18th National & 7th ISHMT-ASMEHeat and Mass Transfer Conference

January 4-6, 2006Paper No: IIT Guwahati, India

Heat Transfer by Impingement of Circular Free-Surface Liquid Jets

John H. Lienhard VDepartment of Mechanical EngineeringMassachusetts Institute of Technology

Cambridge MA 02139-4307 USAemail: lienhard@mit.edu

Abstract

This paper reviews several aspects of liquid jet im-pingement cooling, focusing on research done in ourlab at MIT. Free surface, circular liquid jet are con-sidered. Theoretical and experimental results forthe laminar stagnation zone are summarized. Tur-bulence effects are discussed, including correlationsfor the stagnation zone Nusselt number. Analyti-cal results for downstream heat transfer in laminarjet impingement are discussed. Splattering of turbu-lent jets is also considered, including experimental re-sults for the splattered mass fraction, measurementsof the surface roughness of turbulent jets, and uni-versal equilibrium spectra for the roughness of tur-bulent jets. The use of jets for high heat flux coolingis described briefly.

Nomenclature

A2n constants in Eq. (5).B dimensionless velocity gradient, 2 d

uf

due

dr

Cc contraction coefficient for liquid jets, jet cross-sectional area divided by nozzle area.

Cf skin friction coefficient.d jet diameter, fully contracted (m).

G(Pr) boundary layer function of Prandtl number,Eq. (12).

h local heat transfer coefficient, qw/(Tw − Tf ),(W/m2 K).

h(r) thickness of axisymmetric liquid sheet (m).k thermal conductivity of liquid (W/m·K), or rms

surface roughness (m).k∗ dimensionless surface roughness, k/d.

l distance between nozzle and target plate (m).Nud local Nusselt number based on jet diameter,

qwd/k(Tw − Tf ).p local pressure in liquid (Pa).

p∞ ambient pressure (Pa).pstgn stagnation pressure (Pa).pe(r) pressure distribution along the wall (Pa).P2n Legendre function of 2n order.Pr Prandtl number of liquid.

Q volume flow rate of jet (m3/s).Qs volume flow rate of splattered liquid (m3/s).qw wall heat flux (W/m2).r radius coordinate in spherical coordinates, or

radius coordinate in cylindrical coordinates (m).rh radius at which turbulence is fully developed

(m).ro radius at which viscous boundary layer reaches

free surface (m).rt radius at which turbulent transition begins (m).r1 radius at which thermal boundary layer reaches

free surface (m).Red Reynolds number of circular jet, ufd/ν.

T liquid temperature (K).Tf temperature of incoming liquid jet (K).Tw temperature of wall (K).Tsf liquid surface temperature (K).u, v liquid velocity components in radial, axial di-

rection of cylindrical coordinates (m/s)u′ rms turbulent velocity fluctuation in jet (m/s).uf bulk velocity of incoming jet (m/s).uh velocity of liquid sheet averaged across thick-

ness h (m/s).um free surface velocity of liquid sheet (m/s).

ue(r) radial velocity just above boundary layer (m/s).Vmax centerline velocity of incoming jet (m/s).Wed jet Weber number, ρu2

fd/σ.

Greek Letters

α thermal diffusivity of liquid (m2/s).δ, δt momentum, thermal boundary layer thickness

(m).δrms local rms surface roughness of turbulent jet

(m).η dimensionless wavenumber.θ polar angle of spherical coordinates.ν kinematic viscosity of liquid (m2/s).ξ fraction of impinging liquid splattered, Qs/Q.ρ density of liquid (kg/m3).σ liquid-gas surface tension (N/m).φ velocity potential (m2/s).

1

Figure 1: Laminar impinging jet: Red = 51, 000,d = 5.0 mm, sharp-edged orifice, adiabatic target.

1. Introduction

Liquid jet impingement cooling offers very lowthermal resistances and is relatively simple to imple-ment. Liquid jets are easily created using a straighttube or a contracting nozzle, and this nozzle canbe aimed directly toward the region of a heat load.When the jet strikes the target surface, it forms avery thin stagnation-zone boundary layer which of-fers little resistance to heat flow. Convective heattransfer coefficients can reach tens of kW/m2K.

These high heat transfer coefficients make liquidjet impingement attractive in situations where a highheat load must be removed while maintaining a min-imum temperature or temperature difference withinthe system. For example, in some semiconductorlaser systems, junction temperatures must be heldbelow 150◦C while heat loads may reach 10 MW/m2.Much attention has been given to jet impingementcooling of electronics.

An impinging jet defines its own flow field, oftenwithout the need for added channeling or target mod-ifications. Jets are particularly useful when coolingsystems must not add interfering hardware or makestructural changes to the cooled object. For exam-ple, a fixed nozzle at the end of a processing linecan cool each successive item passed under it; and insome automotive engines, oil jets cool the undersidesof the piston crowns.

Liquid jets can also carry extremely high heatfluxes, if the velocities are such as to produce a highstagnation pressure. Small diameter water jets atspeeds near 130 m/s have removed heat loads of up to400 MW/m2. Liquid jets are well-suited for coolingvery localized, high-flux heat sources.

The jets of interest in the present article are un-submerged jets, those that travel through a gas be-tween the nozzle and the target (Fig. 1). These jetsare only similar to submerged jets in the stagnationregion, and then only when the submerged jet is lessthan about five diameters in length.

2. Stagnation Zone Theory

Near the point of impact, an impinging jet’s fluid

Figure 2: Impinging jet configuration.

flow and heat transfer characteristics are described ingeneral terms by the usual results for the stagnationzone. The flow field can be divided into an outerregion of essentially inviscid flow and an inner viscousboundary layer region.

The analytical solution of the stagnation zoneboundary layer is a classical problem [1, 2, 3], whoseresults depend primarily on the radial velocity gra-dient of the inviscid flow near the stagnation point.To adapt stagnation zone boundary layer results toimpinging jets, this gradient must be determined.Thus, analysis of the stagnation zone requires firsta solution for the inviscid flow of the jet, and thenapplication of the classical boundary solutions for theflow and temperature fields. Together, these lead toexpressions for the wall heat flux and the Nusseltnumber.

2.1 Inviscid Outer FlowThe inviscid flow field of an impinging jet is de-

termined by: a free streamline boundary condition atthe liquid surface; an impermeable wall onto whichthe jet impacts; and assumed forms of the inlet andoutlet velocity profiles (Fig. 2). If the inlet profileis irrotational (e.g., uniform), the velocity field canbe obtained using potential flow theory (∇2φ = 0);otherwise, the Euler equations must be solved.

In all cases, the stagnation-zone flow has radialvelocity distribution at the wall given by

ue(r) = Cr + · · · , r → 0 (1)

for C = (due/dr|r=0) a constant radial velocity gra-dient. This result is necessitated by the kinematicsof any stagnation zone (for irrotational flow, see [4]);the constant C, however, depends on the specific in-viscid flow considered. For later use in heat transferanalyses, it is convenient to nondimensionalize thewall gradient:

B ≡ 2d

uf

due

dr

∣∣∣∣r=0

(2)

2

The inviscid flow near the stagnation point has awall pressure distribution given by

pe(r) = p∞ +ρ

2(u2

f − ue(r)2)

(3)

= pstgn − ρ

2ue(r)2 (4)

for p∞ the ambient pressure and pstgn the stagnationpressure. If the jet is nonuniform, uf refers to thecenterline velocity of the jet away from the target;if surface tension pressure is significant, only Eq. 4applies [5]. Measurements of the wall pressure dis-tribution have often been used to determine ue(r).

The potential flow is independent of Reynoldsnumber and scales with the inlet speed and jet diam-eter. Low levels of turbulence in the incoming jet arelikely to have only slight effects on the mean velocitydistribution outside the wall boundary layer, so theinviscid solutions should apply to either laminar orturbulent jets, if those jets have the specified inlet ve-locity profiles. In addition, the boundary conditionson the free streamline (no shear stress, pressure con-stant at p∞) will apply for steady jets of any density;thus, solutions that have been obtained in the con-text of nonmixing gas jets (no entrainment of sur-rounding fluid) apply equally well to unsubmergedliquid jets. The similarity between submerged jetsand free liquid jets obviously fails once shear layer in-stabilities at the freestreamline cause the submergedjet to begin mixing with the surrounding fluid.

2.2 Uniform velocity-profile circular jetsJets of uniform profile are typical of those cre-

ated by a sharp-edged orifice several diameters down-stream of the outlet and at any Reynolds numberwell above unity. Analysis and experiments on im-pingement of such jets date from the late 1920’s [6].Schach [7, 8] obtained approximate solutions for thevelocity and experimental measurements of wall pres-sure; the solutions and pressure data agreed reason-ably well in the sense of Eq.(3). Subsequent analyti-cal solutions for normal impingement were found byShen [9] and by Strand [10].

More recently, the problem has been revisited indetail by Liu, Gabour, and Lienhard [5], who ex-panded the velocity potential, φ, in a series of Leg-endre polynomials and obtained numerical solutionsfor the coefficients, incorporating the liquid surfacetension. Their result was

φ(r, θ)(2ufd)

=∞∑

n=0

A2n

( r

2d

)2n

P2n(cos θ) (5)

for spherical coordinates (r, θ) with origin at the stag-nation point and polar axis along the axis of thejet. Values of the coefficients are given in Table 1.The velocity along the wall just outside the boundarylayer, ue(r) = ∂φ/∂r|θ=π/2, is then

ue(r)uf

=∞∑

n=1

2nA2n

( r

2d

)2n−1

P2n(0)

= (−A2)( r

2d

)+(

3A4

2

)( r

2d

)3

+ · · ·(6)

Table 1: Coefficients of velocity potential [5].

Wed ∞ 50 25 16.7A2 −1.831 −1.881 −1.944 −2.015A4 2.365 2.858 3.469 4.213A6 0.5906 −0.01553 −0.8006 −1.825A8 −14.81 −19.03 −24.30 −30.78A10 13.35 20.42 29.38 40.61A12 50.74 68.74 91.31 119.3

Table 2: Velocity gradients at the stagnation pointduring laminar circular jet impingement.

Investigators B/2 profile l/d Wed

Schach [8] ≈ 0.88 uniform 1.5 ∞Shen [9] 0.743 uniform 1.5 ∞Strand [10] 0.903 uniform 1.0 ∞Liu, Gabour, 0.916 uniform 1.0 ∞Lienhard [5] 0.981 uniform 1.0 50

1.06 uniform 1.0 251.16 uniform 1.0 16.7

Scholtz and 4.69 parabolic 0.05 ∞Trass [11] to 0.5

From this, the dimensionless stagnation-point veloc-ity gradient is

B = −A2 = 1.831 (7)

for infinite Weber number; lower Weber number val-ues are shown in Fig. 3. Values of B from the variousinvestigations are compared Table 2.

The velocity and pressure distributions along thewall are shown in Fig. 4. The velocity distributionnear the stagnation point can be approximated by alinear distribution

u

uf=

B

2r

d= 0.916

r

d, (8)

and the pressure distribution can be estimated usingthis result and Eq. (3) to within about 10% for r/d <0.5. Heat transfer data suggest that the stagnationzone can reasonably be approximated to extend asfar as r/d ≈ 0.7 from the actual stagnation point.For r/d > 0.7, boundary layer growth must be takeninto account.

2.3 Parabolic velocity-profile circular jetsJets of parabolic profile are created by a laminar

flow issuing from a long circular tube at Reynoldsnumbers below 2000–4000. The parabolic profile willdiffuse toward a uniform velocity profile as the jettravels to the target, if the jet is long enough for vis-cosity to act; if the nozzle is within a few diameters ofthe target, the parabolic distribution should persist.

Inviscid analytical solutions for this case were ob-tained by Scholtz and Trass [11] for nozzle-to-target

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Figure 3: Inviscid stagnation-point velocity gradi-ent, B = 2(d/uf )(due/dr), including effect of Webernumber for uniform velocity profile [5].

Figure 4: Velocity and pressure variation along thetarget plate (y/d = 0) for an inviscid uniform profilejet at several values of Weber number [5].

spacings of 0.05 < l/d < 0.5. To leading order, thevelocity along the wall near the stagnation point forl/d = 0.5 is

ue = 2.323(

Vmax

d

)r, r/d < 0.4 (9)

for Vmax the maximum (centerline) velocity in thenozzle and d the nozzle diameter. To the accuracyof the numerical solutions, the constant in Eq.(9) isunaffected by l/d in the range 0.05 < l/d < 0.5.Scholtz and Trass also obtained experimental resultsfor velocity and pressure distributions over the range0.05 < l/d < 6.0. The experimental results showlittle sensitivity to l/d for l/d > 1. As the nozzleis moved closer than one diameter from the target,constriction alters the flow field downstream of thestagnation point; however, at the stagnation pointitself, the velocity gradient remains unaffected. Theexperiments showed that the stagnation-point pres-sure distribution was within 10% of that given byEqs.(3) and (9) for r/d < 0.4 and all nozzle spac-ings. Note that this is a slightly smaller stagnation

zone than is obtained with a uniform jet.In terms of bulk velocity, uf = Vmax/2, the di-

mensionless velocity gradient of the parabolic jet is

B = 2d

uf

due

dr

∣∣∣∣r=0

= 4.646 (10)

The experimental results generally support this equa-tion for 0.05 < l/d < 6. Most importantly, theparabolic-profile jet has a velocity gradient 2.5 timeshigher than the uniform jet, which raises the heattransfer coefficent substantially.

2.4 Laminar Stagnation-Zone Boundary LayerBoundary layer theory for the stagnation region issummarized in detail in [12]. As we have seen, ue

is linear in r, and the theory shows that the result-ing heat transfer coefficient, h, is independent of r.In other words, the thermal boundary layer has auniform thickness within the stagnation zone. Theuniform value of h also implies that uniform walltemperature and uniform heat flux boundary condi-tions produce an identical heat transfer coefficient.

From the theory, the Nusselt number depends onthe stagnation point velocity gradient as follows:

Nud ≡ hd

k= G(Pr) Re1/2

d

√B (11)

for Red = ufd/ν. The coefficient G(Pr) (discussed inmore detail in [12]) is a function of Prandtl numberwhich can be evaluated numerically. Curve fits canbe constructed within given ranges of Pr [13]. Forthe axisymmetric boundary layer:

G(Pr) ≈

⎧⎪⎨⎪⎩

√2Pr/π

1+0.804552√

2Pr/πfor Pr ≤ 0.15

0.53898 Pr0.4 for 0.15 < Pr < 3.00.60105 Pr1/3 − 0.050848 for Pr ≥ 3.0

(12)Other curve fits have occasionally been applied. Forexample, for 0.7 < Pr < 10: G(Pr) ≈ 0.54 Pr0.37.

Equations (11) and (12) provide theoretical ex-pressions for the stagnation-zone heat transfer co-efficient for any value of B. They apply to eitheruniform wall temperature or uniform heat flux.

3. Laminar Stagnation Zone NusseltNumbers

The laminar theory just described generally agreeswell with experimental results when turbulence iseliminated; however, it must be emphasized that tur-bulence in the impinging jet has been reported toincrease the heat transfer coefficient by 30–150% be-yond that predicted in the following equations.

For high speed jets that are fully contracted, thenozzle-to-target separation has been shown experi-mentally to be unimportant. Nozzle-to-target sepa-ration can be expected to influence laminar jets if:(i) a sharp-edged nozzle is placed too close to thetarget to complete its contraction, so that a uniformprofile is not achieved; (ii) if a tube nozzle is placed

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far enough from the target that viscosity diffuses theparabolic profile toward uniform profile (at the bulkvelocity); or (iii) if the jet velocity is low enough thatgravitational acceleration causes significant variationin jet speed and size.

3.1 Uniform velocity-profile circular jetsFor uniform velocity profile laminar jets, Liu et

al. [5, 14] find

Nud = 0.745 Re1/2d Pr1/3 (13)

This result is based on theory and on experimentswith cold water. It should apply for all Reynoldsnumbers above 300–400 and for liquids with Prandtlnumbers above 3 or so. The Nusselt number within10% of the above value for r/d < 0.7; at larger radii,the heat transfer coefficient decreases rapidly. Thetheoretical lead constant is 0.813, but in the aboveequation it has been adjusted downward (by 9%)to bring it into close agreement with experimentaldata taken in the range 25, 000 < Red < 130, 000using long cold water jets and sharp-edged orificenozzles of 6.3 mm and 9.5 mm diameter. This equa-tion was found to overpredict Nud for water jets ofdj = 2.5 mm by about 15%. A similar equationwas given by Nakoryakov et al. [15] for high Prandtlnumber situations.

For nozzle-to-target separations of less than a fewnozzle diameters, a sharp-edged orifice jet is not fullycontracted, and the velocity profile is not uniform.In that case, heat transfer is likely to be higher thanpredicted here.

Little data is available for lower Reynolds num-bers or for Pr < 1 or Pr > 10. For high Prandtl num-bers, the above equation may be used. For Prandtlnumbers near unity (0.15 < Pr < 3.0), the theoreti-cal results can be applied to obtain:

Nud = 0.729 Re1/2d Pr0.4 (14)

For Pr 1, as for liquid metal jets, theory yields:

Nud = 1.08 Re1/2d Pr1/2 (15)

Convective heat transfer coefficients for liquid metaljets are typically 3 to 8 times greater than those forwater jets of the same diameter size and speed. Suchjets may have value in applications for which veryhigh h is required; for example, jets of liquid galliumhave application to cooling high energy synchrotronx-ray components and particle accelerators [16, 17].

3.2 Parabolic velocity-profile circular jetsFor jets issuing from tubes long enough to pro-

duce a fully developed laminar flow, the velocity pro-file is parabolic if the Reynolds number is below thetransition value of 2000–4000. Theory and experi-mental data by Scholtz and Trass [11] show

Nud = 1.648 Re1/2d Pr0.361 (16)

This equation should apply for 1 < Pr < 10 andReynolds numbers ranging from 100–200 up to 2000–4000. This equation is in good agreement with subli-mation experiments for Pr = 2.45 and 500 < Red <1960, and it appears to be unaffected by nozzle-to-target separation for 0.05 < l/d < 6.1

The size of the uniform-h stagnation zone for thiscase is roughly r/d < 0.15, but the Nusselt numberis within 10% of the stagnation-point value for r/d <0.4; beyond this radius, the Nusselt number usuallydecreases sharply. An exception occurs when l/d ≤0.1, for which case Nud can actually increase withr/d as the edge of the nozzle is approached.

4. Turbulent Jets

The manifolding and piping systems that supplyliquid to nozzles are often turbulent, and unless thenozzle has a very high contraction ratio, this turbu-lence will be carried into the jet formed. A jet issuingfrom a fully developed tube flow without a termi-nating nozzle will also be turbulent if the Reynoldsnumber is above about 4000. Turbulent jets haveelevated heat transfer coefficients owing to both thedirect effect of freestream turbulence on the bound-ary layer and the more indirect effect of a nonuniformvelocity profile on the stagnation-point velocity gra-dient. The increases relative to laminar theory mayrange from 30% to 50%.

4.1 Velocity profile effectsIn contrast to laminar velocity profiles, which

typically vary from uniform to parabolic (with umax

up to 2uf ), the velocity profiles of turbulent jets willlikely vary between a uniform distribution and themildly nonuniform distributions typical of turbulentpipe flows. In the latter case, however, the center-line velocity may still be significantly greater thanthe bulk velocity. For example, at a Reynolds num-ber of 4000 in a circular tube, umax/uf = 1.27, andat Red = 105, umax/uf = 1.18.

Stevens et al. [18] made laser-doppler measure-ments of the radial velocity gradient for several tur-bulent flow nozzles located a distance of one nozzlediameter from a target. For a contoured, convergingnozzle, the gradient found was B ≈ 2.3. This par-ticular nozzle would be expected to have the mostnearly uniform velocity profile, and its stagnation-point gradient is near the uniform-profile theoreticalvalue of B = 1.831. On the basis of laminar theory,the difference in gradients would cause the contourednozzle’s Nusselt number to exceed that of a uniformprofile by about 12%. Corresponding measurementsfor a fully developed pipe nozzle showed B = 3.6 atl/d = 1 [19], well above the uniform value; this dif-ference would increase laminar heat transfer by 41%.One may conclude that variations in B among noz-zles can have significant effects on turbulent jet heattransfer while the nozzle-to-target spacing is small.

1Submerged air jets between 19 and 51 mm diameter wereused; l/d was small enough for these jets to behave as unsub-merged jets near the stagnation point.

5

However, all jets will approach a uniform veloc-ity profile at the bulk velocity when l/d increases,because viscosity tends to eliminate radial gradients.Stevens and Webb [20] found bulk velocity was typi-cally reached within about five diameters downstreamof the nozzle. For long jets, one might infer thatB → 1.83 (uniform profile) and that the observedincreases in heat transfer for long jets result fromturbulence effects.

4.2 Turbulence effectsThe stagnation-zone boundary layer is likely to

remain laminar at the jet Reynolds numbers of in-terest, but turbulence in the impinging jet will tendto disrupt this very thin boundary layer, raising theheat transfer coefficient. This effect is very well docu-mented for the stagnation zone of bodies in gas flows.For example, correlations for the stagnation-zone en-hancement have been made [21, 22], and the effecthas been clearly documented in gas-jet impingement[23], including time-resolved measurements of the tur-bulent heat flux [24]. This literature has been sum-marized by Vader et al. [25].

In contrast, few controlled studies of freestreamturbulence are available for Prandtl numbers aboveunity (or well below unity), and no correlations orquantitative theory for freestream turbulence effectsin those ranges of Pr are known to the author. Un-fortunately, many applications of impinging liquidjets involve these Prandtl number ranges: generalpurpose cold-water jets with 2 < Pr < 10; fluo-rocarbon jets with 10 < Pr < 70 for electronicscooling; oil jets with Pr > 100 for some electricaland machine-tool cooling applications; and liquid-gallium jets with Pr ≈ 0.026 for cooling acceleratoror x-ray components.

Most turbulent liquid jet experiments are basedon water jets with Prandtl numbers between 2.5 and9. The associated correlations have usually fit datato the form suggested by laminar theory, adjustingthe lead constant and Reynolds number exponent asnecessary. The Prandtl number exponent is gener-ally chosen on the basis of the laminar curve-fits toG(Pr). Thus, the independent effects of freestreamturbulence, Pr, and Re are lumped together in suchresults to produce simple engineering equations. Inconsequence, the available correlations for turbulentjet Nusselt number are likely to differ from designconditions if changes in the manifolding or nozzle ar-rangements increase or decrease the turbulence levelrelative to the experiments, or if the Prandtl numberrange is substantially different than the conditions ofthe experiments considered.

One well-defined turbulent nozzle is that of afully developed turbulent pipe flow, as issues from atube or channel of more than about 40 diameters inlength. Turbulence intensities for such channel flowsare generally 4 to 5% in the core of the flow; and ex-tensive experimental and theoretical characterizationis available [26]. Several investigators have adoptedthis nozzle as a standard for turbulent jets. Other

nozzles may be less turbulent than this if they have aa strong and well-contoured contraction at the out-let; however, if a nozzle has flow separation withinit, the turbulence level could be significantly higherthan for a pipe nozzle.

4.3 Nusselt Numbers for Turbulent JetsCircular jets issuing from long tubes will have

fully developed turbulent flow for Reynolds numbersabove a transition value of 2000–4000.

For 4000 < Red < 52000, Stevens and Webb [27]correlated

Nud = 1.51 Re0.44d Pr0.4(l/d)−0.11 (17)

to an average error of 15% and a maximum error of60%. For 16600 < Red < 43700, Pan et al. [28] rec-ommend the following correlation for nozzles whosestagnation-point velocity gradient is known:

Nud = 0.49 Re1/2d Pr0.4B1/2 (18)

These results are both based on cold water jets; thegiven Prandtl number exponents are assumed, butmight be expected to apply in general range of 1 to10. The stagnation zone, of constant Nud, is roughlyr/d < 0.7. For a fully developed tube nozzle at l/d =1, B ≈ 3.6, and the second equation becomes

Nud = 0.92 Re1/2d Pr0.4 (19)

for 16600 < Red < 43700 to an accuracy of about 5%.At higher Reynolds number, both Gabour and

Lienhard [29] and Faggiani and Grassi [30] report astronger dependence of Nusselt number on Reynoldsnumber. This may result from an increasing influ-ence of freestream turbulence; however, further evi-dence is needed to verify that conjecture. For 25,000< Red < 85, 000, Gabour and Lienhard obtain

Nud = 0.278 Re0.633d Pr1/3 (20)

based on experiments with cold water jets having8.2 < Pr < 9.1 and tube diameters between 4.4and 9.0 mm. The uncertainty of the data (at a 95%confidence level) is less than ±10%. Gabour foundthat the effect of changing l/d between 1 and 20 waswithin the uncertainty of the data. The assumedPrandtl number exponent is appropriate for Pr > 3.

Figure 5 shows the stagnation-zone Nusselt num-ber of jets froms from fully developed tube nozzlesfor 500 < Red < 105 and Pr = 8, as predicted byEqs. (16), (17), and (20). (Eq. 19 is quite closeto those shown, and l/d was set to 1 in Eq. 17).The Nusselt number for a sharp-edged orifice nozzle(Eq. 13) is also shown.

For liquid metal jets, the impact of turbulencemay be expected to be small, owing to the highmolecular conductivity relative to the turbulent eddydiffusivity. Lienhard [12] suggested the use of thelaminar flow result, Eq. 15, for such situations. Thisconjecture was recently confirmed experimentally bySilverman and Yarin [31] for turbulent gallium jetsin the range 40, 000 < Red < 110, 000.

6

Figure 5: Stagnation-zone Nusselt number for circu-lar liquid jets at Pr = 8.

4.4 Jets Impinging on Rough WallsThe very thin boundary layer of the stagnation

zone can be disrupted by even small levels of wallroughness. Gabour and Lienhard [29] measured thestagnation-point Nusselt number for fully developedturbulent cold-water jets impinging on surfaces ofvarying roughness. They found that roughness ofonly 28 µm rms height could raise the Nusselt num-ber by up to 50%.

For those experiments, mild steel (1010) surfaceswere scored in a cross-hatch pattern. The depth ofscoring was varied to yield a range of rms rough-ness, 4.7 ≤ k ≤ 28.2 µm, while holding the pat-tern of roughness fixed. Measurements were madefor nozzles of diameter d = 4.4, 6.0 and 9.0 mmand Reynolds numbers from 20,000 to 84,000. ThePrandtl number was held between 8.2 and 9.1. Rough-ness effects on the Nusselt number depended on Red

and the dimensionless roughness k∗ = k/d, as shownin Fig. 6. On these figures, the lowest curves (k∗ <0.0008) are effectively smooth walls; for higher k∗,the Nusselt number is greater than for a smooth sur-face, and the enhancement rises with Red.

Gabour and Lienhard constructed a curve fit forthe threshold level of roughness at which the Nusseltnumber is increased by 10%:

k∗ = 5.95 Re−0.713d (21)

For lower levels of roughness (smaller k∗), the surfacecan be viewed as smooth. Data for other values of Prare not available, but jets of higher Pr liquids wouldbe expected to be more susceptible to wall roughnesseffects, owing to the thinner thermal boundary layer.

The overall scaling of stagnation-point wall rough-ness is different than for standard turbulent bound-ary layers, owing to the nearly zero shear stress of thestagnation zone, and this scaling is as yet not known.

5. Local Heat Transfer Downstream

The flow downstream of the stagnation zone ischaracterized by growing boundary layers and a de-creasing heat transfer coefficient. The boundary lay-ers are laminar near the stagnation zone and undergo

Figure 6: Wall roughness effects on stagnation-zoneNusselt number for circular turbulent liquid jets atPr = 8–9 (data from [29]).

a turbulent transition further downstream. Transi-tion produces a local increase in the heat transfercoefficient, again followed by a declining trend. Ifthe incoming jet is turbulent, local h upstream oftransition can be increased, especially in the acceler-ating flow near the stagnation zone; and the locationof turbulent transition is moved upstream.

Axisymmetric jets spread thinner as they travelradially outward (Fig. 7). The flow perimeter in-creases as 2πr, and inviscid mass conservation re-quires the liquid sheet thickness to decrease as h(r) =d2/8r. This rapid decrease quickly brings the grow-ing boundary layer into contact with the surface ofthe liquid sheet, transforming the flow from a bound-ary layer into a high-momentum viscous film with aradially increasing bulk temperature. Thus, analy-sis and correlation of circular jet heat transfer be-comes complicated, owing to radial changes in theheat transfer mechanism. In addition, any incom-ing turbulence in the jet can greatly disturb the freesurface of the thin liquid sheet, causing splattering ofthe liquid and a very rapid transition to a turbulentfilm flow.

5.1 Flow fieldA laminar axisymmetric jet creates a radial film

that evolves as shown in Fig. 7. The thickness of theliquid sheet initially decreases with radius (as 1/r),but, because viscous drag slows the liquid sheet, itsthickness begins to increase at larger radii. The flowfield may be divided into successive regions:

1. The stagnation zone.

2. The laminar boundary layer region, in whichthe viscous boundary layer thickness, δ, is less

7

Figure 7: Downstream development of an axisym-metric impinging jet.

than the liquid sheet’s thickness, h(r), and therest of the liquid sheet moves at the incomingjet’s speed, uf , and nearly parallel to the wall.

3. The viscous similarity region, in which viscouseffects extend through the entire liquid film[δ = h(r)]. In this region, the surface speed,um, decreases with increasing radius.

4. The region of developing turbulence.

5. The region of fully turbulent flow. This regionmay relaminarize farther downstream, as thefilm speed decreases.

Theoretical solutions for the boundary layer andviscous similarity region were first obtained by Wat-son [32]. Watson’s analysis of the boundary layerregion led to a Blasius type similarity solution forδ < h(r). In the region where δ = h(r), Wat-son obtained an elegant similarity solution that en-compasses the entire thickness of the liquid sheet.Watson also obtained solutions for a turbulent filmflow and for the hydraulic jump; however, these so-lutions are somewhat unsatifactory, as discussed byLiu et al. [14] and by Liu and Lienhard [33]. Wat-son’s laminar results were experimentally substanti-ated by Azuma and Hoshino [34, 35, 36, 37] usinglaser-doppler measurements. Sharan [38] indepen-dently obtained approximate integral-method solu-tions for both regions, and Wang et al. [39] developeda series solution for the boundary layer region.

The various analytical studies are in relativelygood agreement with one another, in spite of minordifferences in the approximations they employ. Toreasonable accuracy, the integral-method results canbe used [14, 38, 40], as we now summarize.

The region of boundary layer behavior, with um(r)= uf , begins at r ≈ 2.23 d. In this region, the bound-ary layer thickness is approximately

δ = 2.679(

rd

Red

)1/2

(22)

and the velocity profile is approximately:

u(r, y) = um(r)[32

y

δ− 1

2

(y

δ

)3]

(23)

The viscous boundary layer reaches the surface of theliquid sheet at a radius ro given by:

ro = 0.1773 Re1/3d d (24)

Beyond ro, the free surface speed decreases as:

um(r) =15

uf d2

h(r) r(25)

where the liquid sheet thickness is

h(r) = 0.1713(

d2

r

)+

5.147Red

(r2

d

)(26)

For r > ro, the velocity profile may be obtained fromthe polynomial approximation (Eq. 23) with um fromEq. (25) and δ = h(r) from Eq. (26).

The radius of onset of turbulence, rt, was corre-lated by Liu et al. [14] as

rt

d= 1200 Re−0.422

d (27)

Liu et al. inferred from heat transfer data the radiusat which turbulence became fully developed, and cor-related it as:

rh

d= 28600 Re−0.68

d (28)

In the fully turbulent region, Azuma and Hoshino[36] found that a 1/7th power velocity distributionworked well. The skin friction coefficient there canbe approximated as [14]

Cf = 0.073(

r

d Red

)1/4

(29)

and the film thickness may be estimated as

h(r)d

=0.02091

Re1/4d

( r

d

)5/4

+ C

(d

r

)(30)

with

C = 0.1713 +5.147Red

rt

d− 0.02091

Re1/4d

(rt

d

)1/4

(31)

5.2 Heat transferExisting analytical solutions for heat transfer as-

sume negligible heat transfer from the liquid surface;in particular, evaporation is neglected. This situa-tion should prevail for a surface temperature, Tsf ,that remains relatively low. If evaporation from thefree surface becomes important, the theoretical ex-pressions for the Nusselt number will underpredict.

Heat transfer in the downstream region has beenmodelled by various investigators, including Chaud-hury [41], Nakaryakov et al. [15], Liu and Lienhard

8

[40], Wang et al. [39], and Liu et al. [14]. Chaud-hury used Watson’s laminar velocity profiles to ob-tain similarity solutions for the liquid temperaturefield along a uniform temperature wall. Wang etal. developed solutions in the boundary layer region(r < ro) that apply to arbitrary distributions of walltemperature and heat flux. Nakaryakov et al. [15]provide theoretical and experimental mass transferresults that are analogous to uniform wall tempera-ture and very high Prandtl number, based on a linearapproximation to the velocity profile within the con-centration boundary layer.

Most experimental data for downstream heat trans-fer are for uniform wall heat flux, and this case hasbeen modelled by Liu and Lienhard [40] and by Liuet al. [14]. Liu and Lienhard separated the laminarflow into several thermal regions for Pr > 1:

Region 1. The stagnation zone.

Region 2. δ < h region: The thermal and viscousboundary layers do not reach the surface; sur-face temperature and velocity, Tsf and um, equalinlet temperature and velocity, Tf and uf .

Region 3. δ = h and δt < h region: The viscousboundary layer has reached the free surface.The velocity outside the viscous boundary layerdecreases with radius, but the surface temper-ature remains at the inlet temperature, Tf .

Region 4. δ = h, δt = h, and Tw < Tsat region:In this region, the thermal boundary layer hasreached the surface. The surface temperatureincreases with radius.

In the laminar regions, Liu and Lienhard appliedthe integral energy equation

d

dr

∫ δt

0

ru(T − Tf ) dy =qw

ρcpr (32)

where T = T (r, y) is the liquid temperature pro-file. In region 2 (before the thermal boundary layerreaches the surface), they approximated the velocityprofile as Eq. (23) and the temperature profile as

T (r, y) − Tw = (Tf − Tw)

[32

y

δt− 1

2

(y

δt

)3]

(33)

for Tw the wall temperature, which increases withradius. While r < ro, the integral solution for thelocal heat transfer coefficient in the boundary layerregion is approximately equal to

Nud = 0.632 Re1/2d Pr1/3

(d

r

)1/2

(34)

which shows the local heat transfer coefficient to de-crease as 1/

√r.

Region 2 ends and region 3 begins where the vis-cous boundary layer reaches the film surface at ro.

In region 3, the integral solution is:

Nud =

0.407 Re1/3d Pr1/3 (d/r)2/3

[0.1713(d/r)2 + (5.147 r/Red d)]2/3[

12

(rd

)2 + C3

]1/3

(35)

where

C3 =0.267(d/r0)1/2[

0.1713 (d/r0)2 + (5.147 r0/Red d)

]2−12

(r0

d

)2

(36)Region 3 ends and region 4 begins where the thermalboundary layer reaches the liquid surface at r = r1;equations defining r1 are given by Liu and Lienhard[40]. In region 4:

Nud =0.25

1PrRed

[1 − ( r1

r

)2] ( rd

)2 + 0.130 h(r)d + 0.0371 h(r1)

d

(37)

where h(r) is given by Eq. (26) above.Region 4 will occur only for Pr less than a criti-

cal value near five2; otherwise, the thermal boundarylayer does not grow fast enough to reach the surfaceof the liquid film, which thickens with increasing ra-dius owing to viscous retardation. Liu et al. foundthat the predictions for regions 3 and 4 are within afew percent of each other, and they suggest that theprediction of region 3 be used for any Pr above unity.Regions 3 and 4 correspond to Watson’s self-similarviscous flow regime.

Liu et al. estimated the fully turbulent heat trans-fer using the thermal law of the wall, from which,using Eq. (29, 30), the Nusselt number is

Nud =qw d

k(Tw − Tf )=

8 RedPr f(Cf , Pr)49 (hr/d2) + 28 (r/d)2 f(Cf , Pr)

(38)In this equation,

f(Cf , Pr) =Cf/2

1.07 + 12.7(Pr2/3 − 1)√

Cf/2(39)

When Pr is well above unity, this simplifies to

Nud = 0.0052 Re3/4d

(d

h

)(d

r

)3/4

×(

Pr

1.07 + 12.7(Pr2/3 − 1)√

Cf/2

) (40)

Comparison of these equations to experimentaldata for a uniform profile laminar jet [14] is shownin Fig. 8, and agreement is seen to be very good. Liuet al. also give integral solutions for Pr < 1.

2Liu and Lienhard [40] gave this value as 4.86. If thehigher-order terms in the integral analysis are retained, thevalue becomes 5.23.

9

Figure 8: Comparison of laminar theory to data [14].

The theoretical solutions should be evaluated us-ing liquid properties at the local liquid temperature(averaged across the liquid thickness). In particular,the liquid viscosity varies strongly with temperaturefor most liquids and may decrease significantly withradius. To use these results in calculating Nud andTw(r), a numerical integration in the radial directionis most expedient [40].

6. Splattering Jets

Turbulence in an axisymmetric jet is carried intothe radially spreading liquid film, where it has twoprimary effects. First, the turbulence tends to in-crease convective heat transfer in the boundary layerdownstream of the stagnation zone and tends to pro-mote turbulent transition of the thin liquid sheet.The turbulent sheet has greater skin friction than alaminar sheet: Measurements of nonsplattering tur-bulent jets [20] show that the liquid surface speedbegins to drop at r ≈ 2.5d, sooner than predicted bylaminar flow theory.

The second effect of jet turbulence is to disturbthe surface of the incoming jet. These disturbancesare carried into the thinning liquid sheet, where theradial spreading can produce a strong increase intheir amplitude. If the initial disturbances are largeenough, the amplified disturbances can cause dropletsto break free from the liquid sheet, resulting in splat-tering (Fig. 9).

Splattering is more important when the jets arelonger and have a higher Weber number, becausethe disturbances reaching the sheet are then larger.Strong splattering can result in atomization of 30 to70% of the incoming liquid (Fig. 10), and becausethe airborne droplets no longer contribute to coolingthe wall, heat transfer far downstream is degraded.On the other hand, splattering has no independentinfluence on heat transfer in the stagnation region,because the droplets break away several diametersdownstream of it.

Figure 9: Splattering turbulent jet showing radiallytravelling waves: Red = 28, 000, ξ = 0.11, fully de-veloped tube nozzle.

Figure 10: Splattering turbulent jet: Red = 48, 300,ξ = 0.31, fully developed tube nozzle.

6.1 The mechanics of splatteringInitial studies of splattering, by Errico [42] and by

Lienhard et al. [43], demonstrated that it is drivenby the disturbances on the surface of the impingingjet. Thus, undisturbed laminar jets do not splatter,unless they are long enough to have significant cap-illary instability. Turbulent jets, on the other hand,develop surface roughness as a result of liquid-sideturbulent pressure fluctuations, and they are highlysusceptible to splattering.

Errico [42] induced splattering for laminar jets bycreating surface disturbances with a fluctuating elec-tric field; these disturbances were carried, as radialwaves, into the liquid film, causing splatter down-stream. When a turbulent jet strikes a target, simi-lar travelling waves originate near the impingementpoint and travel outward on the film (see Fig. 9).When the jet disturbances are sufficiently large, thesewaves sharpen and break into droplets. All observa-tions indicate that the amplitude of the disturbanceson the jet governs splattering. They further indi-cate that splattering is a non-linear instability phe-nomenon, since the liquid film is clearly stable tosmall disturbances but unstable to large ones [44].

Once droplets break away from the liquid sheet,they remain airborne. Lienhard et al. [43] made

10

Figure 11: Measurement of the splattered flow rate.

phase-doppler measurements of the size and speed ofthe departing droplets and found that the dropletstravel at a low angle with nearly the speed of the in-coming jet; thus, they do not fall back into the liquidfilm at an appreciable rate. Their measurements, andstroboscopic observations by Errico [42], show thatdroplets leave the liquid sheet within a fairly narrowradial band. The radius of departure was estimatedto be r/d ≈ 4.5.

6.2 The fraction of liquid splatteredMeasurements of the splattered liquid flow rate

for fully turbulent turbulent jets were first reportedby Lienhard et al. [43] in the form of the ratio ofsplattered flow rate, Qs, to the incoming flow rate,Q (Fig. 11):

ξ =Qs

Q(41)

This quantity is easily measured by capturing the un-splattered liquid and determining its flow rate. Lien-hard et al. also proposed that splattering dependsprimarily on the rms amplitude of jet surface distur-bances. In this model, turbulent pressure fluctuationin the bulk liquid jet create the initial surface distur-bances near the nozzle outlet, which then grow in thestreamwise direction.

The growth of the surface disturbances dependson l/d and the jet Weber number,

Wed = ρu2fd/σ. (42)

In Fig. 12, the amount of splattering, ξ, is shownas a function of nozzle-to-target separation, l/d, forseveral nozzle diameters and Reynolds numbers [45].Each solid line represents data for a narrow range ofWeber numbers, varying by less than ±3% aroundthe stated mean value, a range equal to the experi-mental uncertainty of Wed. As the figure shows, forconstant Wed, ξ is a function of l/d; different curvesare obtained for different Weber numbers, with splat-tering increasing as the Weber number increases.

To study the effect of surface tension directly,a solution of approximately 10% by volume of iso-propanol in water was used, having σ = 0.042 N/m

Figure 12: Splatter as a function of l/d and Wed [45].

(versus 0.072 N/m for pure water). These data showgood agreement with data for water when scaled withthe Weber number (Fig. 13). Splattering increases assurface tension decreases.

At any given Weber number and nozzle-targetseparation, the splatter fraction, ξ, depends extremelyweakly on the Reynolds number, if at all. For exam-ple, in the data set for Wed = 5500, the Reynoldsnumber increases by a factor of 1.5 without any dis-cernible change in the splatter fraction, ξ. In con-trast, a factor of 1.3 increase in the Weber number(from 5500 to 7300) produces significant increase inthe splatter fraction (roughly +25%).

An influence of Reynolds number would be ex-pected to arise primarily from viscous effects nearsolid boundaries, either in setting the pipe nozzle’sturbulence intensity or as an influence of the viscousboundary layer along the target. The stagnation-point boundary layer is extremely thin relative tothe liquid layer, and thus it may have little effecton the surface waves near the region of impinge-ment. The pipe-turbulence variation maybe quan-tified by observing that the ratio of rms turbulentspeed to friction velocity, u′/u∗, is nearly indepen-dent of Reynolds number in fully developed turbulentpipe-flows [45]. Therefore, upon using the definition,u∗ = uf

√f/8, and the Blasius friction factor equa-

tion (f = 0.316Re−1/4d for 4000 < Red < 105), one

findsu′

uf∝ u∗

uf∝√

f ∝ Re−1/8d

11

Figure 13: Splattering as a function l/d and Wed forliquids of different surface tension [45].

This weak dependence of the turbulence intensity onthe Reynolds number may explain the lack of anysignificant dependence of splattering on Red.

6.3 The relation of splattering to jet surfacedisturbances

Referring to Figs. 12 and 13, we see that verylittle splattering occurs close to the jet exit (smalll/d), typically less than 5%. Beyond this region,the amount of splattering at first increases with dis-tance, l/d. Farther downstream, it reaches a plateau.To explain these observations we may refer to directmeasurements of the amplitude of turbulent liquidjet surface disturbances.

Bhunia and Lienhard [46] used a laser-sheet andhigh frequency-response photodiode to measure theinstantaneous diameter of turbulent liquid jets (Fig. 14).They obtained the rms amplitude of jet surface dis-turbances, δrms , at different axial locations along jetsof various Weber numbers (Fig. 15). Starting fromnearly zero near the nozzle exit, δrms initially growsrapidly as the jet moves downstream; farther down-stream the growth rate diminishes and the rms dis-turbance tends to an asymptotic limit. The depen-dence of δrms/d of jet Weber number is also apparent.

This growth of disturbances is the probable causeof the increase in the splatter fraction as the nozzle-to-target separation is increased. The steadily de-creasing rate of amplitude growth results in a plateauof the disturbance amplitude which corresponds tothat in the splatter fraction data.

Direct comparison of the measurements of δrms

to data for ξ and shows a reasonable correlation be-tween the size of δrms and the fraction of liquid splat-tered (Fig. 16). This graph was obtained by plot-ting previously measured splatter fraction ξ values

Figure 14: Laser sheet measurement of jet surfaceroughness [46].

Figure 15: Root-mean-square surface roughnessdata: several runs at fixed Weber number, showingvariability of data [46].

against δrms/d for jets of the same Weber numberand l/d = x/d [46]. Each set of data for a given noz-zle diameter and jet Weber number consists of mea-surements at several different axial locations, x/d.The correlation is reasonably clear, given that bothξ and δrms have significant variability, and it pro-vides further evidence that splattering is due to sur-face disturbances on the jets and is governed by theamplitudes of those disturbances.

For very long, low Wed jets the plateau of splat-tering ends, and ξ again increases with l/d (Fig. 13).This may reflect the appearance of ordinary capillaryinstability on these jets, which have relatively littleturbulence-generated surface roughness.

6.4 The onset of splatteringBhunia and Lienhard [45] defined the onset of

splattering as the point where 5% of the incomingfluid is splattered. For a jet of a given Weber number,the onset point is reached at a certain l/d, denotedas lo/d. A correlation for the onset point data is

lod

=130

1 + 5 × 10−7 We2d

(43)

For low Weber numbers, near 100, turbulent dis-turbances are strongly damped by surface tension,

12

Figure 16: Splatter fraction as a function of rms sur-face roughness [46].

and the observed onset lengths are actually ratherclose to the capillary breakup lengths. In this range,splattering is essentially of drop impingement type.

6.5 The influence of surfactants on splatteringSurfactants lower surface tension by forming a

surface adsorbed monolayer at the liquid surface.When a new liquid surface is formed, some time is re-quired for surfactant molecules to diffuse to the sur-face in sufficient concentration to alter the surfacetension. To study the role of surfactants in splatter-ing, Bhunia and Lienhard [45] used a mixture of ap-proximately 0.2% detergent in water. This reducedthe surface tension of the static solution (liquid sur-face at rest) to 0.027 N/m and corresponded to asaturated concentration of surfactant. They foundthat the splatter fraction for the surfactant-laden jetis identical to that for a pure-water jet of the samevelocity, diameter, and length; in fact, if Wed forthe surfactant jet is calculated on the basis of pure-water surface tension, the curves for the surfactantjets are identical to those of the pure jets. From thestandpoint of splattering, the surface tension of thesurfactant jet is effectively the surface tension of thepure liquid.

To further explore this behavior, Bhunia and Lien-hard [46] measured the surface-roughness evolutionof jets containing surfactant. They found that thesurface roughness of a surfactant-laden jet evolvedat the same rate as for a pure-water jet of the samesize and speed. In other words, surfactants do notdecrease the stability of the turbulent jet surface un-der conditions like these.

A possible cause of this behavior is the finite timerequired to establish a saturated concentration of

Figure 17: Measured power spectra of turbulentliquid-jet surface disturbances. Ordinate is propor-tional to G(k1l) [46].

surfactant on the jet surface. Inside the nozzle, thesurfactant is in the bulk of the liquid. When theliquid exits the nozzle, a new free surface is formedwhich is not initially saturated with surfactant. Dur-ing the time needed for the surfactant to diffuse fromthe bulk to the free surface, the surface remains un-saturated, and in this initial length of the jet, thesurface tension remains near that of pure water.

6.6 Power spectra of surface disturbancesFigure 17 shows the wavenumber spectrum of jet-

surface disturbances. The ordinate is proportional tothe power spectrum of the free surface disturbanceamplitude, G(k1l) [see Eq. (46) below], and the ab-scissa is uk1, where k1 is the wavenumber in the di-rection of the jet axis, u is the free surface velocity,and l is the integral scale of turbulence.

These graphs of power spectrum versus distur-bance wavenumber show that broadband turbulentdisturbances dominate over any single wavenumberdisturbance related to a Rayleigh-type instability. Inaddition, these log-log spectral plots show a por-tion of very nearly linear decrease in the spectralamplitude, characteristic of high wavenumber tur-bulence [47]. Except for measurement locations verynear the nozzle, the slope of this linear portion is−19/3, so that the spectra decay as k

−19/31 .

A model that explains the spectral decay was con-structed by Bhunia and Lienhard [46], who approxi-mated the jet surface as planar. The liquid pressurefluctuations near the free surface are balanced by thesurface tension (Fig. 18), so that the Young-Laplaceequation applies along the plane of the surface

p = −σ

(∂2δ

∂x2+

∂2δ

∂y2

)(44)

Bhunia and Lienhard modelled the liquid-side pres-sure fluctuations as those of a spatially homogeneousturbulence. By treating both p and δ as stationaryrandom functions [48], they obtained the followingequation for the mean-squared amplitude of surface

13

Figure 18: Liquid-jet surface displacement in re-sponse to pressure fluctuation.

disturbances:

δ2 =∫ ∞

−∞

∫ ∞

−∞

1σ2(k2

1 + k22)2

∫ ∞

−∞F (k)dk3dk2dk1

(45)Here, F (k) is the turbulent pressure spectrum, and(k1, k2, k3) are the (x, y, z) components of a wavevec-tor of magnitude k. For isotropic, homogeneous tur-bulence [49],

F (k) ∼ 0.26ρ2u′4l3(kl)−13/3, kl � 1

where u′ is the rms turbulent velocity and l is theintegral scale of turbulence.

To relate these results to the measured one di-mensional spectrum of the surface disturbances, G(η),they introduced the definition

δ2

l2=∫ ∞

0

G(η)dη (46)

where η is the wavenumber of the free surface distur-bances nondimensionalized with the integral scale ofturbulence, l. From Eqs. (45) and (46), it followsthat at high wavenumbers (η � 1) the disturbancespectrum is

G(η) ∼ 0.26 × 2πρ2u′4

σ2

l−13/3

k3p

∫ ∞

−∞

dk3

(k2p + k2

3)13/6

(47)where we have transformed from Cartesian to po-lar coordinates in the wavenumber plane, (k1, k2) →(kp, θ), with η = kpl and kp =

√k21 + k2

2. Upon eval-uating the integral, the one-dimensional spectrum offree surface turbulent disturbances (measured alongany direction θ) is found to be

G(η) ∼ 2.41(

ρ2u′4l2

σ2

)η−19/3, η � 1 (48)

This analysis explains the observed −19/3 slope inthe log-log plots of the disturbance spectra as a con-sequence of the k

−7/31 variation of the one-dimensional

spectrum of the pressure fluctuations and a factor ofk−41 introduced by the derivatives of δ in the capillary

force balance equation.

7. High Heat Flux Jet Arrays

Impinging liquid jets, with their very high heattransfer coefficients, have proven to be quite effec-tive for high heat flux cooling. When heat transfer

is confined to the stagnation region of a jet, a highheat transfer coefficient may be combined with a highstagnation pressure, which raises the liquid satura-tion temperature and facilitates non-boiling heat re-moval. Liu and Lienhard [50] used high speed waterjets to remove fluxes as high as 400 MW/m2 oversmall areas. To obtain high fluxes over larger areas,jet arrays have proven to be useful. Oh et al. [51] andLienhard and Hadeler [52] used an array of fourteen2.78 mm ID nozzles on 10 mm centers to produce47 m/s water jets impinging on an area of 10 cm2.In experimental tests, the jet array removed fluxesof up to 17 MW/m2 by forced convection alone; theassociated heat transfer coefficient was measured tobe 220 kW/m2.

Two practical difficulties arise in experimenta-tion with jets at such high heat fluxes. The firstis that the solid surfaces involved experience veryhigh temperature gradients, and the associated ther-mal stresses are sufficient to induce yielding of themetal [53]. In the jet array experiments, dispersion-strengthened copper plates (alloy C15315) formedthe heat transfer surface; these metal-matrix mate-rials have the strength of stainless steel at high tem-perature while retaining most of the thermal con-ductivity of copper. The second difficulty is to pro-vide test heating at such high fluxes. Heating forthese experiments was produced using 75 µm thickvacuum-plasma-sprayed Ni-80/Cr-20 electrical resis-tance heaters formed directly on the heat transfersurface [54]. These heaters were driven at nearly1000 A and 24 Vdc. The development of these heatersrequired substantial effort, and the technology hasattracted some interest in its own right.

8. Concluding Remarks

Liquid jet impingement remains a rich source ofproblems having both practical value and academicinterest. Relatively simple experiments are possi-ble, as is theoretical analysis using either classicalor modern techniques.

The work summarized above points to a numberof open questions. The spectral analysis of turbu-lent surface disturbances has not been fully explored.Theory remains to be developed, and additional ex-periments can easily be done using inexpensive op-tical equipment together with computer-aided dataacquisition. The role of wall roughness in the stag-nation zone warrants significantly more study. Ad-ditional experimentation is required, and theory forscaling roughness effects is needed. Small diameterlaminar jets have shown deviations from theory ina number of experiments, whereas larger jets tracktheory well. This remains to be explained. Furtherwork on splattering is needed, for example, in thesense of quantifying the downstream mechanisms. Atheory of turbulent jet heat transfer remains miss-ing, if only for the stagnation zone; existing resultsare essentially correlations built from laminar theory.

The research on jets in our lab was, of course,largely carried out by the many students who have

14

worked with me over the years. I wish to thank all ofthem for their efforts and insights, and I particularlywish to acknowledge the contributions of Dr. Xin Liu,Dr. Sourav K. Bhunia, and Dr. Laurette A. Gabour.

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