Oscillation and heat transfer in upward laminar impinging jet flows

am

ai, H

OscillationHeat transfer

etrstigaratferashil. It itduc

values.

sed insuchts, ando theirtter ofurbulenat tran

force due to temperature dependence of the uid density. Whenthe buoyancy force is large, it signicantly affects the ow eldand the on-surface quantities, such as, the heat transfer coefcient,the surface temperature, the surface pressure, and the skin-friction

hether thed direction

some angle, the ow properties vary.Effects of the buoyancy force on the ow behavior and

on-surface quantities, such as, the Nusselt number, thetemperature, and the skin-friction coefcient, are studied by manyresearchers, both for slot jets and for circular jets. Most of thesestudies considered downward impinging jet ows (that is, the jetsare issued along the gravity). Yuan et al. (1988) studied downward,two-dimensional slot jets, impinging onto an isothermal at sur-face. They found that the heat transfer increases with increasingRichardson number, because the buoyancy force causes the wall

q 2/3rd of the presented computational work is carried out during the corre-sponding authors tenure as a Ph.D. student at Yokohama National University, Japan. Corresponding author. Tel.: +81 45 759 2868; fax: +81 45 759 2207.E-mail addresses: [email protected], [email protected]

(C. Shekhar).

International Journal of Heat and Fluid Flow 50 (2014) 316329

Contents lists availab

International Journal of

.eWhen laminar impinging jet ows are subjected to surfaceheating, the post-impingement wall jet experiences a buoyancy

orientation of the jet; that is, depending on wissued in the downward direction, in the upwarhttp://dx.doi.org/10.1016/j.ijheatuidow.2014.09.0010142-727X/ 2014 The Authors. Published by Elsevier Inc.This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/3.0/).ith thejet is, or at

on thesurfacehand, the laminar impinging jet ows are studied because they arepreferred in small-sized and delicate applications, or where theviscosity of the working uid is so large that turbulence productionbecomes practically infeasible. Martin (1977), Jambunathan et al.(1992), Viskanta (1993), and Zukerman and Lior (2006) carriedout thorough reviews on the impinging jet ow studies availablein the literature.

In general, the ow remains unaffected from the buoyancy forcein the forced-convection region of the wall jet, which is the regionwithin a few diameters of the radial distance from the central axis.Afterwards, the buoyancy force becomes effective when themomentum of the radially progressing wall jet diffuses signi-cantly and the local Richardson number becomes sufciently high.

The ow behavior under surface heating also differs w1. Introduction

Impinging jets are extensively utransfer application since long past,industries, in metal processing uniblades and electronic devices. Due timpinging jet ows have been a mamajority of the attention paid to tbecause they yield relatively high he 2014 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/3.0/).

various heat and massas, in paper and textilein cooling of turbinewidespread usage, theconstant study, with at impinging jet ows,sfer rates. On the other

coefcient. The degree to which the buoyancy force can affect aow is determined by the Richardson number: if the Richardsonnumber is small, the buoyancy force is negligible compared tothe inertia force, and therefore the heating does not affect the owproperties any signicantly; whereas the buoyancy force domi-nates if the Richardson number is large. Similarly, if the Richardsonnumber is moderate, the buoyancy and the inertial forces are ofcomparable magnitudes, in which case the ow behavior becomesfairly complex.Natural convectionFlow separation

the surface temperature, and the skin-friction coefcient. The ows slowly approach to statisticallysteady states where oscillation parameters and heat transfer properties tend to stabilize about xedOscillation and heat transfer in upward l

Chandra Shekhar a,, Koichi Nishino ba IHI Corporation, 1, Shin-Nakahara-cho, Isogo-ku, Yokohama 235-8501, JapanbDepartment of Mechanical Engineering, Yokohama National University, 79-5, Tokiwad

a r t i c l e i n f o

Article history:Received 17 December 2013Received in revised form 25 July 2014Accepted 4 September 2014Available online 19 October 2014

Keywords:Upward impinging jetLaminar ow

a b s t r a c t

Upward, laminar, axisymmuid, are numerically invewall jet to prematurely sepzone where the heat transheating-rate dependent Grratio are examined in detaof the ow is moderate, buThe ow oscillation also in

journal homepage: wwwinar impinging jet owsq

odogaya-ku, Yokohama 240-8501, Japan

ic, pipe-issued, submerged impinging jets, with the water as the workingated. The impingement surface is subjected to heating, which causes thee from the impingement surface and turns the following region into a deadrate deteriorates. Effects of (1) the inlet-based Reynolds number, (2) theof number, and (3) the impingement-surface height to the inlet-diametert is found that the separated jet oscillates when the Richardson numberseparates without any oscillation when the Richardson number is large.

es cyclic uctuations in on-surface quantities, such as, the Nusselt number,

le at ScienceDirect

Heat and Fluid Flow

lsevier .com/ locate / i jhf f

al ofNomenclature

b length of the inlet pipeD inlet diameterCf skin-friction coefficient qfilmcfilm0:5qinletW2inlet

@U@z

s

Fr Froude number WinletgD

p~g gravitational acceleration; g j~gj 9:81 m=s2Grq modied Grashof number gbqheaterD

4

kinletc2inletH height of the impingement surface from the upper end

of the inlet pipek thermal conductivity of waterNu Nusselt number qDTTinlet kfilmP pressurePr Prandtl number cinletqinletckinlet where c is the specic heat of

the uid at constant pressureq heat ux at the impingement surfacer radial distance from the central axis

Re Reynolds number WinletDcinletRiq modied Richardson number (for a given heating rate,

q) GrqRe2

t timeT temperatureTlm lm temperature TinletTs2U radial component of the instantaneous ow velocityV azimuthal component of the instantaneous ow

C. Shekhar, K. Nishino / International Journjet to separate from the impingement surface and moves the localuid in the upward direction. Chuo and Hung (1994) studied sim-ilar slot jets, by examining effects of the Reynolds number, theimpingement surface distance from the inlet, and the inlet velocityprole, on the heat transfer rate. Sahoo and Sharif (2004) studiedupward and downward impinging jets, both, and presented thevelocity and the temperature elds. Seban et al. (1978) studiedthe temperature distribution along the axis of symmetry and thepenetration depth of downward, heated air jets discharging intoa colder ambient. All these studies examined only the average owelds and the average heat transfer properties of the ows, withoutpaying any attention to their transient characteristics.

In order to understand mechanism of the ow separation due tothe buoyancy force, Chen et al. (1977) studied a simplied cong-uration of a ow parallel to a horizontal heated surface. They foundthat the heating affects the ow by inducing a pressure gradient inthe streamwise direction. Mori (1961) studied a similar ow, ana-lytically, and found that the skin-friction coefcient in a deceler-ated ow decreases sharply when the surface is subjected toheating, indicating that the ow might have separated from thesurface. These results suggest that a ow in the wall-jet region ofan impinging jet conguration is also prone to separation.

There are only a few studies that focused on unsteady owbehavior of heated impinging jet ows. It is known that, in general,a high degree of unsteadiness is induced when the KelvinHelm-holtz vortices generated in the shear layer of the pre-impingingjet impinge onto a heated surface and then interact with the strat-ied uid in the wall-jet region. In fact, these vortices cause theow to temporarily separate from the impingement surface, even

velocity~VL instantaneous owvelocity; VL j~VLj

U2 V2 W2

pW axial component of the instantaneous ow velocityWinlet bulk-mean ow velocity at the lower end of the inlet

pipez axial distance from the inlet, in the upward directionz axial distance from the impingement surface, in the

downward direction; z* = H zGreek symbolsb thermal expansion coefcient of a uidc kinematic viscosity of a uidq density of a uidw Stokes stream functionh azimuthal direction

Subscriptsinlet a physical quantity dened at the lower end of the inlet

pipelm a physical quantity dened at the lm temperature, Tlms a physical quantity dened at the impingement surfacestag a physical quantity dened at the stagnation point of

the impingement surface

Conventionsat time-averaged value of any physical quantity, a, over

the time-interval 0tar area-averaged value of any physical quantity, a, over the

horizontal circle of radius r, whose center lies on thecentral axis

{a} a physical quantity, a, that is represented in its dimen-sional form

aheater area-averaged value of any physical quantity, a, over the

Heat and Fluid Flow 50 (2014) 316329 317when the impingement surface is not subjected to any heating.Didden and Ho (1985) investigated such ow separations in detail,after creating large KelvinHelmholtz vortices by forcing the inletjet at different pulsing frequencies. Liu and Sullivan (1996) studiedunsteady heat transfer behavior of a circular impinging jet owafter forcing the jet in a similar manner, and found that the vorticescause the heat transfer coefcient to uctuate. Olsson and Fuchs(1998) studied unsteadiness in a circular impinging jet ow, againby exciting the jet at different frequencies. Rohlfs et al. (2012)examined forced axisymmetric ows and suggested that whenthe KelvinHelmholtz vortices generated in the pre-impingementregion impinge onto the surface, they induce secondary vorticesin the wall-jet region, which separate from the surface about theradial location r = 2.1 and yield a local heat-transfer peak similarto that observed in turbulent impinging jet ows. Chung and Luo(2002) studied the unsteadiness in an unforced compressible slotjet where the KelvinHelmholtz vortices generated naturally. Theyobserved organized temperature uctuations at locations veryclose to the impingement surface when the Reynolds number ofthe ow was relatively small. However, the uctuations becameless regular, but remained approximately periodic, when the Rey-nolds number was increased, because it produced stronger vorti-ces. For a smaller impingement surface distance from the inlet,the uctuations became organized again, because the shear layervortices remained weak and underdeveloped when they impingedonto the surface, due to the short traveling distance before theimpingement.

All the aforementioned studies considered downward imping-ing jets, except the brief study of upward slot jets by Sahoo and

entire heater surfacej~aj magnitude of a vector quantity, ~a~a1 ~a2 scalar product of any two vector quantities, ~a1 and ~a2

AbbreviationsPIV Particle Image Velocimetry

impinging jet ows showed development of a heated plume thatseparated from the impingement surface and rose in the upwarddirection, likely due to the buoyancy force that acts on the owin the wall-jet region where the uid becomes warmer after takingheat from the impingement plate. For upward impinging jet ows,however, they observed a usual wall jet that progressed along theimpingement surface, unseparated, despite the controlling param-eters (other than the jet orientation) being same as those in thedownward impinging jet ows. This is likely because upwardimpinging jet ows require stronger heating for the wall jet to sep-arate, which is evident their another observation that the plumeformed even in the upward impinging jet ows when the Richard-son number was increased to 16. These authors also observedsome unsteadiness in the Nusselt number distributions, both inthe upward and the downward impinging jet ows, but did notdiscuss the underlying mechanisms.

The literature survey shows that studies focusing on transientbehavior of heated upward impinging jet ows are very limited.In the present study, which is an extension of our previous studies(Shekhar and Nishino, 2011, 2013), we would thoroughly discussthe heat-induced ow oscillations, its underlying mechanisms,and transient heat-transfer characteristics, by systematically vary-ing the Reynolds number (Re), the Grashof number (Grq), and thesurface-height to the inlet-diameter ratio (H/D), case by case. Wewould also investigate into how the oscillation parameters, suchas, the time period and the amplitude, depend on the Richardsonnumber (Riq), as well as would present time-averaged distributions

al of Heat and Fluid Flow 50 (2014) 316329Sharif (2004). It should be noted that under high heating rates, theow and heat transfer characteristics of upward impinging jetows are likely to differ signicantly from the same of downwardimpinging jet ows, because the heated uid in the latter cases canfreely move in the upward direction, whereas, in the former cases,it would become stagnant against the impingement surface.Srinarayana et al. (2009) studied an upward impinging jet ow ofRe = 50, where the jet was fed into a hotter ow domain. Theyfound that the working uid becomes stratied in the vicinity ofthe impingement surface, which prevents the jet from impingingonto the heater surface, consequently turning the impinging jetow into a fountain ow (the ow prematurely fell down afterslightly spreading at its highest point). Rady (2000) studied anupward slot jet impinging onto an isothermal surface. He foundthat the area-averaged Nusselt number reduces when the temper-ature of the surface is increased. Nada (2009) studied both upwardand downward impinging slot jets, and found that the heat transferin the forced-convection region is lower in upward impinging jetows than the same in downward jet ows. Shekhar and Nishino(2011) studied unsteady ow and heat transfer behavior of anupward circular impinging jet ow, where the ow impinged ontoa at surface subjected to a constant and uniform rate of heating.They found that the heat transfer rate and the surface temperaturein the forced-convection region are not affected by the heating.However, the buoyancy force caused the wall jet to prematurelyseparate from the impingement surface at some radial distance.The separation turned the following region into a dead zone wherethe heat transfer deteriorated. They also found that the separatedow oscillates in a cyclic fashion and causes the ow-separationpoint also to oscillate radially back and forth, which, in turn,induces periodic oscillation in the on-surface quantities. They fur-ther demonstrated that the observed ow separation is a universalphenomenon that occurs whenever the surface heating rate is suf-ciently high. Note that Shekhar and Nishino (2011) did notobserve any KelvinHelmholtz vortex, perhaps because the para-bolic velocity prole that they employed at the inlet did not createa sufciently strong shear layer. The same was found to be the caseeven in the study by Rohlfs et al. (2012). Shekhar and Nishino(2011) indicated that the separated ow oscillates due to a periodicejection of heated uid from the dead zone. In a later study,Shekhar and Nishino (2013) found that the heat transfer rate inthe dead zone decreases when either the surface heating rate isincreased or the Reynolds number is decreased.

Lin and Armeld (2008) also numerically studied similarupward impinging jet ows, but did not observe the ow separa-tion, because they analyzed the ow only for a short time.

When investigating impinging jet ows, some researchers alsotook into account the internal conductive heat transfer that takesplace within the impingement plate itself (Manca et al., 1996;Ruocco, 1997; Bula et al., 2000; Sarghini and Ruocco, 2004), whichbecomes signicant when the impingement plate is sufcientlythick and subjected to discrete heating. These researchers solvedthe soliduid coupled problems and found that, in general, theheat transfer behavior at the soliduid interface are affected byvarious controlling parameters, such as, the heating rate, the regionof the discrete heating, and the material type and the thickness ofthe impingement plate. They also observed a reversal heat transferphenomenon taking place at radially far distances where someheat is transferred back from the uid into the plate, instead ofthe heat being usually removed from the plate, after the wall-jetmomentum signicantly diffuses. Sarghini and Ruocco (2004)solved two-dimensional slot jet coupled problems, with theRichardson number of the ow kept equal to unity, and the uid

318 C. Shekhar, K. Nishino / International Journdensity varying with the temperature. They heated the impinge-ment plate in the central region, with the width of the heated sec-tion being equal to the inlet width. Their results for downwardFig. 1. Schematic diagrams of the ow geometry. The spatial dimensions arenormalized by the inlet diameter, D. Length of the inlet pipe, b/D, is 2.03 in theH/D = 5.95 cases and 4.06 in the H/D = 1.89 and the H/D = 1.02 cases.

of the on-surface quantities, because they are important in practi-cal applications. In this study, we would consider the impingementplate to be very thin, so that when it is heated uniformly, the heatis transferred uniformly even into the ow domain. The studiedimpinging jet system is described in detail in the next section.

2. System details

Schematic top and front views of the ow geometry are shown

Re Grq H/D Riq

C. Shekhar, K. Nishino / International Journal of900 352,800 5.95 0.44600 352,800 5.95 0.98300 352,800 5.95 3.92600 176,400 5.95 0.49600 352,800 5.95 0.98600 705,600 5.95 1.96600 352,800 5.95 0.98in Fig. 1. It consists of a vertical, three-dimensional, cylindricaldomain, with the top, the bottom, and the side walls present. A cir-cular, concentric hole is carved on the bottom surface, in which acircular pipe of inner diameter {D} = 9.85 mm is tightly tted.Water is fed into the ow domain through this pipe, against thegravity, at the uniform temperature {Tinlet} = 21.5 C. The outlet isalso carved on the bottom wall itself, in the form of a concentric,annular hole, which touches the cylindrical side wall. In order toavoid any back ow at the outlet during the computation, an annu-lar cylinder of height 2.03D is also attached to it.

The upward jet impinges perpendicularly onto the top surface,which is at and horizontal. A concentric, uniform, circular heaterof diameter 10.15D is embedded in it, so that the jet impingesdirectly onto the heaters downward facing surface. The upwardfacing surface of the heater, which is exposed to the air, is insu-lated. Moreover, the heater is considered thin, so that the lateralheat-transfer that takes place inside it can be safely ignored.

The whole ow domain is submerged into a large water reser-voir, which is maintained at the constant temperature of 21.5 C(same as the inlet temperature). The upper water surface of thereservoir is considered co-planer with the downward facing sur-face of the heater.

The inlet-based Reynolds number, the Grashof number, and theH/D ratio, each is varied to three values, while keeping the othertwo constant, as summarized in Table 1. The table also containsthe Richardson number, which came out to be the parameter thatprimarily controls the ow oscillation.

Note that the length of the inlet pipe is equal to 2.03D in the H/D = 5.95 case, and 4.06D in the H/D = 1.89 and the H/D = 1.02 cases.The pipe in the latter two H/D cases is kept longer in order to makesure that the ow at its lower end remains unaffected by the pres-ence of the impingement surface. It insures that the computed oweld inside the inlet pipe would closely resemble the realistic ow.

Computed results suggest that the heated upward impinging jetows, in general, can be categorized into three Richardson-numbergroups: Riq < 0.3, 0.3 < Riq < 3, and Riq > 3. We call them low, mod-erate, and high Richardson number ows, respectively, with a cau-tionary note that the boundary values of these groups are nominal.The present study discusses the moderate and the high Richardsonnumber ows, whereas the low Richardson number ows are notconsidered because they remain almost unaffected from theheating.

A ow for (Re, Grq, H/D) = (750, 0, 4) is also computed, in orderto validate accuracy of the computations by comparing the

Table 1The ow conditions considered in the present study.600 352,800 1.89 0.98600 352,800 1.02 0.98calculated pressure on the impingement surface with an experi-mental result available in the literature.

The present study uses two vertical axes, z and z (see the sideview of the ow geometry in Fig. 1). The z axis originates at thegeometric center of the upper end of the inlet pipe and faces inthe upward direction, whereas the z axis originates at the geomet-ric center of the impingement surface and faces in the downwarddirection. Mathematically, z = H z.

2.1. Normalization of the physical quantities

Physical properties of the working uid are normalized withrespect to their values at the inlet temperature. The spatial dimen-sions and the ow velocity components are normalized withrespect to the inlet diameter and the bulk-mean inlet velocity,respectively. Other quantities are normalized as follows:

t ftgfWinletgfDg ; P fPg fPinlet; r0gfqinletgfWinletg

2 ; and

T fkinletgfqgfDg fTg fTinletg;

where the additional subscript r = 0 in {Pinlet, r=0} represents thevalue of {Pinlet} at the central axis.

3. Numerical methodology and validation

3.1. Numerical methodology

The above ow system is solved numerically using the MAC(Marker and Cell) algorithm proposed by Harlow and Welch(1965), on the staggered grid system, in cylindrical coordinate.The governing equations consist of the continuity equation andthe transient momentum and energy equations, as follows:

Continuity equation:

~r ~VL 0 1Momentum equation:

@~VL@t

~VL ~r~VL ~rP 1Re~r c~r~VL q

~g=g Fr2

2

Energy equation:

@T@t ~VL ~rT 1Re Pr

~r k~rT

3

where ~r is the gradient operator. The momentum equation issubstituted into the continuity equation to derive the pressure Pois-son equation.

The density, the kinematic viscosity, and the thermal conductiv-ity of water vary by 4.5%, 84%, and 19.2%, respectively, when thetemperature is increased from 0 C to 100 C (Rika Nenpyo, 2009;Ziebland, 1981).

Since the density varies by less than 5%, it is incorporated in themomentum equation by means of the Boussinesq approximation.The density is considered constant at the inlet temperature inthe remaining terms of the governing equations. Note that the spe-cic heat of the water varies only by 0.9% in the 0100 C temper-ature range (Rika Nenpyo, 2009), and therefore it is also assumedconstant.

Transient, three-dimensional computations are carried out inthe ow domain, which includes the impingement chamber, theinlet pipe, and the annular outlet (see Fig. 1). The three-dimension-

Heat and Fluid Flow 50 (2014) 316329 319ality of the computational domain is retained, in order to captureany azimuthal ow motion that the surface heating may induceat any later time. However, since the ow domain and the initial

a dened form after applying the LHospitals rule, together withusing the relations lim

r!0@U@h V and limr!0

@V@h U (Constanttinescu

and Lele, 2001), where U and V are the radial and the azimuthalvelocity components, respectively.

In order to supply boundary conditions to the pressure Poissonequation on the solid walls and at the inlet, pressure gradients nor-mal to these boundaries are derived from the momentum equation.

scheme, proposed by Ruuth (2006), which has strong stability pre-

where the calculated Nusselt number (Nu) was 1.138, 1.064,

al of Heat and Fluid Flow 50 (2014) 316329and the boundary conditions (which are described later in this sec-tion) are axisymmetric, the computations are carried out only in anangular slice of the physical domain, as marked in the top view ofFig. 1.

3.1.1. DiscretizationIn the azimuthal direction, uniform grids are used, because the

ow geometry is axisymmetric. In the other two directions, how-ever, non-uniform grids are used: ne grids are used near the solidboundaries and the central axis, in order to adequately resolve highgradients of the local velocity and the local temperature that areknown to exist in these regions. In the inner region where the gra-dients happen to be moderate, coarser grids are used, in order toreduce computational requirements. The grid-spacing in the azi-muthal direction, {Dh}, is equal to 5, and its ranges in the radialand the axial directions are, respectively, Dr = 0.0300.132 andDz = 0.0100.153.

The convection term of the energy equation is discretized usinga hybrid-scheme, which is partially central-difference and partiallyupwind in nature, as follows:

@T@r

hybrid

@T@r

CD2

a Dr2 @

2T@r2

!CD2

4

The subscript CD2 implies that the term is discretized using thestandard, second order central difference scheme. The second termin the right hand side controls the amount of the numerical viscos-ity that can be introduced in the scheme (Sengupta, 2004) (somenumerical viscosity is necessary, in order to remove some non-physical oscillations that appear in the calculated temperatureeld when we discretize the convection term with the central dif-ference scheme). When |a| = 0 and |a| = 1, the hybrid scheme yieldsthe central difference and the upwind schemes, respectively. Thesign of the coefcient a is opposite to the local ow velocity. Testcomputations are carried out for |a| = 0.1, 0.3, 0.5, 0.7, and 0.9,where it is found that the obtained temperature eld exhibits thenon-physical oscillations in the impingement region when |a| issmall but, the boundary layer temperature gets extra smoothedwhen |a| is large. Therefore, |a| = 0.5 is nally chosen as a compro-mise, which is the smallest value for which the oscillation-freeresults could be obtained.

As for the convection term in the momentum equation, it is suc-cessfully handled using the standard second-order central differ-ence scheme itself, because they are discretized on the staggeredgrids, which has some inherent numerical viscosity.

The rest of the spatial derivative terms of the governing equa-tions are also discretized using the central difference scheme.

3.1.2. Boundary conditionsAt the azimuthal boundaries of the computational domain

(which are marked as B1 and B2 in the top view of Fig. 1), the peri-odic boundary condition is used; whereas, the no-slip boundarycondition is used at the solid surfaces. All the solid surfaces, exceptthe heater surface where constant heat ux condition is imposed,are considered thermally insulated. At the inlet, the radial andthe azimuthal velocity components are kept zero, whereas theaxial velocity component increases with time in an asymptoticfashion, while its radial distribution maintains a parabolic shape.The axial velocity on the central axis reaches 99% of the nallydesired value in one second.

At the outlet, the non-reective, Orlansky boundary condition(Orlansky, 1976) is enforced. Moreover, the radial velocity compo-nent is set equal to zero on the central axis. It should be noted that

320 C. Shekhar, K. Nishino / International Journthe viscous term in the radial component of the momentum equa-tion takes the undened form of 00 on the central axis, due to thecoordinate system of our choice being cylindrical. It is brought into1.000, and 0.948 times for the four grid spacings, respectively.These differences, however, reduced fast with the increasing radialdistance and completely vanished by the distance as far as r = 1.

Table 2The stagnation point Nusselt number in the different Re cases, as computed from acorrelation (Corr.) proposed by Scholtz and Trass (1970a,b) (see Eq. 5) and thatcomputed in the present study. The quantity in the last column represents thepercentage difference between the two.

Re Grq H/D Nustag (Corr.) Nustag (Comp.) DNustag (%)

900 352,800 5.95 99.8 114.7 +14.9serving properties. It should be noted that while advancing fromthe current time step to the next time step, boundary conditionsneed to be applied even at the intermediate stages of the RungeKutta schemes. These intermediate boundary conditions arederived based on a method proposed by Pathria (1997), whichmaintains the order of accuracy of the RungeKutta schemes.

Test computations are carried out by varying the time-step to10 ls, 15 ls, 20 ls, and 25 ls, for the (Re; Grq; H/D) = (300,600,900;352,800; 5.95) cases. It is found that the maximum time-step valuefor which the iterative computations converged is 20 ls. Therefore,this time-step is nally chosen in this study.

The computations are carried out using a self-developed code inthe C programming language. The code is parallelized for shared-memory processors, using OpenMP, and run on a 2.83 GHz, 64bit,eight-processor computer. In each of the studied cases, the owis computed for more than t = 1096. The total computation timeconsumed to obtain the results is more than one year.

3.2. Validation of the methodology

Our previous study (Shekhar and Nishino, 2011) examined thegrid-dependency of the computed results by comparing radial dis-tributions of the Nusselt number at four different grid spacings,which are, respectively, 1.4, 1.2, 1.0, and 0.8 times the one that isnally chosen for the computations. The grid spacings were variedin all the three directions. The test computations were carried outfor the (Re, Grq, H/D) = (600, 352,800, 5.95) ow. The obtainedresults revealed strong grid dependency at the stagnation point,At the outlet, constant pressure condition is applied, with the pres-sure {P} set equal to qinletgfH 2:03Dgwhich assumes that the waterlevel in the reservoir is coplanar with the impingement surface.

3.1.3. Initial conditionAt t = 0, the working uid is considered perfectly stand-still,

with the pressure inside the ow domain set according tofPg qinletgfzg. The initial temperature of the water is set equalto the reservoir water temperature.

The discretized momentum equation is marched forward intime using a third-order accurate RungeKutta scheme proposedby Arnold (1998), which is dedicated to solve stiff differentialalgebraic equations. Similarly, the time marching of the energyequation is done with another third-order accurate RungeKutta600 352,800 5.95 81.5 85.4 +4.8300 352,800 5.95 57.6 51.9 9.9

The stagnation-point Nusselt number, Nustag, is compared witha correlation proposed by Scholtz and Trass (1970a,b), which canbe given as follows:

and the experimentally-obtained velocity components for thesame ow. Some of these comparison plots are reproduced herefor the completion purposes. At rst, the z versusW plots are com-

Fig. 2. It should be noted that the radial locations in the comparison

Fig. 2. Axial velocity comparison between the computed (Comp.) and the ParticleImage Velocimetry (PIV) results, along the central axis, for the (Re, Grq, H/D) = (600,176,400, 5.95) ow.

Fig. 5. Surface pressure comparison between experimental results (Exp.) obtainedby Scholtz and Trass (1970b) and the computed results (Comp.) at t = 228.5, for the(Re, Grq, H/D) = (750, 0, 4) ow.

C. Shekhar, K. Nishino / International Journal of Heat and Fluid Flow 50 (2014) 316329 321Nustag 1:648Re0:5Pr0:361: 5The Nustag values obtained from the correlation and that com-

puted in the present study (along with the differences DNustagbetween the two) are summarized in Table 2, for the (Re; Grq; H/D) = (900,600,300; 352,800; 5.95) ows. The table shows a reason-ably good agreement between the two studies, especially when wekeep in mind that the stagnation point is the most sensitive loca-tion with respect to the grid spacing.

Our previous study (Shekhar and Nishino, 2011) experimentallyconrmed the ow separation phenomenon, the heat accumula-tion in the dead zone, and the oscillatory nature of the separatedow. The study also qualitatively validated the computed oscilla-tion phases with the experimentally observed ones, for the (Re,Grq, H/D) = (600, 176,400, 5.95) ow. The particle image velocime-try (PIV) technique was used for the velocity measurements,whereas the laser induced orescence technique was used to visu-alize the temperature eld.

In the same study, the accuracy of the computation programwas further validated by quantitatively comparing the computedFig. 3. Axial velocity comparison between the computed (Comp.) and the PIV

Fig. 4. Radial velocity comparison between the computed (Comp.) and the PIplots are well before the ow-separation point (which is aboutr = 3); and, therefore, the local on-surface quantities remain almostunaffected from the oscillation of the separated ow.

The above gures conrm that the computed results match wellwith the experimental results.

Radial distribution of the computed surface pressure att = 228.5 is compared with experimental results by Scholtz andTrasss (1970b), in Fig. 5, for the (Re, Grq, H/D) = (750, 0, 4) ow.The gure conrms that the two results agree well with each other.

4. Results and discussion

The computed results show that the ows remain axisymmetricand do not develop any azimuthal velocity component. Thepared in Fig. 2, at a random time instant sufciently after the onsetof the ow separation. It is followed by comparisons of (1) theradial distributions of W, at three different axial locations and (2)the axial distributions of U, at three different radial locations, inFigs. 3 and 4, respectively, at the time instant same as that inmeasurement results, for the (Re, Grq, H/D) = (600, 176,400, 5.95) ow.

V measurement results, for the (Re, Grq, H/D) = (600, 176,400, 5.95) ow.

vortices that are marked in Fig. 6(a) and the heat accumulation inthe dead zone (see Fig. 6(b)) play dominating roles in determiningdynamics of the ow oscillation (which is described in Section 4.1).

The oscillating ow causes the ow-separation point also tooscillate radially back and forth, which induces oscillation evenin the on-surface quantities, as evident from the Nusselt numberversus time plots at six different radial locations in Fig. 7. The g-ure also shows that (1) the oscillation is triggered only after theow develops through some initial settling time interval 0600

Fig. 6. Instantaneous (a) velocity streamlines (line contours) and velocity magnitudesThe

Fig. 8. Area-and-time-averaged Nusselt number plots against the averaging time-interval itself. The area averaging is done over the entire heater surface, and thetime averaging is done in the interval 0t.

322 C. Shekhar, K. Nishino / International Journal odownward separation of the wall jet is demonstrated by plottingthe instantaneous velocity streamlines for the (Re, Grq, H/D) =(600, 352,800, 5.95) ow (where Riq = 0.98) at t = 725, in Fig. 6(a).In order to plot the streamlines, the Stokes stream function, w, iscalculated in the axisymmetric plane by integrating the radialvelocity component, U, as follows:

wr; z rZ Hz

Udz 6

This stream function calculation assumes that the stream func-tion is zero at the stagnation point of the impingement surface(that is, at (r, z) = (0, H)). In Fig. 6(a), line contours of the calculatedstream function is plotted at constant intervals, so that the linedensity at a given radial distance from the central axis is propor-tional to the local velocity magnitude (the same line density atsmaller and larger radial distances indicate larger and smallervelocity magnitudes, respectively). The gure also contains gray-scale contours of the ow velocity, VL, in order to provide a clearerpicture of its variations.

The computed temperature eld for the same ow at the sametime instant is plotted in Fig. 6(b), along with some pressure con-tour lines. Both Fig. 6(a) and (b) mark the forced-convection regionand the dead zone, and conrm the large heat accumulation andhigh pressure in the dead zone.

It should be noted that the lines of action of the buoyancy andthe wall-jet inertia forces are perpendicular to each other. Despitethat, they strongly interact with each other through some vortices

contours) distributions, at t = 725, for the (Re, Grq, H/D) = (600, 352,800, 5.95) ow.that they induce in the course of the ow. It is found that the two

Fig. 7. Time history of the Nusselt number at different radial locations, for the (Re,Grq, H/D) = (600, 352,800, 5.95) case (where Riq = 0.98).(grayscale contours) and (b) pressure (line contours) and temperature (grayscalegures also mark the forced-convection region and the dead zone.

f Heat and Fluid Flow 50 (2014) 316329and (2) the oscillation is prominent only at r = 2.28, which is closeto the ow-separation point. However, at r = 1.02, 1.21, 1.48, and2.92, which are the locations far from the ow-separation point,the on-surface quantities are very less affected by the ow oscilla-tion, irrespective of whether these points lie in the forced-convec-tion region or in the dead zone.

Fig. 7 further shows that the oscillation at r = 2.28 is relativelysmall until t 1100, but, afterwards, it increases signicantly. Sim-ilarly, at r = 1.77, the oscillation remains almost absent untilt 1500, but then it gradually increases as the time passes further.The pattern remains the same even at the radial locations closer tothe central axis (despite the local amplitude of the oscillation atthese places being much smaller). These observations suggest thatthe ow-separation point slowly shifts towards the central axisand induces oscillation even at those places which were previouslyunaffected; that is, the dead zone gradually expands towards thecentral axis and the forced-convection region shrinks. The oscilla-tion and the gradual inward radial expansion of the dead zonebecomes more evident in the upcoming Fig. 10(a)(d).

Fig. 7 also reveals that the heat transfer rate in the forced-convection region gradually decreases with time (see the plots at

the present case, the ow would turn into a fountain ow and

al ofr = 1.02, 1.21, 1.48, and 1.77). The local skin-friction coefcient alsoexhibits a similar variation pattern. These decrements are causedby gradually increasing bulk-mean uid temperature of the owdomain (which stops only after the ow reaches statistically steadystate). The warmer surrounding uid opposes the colder pre-impingement jet (which enters into the ow domain at a constanttemperature) with increasingly larger downward buoyancy force.It reduces the impingement velocity, which, in turn, reduces thewall-jet momentum in the post-impingement region.

The expected temperature increment in the forced-convectionregion due to the reduced wall-jet momentum is found to benegligible.

Note that it is the gradually decreasing momentum of the walljet that is also responsible for the aforementioned radially inwardexpansion of the dead zone. This is because a wall-jet with smallermomentum can remove the heater supplied heat only until asmaller radial distance (Shekhar and Nishino, 2011, 2013), which,in turn, would cause the wall jet to separate radially closer to thecentral axis.

The gradual inward shift of the ow-separation point, alongwith the gradually reducing heat transfer rate in the forced-convection region, makes the area-averaged heat transfer ratedecrease with time. It is made evident by plotting the area-and-

time-averaged Nusselt number, Nutheater , against the time intervalof the averaging, for the (Re, Grq, H/D) = (600, 35,2800, 5.95) ow,in Fig. 8 (the gure also contains two additional plots, which wouldbe discussed later). The plotted Nusselt number is rst area-averaged over the entire heater surface, and then time-averaged

over the interval 0t. The gure reveals that the Nutheater plot asymp-totically converges to a smaller value, indicating that the oscillationand the heat transfer properties of the ow are reaching statisticallysteady state conditions. Since these properties are determined bythe ow dynamics, the convergence further suggests that otherow properties are also likely reaching statistically steady states.

Fig. 8 further shows that, after t = 1000, the Nutheater plot changeswith only a marginal rate, indicating that the time t = 1000 issufcient for the on-surface quantities to almost reach their statis-

tically steady state values. It is found that the Ctf ;heater and the Tts;heater

plots also behave in qualitatively similar ways (except that the

T ts;heater plot converges to a larger value). Therefore, the values aver-aged over such large time intervals can be used in applicationdesigning. In the upcoming Section 4.4, radial distributions of thesequantities are presented at t = 1096, for all the studied cases.

When the averaging area is small and well excludes theow-separation point, the averaged Nusselt number plot decreasesapproximately linearly, with a much slower rate than when theaveraging area includes the ow-separation point (the decrementrate in the latter case is comparable to the (Re, Grq, H/D) = (600,352,800, 5.95) plot in Fig. 8). This is because when the ow-separation point is excluded, the averaged value is affected onlyby the Nusselt number decrement in the forced-convection region,which is linear and small compared to the decrement due tothe gradual shrinkage of the forced-convection region (as thedead zone expands radially inward). The averaged skin-frictioncoefcient plot also behaves in a qualitatively similar way; butthe variation in the averaged temperature plot is negligible, asexpected because the temperature in the forced convection regionremains almost unchanged.

In large Richardson number ows, such as, in the (Re, Grq, H/D) =(300, 352,800, 5.95) case (where Riq = 3.92), the buoyancy forceoverwhelmingly dominates over the inertia force, which causesthe oscillation to completely cease after some time, but the ow

C. Shekhar, K. Nishino / International Journseparation point keepsmoving radially inward. Thismovement rateis faster than the same in moderate Richardson number ows, aswould not impinge onto the heater surface at all, similar to thatobserved by Srinarayana et al. (2009).

The H/D in the aforementioned cases was 5.95, which is largeenough to let the lower section of the separated ow move freely.However, when the H/D is reduced signicantly, it is found thatthe separated ow interacts strongly with the base surface (seeFig. 1) and experiences large drag force. As a consequence, the owoscillation either slows down, or even completely stops if theH/D issufciently small (because the drag force in this case becomes verylarge). The latter is found to be the scenario in the studied (Re, Grq,H/D) = (600, 352,800, 1.02) case. Note that the drag force does notmuch affect the rate with which the ow-separation point shiftsradially inward, as evident from comparison of the (Re, Grq, H/D) =(600, 352,800, 1.02) and (Re, Grq, H/D) = (600, 352,800, 5.95) plotsin Fig. 8. The gure also shows that the Nutheater magnitude in thepresent case is larger, because this jet impinges onto the surfacewith a larger velocity, which is a result of its smaller viscousdiffusion due to the smaller path length between the inlet and theimpingement surface.

The oscillation and heat transfer properties of all the studiedows are quantied in the upcoming Section 4.3.

Note that the separated ow in a large H/D case penetratesincreasingly deeper in the downward direction, because its tem-perature remains almost constant and smaller than the bulk-meantemperature of the ow domain, and therefore the downwardbuoyancy force that acts on it gradually increases with the increas-ing bulk-mean temperature. If the separated ow starts to stronglyinteract with the base surface before the bulk-mean temperaturereaches statistically steady state, the oscillation would rst slowdown and then stop eventually. Otherwise, the oscillation wouldcontinue with statistically xed properties.

In a relatively open ow domain, the oscillation is more likely toachieve statistically steady state before the separated ow startsinteracting with the base surface. This is because the heat wouldleave such a ow domain with a higher rate, which will causesmaller increment in the bulk-mean temperature. As a result, theseparated ow would penetrate less in the downward direction.If the base surface (see Fig. 1) is removed from the ow domain,the separated ow would always be free to move and theoscillation would continue forever.

Here, it also worth mentioning that, in a larger ow domain, therate with which themean ow-separation pointmoves towards thecentral axis would reduce. This is because the heat convected fromthe dead zone in a larger domain mixes with a larger amount of theworking uid, and therefore the bulk-mean uid temperature andthe adverse buoyancy force that acts on the pre-impingement jetincrease with a marginal rate only. The reduced rate of theseparation point movement is evident from a previous study byShekhar and Nishino (2011), who showed that when the owdomain in the (Re, Grq, H/D) = (600, 176,400, 5.95) case is widenedto 1.83 times the present one, the mean ow-separation distanceremained almost constant.

4.1. Dynamics and mechanism of the oscillation

When the ow starts, the momentum of the radially advancingwall jet diffuses after some distance and becomes unable toevident from comparison of the Nutheater plots for the present caseand the (Re, Grq, H/D) = (600, 352,800, 5.95) case, in Fig. 8. Note thatthe plot in the present case also tends to converge asymptotically. Ifthe Richardson number of a ow is signicantly larger than that in

Heat and Fluid Flow 50 (2014) 316329 323efciently convect away the incoming heat from the heater. As aresult, the local uid temperature near the edge of the heater

al o324 C. Shekhar, K. Nishino / International Journgradually increases. When the colder wall jet encounters thisheated region, the wall-jet falls slightly downward due to its largeruid density, leading to a mild ow separation, which, in turn, fur-ther reduces the heat convection from the heated region. Conse-quently, the local heat accumulation accelerates, which causesthe ow to separate at a larger angle from the impingement sur-face. The aforementioned downward fall of the colder wall jetcan also be perceived as if it is forced in the downward directionby the warmer uid in the dead zone. It should be noted that itis this downward force that drives the ow oscillation.

The ow oscillation is demonstrated by plotting the velocitystreamlines in the axisymmetric plane of the (Re, Grq, H/D) =(600, 352,800, 5.95) ow, at eight successive time instants of anoscillation cycle, in Fig. 9(a)(h), respectively. Each of these plotscontains line contours of the Stokes stream function at constant

Fig. 9. (a)(h) Instantaneous velocity streamlines and temperature grayscale contours ofof an oscillation cycle. T 0 and T 0.12 are represented by the bright and the dark graintermediate grayscales. For legend, see Fig. 6(b).f Heat and Fluid Flow 50 (2014) 316329intervals. The same gures also contain grayscale contours of theinstantaneous temperature elds at the same time instants. Thethe temperature T 6 0 and TP 0.1 are represented by the brightand the dark grayscales, respectively, whereas its values in therange 0 < T < 0.12 are represented by the linearly increasing gray-scales. The gures also mark the instantaneous movement direc-tion of the ow-separation point by placing a horizontal arrowabove the separation point.

Fig. 9(a)(h) readily show that the separated ow reattachesback with the impingement surface at some larger radial distance(the reattachment point is marked in Fig. 9(a)). Since the separatedow keeps convecting away a portion of the heated uid (from thedead zone) along with itself, the heated uid encounters the colderuid that is lying in the lower section of the ow domain, aftersome time. The colder uid exerts buoyancy force on the heated

the (Re, Grq, H/D) = (600, 352,800, 5.95) ow, at successively increasing time instantsyscales, respectively, whereas 0 < T < 0.12 are represented by the linearly increasing

al of Heat and Fluid Flow 50 (2014) 316329 325C. Shekhar, K. Nishino / International Journuid in the upward direction, causing it to return back and reattachwith the impingement surface, approximately enclosing the deadzone inside itself.

Fig. 9(a) shows the presence of two vortices, which are markedas Vortex-1 and Vortex-2. The separated wall jet directly inducesVortex-1, which, in turn, induces Vortex-2. It also induces a veryweak vortex inside the enclosed dead zone. As the dead zone accu-mulates more heat with time, the heated uid breaks Vortex-1 intotwo parts and enforces itself between them (see Fig. 9(b) and (c)).We call the newly formed outer vortex Vortex-1(a), while keep call-ing the inner vortex Vortex-1. In this process of the vortex breakup,the separation point moves towards the central axis, which causesstronger interaction between Vortex-2 and the pre-impingementupward jet. Since the upward jet opposes the internal ow motionof Vortex-2, the latter gradually fades out and eventually vanishes(see Fig. 9(d)). In this course of the evolution, the ow inducessome additional weak vortices, as well (such as, in Fig. 9(c)), butthey do not inuence the oscillation dynamics any signicantly.

Vortex-1 and Vortex-1(a) stretch the warm uid of the dead zonepresent between themselves in the downward direction (seeFig. 9(d)). In this process, the lower section of the warm uid grad-ually mixes into Vortex-1 (see Fig. 9(e)), while the wall jet keepsseparating at a constant radial distance. The mixing removes a sig-nicant amount of heat from the dead zone, which causes the walljet momentum to dominate the local ow dynamics and push theow-separation point to a radially farther distance (see Fig. 9(f)). Aportion also breaks off from the lower part of the downwardstretched uid (see Fig. 9(f)).

As the ow-separation point is pushed radially outward, theow separates at increasingly smaller angle and with largervelocity. The larger ow-separation velocity strengthens Vortex-1,

Fig. 10. Time histories of (a) the Nusselt number, (b) the surface temperature, (c) the352,800, 5.95) ow.which, in turn, induces a new instance of Vortex-2 between itselfand the pre-impingement jet (see Fig. 9(g)). After some time whenthe heater-supplied heat accumulates signicantly in the deadzone, it starts pushing the ow-separation point radially inward,again (see Fig. 9(h)).

Note that during the evolution of the ow from Fig. 9(b) untilFig. 9(g), the outer vortex, Vortex-1(a), remained approximatelystagnant adjacent to the upward-bent section of the separatedow, while its strength gradually diffused. In Fig. 9(h), the vortexvanishes completely. Further note that the ow eld in Fig. 9(h)is similar to that in Fig. 9(a). As the time passes, the oscillationcycle repeats itself.

The oscillation dynamics and the underlying mechanismremain qualitatively the same for all the moderate Richardsonnumber ows.

surface pressure, and (d) the skin-friction coefcient, for the (Re, Grq, H/D) = (600,

Table 3Time-averaged values of the oscillation parameters and the heat transfer coefcient,for all the studied ows. Here, the abbreviations Sep.Dist., Amp., and T.P. imply theow-separation distance, the amplitude, and the time period of the oscillation,respectively.

Re Grq H/D Riq Sep:Dist: 1096 Amp: 1096 T:P: 1096 Nu1096heater

900 352,800 5.95 0.44 4.2 0.49 285 19.9600 352,800 5.95 0.98 2.4 0.27 112 11.2300 352,800 5.95 3.92 0.8 0 0 5.8

600 176,400 5.95 0.49 3.1 0.31 210 13.8600 352,800 5.95 0.98 2.4 0.27 112 11.2600 705,600 5.95 1.96 1.8 0.17 72 9.9

600 352,800 5.95 0.98 2.4 0.27 112 11.2600 352,800 1.89 0.98 2.6 0.22 138 12.7600 352,800 1.02 0.98 2.3 1 1 11.7

4.2. Oscillation of the on-surface quantities

The radially back-and-forth oscillation of the on-surface quantitiesis demonstrated by plotting time histories of the Nusselt number, thesurface temperature, the surface pressure, and the skin-friction coef-cient, in Fig. 10(a)(d), respectively, for the (Re, Grq, H/D) = (600,352,800, 5.95) ow. These gures readily conrm that (1) as the timepasses, the mean ow-separation point slowly shifts towards thecentral axis and (2) at locations far from the ow-separation point(both in the forced-convection region and in the dead zone), theon-surface quantities are very less affected by the oscillation.

The Nusselt number (see Fig. 10(a)) is the largest at the centralaxis, but gradually decreases with the radial distance. About r = 2.5,

326 C. Shekhar, K. Nishino / International Journal owhere the Nusselt number decreases abruptly from 10 to 0, due tothe ow separation. In Fig. 10(a), the Nu = 10 contour line approx-imately marks the locus of the separation points. Similarly, the sur-face temperature (see Fig. 10(b)) abruptly increases across theseparation point and becomes signicantly higher in the deadzone, as expected. The separation points can be clearly identiedeven in the surface pressure and the skin-friction coefcient plots(see Fig. 10(c) and (d), respectively).

A careful observation of Fig. 10(c) further reveals that when theow separation distance is about the maximum during an oscilla-tion cycle, the pressure magnitude in the forced-convection regionabruptly increases, which appears like a narrow, dark ridge in thetime-history plot. Note that the wall jet velocity signicantlyincreased at such time instants (see Fig. 9(g) and (h)).

4.3. Quantication of the oscillation parameters and the heat transfercoefcient

Time-averaged values of the oscillation parameters are summa-rized in Table 3. The averaging is done in the non-dimensional timeinterval 01096. In the (Re, Grq, H/D) = (300, 352,800, 5.95) case(where the Riq = 3.92), both the amplitude and the time period ofthe oscillation are set to zero, because the ow oscillation in thiscase ultimately ceases. Similarly, in the (Re, Grq, H/D) = (600,352,800, 1.02) case, the two quantities are set to 1, because theoscillation never started in this case.

In order to demonstrate that the oscillation properties of largeH/D ows can be described by the Riq only (instead of individualmagnitudes of the Re and the Grq), the oscillation parameters forall the studied H/D = 5.95 ows are plotted against Riq, in Fig. 11.The gure readily shows that the mean ow-separation distanceasymptotically decreases with increasing Riq, because a given owcan remove an increased supply of the heat only until a smallerradial distance (the same is true when a smaller Re jet is used toremove a given heat supply). Fig. 11 further shows that the time-period and the amplitude also decrease with increasing Riq insimilar asymptotic ways.Fig. 11. The time-averaged values of the ow-separation distance (Sep.Dist.), theamplitude (Amp.), the time period (T.P.), and the area-averaged Nusselt numberNuheater plotted against the Richardson number, Riq. H/D = 5.95, in all the cases.The Nu1096heater values are also summarized in Table 3, and plottedin Fig. 11. The plot shows that the averaged heat-transfer coef-cient follows the same decreasing trend, as the mean ow-separa-tion distance and the oscillation parameters did.

Since the wall jet does not separate in very small Richardsonnumber ows, the mean ow-separation distance and the oscilla-tion parameters can be assumed 1, as rightly indicated by sharpincrements of the corresponding plots in Fig. 11 when the Riqreduces below unity. The averaged Nusselt number can also beobserved to have increased in a similar way, but note that its incre-ment would slow down and converge to a constant value if the Riqkeeps reducing. Further note that the converged value woulddepend only on the Reynolds number of the ow.

On the other hand, when the Richardson number is very large,the jet ow would turn into a fountain ow where the ow-sepa-ration distance, the oscillation parameters, and the averaged Nus-selt number, all can be assumed zero, as also indicated by thecorresponding almost attened plots in Fig. 11 when the Riqincreases beyond 2.

Reynolds number dependence of the Nu1096heater can be understoodfrom Table 3, which shows that the averaged Nusselt numberincreases by about 90% and 250% when the Re is increased from300 to 600 and from 300 to 900, respectively (while keeping theGrq and H/D constant at 352,800 and 5.95, respectively).

The same table also reveals a severely negative impact of theincreased heating rate on the heat transfer coefcient: when theGrq is increased from 176,400 to 352,800 and from 176,400 to705,600, the heat transfer rate decreases by about 19% and 28%,respectively, because a higher heating rate creates a larger deadzone.

As for dependence of the ow and heat transfer properties onH/D, both the mean ow-separation distance and the averagedheat-transfer coefcient increase when the H/D is reduced from5.95 to 1.89 (while keeping the Re and the Grq constant at 600and 352,800, respectively). This is because the jet in the latterH/D case impinges onto the surface with a higher velocity. Sincethe higher impingement velocity is equivalent to a larger localReynolds number in the wall-jet region, the time period alsoincreases, as it did when the Reynolds number of the owincreased from 300 to 600, and then to 900, in Table 3. As for theamplitude, it slightly decreases (although it is expected to increaseas per the trend in Table 3), perhaps because the lower part of theseparated jet starts interacting with the base surface.

When the H/D is further reduced to 1.02, the local Reynoldsnumber in the wall-jet region increases further. Despite that, themean ow-separation distance decreases, because the wall-jetinertia is now also opposed strongly by the drag force that the basesurface exerts on the lower section of the separated ow (note thatthe separated ow in this case impinges onto the base surface witha signicant velocity, and therefore the drag force is large). Theheat-transfer coefcient also decreases, because the heat convectedfrom the dead zone by the separated ow reduces in the absence ofthe oscillation.

4.4. Radial distributions of the on-surface quantities

4.4.1. Nusselt number distributionsThe radial distributions of Nu1096are plotted for the (Re; Grq;

H/D) = (300,600,900; 352,800; 5.95), (600; 176,400, 352,800,705,600; 5.95), and (600; 352,800; 5.95,1.89,1.02) ows inFig. 12(a)(c), respectively. Fig. 12(a) shows that the heat transfercoefcient greatly varies with the Reynolds number, as expected,with the largest difference occurring in the forced convection

f Heat and Fluid Flow 50 (2014) 316329region. The large deterioration of the heat transfer rate in the deadzone can also be conrmed from a sudden drop of the distributionproles at the respective ow-separation points (see Table 3),

al ofC. Shekhar, K. Nishino / International Journespecially in the Re = 300 and the Re = 600 cases where the prolesbecome approximately zero in the dead zone. On the other hand,Fig. 12(b) shows that the heat transfer in the forced-convectionregion remains almost independent from the heating rate, whereassize of the dead zone increases with increasing heating rate. Itshould be noted that it is this size increment that is primarilyresponsible for the corresponding differences observed in Nu1096heaterin Table 3. Fig. 12(c) shows that the heat transfer rate increasesat all the radial locations when the H/D is reduced from 5.95 to1.89, due to the increased impingement velocity. However, whenthe H/D is further reduced to 1.02, the increment in the forced-convection region is only subtle. In the dead zone, on the otherhand, the rate signicantly decreases, instead, in the absence ofthe ow oscillation. Note that it is this decrement that reectedin the unexpected reduction in the corresponding Nusselt numberfrom 12.7 to 11.7, as listed in Table 3.

4.4.2. Surface temperature distributionsThe radial distributions of the T1096s are plotted in Fig. 13(a)

(c), for the (Re; Grq; H/D) = (300,600,900; 352,800; 5.95), (600;176,400, 352,800, 705,600; 5.95), and (600; 352,800;5.95,1.89,1.02) ows, respectively.

Fig. 12. Radial distributions of Nu1096 in the (a) (Re; Grq; H/D) = (300, 600, 900;352,800; 5.95), (b) (Re; Grq; H/D) = (600; 176,400, 352,800, 705,600; 5.95), and (c)(Re; Grq; H/D) = (600; 352,800; 5.95,1.89,1.02) cases.Heat and Fluid Flow 50 (2014) 316329 327Fig. 13(a) shows that the surface temperature is the largest inthe Re = 300 case, and it greatly decreases with increasing Re. Thedifference is relatively small in the forced-convection region, butvery large in the dead zone, as expected, because the natural con-vection properties in the dead-zone are determined by the Rich-ardson number Riq / 1Re2 : The local high temperature maybecome a matter of concern in practical applications, because itmakes the ow prone to boiling. In Fig. 13(b) where the heatingrate is successively increased (while keeping the Reynolds numberand the H/D constant), the temperature prole in the forced-con-vection region remains almost unaffected. In the dead zone, how-ever, the heights of the proles vary, but only moderately. Notethat these temperature differences are signicantly large whenthey are considered in dimensional form.

When the H/D is decreased from 5.95 to 1.89 (see Fig. 13(c)), thetemperature decrement due to the larger impingement velocity inthe latter case remains subtle in the forced-convection region,despite the clear enhancement in the corresponding local heattransfer rate (see Fig. 12(c)). In the dead zone, however, the tem-perature decrement is clearly evident. When the H/D is furtherreduced to 1.02, the dead-zone temperature increases, whichreconrms the favorable role of the ow oscillation in efciently

Fig. 13. Radial distributions of T 1096s in the (a) (Re; Grq; H/D) = (300,600,900;352,800; 5.95), (b) (Re; Grq; H/D) = (600; 176,400, 352,800, 705,600; 5.95), and (c)(Re; Grq; H/D) = (600; 352,800; 5.95,1.89,1.02) cases.

al o328 C. Shekhar, K. Nishino / International Journremoving the heat from the dead zone. The H/D = 1.02 prolealso shows the presence of a local minimum about the locationr = 4. The minimum appears because, after the separated owimpinges onto the bottom surface, it deects back and reattacheswith the impingement surface about the radial location ofthe minimum with a signicant velocity. The correspondingenhancement in the local heat transfer rate can also be conrmedin Fig. 12(c).

4.4.3. Skin-friction coefcient distributionsThe radial distributions of C1096f are plotted in Fig. 14(a)(c), for

the same (Re; Grq; H/D) = (300,600,900; 352,800; 5.95),(600; 176,400, 352,800, 705,600; 5.95), and (600; 352,800;5.95,1.89,1.02) ows, respectively. The gures show that all theplots, rst, increase with the radial distance, from zero on the cen-tral axis to a large maximum at the location r 0.4. Afterwards,they asymptotically decrease at the larger radial distances andbecome almost zero in the dead zone.

Fig. 14(a) further shows that the skin-friction coefcient in theforced-convection region is inversely proportional to Re. Fig. 14(b),on the other hand, shows that the proles in the forced-convectionregion are unaffected from the surface heating. Similarly, Fig. 14(c)shows that when the H/D is reduced to from 5.95 to 1.89, the

forced-convection region.The separated ow oscillated whenever an upward impinging

Fig. 14. Radial distributions of C1096f in the (a) (Re; Grq; H/D) = (300,600,900;352,800; 5.95), (b) (Re; Grq; H/D) = (600; 176,400, 352,800, 705,600; 5.95), and (c)(Re; Grq; H/D) = (600; 352,800; 5.95,1.89,1.02) cases. jet was heated moderately. The moderate heating ensures that

the wall-jet inertia and the buoyancy force are of comparablemagnitude. The ow characteristics in the dead zone and theoscillation properties could be explained in terms of the Richardsonnumber (Riq) alone. When the Richardson number was moderate(0.3 < Riq < 3), the separated ow oscillated with time, in all theH/D 1.89 cases. The oscillating ow also caused the ow-separation point to move radially back and forth, which, in turn,induced oscillation in the on-surface quantities. With time, thebulk-mean temperature of the ow domain increased, whichopposed the colder, pre-impingement jet with a graduallyincreasing downward buoyancy force. As a result, the wall-jetinertia also decreased in the post-impingement region and causedthe mean ow-separation distance to gradually decrease withtime. The increasing bulk-mean temperature also caused theseparated ow to penetrate gradually deeper downward. Whenthe bulk-mean temperature approached statistically steady stateafter a sufcient time, the mean ow-separation distance, theoscillation parameters, and the penetration depth, all tended tostabilize to xed values, and hence the area-averaged value ofthe heat transfer rate as well.

When the Richardson number of the ow was large, the buoy-ancy force overwhelmingly dominated over the wall-jet inertiamaximum at the r 0.4 location becomes higher, due to theincreased impingement velocity. In fact, the prole in the H/D = 1.89 case is little higher even at other locations in the forcedconvection region. When the H/D is further reduced to 1.02, themagnitude of the maximum peak increases little more, butthe increment remains subtle at other locations of the forced-convection region.

Note that, although the skin-friction coefcient in thedead-zone becomes zero at far radial distances from the meanow-separation point, as expected, it behaves differently aboutthe ow-separation point itself (see Fig. 14(c)): the magnitudeincreases when the H/D is reduced from 5.95 to 1.89, due to theincreased impingement velocity, but it decreases when the H/D isfurther reduced to 1.02, because the ow now separates from theimpingement surface almost perpendicularly, whereas it separatedat varying angles (due to the oscillation) in the larger H/D cases.The perpendicular separation decreased the magnitude of thetime-averaged skin-friction coefcient near the ow-separationpoint.

Although not presented, but it worth mentioning that all theskin-friction coefcient distributions are found to almost coincidein the forced-convection region when they are normalized withrespect to the corresponding peak values at r 0.4.

5. Conclusions

Oscillation and heat transfer characteristics of axisymmetric,submerged, upward, laminar impinging jet ows were numericallystudied, after categorizing them based on Re, Grq, and H/D. Theresults showed that the surface heating causes the wall jet toseparate prematurely, which, in turn, turns the following regioninto a dead zone where the local heat transfer coefcient and thelocal skin-friction coefcient becomes almost negligible. The dete-riorated heat transfer signicantly increased the local temperaturein the dead zone, which is a matter of concern in practicalapplications, as it makes the ow prone to boiling. On the otherhand, the surface heating did not affect the ow properties in the

f Heat and Fluid Flow 50 (2014) 316329and stopped the oscillation. In case of a larger Richardson numberow, the heating would stratify the working uid even in theimpingement region, which, in turn, would completely prevent

the jet from impinging onto the heater surface, in which case theimpinging jet ow would turn into a fountain ow.

When the mean ow-separation distance, the amplitude andthe time-period of the oscillation, and the area-averaged heattransfer rate were plotted against the Richardson number, all theplots reduced asymptotically with increasing Richardson number.

When the H/D of the ow was reduced, the jet impinged ontothe impingement surface with a larger velocity (due to the pre-impingement jets smaller viscous diffusion), which increased theheat transfer coefcient and the skin-friction coefcient in theforced-convection region, but the local temperature remainedalmost constant. In the dead zone, on the other hand, the ow prop-erties were affected by both thewall-jet momentum and the degreeof interaction between the separated ow and the base surface.When the interaction was weak, the heat transfer coefcient andthe skin friction coefcient, both increased with the increasedimpingement velocity, whereas the surface temperature decreased.However, when the interaction was large, such as in the H/D = 1.02

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Oscillation and heat transfer in upward laminar impinging jet flows1 Introduction2 System details2.1 Normalization of the physical quantities

3 Numerical methodology and validation3.1 Numerical methodology3.1.1 Discretization3.1.2 Boundary conditions3.1.3 Initial condition

3.2 Validation of the methodology

4 Results and discussion4.1 Dynamics and mechanism of the oscillation4.2 Oscillation of the on-surface quantities4.3 Quantification of the oscillation parameters and the heat transfer coefficient4.4 Radial distributions of the on-surface quantities4.4.1 Nusselt number distributions4.4.2 Surface temperature distributions4.4.3 Skin-friction coefficient distributions

5 ConclusionsReferences