published 8 September 2000 doi 101098rspa20000606456 2000 Proc R Soc Lond A

C T Avedisian and Z Zhao The circular hydraulic jump in low gravity

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101098rspa20000606

The circular hydraulic jump in low gravity

By C T Avedisian an d Z Z haoySibley School of Mechanical and Aerospace Engineering

Cornell University Ithaca NY 14853-7501 USA

Received 9 September 1999 accepted 5 January 2000

An experimental study of the circular hydraulic jump (CHJ) was carried out toshow how reducing gravity to 10iexcl2 times that of the Earthrsquos normal gravity anotectsthe CHJ diameter and the downstream regow patterns Water was the working reguidMeasurements of the CHJ diameter and the shape of the free liquid surface acrossthe jump were made in low gravity using the downstream reguid height and the regowrate as the parameters Comparisons are made with normal-gravity regow conditionsA drop tower was used to create low gravity The measurements are compared withan existing theory for predicting how gravity anotects the CHJ diameter

Results show that the steady-state CHJ diameter is larger at low gravity thannormal gravity As the downstream reguid height is increased the CHJ diameter atnormal gravity moves inward but at low gravity the CHJ diameter does not show asimilar inreguence of downstream reguid height The curvature across the CHJ is higherat normal gravity than at low gravity and the length of the transition zone from theupstream to downstream height increases as gravity is reduced Waves in the down-stream regow are observed for all low-gravity regow conditions but at normal gravitythey were observed only for selected regow conditions suggesting that the inreguence ofsurface tension and viscosity dominate at low gravity Measured normal-gravity andlow-gravity CHJ diameters were bounded by theoretical predictions which assumeeither fully developed or developing regow conditions for the upstream thin shy lm

Keywords hydraulic jump jet impingement low gravitywaves liquid macr lm impingement cooling

1 Introduction

Impingement of a circular liquid jet onto a solid surface is important in a variety ofcooling processes impingement cooling of electronic devices materials processing inmanufacturing laser mirrors aircraft generator coils and vapour absorption refriger-ation cycles A feature of impinging liquid jets is their potential for dissipating highheat transfer rates because of the small liquid shy lm thickness and high regow velocityFor example the highest steady-state heat reguxes reported in any conshy guration wereachieved by a water jet impinging onto a plasma heated surface (Liu amp Lienhard1993a)

General features of a liquid jet impinging onto a solid surface are shown in shy g-ure 1 During radial spread away from the stagnation point a circular hydraulicjump (CHJ) may form The lower regow velocity downstream of the CHJ relative to

y Present address Broadcom Corp 2099 Gateway Place Suite 700 San Jose CA 95110 USA

Proc R Soc Lond A (2000) 456 21272151

2127

creg 2000 The Royal Society

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2128 C T Avedisian and Z Zhao

Dj

Uj

r0

rrh Lj

h yen

L

liquid jet

hydraulicjump

stagnationregion

V Pr

Vr1 P1

h(r)

boundary layer

t w

control volume

Figure 1 Schematic of jet impingement structure

the upstream velocity in the thin shy lm motivates predicting the parameters uponwhich the CHJ depends because of the potential for reduced heat transfer in thedownstream regow in applications where a cold impinging jet is used to cool a hottarget plate (see for example Womac et al 1993)

In this paper we present new data and observations which show the inreguence oflowering gravity on the CHJ The parameters are the liquid height downstream ofthe CHJ the volumetric regow rate and the jet orishy ce opening The magnitude of`lowrsquo gravity employed in the present study is nominally 002 that of Earthrsquos normalgravity which is achieved by carrying out the experiments in a laboratory frame thatis literally dropped or in freefall Specishy c objectives were the following (1) obtainphotographic documentation to show how a CHJ is anotected by reducing gravity (2)measure the CHJ diameter and shape of the liquid surface in the transition zonein low gravity and compare the results with normal-gravity observations and (3)compare the measurements with prior analysis of jet impingement as applied to lowgravity

Gravity anotects the location of the CHJ the curvature of free surface across thejump and the length of the transition zone across the jump (Lj in shy gure 1) Theseenotects are most directly shown by considering the case of an inviscid reguid (x 2 reviewssome prior analyses of a CHJ) and a CHJ that is a discontinuity in the regow (Lj = 0)Middleman (1995) provides details and only a brief summary is given here A momen-tum balance on a control volume (see shy gure 1) across a CHJ relates the change inmomentum of the reguid to the change of the force of the pressure due to the dinotering

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The circular hydraulic jump in low gravity 2129

shy lm thicknesses upstream and downstream of the jump (see the Nomenclature for adeshy nition of symbols)

Z h1

0

V 2r1 dz iexcl

Z h

0

V 2r dz =

Z h

0

P dz iexclZ h 1

0

P1 dz (11)

Since we are considering a frictionless reguid (analyses which include viscosity arediscussed in x 4 c) equation (11) can be integrated directly because Vr and Vr1 areindependent of z for the frictionless case Combining the result with mass conserva-tion gives

plusmn r2j Uj

2rh

sup22plusmn1

hiexcl 1

h 1

sup2= 1

2Gg0(h2

1 iexcl h2) (12)

where G = g=g0 A well-known relationship between the ratio of downstream toupstream shy lm thickness follows by expressing equation (12) in non-dimensional form

H 1 =H = 12fp

1 + 8Fr iexcl 1g (13)

where the Froude number (Fr) based on the average velocity (U ) of the liquid shy lmjust upstream of the jump is U2=Gg0h and mass conservation was used to relate Uj

to U A relationship between the CHJ diameter and regow parameters follows fromequation (12) by substituting for h the expression obtained from a mechanical energybalance applied across the CHJ (ie h = r2

j =2r h )

D h =1

H 1(Frj iexcl 1

2) (14)

where Frj denotes the grouping of terms U2j =Gg0h 1 and from mass conservation

Frj = Fr(4D2h H3=H 1 ) Equation (14) shows the inreguence of gravity on D h for the

inviscid model for given h 1 D h 1=G When G = 0 D h should be inshy nite (whichmeans that a CHJ will not form and the regow is `supercriticalrsquo throughout in thesense that Fr frac34 1 everywhere) More practical is that if 0 lt G lt 1 D h should belarger in low gravity than normal gravity

Gravity also anotects the free surface radius micro (see shy gure 1) across the transitionzone of the jump To consider these inreguences the assumption of the CHJ as adiscontinuity has to be relaxed In general terms the radius of curvature of thefree surface in the transition zone across the CHJ is determined by balancing thehydrostatic pressure in the downstream shy lm with the surface tension force at theliquidair interface In non-dimensional form

middotmicro ordm 1=Bo (15)

where the Bond number is deshy ned as Bo sup2 raquo Gg0h 1 rj=2 frac14 and middotmicro sup2 micro=rj From equa-tion (15) middotmicro 1=G whereby decreasing G increases the radius of curvature and thetransition to the downstream regow should be more gradual in low gravity comparedwith G = 1 Associated with the increase of middotmicro as G is lowered is a lengthening ofthe transition region across the jump (Lj in shy gure 1) For planar hydraulic jumpsHager (1993) shows experimentally that Lj is related to Fr as

Lj = C1 + C2 tanh(C3

pFr) (16)

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2130 C T Avedisian and Z Zhao

where the hyperbolic tangent form was found to best represent the measurementswhich also determine the constants Equations (15) and (16) together show thatreducing gravity should make the transition across the jump more gradual

From the above equations two non-dimensional groups anotect the CHJ when vis-cosity is negligible the Froude number and the Bond number The Reynolds num-ber enters when viscous enotects are important Parameters upon which these non-dimensional groups depend can be adjusted in many ways to give the same dynamicenotect and except for G they have all been investigated in past work For exampleincreasing the upstream velocity produces the same enotect on the CHJ position asdecreasing the downstream shy lm height increasing the radius of curvature across thejump can be accomplished by increasing the reguid surface tension or lowering thedownstream shy lm height and so forth As will be shown several unique enotects ariseby lowering gravity that are not readily explainable by existing theories

2 Brief review of prior work

An extensive literature exists on jet impingement in general and the hydraulic jumpin particular Standard reguid mechanics textbooks provide basic formulations forplanar inviscid hydraulic jumps (see for example Allen amp Ditsworth 1972 Streeteret al 1998) Examples of recent studies that extend these treatments include workby Hornung et al (1995) who include vorticity generation in the downstream regowof a planar jump and Higuera (1994) who developed a model for a planar jumpincluding regow in the transition region taking into account surface tension for thelimit of inshy nite Reynolds number

Nirapathdongporn (1968) reviews the literature on the CHJ prior to 1968 andMiddleman (1995) provides an excellent pedagogical review of CHJ analyses Oneof the earliest theories of a CHJ was by Tani (1949) (with the basic analysis laterincluded in the study by Bouhadef (1978)) who analysed boundary layer regow of aviscous reguid and identishy ed regow conditions where dh=dr (shy gure 1) is inshy nite Beforereaching this condition Tani (1949) speculates that the regow separates and displacesthe shy lm upward Errico (1986) also reviews some of the theories for the CHJ andpresents measurements at G = 1 that show various levels of agreement with anal-yses Watson (1964) derives a self-similar solution to the boundary layer equationsupstream of the jump assumes inviscid regow downstream of the jump and applies themomentum balance (equation (11)) to derive an expression for the CHJ diameterfor the limit of inshy nite Froude number Bowles amp Smith (1992) extend Watsonrsquosanalysis to include surface tension the formation of small standing waves upstreamof the jump and the cross-stream pressure gradient due to streamline curvature Thenumerical solution by Chaudhury (1964) accounts for the enotect of turbulence andheat transfer and predicts both the CHJ diameter and shy lm thickness downstreamof the jump Experiments reported by Craik et al (1981) are noteworthy for theinsights they provided on the regow structure within and behind a CHJ Back-regowseparation and eddies were observed near the CHJ The role of surface tension onstabilizing a CHJ and promoting separation of the regow at G = 1 was examined inexperiments by Liu amp Lienhard (1993b)

The enotect of gravity has not specishy cally been addressed in prior experiments onthe CHJ The closest one comes is the experimental study of Labus (1976) on jetimpingement regows at G frac12 1 carried out in a drop tower The focus was on studying

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The circular hydraulic jump in low gravity 2131

the shapes assumed by a free liquid surface rolling onot of the edge of a pedestal onwhich a liquid jet impinged in a G frac12 1 environment Labus (1976) noted that noCHJ occurred during any of the G frac12 1 tests but that they were a common occur-rence at G = 1 for nominally the same upstream regow conditions Reducing gravitythus appeared to push the CHJ onot the target plate for the regow conditions examinedConcerning analysis a number of numerical and analytical studies specishy cally refer-enced the condition G = 0 Simplishy ed one-dimensional analysis of jet impingementregows (Thomas et al 1990) and direct numerical simulations (Rahman et al 1990)predict that the CHJ diameter is inshy nite or equivalently that a CHJ does not formwhen G = 0 which is consistent with equation (14)

3 Experimental design

A low-gravity environment for the present experiments was created by using a droptower in which the regow loop and associated instrumentation are dropped Low gravityis created in the moving frame of reference during the packagersquos downward regightBecause we had a limited period of free-fall the simplest way to examine gravityrsquosenotect on the jet impingement regow structure and the CHJ diameter was to shy rstestablish a CHJ at G = 1 then with cameras in operation rapidly reduce gravityIn this way we could observe the transient nature of the adjustment of the CHJto the change in gravity as well as the steady-state structure of the CHJ in thelow-G environment The package containing the regow loop was released into free-fallby deactivating a magnet which initially held the package to the support ceilingThe general drop tower facility was described previously (Avedisian et al 1988) Itprovides free-fall over a 76 m distance of an instrumentation package which housesthe experiment to give an experimental run time of about 125 s

The gravity level in the moving frame of reference increases from zero immediatelyupon release of the package because of air drag Measurements of G within the fallingpackage used in this study showed G reaching 005 after 12 s An average value forthe 12 s duration of the free fall is 002 and this is the value which we mean by theterms `low gravityrsquo and `G frac12 1rsquo

A schematic of the regow loop is shown in shy gure 2 The components are the follow-ing recirculating pump plenum containing beads a regow straightener (to dampenthe inlet regow) and a removable orishy ce plate attached at one end Plexiglas chamberwithin which the target plate is mounted regow control valve and cameras and light-ing The height of the liquid level downstream of the jump is controlled by partlysubmerging the target plate and raising the water level to the desired height above theplate The experimental conditions examined were the following The downstreamreguid height h 1 was 20 40 60 100 or 150 mm The liquid regow rates variedbetween 239 and 265 ml siexcl1 To create laminar jets a sharp-edged orishy ce was usedOrishy ce diameters of 122 256 and 383 mm were used They were machined into astainless steel plate that was bolted to the bottom of the plenum Figure 3 showsthe design The plenum chamber (shy gure 4) was mounted such that the orishy ce was762 cm above the target surface

The target plate was a 635 mm thick and 23 cm diameter Pyrex glass disc Theglass disc was in turn attached to a Plexiglas support disc 19 cm thick and 23 cmin diameter and mounted to one end of a circular Plexiglas cylinder that was thenbolted to the containment vessel and sealed with o-rings A mirror tilted to 45macr

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2132 C T Avedisian and Z Zhao

steelframe

105 mmlens

35 mmcamerabody

28 mmlens

CCDcamera

electromagnetsteel plate

lenssupportbracket

aluminium topplate

pump

flowcontrolvalve

drain valve

aluminiumplate

electronicflowmeter

halogenlamp

lenshydraulicjump

plenum

liquid jetwater

level

aPlexiglasside

walls

hyen

O-ringseal

aluminiumsupportbracket

mirror

Figure 2 Schematic of deg ow loop mounted within the steel frame that is physically dropped tocreate low gravity

under the tube allowed for observations of the underside of the jet This view was aprimary one to measure the CHJ radius Additional viewing angles were along theplane to show the cross-sectional proshy le shape of the liquid shy lm across the CHJ andan angled top view for global features of the CHJ The working reguid was water atroom temperature in all the experiments

To keep the design relatively simple the primary means of data acquisition wasphotographic A 35 mm Nikon F3 camera with MD-4 motor drive (operated at shy veframes per second) and with a 105 mm NIKKOR macro lens attached to it wasused to record images at all three camera positions Video images using a COHUCCD camera (30 images per second) were taken with a 28 mm NIKKOR macrolens to record the evolution of the CHJ in the transition to low gravity All cam-eras and lenses were securely mounted to prevent damage due to the shock of theimpact Lighting was provided by halogen lamps (GE FHX 13 W bulbs) The CHJdiameter was measured from video images of bottom views using a `video caliperrsquo(Video Caliper 306 Colourado Video Inc) placed on the video image that was cal-ibrated with a precision ruler (Schaedler Quinzel Inc USA) which was added tothe image to calibrate (to sect158 m m) the voltage output from the caliper Side viewswere used to obtain the cross-sectional shape of the liquid shy lm across the CHJ Theside-view images were fed into a MAC-based data acquisition system (AutomatixImage Analysis Program) and an operator-selected grey scale was used to identifythe various boundaries involved A 1905 mm diameter ball-bearing image convertedthe side-view pixel count to length with a precision of about sect66 m m

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The circular hydraulic jump in low gravity 2133

Dj(see fig 4)

O-ring

7620

15240

water inlet

flange

weld

glass beads (35 diameter)

20 mesh screen

PVC sleeve

orifice plate (see fig 4)

4445

3975

Figure 3 Plenum chamber design (dimensions in mm not to scale)

481

5715

6350

55deg

6-32 clearance(6 required)

O-ring grooveDj (= 122 256 383)

Figure 4 Design of sharp-edged orimacrce for producing laminar liquid jets(dimensions in mm not to scale)

4 Discussion of results

(a) Presentation of photographs and data

In this section we present the main experimental results of the photographic docu-mentation of the CHJ at low G and quantitative measurements of the CHJ diameterFurther discussions about physical mechanisms for some of the observations are givenin x 4 b

Figure 5 is a photograph of a CHJ at G = 1 formed by impingement of a laminarwater jet onto the target surface for the following conditions h 1 = 4 mm (ie thetarget plate was submerged to this depth) Dj = 256 mm and a volumetric regow rate

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2134 C T Avedisian and Z Zhao

Figure 5 Photograph of a laminar water jet striking the target surface (Dj = 254 mmQ = 957 ml s iexcl 1 h 1 = 4 mm) at G = 1 The water jet is the rod-like structure The CHJis the dark outer ring bounding the upstream thin macrlm

Figure 6 Photograph of the water jet in macrgure 5 taken 200 ms after the period of low gravitybegins The transition zone upstream of the CHJ has considerably thinned indicating reducedcurvature of the free surface A humprsquo or large wave is visible immediately downstream

(Q) of 957 ml siexcl1 The laminar liquid jet is the rod-like cylinder in the central portionof the photograph The CHJ is the dark circular region that surrounds the bright thinshy lm The liquid shy lm abruptly increases across the CHJ to match the downstreamregow height which is a controlled variable Waves are faintly visible in the upstreamregow just before the CHJ The downstream regow shows some mild disturbances butno clearly distinguished wave patterns

When G is suddenly reduced shy gures 6 and 7 show how the G = 1 CHJ of shy gure 5reacts Figure 6 was taken 200 ms after the period of free-fall In the vicinity of thestagnation point the liquid continues to spread radially outward in a thin shy lm butat the start of the CHJ in G frac12 1 the curvature of the liquidair interface across thejump decreases (or micro in shy gure 1 increases) and a `humprsquo or large wave forms in thedownstream regow followed by a a regular pattern of waves in the downstream regow

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The circular hydraulic jump in low gravity 2135

Figure 7 Photograph of the water jet in macrgure 5 taken 400 ms into the G frac12 1 period

These waves were commonly observed in all of the low-gravity conditions and in thedownstream regow at G = 1 for the smallest h 1

To better show the transition zone of the CHJ subsurface photographs were takenby viewing from the side through the transparent wall of the containment tankFigure 8 shows six photographs in series of the cross-section of the CHJ for h 1 =4 mm spaced 200 ms apart The photographs are aligned with the jet axis to showthe changes in the CHJ during the transition to low G Figure 8a shows the jump atG = 1 and shy gure 8bf shows the regow situation at low G As shown in shy gure 8bf thetransition across the free surface to the downstream regow is more gradual at low Gthan G = 1 (shy gure 8a) which is consistent with the force balance of equation (15)The lengthening of the transition zone is also evident

Figure 9 shows measurements of the shape of the free surface across the CHJdigitized from the photographs in shy gure 8 Taking the CHJ to be the radial loca-tion where the shy lm just starts to increase the arrows mark the `toersquo of the CHJAs is evident the transition across the CHJ becomes more gradual when G isreduced

A series of photographs to show the evolution of a CHJ at four dinoterent down-stream heights for shy xed regow rate is presented in shy gures 10 and 11 From shy gure 10(h 1 = 2 mm) the increase of the CHJ diameter is clear for G frac12 1 compared withG = 1 as shown by comparing parts (a) and (c) and parts (d) and (g) of shy gure 10A large wave or `humprsquo is visible at t = 02 s in shy gure 10b Waves or ripples in thedownstream regow are evident in shy gure 10c For h 1 = 4 mm (shy gure 10dg) the CHJdiameter at G = 1 is smaller than for h 1 = 2 mm at G = 1 which is consistent withequation (15) in that D h 1=H 1 For G frac12 1 the CHJ diameter for h 1 = 5 mm(shy gure 10g) is only slightly larger that for h 1 = 2 mm (shy gure 10c) For h 1 = 10 mma CHJ did not form at G = 1 as shown in shy gure 11 On lowering G for the regowconditions shown in shy gure 11 a CHJ appeared

Measurements of the evolution of CHJ diameter are shown in shy gures 12 and 13The downstream height (h 1 = 4 mm) and nozzle diameter were shy xed and only theregow rate was changed For the four regow rates shown the CHJ diameter was largerin G frac12 1 than in G = 1 For the conditions shown in these shy gures a CHJ existedat G = 1 This enotect is also seen in shy gure 9 For all such cases a transient period

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2136 C T Avedisian and Z Zhao

jet

(a)

(b)

(c)

(d)

(e)

( f )

h yen

Figure 8 Side-view photographs of a water jet showing a cross-section of the liquid macrlm acrossthe CHJ for h 1 = 4 mm and Q = 632 ml s iexcl 1 and Dj = 122 mm (a) G = 1 (b)(f ) G frac12 1The time between each photograph is 200 ms The dashed line shows the position of the waterjet

of about 200 ms was sumacr cient to establish steady regow conditions in G frac12 1 after thesudden change in G Thereafter the CHJ diameter was constant in G frac12 1 as shownin shy gures 12 and 13

The increase of the CHJ diameter at G frac12 1 relative to G = 1 is clearly illustratedby the three bottom-view photographs shown in shy gure 14 The dark circular zonein the centre of the photographs is the nozzle exit viewed through the underside ofthe transparent target plate In these bottom-view photographs the G frac12 1 CHJ

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The circular hydraulic jump in low gravity 2137

3

1

2

0

Y (

mm

)

3

1

2

0

Y (m

m)

(a)

(b)

time = 0 ms

time = 200 ms

3

1

2

0

Y (

mm

)

(c) time = 400 ms

3

1

2

0

Y (m

m)

(d ) time = 600 ms

3

1

2

0

Y (

mm

)

(e) time = 800 ms

3

1

2

0

Y (m

m)

( f ) time = 1000 ms

5 10 15X (mm)

20 25

Figure 9 Measurements of the promacrle shape across the CHJ digitized from the photographs ofmacrgure 8 The macrgures are lined up with the CHJ indicated by the arrow in each macrgure

boundary is revealed at t = 04 s as the central circular zone of relatively uniformscattered light The pattern of circular waves downstream of the CHJ is also visiblein shy gure 14 at t = 04 s

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2138 C T Avedisian and Z Zhao

40 mm

t = 0 s G = 1

t = 0 s G = 1

t = 02 s G ltlt 1

t = 02 s G ltlt 1

t = 04 s G ltlt 1

t = 04 s G ltlt 1

(b)

(e)

(a)

(d)

(c)

( f )

t = 06 s G ltlt 1

( g)

Figure 10 Photographic series of evolution of a CHJ during the transition from G = 1 to G frac12 1in time-intervals of 200 ms with Q = 632 ml s iexcl 1 and Dj = 122 mm (a)(c) h 1 = 2 mm (d)(g)h 1 = 4 mm During the transition the deg ow downstream of the CHJ thins indicating reducedcurvature of the free surface and a large wave or humprsquo is visible (b) (f) which propagates outof the macreld of view and leaves behind a series of concentric waves or ripples in the downstreamdeg ow

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The circular hydraulic jump in low gravity 2139

25 mm

t = 0 s G = 1 t = 0 s G = 1

t = 02 s G ltlt 1 t = 02 s G ltlt 1

t = 04 s G ltlt 1 t = 04 s G ltlt 1

(b) (e)

(a) (d)

(c) ( f )

Figure 11 Photographic series showing development of a CHJ in G frac12 1 when a CHJ did not format G = 1 because of the large downstream deg uid height (Q = 632 ml s iexcl 1 and Dj = 122 mm)(a)(c) h 1 = 10 mm (d)(f ) h 1 = 15 mm

For the largest values of h 1 studied 10 mm and 15 mm a CHJ did not existat G = 1 for Dj = 122 mm but was drawn out of the regow after the transition toG frac12 1 (see shy gure 11) Furthermore the CHJ diameter did not attain a steady valueduring the period of the experiment Figure 15 shows the evolution for the conditionsof shy gure 11df There was a brief period in which the liquid thinned considerablythen gave way to the CHJ After the CHJ appeared (shy gure 11e) the CHJ diameternever attained a steady-state value during the period of the experiment as shown inshy gure 15

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2140 C T Avedisian and Z Zhao

40

10

15

20

25

30

35

12

time (s)

2rh

(mm

)

2rh (G ltlt 1)

G = 1 G ltlt 1

- 02 0 02 04 06 08 1

transientperiod

h yen = 4 mm Q = 239 mlh yen = 4 mm Q = 632 ml

Figure 12 Evolution of the CHJ diameter (2rh ) showing the indeg uence of deg ow rate (Q = 239and 632 ml s iexcl 1 ) for h 1 = 4 mm and Dj = 122 mm The transient period indicates the timeover which the deg ow adjusts to the rapid change of G

The variation of CHJ diameter with h 1 for a regow rate of 632 ml siexcl1 is shownin shy gure 16 The lines shown in this shy gure are to suggest trends For h 1 gt 8 mmat low gravity the CHJ diameter was unsteady as shown in shy gure 15 For suchconditions we took an average value of the CHJ diameter in the interval of 400600 ms(see shy gure 15) As pointed out previously and quantitatively shown in shy gure 16D h G = 1 lt D h G frac12 1 All of the G = 1 measurements (shy gures 12 and 13 show typicaldata) follow the trend that D h decreases as H 1 increases However at G frac12 1the CHJ diameter appears to be independent of H 1 as shown in shy gure 16 Anexplanation for these trends is onotered in the next section

How the CHJ evolves for the lowest regow rates examined 239 ml siexcl1 and thesmallest downstream heights 2 mm is shown in shy gure 17 When h 1 = 2 mm theCHJ at G = 1 is quite distinct as shown in shy gure 17a However surface wavesare observed even at normal gravity for these conditions in shy gure 17 The tran-sition to G frac12 1 (shy gure 17b c) thins the transition zone and the CHJ increasesslightly Increasing h 1 to 4 mm while maintaining Q at 239 ml siexcl1 moves the CHJclose to the stagnation point as shown in shy gure 17d and considerable turbulenceis observed in the downstream regow However in G frac12 1 the turbulent agitation inthe downstream regow gives way to the regular pattern of circular waves as shownin shy gure 17f Lowering gravity clearly enotects the turbulent agitation in the tran-sition and downstream regow it eliminates it as shown in shy gure 17d (G = 1) and17f (G = 002)

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The circular hydraulic jump in low gravity 2141

100

0

20

40

60

80

12

time (s)

2rh

(mm

)

G = 1 G ltlt 1

- 02 0 02 04 06 08 1

transientperiod

Q = 2647 ml s- 1

Q = 993 ml s- 1

Figure 13 Evolution of the CHJ diameter (2rh ) showing the indeg uence of deg ow rate (Q = 993and 2647 ml s iexcl 1 ) for h 1 = 4 mm and Dj = 256 mm The transient period indicates the timeover which the deg ow adjusts to the rapid change of G

(b) Explanation of results

In this section we explain qualitatively the results of the previous subsection Theseinclude the CHJ diameter proshy le of the liquid shape across the jump the length ofthe jump and downstream regow structure as G is reduced Because a complete theoryhas not yet been developed for the CHJ that includes all of the enotects observed thediscussion below is in many cases speculation about the mechanisms leading to thephenomena observed when G is reduced

The inreguence of G is traced to the hydrostatic pressure (P1) in the downstreamshy lm This pressure balances the momentum of the upstream reguid and surface tensionin the transition zone Since P1 g0Gh 1 reducing G or h 1 will reduce the force thatcounterbalances reguid momentum The result is a repositioning of the CHJ to a largerdiameter where the velocity is lower because the velocity is inversely proportionalto r h This enotect is evident from the results presented in the previous subsectionFigure 14 shows the shift of the CHJ due to lowering G and shy gures 12 and 13 showquantitative measurements of the CHJ diameter increasing as G changes from G = 1to G = 002

Within the low-gravity environment of G ordm 002 D h showed no deshy nite trendwith h 1 as shown in shy gure 16 whereas at normal gravity the trend of equation (14)is followed The disagreement of the low-gravity measurements with the trend ofequation (14) can be explained by reconsidering the momentum balance which ledto equation (12) but this time assuming that viscous enotects dominate and gravity

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2142 C T Avedisian and Z Zhao

t = 0 s G = 1

t = 04 s G ltlt 1

t = 02 s G ltlt 1

40 mm

waves

CHJ

(a) (b)

(c)

Figure 14 Series of three bottom-view photographs (viewed through the underside of the trans-parent target plate) of the water jet during the transition from G = 1 (t = 0 s) to G frac12 1(t = 02 and 04 s) Ripples in the downstream deg ow are visible for G frac12 1 The increase of theCHJ diameter is apparent (Dj = 256 mm Q = 993 ml s iexcl 1 h 1 = 4 mm)

is negligible For this situation we seek to determine how D h scales with parameterswhen viscosity dominates over gravity in the CHJ

The counterpart of equation (12) when drag at the plate in the transition zoneof the CHJ balances reguid inertia is obtained by a force balance around the controlvolume shown in shy gure 1 as

r2j Uj

2rh

21

hiexcl 1

h 1

rh

rh + Lj

ordm 1

2

Lj(2r h + Lj)

raquo r h

frac12 w (41)

For the wall shear stress we use a rough scaling for a Newtonian reguid as a meanvalue between the velocity gradients at the wall upstream and downstream of theCHJ

frac12 w = middotVr

zordm 1

2middot

Vr

z rh

+Vr

z rh + Lj

ordm 12middot

Vr

h+

Vr1

h 1 (42)

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The circular hydraulic jump in low gravity 2143

50

0

10

20

30

40

12

time (s)

2rh

(mm

)

G = 1 G ltlt 1

- 02 0 02 04

CHJ first appears

06 08 1

Q = 632 ml s - 1 h yen = 15 mm

2rh (G ltlt 1)

thinning ofupstream region

Figure 15 Evolution of the CHJ diameter (2rh ) for conditions when a CHJ did not existat normal gravity but was formed after reducing G (h 1 = 15 mm Q = 632 ml s iexcl 1 andDj = 122 mm) The CHJ diameter is transient throughout the period of the experiment whichis 12 s A CHJ diameter for G frac12 1 was averaged over the measurements from 400 ms to 600 msinto the period of low gravity

With mass conservation the velocity components in the above equation are elimi-nated to give a relationship between the shear stress and parameters as

frac12 w = 14middot

Ujr2j

rh H2

plusmn1 +

h2

h21

r h

r h + Lj

sup2 (43)

Substituting equation (43) into equation (41) and expressing the result in non-dimensional form gives

1 iexcl H

H 1

D h

D h + 2middotLj

=middotLj(D h + middotLj)

HRej

plusmn1 +

H2

H21

D h

D h + 2middotLj

sup2 (44)

It is reasonable to assume that H=H 1 frac12 1 and if also D h frac34 middotLj then equation (44)simplishy es to

D h ordm HRej=middotLj (45)

To complete the scaling and arrive at the desired result we need to know how theupstream shy lm thickness H depends on D h Middleman (1995) has shown that anenergy balance in the upstream shy lm in which the reguid kinetic energy is balanced byviscous dissipation when gravity is negligible yields

H =1

2D h

1

K(1 iexcl (8=Rej) copy (D h H)) (46)

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2144 C T Avedisian and Z Zhao

60

0

20

40

16

h yen (mm)

2rh

(mm

)

0 2 4 6 8 1210 14

Q = 632 ml s- 1

G ltlt 1G = 1

no jump observed at G = 1

Figure 16 Variation of CHJ diameter with h 1 for a given deg ow rate (Q = 632 ml s iexcl 1 andDj = 122 mm) The normal-gravity data follow the trend of equation (14) but the low-gravitymeasurements show no clear variation of the CHJ diameter with downstream deg uid height

where K is a factor (unity for a frictionless reguid) which depends on the form ofthe velocity proshy le in the upstream shy lm and copy is a dissipation function (zero for africtionless reguid) Because copy depends only on D h and H equation (46) implies arelationship for H as H = f1(Rej D h K) which when combined with equation (45)implies an equation for D h in terms of parameters as

D h = f2(Rej middotLj K) (47)

Equation (47) does not contain the downstream shy lm height H 1 Thus we see thatif by reducing gravity viscous enotects dominate then the dependence of the CHJdiameter on H 1 is removed It is thus not surprising that D h shows no dependenceon h 1 in low gravity which is consistent with the data shown in shy gure 16

The radius of curvature across the jump and downstream regow pattern are alsoinreguenced by G The enotect is again traced to the hydrostatic pressure in the down-stream shy lm Concerning the radius of curvature surface tension exerts a force whichpresses on the free liquid surface and the radius of curvature increases to balancethe lower hydrostatic pressure brought about by reducing G Concurrently there isa more gradual transition from the upstream to the downstream regow as seen in theside-view photographs in shy gure 8 and the digitized images in shy gure 9 Along withthe increase in curvature across the CHJ and a more gradual transition to the down-stream regow is a lengthening of the transition scale Lj in shy gure 1 This is evident bycomparing shy gures 5 and 7 for G = 1 (shy gure 5) and G = 002 (shy gure 7) The increaseof Lj brought about by decreasing G is consistent with equation (16) which wasdeveloped for planar jumps

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The circular hydraulic jump in low gravity 2145

t = 0 s G = 1 t = 0 s G = 1

t = 02 s G ltlt 1 t = 02 s G ltlt 1

t = 04 s G ltlt 1 t = 04 s G ltlt 1

(b) (e)

(a) (d)

(c) ( f )

40 mm

Figure 17 Photographs showing the development of the CHJ for the lowest deg ow rate examinedQ = 239 ml s iexcl 1 For h 1 = 2 mm (a)(c) circular waves are visible in the downstream deg ow atnormal gravity (a) showing that these waves are not unique to the low-gravity environmentFor h 1 = 4 mm (d)(f ) the CHJ diameter has moved very close to the jet and the downstreamdeg ow shows considerable agitation (d) When gravity is reduced ((d) (e)) the downstream distur-bances cease and give way to the regular pattern of circular waves shown in (f ) (Dj = 122 mmQ = 239 ml s iexcl 1 )

For all of the regow conditions at low gravity waves in the downstream regow wereobserved at low gravity as shown in shy gures 7 10c f 14 and 17 However only forthe smallest downstream depth 2 mm were they seen at normal gravity as shownin shy gure 17a Several mechanisms could promote formation of these waves Theseinclude sudden reduction of the hydrostatic pressure brought about by dropping the

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2146 C T Avedisian and Z Zhao

(a)

(b)

downward thrustas G is reduced

rh(G = 1) Lj(G = 1) Lj(G ltlt 1)

rh(G ltlt 1)

Lj

hump

hyen

h yendeflected flow

vortex

hump

Figure 18 Schematic illustration of deg ow situations in the transition zone that could create wavesin the downstream deg ow (a) with rapid reduction of G surface tension pulls downward on thesurface which increases the radius of curvature and increases the transition length (b) deg ow pasta wall vortex

package with the result described above that the force due to surface tension at thefree surface exerts a downward thrust across the transition zone or separation andrecirculation in the downstream shy lm causing the oncoming regow to be deregected awayfrom the target plate There are several mechanisms which can create waves in theregow downstream of a CHJ These include surface tension and formation of a vortexin the transition zone

When G is suddenly reduced surface tension will pull downward on the free surfaceand create ripples that propagate outward Figure 18a is a schematic illustration ofhow this might be viewed The close-up photographs of shy gures 57 show developmentof such a wave at the CHJ boundary The hump just downstream of the CHJ inshy gure 6 400 ms after low gravity begins is the shy rst evidence of it and shy gures 710b e and 14 show additional photographs Figure 10b e also shows a single largewave as it is just moving beyond the CHJ boundary

A vortex which forms in the transition zone can deregect the regow away from thewall Figure 18b illustrates this situation schematically The enotect of varying h 1on vortex formation and the consequences it has on the transition regow structurefor normal-gravity CHJs have been discussed by Liu amp Lienhard (1993b) At normalgravity aspects such as regow reversal surface rollers and separated eddies are observedto form if the hydrostatic pressure is considered to increase when h 1 is increasedWithin low G the hydrostatic pressure in the downstream regow is already small sothat the regow structure in the transition should be similar to the smooth-jump low-h 1 cases discussed previously (Liu amp Lienhard 1993b) In fact all of the low-gravityconditions examined here showed a smooth transition across the CHJ

Associated with formation of waves in the regow downstream of a CHJ at low gravityare the wave speed and wavelength Inviscid deep-water theory is a useful perspec-tive from which to understand the mechanisms which determine these quantities

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The circular hydraulic jump in low gravity 2147

Waves for which surface tension controls propagation (ie `ripplesrsquo) have a smallerwavelength than `gravityrsquo waves from the inviscid theory (Milne-Thompson 1979) Inlow gravity if surface tension dominates we would then expect that para G frac12 1 lt para G = 1The evidence for this trend in our experiments is circumstantial because we couldnot measure the wavelength with sumacr cient accuracy However we see from shy gure 8athat the free surface is not as wavy at G = 1 as at G frac12 1 as also shown in shy gures 7and 8d e From this we conclude that wave propagation in the downstream regow atlow gravity is most probably controlled by surface tension

For large downstream reguid shy lm thicknesses the response time of the CHJ to thereduction in gravity was longer than the experimental run time of 12 s Figure 15shows an example of the transient response of the CHJ diameter for h 1 = 15 mmThe reason for the long adjustment time of the CHJ for the largest h 1 examined maybe traced to the characteristic response time of the reguid to a sudden disturbanceThis time is of the order of ( macr 2=cedil ) where macr is a characteristic reguid length-scaleThough a value for macr is open to question if we assume that h 1 is a reasonablecharacteristic length then increasing the downstream reguid height by a factor of twowould quadruple the response time The largest heights examined showed a transientprocess that was longer than the experimental time (eg shy gure 15) in contrast tothe smaller values of h 1 in which the CHJ adjusted to the sudden change in G overtimes which were shorter than the experimental time (eg shy gures 12 and 13)

(c) Comparison of measured CHJ diameter with theory

The availability of theories for the CHJ motivated us to compare our measure-ments for steady-state regow conditions with predictions from one of them The mostelementary analysis of a CHJ neglects viscosity as noted in x 1 Equations (12)(14)result from making this assumption The next level of complexity is to include vis-cous enotects Watson (1964) was one of the shy rst to do this A boundary layer modelpredicts the upstream velocity proshy le which was found to be self-similar at someradial distance away from the stagnation point This theory has the essential fea-tures of more advanced treatments (see for example Bowles amp Smith 1992 Higuera1994) and is applied to the present data

Watson (1964) considered two situations a fully developed boundary layer (r gt r0

in shy gure 1) and a developing proshy le for the regow in the upstream thin shy lm r lt r0

(shy gure 1) The anslysis assumes that h21 frac34 h2 it ignores viscous enotects in the

downstream regow and a similarity solution is derived to the boundary layer equationsfor the regow in the thin shy lm upstream of the CHJ Middleman (1995) expressed thesolution in terms of the following non-dimensional variables

Rej sup2 4Q

ordm Dj cedil d h sup2 D h Re

iexcl1=3j middotH sup2 H 1 Re

iexcl1=3j Y sup2 d h

middotH

Frj

+1

2d hmiddotH

In terms of Y and dh the results are as follows

Y =026

d3h

+ 0287(48)

when r lt r0 (`developingrsquo regow) and

Y = 1 iexcl 102d3=2h (49)

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2148 C T Avedisian and Z Zhao

10

00001

0001

1

01

001

10

Y

01 1

dh

eqn (48)

G = 1

eqn (49)

G ltlt 1

inviscid limit

Figure 19 Comparison of the CHJ measurements with the theory of Watson (1964)cast in the variables of Middleman (1995) (equations (48) and (49))

when r gt r0 (`fully developedrsquo regow) The inviscid limit is d h 0 and Y = 1 inequation (49) (remembering that the approximation h2

1 frac34 h2 was applied) Y andd h are known from the experimental results 2r h (the CHJ diameter) is measuredand h 1 and Q are control parameters (Uj = Q=[ 1

4ordm D2

j ]) of the experiment For thewater kinematic viscosity at 300 K cedil ordm 895 pound 10iexcl7 m2 siexcl1 (Keenan et al 1969)An average G of 002 is used over the free-fall distance

Figure 19 shows the variation of Y with d h The inviscid limit does not predictour measurements The measurements are bounded by the developing regow (equa-tion (48)) and fully developed regow limit (equation (49)) for both the normal- andlow-gravity data The normal-gravity measurements are reasonably well predictedby assuming the regow to be developing in the upstream shy lm in that the boundarylayer (shy gure 1) has not yet reached the free surface On the other hand the low-gravity data shown in shy gure 19 are closer to the fully developed regow prediction ofequation (49) though still not quantitatively predicted by that limit This generalresult is consistent with the increased upstream regow length (ie larger CHJ diame-ter) when gravity is lowered With the larger regow length at low gravity the regow inthe upstream thin shy lm has a chance to become more developed The fact that thelow-gravity data are bounded by the developing and fully developed limits showsthat the analysis which leads to equations (48) and (49) is a reasonable model forthe conditions of our experiments

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The circular hydraulic jump in low gravity 2149

The agreement shown in shy gure 19 could be viewed as unusually good given whatis not included in the analysis the CHJ is treated as a discontinuity in the analysiswhereas in reality the transition to the downstream regow occurs over a shy nite lengthwhich is more extended in low gravity there are waves in the downstream regow theregow downstream of the CHJ may separate and form a circulation eddy and vorticitygeneration at the CHJ is not included Some of these aspects have been consideredindividually in prior studies as outlined in x 2 Including all of them at once is achallenge for future work Finally if G were identically zero analyses by Thomas etal (1990) and Rahman et al (1990) as well as equation (14) predict that a CHJwould not form While we do not achieve this limit in our study (G ordm 002 as notedpreviously) the larger CHJ diameter at low gravity compared with normal gravityis consistent with these predictions

5 Summary

The results are summarized as follows

(1) A steady CHJ can be established in low gravity (G 6= 0) and the CHJ diameterat low gravity is larger than at normal gravity for the same jet conditions

(2) The response time of the CHJ to a sudden transition in gravity is less than200 ms when a jump exists at normal gravity for the regow conditions examined

(3) The curvature of the free surface across the CHJ decreases when gravity isreduced and the transition to the downstream regow is more gradual than atnormal gravity

(4) Suddenly reducing gravity produces a large wave or hump just downstream ofthe transition zone across the CHJ which propagates into the downstream shy lmand concentric waves form in the downstream regow at low gravity

(5) The variation of the CHJ diameter with downstream shy lm thickness at normalgravity qualitatively (but not quantitatively) follows the trend of the inviscidtheory at normal gravity but the trend is not followed for the low-gravitymeasurements

(6) Measured CHJ diameters are bounded by the two limits of fully developedregow and developing regow with the normal-gravity measurements reasonablywell predicted by assuming developing regow in the upstream shy lm and the low-gravity measurements falling between the two limits

We thank Jun-Eu Tang and Hanwook J Kim of Cornell for helping with the experimentsand Jack Salzman (NASA) for drawing the authorsrsquo attention to Labus (1976) This study wassupported in part by NASA under grant NAG 3-1627 (Bradley Carpenter Program Director andDavid Chao Project Monitor) and by the Semiconductor Research Corporation (task 96-004)

Nomenclature

Bo Bond number (sup2 raquo Gg0h 1 rj=2frac14 )Dj orishy ce diameter shy gure 1

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2150 C T Avedisian and Z Zhao

D h sup2 r h =rj

d h sup2 D h Reiexcl1=3j

Fr Froude number based on average liquid shy lm velocity just before CHJ(sup2 U 2=Gg0h)

Frj Froude number based on average velocity of impinging jet(sup2 U 2

j =Gg0h 1 )

g local gravitational acceleration within free-fall packageg0 Earthrsquos normal gravity (= 98 m siexcl2)G sup2 g=g0

h 1 liquid shy lm thickness immediately downstream of the CHJ (see shy gure 1)h liquid shy lm thickness immediately upstream of the CHJ (see shy gure 1)H sup2 h=rj

H 1 sup2 h 1 =rj

middotH sup2 H 1 Reiexcl1=3j

Lj length of jump (shy gure 1)middotLj Lj=rj

P reguid pressure in liquid shy lm immediately upstream of the CHJP1 reguid pressure in liquid shy lm immediately downstream of the CHJQ volumetric regow rateRej Reynolds number (sup2 2Q=ordm rj cedil )rj radius of orishy ce (= Dj=2)r h radius of the CHJU average velocity of liquid shy lm immediately upstream of the CHJUj average velocity of jet at orishy ceVr1 local velocity within liquid shy lm immediately downstream of the CHJ

(see shy gure 1)

Vr local velocity within liquid shy lm immediately upstream of the CHJ (seeshy gure 1)

Y sup2 (d hmiddotH=Frj) + (1=2d h

middotH)z coordinate normal to the target platemicro radius of curvature of the free surface in the transition zone across the

CHJ (see shy gure 1)

middotmicro sup2 micro=rj

para length of waves in the downstream regowraquo liquid densityfrac14 surface tensioncedil liquid viscosity

References

Allen T amp Ditsworth R L 1972 Fluid mechanics p 291 New York McGraw-Hill

Avedisian C T Yang J C amp Wang C H 1988 On low gravity droplet combustion Proc RSoc Lond A 420 183200

Bowles R I amp Smith F T 1992 The standing hydraulic jump theory computations andcomparisons with experiments J Fluid Mech 242 145168

Proc R Soc Lond A (2000)

on November 10 2014rsparoyalsocietypublishingorgDownloaded from

The circular hydraulic jump in low gravity 2151

Bouhadef M 1978 Etalement en couche mince drsquo un jet liquide cylindrique vertical sur un planhorizontal Z Angew Math Phys 29 157167

Chaudhury Z H 1964 Heat transfer in a radial liquid jet J Fluid Mech 20 501511

Craik A D D Latham R C Fawkes M J amp Gribbon P W F 1981 J Fluid Mech 112347362

Errico M 1986 A study of the interaction of liquid jets with solid surfaces PhD thesis Universityof California San Diego

Hager W H 1993 Classical hydraulic jump free surface promacrle Can Jl Chem Engng 20536539

Higuera F J 1994 The hydraulic jump in a viscous laminar deg ow J Fluid Mech 274 69

Hornung H G Willert C amp Turner S 1995 The deg ow macreld downstream of a hydraulic jumpJ Fluid Mech 287 299316

Keenan J H Keys F G Hill P G amp Moore J G 1969 Steam tables p 114 Wiley

Labus T L 1976 Liquid jet impingement normal to a disk in zero gravity PhD thesis Universityof Toledo

Liu X amp Lienhard J H 1993a Extremely high heat deg uxes beneath impinging liquid jets JHeat Transfer 115 472476

Liu X amp Lienhard J H 1993b The hydraulic jump in circular jet impingement and in otherthin liquid macrlms Exp Fluids 15 108116

Middleman S 1995 Modeling axisymmetric deg ows ch 5 Academic

Milne-Thompson L M 1979 Theoretical hydrodynamics 5th edn p 448 London Macmillan

Nirapathdongporn S 1968 Circular hydraulic jump MEng thesis Asian Institute of TechnologyBangkok Thailand

Rahman M M Faghri A amp Hankey W L 1990 New methodology for the computation ofheat transfer in free surface deg ows using a permeable wall Numer Heat Transfer B 18 2341

Streeter V L Wylie E B amp Bedford K W 1998 Fluid mechanics 9th edn pp 372 610612New York McGraw-Hill

Tani I 1949 Water jump in the boundary layer J Phys Soc Jap 4 212215

Thomas S Hankey W L Faghri A amp Swanson T 1990 One-dimensional analysis of thehydrodynamic and thermal characteristics of thin macrlm deg ows including the hydraulic jumpand rotation J Heat Transfer 112 728735

Watson E J 1964 The radial spread of a liquid jet over a horizontal plane J Fluid Mech 20481499

Womac D J Ramadhyani S amp Incropera F P 1993 Correlating equations for impingementcooling of small heat sources with single circular liquid jets J Heat Transfer 115 106115

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