Flow Structures and Heat Transfer in Submerged and Free Laminar Jets
Conference Paper · August 2014DOI: 10.1615/IHTC15.kn.000028 CITATIONS 13 READS 1,637 4 authors:
Some of the authors of this publication are also working on these related projects:
containmentFoamView project
Cooling of electrical enginesView project Reinhold Kneer
RWTH Aachen University 307PUBLICATIONS 3,125CITATIONS
SEE PROFILE
Herman D. Haustein Tel Aviv University 44PUBLICATIONS 325CITATIONS SEE PROFILE Claas Ehrenpreis RWTH Aachen University 15PUBLICATIONS 54CITATIONS SEE PROFILE Wilko Rohlfs University of Twente 73PUBLICATIONS 690CITATIONS SEE PROFILE
IHTC15-KN11
FLOW STRUCTURES AND HEAT TRANSFER IN SUBMERGED AND FREE
LAMINAR JETS
R. Kneer,1,∗H. D. Haustein,2C. Ehrenpreis,1 W. Rohlfs1
1Institute of Heat and Mass Transfer, RWTH Aachen University, Aachen, 52056, Germany 2School of Mechanical Engineering, Faculty of Engineering, Tel Aviv University, Ramat Aviv, Tel Aviv
69978, Israel
ABSTRACT
Impinging jets are central to a large amount of industrial processes and applications, and as such have been investigated extensively. Despite this fact, review of the literature shows that a comprehensive description for the flow and heat transfer of the basic underlying laminar single-jet case has not been given. Thereby the present study attempts to collect the understanding gained by previous studies, together with that obtained in recent studies by the authors, in a logical manner (following the flow), in order to formulate such a description. While not necessarily complete, more complex aspects such as turbulence, confinement and dissipation are not addressed, the present study takes a major step towards a unified description of this fundamental flow-configuration.
KEY WORDS:Convection, Two-phase/Multiphase, Heat transfer enhancement
1. INTRODUCTION
Impinging jets are fluid flows released against a surface and are characterized by high heat and mass transfer rates, making them an attractive technique in heating, cooling and drying processes. Their industrial application ranges widely, including gas turbine or piston engine cooling, cooling and lubrication in cutting or grinding, steel quenching or glass tempering, cooling of high energy-density electronic components, drying of paper or textiles [11, 74]. Thus, it is not surprising that jet-impingement cooling has been of interest in the field of fluid dynamics and thermal engineering with many experimental and numerical investigations over the last decades focused on controlling and improving heat transfer characteristics in single-jet and jet-array configurations. A major distinction between two types of jets can be made, if a liquid is used as the working fluid: Submerged jets in which case a liquid issues into a liquid and free-surface jets in which a liquid jet enters a gaseous atmosphere. Besides this general distinction, flows can be defined by a few basic parameters such as the mass flow rate, jet diameter, distance between nozzle exit and impingement surface, and working fluid. However, more complicated effects have also been examined such as nozzle geometries, oscillating flow conditions, liquid extraction, jet-inclination, crossflow, or impingement surface modifications.
Comprehensive surveys focussing on submerged jets are presented by various authors. Martin [44] reviewed the heat transfer characteristics of submerged jets (correlations are given based on theoretical and experimental investigations), covering the local and integral heat transfer in single nozzle and nozzle-array configurations. In Viskanta [68], the fundamentals of submerged jet impingement are clarified through influencing parameters, whilst more complex aspects, such as entrainment, geometric parameters or jet outlet conditions are discussed
separately. Alternatively, Polat et al. [49] and Zuckerman and Lior [74] focus on numerical studies of laminar and turbulent impinging submerged jets, discussing method specific considerations, e.g. turbulence models and wall functions.
A review on free-surface jet impingement is given by Lienhard [33], who summarized analytical solutions for the heat transfer in the stagnation and wall-jet zone for laminar jets and correlations based on experimental studies for turbulent jets. A review on free-surface jet array heat transfer correlations is presented in a recent study by Bhunia and Chen [3].
In the present study it will be shown that despite the complex physics under application-relevant conditions such as turbulence, complicated geometries and flow interactions, most of the basic underlying mechanisms describing the flow can be understood by analyzing single laminar jets. Rather than providing a review of the extensive literature on the topic, the goal of this article is to clarify the governing physical mechanisms and to summarize the scalings and similarities in submerged and free-surface jet impingement. Results of axisymmetric direct numerical simulations and experiments are used throughout the article. More detailed information on the numerical set-up is provided in Rohlfs et al. [56–58] and on the experimental set-up in Haustein et al. [26, 27].
The structure of this article follows the jet in the direction of flow from the nozzle exit up to the hydraulic jump (in case of a free-surface jet) as depicted in Fig. 1. After emerging from the nozzle with a specific ve-locity profile, the jet travels as a free jet towards the wall, during which its veve-locity profile evolves subjected to internal viscous relaxation and external shear (section 2.2). Due to surface tension forces, the free-surface jet is also subjected to the Rayleigh-Plateau instability, which, given time to develop, causes oscillating flow conditions and jet break-up for large nozzle-to-plate distances (subsection 2.3a). For submerged jets, strong shear forces at the outer edge can lead to the formation of Kelvin-Helmholtz vortices, which influence the heat transfer as they impinge on the wall (subsection 2.3b). Before the jet enters the stagnation zone, a transition occurs in which the force balances change from a viscosity-dominated flow regime to a potential-type flow regime (section 3). In the stagnation zone itself, the axial momentum is converted through pressure into radial momentum causing a strong radial velocity component in the vicinity of the wall. This flow acceleration sig-nificantly depends on the impinging velocity profile and thus on the nozzle velocity profile and its evolution. This radial acceleration thins the boundary layer causing locally increased heat transfer (section 4.2). Beyond the stagnation zone, a configuration-specific wall jet is formed: For the submerged jet, the heat transfer in the wall jet region is influenced by large scale vortices impinging and travelling along the wall (section 5.2); while for the free-surface jet, theoretical analysis reveals a regime transition and thus a different scaling compared to the impingement region (section 5.1). In the latter case, the wall jet is subject to a hydraulic jump (section 5.3), which is a sudden recovery from a super-critical state of the flow (shooting). This jump is accompanied by a significant reduction in heat transfer and is especially important to jet-array cooling.
The present study takes the approach of comparing the results to literature (where possible), from which a picture of the fundamental physics is drawn in order to formulate a unified description of laminar impinging jets.
2. FREE JET DEVELOPMENT 2.1 Exit velocity profile
Nozzle type and geometry directly dictate the nozzle exit velocity profile and have thus a strong influence on the entire impinging jet. While short or orifice type nozzles result in a more or less uniform axial velocity profile, longer nozzles allow for a fully developed profile. In the laminar case, this leads to a parabolic shaped profile. Furthermore, it has been found in some cases that the heat transfer is not only affected by the well-known dimensionless parameters, but also by the nozzle-diameter itself [21], indicating that further effects play an important role. Similarly, it has been shown that the shape of the nozzle edge has a strong effect [34]. In the
D Free-surface jet
Submerged jet
exit velocity profile
free jet region
transition to wall zone Kelvin-Helmholtz instability
wall jet region
hydraulic jump nozzle
stagnation zone wall jet region
x x’ r nozzle-to-plate distance H x´ r I II III IV rv h(r) u(r,x) Rayleigh-Taylor instabilities
imposed heat flux or imposed temperature
Fig. 1 Schematic illustration of submerged and free-surface jet impingement. Zones indicate regions where different physical mechanisms govern the fluid motion. For the free-surface case the upper insert shows the development from the stagnation point flow to the wall-jet including the regions defined by Liu and Lienhard [36, 37].
present study nozzle shapes are not considered, as rather typical velocity profiles are used as inlet conditions in the numerical simulations.
2.2 Self-similarity in laminar undisturbed jets
The velocity profile emerging from the nozzle undergoes a development while travelling towards the impinge-ment wall, depending on ambient boundary conditions. For example, in submerged jets, where the surrounding liquid decelerates the jet rapidly due to high shear forces, heat transfer correlations usually contain a strong dependence on the nozzle-to-plate distance of the type(H/D)γ whereγ varies from a value of γ = −0.191 toγ = −0.55 [40, 68]. While for free-surface jets this influence is found to be much weaker (γ = −0.0336 toγ = −0.105 [22, 64]) and in the limiting case of low jet velocities, gravitational acceleration influences jet speed and diameter [33]. It will be shown here that just the nominal nozzle-to-plate distance is not sufficient to describe the jet velocity profile.
Recently the authors have shown analytically and numerically a self-similar behavior of free- and submerged jets [56] as will be recalled briefly. Starting with the axisymmetric dimensionless form of the axial momentum conservation equation u∗∂u∗ ∂x∗ + v∗ H D ∂u∗ ∂r∗ = −∂x∂p∗∗ +D · ReH 1 r∗ ∂ ∂r∗ r∗∂u∗ ∂r∗ +HD∂∂x2u∗2∗ , (1)
where the equation is normalized by a reference velocity, a dynamic pressure and two different length scales (u= u∗· U∞, p= p∗· ρU∞2 , x= x∗· H, and r = r∗· D). It is seen that in free-surface jets, the shear at the outer edge of the jet and pressure gradient are negligible. In addition, radial viscosity forces dominate, such that the jet undergoes relaxation, consequently leading to
u∗∂u∗ ∂x∗ = H D · Re 1 r∗ ∂ ∂r∗ r∗∂u∗ ∂r∗ . (2)
In this equation, a single parameter is found to be important for the development of the free jet’s velocity profile, i.e. H/(D · Re), where the Reynolds number is defined by the average velocity of the jet U∞as Re= u∞D/ν. This scaling parameter is well-known for the development length of laminar pipe and channel flows (for a review see Durst et al. [15]) and also for the draw-down of a vertical free-surface jet [45]. Consequently, the development of the submerged and free-surface jet should also be described by this scaling parameter, rather than by the pure geometrical nozzle-to-plate distance (H/D). Note that this scale can also be seen as the ratio of time available for relaxation (time of flight) to inherent viscous time scale (tνt = x/u · ν/D2 → x∗ = x/(D · Re)). The implications of this scaling parameter to the evolution of the free-surface and submerged jet before impingement are shown in Fig. 2.
1.5 1.0 0.5 0 0.5 1.0 1.5 r/D parabolic
exit velocity profile:
0.1∙u∞ 0.1∙u∞ 0.05∙u∞ 0.05∙u ∞ 0.95∙u∞ 0.9∙u∞ 2 1 0 0 u/u∞ 0.01 0.02 0.03 0.04 x/(D∙Re) submerged jet, uniform profile free-surface jet submerged jet x/D 6∙D 12∙D 18∙D 24∙D submerged jet 0 0.5 0.5 r/D x/D free-surface jet 8∙D 12∙D 18∙D 24∙D centerline velocity parabolic profiles: a) b) c) uniform parabolic
Fig. 2 Velocity profile of free jet flows (Re= 540) for the submerged (left image) and free-surface configura-tion (center image) obtained from radial symmetric direct numerical simulaconfigura-tions [57]. The right image shows the development of the centerline velocity. Note that the axial and the radial direction are not scaled equally. According to Viskanta [68] the submerged jet prior to impingement can be divided into several regions in axial direction: the potential core zone (where the nozzle exit velocity is sustained), the developing zone, and the de-veloped zone (characterized by self-similarity of the velocity profile). The left side of Fig. 2a shows simulation results of a laminar jet (Re= 540), where the potential core penetrates up to 17 diameters. Using a criterion of0.90 · U∞the potential core length is given by H/(D · Re) ≈ 1/30. However, this simple analysis is not valid for higher Reynolds numbers, where vortices (emerging due to Kelvin-Helmholtz instability) disrupt the smooth velocity profile. According to Livingood [39], the length of the potential core zone is 6-7 diameters from the nozzle exit, independent from Reynolds number. For a parabolic outlet profile, the center velocity starts to decay already at the nozzle exit (see Fig. 2a, right side), such that a simple definition of the potential core zone cannot be applied. The developed velocity profile (after approximately6 · D) in this case has been suggested to be self-similar [60] and best fitted by a Gaussian distribution [68] with a linear jet broadening. However, present simulation results show that the jet broadening rather follows a parabolic shape [57]. Figure 2b) shows the case of a free-surface jet, which also can be subdivided into two major regions: the developing and the developed zone. In the developing zone, the inner core of the jet decelerates while the outer part accelerates.
Assuming no loss in mass and momentum, the center velocity decays from2 · u∞ to4/3 · u∞(see Fig 2c), where u∞is the average velocity of the parabolic profile. Simultaneously, the jet contracts such that the radius reduces to3/4 · r [25]. Using the scaling parameter x/(D · Re), self-similarity in the decay of the centerline
velocity, uaxcan be found: uax u∞ = 4 3 + 2 3exp −mD · Rex (3)
The exponent m can be determined analytically only at the nozzle exit where it takes a value of m = 24. Further downstream the exponent increases such that an average decay described by a parameter value of m = 40 ± 1 gives reasonable agreement with numerical data [57]. After the developing region, the velocity profile of the jet is uniform. Using uax/u∞= 0.05 · (2 − 4/3) + 4/3 as a criterion for the developed profile, eq. (3) yields a development length of x/(D · Re) ≈ 0.077, which agrees well with values suggested for the entrance length of laminar pipe flows [15], where the opposite transition from a uniform velocity profile into a parabolic one occurs. Note that also the centerline velocity of the parabolic submerged jet in Fig. 2c shows the same decay up to x/(D·Re) ≈ 0.02. Beyond that location, the centerline velocity is affected by the submerged configuration, resulting in a faster loss of momentum compared to the free-surface jet configuration.
2.3 Instabilities in the issuing jet
Under realistic conditions, perturbations at the nozzle exit cause significant deviations from the idealized de-scription given above. These perturbations increase the complexity of the flow, thereby reducing its level of self-similarity.
Rayleigh-Plateau Instability in axisymmetric free-surface jets From Rayleigh’s theory for jets and cylin-drical surfaces [52], it is known that an axisymmetric liquid cylinder (jet) is unstable against infinitesimal disturbances leading to a break-up into droplets above a critical length (Plateau-Rayleigh instability). The break-up is caused by surface tension effects and can be explained by a balance of the surface tension forces in radial direction resulting from the curvature in axial and angular direction [55]. Linear stability analysis reveals a critical length of the filament of Lcrit.= 2πR, and a wavelength with highest amplification, Lmax= 4.55D [45], regardless of fluid viscosity. According to the inviscid theory [52], the break-up depends on the capillary time-scale tcap = ρD3/σ, which can be converted to a dimensionless break-up length scale (disturbances travel with the velocity of the jet) L∗/D = CU2ρD/σ, or to:
L∗
D · Re = C
ρν2
σD = C · Oh, (4)
where Oh denotes the Ohnesorge number Oh = νρ/(Dσ). For the parameter C, different values can be found in literature (5 ≤ C ≤ 13, [45, 48]), due to the strong dependency on initial disturbances in the system. Thus, the correct scaling of those initial perturbations (which are generally unknown) is necessary for a self-similar description. A simple separation of scales in axial and radial direction as shown for the unperturbed case by x/D· Re is not valid any more, due to the additional coupling of the critical streamwise length to the radius of the jet.
N
Figure 3 (right plot) shows the dependence of the dimensionless break-up length L∗/(D · Re) on Ohnesorge number. In addition, the break-up length is shown in dimensional form for technically relevant conditions with water as a working fluid (plate distances of H/D= 15, jet diameter of 1 mm at a Reynolds numbers below 800, if C = 5 is applied). Note that only one regime of jet break-up has been presented here. For high Ohnesorge numbers, wind induced break-up is expected according to the theory of Weber [71].
Kelvin-Helmholtz instability in axisymmetric submerged jets As stated earlier, submerged jets are much stronger influenced by secondary structures. Large scale vortices arise at the outer edge of the jet caused by a Kelvin-Helmholtz-like instability [28]. Due to the dependence of the instability on the shear rate, the velocity
Re = 590 Re = 760 Re = 1020 L/D = 76; L/(D Re) = 0.12 L/D = 93; L/(D Re) = 0.12 L/D = 114; L/(D Re) = 0.11
dimensional jet diameter (water) [mm]
10−1 10 101 0.002 0.003 0.005 0.01 0 0.05 0.1 0.15 Oh L * /(D Re) 0 C = 5 C = 13
Fig. 3 Plateau-Rayleigh break-up: Water emerging from a nozzle with a diameter of D= 0.6mm (left). Critical dimensionless break-up length as dependent on Ohnesorge number (bottom scale) or on the dimensional jet diameter for standard properties of water (top scale), withνH2O= 10−6m2/s, ρH2O= 1000kg/m3, σH2O= 0.07N/m (right).
profile (from uniform to parabolic) has a strong effect on the growth rate and the scale of the vortices. Thus, two different length scales play an important role. First, the momentum boundary layer thicknessθ at the nozzle edge, which is important for the initial development of the shear layer. Second, the jet diameter D which typically scales the vortices further downstream [67]. Consequently, the frequency of highest amplification changes due to vortex pairing from a high frequency mode (disturbances having a size of the order of θ, Stθ = fθ
U ≈ 0.017) [28] to a low frequency mode at the end of the potential core (disturbances with a size
of the order of D, St= fDU = 0.3 to 0.5). This vortex formation process and thus the formation of coherent structures can be influenced by an external excitation of the flow [7, 30]. The influence of these large scale structures on heat transfer is presented and explained in section 5.2.
3. TRANSITION TO WALL ZONE
The simplified balance of viscous and inertia forces in the free jet region is valid as long as the jet flow does not “sense” the impinging wall. In the near wall region, inertia is converted to static (stagnation) pressure, such that the influence of the pressure gradient in axial direction (∂p∗/∂x∗) increases, while the relative influence of viscous forces reduces. Consequently, the force balance in the near wall region simplifies to
u∗∂u∗ ∂x = −
∂p∗
∂x∗. (5)
This potential type flow equation carries no characteristic length scale, neither H nor D. The transition from viscous/inertia dominance in the free jet region to an pressure/inertia dominance in the near wall region, changes the relevant importance of each term in the momentum equation (1).
In Rohlfs et al. [56] two distances are introduced: The effective nozzle distance N , representing the free trav-elling distance of the jet, where it is exposed primarily to viscous relaxation and the effective plate distanceΠ, representing the initiation of impingement. It was further found that a linear relation between the two distances exists as shown in Fig. 4, left.
As the nozzle distance N is the point at which the flow begins to sense the influence of the wall, the correspond-ing velocity profile can be attributed to the inlet boundary conditions for the near wall region and impcorrespond-ingement zone. For validation, the center-line velocity at this position has been examined for different Reynolds numbers and nozzle-to-plate distances using DNS data, see Fig. 4, right. Comparing the results for impinging jets (sym-bols) to the development of the free jet (lines) shows that the corresponding velocity profiles at the position N are very similar. This implies that the jet has not yet been significantly influenced by the impingement wall,
dimensionless center velocity: u
Ν
/u∞
dimensionless nozzle distance: Ν/(D·Re) 0.4 0.6 0.8 1.0 1.2 1.4 1.6 2.0 1.8 0 0.05 0.10 0.15 0.20 0.25
free-surface jet, DNS results submerged jet, DNS results
centerline velocity free jet:
free-surface jet
submerged jet
dimensionless nozzle distance: Ν/(D∙Re)
dimensionless plate distance: Π / D
0 0.05 0.10 0.15 0.20 0.25 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5
free-surface jet, DNS results submerged jet, DNS results Π / D= 9.75 ∙ (Ν /(D∙Re)) + 1.03 Π / D= 12.36 · (N/(D·Re)) + 1.18
Fig. 4 Characteristics of free-surface and submerged jets in the near wall region: Dimensionless plate distance vs. dimensionless nozzle distance (left plot). The solid and the dashed curves are least square fits to the nu-merical data points. Dimensionless axial velocity at the entrance of the impingement zone, N/(D · Re) (right plot). The solid and the dashed curves represent the development of a corresponding free jet.
validating the choice of the transition criterion and suggesting that this position is a well-suited location to evaluate the velocity profile as arrival boundary condition for the impingement region.
4. HEAT TRANSFER IN THE STAGNATION ZONE
The heat transfer characteristics for jet-impingement are often expressed in dimensionless form by the nozzle-diameter based Nusselt number
Nu(r) = hq(r)D
λ , (6)
where hq= qW/(TW(r) − Tref) is the heat transfer coefficient and λ the thermal conductivity of the fluid. As reference temperature, either the temperature of the jet flow or the adiabatic wall temperature can be chosen. In the numerical simulation results presented, dissipative heat generation is not accounted for, which is equivalent to the use of the adiabatic wall temperature as reference.
Heat transfer is typically described by a correlation of the form Nu = C · P rm· Ren·H
D γ
, (7)
where P r is the Prandtl number defined as P r = ν/a denoting the ratio between viscous diffusivity ν and thermal diffusivity a.
4.1 Prandtl number dependency
The dependency of Nusselt number distribution on Prandtl number has been intensively studied experimentally by Sun et al. [65, 66], who found for P r = 0.7 to P r = 348 values of C = 1.38, m = 0.329 and n = 0.489. The value of the coefficient m is close to the analytic value of m = 1/3, which has also been confirmed by other experimental studies [42]. Based on exact solutions for stagnation flows, White [72] presents an analytical Prandtl number dependency (following the derivations of Homann[29] and Sibulkin [62]), which is given by a
step-wise approximation in Liu et al. [35] as: G(P r) = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ √ 2P r/π 1+0.804552√2P r/π P r ≤ 0.15 0.53898P r0.4 0.15 ≤ P r ≤ 3.0 0.60105P r1/3− 0.050848 P r ≥ 3.0 (8)
UsingG(P r) to rescale the numerical simulation results for free [56] and submerged jets with Prandtl num-bers ranging from P r = 0.07 to P r = 1300 results in an unified curve (as shown in Fig. 5), where the spread of symbols in vertical direction represent solutions for different Prandtl numbers. Although the Prandtl number ranges over five orders of magnitude, deviations from the unified curve are below 5 percent. This analytical dependency is strictly speaking only valid for the velocity field at the stagnation point (r= 0). How-ever, the simulation results reveal a good agreement for all Prandtl numbers in the entire impingement region, r/D < 0.5. For larger radial distances, the flow behavior changes from a stagnation flow to a wall jet, conse-quently resulting in a different Prandtl number dependence. For example for Prandtl numbers of P r= 7 and below, deviations from the self-similar behavior are found to appear beyond r/D > 0.6 [56]. As the Prandtl dependency is strongly influenced by the thermal boundary layer development, analytical formulations become a more sophisticated task.
4.2 Dependence of the stagnation region heat transfer on the incoming jet velocity
The heat transfer in the stagnation zone is expected to be dictated by the approaching velocity profile, whose dependency on the dimensionless nozzle distance was shown in Fig. 4 (right) by the decay of the centerline velocity uN/U∞. Note that a unique relation between the centerline velocity and the relaxation of the velocity profile exists for a given velocity profile at the nozzle exit. The dependency between this centerline velocity and the heat transfer in the stagnation zone (represented by the reduced Nusselt number, Nu/(G(P r) · Re1/2), averaged over r/D≤ 0.5) is shown in Fig. 5 (left), yielding a linear relation. This relation later is used to find a direct relation to nominal conditions shown in Fig. 5, right.
For the submerged case, the dimensionless velocity is not bounded to a lower value, whereas for the free-surface case, uN/u∞ ranges from 4/3 (impingement of a uniform velocity profile) to 2 (impingement of a parabolic profile). Consequently, the reduced stagnation zone Nusselt number ranges from approximately 1.8 to 2.8, relating to a uniform (high N/(D · Re)) or a parabolic velocity profile (low N/(D · Re)), accordingly.
4.3 Self-similarity in the stagnation region heat transfer
The previous subsection revealed a direct dependency between stagnation region heat transfer and dimen-sionless center velocity before impingement, which itself depends on the dimendimen-sionless travelling distance N/(D · Re). Using the dependency found between the plate distance Π and the travelling distance N, a di-rect relation between the nozzle-to-plate distance H/(D · Re) and the heat transfer in the stagnation region can be obtained. Figure 5 (right plot) shows this dependency for the free-surface and the submerged case, whereby the Reynolds number and the nozzle-to-plate distance have been varied between73 ≤ Re ≤ 2000 and4 ≤ H/D ≤ 110, respectively. The curves show that the dimensionless nozzle-to-plate distance is the major influencing parameter, despite the specific values of Re and H/D. Although Re and H have been varied independently, the numerical solutions in terms of Nu/(G(P r) · Re1/2) coincide for the same value of H/(D · Re). For low nozzle-to-plate distances, both flow configurations lead to the same stagnation re-gion heat transfer. Contrary, for large nozzle-to-plate distances, the free-surface jet depicts a convergence to Nu/(Re1/2G(P r)) ≈ 1.8 while the Nusselt number for the submerged jet decreases continuously. The results
for the average heat transfer coefficient are in agreement with the numerical results of Saad et al. [59], who also compared the results for low nozzle-to-plate distances to the experimental work of Scholtz [61].
0.5 1.0 1.5 2.0 2.5 3.0 Nu/(G(Pr)·Re 1/2 ) 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
dimensionless centre velocity: uΝ/u∞
free-surface jet, DNS results submerged jet, DNS results
f(u /u ) = 1.562 · (uΝ/u∞) - 0.3 f(u /u ) = 1.232 · (uΝ/u∞) + 0.338 Ν∞ Ν∞ 0.5 1.0 1.5 2.0 2.5 3.0 10-3 10-2 10-1 1 H/(D∙Re) Nu/(G(Pr)∙Re 1/2 )
free-surface jet, DNS results submerged jet, DNS results free-surface jet, eq. (9) submerged jet, eq. (10)
Fig. 5 Comparison of free-surface and submerged jets in the near wall region for the parameter range0.07 < P r < 1307, 73 < Re < 2000, and 4 < H/D < 110: Relation between centerline velocity and Nusselt number (left) and relation between the nozzle-to-plate distance and the stagnation region Nusselt number averaged over r/D ≤ 0.5.
An interesting effect is found for0.008 ≤ H/(D · Re) ≤ 0.05 (see Fig. 5), where the stagnation region heat transfer for the submerged jet surpasses the value for the free-surface jet. To identify the root cause, the radial distribution of the Nusselt number is shown for three specific cases, both, for submerged and free-surface jets in Fig. 6 (left). For the lowest value of H/(D · Re) = 0.002, the heat transfer distribution is identical up to r/D < 2 due to the low plate distance and the identical velocity profile in the potential core region of the free jet. For H/(D · Re) = 0.0152, the submerged jet overcomes the free-surface jet in the stagnation region r/D < 0.55. For this case, the velocity distribution at the effective wall distance of Π = 1.25 · D is shown in the upper right plot, revealing the identity of the centerline velocity for both jet configurations. Although the velocity decreases for both cases in radial direction, a slightly higher gradient is found for the submerged jet configuration. This difference in the incoming velocity profile is reflected in the pressure distribution at the wall, revealing the same pressure at the stagnation point and a higher pressure gradient for the submerged jet (r/D < 0.65). The higher heat transfer coefficient of the submerged jet case can be attributed to the higher pressure gradient, which was also shown in Rohlfs et al. [58] to increase the heat transfer at the impinging wall.
The profile of the free-surface jet in Fig. 6 for H/(D · Re) = 0.0691 is characterized by a distinct peak around r/D≈ 0.6. Such peaks in the Nusselt distribution are well-known to occur in submerged jets for small H/D ratios [20, 40, 47]. To the author’s knowledge, only two previous experimental studies show the possible existence in free jets (Baonga et al. [2] and in Fig 12a of [26]). Several plausible explanations for the peak in submerged jets have been proposed such as local flow acceleration [20, 31], the penetration of turbulence into the boundary layer [40, 47], or the impingement of large vortices on the wall [31]. In a numerical study by Saad et al. [59], two different inlet velocity profiles – parabolic and uniform – have shown a strong effect on the heat transfer characteristics. In a former study by the authors [58], the origin of the inner peak was associated with the occurrence of local flow acceleration and is not associated with the occurrence of turbulence. An increase of the radial acceleration in radial direction, causing an inflow towards the wall, was found to be the key explanation for the inner peak. As this acceleration is caused by the pressure gradient in the stagnation region, the shape of the velocity profile at the nozzle exit was found to dictate whether a local peak exists or not. This also explains, why the inner peak has been previously associated with higher Reynolds numbers and lower nozzle-to-wall distances, where the required form of the pressure gradient is met.
0 0.6 1.2 1.8 u(Π = 1.25·D) /u∞ H/(D·Re) = 0.0152 free-surface jet submerged jet 0 0.5 1.0 1.5 2.0 2.5 3.0 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 r/D Nu/(Re 0.5 ·G(Pr)) 0 0.2 0.4 0.6 0.8 1.0 0 0.2 0.4 0.6 0.8 1.0 r/D H/(D·Re) = 0.0152 0 0.4 0.8 1.2 1.6 p/(ρ·u 2) H/(D·Re) = 0.0152 H/(D·Re) = 0.002 H/(D·Re) = 0.0691 jet surface
Fig. 6 Root cause for the higher heat transfer coefficient of the submerged jet compared to the free-surface jet. Distribution of the reduced Nusselt number (left), radial velocity profile before impingement (right top), and radial pressure distribution at the wall (right bottom).
4.4 Correlation for the stagnation region heat transfer coefficient
Based on the data points obtained from numerical simulations and shown in Fig. 5 (right plot), a comprehensive correlation for the stagnation region heat transfer coefficient has been deduced. Due to the transitional shape for the free-jet case a dependency of (H/(D · Re))γ is not appropriate. In Rohlfs et al.[56], a step-by-step deduction of a heat transfer correlation function was presented on the basis of physical relations shown in Figs. 4 to 5. Nu Re1/2G(P r) = 1.76 4 3+ 2 3e−34.5 H D·Re − 0.62. (9)
The present correlation is proposed as a physically derived alternative to previous correlations, covering the majority of flow conditions in the laminar range (0.07 < P r < 1307, 73 < Re < 2000, and 4 < H/D < 110). The reduced Nusselt number cannot be compared directly to values found in literature, due to the more complex Prandtl number dependency employed here. IfG(P r) = P r1/3 is applied, the correlation can be compared to literature values, but the data-spread for different Prandtl numbers is strongly increased. For this common Prandtl dependency, the values for the parameter C in the general correlation (7) range from approximately C = 1 for a uniform impinging profile (uN/u∞ = 4/3) to C = 1.6 for a parabolic profile (uN/u∞ = 2).
A comparison to literature reveals that these values cover almost the entire range found experimentally. Ma et al. [43] have found a constant of C = 0.967 (with a Reynolds exponent of n = 0.510) for Reynolds numbers in the range of 200 to 2000 and nozzle to plate distances of6 ≤ H/D ≤ 20. Liu et al. [37] on an analytical basis found a coefficient of C = 0.797 for a uniform velocity profile of the jet, and Lienhard [33] a value of C = 0.745. On the other hand for pipe-type nozzles, Ma et al. [43] present a coefficient of C = 1.27, which is more or less the center value of the distribution given here. Liu et al. [35] using the theoretical and experimental data by Scholtz and Trass [61] found a coefficient of C = 1.648 (with a Prandtl dependency of 0.361) for a parabolic velocity profile, which is very close to the value found here.
Using a similar methodology for submerged jets yields the following correlation function Nu
Re1/2G(P r) = 2.31 · e−10.2 H
D·Re + 0.51, (10)
5. WALL JET 5.1 Free-surface jets
Flow characteristics Beyond the stagnation region, the flow on the impingement wall changes from the characteristic stagnation point flow to a wall jet. This transition has been analytically investigated by Watson [70], who divided the flow into different regimes. Based on this separation, Watson derived analytical equations for the radial development of the surface h and the surface velocity Us. An important criteria for the separation is the viscous boundary layer, δν, which grows from the stagnation point until it reaches the free-surface (δν = h) and consequently covers the entire flow. As a result, the velocity profile changes from the radial Blasius-type solution to a different similarity profile, which consequently effects the scaling behavior in the two regions. In the following, simulation results are presented in which the Reynolds number is always based on a parabolic velocity profile. Figure 7 (upper left plot) shows velocity profiles obtained from DNS of the wall jet for two different Reynolds numbers (but identical dimensionless nozzle-to-plate distance H/(D · Re)). The dots indicate positions of highest radial velocity. In the plot, the dimensionless radial distance is r/D, the axial distance is x/D· Re1/2, and the dimensionless velocity is u/U0. The profiles exhibit a self-similar behavior in the stagnation region for x < x(umax). However, beyond this region, the profiles diverge, revealing a loss of self-similarity with the applied scaling. In Fig. 7b, the same velocity profiles are shown with a different scaling (radially r/(D · Re1/3) and axially x/D · Re1/3). With this scaling, self-similarity of the velocity profiles and the liquid height in the wall jet region is obtained, while in the stagnation region it is lost. Note that the additional dotted lines show a well-known parabolic velocity profile u= umax 3/2(x/δ) − 1/2(y/δ)3 , used in Liu and Lienhard [36].
In Fig.7 (left center plot), the dimensionless nozzle-to-plate distance is held constant, such that a similar ve-locity profile impinges. Contrary, in Fig. 7 (left bottom plot), veve-locity profiles for the same Reynolds number, but for different nozzle-to-plate distances are shown, illustrating the influence of the impingement velocity profile. For a small nozzle-to-plate distance and thus an impinging parabolic radial velocity profile (grey line), the velocity in the impingement region is different compared to the one for a large nozzle-to-plate distance. However, further downstream (for larger radial distances), the profiles converge as the ratio between mass and momentum is equal for the two cases. Note that the jet diameters and average jet velocities before impingement are different due to a different extent of viscous relaxation and jet contraction. Consequently, the jet flow could be attributed to different values of the Reynolds number.
Heat transfer characteristics According to the different hydrodynamic regimes, the heat transfer character-istics of the wall jet can be subdivided into different regions as shown in the insert in Fig. 1 [36, 37, 41, 42], allowing for a piecewise description of the local heat transfer coefficient which will be evaluated here by DNS. Due to the additional Prandtl number dependency, a general description for the heat transfer cannot be derived. This is possible only for more distinct sub-regions, for which the following distinction has been made [36, 37] (Please note that the correlations only refer to the high Prandtl number case, P r >1):
I The stagnation region (discussed in section 4)
II δ < h: Thermal and viscous boundary layers are smaller than the film thickness, h. Thus, the surface temperature and velocity are equal to the inlet values. The thermal boundary layer is assumed to grow according toδ ∼ r0.5, such that an analytical solution for the local Nusselt number based on a third-order polynomial approximation of the velocity and temperature profile [36] reveals a decrease of the Nusselt number with1/√r.The transition between region II and III is obtained for a constant value of r0/(DRe1/3) = 0.1773, where the viscous boundary layer reaches the film surface.
III δν= h, δt< h: The viscous boundary layer has reached the free surface and the velocity decreases with the radius. The surface temperature is still unaffected. Due to the decreasing surface velocity in region
0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 Nu/(Re 1/2 ·G(Pr)) 0 1 2 3 4 5 r/D Re = 2000, H/D = 8 Re = 1000, H/D = 4 Re = 1000, H/D = 27.4 0 2 4 6 8 10 12 0 0.1 0.2 0.3 0.4 0.5 Nu/(Re 1/3 ·G(Pr)) r/(D∙Re1/3) 0.5 1.0 1.5 2.0 0 0.1 0.2 0.3 0.4 0.5 r/(D∙Re1/3) x‘∙Re 1/3 /D 0 Re = 2000 Re = 1000 2 4 6 8 0 1 2 3 4 5 r/D x‘/D∙Re 1/2 0 Re = 2000 Re = 1000 surface r0 0 0.5 1.0 1.5 2.0 0 0.1 0.2 0.3 0.4 0.5 r/(D∙Re1/3) x‘∙Re 1/3 /D H/(D∙Re) = 0.004 H/(D∙Re) = 0.027 Nu G(Pr)∙Re1/2= 1.648 Pr1/3 G(Pr) Nu G(Pr)∙Re1/2= 0.632∙√D/r Pr1/3 G(Pr) Nu G(Pr)∙Re1/2= 0.745 Pr1/3 G(Pr) heat transfer correlations according to Lienhard, 2006:
heat transfer correlations according to Lienhard, 2006, eq. (35) equation for h according
to Watson, 1964
Fig. 7 Left: Surface topology and radial velocity profiles in the wall jet region adopted from Rohlfs et al. [56]. Top plot: Scaling by r/D and x/D · Re1/2revealing self-similarity of the near wall velocity field in the stagnation region for two different Reynolds numbers. Center and bottom plot: Scaling by r/(D · Re1/3) and x/D · Re1/3revealing self-similarity in the wall jet region. Right: Local heat transfer distribution: Top plot:
Scaling by r/D and Nu/(Re1/2G(P r)) showing self-similarity in the stagnation region. Grey areas mark correlations [33] for the stagnation region heat transfer for a parabolic (high value) and a uniform (low value) impinging velocity profile for7 ≤ P r ≤ 1302. The dashed lines reveal the slope Nu ∼ 1/√r. Bottom plot: Scaling by r/(D · Re1/3) and Nu/(Re1/3G(P r)) revealing self-similarity in the developed wall jet region.
III [70], the velocity and temperature profiles change their scaling behavior, such that the scaling of the Nusselt number distribution changes also from Re1/2to Re1/3
IV δ = h, δt< h: Both boundary layers have reached the free surface and the surface temperature increases with the radius. This region will only occur for P r <5.
V A hydraulic jump occurs (for a discussion on the hydraulic jump see section 5.3)
Note, that in region II and III, the same Prandtl dependency holds, while in region IV a more complex Prandtl dependency exists. In Liu and Lienhard [36], a validation of the analytic model has shown very good agreement for Reynolds numbers above3.56 · 104.
Figure 7 (upper right plot) shows the radial distribution of the reduced Nusselt number (Nu/(Re1/2G(P r))) obtained from the fully resolved simulations for two different Reynolds numbers, two different nozzle-to-plate
distances, and Prandtl numbers from 7 to 1302. As described in the previous subsection, the heat transfer in the stagnation region scales very well with Re1/2G(P r), such that the solutions for Re = 1000 and Re = 2000 overlap, while the solutions for the two nozzle-to-plate distances (H/(D · Re)) clearly differ. Comparing the profiles to the analytic functions given by Liu and Lienhard for region II (Nu ∼ 1/√r), the Nusselt number decrease is more pronounced than it was found by Liu and Lienhard. In the wall jet region, the Nusselt number for the two different Reynolds numbers mildly diverge, though this implies a loss of the validity of the proposed scaling.
In Fig. 7 (lower right plot) the scalings are changed according to the similarity solution found by Watson [70], e.g. the reduced Nusselt number becomes Nu/(Re1/3G(P r)) and the radial distance r/(D · Re1/3). With this scaling, the radial profiles for the two different Reynolds numbers converge in the developed wall jet region r/(D · Re1/3) > 0.1 but differ in the stagnation zone. A comparison to the solution of Liu and Lienhard for
region III, shows that the slope for r/(D · Re1/3) > 0.25 is well captured. Contrary, the trend predicted for 0.1773 < r/(D · Re1/3) < 0.25 does not agree. The disagreement is caused by the assumed profile of the fluid
height, h, shown by the dashed line in Fig. 7b and c. Thus, a better agreement between the analytically derived function of Liu and Lienhard [36] and the simulation results can be expected for a better approximation of the liquid surface height.
5.2 Submerged jets: Influence of large-scale structures
0 1.0 2.0 3.0 4.0 4.5 4.0 x/D [-] r/D [-] 3.5 3.0 2.5 2.0 1.5 1.0 3.0 4.5 4.0 x/D [-] r/D [-] 2.0 1.0 4.0 4.5 4.0 x/D [-] r/D [-] 3.0 2.0 SV - Secondary vortex SZ RZ SZ - Separation zone RZ - Reattachement zone PV - Primary vortex SV SV PV PV PV Streamlines Streamlines Vorticity
Fig. 8 Instantaneous velocity field of the disturbed flow profile[58]. Background and upper inset: streamlines; lower inset: iso-vorticity plot.
A further characteristic of the heat transfer of submerged jets is the occurrence of an outer peak in the Nusselt number profile, which is found experimentally to occur in the transitional/weakly turbulent regime
1 2 3 v [10 -5 m/s] 0 1 2 3 4 Radial position r/D [−] 10 20 30 40 Nusselt number [−] wall distance: 2.5μm time-averaged value range of oscillations 0 1 2 3 4 Radial position r/D [−] 4
Fig. 9 Influence of large scale vortices on the heat transfer of laminar submerged jets. Left plot: Nusselt number shown for a constant heat flux boundary condition and P r= 0.72. Right plot: Time averaged value of the axial velocity in the vicinity of the wall.
(Re ≈ 3000) [1]. Thus, the peak is commonly associated with a transition of the wall-jet boundary layer to tur-bulence. In contrast to this explanation, simulations by Chung et al. [13] have shown that the outer peak occurs instantaneously when large scale laminar vortices (primary vortices, resulting from the KelvHelmholz in-stability as explained in section 2.3) pass by, generating secondary vortices near the wall. These instantaneous peaks in the Nusselt number were found for Reynolds numbers as low as 500. The study showed a correlation between the location of the instantaneous peak in the Nusselt number and the point of flow reattachment, in front of the secondary (near-wall) vortex. Similarly, Hadzibdic and Hanjalic [23] concluded that this peak, ob-served in a time-averaged way atRe = 20, 000, is caused by the same flow patterns. The authors have studied in further detail this vortex-wall interaction and its influence on the average heat transfer [58], which will be briefly summarized in the following.
Figure 8 shows DNS results of an instantaneous flow field (streamline and vorticity plot) for a submerged jet with a Reynolds number of 1800 and a nozzle-to-plate distance of H/D = 4.5. To support the generation of Kelvin-Helmholtz vortices, a small perturbation of3% at resonance frequency (f = 100Hz) is imposed on the nozzle exit velocity. Within a distance of about x/D = 1.5 from the nozzle exit, the primary vortex (PV) is established and travels towards the wall. Thereby, the vortex generates a local contraction (and expansion) in the potential core of the incoming jet leading to a pulsating flow at the wall. As the vortex approaches the wall, its path begins to diverge from the axis and the vortex travels radially, slowing down and increasing its maximal vorticity. The minimum wall distance of the primary vortex is at about r/D = 1. Shortly after this point, this primary vortex generates a secondary vortex (SV) located ahead of it, closer to the wall in the boundary layer. The initiation of the secondary vortex is illustrated in the streamline plot (top inset of Fig. 8) between r/D = 1.2 and 1.4 by the diverging streamlines. As the secondary vortex grows and travels along the wall, its center begins to move slightly away from the wall together with the primary vortex. When the secondary vortex becomes large (comparable in size to the primary vortex) and reaches a radial distance of about r/D = 2.1, the pair of vortices suddenly separates from the wall in a mild “eruption” [14] as shown in the vorticity plot (lower inset of Fig. 8). Beyond this point, the pair of vortices travels away from the wall, constantly decreasing in vorticity.
This complex steady-periodic flow results in a periodic Nusselt number distribution, which exhibits local in-stantaneous maxima and minima as shown by the grey area in Fig. 9 (left). The most excessive extrema in the instantaneous heat transfer distributions are observed for1.4 < r/D < 2.6. They are clearly related to the activity of the secondary vortices travelling along the wall (see Fig. 9, right). The time-averaged Nusselt number over one period shown by the black line in Fig. 9 reveals that a peak is clearly visible in this region also in the time-averaged form. By comparison to the time-averaged velocity normal to the wall, it is clear that this peak in heat transfer is again related to a peak in inflow of colder fluid, which can be seen in the reattach-ment zone located ahead of the secondary vortex (in Fig. 9, right). In a time-averaged way, the outflow and inflow associated with the separation and reattachment zones, should cancel out, but this balance is disturbed by the vortex detachment from the wall in the vicinity of r/D≈ 2.2. This generates a time-averaged minimum
and maximum of the inflow around this location such that a local peak in the heat transfer coefficient can be observed.
5.3 The hydraulic jump in free-surface jets
Hydrodynamics Researchers have been fascinated by the hydraulic jump ever since the early work of Rayleigh [53], however, “genuine understanding of the phenomenon, is very sparse” [18], especially “con-sidering how simple and common the circular hydraulic jump appears to be, it is surprising that a satisfactory systematic theory does not exist” [69]. For example, the hydraulic jump has been described as a shock (wave) or river bore, white or black hole and as a travelling or standing wave [6, 10, 51, 53]. In addition numerous modelling attempts have been made employing viscous-inviscid flow, as well as shallow water, gas and bound-ary layer theory [4, 12, 73]. The inviscid theory seems to underestimate the hydraulic jump radius and predict a strong dependency on incoming jet diameter, which has not been observed experimentally [70]. Considering viscosity, Bohr et al. [5] present an approach for the location of the hydraulic jump as
Rj = Q5/8ν−3/8g−1/8or: Rj/D = Re3/8F r1/8, (11) where Rjis the jump radius, Q is the volume flow rate,ν is the dynamic viscosity and g is the gravitational ac-celeration. Although good agreement with experiments has been demonstrated under unconfined configurations [24, 50], even surface roughness and plate edge chamfer affect this value [16, 50], while mild confinements significantly reduce the jump radius. Thus, the prediction by Bohr et al. [5] can be considered as an upper limit to the hydraulic jump, while most practical configurations will have a measure of confinement and display up to an order of magnitude lower radii [2, 17]. Under strong confinement, the authors observed that the flow rate dependency given by eq. (11) is preserved, despite jump radii being up to an order of magnitude smaller. Conversely, in turbulent flows, different jump radii and a higher dependency on flow rate have been found experimentally (Rj ∼ Q0.5) [38, 63].
Following the analysis of Rayleigh [53], in assuming mass and momentum conservation across the jump (shock) leads to the relation:
s h = 1 2 1 + 8F r2 h− 1 with: F rh= uj gh3, (12)
where s is the downstream depth (post-jump), h is the upstream depth (pre-jump), and ujthe nominal velocity of the jet. As noted by Liu & Lienhard [38], most expressions for circular hydraulic jumps (including Watson [70]) are a variation of this equation. Their comparison with experiments also demonstrated that this value was somewhat idealized – typically the jump ratio is about 80% of that predicted by eq. (12).
Experimental studies show that typical values of h do not vary significantly (for water from around 0.1mm to 0.3mm over two orders of magnitude in Reynolds [2]), although eq. (12) suggests a strong dependence between downstream depth and jump radius. Using flow-visualization methods it has been observed that not only does the downstream depth play a significant role on jump location, but also on its form. Therein, several types of jumps have been identified which are, in order of increasing depth: For low downstream depths, a single attached vortex occurs near the wall and the high speed main flow is forced upwards (regime A). At higher downstream depths this upwards thrust leads to the generation of a secondary counter-rotating vortex – termed the type B jump. This regime can be split into a further sub-section, wherein at even greater depths the hydraulic jump is conducted in two steps – one similar to that described above and then a second (regime C). Finally, if a critical depth is reached, the hydraulic jump is lost and a “flooded” jet impingement is observed (i.e. a “plunging jet”). This type of flow is often accompanied by air entrainment by the incoming jet causing bubbles in the downstream flow, which can be considered an intermediary step towards a full submerged configuration (regime D). Under turbulent conditions, the wall-jet and jump surface is significantly roughened, and often accompanied by additional gas (bubble) entrainment (regime E).
Recent work on the hydrodynamics of hydraulic jumps has focused on the modes of symmetry breaking (n-polygonal jump, n ≥ 2, [18]), the importance of surface tension and curvature on type I jumps [9] and the wave behavior before and after the jump [50, 69]. Regarding the latter, some disagreement seems to exist with respect to the wave velocity before the jump (in the so-called super-critical zone). While theory predicts a slower wave velocity than the average flow (linear surface waves [69]), experimental observation suggests a 3-4 times faster velocity [50]. Observations of standing waves near the jump by the authors, agree in wavelength and velocity with the capillary wave theory (see Fig. 10), raising the need to consider this theory as well.
0 1 2 3 0.25 5.25 10.25 15.25 20.25 25.25 Cross-secon, x [mm] 0 1 2 3 0 5 10 15 20 25 30 A B E cross-section, x [mm] impinging jet suction S = 14.5 mm liquid thickness, δ [mm] D C
Fig. 10 Hydraulic jump under various configurations: Types of jump in order of increasing depth or flow rate adopted from [10, 38] (left). Photo of two-by-two jet array with liquid suction (top right image) and measured liquid film thickness profiles (D = 1.6mm; upper plot: Re = 1800, H/D = 11, S/D = 9; lower plot: Re = 1350, H/D = 11, S/D = 4).
Heat transfer For heat transfer, two important aspects of the hydraulic jump must be considered: first, a heat transfer reduction due to major near-wall velocity reduction, and, second, the influence of confinement on the location of this heat transfer reduction. The latter comprises of two separate effects – as the jump occurs beyond a super-critical region no information is transferred upstream and heat transfer can be predicted up to the location of the jump using the presented laminar theory, without considering downstream conditions (see section 5.1). On the other hand the region beyond the jump not only represents the area of reduced heat transfer but also the area of highest temperature under uniform heat flux, leading to reduced thermal uniformity and risk of failure - vital to high-power electronics cooling. In practical applications such as free-jet arrays, additional considerations become crucial: local flooding of the impingement area (drainage limitations), impingement reducing cross-flow and average heat transfer level and uniformity. Despite a large number of heat transfer studies on such arrays, due to the complexity of the flow (pre- and post-impingement jet interaction, drainage, heater and jet size dependencies - especially at the micro-scale), a consistent description has not yet been found [19, 54]. Until better prediction and understanding of the hydraulic jump is obtained, its reduction to a minimum will greatly benefit jet-array heat transfer. One of the aims of the present study is to clarify the causes of complexity, to show how to reduce them and also to show how single-jet understanding can be carried over into the application-relevant array. One of the most significant problems in jet arrays is that of cross-flow and drainage (flooding). Both of these effects reduce the heat transfer and more importantly, its
uniformity. In recent studies spear-headed by researchers at IBM [8, 46], liquid extraction points have been implemented between the jets (in a submerged jet array), leading to an increase of the effective cooling and to a several times increased heat transfer. By applying this configuration to free-surface jets the authors were able to avoid flooding completely, study very low jet-plate spacings, and reduce the area taken up by the inter-jet hydraulic jump significantly, in a 2x2 (basic array module) configuration [27], see Fig. 10 top right. Following the observations of Kate [32], the so-called standing fountain regime was obtained, and liquid film thickness measurements were performed even under the unstable standing fountain regime (see Fig. 10 bottom right). Within the latter regime the inter-jet hydraulic jump takes up a very small area (order of one jet diameter, when inter-jet liquid extraction is present), its location is unstable and oscillates, thereby increasing the heat transfer in this degraded area.
6. CONCLUSIONS
Impinging jets are central to a large amount of industrial processes and applications, and as such have been investigated extensively. Despite this fact and the numerous experimental, numerical and analytical studies on this topic, comprehensive descriptions of the basic underlying laminar single-jet (in terms of hydrodynamic and heat transfer) are still incomplete. Thus, the present study attempts to collect the understanding gained by previous studies, together with that obtained in recent studies by the authors in a logical manner by following the flow. While the description given is not necessarily complete – more complex aspects such as turbulence, radial asymmetry and dissipation are not addressed – the present study takes a major step towards a unified description of this fundamental flow-configuration. To do this, direct numerical simulation supported by ex-perimental observations are used to gain deeper insights into the physics of the problem. Thus, the root cause of different phenomena such as heat transfer peaks due to local flow acceleration or local vortices and the influence of nozzle-to-plate distances are revealed. The side-by-side analysis of free-surface and submerged jets clarifies differences and similarities, thus maximizing the understanding that can be carried-over between the two flow configurations.
The key findings of the present study are highlighted along the path of the flow: Beginning with a velocity profile emerging the nozzle, two limiting cases (parabolic and uniform profiles) have been examined in the present study. While the jet travels towards the wall, the velocity profile flattens due to viscous relaxation, which is strongly dependent on the flow configuration (submerged jet or free-surface jet). The relaxation pro-cess scales similarly to the velocity profile development at the pipe inlet, allowing to formulate a self-similar description of the velocity profile evolution. As the jet travels, instabilities may occur under certain conditions – though they are highly sensitive to the noise-level at the nozzle exit. For free-surface jets this is typically the Rayleigh-Plateau (at long travel distances), while for submerged jets the high-shear at the jet boundary leads to rapid development of Kelvin-Helmholtz vortices under much more common/ mild conditions. These instabilities introduce new length and time scales and thus reduce the extend of self similarity in the system. Using the scaling identified for the idealized case reveals the location at which the jet begins to significantly sense the wall, e.g. transition to the stagnation zone. Beyond this location, the flow can be well approximated by a potential-type equation. The shape of the velocity profile at this specific location is found to directly dictate the local heat transfer distribution within the stagnation zone (r/D≤ 0.5). If the approaching velocity profile is more or less uniform (either due to the relaxation process in case of free-surface jets or due to a uniform velocity profile at the nozzle exit), an off-center peak in the heat transfer distribution of submerged and free-surface jets can occur. With a previously proposed Prandtl number dependency and the identified scaling, a physically-driven correlation for the heat transfer of free-surface jets covering a wide range of laminar conditions is proposed (eq. 10). Beyond the stagnation region in the wall-jet region the scaling behavior changes for free-surface jets, such self-similarity is obtained if the radial coordinate is additionally scaled by Reynolds number (see Watson [70]). This scaling behavior is also found in the piece-wise theory of Liu and Lienhard [35], whose assumptions were validated in the present study against detailed simulations. In the submerged jet
case, the commonly observed secondary off-center peak in the heat transfer, often associated with transition to turbulence, has been shown to occur due to an temporal asymmetry of the vortex-wall interaction. Finally, it was shown that major difficulties still exist in predicting the behavior and location of the hydraulic jump, and a method for reducing its negative influence in jet arrays was presented.
We gratefully acknowledge the financial support from the Deutsche Forschungsgemeinschaft (grant number DFG KN 764/15-1).
REFERENCES
[1] Angioletti, M., Tommaso, R. D., Nino, E., and Ruocco, G., “Simultaneous visualization of flow field and evaluation of local heat transfer by transitional impinging jets,” Int. J. Heat Mass Tran, 46(10), pp. 1703 – 1713, (2003). [2] Baonga, J., Louahlia-Gualous, H., and Imbert, M., “Experimental study of the hydrodynamic and heat transfer of
free liquid jet impinging a flat circular heated disk,” Applied Thermal Engineering, 26(11-12), pp. 1125 – 1138, (2006).
[3] Bhunia, A. and Chen, C. L., “Pressure drop in generating free-surface liquid microjet array from short cylindrical orifices,” J. Fluids Eng., 133(6), pp. 061103, (2011).
[4] Birkhoff, G. and Zarantonello, E., Jets, wakes, and cavities, Applied mathematics and mechanics, Academic Press, (1957).
[5] Bohr, T., Dimon, P., and Putkaradze, V., “Asymptotic properties of the circular hydraulic jump,” J. Fluid Mech., 254, pp. 635, (1993).
[6] Bohr, T., Putkaradze, V., and Watanabe, S., “Averaging theory for the structure of hydraulic jumps and separation in laminar free-surface flows,” Phys. Rev. Lett., pp. 1038–1041, (1997).
[7] Broze, G. and F., H., “Nonlinear dynamics of forced transitional jets: Periodic and chaotic attractors,” J. Fluid Mech., 263, pp. 93–132, (1994).
[8] Brunschwiler, T., Rothuizen, H., Fabbri, M., Kloter, U., Michel, B., Bezama, R. J., and Natarajan, G., “Direct liquid jet-impingment cooling with micron-sized nozzle array and distributed return architecture,” Thermal and Thermomechanical Phenomena in Electronics Systems, ITHERM, pp. 196–203, (2006).
[9] Bush, J. W. M. and Aristoff, J. M., “The influence of surface tension on the circular hydraulic jump,” J. Fluid Mech., 489, pp. 229–238, (2003).
[10] Bush, J. W. M., Aristoff, J. M., and Hosoi, A. E., “An experimental investigation of the stability of the circular hydraulic jump,” J. Fluid Mech., 558, pp. 33–52, (2006).
[11] Cho, H. H., Kim, K. M., and Song, J., (2011). Cooling Systems: Energy Engineering and Applications. Nova Science Publishers, Inc., Ch. Applications of impingement jet cooling system, pp. 37–67.
[12] Chow, V. T., Open-channel hydraulics, McGraw-Hill civil engineering series, McGraw-Hill, (1973).
[13] Chung, Y., Luo, K., and Sandham, N., “Numerical study of momentum and heat transfer in unsteady impinging jets,” Int. J. Heat Fluid Flow, 23(5), pp. 592 – 600, (2002).
[14] Doligalski, T. L., Smith, C. R., and Walker, J. D. A., “Vortex interactions with walls,” Anu. Rev. Fluid Mech., 26, pp. 573–616, (1994).
[15] Durst, F., Ray, S., ¨Unsal, B., and Bayoumi, O. A., “The development lengths of laminar pipe and channel flows,” J. Fluids Eng., 127, pp. 1154–1160, (2005).
[16] Ead, S. and Rajaratnam, N., “Hydraulic jumps on corrugated beds,” J. Hydraul. Eng., 128(7), pp. 656–663, (2002). [17] Ellegaard, C., Hansen, A. E., Haaning, A., and Bohr, T., “Experimental results on flow separation and transitions
in the circular hydraulic jump,” Phys. Scr., 1996(T67), pp. 105, (1996).
[18] Ellegaard, C., Hansen, A. E., Haaning, A., Hansen, K., Marcussen, A., Bohr, T.and Hansen, J. L., and Watanabe, S., “Cover illustration: Polygonal hydraulic jumps,” Nonlinearity, 12(1), pp. 1–7, (1999).
[19] Fabbri, M. and Dhir, V. K., “Optimized heat transfer for high power electronic cooling using arrays of microjets,” J. Heat Transfer, 127(7), pp. 760–769, (2005).
[20] Gardon, R. and Akfirat, J., “The role of turbulence in determining the heat-transfer characteristics of impinging jets,” Int. J. Heat Mass Tran, 8(10), pp. 1261 – 1272, (1965).
[21] Garimella, S. V. and Nenaydykh, B., “Nozzle-geometry effects in liquid jet impingement heat transfer,” Int. J. Heat Mass Tran, 39(14), pp. 2915 – 2923, (1996).
[22] Grassi, W. and Magrini, A., “Effect of the free jet fluid-dynamics on liquid jet impingement heat transfer,” 9th National Heat Transfer Conference of Italy, (1991).
[23] Hadziabdic, M. and Hanjalic, K., “Vortical structures and heat transfer in a round impinging jet,” J. Fluid Mech., 596, pp. 221–260, (2008).
[24] Hansen, S. H., Hørl¨uck, S., Zauner, D., Dimon, P., Ellegaard, C., and Creagh, S. C., “Geometric orbits of surface waves from a circular hydraulic jump,” Phys. Rev. E, 55, pp. 7048–7062, (1997).
[25] Harmon, D. B., “Drop sizes from low speed jets,” J. Franklin Inst., 259(6), pp. 519 – 522, (1955).
[26] Haustein, H., Tebr¨ugge, G., Rohlfs, W., and Kneer, R., “Local heat transfer coefficient measurement through a visibly-transparent heater under jet-impingement cooling,” Int. J. Heat Mass Transfer, (2012).
[27] Haustein, H. D., Joerg, J., Rohlfs, W., and Kneer, R., “Influence of micro-scale aspects and jet-to-jet interaction on free-surface liquid jet impingement for micro-jet array cooling,” Thermal and Thermomechanical Phenomena in Electronics Systems, ITHERM, (2014).
[28] Ho, C.-M. and Huerre, P., “Perturbed free shear layers,” Ann. Rev. Fluid Mech., 16, pp. 365–424, (1984).
[29] Homann, F., “Der Einfluss großer Z¨ahigkeit bei der Str¨omung um den Zylinder und um die Kugel,” J. Appl. Math. Mech., 16(3), pp. 153–164, (1936).
[30] Hussain, A. K. M. F. and Zaman, K. B. M. Q., “Vortex pairing in a circular jet under controlled excitation. part 2. coherent structure dynamics,” J. Fluid Mech., 101(3), pp. 493–544, (1980).
[31] Kataoka, K., Suguro, M., Degawa, H., Maruo, K., and Mihata, I., “The effect of surface renewal due to largescale eddies on jet impingement heat transfer,” Int. J. Heat Mass Transfer, 30(3), pp. 559 – 567, (1987).
[32] Kate, R. P., Das, P. K., and Chakraborty, S., “An experimental investigation on the interaction of hydraulic jumps formed by two normal impinging circular liquid jets,” J. Fluid Mech., 590, pp. 355–380, (2007).
[33] Lienhard, J. H., “Heat transfer by impingemnt of circular free-surface liquid jets,” 18th National & 7th ISHMT-ASME Heat and Mass Transfer Conference, Jan. 4-6, Guwahati, India, (2006).
[34] Lienhard, J. H. and Lienhard, J. H., “Velocity coefficients for free jets from sharp-edged orifices,” J. Fluids Eng., 106, pp. 13–17, (1984).
[35] Liu, X., Gabour, L. A., and H., L. J., “Stagnation-point heat transfer during impingement of laminar liquid jets: Analysis including surface tension,” J. Heat Transfer, 119(3), pp. 99–105, (1993).
[36] Liu, X. and Lienhard, J. H., “Liquid jet impingement heat transfer on a uniform flux surface,” Heat Transfer Phenomena in Radiation, Combustion, and Fires, (1989).
[37] Liu, X., Lienhard, J. H., and Lombara, J. S., “Convective heat transfer by impingement of circular liquid jets,” J. Heat Transfer, 113(3), pp. 571–582, (1991).
[38] Liu, X. and Lienhard, J.H., V., “The hydraulic jump in circular jet impingement and in other thin liquid films,” Exp. Fluids, 15(2), pp. 108–116, (1993).
[39] Livingood, J. N. B. and Hrycak, P., “Impingement heat transfer from turbulent air stream jets to flat plates – a literature survey,” , Tech. rep., NASA Technical Memorandum, (1973). NASA TM X-2778.
[40] Lytle, D. and Webb, B., “Air jet impingement heat transfer at low nozzle-plate spacings,” Int. J. Heat Mass Transfer, 37(12), pp. 1687 – 1697, (1994).
[41] Ma, C., Zhao, Y., Masuoka, T., and Gomi, T., “Analytical study on impingement heat transfer with single-phase free-surface circular liquid jets,” J. Therm. Sci., 5(4), pp. 271–277, (1996).
[42] Ma, C. F., Gan, Y., Tian, Y., Lei, D., and Gomi, T., “Liquid jet impingement heat transfer with or without boiling,” J. Therm. Sci., 2, pp. 32–49, (1993).
[43] Ma, C. F., Zheng, Q., and Ko, S. Y., “Local heat transfer and recovery factor with impinging free-surface circular jets of transformer oil,” Int. J. Heat Mass Transfer, 40(18), pp. 4295–4308, (1997).
[44] Martin, H., “Heat and mass transfer between impinging gas jets and solid surfaces,” Adv. Heat Transfer, 13, pp. 1–60, (1977).
