summarize the configurations of Case 2 and Case 3, and Figure4.5 shows the position of deepest point of dimple for different cases One

may note that both Dx and Dz for Case 2C.2* (Table 4.1) and Case 3C.2*

(Table 4.2) are zero; the dimple in these cases are called symmetric (or

X Z

2H

L

W

Flow direction

Figure 4.3: Computational domain and dimpled plate

general or ordinary spherical) dimple Since the pressure gradient is normal

to the spanwise direction, asymmetric dimples whose spanwise skewness are

negative should be considered equivalent to that with positive Dz Thus only the cases with positive Dz are investigated in this study.

The mean bulk velocity U b is obtained to define Re 2H for each case,

whose range is 4, 000 < Re 2H < 6, 000. To maintain uniformity inthe interpretation of data, all results are non-dimensionalized by friction

velocity in the channel u τ Each calculation is initiated with initial guess ofvelocity and temperature fields and allowed to develop under the prescribedpressure gradient and boundary conditions

The global thermo-aerodynamic parameters in terms of friction

coef-ficient ratio C f /C f 0 , Nusselt number ratio Nu/Nu0, area goodness factor

both temporal and spatial averaging, where the subscript ‘0’ denotes the

Figure 4.4: Arrangement of dimples on bottom channel wall

Table 4.1: Different configurations of dimples for Case 2 at h/D = 10%: ‘C’

stands for the case where dimple’s deepest point is skewed in streamwise

centerline (Dz = 0%), ‘S’ stands for the case where dimple’s deepest point

is on offset side of centerline (Dz = 15%); * stands for symmetric dimple,

cases without * are asymmetric dimples

Table 4.2: Different configurations of dimples for Case 3 at h/D = 15%.

The meanings of ‘C’, ‘S’ and * are the same as in Table 4.1

2C.1 2C.2 2C.3

2S.1 2S.2 2S.3

2C.4 centerline

Figure 4.5: Deepest point of dimple for different configurations

numerical results of flat plate case Note that the numerical results offlat plate has been interpolated to the counterparts at the same Reynolds

number Re 2H as the dimpled case by the trend of Eqs (2.25) and (2.26).All these factor ratios are calculated in order to quantitatively assess the

performances such as the pressure loss C f /C f 0 and heat transfer capacity

4.3 Global thermo-aerodynamic performance

on pressure loss have even larger variations (ranging between 1.0 and5.0) Such wide variation of results can be possibly attributed to differentdimple’s geometry, measurement conditions, numerical turbulent models,and even evaluation methods This observation motivates us to carry out

a more systematic study of the performance of symmetric dimples withdifferent depth ratio

To clarify the effects of depth ratio, five different cases (h/D = 0%, 5%, 10%, 15%, 20%; D = 5; d/D = 0.8) are simulated It should be noted that h/D = 0% denoted smooth flat plate.

Friction factor ratio C f /C f 0 and heat transfer rate augmentation

of depth ratio, Nu/Nu0 shows higher rate of increase at first (0%≤ h/D ≤10%), then almost reaches asymptotic value of about 1.7–1.9 for h/D ≥10% The thermal performance obtained are compared with the

experimental results in Burgess and Ligrani (2005) as shown in Figure 4.7

It is found that both the trend and level of heat transfer augmentationagree basically well with that of Burgess and Ligrani (2005) Although thetrend of approaching an asymptotic limit at high depth ratio found in thepresent results does not exist in the results in Burgess and Ligrani (2005),such trend is consistent with the findings (Lee et al., 2008) that strongrecirculation in deep dimple restricts further heat transfer enhancement.

It can also be found that C f /C f 0 grows monotonically with the increase

of depth ratio Though quantitatively it does not match well with mental results because that friction strongly depends on Reynolds number,dimple’s geometry and measurement conditions, its nonlinear increasingtrend still qualitatively matches the conclusion obtained by Burgess andLigrani (2005)

experi-When depth ratio h/D increases, Ga/Ga0 and Gv/Gv0 first increase

to be followed by a decrease (the critical depth ratio for the change in trend

is about 10%) As such the area/volume goodness factors for deep dimple

can be lower than that for the shallow dimple (see Figure 4.6(b)) The

results indicate that for symmetric dimple, the optimum depth ratio h/D

is 10%

4.3.2 Asymmetric dimple

Figure 4.8 shows the pressure loss and heat transfer capacity for Case 2

(h/D = 10%, Table 4.1) For Case 2C (Dz = 0%), the heat transfer ratio

Figure 4.7: Comparison with experimental results in Burgess and Ligrani(2005)

loss ratio C f /C f 0 first increases and then decreases with an increase of Dx For Case 2S (Dz = 15%), the heat transfer ratio Nu/Nu0 and the pressure

loss C f /C f 0 both first increase and then decrease with an increase of Dx The pressure loss ratio C f /C f 0 for all asymmetric dimple except for Case

2S.2 (Dx = 0%, Dz = 15%) are lower values than that of symmetric dimple (Case 2C.2*) It is also shown that both C f /C f 0 and Nu/Nu0for Case 2S.1

and 2S.2 (Dx = −15% and 0%, Dz = 15%) indicate an increase compared

to the counterpart of Case 2C (Dz = 0%) One may note that asymmetric dimple of Case 2C.4 (Dx = 17.5%, Dz = 0%) is intentionally carried out since the trend of increasing Nu/Nu0 and the slightly decreasing trend

of C f /C f 0 with the increase of Dx (and Dz = 0%) are found; both the

observed trends are conclusive for an effective heat transfer augmentation

with minimal/decreasing pressure loss Results for Case 2C.4 (Dx = 17.5%,

Dz = 0%) show, however, that the trend of increasing Nu/Nu0 has tapered

off while the C f /C f 0 trend has reversed and actually increased

Figure 4.8: Friction and Nusselt number ratios for Case 2 at h/D = 10%

The area goodness factor ratio Ga/Ga0 increases with an increase of

Dx for both Case 2C (Dz = 0%) and Case 2S (Dz = 15%), as shown

in Figure 4.9(a) Meanwhile, Ga/Ga0 for the arrangement for Case 2S

(Dz = 15%) exhibits broadly a lower quantity than the counterpart of Case 2C (Dz = 0%) Compared to the symmetric dimple (Dx = 0%, Dz = 0%),

it is also clear that asymmetric dimple properly chosen like Case 2C.3

(Dx = 15%, Dz = 0%) and even Case 2S.3 (Dx = 15%, Dz = 15%) have

better performance One may note, however, that the above mentioned

Case 2C.4 (Dx = 17.5%, Dz = 0%) specially carried out indicates a leveling

off of the Ga/Ga0 quantity which concurs with the trends observed for

Figure 4.8 and discussed above

The variation of volume goodness factor ratio Gv/Gv0 versus Dx in Figure 4.9(c) shows a fairly similar pattern as that for Ga/Ga0: it increases

with Dx for both Case 2C (Dz = 0%) and Case 2S (Dz = 15%) Case 2C.4 (Dx = 17.5%, Dz = 0%) shows a slight further improvement of the

Figure 4.9: Area and volume goodness factor ratios

Closer examination reveals that Ga/Ga0 and Gv/Gv0 for both

asym-metric dimple at Dx = 15% & Dz = 0% (Case 2C.3) and Dx = 17.5%

& Dz = 0% (Case 2C.4) are approximately up to 3.8% and 4.1% better

than the symmetric dimple (Case 2C.2*) The results evaluated in terms of

(Dx = 15%, Dz = 0%) and 2C.4 (Dx = 17.5%, Dz = 0%) yield the most superior thermo-aerodynamic performance for h/D = 10% regarding

the enhancement of a heat exchanger matrix’s compactness as previouslymentioned Equally important is the implication that the performance has

shown an asymptote as Dx increases from 15% to 17.5%, and is unlikely

to increase further It is interesting to note that while the heat transfercharacteristic is enhanced for Cases 2C.3 and 2C.4, there is a correspondingslight decrease in pressure loss by approximately 1%, compared to thesymmetric dimple.

Figure 4.10 shows the thermo-aerodynamic parameter ratios for Case 3

(h/D = 15%, Table 4.2) It is noted that both the pressure loss ratio

C f /C f 0 and the Nusselt number ratio Nu/Nu0 increase with the increase

of Dx for Case 3C (Dz = 0%) However C f /C f 0 and Nu/Nu0 increase

first to be followed by a slight decrease with a increase of Dx for Case 3S (Dz = 15%) It also shows that both C f /C f 0 and Nu/Nu0 for Case

3S (Dz = 15%) are higher than their respective counterpart in Case 3C (Dz = 0%).

The ratio of area goodness factor Ga/Ga0 increases with an increase

of Dx both for Case 3C (Dz = 0%) and Case 3S (Dz = 15%) as shown in Figure 4.9(b) Meanwhile, Ga/Ga0 for Case 3C.3 (Dx = 15%, Dz = 0%) is obviously higher than that for Case 3S.3 (Dx = 15%, Dz = 15%) On the other hand, as plotted in Figure 4.9(d) the ratio of volume goodness factor

with the increase of Dx Additionally, both Ga/Ga0 and Gv/Gv0 for

the asymmetric dimple configurations are higher than those for symmetric

dimple (Case 3C.2*) except for the Case 3C.1 (Dx = −7.5%, Dx = 0%) Further examination of Figures 4.9(b) and 4.9(d) suggests that the area

Figure 4.10: Friction and Nusselt number ratios for Case 3 at h/D = 15% and volume goodness factors are maximized for the Case 3C.3 (Dx = 15%,

Dz = 0%) among the present six different configurations The increments

of Ga/Ga0 and Gv/Gv0 are approximately +8% and +20%, respectively,

compared to the case of Dx = 0%, Dz = 0% corresponding to the symmetric dimple without skewing (Case 3C.2*) The analysis of Ga/Ga0

and Gv/Gv0 indicate that the configuration of asymmetric dimple with

Dx = 15% and Dz = 0% yields the most superior thermo-aerodynamic

performance in regard to the enhancement of compactness of a heat

exchanger; this is true for both h/D = 10% (Case 2) and h/D = 15%

(Case 3) By comparing Case 2C.3 or Case 2C.4 in Figures 4.8 and 4.9(c)with Case 3C.3 in Figures 4.10 and 4.9(d), the most concerned parameter

for Case 3C.3 This may suggest that shallow asymmetric dimple exhibitslightly better characteristics The results also show that asymmetryachieved by shifting the deepest position downstream is more effective than

that by shifting upstream.

4.3.3 Effect of asymmetry versus effect of depth

To further compare the performance of asymmetric dimple and symmetric

dimple, C f /C f 0 , Nu/Nu0, Ga/Ga0 and Gv/Gv0 of asymmetric dimple

(h/D = 10%, Case 2C.4; h/D = 15%, Case 3C.3) are plotted in Figure 4.6

with symmetric dimple and represented in Figure 4.11 It is shown that

shallow asymmetric dimple (h/D = 15%, Case 3C.3) generates much higher

heat transfer and much less pressure loss than deep symmetric dimple

(h/D = 20%) Quantitatively, one can also see the volume goodness factor ratio Gv/Gv0 of deeper asymmetric dimple (Case 3C.3, at h/D = 15%)

is increased by about 9% compared to that of shallower symmetric dimple

(Case 2C.2*, at h/D = 10%) It can be concluded that skewing the center

of shallow dimple (h/D < 20%) in the downstream direction provides a

more efficient way to enhance heat transfer efficiency than only increasing

its depth ratio (h/D ≥ 20%).

After the flow shows a statistically stationary state, temporal averaging ofvelocity as well as temperature field, friction and heat transfer coefficientsare performed during a typical sampling time interval, taken as 40 non-dimensional time units

Cf/Cf0of asymmetric dimpleNu/Nu0of asymmetric dimple

(b)

Figure 4.11: Depth ratio effects on asymmetric dimple’s performance

4.4.1 Mean flow field patterns

Figures 4.12(a)–(c) show the streamline patterns in the vicinity of the

dimpled wall (y+ = 1.5) for Cases 2C.1 to 2C.3 (Dx = −15% to 15%,

Dz = 0%) Figure 4.12(b) shows two stagnation points surrounded by weak

vortex motion at the spanwise edges of the symmetric dimple (Case 2C.2*).For the asymmetric dimple whose center is skewed to upstream direction(Figure 4.12(a), Case 2C.1), the flow separates earlier and reattaches on thewall later compared to the flow for symmetric dimple (Figure 4.12(b), Case2C.2*) This leads to larger region of reverse flow and stronger vortex

at stagnation points As shown in Figure 4.12(c), when the dimple’scenter is skewed in the downstream direction (Case 2C.3), separation offlow is delayed and the reattachment line shifts upstream, thus the reverseflow region is suppressed Furthermore, the stagnation points and vortexmotion disappear for Case 2C.3 On the whole, the streamline patterns forthese dimples (Case 2C, Figures 4.12(a)–(c)) are fairly symmetric aboutthe streamwise centerline

Figures 4.12(d)–(f) show the streamline patterns in the vicinity of the

dimpled wall (y+ = 1.5) for Case 2S.1 to 2S.3 (Dx = −15% to 15%,

Dz = 15%) The streamline patterns pertaining to these dimples are

obviously asymmetric Figures 4.12(d)–(e) show that the vortex structuresare stretched to one side of dimple, and the rotating feature of flow atthis side are increased while the rotating at the other side practicallydisappears Though the streamlines are extremely distorted, no obvious

(f) Case 2S.3

Figure 4.12: Flow patterns for Case 2 (h/D = 0.1), the fluid flows from left

to right (red dots refer to the deepest point of dimple, the same hereinafter)

vortex motion appears for Case 2S.3 as shown in Figure 4.12(f) On thewhole, the structure of the large scale flow inside the asymmetric dimplewhose center is skewed in spanwise direction (Case 2S) has a spiral shapeflow.

Figures 4.13(a)–(c) show the patterns of the streamlines in the vicinity

of the dimpled wall (y+ = 1.5) for Case 3C.1 to 3C.3 (Dx = −7.5%

to 15%, Dz = 0%) Reverse flow regions are indeed larger than those

found for Case 2C in Figures 4.12(a)–(c) as expected since the dimple isdeeper Additionally, Figure 4.13(a) shows two pairs of separation andreattachment lines inside the dimple These lines indicate that there arethree rotating flows inside the dimple (one is the main, and the other twoare secondary), which do not exist in Case 2 It is observed that the

reattaching line shrinks to a point for Case 3C.3 (Dx = 15%, Dz = 0%)

as shown in Figure 4.13(c) Similar to the findings in Figures 4.12(a)–(c),

increasing Dx (i.e skewing the center of dimple in downstream direction)

delays flow separation and shifts the reattachment position upstream, thusreducing the area of recirculation flow region On the whole, the streamlinepatterns are symmetric about the streamwise centerline for Case 3C

Figures 4.13(d)–(f) show the streamline patterns in the vicinity of the

dimpled wall (y+ = 1.5) for Case 3S.1 to 3S.3 (Dz = −7.5% to 15%,

Dz = 15%) Similar to those observed in Figures 4.12(d)–(f), here both

the flow paths and the separation/reattaching lines are asymmetric about

(f) Case 3S.3

Figure 4.13: Flow patterns for Case 3 (h/D = 0.15), the fluid flows from

left to right