Thermal performances of corrugated channel with skewed wall waves at rolling and pitching conditions

International Journal of Heat and Mass Transfer 55 (2012) 4548–4565

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Thermal performances of corrugated channel with skewed wall waves at rolling and pitching conditions Shyy Woei Chang a,⇑, Bo Jyun Huang b a b

Thermal Fluids Laboratory, National Kaohsiung Marine University, No. 142, Haijhuan Rd., Nanzih District, Post Code: 811 Kaohsiung, Taiwan, ROC Department of Marine Engineering, National Kaohsiung Marine University, Taiwan, ROC

a r t i c l e

i n f o

Article history: Received 18 January 2012 Received in revised form 29 March 2012 Accepted 29 March 2012 Available online 21 May 2012 Keywords: Thermal performance Pitching, Rolling Wavy channel

a b s t r a c t This experimental study investigates the thermal performances of a narrow wavy channel which swings about two orthogonal axes under single and compound rolling and pitching oscillations. Full-field Nusselt number (Nu) images over the wavy channel wall are detected at static and swinging conditions by infrared thermography method and examined comparatively to highlight the influences of rolling and pitching oscillations on heat transfer performances. A set of selected heat transfer data illustrates the individual and interdependent influences of rolling and pitching oscillations on the detailed Nu distributions and the area-averaged Nusselt numbers (Nu). Pressure drop coefficients (f) for isothermal flows and thermal performance factors (TPF) at static and swinging conditions are subsequently analyzed. For this particular channel configuration, the single rolling or pitching oscillation elevates Nu from the static references; whereas the synergistic effects of compound rolling and pitching oscillations with either harmonic or non-harmonic rhythms suppress the beneficial heat transfer impacts by single rolling or single pitching oscillations. Buoyancy effects in isolation elevate Nu but are weakened as the relative strength of swinging force enhances. Pressure drop coefficients (f) consistently increase as the relative strength of swinging forces increases. At the expense of increased pressure drop penalties for heat transfer augmentations by swinging oscillations, the thermal performance factor (TPF) respectively increases and decreases as Reynolds number (Re) increases with laminar and turbulent reference conditions. Empirical heat-transfer and pressure-drop correlations are generated to permit the evaluations of individual and interactive effects of single and compound swinging force effects with buoyancy interactions on Nu and f coefficients. By the aid of these Nu and f correlations, the favorable and worse operating conditions in terms of the swinging parameters for the TPF properties of this corrugated wavy channel are identified. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction While the shipping machineries widely deploy the plate type heat exchanger to utilize skew ribs for Heat Transfer Enhancement (HTE) in the central cooling system, the compact heat exchanger constructed by narrow channels with inclined wall waves is proposed as an alternative HTE solution [1,2]. This is an attempt to moderate the penalty of pressure drops from surface ribs but maintain the similar HTE impacts by constructing the stack of channel walls as the skewed sinusoidal waves [1,2]. Early studies had examined the thermal performances of corrugated channels [3,4] which confirmed the noticeable HTE impacts by longitudinal vortices, regional separations and flow reattachments. Further research progresses toward the saw-tooth wavy channel discovered the con-

⇑ Corresponding author. Tel.: +886 7 6126256; fax: +886 7 3629500. E-mail address: [email protected] (S.W. Chang). 0017-9310/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2012.03.076

siderable pressure drops which neutralized the accompanying HTE effects but stimulated the development of sinusoidal wavy channels [5,6]. In a channel with undulant walls, the vortical cells in the wavy trough could be agitated by increasing Re and led to the enhanced macroscopic mixing between the near-wall and core fluids due to the shear layer instability [7] and the self-sustained oscillations of these vortical cells [5,8,9]. Such macroscopic mixings played as the dominant HTE mechanism by enhancing the heat and momentum transportations between wavy walls and core fluids. For furrowed wavy channels, the convergent-divergent flow pathway repeatedly generated the streamwise adverse pressure gradients to trigger flow separations and reattachments with reverse flows in the bulges [10]. By way of Re increase to transit the flow regime from laminar to transitional flow, the unsteady selfsustained oscillatory state emerged in a furrowed wavy channel. Further increase of Re above than the transitional value, the core fluid replenished the near-wall fluid due to the destabilization of

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Nomenclature English symbols Bu Buoyancy number BuR = b(Tw  Tb)Ro2(HR/d), BuP = b(Tw  Tb)Pi2(HP/d) Cp specific heat of fluid (m2 s2 K1) d hydraulic diameter of test tube (m) f Pressured drop coefficient ¼ DP=ð0:5qW 2m Þðd=4LÞ fanning friction factors in plain tube (laminar flow f1 = 16/Re, turbulent flow = 0.079Re0.25) fr swinging frequency (s1) H swinging arm (m) dH channel height (m) k thermal conductivity of fluid (W m1 K1) L channel length (m) _ m mass flow rate of coolant flow (kg s1) Nu swinging Nusselt number = qd/(Tw  Tb)k Nu0 static Nusselt number Nu1 Nusselt numbers in plain tube (48/11 for laminar flow and Dittus–Boelter correlation for turbulent flow) Nu area-averaged Nusselt number over wavy wall Pi pitching number = XP d/Wm DP pressure drop between channel entry and exit (N m2) Pr Prandtl number = lCp/k q convective heat flux (W m2) Re Reynolds number = qWm d/l Ro rolling number = XR d/Wm S wave-wise coordinate (m) Sp peripheral length of transverse section of test channel (m) SR wave-wise length of skewed wall wave (m) T local fluid temperature (K) Tb fluid bulk temperature (K) Tw wall temperature (K)

hydrodynamic boundary layer. The onset of such wall-to-core fluid mixing was accompanied by the roller vortices in the free shear layer to stimulate small oscillations as the sustainable mechanism to enhance macroscopic mixing for further HTE impacts [9]. With corrugated wavy channels, the trapped vortical cell in each wavy trough was induced by the serpentine flow pathway, but smaller than that in the furrowed wavy channel [9]. The enhanced wallto-core mixings by raising Re was generated by the large scale vortical oscillations which led to the upstream shifts of reattachment points to trigger the injection of free-stream fluids into the recirculation cell along with the fluid-ejection from the recirculation cell into the core flow in return [9]. This type of wall-to-core mixings was dynamic and promoted the HTE impact further [9]. With turbulent flows, the primary HTE mechanism in the up-slope region of the sinusoidal wavy wall was identified as the vortex stretch accompanying with the near-wall flow acceleration, together which the vortices could be intensified by increasing the amplitude of wall wave [11]. The Nu imprints over the wavy wall were considerably affected by flow separations with the Nu peak developed near the wavy crest where the inviscid free-stream velocity reached local maximum [12]. In this regard, the maximum Nu and friction drag on the sinusoidal wavy wall were respectively raised about four and three times of the smooth-wall references when the wave amplitude-to-pitch ratios was in the range of 0.01 to 0.1 [11]. In the layer which separated the recirculating cell from turbulent core, the intensified Reynolds stresses generated the higher streamwise velocity gradients at the wavy wall and reduced the size of the recirculating flow cell from the laminar-flow counterpart; which consequently shortened the axial span of minimum Nu in the bulge of the wavy channel [13]. These augmented

TPF ~ V ~ V W Wm } R x, y X, Y ~i;~j; ~ k OXYZ

thermal performance factor = (Nu/Nu1)/(f/f1)1/3 fluid velocity vector (m s1) ! Dimensionless flow velocity ¼ V =W m channel width (m) mean flow velocity of wavy channel (m s1) pitching function rolling function spanwise and streamwise coordinates (m) dimensionless spanwise and streamwise coordinates = (x/d, y/d) unit vector of oxyz reference frame swinging coordinate frame

Greek symbols a attack angle of skewed wall waves (degree) b thermal expansion coefficient of coolant (K1) bDT density ratio of coolant due to referenced temperature difference = b(Tw  Tb) C dimensionless time scale = XR,P  t g dimensionless fluid temperature = (T  Tb)/(Tw  Tb) k pitch of channel wall waves (m) l fluid viscosity (kg m1 s1) q fluid density (kg m3) X angular velocity of swinging channel (s1) Subscripts P pitching oscillation R rolling oscillation 0 static condition

Reynolds stresses tended to diminish the viscous sublayer, increase the gradients of wall temperature, and produce local Nu peak shortly behind the separation point. The HTE performances of wavy walls were further enhanced for turbulent flows. The turbulent velocity and temperature fields over the sinusoidal wavy wall were experimentally examined using digital particle image velocimetry (PIV) and liquid crystal thermometry (LCT) [14,15]. As fluids were partially decelerated by wall shears and heated by the hot wall during the streamwise convection, the low-momentum flow structures at high temperatures emerged. While the low-momentum hot fluids surged from the heated wavy wall, the large-scale longitudinal flow structures, which carried the bulk of kinetic energy in the momentum and scalar fields over the wave-wise sectional plane, replaced the high-momentum cold fluid directing toward the wavy wall. These thermal flow structures were periodic in spanwise direction and stretched in streamwise direction, leading to the complex three dimensional flows. As well as a reconfirmation, the Heat Transfer Enhancements over a wavy wall were the result of unsteady interactions between the core fluid and the boundary-layer fluid through the shear layer destabilization and self-sustaining oscillations [14]. By raising the buoyancy levels to the mixed convection regime in the wavy channel, a meandering of the scalar plume induced by longitudinal flow structures was observed and caused the spanwise spreading of the mean scalar field [15]. Consequently, the spanwise scalar transport such as the turbulent heat flux was greatly enhanced from the isothermal condition. Together with the enhanced vertical transport by raising buoyancy levels and the enhanced spanwise transport by the longitudinal flow structures emerged in mixed-convection regime, the momentum transports of mixed convective flow in the wavy chan-

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nel was increased from the isothermal conditions [15]. As an attempt to further improve the thermal performances of wavy channels, this research group comparatively examined the performances of heat transfer and pressure drop in two types of furrowed channels with transverse and skewed sinusoidal wavy walls [1]. As the skewed sinusoidal wall-waves tripped strong cross-plane secondary flows in the form of two-pair rolling vortices [2], the Nu levels over the skewed wall waves were elevated from those on the transverse wall waves. Due to the shear layers generated by these adding cross-plane secondary flows, the high turbulence intensities developed along the channel perimeter with streamwise peaks at the crests of the skewed wavy channel [2]. In the Re range of 5000–30000, the area-averaged Nusselt numbers over the skewed wavy wall were raised to 4.12–4.03 times of Dittus–Boelter levels [16]. As the HTE impacts between the skewed wavy channel [1,2] and the ribbed channels [17,18] were similar, the thermal performance factors obtained from the skewed wavy channel were improved from the ribbed channels due to the reduced pressured drop coefficients. While the thermal performances of the wavy channel with skewed wall waves were observed at the static conditions, it is essential to assess the impacts of rolling and pitching motions on its heat-transfer and pressure-drop properties prior to the on board applications. Ships roll and pitch at low frequencies on open sea. The classification society guidelines for maximum across-the-deck and normal-to-deck accelerations formulate the maximum swinging accelerations for a sea-going ship. As a typical example, the ABS guidelines with maximum roll angle of 23.8° for a navigating ship give rise to the maximum across-the-deck and normal-to-deck accelerations in the respective ranges of 0.54–0.59 and 1.05–1.15 times of gravitational accelerations (g) [19]. These swinging acceleration fields impose periodical body forces to affect the transport phenomena of working fluids in the on-board thermal fluid machineries. Particular attentions toward the nuclear powered ship addressed the rolling effects by ocean waves on the natural flow circulation for ensuring the safety of reactor. The thermal hydraulic performance of a marine reactor was characterized by the overlapping of flow oscillations caused by heaving motion and the self-excited oscillations, together to produce complex unstable oscillatory flows [20,21]. The experimental study of natural circulation flow subject to rolling motions revealed that the mass flow rate of coolant circulation was altered periodically owing to the repeated inertial forces adding by rolling accelerations. As a result, the heat transfer rates in channel core were enhanced; but the single-phase natural circulation flow constantly fluctuated at the period similar to the rolling period [22,23]. While the rolling motion was beneficial for HTE performances, the friction resistances were increased by flow fluctuations [23]. To resolve the heat transfer properties of enhanced channel flows with particular cooling applications to shipping machineries, the HTE impacts for a number of cooling configurations at rolling and pitching conditions were examined [19,24,25]. The non-dimensional flow parameters which governed the momentum and energy transportations for a flow system subject to rolling and pitching oscillations were previously identified by deriving the dimensionless momentum and energy equations with the fluid motions referred to the coordinate system that swung synchronously with the swinging flow system [24]. To emulate the sea-going conditions, these experimental studies, as well as the present study, performed heat transfer tests with the maximum rolling or pitching angle of 22.5° at the maximum swinging frequency of 1 Hz, giving the maximum rolling and pitching accelerations of 0.15 and 0.3g respectively. With the airflow convected through the swinging square duct fitted with the smooth-walled twisted tape, the heat transfer levels for this swinging swirl duct were initially reduced from the static references; but recovered subsequently to lead heat transfer improve-

ments form the static conditions as the relative strength of swinging force kept increasing [24]. While the swinging buoyancy interaction generally improved heat transfer performances [19,24,25], its impact was moderated as the relative strength of swinging force enhanced. But, the synergistic effects of compound rolling and pitching motions with either harmonic or non-harmonic oscillations suppressed the HTE impacts attributed from the adding single rolling and single pitching effects [19,24,25]. These experimental results reported in [19,20–25] enlighten the influences of rolling and/or pitching oscillations on heat transfer properties as the results of modified flow structures by the swinging forces with or without buoyancy interactions. However, none of these previous works reported the full-field heat transfer data subject to the effects of pitching and rolling oscillations. This experimental study adopts the infrared thermography method to measure the full-field Nu distributions over the wavy channel wall at the test conditions with single and combined rolling and pitching oscillations. Directions of swinging vectors for the rolling and pitching motions are both orthogonal to the mainstream direction of the coolant flow. Together with the heat transfer data, the pressure drop measurements at the static and swinging test conditions are analyzed to evaluate the thermal performance factors (TPF) of the present test channel. It is with the combined effects of the skewed wall-waves and the rolling and pitching motions that the present investigation is considered. Two sets of empirical correlations are developed to evaluate the individual and interactive effects of single and compound swinging forces with and without buoyancy interaction on area-averaged Nusselt numbers and pressure drop coefficients for the present wavy channel.

2. Experimental methods 2.1. Swinging rig and test section This experimental test rig facilitates rolling and/or pitching oscillations at the controllable angular rolling (XR) and pitching (XP) velocities, Fig. 1(a). The rolling and pitching oscillations are converted from the reciprocations generated by two sets of crank-wheel mechanisms. Each crank-wheel assembly is individually driven a DC motor via a reduction gear box with adjustable speeds. The wavy channel test module is vertically mounted on the platform of the swinging test rig with the rolling and pitching vectors orthogonal to the main flow direction. The rotating radii of rolling (HR) and pitching (HP) oscillations from the swinging centers to the geometric center of the wavy channel are 475 and 790 mm respectively. The test coolant (aerated water) is fed from a gravity tank. Prior to entering the test module, the test coolant is channeled through a needle valve and the volume flow meter to control and measure the flow rate. During heat transfer tests, the coolant flow rates are frequently adjusted to compensate the variations of fluids properties due to temperature variations. In addition, the variation of the pressure drop properties of the test channel due to the swing motions also affects the coolant flow rate through the test channel, thus affecting Re of the test channel. A buffer chamber is installed between the flow meter and the test channel to provide the damping effect for absorbing the pressure waves generated by the swinging motions. When the swinging condition varies, the needle valve which controls the coolant flow is accordingly adjusted to compensate the variations of pressure drop properties in the test channel by the swinging forces. The deviations of Reynolds number (Re) from the targeting values at channel entrance are maintained within ±1%. The metered coolant flow is then channeled through a divergent channel adjoining the tested narrow wavy channel. To measure the full-field wall temperatures (Tw) over the heated wavy wall at both static and swing-

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Fig. 1. (a) Experimental rig (b) Test module.

ing conditions, the infrared camera (IC) is installed on the swinging test platform opposite to the test module. The optical trajectory of this IC is normal to the scanned surface. The scan rate of this IC for a thermal image is 30 frames per second. Such scan rate of 0.033 seconds per snapshot is considered sufficient to detect the detailed Tw distributions for each swinging heat transfer test at the highest frequency (fr) of 1 Hz. The Tw signals scanned by IC and the thermo-

couple measurements for the fluid entry and exit temperatures through the Fluke Net-Daq 2640A data logger are transmitted to the computer. The pitching and rolling frequencies are converted from the rotating speeds of the rotors detected by two sets of optical pick-up and encoder units. Ten sets of repeated tests at the steady states for each targeting condition are performed to acquire several wall temperature snapshots. The averaged Tw image from

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these snapshots is recorded for the subsequent evaluation of the temporally averaged full-field Nu distribution. Balance weights are installed on the corners of the swinging platform and the two driven wheels connecting with the motors to ensure the dynamic balance. Fig. 1(b) depicts the constructional details of the test module. This corrugated test channel is constructed by two opposite inlined undulant walls with skewed sinusoidal waves and two narrow sidewalls. Upstream and downstream the undulant test section, the divergent entry (1) and convergent exit (2) plenum chambers are installed, respectively. In the divergent entry plenum chamber (1), the honeycomb and meshes are fitted to stratify the coolant flow. The entry (3) and exit (4) Teflon flanges are machined with the identical sectional contours matching the peripheral contours of the entrance and exit sections of the corrugated test channel. The end sections of the divergent entry (1) and the convergent exit (2) plenum chambers, which connect the wavy test section with the width and height of 80 mm and 10 mm respectively, envelop the sectional contour of the present corrugated wavy channel. The area ratio between the entry/exit plenum chamber (1)(2) and the sectional area of the present wavy channel is 1. This configuration emulates the abrupt flow entry condition with the thermal and hydraulic boundary layers developing simultaneously. A 50 mm thick Teflon undulant back wall (5) and its opposite wavy stainless steel heating foil (6) formulate the tested wavy section. Another stainless steel heating foil (7) with the identical wave form attaches on the undulant side of the Teflon back wall (5). Each of the heating wavy foils is forged to the required wave form from a 0.1 mm thick, 80 mm wide (W) and 470 mm long (L) stainless steel foil. The amplitude (a), wave pitch (k) and the angle of attack (a) for these skewed wall-waves are 2.4 mm, 12 mm and 45° respectively. As indicated in Fig. 1(b), the scanned region covers the entire channel width and the streamwise length of 430 mm. The sinusoidal waviness gives rise to the amplitude to pitch ratio of 0.2. With the sectional peripheral length (Sp) of 234 mm at the channel entrance, the hydraulic diameter (d) of this wavy channel is 13.7 mm. This hydraulic diameter is selected as the characteristic length to evaluate Nu, f and the non-dimensional flow parameters. The wavy heating foil (6) is secured within the front Teflon frame (8) to enable the optical assess of wall temperature (Tw) scans by the infrared radiometer. Each full-field Tw image over the wavy heating foil (6) is scanned by a calibrated two-dimensional infrared radiometer (IR) (9) which completes a 239  255 matrix scan in 0.3 s. The precision of the present IR system for the temperature measurements is 0.1°C. The back scan surface of the wavy heating foil (6) is painted black to enhance its emission. The in-house emissivity coefficient of the present IR for this test set up is adjusted to 0.81 so that the differences between the Tw measurements detected by IR and by the calibrated thermocouples in the scanned Tw range are less than 0.3°C. The two ends of each stainless steel heating foil (6) or (7) are sandwiched between the two-piece entry (10) and exit (11) copper plates. Electrical cables are connected with the pair of copper plates (10)(11) to complete the electrical heating circuit in series. The adjustable high-current, low voltage power supply unit feed the required heater power through the two stainless steel wavy foils (6)(7) to regulate the basically uniform heat flux over each wavy wall. The range of heating powers is 750–6500 W. Two Teflon channel sidewalls (12)(13) with the waviness matching the front and back wavy foils (6)(7) set the channel height (H) of 10 mm. This corrugated wavy channel is characterized by four dimensionless geometric parameters, namely the channel length (L) to channel width (W) ratio of 470 mm/80 mm = 5.875, the channel width (W) to channel height (dH) ratio of 80 mm/10 mm = 8, the wave amplitude (a) to channel height (dH) ratio of 2.4 mm/10 mm = 0.24 and the wave amplitude (a) to pitch (k) ratio of 2.4 mm/12 mm = 0.2. The channel aspect ra-

tio (W/dH) of 8 typifies the core stacks of an on-board plate-type compact heat exchanger. The pressure drops (DP) across the corrugated wavy section are detected by the micro-manometer (14). The precision of this micro-manometer (14) is 0.01 mm-H2O, which connects with two pressure taps (15)(16) installed on the channel sidewall at the immediately entry and exit of the corrugated wavy section. The pressure drop tests at the static and swinging conditions are individually performed with isothermal flows. The entire test assemblies are tightened by four draw bolts through the positioning holes on the entry and exit flanges (3)(4). As indicated by Fig. 1(b), the origin of the present x–y coordinate system is located at the entry corner of test channel. With 45° inclination from the y axis, the wave-wise S coordinate is also employed, Fig. 1(b). At the geometric center of the flow entry plane of the corrugated wavy section, a K-type thermocouple (17) is installed to measure the fluid entry temperature. Five x-wise K-type thermocouples (18) with equal intervals alongside the spanwise centerline on the flow exit plane of the corrugated wavy section detect the fluid exit temperatures. The measured fluid exit temperature is determined as the average of these five temperature measurements from these thermocouples (18). 2.2. Dimensionless parameters and data processing In order to identify the controlling flow parameters for heat transfer in a swinging channel subject to rolling and/or pitching motions, the fluid motion is referred to a coordinate system that swings synchronously with the rolling and/or pitching channel [19,24,25]. Due to the scalar invariance, the energy equation referring to a swinging coordinate system is not modified from that described by a static frame of reference. But the momentum equation described by a static coordinate system is modified when the fluid motion is referred to a swinging coordinate system [19,24,25]. By referring the fluid motion to a swinging coordinate system and ~ as ! introducing the dimensionless flow velocity (V) V =W m and the dimensionless fluid temperature (g) as (T  Tb)/(Tw  Tb), the momentum and energy equations in dimensionless forms were derived as Eq. (1) and (2) respectively [19,24,25].

e @V er e ÞV e ¼ r ~P eþ 1 r e ~ 2V þ ðV @C Re * * e  2ðRo sin XR t i þRo sin XP t k Þ V ( ) 2 * * Dq=qXP HP d Dq=qX2R HR d 2 2 þ ðsin XP tÞ i þ ðsin XR tÞ k W 2m W 2m

ðRo þ PiÞ

Dg 1 ~2 r g ¼ Dt RePr

ð1Þ

ð2Þ

All the symbols in Eqs. (1) and (2) are referred to the nomenclature section. In Eqs. (1) and (2), the dimensionless time scale (C) is defined as XR,P  t and the reference fluid velocity (Wm) is selected as the mean flow velocity at the channel entrance. Referring to the present coordinate system shown by Fig. 1(b), * * the unit vectors along X and Z axes are indicated as i and k in Eq. (1) to respectively feature the rolling and pitching motions. The rolling (Ro = XRd/Wm) and pitching (Pi = XPd/Wm) numbers in Eq. (1) stand for the relative strength of the swinging forces originated from the Coriolis force components. With heat transfer, the distribution of fluid density becomes a spatial function as the local fluid temperature is no longer isothermal. The volumetric body forces adding by the various swinging accelerations in a heated swinging channel become hydro-dynamically coupled with the surface forces to affect the fluid motion. Following the Boussinesq approximation to uncouple the hydrostatic and

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hydrodynamic buoyancy interactions with the rolling and pitching accelerations, two additional buoyancy terms are generated as the final two components in Eq. (1). In a swinging channel within which the flow is virtually unsteady, the density perturbation from the reference temperature Tb*are coupled with the * swinging accelerations (X2R;P HR;P sinðXR;P tÞ i ; j ) to feature the unsteady mode of buoyancy interactions. As a convenient measure to index the relative strength of the swinging buoyancy force for experiments, the density ratio (Dq/q) is evaluated from the wall-to-fluid temperature difference where the reference fluid temperature is selected as the local fluid bulk temperature (Tb). However, for the convenience of data presentation, the combination of Dq=qXR;P2 HR;P =W 2m is replaced by BuR = Dq/qRo2(HR/d) and BuP = Dq/qPi2(HP/d), which are respectively referred to as the rolling and pitching buoyancy numbers. The present eccentricity ratios for rolling and pitching oscillations (HR,P/d) are respectively 35 and 58. The Reynolds (Re = qWm d/l) and Prandtl (Pr = lCp/ k) numbers carry their usual physical identities. The local Nusselt number (Nu) of the swinging wavy channel is therefore parametrically related to the dimensionless flow parameters appeared in Eqs. (1) and (2) as well as the boundary conditions formulated by this particular test section. As the variation of Pr in the tested Tb range is less than 16.1% for this study, the local Nusselt number is parametrically controlled by Re, Ro, Pi, BuR,P for this particular set of boundary conditions, including the geometry of wall waves, the thermal boundary condition at the wall-fluid interface, the flow entry condition and the shape of swinging duct. The isolated effects of each controlling flow parameter, namely Re, Ro, Pi and/or BuR,P, are examined by systematically varying a flow parameter with the others remain unchanged. The corresponding Nu or f variations to the systematic changes of Re, Ro, Pi or BuR,P reveal the individual effects in association with the varying parameter. The local Nu is experimentally evaluated as qd/k(TwTb) where q and k are the convective heat flux and the thermal conductivity of fluid respectively. The local convective heat flux (q) is obtained by subtracting the heat loss flux from the total heat flux generated over the wavy foil. The heat loss flux (qloss) is determined using a set of results acquired from the heat loss calibration tests. To perform the heat loss calibration tests, thermal insulation material is filled in the test wavy section. The supplied heat flux is balanced with the heat loss flux at the steady state condition. Based on this set of heat loss test results, qloss is correlated as kL  ðT w  T 1 ) in which kL, T w and T1 are the heat loss coefficient, the averaged wall temperature scanned by the infrared camera and the ambient temperature. Following the swinging frequencies for heat transfer tests, the heat loss coefficients (kL) are individually determined at static, the single rolling/pitching and the combined rolling and pitching test conditions. The maximum percentage of heat loss flux is about 8% at the highest swinging frequencies with combined rolling and pitching oscillations. With the information of local qf distribution over the wavy wall, the streamwise variation of the net heating power is subsequently calculated. Local fluid bulk temperatures can be determined using the enthalpy balance method by means of a sequential integration of local enthalpy from the axial location yi to the subsequent downstream location yi+1. In this regard, the fluid bulk temperature at location yi+1 is approximated as T b ðyiþ1 Þ _ p Þ where Sp is the sectional ¼ T b ðyi Þ þ qðyi Þ  ðyiþ1  yi Þ  Sp =ðmC _ is the mass flow rate of coolant through heating perimeter and m the test section. This calculating process starts from the flow entry where the inlet fluid bulk temperature is measured. To verify the calculating process for Tb, the calculated exit fluid bulk temperature and the measured Tb at the channel exit are compared. Experimental raw data are collected only if the differences be-

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tween the calculated and measured exit fluid bulk temperatures are less than ±10%. 2.3. Experimental program Initially, the full-field Nu0 distributions were measured at the static conditions with fixed Re of 350, 750, 1100, 2200, 3000, 3700, 4400, 5500. The characteristic pattern of Nu0 distributions and the heat transfer correlation for the area-averaged Nu0 (Nu0 ) were treated as the static heat transfer references against which the swinging heat transfer data were compared. A series of swinging heat transfer experiments under single and compound modes of swinging oscillations were subsequently performed. The parametric analysis using the swinging heat transfer data was then carried out with the attempt to reveal individual and combined effects of Re, Ro, Pi and BuR,P on Nu under single and compound modes of swinging oscillations. The physically consistent Nu correlation was accordingly generated. The second phase of experiments measured the isothermal pressure drop (f) coefficients at the same Re, Ro and Pi as those tested for heat transfer measurements with single and compound swinging oscillations. The f correlation using Re, Ro and Pi as the controlling variables was generated and the thermal performance factors (TPF) defined as ðNu=Nu1 Þ=ðf =f1 Þ1=3 were evaluated. The references of Nu1 and f1 were selected as the plain tube values with individual definitions for laminar and turbulent flows as defined in the nomenclature. Finally the parametrical presentation of the Heat Transfer Enhancements, the pressure drop augmentations and the TPF variations at single and compound swinging conditions with and without buoyancy interactions was discussed with the aim to identify the favorable and worse operating conditions for this particular channel geometry. Each set of swinging heat transfer or pressure drop data was generated at the fixed Re for three different swinging frequencies (fr) of 0, 0.5, 0.64 and 0.83 Hz. These swinging frequencies were used individually at the single rolling or pitching oscillations and at the compound rolling and pitching oscillations with harmonic and non-harmonic rhythms. At the non-harmonic swinging conditions, the rolling and pitching frequencies were different so that the rolling and pitching motions were out of phase in a periodical manner even if the starting phase angles for rolling and rolling motions were identical. For heat transfer tests, four different heater powers were used to raise Tw at the hottest spot on the wavy wall to the levels of 45°, 50°, 55° and 60°C at each set of predefined Re, Ro and Pi. It took about 45 min to achieve the quasi-steady state after the heating power, the swinging frequency or the flow rate was adjusted. The quasi-steady state was assumed when the variations of the time-averaged Tw at several locations on the scanned wavy wall remained within ±0.3°C. Having satisfied the quasi-steady state assumption, the on-line data acquisition and storage system was activated to capture and store several instant Tw scans for a period of 20 s. The measured Tw matrix averaged from these thermal images were used to evaluate the full-field Nu distribution at each swinging test condition. Table 1 lists the ranges of the controlling dimensionless parameters. It is worth noting that the density ratio (Dq/q) quoted in Table 1 are post-processed as the area-averaged value using the scanned Tw and the axial Tb values determined from the enthalpy accountancy routine. The experimental uncertainties for the dimensionless parameters generated by this study were estimated in accordance with the policy specified by ASME J. Heat Transfer [26]. The major source attributing to experimental uncertainties for these dimensionless parameters was the temperature measurement as the fluid properties were calculated from the fluid bulk temperature. With the heater powers, the wall-to-fluid temperature differences and the pressure drops across the test section in the respective

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Table 1 Ranges of dimensionless parameters tested. Dimensionless parameter

Range

Reynolds number (Re = qWm d/l) Rolling number (Ro = XRd/Wm) Pitching number (Pi = XPd/Wm) Area-averaged density ratio [Dq/q = b(Tw  Tb) = bDT] Buoyancy number [BuR = b(Tw  Tb)Ro2(HR/d), BuP = b(Tw  Tb)Pi2(HP/d)]

350–5500 0–3.1 0–3.1 0.0008–0.004 0.001–1.815

ranges of 750–6500 W, 31.1–61.1 K and 27.26–1354.75 N m2, the maximum uncertainties for Nu, Re, Ro, Pi, BuR, BuP and f were about 9.1%, 5.9%, 2.1%, 2.2%, 9.8%, 9.7% and 6.3% respectively. 3. Results and discussion 3.1. Heat transfer results The vortex structures in the static wavy channel and the crossplane secondary vortices, which are both dependent on Re, characterize the heat transfer properties. Fig. 2 depicts the detailed Nu0 distributions over the static wavy wall at Re = (a) 1100, (b) 2200, (c) 3000, (d) 3700, (e) 4400 (f) 5500. The variations of Nu0 distribution over the present wavy wall in response to the increase of Re shown by Fig. 2 clearly demonstrate the upstream extension of the high Nu0 region as Re increases. This particular Re-driven heat transfer variation is revealed as the result of the complex vortex transitions along a corrugated wavy pathway [27]. Following a steady flow entry regime, the two-dimensional vortex tubes in the corrugated channel with transverse wall waves are initially tripped, after which the downstream flows become unsteady [27]. Further downstream, these vortex tubes distort into the coherent threedimensional vortexes which stretch considerably in streamwise direction. Increase of Re advances the aforementioned transition process spatially. At the higher Re, the rapid growth of instability can even diminish the two dimensional vortex tubes and triggers the elongated vortex tubes which stretch into smaller and discon-

nected structures in further downstream region [27]. The wallwave induced HTE imprints thus extend upstream as Re increases as shown by Fig. 2 and the area-averaged Nusselt number (Nu0 ) increases with the increase of Re consistently. In addition, the skewed wall waves in present corrugated wavy channel trip strong crossplane secondary flows [2] which generate the S-wise (wave-wise) Nu0 decay along each skewed wall wave, Fig. 2. The high Nu0 region emerges over the obtuse region on the wavy wall at each Re tested. Accompanying with the wave-wise and x-wise (spanwise) Nu0 variations, the y-wise (streamwise) Nu0 variations follow the typical undulant pattern with the higher Nu0 over the up-slope region for each wave pitch. In this respect, the detailed streamwise, wavewise and spanwise heat transfer variations are examined by comparing the y-wise, S-wise and x-wise Nu0 profiles obtained at the various Re in Fig. 3. The centerline X-wise Nu0 profiles collected in Fig. 3(a) at the Reynolds numbers between 350 and 5500 depict the characteristic undulant pattern of streamwise heat transfer variations. As shown by Fig. 3(a) for each wave pitch, the high heat transfer region develops along the up-slope wave to respond the flow reattachment upstream the wave crest; whereas the lower Nusselt numbers occur along the down-slope wave where the separation flow cell develops [27]. The minimum and maximum Nu0 peaks arise at about 0.55 and 0.8 wave-pitch (k) respectively. Cross examining the waveforms for the centerline X-wise Nu0 profiles at the various Re in Fig. 3(a), a minor Nu0 bump downstream the minimum Nu0 spot gradually emerges as Re increases above than 750. Such transition of the pitch-wise Nu0 waveform takes place after several wave numbers and shifts upstream as Re increases. As Re increases from 350 to 5500, all the x-wise, S-wise and y-wise Nu0 profiles collected in Fig. 3 show no sign of apparent Nu0 jump from laminar to turbulence. As reported in [9,27], the flow structures in wavy channels at low Re cannot be assimilated to the typical laminarto-turbulent transition but rather as the onset jump of vortex transformation along the wavy pathway which does not necessary imply turbulence. The four S-wise Nu0 profiles sectioned form one wave pitch of wave 13 at 0, 0.25, 0.5, 0.75 k with Re = 370–5500 are compared

Fig. 2. Detailed Nu0 distributions over the static wavy wall at Re = (a) 1100, (b) 2200, (c) 3000, (d) 3700, (e) 4400 (f) 5500.

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Fig. 3. (a) Centerline x-wise (b) S-wise (c) y-wise Nu0 profiles at Reynolds numbers of 350–5500.

in Fig. 3(b). With the sufficient flow momentum to trip the sectional vortices by the skewed wall waves at the higher Re, the convective bulk stream takes the swirling form. The complex vortex rolls induced by the streamwise wall-waves merge into the swirling mainstream to affect the heat transfer properties. As the crossplane vortical flows tripped by the skewed wall waves direct the near wall flows from the obtuse edge toward the acute edge, the attendant obtuse-to-acute wall temperature rise at each k location is produced. While the S-wise Nu0 profiles are rather flat at Re=350 in Fig. 3(b) due to the lack of flow momentum to trip the cross-plane vortical flows, the oscillating Nu0 profiles tripped by the streamwise undulation are observed in Figs. 3(a) and 3(c). With ReP750, the Swise obtuse-to-acute Nu0 decay emerges for this particular channel configuration. As Re increases from 750–5500 to enrich the flow momentum for tripping the cross-plane vortical flows, the degree of S-wise obtuse-to-acute Nu0 decay at each k location is accordingly enhanced, Fig. 3(b). However, the S-wise Nu0 increase over a short span of 0–0.15 S/SW at k = 0 and 0.75 where corresponds to the up-slope wave adjacent to the obtuse edge is consistently shown by Fig. 3(b). This particular result is expected to reflect the local development of counteracting small-scale vortical cells alongside the obtuse edge on the up-slope wave. The flow structure of one vortex pair developed over the cross-plane of the convergentdivergent wavy channel [2] is likely yielded to the multiple vortex pairs in the present narrow corrugated wavy channel with AR = 8. In view of the y-wise Nu0 profiles at k = 0, 0.25, 0.5, 0.75, the oscillating Nu0 waveforms are induced by the streamwise undulation of channel wall at Re = 350–5500. The amplitude of such y-wise Nu0 oscillation is amplified as Re increases. As the in-trough separation vortices occupy the down-slope wall-waves along the undulant channel perimeter, the high Nu0 regions also emerge over the upslope waves to oppose the Nu0 decrease along the down-slope

waves for each y-wise Nu0 profile, Fig. 3(c). With the presence of the cross-plane sectional vortices tripped by the skewed wall waves, the general pattern of obtuse-to-acute Nu0 decay is followed by all the y-wise Nu0 oscillations at k = 0, 0.25, 0.5, 0.75, Fig. 3(c). The relative HTE impacts for this narrow corrugated wavy channel at static conditions are indicated as the ratios of the present area-averaged Nusselt number (Nu0 ) to the developed-flow heat transfer references in a plain tube (Nu1), namely the Dittus-Boelter level [28] for turbulent flows and 48/11 for laminar flows. The variations of Nu0 and Nu0 =Nu1 against Re are respectively shown by Fig. 4(a) and (b). In Fig. 4(b), the Nu0 =Nu1 ratios for a number of passive HTE devices [1,29–31] are included for comparison. As depicted by Fig. 4(a), the present Nu0 increases consistently as Re increases from 350 to 5500. There is no apparent Nu0 jump to reflect the typical laminar-to-turbulent transition due to the improved fluid mixing by the flow unsteadiness in association with the spatial transition of complex vortex tubes in a wavy channel [27]. As Pr variation is negligible for this investigation, the Nu0 correlation is correlated by Re using the experimental data collected in Fig. 4(a) as:

Nu0 ¼ 0:441  Re0:674

ð3Þ

Discrepancies between the correlated Nu0 by Eq. (3) and the experimental data are less than ±10%. Eq. (3) is used to evaluate the static heat transfer references against which the area-averaged Nusselt numbers obtained at the swinging conditions (Nu) are compared to enlighten the swinging effects on heat transfer properties. With a plain tube, the developed-flow Nusselt numbers (Nu1) remain constant for laminar flows, whereas the Re exponent in Nu1 correlation for turbulent flows is typical of 0.8. In the Re range which gives rise the laminar flow condition in plain tube, the HTE ratios

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Fig. 4. Variations of (a) Nu0 (b) Nu0 =Nu1 against Re.

of Nu0 =Nu1 for this corrugated wavy channel thus increases with the increase of Re. For the Re range suitable of Dittus–Boelter correlation, the Re exponent of 0.674 in Eq. (3) is less than 0.8, leading to the reduced Nu0 =Nu1 as Re increases. With the plain tube, there is no Pr effect on laminar Nu1 but the Dittus–Boelter correlation features the Pr1/3 impact on Nu1 for turbulent flows. With the Pr effect including in the turbulent Nu1, the turbulent Nu0 =Nu1 ratios obtained from the present corrugated wavy channel using water as the coolant are close to those detected from the convenient-divergent furrowed channels with dry air as the coolant [1], Fig. 4(b). But the noticeable laminar Nu0 =Nu1 differences between the present corrugated wavy channel and the furrowed wavy channels [1] with different test coolants are observed, Fig. 4(b). In this regard, the laminar Nu1 reference of 48/11 discards the Pr effect on heat transfer properties in plain tubes. While the turbulent Nu0 =Nu1 between the wavy channels with water and dry-air [1] flows show the favorable agreements in Fig. 4(b), the different laminar Nu0 =Nu1 are observed when the test coolants are different. This particular discrepancy enlightens the Pr effect for the wavy channel even if the Reynolds number is as low as 350. By assuming the similar Pr effect over the Re range of 350-5500 for these wavy channels, the plot of (Nu0 =Nu1 Þ=ðPr air =Pr test-coolant Þ1=3 versus Re in Fig. 3(b) depicts the favorable agreements between the present corrugated wavy channel and the furrowed wavy channels [1] with different test coolants. This ensured Pr effect follows the typical impact on Nu0 for these wavy channels in the Re range of 350–5500. Nevertheless, with Nu1 references selected as 48/11 and the Dittus–Boelter levels, the present laminar and turbulent Nu0 =Nu1 ratios fall in the respective ranges of 5.34–18.02 and 3.97–3.67 with 3506Re65500. While the HTE ratios for the narrow ribbed channels with AR = 8 [29] are less than those measured from the square ribbed channels [30] due

to the moderated rib flows in the narrow channels [29], the present turbulent Nu0 =Nu1 ratios are raised from the ribbed narrow channels with V-ribs [29] and approaching the HTE ratios obtained from the square channels fitted with the continuous and broken V-ribs [30], Fig. 3(b). The present Nu0 =Nu1 ratios still remain less than those developed in the rectangular channel roughened by the compound V-ribs and scale imprints [31]. The heat transfer properties in the wavy channel respond closely to the spatial transitions of the complex vortex structures which are also affected by the swinging oscillations. Fig. 5 depicts the detailed Nu distributions over the swinging wavy wall at (a) single rolling (b) single pitching oscillations and at the compound rolling and pitching oscillations with (c) non-harmonic (d) harmonic rhythms. This set of test results is selected at Re = 4400 with the buoyancy parameter (fluid density ratio = bDT) fixed at about 0.0023. Relative to the present swinging axes, the Coriolis forces generated by the rolling and pitching oscillations are periodically directed toward the two opposite wavy walls and in the y-wise direction toward the obtuse and acute edges of the swinging channel, respectively. At this set of particular swinging conditions, all the area-averaged Nusselt numbers over the swinging wavy wall (Nu) are raised from the static Nu0 level, Fig. 5. A general tendency of the wave-wise and upstream expansions of high Nu regions is commonly observed at the various swinging conditions, leading to the heat transfer augmentations from the static references, Fig. 5(a)–(d). With single rolling or single pitching oscillations, the increase of Ro or Pi consistently raises Nu from the static Nu0 reference, Fig. 5(a) and (b). With the compound swinging oscillations as depicted by Fig. 5(c) and (d), the further wave-wise and upstream extensions of the high Nu region by raising the relative strength of rolling and/or pitching forces elevate the Nu levels from

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Fig. 5. Detailed Nu distributions over swinging wavy wall with Re = 4400, bDT = 0.0023 at (a) single rolling (b) single pitching oscillations and at compound rolling and pitching oscillations with (c) non-harmonic (d) harmonic rhythms.

the single swinging conditions. But the Nu level obtained at each compound swinging condition is less than the additive value from the individual Nu levels obtained at the associated single pitching and single rolling conditions. The compound rolling and pitching oscillations which induce the periodical Coriolis forces in both directions across the channel-height and channel-width tend to weaken the combined rolling and pitching effects. This synergetic effect by the compound rolling and pitching oscillations will be later examined. The cross examination of the various swinging impacts on heat transfer properties reveals the subtle differences between the Ro and Pi impacts at the various swinging conditions, Fig. 5. The rolling oscillations with the Coriolis forces directing toward the two opposite swinging wavy walls provide the lesser degrees of heat transfer impacts than the pitching counterparts. The present rolling scenario with the Coriolis force directed toward the two opposite wavy walls is similar to the condition of the rotating duct with high channel aspect ratio (AR). In [32–34], the Coriolis force effects on heat transfer performances in the rectangular channels with the higher AR are found to be weakened from those developed in the rotating channels with the lower AR in which the

direction of the Coriolis forces is similar to the present pitching scenarios. Fig. 6 compares (a) centerline x-wise (b) wave-wise (c) y-wise Nusselt number profiles obtained at the static condition, the single rolling condition of Ro = 0.13, the single pitching condition of Pi = 0.13 and the compound rolling and pitching condition of Ro = 0.13, Pi = 0.13 with Re = 5500 and bDT at about 0.0026. To reflect the expanding high Nu regions over the wavy wall as a typical swinging force effect, the local Nusselt numbers and the amplitudes of the undulant Nu profiles are enhanced from the static references by adding the periodical Coriolis forces, Fig. 6. While the Nu levels obtained at the single pitching condition of Pi = 0.13 are slightly higher than those obtained at the single rolling condition of Ro = 0.13, the Nusselt numbers obtained at the compound swinging condition are further elevated from each single swinging levels. Even with the periodical Coriolis forces directing toward the two wavy walls or/and toward the obtuse and acute edges of each swinging channel, the wave-wise Nu decays and the undulant Nu waveforms are still evident in Fig. 6(b) and (c) respectively. The dominant flow physics characterizing the heat transfer properties

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Fig. 6. Comparison of (a) centerline x-wise (b) wave-wise (c) y-wise Nu profiles obtained at Ro = 0.13, Pi = 0, Ro = 0, Pi = 0.13 and Ro = 0.13, Pi = 0.13 with Re = 5500 and bDT at about 0.0026.

over the swinging wavy wall still remain as the spatial transition of the complex vortex structures and the cross-plane secondary vortices, which are simultaneously induced by the skewed wall waves. However, the addition of periodical Coriolis forces by swinging oscillations tends to enrich the vorticity, leading to the elevations of Nu by expanding the higher Nu regions over the swinging wavy wall. As an attempt to illustrate the swinging buoyancy effect on heat transfer performances, the Nu distributions over the swinging

wavy wall at three ascending buoyancy (bDT) levels for (a) Ro = 0.3 (b) Pi = 03 (c) Ro = 0.3, Pi = 0.3 with Re = 3000 are collected in Fig. 7. Unlike the typical Coriolis force effects shown in Fig. 5 which depicts the noticeable variations in the pattern of Nu distribution by varying Ro or/and Pi, the patterns of Nu distributions for Fig. 7(a), (b) or 7(c) remain similar at three different buoyancy levels. For the parametric conditions investigated by this study, the flow structures developed in the swinging wavy channel are not likely to be

Fig. 7. Swinging buoyancy effects on Nu distributions over wavy wall at (a) Ro = 0.22 (b) Pi = 0.22 (c) Ro = 0.3, Pi = 0.3 with Re = 3000.

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noticeably altered by adjusting the buoyancy level. With the relative strengths of the swinging buoyancy force in the present BuR and BuP ranges, the buoyancy forces could locally affect the nearwall fluid temperature gradients in the manner to raise the local Nusselt numbers over the swinging wavy wall. As a result, the local Nusselt numbers and Nu obtained at each swinging condition are consistently elevated by raising the buoyancy level, which indicates the favorable buoyancy effect for improving the heat transfer performances. The following parametric analysis examines the area-averaged heat transfer properties over the wavy wall at various swinging conditions with the specific aims to reveal the heat transfer modification from the static condition and to generate the Nu correlation to represent the entire set of heat transfer data. The heat transfer modification by swinging forces is indexed as the ratio of Nu to Nu0 obtained at the same Re. Initially, the effects of single rolling or single pitching oscillations on the area-averaged heat transfer properties are examined by plotting the variations of Nu=Nu0 against BuR,P at (a) single rolling conditions with fixed Ro (b) single pitching conditions with fixed Pi, Fig. 8. Our previous works [19,24,25] reported that the normalized Nusselt number (Nu=Nu0 ) is independent of Re at any tested swinging condition as the Re impact on Nu correlations follows the functional structure implanted in Nu0 correlation which is Re0.674 for the present corrugated wavy channel. This observation is reconfirmed after examining all the Nu=Nu0 data trends obtained at fixed Ro or Pi but with different Re. As typified by Fig. 8(a) and (b), the three sets of Nu=Nu0 variations against BuR,P at the fixed Ro or Pi with a number of Re collapse into three tight data trends, indicating the diminished Re impact on Nu=Nu0 Each of these tight Ro or Pi controlled Nu=Nu0 series increases as BuR or BuP increases. The swinging buoyancy effect improves heat transfer performances at all the single rolling or single pitching conditions. But the slope of each Ro or Pi controlled Nu=Nu0 series shown in Fig. 8(a) or (b) decreases as Ro or Pi increases. Such reduced slopes by increasing Ro or Pi indicate the improving BuR,P effects on Nu=Nu0 are weakened as the relative strengths of the swinging forces enhance. The extrapolation of each Ro or Pi controlled Nu=Nu0 series toward the asymptotic condition of BuR,P = 0 recovers the so-called ‘‘zero-buoyancy’’ swinging heat transfer level. This is illustrated by the liner correlating lines displayed in Fig. 8 where the Ro or Pi controlled Nu=Nu0 series are extrapolated back to the limiting conditions of BuR,P = 0. The zero-buoyancy Nu=Nu0 values are inferred from the intercepts of these liner correlating lines, Fig. 8. Clearly, as typified by the various data trends shown in Fig. 8, the zero-buoyancy Nu=Nu0 levels are the functions of Ro or Pi at the single swinging conditions. Led by the consistent linear varying trends of Nu=Nu0 against BuR,P as typified by Fig. 8, the Nu=Nu0 correlation for single swinging oscillation is selected as

Nu=Nu0

Single swinging oscillation

4559

¼ UR;P fRoorPig þ uR;P fRoorPig  BuR;P

ð4Þ

In Eq. (4), UR,P and uR,P are respectively the intercept and the slope of each Ro or Pi controlled Nu=Nu0 series, which are the functions of Ro or Pi. The physical implications for UR,P and uR,P are the zero-buoyancy Nu=Nu0 level and the degree of buoyancy impact on Nu=Nu0 levels respectively. As uR,P, which stands for the degree of buoyancy impact, varies with Ro or Pi, the impacts of BuR,P on Nu=Nu0 are interdependent with Ro or Pi.To examine the effects of single swinging oscillations on Nu=Nu0 at the isothermal conditions and the interdependent Ro or Pi impacts on the swinging buoyancy effects, Fig. 9(a)–(d) depict the variations of the extrapolated zero-buoyancy Nu=Nu0 levels quoted as UR,P and the uR,P values against Ro or Pi, respectively. For this particular channel geometry, the zero buoyancy Nu=Nu0 values increases linearly from the static datum of Nu=Nu0 = 1 as Ro or Pi increases, Fig. 9(a) and (b). Each mode of single rolling or pitching oscillations consistently improves heat transfer performances from the static references. The physical constraint of UR,P = 1 as Ro = Pi = 0 recovers the static heat transfer results correlated by Eq. (3). Based on the data trends depicted by Fig. 9(a) and (b), the zero-buoyancy Nu=Nu0 levels at the single rolling and single pitching conditions, namely the UR and UP values, are well correlated by Eqs. (5) and (6).

Nu=Nu0 Single rolling oscillations with zero buoyancy level ¼ UR ¼ 1 þ 0:235  Ro ð5Þ Nu=Nu0 Single pitching oscillations with zero buoyancy level ¼ UP ¼ 1 þ 0:239  Pi ð6Þ The functional structure for uR,P correlations is justified by the data trends shown in Fig. 9(c) and (d). As indicated by Fig. 9(c) and (d), uR,P values follow the exponential like decay as Ro or Pi increases. Although all the uR,P values in the test Ro or Pi range of 03.1 still remain positive, indicating the improving buoyancy impact on heat transfer performances, but such favorable buoyancy impacts decay rather fast as Ro or Pi increases. Nevertheless, all the uR and uP values can be well correlated by Eqs. (7) and (8).

uR Single rolling oscillations ¼ 0:58 þ 235:4  eð7:565RoÞ uP

Single pitching oscillations

¼ 0:43 þ 211:1  eð8:484PiÞ

ð7Þ ð8Þ

The Nu=Nu0 correlation for single rolling or pitching oscillation is generated as R {Ro, BuR} and } {Pi, BuP} by substituting Eqs. (5)– (8) into (4).Having determined the correlations of R {Ro, BuR} and } {Pi, BuP} to evaluate the area-averaged heat transfer properties at the single mode swinging conditions, the general heat transfer correlation involving the compound effects of the rolling and pitching oscillations is in subsequent pursuit. As the convenient

Fig. 8. Variations of Nu=Nu0 for various Re against (a) BuR at single rolling conditions with fixed Ro (b) BuP at single pitching conditions with fixed Pi.

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Fig. 9. (a)(b) variations of zero-buoyancy Nu=Nu0 levels against Ro or Pi (c)(d) variations of uR,P against Ro or Pi, (e) variations of f against Ro  Pi (f) variation of n against Ro  Pi  bDT (g)comparison of experimental and calculated Nu=Nu0 .

measures, the Coriolis force effects by the combined rolling and pitching oscillations at the isothermal conditions and the associated swinging buoyancy effects are respectively quoted as the functions of f {Ro  Pi} and n {Ro  Pi  bDT}. With any single rolling or pitching condition, the dependent variables in f and n functions, namely Ro  Pi and Ro  Pi  bDT, approach zero to give rise the vanished f and n values for recovering the R or } correlation. The functional values of f and n can thus stand for the compound rolling and pitching effects on the zero-buoyancy Nu=Nu0 values and on the swinging buoyancy impacts at the compound swinging conditions, respectively. By assuming the f and n functions are additive to the R or } correlation, the Nu=Nu0 correlation for approximating the Nu=Nu0 level at both single and compound swinging conditions takes the general form of: Nu=Nu0 ¼ RfRo;BuR g þ }fPi;BuP g þ ffRo  Pig þ nfRo  Pi  bDTg

ð9Þ

The values of f function in Eq. (9) are obtained by subtracting the zero-buoyancy R and } values from the extrapolating zero-buoyancy Nu=Nu0 levels at the compound swinging conditions with vanished bDT. The values of n function in Eq. (9) are consequently determined by subtracting the R, } and f values from the experimental Nu=Nu0 data obtained at the compound swinging conditions with finite bDT values. An additional boundary condition is set to give rise the result of Nu=Nu0 = 1 as Ro = Pi = bDTRoPi = 0. Fig. 9(e) and (f) respectively depict all the varying manners of f values against Ro  Pi and n values against Ro  Pi  bDT. The reason-

able convergences of all the f and n values into the tight data trends are shown by Fig. 9(e) and (f). As Ro  Pi increases, the f value is initially reduced but followed by a subsequent tendency of recovery, Fig. 9(e). For the present compound swinging test conditions, all the f values remain negative. The synergetic rolling and pitching effects caused by the interactive Coriolis forces directing toward the two opposite wavy walls and toward the obtuse and acute channel edges at either harmonic or non-harmonic rhythms tend to suppress the beneficial heat transfer impacts by single rolling or pitching oscillations. The values obtained at the compound swinging test conditions are lesser than the addition of their individual R and } values. In view of the n values obtained at all the compound swinging conditions shown by Fig. 9(f), the increase of bDT  Ro  Pi incurs the consistent n reduction. This will be later illustrated in the section examining the thermal performance factors (TPF). In compliance with the data trends depicted by Fig. 9(e) and (f), the f and n values are respectively correlated by Eq. (10) and (11).

f ¼ 1 þ 0:196  Ro  Pi þ 0:0122  ðRo  PiÞ2

ð10Þ

n ¼ 28:36  Ro  Pi  bDT

ð11Þ

The heat transfer correlation that permits the evaluation of individual and interdependent effects of the single and compound swinging effects with and without buoyancy interactions on Nu is obtained by substituting R and } functions as well as Eqs. (10), (11) into Eq. (9). The comparison of the calculated and experimental

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results in terms of Nu=Nu0 is indicated by Fig. 9(g). The maximum discrepancy of ±20% between the experimental and correlating results is achieved for 96% of the entire heat transfer data generated. With present test conditions, the range of Nu=Nu0 falls between 1.1–2.58. In terms of Nu=Nu1 with Nu1 referring to the laminar and turbulent heat transfer levels for developed flows in plain tube, the HTE index are raised to 4.13–20.28. Even with the prevailing swinging force effects in the corrugated wavy channel, the favorable HTE impacts are ensured.3.2. Pressure drop coefficients and thermal performance factors The friction and form drags caused by the complex vortical structures and the adverse pressure gradients in a wavy channel elevate the pressure drop coefficients from the plain-tube f1 references, namely the Blasius equation level and 16/Re for turbulent and laminar flows respectively. Fig. 10(a) depicts the variations of pressure drop coefficient (f0) and the normalized pressure drop coefficient (f0/f1) against Re at static conditions. The f0 coefficients follow the typical exponential decay as Re increases so that the f0 correlation for present wavy channel is accordingly generated as Eq. (12).

F 0 ¼ 3:1  Re0:306

4561

ð12Þ

The maximum discrepancies between the calculated f0 values by Eq. (12) and the experimental f0 data are less than ±20% for the entire test results. As previously described, the HTE benefits generated by this corrugated wavy channel are accompanying with the augmentations of pressure drops from the smooth-walled references which are indexed as the ratios of f0/f1. Fig. 10(a) also indicates the comparison of the pressure-drop penalties among several passive HTE measures by separately plotting f0/f1 against Re with different f1 references for laminar and turbulent flows. For the present wavy channel, the f0/f1 ratios are respectively increased and reduced as Re increases due to the different f1 references selected for laminar and turbulent flows, Fig. 10(a). The f0/ f1 ratios in the Re ranges of 350–2000 and 2500–5500 are in the respective ranges of 19.58–40.76 and 24.5–24.24 for the present corrugated wavy channel. However, unlike the furrowed wavy channel, the serpentine turnings of the main stream in a corrugated wavy channel result in the constant changes of flow momentum. Therefore the serpentine flow pathway incurs the additional pressured drops to counteract the periodically varying body forces

Fig. 10. (a) f0 and f0/f1 versus Re at static conditions (b)(c) f and f/f0 against Ro or Pi at single swinging conditions (d) variations of U against Ro  Pi (e) comparison of experimental and calculated f/f0.

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arose from the constant changes of flow momentum. The f0/f1 ratios obtained from the present corrugated wavy channel are noticeably raised from the f0/f1 levels measured from the furrowed wavy channels [1] and the ribbed channels [29,30]. But the f0/f1 ratios for this corrugated wavy channel remain less than those measured from the rectangular channel with the compound V-ribs and scale imprints [31], Fig. 10(a). At swinging conditions, the vortical structures in the corrugated wavy channel interact in the periodical manners to raise the pressure drops further. This is initially examined by plotting the f coefficients obtained at the single rolling or single pitching conditions against Ro or Pi for all the Re tested, Fig. 10(b). The f factors consistently increase from the f0 references as Ro or Pi increases. Cross examining the f levels between the rolling and pitching channels as compare by Fig. 10(b), the f values raised by pitching actions are generally higher than the rolling counterparts due to the different orientations of the swinging Coriolis forces. By normalizing the swinging f factors with the f0 references, the various f/f0 ratios obtained at different Re in the range of 350–5500 collapse into the tight Ro or Pi driven data trend, Fig. 10(c). With the possible isolation of the Re impact from the f/f0 ratios at all the swinging test conditions, the quest for seeking the f correlation is considerably simplified by formulating the f correlation as Eq. (13).

f =f0 ¼ Rf fRog þ }f fPig þ UfRo  Pig

ð13Þ

The rolling (Rf ) and pitching (}f ) functions in Eq. (13) are obtained by correlating the data trends depicted in Fig. 10(c). Following the similar procedures generating the Nu=Nu0 correlations, the interactive impacts of the compound rolling and pitching oscillations on f/f0 are expressed as the collect term of U function which

varies with Ro  Pi. As shown by Fig. 10(d), the U values also decrease consistently as Ro  Pi increases. The heat transfer impediments by the synergetic rolling and pitching effects at the compound swinging conditions with either harmonic or non-harmonic rhythms are accompanied with the reduced pressure-drop penalties. The f/f0 ratios obtained at the compound swinging test conditions are lesser than the sum of their individual Rf and }f values. After substituting the correlated Rf ; }f and U functions into Eq. (13), the empirical f/f0 correlation that permits the evaluations of the individual and interdependent swinging forces effects at the single and compound swinging conditions is generated as Eq. (14) 2

f =f0 ¼ 1:24  Ro2 þ 0:2002  Ro þ 1 þ 1:61  Pi þ 0:238  Pi þ 1  1:16  Pi  Ro  1

ð14Þ

As the attempt to examine the success of Eq. (14) for correlating the experimental f/f0 data, Fig. 10(e) compares all the correlated and experimental f/f0 values. The maximum discrepancy of ±30% between the experimental and correlating f/f0 values is obtained for 90% of the f/f0 data generated. Having generated the empirical correlations for Nu=Nu0 and Nu0 ; f =f0 and f0, the parametric analysis aimed at disclosing the thermal performances of this corrugated wavy channel at the various swinging test conditions is attempted using the heat-transfer and pressure-drop correlations generated. Initially, the comparisons of thermal performance factors (TPF) defined as ðNu0 =Nu1 Þðf =f1 Þ1=3 for a number of HTE devices [1,30,31] at the static conditions are separately performed with two Re ranges for defining the laminar and turbulent Nu1 and f1 references, Fig. 11(a). The present TPF factors evaluated at the tested Reynolds numbers reflect the relative efficiency of heat transmission compared at the same pumping power consumption.

Fig. 11. (a) variations of TPF against Re at static conditions (b) variations of TPF against Ro or Pi at single swinging conditions with isothermal flows.

S.W. Chang, B.J. Huang / International Journal of Heat and Mass Transfer 55 (2012) 4548–4565

Due to the different Nu1 and f1 references selected for laminar and turbulent Re ranges, the data trends of TPF against Re are respectively increased and decreased as Re increases at the laminar and turbulent reference conditions, Fig. 11(a). The similar Re-driven TPF data trends are also observed for the other passive HTE devices collected in Fig. 11(a). With laminar reference conditions, the TPF values for present corrugated wavy channel, the furrowed wavy channel with skewed wall waves [1] and the rectangular channel with V-ribs and scale imprints [31] are in the similar levels, but are raised from the TPF values collected with the ribbed channels [29,30], Fig. 11(a). With turbulent reference conditions, the elevated f/f1 ratios for the present corrugated wavy channel undermine the TPF properties. The TPF for the present corrugated wavy channel at the turbulent reference conditions are close to those detected from the ribbed channels [29,30] but are less than the TPF values detected from the furrowed wavy channel with skewed wall waves [1] and the rectangular channel with V-ribs and scale imprints [31]. With the laminar and turbulent reference conditions at bDT = 0, the TPF values at the static conditions fall in the respective ranges of 2.36–5.23 and 1.35–1.26 for the present corrugated wavy channel with skewed wall waves. At the single swinging test conditions, both Nu and f are raised by increasing Ro or Pi but with different paces. As indicated by Fig. 11(b), the TPF values follow the similar trend of exponential like decay as Ro or Pi increases at the laminar reference conditions; but remain at the similar levels for the turbulent reference conditions, Fig. 11(b). Nevertheless, there are buoyancy effects on Nu proper-

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ties and the synergetic rolling and pitching effects at the compound swinging conditions to affect both Nu and f performances. The manners of TPF variations at the compound swinging conditions involving buoyancy impacts can be disclosed by evaluating the f and Nu levels at the various Ro, Pi and bDT values using the f and Nu correlations. As a set of illustrative example, the parametric presentations of (a) Nu (b) f (c) TPF at bDT = 0 and 0.004 with Re = 2000 and 3000 are depicted by Fig. 12. The favorable operating conditions with the highest Nu and TPF but the lowest f; as well as the worse operating conditions with the lowest Nu and TPF but the highest f are individually marked in each plot of Fig. 12. The manners for which the Nu, f and TPF vary against Ro and Pi at the compound swinging conditions are clearly visible in Fig. 12. The overall review for the plots collected in each row of Fig. 12 enlighten the similar patterns of Nu, f and TPF distributions at different Re. This similarity is confirmed by reviewing all the test results expressing by the manners constructing Fig. 12 to reassure that the Re impacts can be isolated from the swinging forces effects on Nu, f and TPF. While the single Ro and pitching effects are specified by the data variations along the vertical and horizontal axes of each plot in Fig. 12, the interior of each plot reflects the operating conditions subject to the synergetic rolling and pitching effects on Nu, f and TPF. As shown by Fig. 12(a), the Nu isolines over the interior region on the Nu contours are moderated at the combined Ro and Pi test conditions. The increase of buoyancy level from the zero-buoyancy condition yields the patterns of Nu distributions over the interior region toward the loop form, Fig. 12(a). By way of increasing bDT

Fig. 12. Variations of (a) Nu (b) f (c) TPF with Ro and Pi at bDT = 0 and 0.004 with Re = 2000 and 3000.

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Fig. 13. Variations of TPF against Ro and Pi with ascending bDT at Re = (a) 500 (b) 5000.

from 0 to 0.004, the loci with low Nu drift away from the single rolling (Pi = 0) and single pitch (Ro = 0) axes toward the combined rolling and pitching conditions into the interior of the Nu plot in Fig. 12(a). However, in view of the f plots compared by Fig. 12(b), neither Re nor bDT can cause the noticeable modification for the distributing pattern of the f contours. The f factor keeps increasing along both Ro and Pi axes with the lower values developed at the compound swinging conditions over the interior of each f plot. Driven by the noticeable buoyancy impacts on the Nu distributions as seen in Fig. 12(a), the sensible variations of TPF distributions are observed in Fig. 12(c) for both laminar and turbulent reference conditions. As bDT increases, the high TPF region systematically emanates along the Ro and Pi axes from the locations near the static conditions; whereas the worse operating conditions with low TPF are shifted toward the interior of the TPF plot at the combined rolling and pitching conditions, Fig. 12(c). To further examine the buoyancy effects on the varying manner of TPF at the various swinging conditions, Fig. 13 depicts the variations of TPF against Ro and Pi with ascending bDT at Re = (a) 500 (b) 5000. The increase of Re reduces the TPF levels but has diminished impact on the pattern of TPF distributions, Fig. 13. As bDT systematically increases, the high TPF zone gradually stretches along the single rolling and single pitching axes to reflect the growing heat transfer impediments by the synergetic rolling and pitching effects at the combined swinging test conditions, Fig. 13(a) and (b). As a result, the worse operating conditions with the lower TPF values are shifted toward the combined rolling and pitching conditions with the reduced Ro and Pi values. Nevertheless, the favorable operating conditions still remain close to the static conditions at each Re tested, Fig. 13. For this particular channel geometry, the favorable operating conditions with the higher TPF emerge at about Ro = 0.3, Pi = 0.2 for all the swinging conditions. As the TPF values for the present corrugated wavy channel at all the single and combined swinging test conditions are above than unity with considerable

HTE benefits, the present HTE channel appears as a feasible approach for on-board deployment. 4. Conclusions This experimental study investigates the properties of heat transfer, pressured drop and thermal performance factor of a corrugated wavy channel subject to single and combined rolling and pitching oscillations. As the first attempt to this heat transfer community, the detailed Nusselt number distributions over the wavy channel wall at static and swinging conditions are generated and analyzed to disclose the isolated and interdependent Ro, Pi, BuR, Bup effects on Nu, f and TPF. With the empirical correlations for determining Nu and f, the favorable and worse operating conditions for this channel geometry under the simulating sea-going conditions are identified. Based on the research findings, the following concluding remarks are summarized to aid in the on-board deployment of compact heat exchangers. 1. The HTE imprints triggered by the skew wall-waves extend upstream as Re increases, leading to the consistent Re-driven Nu0 increases. The strong cross-plane secondary flows cause the wave-wise Nu0 decay along each skewed wall wave, which superimposes with the streamwise undulant Nu0 waves with the higher heat transfer rates along the up-slope wave relative to the down-slope counterparts. Relative to the laminar and turbulent Nu1 references, the Nu0 =Nu1 ratios are raised to the respective ranges of 5.34–18.02 and 3.97–3.67 with 3506Re62200 and 30006Re65500. Eq. (3) is generated to evaluate Nu0 with Re as the controlling variable. 2. With single rolling and pitching oscillations, the periodical Coriolis force effects induced by the swinging motions modify the patterns of Nu distributions over the wavy wall in the

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3.

4.

5.

6.

manner to raise Nu as Ro or Pi increases. With present single swinging test conditions, the Nu=Nu0 ratios increase consistently by raising Ro or Pi at the isothermal conditions. While the pattern of Nu distributions over the wavy wall is not noticeably modified by varying BuR and BuP over the present test ranges, the swinging buoyancy effect affects the near-wall fluid temperature gradients in the manner to raise the local Nusselt numbers over the swinging wavy wall, indicating the improving buoyancy effect on Nu. Nevertheless, with the prevailing Coriolis forces in both rolling and pitching orientations at the compound swinging conditions, the synergetic rolling and pitching forces interact with the buoyancy mechanisms to suppress the beneficial swinging buoyancy impacts. Together with the suppressed HTE impacts by the combined swinging force effects, the range of Nu=Nu0 and Nu=Nu1 fall respectively between 1.1–2.58 and 4.23–20.28. The Nu correlation that permits the evaluations of the individual and interdependent Ro, Pi, BuR and BuP effects at both single and combined swinging conditions is generated. The pressure drop coefficients increase consistently by increasing Ro or Pi at the single swinging conditions. The synergetic rolling and pitching actions at the compound swinging conditions moderate the Ro or Pi driven f augmentations so that the f/f0 ratios obtained at the compound swinging test conditions are lesser than the sum of their individual Rf and }f values. A set of f0 and f /f0 and correlations is generated to enable the evaluation of f coefficients at both single and compound swinging conditions for isothermal flows. As a first attempt to this heat transfer community, the TPF properties for the swinging channel are examined parametrically. As bDT systematically increases, the high TPF zone gradually stretches along the single rolling and single pitching axes to reflect the growing heat transfer impediments by the synergetic Ro and Pi effects at the combined swinging test conditions. While the high TPF region systematically emanates along the Ro and Pi axes from the locations near the static conditions as bDT increases, the worse operating conditions with low TPF are shifted toward the combined rolling and pitching conditions with the reduced Ro and Pi values. The increase of Re reduces the TPF levels but has diminished impact on the pattern of TPF distributions. With the present corrugated wavy channel, the favorable operating conditions with the higher TPF emerge at about Ro = 0.3, Pi = 0.2 for all the swinging conditions. As all the TPF values at the single and combined swinging test conditions are above than unity with considerable HTE benefits, the present corrugated wavy channel appears as a favorable HTE measure for on-board applications.

[3] [4]

[5]

[6]

[7] [8] [9]

[10] [11] [12]

[13]

[14] [15] [16] [17]

[18]

[19]

[20] [21] [22]

[23]

[24] [25]

[26]

[27] [28] [29]

Acknowledgement The research facilities and numerical software were respectively supported by National Science Council, Taiwan, under NSC 99-2221-E-022-015MY3 project.

[30]

[31]

[32]

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