flow structures and heat transfer on dimpled surfaces · flow structures and heat transfer on...
TRANSCRIPT
FLOW STRUCTURES AND HEAT TRANSFER ON DIMPLED SURFACES
Johann Turnow, Valery Zhdanov, Egon HasselInstitute of Technical Thermodynamics
University of RostockAlbert-Einstein-Str. 2, 18059 Rostock
Nikolai KornevInstitute of Modeling and Simulation
University of RostockAlbert-Einstein-Str. 2, 18059 Rostock
ABSTRACTVortex structures and heat transfer enhancement mecha-
nism in a turbulent flow over a staggered dimple array in a nar-row channel have been investigated using Large Eddy Simu-lation (LES), Laser Doppler Velocimetry (LDV) and pressuremeasurements for ReD = 10000 and ReD = 20000. The flowand temperature fields are captured by LES using dynamicmixed model applied both for the velocity and temperature.Simulations are validated by comparison with experimentaldata obtained for smooth and dimpled channels. Experimentsand LES show that the time averaged fields are symmetric inspanwise direction for each dimple. The flow inside of thedimple is chaotic and consists of small eddies with a broadrange of scales where coherent structures are hardly to de-tect. For both Reynolds numbers it was found that the dimplepackage with the depth t to diameter D ratio of t/D = 0.26provides the highest heat transfer and thermo-hydraulic per-formance. Proper Orthogonal Decomposition (POD) methodis applied to resolved LES fields to identify spatial-temporalstructures hidden in the random fluctuations of pressure andvelocity fields.
INTRODUCTIONAt present several methods of heat transfer enhancement
like ribs, fins or dimples have been thoroughly investigatedwith the aim to improve the heat exchange at a minimumhydraulic loss. It has been shown by many authors (seeLigrani P.M. (2003)) that concave formed dimples incomparison to other conventional methods show the bestthermo-hydraulic performance defined as the ratio betweenthe heat exchange and the pressure loss increases. Dependingon the dimple/channel configuration and on the flow prop-erties, the spatially averaged Nusselt number Nu referred tothat of a smooth channel Nu0 varies from Nu/Nu0 = 1.1to Nu/Nu0 = 2.5. The relative pressure resistance increasevaries in a range of Cp/Cp0 = 1.05− 3.5 which is fairly low
compared to other methods.High values of the thermo-hydraulic performanceNu/Nu0/(Cp/Cp0)1/3 are attained on dimples becauseof an efficient way of vortex generation. The vortices on ribsand fins, which are necessary to mix the fluid, are createdthrough the flow separation on protruding elements whatresults in a high pressure drag. The vortices on dimpledsurfaces are created inside of concave cavities. Since thereis no blockage of the channel the additional resistance isminimum. Because the vortices play a significant role in theconvection enhancement, their formation was in the focusof many studies. Unfortunately, in most investigations mainattention is paid to time averaged values whereas the flowstructures within the cavities and their contribution to theheat transfer mechanism remain still unclear and are notcompletely understood.
The most detailed experimental study of unsteady flowcharacteristics on dimpled packages has been performed byLigrani P.M. (2003). Flow visualization by smoke patterns re-vealed a primary vortex pair and additionally two secondaryvortices arise at the spanwise edges of each dimple which en-hance mixing process. Unfortunately, these first valuable ob-servations of an array of dimples have never been quantifiedproperly and analyzed using modern non-intrusive measure-ment techniques and advanced numerical technologies likeLES and DNS.The objective of this investigation is to clarify the role of thevortex formations with respect to the heat transfer on stag-gered dimple packages using LES and time resolved pressureand velocity measurements. In our previous work Turnow J.(2010) vortex structures have already been investigated for asingle spherical dimple. LES simulations revealed formationof asymmetric structures with an orientation switching be-tween two extreme positions. The emphasis of the presentwork is identification of vortex structures on dimple pack-ages. Since the augmentation of heat transfer is documentedfor dimples with a depth to diameter ratio of t/D ≥ 0.22 the
1
range of t/D between 0.195 and 0.326 is investigated at tur-bulent flow regimes at ReD = 10000 and ReD = 20000.
EXPERIMENTAL SETUPExperimental investigations have been performed in a
closed water loop channel schematically presented in Figure1. Main parts of the test rig are the head tank with an overflowweir inside, a settling chamber, the converging nozzle (con-traction) followed by the test section and a back tank witha metering system. Further a water reservoir, main pump,pipelines and four controlling valves are placed between eachdevice to ensure stable water levels. Water is pumped fromthe reservoir into the head tank by the main pump which canbe additionally tuned to control the mass flow rate. Insightthe main tank a weir is installed to reduce turbulence causedby the water pump. In addition an overflow pipe is placedat a predefined height into the head tank to provide a con-stant water level. The advantage of this set-up construction isthat a constant water level is ensured in the head tank provid-ing flow with a constant flow rate and a low level of turbu-lence inside the test section. The rectangular test channel sec-tion behind the converging nozzle has the dimension in termsof channel width B, channel height H and total length L of200mm×15mm×1340mm. To provide visual access the testsection was made of acrylic glass. Thirty rows of dimpleswith a print diameter of D = 23mm and depth of t = 6mmwere drilled in a staggered arrangement and then polished bya CNC machine on the lower channel wall. The first row wasplaced at 500mm behind the inlet of the test section (see Fig-ure 2). To measure the static pressure loss, twelve bore holeswith an inner diameter of 1.5 mm were drilled along the cen-terline of the test section. The pressure resistance coefficientCp for both smooth and dimpled channels is calculated as thepressure difference between the inlet and outlet of the dimplepackage.
LARGE EDDY SIMULATIONLarge Eddy Simulations (LES) have been performed
based on a 3-D finite volume method. The filtered transportequations are solved on a non staggered Cartesian grid, thediscretisation in space and time of the quantities at the cellfaces is of second order using central differencing scheme.The LES equations are obtained by filtering the continuityequation, the Navier-Stokes equations and the transport equa-tion for the nondimensional temperature θ at the filter width∆:
∂ ui
∂ t+
∂ u jui
∂x j=− 1
ρ
∂ P∂xi
+∂
∂x j
[ν
(∂ ui
∂x j+
∂ u j
∂xi
)− τi j
],
(1)
∂ θ
∂ t+
∂ u jθ
∂x j=
∂
∂x j
[ν
Pr∂ θ
∂x j− JSGS
j
]. (2)
The unclosed subgrid stress tensor τi j = uiu j − uiu j and thesubgrid contribution to the scalar dynamics JSGS
j = θu j− θ u j
are modeled in terms of the filtered quantities ui and θ ,usingthe localized dynamic mixed model (LDMM) (Zang, 1993).In Eq. 2 the non-dimensional temperature θ is defined as(θ = (T − Tlowerwall)/(Tlowerwall − Tupperwall)) and furthertreated as a passive scalar without buoyancy effects. Themolecular Prandtl number Pr was set to Pr = 0.7, whereas inLES the turbulent viscosity µt and the turbulent Prandtl num-ber Prt are determined dynamically using LDMM. To ensurethe numerical stability of the DMM a clipping procedure de-rived from a rigorous mathematical analysis based on Taylorseries approximation is applied (see Kornev (2007)).
Since the whole dimple package starting from the first row
D
L
L
t
H
Figure 2: Computational domain of the channel withdimples at the lower wall
can not be simulated by LES due restricted computer re-sources only a part of dimples located within the full devel-oped flow was considered (see Figure 2). It should be notedthat the domain size should be chosen carefully to capture typ-ical velocity and temperature structures. In the most of previ-ous works time resolved calculations of heat exchangers withdimples or protrusions have been performed for a half or foronly one whole dimple with enforced periodic conditions instreamwise and spanwise directions. This domain size is notsufficient because the typical vortex structures can be largerthan the dimple diameter. In this work a large domain with thedimension 4.66H×H×4.66H, where H is the channel height,including several dimples was chosen to guarantee capturinglargest flow structures. The diameter D of the dimple with asharp edge was kept constant at D = 23 mm. Three differentdimple depths t of t/D = 0.196, t/D = 0.26 and t/D = 0.326were studied. Periodic boundary conditions were applied instreamwise and spanwise directions. No slip wall conditionwere enforced on the lower and the upper solid walls for thevelocity and fixed values were set for the temperature at thelower (hot surface, θ = 1) and the upper (cold surface, θ = 0) channel wall. To drive the flow with a constant massflow ratethe overall losses are calculated within the domain and simplyadded as the driving force to the momentum equation.
RESULTSValidation
For validation, reference and for establishment of thegrid independency a series of calculations of a turbulent chan-nel flow with the dimension of 4.66H × H × 4.66H with
2
Head tank
Contraction
Test section Back tank
Settling chamber
Figure 1: Flow chart and sketch of the experimental bench
smooth walls have been carried out first. Two Reynolds num-bers ReD = UbulkD/ν = 10000 and ReD = 20000 were con-sidered which are equivalent to the Reynolds numbers ReH =UbulkH/ν = 6521 and ReH = 13042 based on the channelheight H. The total number of grid points used for the channelflow is 256×80×256 ≈ 5.25 ·106. In spanwise and stream-wise direction homogeneous grid stretching is used normal tothe wall with the first grid point is placed at y+ = 0.4. Ta-ble 1 summarizes the results for the skin-friction coefficientC f obtained from numerical simulation and our pressure mea-surements. The skin-friction coefficients obtained from ex-
Table 1: Comparison of the skin-friction coefficient C fobtained from LES and experiment with empiric cor-relations from Dean for a turbulent channel flow atReH = 6521 and ReH = 13042.
C f Dean
ReH = 6521LES 0.0671
0.06835Exp 0.0692
ReH = 13042LES 0.0591
0.05748Exp 0.0602
periment and simulations show very good agreement with theempirical correlation given by Dean C f = 0.06138 Re−1/4.Comparison of LES data for the heat transfer rates with theempiric correlations of Gnielinsky
Nu =(ξ/8) RePr
1+12.7√
ξ/8 (Pr2/3−1)(1+(dh/l)2/3) (3)
and Dittuis-Boelter Nu = 0.023 Re0.8 Pr0.4 is summarized inTable 2. The discrepancy does not exceed 1% what under-
Table 2: Comparison of Nusselt number Nu of LESand empiric correlations from Gnielinsky and Dittuis-Boelter for a turbulent channel flow at ReH = 6521 andReH = 13042.
Nu Boelter Gnielinsky
ReH = 6521 LES 23.78 22.39 24.04
ReH = 13042 LES 38.4 38.97 38.87
lines the accuracy of the present calculations. On the basisof the turbulent channel flow calculations, a block structuredcurvilinear grid consisting of 13’196’000 cells was chosen forfurther investigations of the dimpled channel shown in Figure2. To escape numerical errors caused by large aspect ratiosof grid cells, mesh motion functionality based on a grid diffu-sion equation was applied. At the first step a block-structuredgrid is generated for flat channel and then a mesh diffusionwas used to stretch the mesh onto the dimpled surface. As aresult, the grid lines nearly follow the streamlines inside thedimples and large aspect ratios are avoided. Special attentionis paid to dimple edge resolution to reproduce the flow sepa-ration and shear layer gradients with a proper accuracy. Thecondition y+ ≤ 1 for the first grid point was satisfied even atthe highest Reynolds number to ensure a correct estimation ofthe local heat flux. Validation for a spherical dimple packageis presented in Fig. 3. Mean profiles of velocity and rms val-ues for ReH = 6521 show a good agreement between exper-iment and simulation. At ReH = 13042 the rms values fromLES are twice as high as experimental ones. Most probablythe reason is the non sufficient accuracy of measurements atthis Reynolds number. Indeed, for the smooth channel flow at
3
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4y/H
−0.4−0.2
0.00.20.40.60.81.01.2
(〈U〉,u
rms)/u
b
ReH = 6521
Exp 〈U〉Exp urms
LES 〈U〉LES urms
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4y/H
−0.4−0.2
0.00.20.40.60.81.01.2
(〈U〉,u
rms)/u
b
ReH = 13042
Figure 3: Velocity and rms profiles across the channel fordimples t/D = 0.26 at two different Reynolds numbersin the fully developed package flow. y/H = 1.4 is thedimple bottom
ReH = 13042 the rms values from LES reveal a high peaknear the wall with a strong decrease towards the center ofthe channel. In contrast, the experimental rms values showno tendency to decrease far from the upper wall being nearlyconstant. Moreover, they decrease inside the dimple whichnormally acts as turbulence promoter.
Vortex structuresVarious visualizations and numerical simulations of dim-
ple packages show clearly that at small Reynolds numbersand shallow dimples the typical vortex structure is symmetric.For instance, visualizations using smoke wires performed byLigrani P.M. (2003) at ReH = 1250 for dimples t/D = 0.2 re-vealed formation of one symmetric primary vortex pair in thecenter of the dimple and two secondary vortex pairs at eachside in spanwise direction. It was reported that the formationof structures, ejection of fluid from the dimple and structuresadvection show periodic behaviour in streamwise direction.
At large Reynolds numbers commonly used vortex iden-tification methods like λ2 or Q criteria applied both for sin-gle dimple and dimple package reveal that the flow insidethe cavities is dominated by small scale eddies. While fora single dimple it was possible to identify coherent structuresby phase averaging (see Turnow J. (2010)), the flow on thedimple package is proved to be fully stochastic without anypronounced coherent structures. Any kind of order in vortexstructures observed in a single dimple is destroyed in pack-age due to irregular perturbations coming from dimples lo-cated upstream. However, some instantaneous cell patternscaused by dimples can be recognized in the streamline pic-tures averaged over 0.39s (see Figures 4 and 5 ). Analysis of
pictures shows the mix of instantaneous symmetric and asym-metric structures. For instance, the streamline behaviour inthe deep dimples (see Fig. 5) indicates the presence of anasymmetric vortex structure whereas the top dimple is filledwith a symmetric one. Opposite flow topology in the top andthe lowest dimples can be recognized in Figure 4. The flowtopology is steadily changed in time showing no stable pat-tern where within the dimples it consists of many small scaleeddies primarily arising in the shear layer shedding from theleading dimple edge. At the shedding point the vortices arenearly perpendicular to the local velocity vector, showing nostable preferential direction already at the place of their cre-ation. Their further evolution is strongly influenced by in-teraction with irregular vortices coming from the dimples lo-cated upstream. One additional weighty argument in favourof absence of stable structures inclined to the incident flowis the nearly symmetric distribution of time averaged velocityprofiles at y/H = 0.9 in transversal direction (see Figure 6).If any steady asymmetric structures exists as predicted fromURANS calculations, the velocity distribution should be alsoasymmetric in experiments and LES within each dimple.
Figure 4: Time averaged streamlines at t+ = tUb/L = 5for dimple package with the depth to diameter ratio oft/D = 0.26, ReH = 13042.
Figure 5: Time averaged streamlines at t+ = 5 for dim-ple package with the depth to diameter ratio of t/D =0.326, ReH = 13042.
4
−4 −3 −2 −1 0 1 2 3 4z/H
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
〈U〉[m/s
]
Exp ReH = 6521
Exp ReH = 13042
LES ReH = 6521
LES ReH = 13042
Figure 6: Time averaged velocity profiles at y/H = 0.9in spanwise direction at t/D = 0.26 using LES methodand experiment
Heat transferTable 3 illustrates the influence of the relative depth on
hydraulic loss and heat transfer enhancement.
Table 3: Pressure resistance coefficient Cp and inte-gral Nusselt number Nu for different dimple depths atReH = 6521 and ReH = 13042.
t/D 0.195 0.26 0.315
ReH = 6521Nu/Nu0 1.62 1.93 1.46
Cp/Cp0 2.4 2.89 3.41
ReH = 13042Nu/Nu0 1.75 2.01 1.52
Cp/Cp0 2.83 3.03 3.81
The integral Nusselt number Nu/Nu0 = 1.93 atReH = 13042 from LES agrees well with the value ofNu/Nu0 = 1.83 given by Ligrani et al. at ReH = 10200at t/D = 0.22. A significant discrepancy between LESand our measurements was documented for the pressureresistance coefficient Cp. Experimental pressure coefficientreferred to the flat channel is Cp/Cp0 = 2.14 at ReH = 6521and Cp/Cp0 = 2.81 at ReH = 13042 which is smallerthan Cp/Cp0 = 2.89 at ReH = 6521 and Cp/Cp0 = 3.03 atReH = 13042 obtained from numerical simulations.The mostprobable reason of this discrepancy is that the experimentaldimple package has a finite length including a transitionalsection where the flow is far from the developed state. In LESthe dimple package is treated as infinite one in all directionswith full developed flow. That is why the pressure drop issufficiently larger in simulations.
Calculation of the thermo- hydraulic performanceNu/Nu0/(Cp/Cp0)1/3 shows the best results for dim-ples with the depth to diameter ratio of t/D = 0.26 what isin accordance with the expected one from literature survey.This result can be explained as follows. The flow inside thedimple is characterized by the strong reverse flow where thetime averaged recirculation zone occupies nearly 80% of thewhole cavity. The main flow separating from the leading edgereattaches at the windward side of the dimple and branchesout. A part of the fluid is directly ejected back into the mainflow producing vortex shedding at the downstream edgewhich results in a high level of mixing in the channel in frontof the following dimple as predicted in the rms values inFigure 3. The highest rate of the heat transfer takes placewithin the reattachment zone where the cold flow is comingfrom the main flow. Inside the recirculation zone velocityis substantially decreased and thus the local heat fluxes aregetting lower compared to a smooth channel. Additionalenhancement of heat transfer is found due to variation of thereattachment line caused by shear layer fluctuations. The hotfluid is driven out of the dimple in transverse direction andleave the dimple on one or both dimple sides depending onthe instantaneous flow topology (symmetric or asymmetric).Surprisingly, the outcoming fluid is not entering the followingdimple as it might be expected, but due to its angle of outflowit moves directly into the main channel flow enhancing mix-ing process. The vortices arising in a dimple with t/D = 0.26are stronger than these in the shallow dimple at t/D = 0.195.Therefore, both the hydraulic losses and the heat flux isincreased (see Table 3). However, the further increase of thedepth from t/D = 0.26 to t/D = 0.326 is followed by theheat exchange reduction. The reason for this phenomena canbe explained by the decrease of the fluctuations caused byvortices of the shear layer when the dimple depth exceeds acertain threshold. The vortices generated in the shear layerover the dimple induce the fluctuations, driving the hot fluidfrom the surface and enhancing the mixing inside the dimple.The induced fluctuations depend both on the vortex intensityand on the ratio of the vortex scale to the dimple depth, i.e.on the distance between the dimple bottom and the vortexcentres. In the case of t/D = 0.26 this ratio is sufficientlylarger than in the case t/D = 0.195 whereas the vortexintensities are comparable. In deep dimples the recirculationzone is more stable. The hot fluid stays longer within therecirculation zone weakening the heat exchange. As a result,both the ratio Nu/Nu0 and the thermo-hydraulic performancedefined as Nu
Nu0/
CpCp0
have a maximum at t/D = 0.26. Thusthe optimal dimple should have a restricted depth to diameterratio t/D. From one side, it can not be too small, becausethe creating vortices are not strong enough to enhance themixing between the hot and cold fluid especially behind thedimple. From the other side, it can not be too large fromtwo following reasons. First, the increase of t/D leads to adrastic decrease of the thermo-hydraulic performance, what iswell known from previous investigations. Second, as shownin this paper, the heat exchange is getting worse when t/Dexceeds a certain threshold, since the shear layer vortices arenot able to provide efficient mixing in a large relatively stablerecirculation zone.
5
PROPER ORTHOGONAL DECOMPOSITIONPOD technique originally proposed by Lumley (1970)
for turbulent flows is based on the Karhunen-Leove decompo-sition for identification of spatial structures and estimation oftheir contributions to the total energy of the flow field. Thedecomposed fields are represented in the sum of products ofspace dependent functions Φ
(n)i (x) and time dependent coef-
ficients a(n)(t).
ui(x, t) =N
∑n=1
a(n)(t)Φ(n)i (x) (4)
For application of POD on vector fields the snapshot methodproposed by L. (1987) is used for decomposition of the veloc-ity field u(x, t) and pressure field p(x, t). The turbulent fieldsui(x, t) and p(x, t) are taken from numerical calculations atequidistant time instants using 3500 samples at ∆t+ = 0.0126.The time averaged mean fields are subtracted from each fieldto consider only the fluctuations. In case of dimples, wherethe flow is dominated by eddies with a wide range of smallscales, a rapid decay of the energy content within the firstmodes could not be observed. The first 190 modes representabout 49.95% of the total energy at ReH = 13042. Moreover,the energy of the first modes are nearly equal to each otherfor both investigated Reynolds numbers. Since the stream-lines of the POD modes for ReH = 6521 look similarly toReH = 13042 for dimples with a ratio of t/D = 0.26 onlystreamlines of the first mode of the velocity field Φ(1) obtainedfor ReH = 13042 and t/D = 0.26 is presented in Figure 7. The
U
Figure 7: Streamlines of the first spatial mode Φ(1) ob-tained from POD of the velocity field at ReH = 13042for dimples with a ratio of t/D = 0.26.
first POD mode Φ(1) reveals large rotating structures taking itsorigin inside the cavities and penetrating into the main channelflow. Topologically this structure is very similar to the tornadolike structure proposed by Kiknadze from intuitive considera-tions at the beginning of the dimple developments in the lateeighties. The first mode involves the fluid motion out of thedimple which results in a significant enhancement of the heattransfer. Furthermore, the mode axis is aligned with the mainflow direction minimizing the pressure loss in heat exchangerchannel. This is an additional explanation why the increase of
the heat exchange in dimpled exchangers occurs at minimumhydraulic losses.
CONCLUSIONExperimental and numerical investigations have been
performed for turbulent flow over spherical dimples in a stag-gered arrangement inside a narrow channel. It was shownthat in contrast to the single dimple case, any coherent vortexstructures are hardly to detect. The flow is chaotic and con-sists of small eddies with a broad range of scales. Snapshotsof streamlines show some kind of cell pattern correspondingto dimple arrangement. The instantaneous flow within eachdimple can have both symmetric and asymmetric forms fordimples at a ratio of t/D = 0.26. The forms are changedsteadily in time revealing no stable configurations. Ejection ofthe heated fluid into the main channel flow occurs at the span-wise edges what is in accordance with visual observations ofLigrani P.M. (2003). The reason of the discrepancy betweenthe flow topologies for a single dimple and dimpled package isthe high level of perturbations coming into each dimple fromdimples upstream. They prevent creation and evolution oflarge scale coherent structures. For both Reynolds numbersit was found that the dimple package with the depth t to diam-eter D ratio of t/D = 0.26 provides the highest heat transferand thermo-hydraulic performance. At t/D smaller than 0.26the vortices arising in the cavity are not strong enough to mixthe hot and cold fluid effectively, what results in a weak heattransfer enhancement. When t/D is larger than 0.26 a rela-tively stable recirculation zone arises in the dimple. The vor-tices of the shear layer are not able to destroy the recirculationzone and hot fluid is kept longer in this zone weakening theheat flux. Proper Orthogonal Decomposition (POD) methodis applied to resolved LES fields to identify spatial-temporalstructures hidden in the random fluctuations of pressure andvelocity fields. The first POD mode looks similar to tornadolike structures proposed by Kiknadze from intuitive consider-ations. This structure with a strong inner rotation and largeenergy content is aligned with main flow direction.
ACKNOWLEDGMENTThis work was supported by the German Scientific Foun-
dation (DFG). Numerical calculations have been performed atthe High-Performance Computing center (HLRN) for north-ern Germany. We thank scientific group of Prof. Leder fortheir help in LDV measurements.
REFERENCESL., Sirovich 1987 Turbulence and the dynamics of coherent
structures. part i: coherent structures. part ii: symmetriesand transformations. part iii: dynamics and scaling. Quart.Appl. Math. 45, 561590.
Ligrani P.M., Oliveira M.M., Blaskovich T. 2003 Comparisonof heat transfer augmentation techniques. AIAA J. 41 (3),337–362.
Turnow J., Kornev N., Isaev S. Hassel E. 2010 Vortex mecha-nism of heat transfer enhancement in a channel with spher-ical and oval dimples. Heat and Mass Transfer 47 (3), 301–313.
6