Second-moment closure simulation of flow and heat transfer in a gas-droplets turbulent impinging jet

Second-moment closure simulation of flow and heat transfer in a gas-droplets turbulent impinging jet

International Journal of Thermal Sciences 60 (2012) 1e12 Contents lists available at SciVerse ScienceDirect International Journal of Thermal Science...

807KB Sizes 0 Downloads 299 Views

International Journal of Thermal Sciences 60 (2012) 1e12

Contents lists available at SciVerse ScienceDirect

International Journal of Thermal Sciences journal homepage:

Second-moment closure simulation of flow and heat transfer in a gas-droplets turbulent impinging jet Maksim A. Pakhomov, Viktor I. Terekhov* Laboratory of Thermal and Gas Dynamics, Kutateladze Institute of Thermophysics Siberian Branch of Russian Academy of Sciences, 630090, 1, Acad. Lavrent’ev Avenue, Novosibirsk, Russia

a r t i c l e i n f o

a b s t r a c t

Article history: Received 14 October 2011 Received in revised form 30 May 2012 Accepted 30 May 2012 Available online 4 July 2012

The numerical model for description of flow dynamics and heat transfer in an impinging axisymmetric gas-droplets jet is presented. The Eulerian model uses for computations of the impinging gas-droplets jet. In this paper the two-phase turbulent jet is numerically predicted by the set of axisymmetic Reynolds averaged NaviereStokes equations. The flow structure and heat transfer in the gas-droplets impinging spray with low mass fraction of droplets (ML1 1%) is studied numerically. Gas phase turbulence is modeled with the use of Reynolds stress transport model for two-phase flow. Droplets addition causes a significant increase in heat transfer intensity (almost twice) in comparison with a single-phase impinging air jet in the stagnation zone. In the region of wall jet the heat transfer intensity in the two-phase impinging jet decreases and approaches the value of a single-phase impinging jet. Ó 2012 Elsevier Masson SAS. All rights reserved.

Keywords: Impinging two-phase jet Heat transfer Turbulence modeling Second-moment closure Two-way coupling

1. Introduction Flow and heat transfer due to impinging single-phase and gasdroplets jets is a subject of considerable interest from theoretical and practical points of view. There are numerous papers dealing with this problem both numerically and experimentally. A number of reviews appeared (see Refs. [15]). Description of the flow and heat and mass transfer in the stagnation zone of impinging jet is an interesting problem for many practical applications (spray painting, spray cooling of electronics packages and gas turbine blades, drying and quenching of various materials, etc.). The flow in a round impinging jet is very complex and three areas can be identified in the impinging jet: zone of free submerged jet (I), region of stagnation (II) and area of propagation of radial wall jet (III) (see Fig. 1). Here xH =R ¼ 10e12 is the length of potential core of the free jet. These are illustrated in Fig. 1. The free jet region (I) is the region that is largely unaffected by the presence of impingement surface; this exists at the distance of about 1.5 pipe diameters from the impingement surface. The flow in stagnation region (II) is nearly irrotational and there is a large total strain along the streamline. It turns in the radial direction with substantial curvature. The rate of local heat and mass transfer in this zone has

* Corresponding author. E-mail address: [email protected] (V.I. Terekhov). 1290-0729/$ e see front matter Ó 2012 Elsevier Masson SAS. All rights reserved.

the highest value. At the stagnation point where the mean velocity is zero and within this zone the jet is deflected into the wall jet flow. The wall jet zone (III) extends beyond the radial limits of the stagnation region. Turbulent impinging jets have complex features due to entrainment, stagnation and high streamline curvature. A brief review of some of the previous efforts concerning impinging air jet heat transfer and RANS simulations follows below. The researches of [69] have revealed disadvantages of various ke3 models. The two-equation models have some well-known shortcomings. These models do not describe turbulent stress anisotropy, and, moreover, lead to considerable errors in modeling of strongly non-equilibrium flows with high velocity gradients due to strongly curved flows, etc. The two-equation k3 models can be applied only in modeling of quasi-equilibrium flows characterized by approximate equality between production and dissipation of the turbulence energy. These models overpredict the values of turbulent kinetic energy (TKE) at the critical point almost by 75%, what leads to an increase in Nusselt number of up to 100% [6e8]. The spreading of wall jet and temperature along its axis are also calculated incorrectly. The velocity is underpredicted near the wall and it is overpredicted in the outer zone of the wall jet. For more accurate consideration of streamline curvature in the stagnation zone and for calculation of TKE generation different corrections are used: KatoeLaunder modification [10] and Durbin’s correction [11]. The poor performance of two-equation models is


M.A. Pakhomov, V.I. Terekhov / International Journal of Thermal Sciences 60 (2012) 1e12

Nomenclature AR B b1D

anisotropy ratio slot width (m) diffusion parameter of injection, determined with the use of saturation curve d droplet diameter (m) drag coefficient of evaporating droplet, written with CD the taking into account the deviation of the Stokes law specific heat capacity (J kg1 K1) CP D diffusion coefficient (m2 s1) d droplet diameter (m) H the distance between the pipe exit cross-section and impinging flat plate (m) H/(2R) dimensionless distance between pipe exit crosssection and impinging flat plate J mass flux of steam from the surface of evaporating droplet (kg m2 s1) K mass concentration in the binary vaporegas mixture mass vapor concentration at the “vaporegas mixture KV edroplet” interface, corresponding to saturation parameters at droplet temperature TL KV0 mass concentration of steam far from the droplet k ¼ hui ui i=2 turbulent kinetic energy (TKE) (m2 s2) L the latent heat of vaporization, (J kg1) M mass fraction in the triple vaporegas-droplets system droplets mass fraction ML Nu Nusselt number P pressure (Pa) production of turbulent kinetic energy (m2 s3) Pk Pr Prandtl number R pipe radius (m) R specific gas constant (J kg1 K1) ! ! ReL ¼ j U  U L jd=n Reynolds number of dispersed phase r radial coordinate (m) Sc Schmidt number Sh Sherwood number diffusional Stanton number StD T temperature (K) component of averaged velocity (m s1) Ui friction velocity (m s1) U* mean droplet velocity in the wall region (m s1) UWL DU slip velocity between droplet and gas phase (m s1) 0 0 hu i; hv i intensity of velocity pulsation in axial and radial directions (m s1) turbulent Reynolds stresses (m2 s2) huvi0 turbulent heat flux (m K s1) huj ti turbulent diffusion flux (m s1) huj kV i huLi uLj i kinetic stresses in the dispersed phase (m2 s2) x axial coordinate (m) length of initial region of the jet (m) xH y distance normal to the wall (m)

not surprising since they were developed and tested for the flows parallel to the wall. Physical phenomena involved in impinging flows on a solid surface are substantially different and therefore, these models do not provide the required accuracy at simulation of complex flows, e.g., impinging jets. All two-equation models use a single-point approach that cannot represent the non-local effects of pressure-reflection that occur near the solid boundaries. Now there are two main ways to model these complex turbulent flows: large eddy simulation (LES) and second-moment closure

yþ y0 We We*

dimensionless distance from the wall dimensionless transverse coordinate from the wall, 2=3 W ¼ 1 þ 2ReL =3 Weber number critical Weber number

Greek F ¼ MLr/rL volumetric concentration of dispersed phase GE turbulence macroscale (m) UtL time of interaction with temperature pulsations of the carrying flow (s) ULag Lagrangian integral time scales (s) UE Eulerian integral time scales (s) U3 time microscale of turbulence (s) a heat transfer coefficient of evaporating droplet (W m2 K1) b mass transfer coefficient (m s1) 3 dissipation of the turbulent kinetic energy (m2 s3) m dynamic viscosity (kg m1 s1) n kinematic viscosity (m2 s1) turbulent heat flux (K m s1) hqL uLj i 2 droplet temperature fluctuations (K2) hqL i r density (kg m3) s surface tension (N m1) s dynamic relaxation time of droplets (s) sf turbulent time scale (s) sQ thermal relaxation time of droplets (s) Superscripts * parameter on the droplet surface Subscripts 0 parameter under conditions at the stagnation point 1 parameter under initial conditions A air L droplet P particles (without evaporation process) T turbulent parameter V steam W parameter on the wall condition m mean-mass parameter Acronyms AR anisotropy ratio CV control volume DNS direct numerical simulation LES large eddy simulation LRN low Reynolds number RANS Reynolds averaged NaviereStokes SMC second-moment closure TKE turbulent kinetic energy

(SMC). In the LES method all turbulent scales above the cell size are computed directly by equations. The sub-grid scales are modeled with the use of various approaches. The LES computations give very detailed pattern of flow and heat transfer in the impinging jets. But now this method has relatively high cost for computations. More accurate modeling of the turbulent flow and then two-equation models can be based on the solution to the system of transport equations for the components of Reynolds stresses. The SMC of turbulence modeling predicts the turbulent Reynolds stresses

M.A. Pakhomov, V.I. Terekhov / International Journal of Thermal Sciences 60 (2012) 1e12

Fig. 1. The scheme of round impinging jet.

directly from differential equations. The SMC allows us to compute the flow with anisotropic and non-equilibrium effects. It is much more detailed than the two-equation models and has the lower computational cost than LES. One of the ways to enhance heat transfer in the jet impingement is the use of gas-droplet mist impinging jet [12e22]. The mist impinging jets are commonly employed in many industrial applications, e.g., spray cooling of heated surface coatings, electronic components and gas turbine blades, spray coating and painting processes, etc. Interaction between the axisymmetric gas-droplets impinging jet with flat surface is studied theoretically in [12]. The cases of mono- and polydispersed jets are examined. Considerable heat augmentation was shown in comparison with a single-phase flow. Papers [15,16,18] deal with experimental [15] and numerical [16,18] investigations of steam-droplets slot jet impingement heat transfer on the flat and concave surfaces. The impinging two-phase jet is numerically simulated with the use of the CFD-package “FLUENT” with application of EulerianeLagrangian method. Gas turbulence is described by the k3 model. The case of low initial mass concentration of water droplets (ML1 < 2%) and mean arithmetic diameter (d1 ¼10 mm) is considered. They concluded that stagnation point heat transfer is enhanced over 200% by the addition of 1.5% of mist. Heat transfer in a round impinging gas-droplet jet is studied in [17] with application of commercial CFD-package “FLUENT” with the use of RNG k3 model. Droplets trajectories are calculated by the Lagrangian approach. The influence of the main thermal- and gasdynamic parameters of the two-phase flow (such as mass concentration of droplets, their initial size, gas velocity, pipe diameter, and pipe exit-to-plate distance) on heat transfer is examined. It shows that in the gas-droplets flow heat transfer increases by the factor of about 1.5 in comparison with the singlephase impinging jet. Perhaps, the only one experimental study of the gas-droplet impinging jets under conditions of impinging surface cooling without boiling is [19]. Heat transfer increases significantly (several times) with addition of droplets in comparison with the singlephase impinging jet of air. The gas-droplet impinging jet is investigated in [21,22]. The authors use the Eulerian approach [23e27] with application of the modified ke3 model [28]. The modification takes into account the presence of evaporating droplets. The flow and heat transfer is performed for the confined and unconfined configuration. Addition of droplets causes significant increase in heat transfer intensity around the stagnation point in comparison with the single-phase air impinging jet. The presence of the confined upper surface decreases wall friction and heat transfer rate, but a change in friction and heat transfer coefficients at the stagnation point is


insignificant. The effect of confinement on heat transfer is observed only at very small distances between pipe exit and target surface (H/(2R) < 0.5) both in the single-phase and mist impinging jets. In the literature there are no works dealt with modeling of gasdroplet impinging jets with application of SMC. The only exclusion is [18], where the one-component steam-droplet impinging jet is studied with the use of SMC. In the paper the two-phase (droplets and vaporegas mixture) and two-component (air and steam) systems are considered. This should be emphasized because in the case of two-component mixture it is necessary to solve equation of diffusion of vapor to the binary air and steam flow. In the study the possibility of computations of gas-droplets mist impinging turbulent jet with the use of SMC and the effect of droplets evaporation on the flow and heat transfer are presented. The present paper continues latter numerical simulations of heat transfer in the gas-droplet jet impingement [21,22]. 2. Mathematical model The motion of the gas-droplet impinging jet consisting of incompressible viscous flow and small droplets is considered. The droplets behavior in turbulent fluid and their back action on the flow is determined by drag, gravity force, turbulent transport and turbulent diffusion, while the forces associated with displaced masses and the Basset force can be neglected. In order to account for the interaction between phases, i.e., momentum exchange and heat and mass transfer, the conservation equations have to be extended by appropriate source/sink terms. The modeling of dispersed phase is accomplished by the Eulerian approach that treats the particulate phase as a continuous medium with properties analogous to those of a fluid [29]. The technique involves the solution to the set of RANS equations in addition to those of the gas phase. The Eulerian or two-fluid approach is used for computation of two-phase flows [2127,29e32]. In the two-fluid approach both phases are considered as interacting continua. Properties such as the mass of particles per unit volume are considered as a continuous property and the particle velocity is the averaged velocity over an averaging control volume. Also the interfacial transfer of mass, momentum and energy requires averaging over the computational control volume. The dispersed phase is modeled by the Eulerian approach based on kinetic equations for one-point probability density function (PDF) of particle coordinate, velocity, and temperature in the turbulent Gaussian fluid flow fields [23e27]. The PDF method is a powerful tool used in continuum mechanics for developing hydrodynamic models. In particular, introducing the PDF permits to proceed from the dynamic stochastic description of separate particles to the statistical modeling of the behavior of a particle ensemble immersed into the turbulent flow. The PDF modeling approach starts from the Lagrangian equations for individual particles, and the kinetic equation is derived that governs the PDF of particle coordinate, velocity, temperature, size, and other variables of interest. From the kinetic equation, we can gain a set of continuum conservation equations for the statistical moments of the PDF. These conservation equations govern the averaged properties of the dispersed phase in the framework of the Eulerian modeling formalism. Initially this model was developed for description of turbulent flows with solid particles without heat transfer between the phases. The authors used this model successfully for simulations of gasdroplet jet with droplets evaporation (see, for example, [21,22]). Gas phase turbulence is modeled by the SMC model of [33]. The back effect of the dispersed phase on transport process in the gas phase is taken into account in this paper. The terms describing the effects of the dispersed phase on the gas phase in


M.A. Pakhomov, V.I. Terekhov / International Journal of Thermal Sciences 60 (2012) 1e12

these equations are referred to as the two-way coupling. The main basic of the two-way coupling is an existence of relative motion between particles and the carrier fluid that leads to an extra dissipation or generation of gas turbulence energy. The two-way coupling problem is as follows: carrier phase turbulence modifies particle behavior that, in turn, modifies turbulence of the gas phase. The feedback effect is the reduction of gas turbulence by interaction of droplets with turbulent eddies [29,30]. The two-way coupling terms can be evaluated by considering the jump condition at the interfaces. Volume concentration of dispersed phase F1 ¼ ML1r/rL < 104 is assumed to be sufficiently small so inter-particle collisions can be neglected and initial droplet diameter is d1 100 mm. No coalescence occurs in the flow because the amount of the dispersed phase is small, and droplets never undergo the brake-up [21,22,34]. The preliminary calculations with application of one-way and two-way couplings showed that within the given range of initial volumetric concentrations and sizes of droplets the feedback onto the mean motion of the gas phase will be insignificant (less than 3%). The influence on turbulence is more obvious (the maximum is up to 20%) and it cannot be neglected. To make model writing more general and for the case of high droplet concentrations, the effect of two-way coupling model on the mean flow and turbulence of the carrier phase is taken into account in the current study. The Weber number based on fluid flow conditions is ! ! We ¼ rj U  U L j2 d=s  1, and the Weber number based on the ! wall conditions is WeW ¼ rLW j U WL j2 dW =sW < 50. The mechanism of interaction between the droplet and heated wall is beyond the limits of the current research; we consider only heat transfer between the heated wall and deposited droplets. The main assumption of the work is the idea that the particles deposited on the wall from the two-phase flow evaporate momentarily, always leaving the wall surface dry. This assumption is quite valid for heated surface and it is used if the difference between wall temperature TW and droplet temperature TWL is TW  TWL  40 (see [34]). Concentration of particles decreases steadily in the downstream direction due to their deposition onto the plate surface and due to wall jet expansion. The droplet temperature is assumed to be uniform over the droplet radius, and droplets are considered spherical. The system of governing mean equations for the gas phase. The system of Reynolds averaged NaviereStokes axisymmetric equations, is shown as [21,22]:


vUj 6J ¼ F d vxj

  v U i Uj vðP þ 2k=3Þ v ¼  þ rvxi vxi vxj


  vU n i  ui u j vxj

!  ðUi  ULi Þ



(2)     n vT   vðUi TÞ v ðC  CPA Þ vKV vT ¼  uj t þ DT PV vxi vxi vxi vxi Pr vxi CP 6F  ½aðT  TL Þ þ JL rCP d

The continuity (1), impulses (2), energy (3), and diffusion (4) equations involve the source and sink terms that model the effect of droplets on transport processes in the mean motion. Here s ¼ rL d2 =18mW is the relaxation time of droplets. Turbulent heat and diffusion flows in the gas phase are determined according to Boussinesq hypothesis:

  n vT ; uj t ¼  T PrT vxj

  n vKV uj kV ¼ T ScT vxj

The values of turbulent Prandtl and Schmidt numbers in this study are taken equal: PrT ¼ ScT ¼ 0.85 as in [21]. The second-moment closure model for the two-phase flow consists of the system of equations for the second moments in the single-phase impinging flow and equation of TKE dissipation [33] with consideration of the effect of the dispersed phase on gas turbulence [24]. The method allows consideration of turbulence attenuation as a result of additional dissipation; however, it does not describe the mechanism of turbulence enhancement due to formation of an unsteady vortex wake. The second-moment closure transport equations and equation of dissipation rate of the gas turbulence are given as

   v Uj ui uj ¼ dij þ Pij þ Aij  3 ij  AP vxj


" !2 #    2 v Uj 3 3 vUi 3 v k v3 C3 hul uk i  3P  C3 2  ¼ C3 1 Pk þ nT vxj k vxj k vxk 3 vxl (7) Last terms in Eqs. (6) and (7) include an additional summand for description of interfacial interaction. The Eq. (6) together with the standard terms describes turbulent diffusion, production of turbulent stresses from the shear averaged motion, redistribution of pulsation energy between different components of velocity fluctuations caused by correlation of pressure and deformation rate pulsations, and viscous dissipation, including interfacial interaction AP. The diffusion term (turbulent mixing at interaction of fluctuation velocity components) takes form

dij ¼ CS

  v k v  ui uj hul uk i vxk 3 vxl

Production of turbulent stresses from the averaged motion (intensity of energy transfer from averaged to fluctuation motion)

   vUi vUj  Pij ¼  hui uk i þ uj u k vxk vxk The pressureestrain term considering energy exchange between different components of pulsations due to pressureedeformation rate correlations takes form [33]

Aij ¼ C1

     2 1 ui uj  dij k  C2 Pij  Pkk dij þ AW ij k 3 3 3


where the near-wall effects in Eq. (8) are written as

   n vKV  vðUi KV Þ v 6J F ¼  uj kV þ d vxi vxi Sc vxi


W W AW ij ¼ fij1 þ fij2

r ¼ P=ðRTÞ


fW ij1 ¼ CW1 fy hul um inl nm dij  hui um inm nj 



3 2

 3 u um nm ni 2 j


M.A. Pakhomov, V.I. Terekhov / International Journal of Thermal Sciences 60 (2012) 1e12

  vUl 1 kfy hul um i nq nq dij  3ni nj fy  CW2 vxm   vUk 3 vUi 3 vUj  nl nk alm dij  nl nj alm  nl ni alm vxm 2 vxm 2 vxm    vUl 1 2 nl nm ni nj  nq nq dij þ CW2 kfy  3 vxm

fW ij2 ¼ CW2

aij ¼ hui uj i=k  ð2=3Þdij is anisotropic stress and fy ¼ k3=2 =ðCl y3 Þ is the function of turbulent scale of length. The isotropic concept is used for dissipative term:3 ij hð2=3Þdij 3 . Other constants in Eqs. (6) and (7) look like [36]: CS ¼ 0.18, 1 2 ¼ 0:1; CW2 ¼ 0:4, C1 ¼1.8, C2 ¼ 0.6, CW1 ¼ 0.5, CW2 ¼ 0.08, CW2 C3 1 ¼1.44, C3 2 ¼ 1.92 and C3 ¼ 0.18. Here Cl ¼ 2.5, Pk ¼ nT ððvUi =vxj Þ þ ðvUj =vxi ÞÞvUi =vxj is the production of turbulent kinetic energy. The detailed description of the SMC model is presented and tested in [6,31]. Last summands in the right part of Eqs. (6) and (7) take into account the back effect of dispersed phase on transport processes. Terms AP and 3 P determine additional dissipation of gas turbulence due to pulsation motion and take the form of [24]

AP ¼

2rL F


ð1  fu Þhui ui i;



2rL 3


½Fð1  f3 Þ


Here fu ¼ 1  expðU3 L =sÞ and U3 L is time of particle interaction with turbulent vortex [23]

( 3L





2ðhu2 iÞ1=2 $ULag

v rL ULj 6J F ¼  vxj d      v rL FULj ULi v rL F uLi uLj r þ ¼ FðUi  ULi Þ L þ FrL g s vxj vxj   1 v rL DLij F  s vxj     v rL FULj TLi r v  r F quLj þ ¼ FðTi  TLi Þ L sQ vxj L vxj

Q 1 v rL DLij F  sQ vxj


vTL vr


¼ a T  T *  JL

and equation of vapor mass conservation on evaporating surface of the droplet

J ¼ JKV*  rD

vKV* vr


where KV* is vapor concentration at the “vaporegas mixtureedroplet” interface, corresponding to saturation parameters at droplet temperature TL. Considering the diffusion Stanton number StD takes form

StD ¼ rD

i vKV* h ! ! * = rð U  U L Þ KV  KV0 vr

Eq. (13) can be written as


where G ¼ is macroscale of turbulence, ULag ¼ 0.608UE and UE ¼ 0.22k/3 are Lagrangian and Eulerian time macroscales of turbulence. Here f3 ¼ 1  expðT3 =sÞ, g3 ¼ U3 =s  f3 where U3 ¼ ð15n=3 Þ1=2 is time microscale of turbulence [24]. Mean equations for the dispersed phase. The system of mean equations describing momentum and energy transport in the dispersed phase takes form [21,22]

Kinetic stresses model for dispersed phase. Equations of the kinetic stresses, turbulent heat flux and temperature fluctuations in the dispersed phase that used in the system of Eqs. (11) and (12) have the form [24]. Heat and mass transfer on the surface of evaporating droplet. The system of Eqs. (1)e(12) is supplemented by the equation of heat transfer at the interface under the condition of constant temperature over the droplet radius

! ! J ¼ StD rð U  U L Þb1D


UE ; j U  U L jUE  GE ! ! ! ! GE =j U  U L j; j U  U L jUE >GE



NuP ¼ aP d=l ¼ 2 þ 0:6ReL Pr 1=3 and ShP ¼ bd=D 1=2

¼ 2 þ 0:6ReL Sc1=3 : Here NuP and ShP are Nusselt and Sherwood numbers of a nonevaporating particle and b is mass transfer coefficient. Taking into account that the diffusion Stanton number can be determined from expression StD ¼ StP =ðReL ScÞ, then the final form of Eq. (14) is

s s where DLij ¼ sðhuLi uLj i þ gu hui uj iÞ, DQ Lij ¼ Q huLj tL i þ gut huj ti are turbulent diffusivity tensor and particle turbulent heat transport tensor [24], sQ ¼ CPL rL d2 =12lY is the thermal relaxation time, 1=2 Y ¼ ð1 þ 0:3ReL Pr 1=3 Þ, and gut ¼ UtL =sQ  1 þ expðUtL =sQ Þ is coefficient of particle involvement into pulsations of gas phase temperature. In the first approximation, let us take UtL zU3 L according to [21].Correlation functions fu, f3 , g3 , fut, and gu are the coefficients of droplet involvement into the macropulsation and micropulsation motion of the gas phase. In detail their assumptions used for their derivation are presented in [23,24,27].

! ! 1=2 2 þ 0:6ReL Sc1=3 rð U  U L Þb1D =ðReL ScÞ:

According to [34], heat transfer coefficient for evaporating droplets a is connected with the similar coefficient of nonevaporating particles aP via the following relationship

a ¼ (12)

(14) KV* Þ

where b1D ¼  KV 0 Þ=ð1  is diffusion parameter of injection, determined with the use of saturation curve. Equations of heat and mass transfer from the surface of non-evaporating sphere have the form of the known equations of [35]

J ¼ (11)


aP 1 þ CP ðT  TL Þ=L

The equation of material balance for the binary vaporeair mixture is: KA þ KV ¼ 1. For the triple mixture of vaporegaseliquid it is written as: MA þ MV þ ML ¼ 1. The link between the values of mass concentrations of mixture components Ki and Mi can be written as: KV [ MV/(MA þ MV); KA ¼ MA/(MA þ MV) ¼ 1eKV


M.A. Pakhomov, V.I. Terekhov / International Journal of Thermal Sciences 60 (2012) 1e12

border. At the outer edge of computational domain (the wall jet periphery) conditions vf=vr ¼ 0 for all variables are set. The components of Reynolds stresses are determined at the same points on the control volume surface as the corresponding components of averaged gas velocities by the method of [33]. 4. The validation analysis for the single-phase impinging turbulent jet

Fig. 2. Grid convergence test. 1 e 100  128 control volumes, 2 e 200  256, 3 e 400  400.

3. Boundary conditions and numerical realization Numerical solution is obtained by the method of finite volumes at the staggered grid. For convective terms of differential equations the QUICK procedure is used. For diffusion terms flows secondorder central differences are applied. The pressure field is corrected by the finite-volumetric agreed SIMPLEC procedure. The computational domain is a cylinder with a size of 20R in the radial direction and height H. The computational grid, non-uniform both in axial and radial directions, is applied. The first cell is located at distance yþ ¼ yU*/n ¼ 0.3e0.5 from the wall, where U* is the friction velocity, obtained for the flow in the pipe. At least 10 CVs has been generated to be able to resolve the mean velocity field and turbulent quantities in the viscosity-affected near-wall region (yþ < 10). Grid sensitivity studies are carried out to determine the optimum grid resolution that gives the mesh independent solution. Fig. 2 shows a comparison of these results, where r 0 ¼ r=ð2RÞ is dimensionless radial coordinate. For all further investigations, a grid with 200  256 control volumes in axial and radial directions in these predictions has been used. The grid convergence is checked by three cases: 100  128 control volumes, 200  256, and 400  400. The outcome with the 200  256 CVs find to be satisfactory, as shows in Fig. 2. Differences in the value of heat transfer calculated for a gasdroplets jet flow were less than 1%. The solutions presented here are considered grid independent. The maximum error emax has been defined as

emax ¼ max fni  fn1  106 i i ¼ 1; N

where N denotes the total number of control volumes, subscript i denotes the specific CV, superscript n denotes the iteration level and 4 denotes all flow variables. The results of preliminary calculations for a single-phase flow in the pipe with the length of 150R are used for the gas phase velocity and turbulence on the pipe edge. This is enough to achieve the fully developed turbulent gas flow. The symmetry conditions are set on the jet axis for gas and dispersed phases. The no-slip conditions are set on the wall for the gas phase. Boundary conditions on the outer border of jet and impinging surface for dispersed phase correspond to the conditions of “absorbing surface” [23,27], when droplets do not return to the flow after the contact with the wall or external

For comparative analysis in the case of single-phase air jet the experimental data on flow [36] and heat transfer of the impinging air jet [37e40], LES predictions [41] and v2f model [42] are used. Turbulent parameters. Distributions of radial (parallel to wall) and axial (normal to wall) fluctuations of gas velocity are shown for 2 , H/(2R) ¼ 2 in Figs. 3 and 4, respectively, where hv0 i ¼ hv2 i=Um1 0 2 2 0 hu i ¼ hu i=Um1 , and y ¼ y=ð2RÞ. Good agreement between measurements of [36] and calculations of the current study on the value and position of velocity pulsation maximums hv0 i and hu0 i is also observed at distribution of turbulence intensity in the outer zone of the wall jet. The deviation of numerical results from experimental data is shown in the wall area in Figs. 3 and 4. In general, the authors’ calculations predict a lower value of gas turbulence in the wall area than the measurements of [36]. The difference between the experiments and predictions is observed only in the wall zone. Similar results are presented in [6] in calculations of impinging jets with application of SMC. The secondmoment closure does not capture the near-wall peak of both axial and radial fluctuating velocity found in measurements of [36]. In general the use of the SMC model leads to small overestimation of both components of fluctuating velocity excluding the near-wall zone. The profiles of Reynolds stresses at different distances from the 2 , stagnation point of impinging jet, where huvi0 ¼ 100huvi=Um1 are shown in Fig. 5. It is necessary to note a good agreement between the results of predictions by the SMC and experiments of [36] both in the zone of gradient flow and in the region of wall jet development. Results of calculations by the ke3 model [28] with consideration of Durbin’s correction [11] cause significant quantitative deviation from data of [41] in the whole computation domain. Heat transfer. Results of heat transfer predictions at the stagnation point are compared to the available experimental data sets in Fig. 6, where H 0 ¼ H=ð2RÞ and Nu00 ¼ Nu0 =ðRe0:5 Pr0:3 Þ. Points are the available experimental data sets [37e40], curves (1e5) are numerical computations. Results of calculations by empirical formula [43] are shown additionally in Fig. 6 (line 6):

Nu0 ¼ 1:6Re1=2 Pr 1=3 ðH 0 Þ0:11 ; H 0  6:2 Nu0 ¼ 5:25Re1=2 Pr 1=3 ðH0 Þ0:77 ; H0 >6:2


Significant difference in results of predictions by the wellknown k3 model [44] (line 1) with data of author’s simulations and other studies is observed. The predictions with the use the model by [33] (lines 4) show better agreement with measurements of [37e40]. If one wants to use this model for design purposes, the efficiency of the cooling system would be overestimated. The design optimization process would also fail completely and the k3 model predicts two optimal distances of H’ ¼ 2  2:5 and 6, in disagreement with the experiments. Measurements of [37e40] have indicated that the stagnation Nusselt number exhibits a maximum value at H 0 z6. The testing simulations confirm this finding. The good agreement (difference does not exceed 20e25%) between data of other authors and calculation results of [21] (3) is observed. Computations by the SMC and v2f models give close results.

M.A. Pakhomov, V.I. Terekhov / International Journal of Thermal Sciences 60 (2012) 1e12








Fig. 3. Distributions of radial fluctuations of gas velocity downward from stagnation point. Re ¼ 2.3  104 and H/(2R) ¼ 2. Points are measurements [36], lines are simulations of the current study. (a) r/(2R) ¼ 0, (b) 0.5, (c) 1, (d) 2, (e) 2.5, (f) 3.

of TKE is observed at H 0 z6 and is higher than that of H 0 ¼ 2. These results are not presented here and qualitatively coincide with the experiments of [39]. Deviations become obvious only for small distances between the pipe edge and target surface H 0 < 1. Simulations by the LES (2) predict the lower values of heat transfer for all distances from the pipe edge to impinging surface than the results obtained via the SMC model of turbulence, excluding H 0 ¼ 7e9. In general, computations agree quantitative with measurements and computations of [36e43] in a wide range of distances between the pipe edge and impinging surface. Therefore, the RANS and SMC

The anomalously high heat transfer at stagnation point predicted by the k3 model can be attributed to erroneous physics. The k3 model produces a spurious high value of turbulence (up to 75%) and this leads to the excessive value of predicted Nusselt number (up to 100%) (see Fig. 6). The impinging jet potential core of the free round jet contains relatively low level of turbulence and these should remain at the stagnation point. The v2f and SMC models computations are consistent with this expectation [2,5]. The locus of maximum value of TKE lies around the stagnation point for the k3 model and at r/(2R) z 2 for the v2f and SMC models. The value





Fig. 4. Profiles of axial fluctuations of gas velocity in the near-wall jet. (a) r/(2R) ¼ 0.5, (b) 1, (c) 2.5, (d) 3. Conditions of the numerical simulation and indication of lines correspond to Fig. 3.


M.A. Pakhomov, V.I. Terekhov / International Journal of Thermal Sciences 60 (2012) 1e12





Fig. 5. Turbulent shear stresses distributions in the near-wall jet. (a) r/(2R) ¼ 0.5, (b) 1, (c) 2.5, (d) 3. Calculation conditions correspond to Fig. 3. Points are measurements [36], dashed line e calculation of [21], solid lines e authors’ simulations by SMC model.

model allow the correct quantitative description of complex regularities of heat transfer in the impinging jet flows. All the above mentioned is the reliable basis for numerical simulations of the gasdroplets impinging jet. The SMC captures the main features of heat transfer at the stagnation point for various pipe exit cross-section and impinging surface. The SMC and v2f models predict close results in calculations of heat transfer around the stagnation point. The difference between these models is less than 10e15%. The v2f model is simpler than SMC for computational realization. In literature there are several papers presenting the Reynolds stresses transport models for two-phase flows and describing the sink terms with consideration of dispersed phase effect on carrier phase turbulence (see, for example Refs. [24,31]). The authors do not know the papers on modeling of two-phase flows with the use of modified v2f model.

Fig. 6. Dependence of stagnation Nusselt number on distance between pipe edge and flat surface for the impinging jet at Re ¼ 2.3  104. Points are experimental data sets of [37e40], lines are predictions: 1, ke3 model [25]; 2, LES [41]; 3, ke3 model [44]; 4, authors’ computations by SMC model; 5, v2f model [42]; 6, calculation by empirical formulas (18) of [43].

5. Numerical results for gas-droplets mist impinging jet and discussion All computations are carried out for the monodispersed mixture of water droplets and air under the atmospheric pressure. The pipe diameter is 2R ¼ 20 mm. The mean velocity of gas flow on the axis of the pipe exit cross-section is Um1 ¼ 20 m/s, Reynolds number for the gas phase is Re ¼ 2RUm1/n ¼ 2.6  104. The dispersed phase is given as uniform distribution of parameters over the cross-section on the pipe edge. The initial velocity of dispersed phase is UL1 ¼ 0.8Um1. The initial size of droplets varies within d1 ¼ 0e100 mm, and its mass concentration varies within ML1 ¼ 0e1%. The wall temperature is TW ¼ 373 K, and initial gas and droplets temperatures are T1 ¼ TL1 ¼ 293 K. The distance between the pipe edge and target surface changes within H/(2R) ¼ 1e10. Flow structure in the mist impinging jet. Axial mean velocity distributions of gas and droplets are shown in Fig. 7. Here U’ ¼ U=Um1 and UL’ ¼ UL =Um1 are dimensionless axial velocities of gas (solid lines) and droplets (dashed lines), respectively and r’ ¼ r=ð2RÞ is dimensionless radial coordinate. Initially at the pipe edge at x/(2R) ¼ 0 the fully developed profile of gaseous phase and uniform distribution of dispersed phase are set (see Fig. 7a). In this position the impingement surface does not effect the jet and spreading, and mixing of the jet coincides with regularities of the free jet [45]. Then, the gas velocity reduces because of deceleration while approaching the impinging surface, and the droplet velocity increases due to their inertia (see Fig. 7aec). However, the value of droplet velocity is still slightly lower than the gas phase velocity (see Fig. 7). In the wall area at x/(2R) ¼ 1.9 (see Fig. 7d) the droplet velocity is higher than the velocity of particles within the whole area of the figure. The mean axial gas and droplet velocities are maximal at the stagnation point. With increasing radial distance from the stagnation point both axial velocities decrease. Axial mean slip velocityDU ¼ ðU  UL Þ=Um1 is shown in Fig. 8. The results are plotted for four distances downstream the pipe exit cross-section. Fig. 8 presents the radial variation of difference between the local axial mean gas and droplet velocities. Axial mean slip velocity is calculated by the SMC (solid curves) and ke3 (dashed

M.A. Pakhomov, V.I. Terekhov / International Journal of Thermal Sciences 60 (2012) 1e12




b c d

Fig. 7. Distributions of phase velocities downward the pipe edge over the jet crosssection. Solid lines are computations for the gas phase, dashed lines are simulations for droplets. Re ¼ 2.66  104, H/(2R) ¼ 2, ML1 ¼1%, d1 ¼ 50 mm. (a) x/(2R) ¼ 0, (b) 0.5, (c) 1.5, (d) 1.9.

curves) models. Results of Fig. 8a show that there is no difference between these two models in the initial cross-section. Downstream the pipe exit the difference is observed (see Fig. 8bed). Predictions demonstrate the following trend. The magnitude of the mean slip velocity calculated with the use of ke3 model is higher in the stagnation region than that obtained by the use of SMC. The main reason for this fact is as follows: the value of mean gas velocity is overpredicted by the ke3 model. A lack of two-equation models is common and this fact is revealed in [8] only for the single-phase impinging jet. The magnitude of droplet velocity predicted with the use of these models is approximately equal. Simulations by both turbulence models predict qualitatively similar results, but maximal quantitative difference in the zone of wall jet reaches almost 100%. Predictions by the ke3 models overpredict the value of DU at all distances from the pipe edge to the impinging surface and it is underpredicted at the end of stagnation zone ðr 0  0:5Þ. The profiles of Reynolds stress along the transverse coordinate are shown in Fig. 9a, where y0 ¼ y=ð2RÞ ¼ ðH  xÞ=ð2RÞ is


b c


Fig. 8. Profiles of slip velocity downstream from the pipe edge over the radial coordinate. The conditions of predictions are the same as for Fig. 6. Solid lines are SMC results, dashed lines are results obtained with the use of ke3 model. (a) x/(2R) ¼ 0, (b) 0.5, (c) 1.5, (d) 1.9.


Fig. 9. Reynolds stresses (a) and anisotropy ratio (b) profiles in the zone of development of wall jet at Re ¼ 2.66  104, H/(2R) ¼ 2, ML1 ¼1%, d1 ¼ 50 mm. 1 e r/(2R) ¼ 0.5, 2 e 1, 3 e 2, 4 e 3.

2 dimensionless distance normal to the plate and hu0 v0 i ¼ huvi=Um1 is turbulent Reynolds stress. The maximal value of Reynolds stress is observed at distance r/(2R) z 3 from the stagnation point. In the wall zone there is the local maximum that corresponds approximately to the outer boundary of the wall jet. These conclusions correspond qualitatively to the data of known measurements [36] for the single-phase impinging jet. Line 1 differs significantly from curves 2 to 4 and it represents calculations of Reynolds stresses inside the free shear layer at y0 ¼ 0:5; therefore, it does not equal zero. Variation of gas phase anisotropy ratio AR in the impinging jet is shown in Fig. 9b. Here, AR ¼ hv0 2i=hu0 2i, where hv0 2i is radial (parallel to wall) and hu0 2i is axial (normal to wall) pulsations of the gas phase velocity. In the gradient area at r/(2R) ¼ 0.5 (line 1) significant anisotropy of velocity pulsations is observed (AR z 3.5). While moving downward from the stagnation point (in the zone of wall jet development) the anisotropy level decreases, and at distance r/(2R) ¼ 3 (line 4) AR z 1.8. It should be noted that the ke3 turbulence models cannot predict accurately distributions of components of Reynolds stresses in the impinging jet. The turbulent impinging jets have complex features due to entrainment and mixing with the quiescent medium, stagnation and high streamline curvature. Some problems of the ke3 models are shown in papers [6e9]; most importantly they obtained substantial overprediction of the turbulence level and heat transfer in the stagnation region with the widely used ke3 turbulence models.


M.A. Pakhomov, V.I. Terekhov / International Journal of Thermal Sciences 60 (2012) 1e12

Heat transfer in the mist impinging jet. Distributions of Nusselt number along the radial coordinate are shown in Fig. 10. Simulations are carried out by the SMC model (solid lines) and ke3 model (dashed line). Computation with the SMC approach predicts not only the quantitative, but also the qualitative difference in distribution of heat transfer along the radial coordinate in comparison with calculation by the ke3 model. The calculations of [21] with the use of the ke3 model [28] cannot describe correctly the value and position of local minimum and secondary maximum of heat transfer in the two-phase impinging jet. An increase in mass concentration of droplets intensifies heat transfer between the jet and wall surface because of evaporation processes (see Fig. 10a). The most significant double increase in heat transfer intensity is observed in the stagnation zone of r/ (2R) < 2. Then, the magnitude of heat transfer almost corresponds to the similar value for the single-phase impinging jet because of a decrease in dispersed phase concentration at jet expansion and droplets evaporation near the heated wall. An influence of variation in the initial size of droplets on heat transfer is more complex (see Fig. 10b). The heat transfer intensity for smallest particles d1 ¼10 mm (curve 2) is higher around the stagnation point than that for the larger droplets (see curves 3 and 4). It is caused by a significant growth of the interface for the fixed value of droplets mass fraction. But the magnitude of heat transfer for smallest droplets decreases substantially along the radial



Fig. 10. The effect of dispersed phase on heat transfer in the impinging gas-droplets jet at H/(2R) ¼ 2 j Re ¼ 2.66  104. Solid lines, predictions of the current study; dashed lines, calculation by model [21]. (a): d1 ¼ 50 mm, 1 e ML1 ¼ 0, 2 e 0.5%, 3 e 1%; (b): ML1 ¼1%, 1 e d1 ¼ 0 mm, 2 e 10, 3 e 50, 4 e 100.

Fig. 11. Heat transfer at stagnation point of the gas-droplets flow for various Stokes number. Solid lines are simulations of this work, dashed lines are calculations by model of [21]. Re ¼ 2.66  104, H/(2R) ¼ 2. 1 e ML1 ¼ 0.2%, 2 e 0.5, 3 e 1.

coordinate. It is clear that the small droplets evaporate faster than larger droplets and many small droplets evaporate in the stagnation area. This leads to the fact that in the region of wall jet development the mass concentration of droplets is significantly lower due to intensive evaporation process and wall jet entrainment with the ambient medium, and therefore the heat transfer rate is also diminished. The value of Nusselt number for the largest droplets d1 ¼100 mm (curve 4) near the stagnation point is less than (up to 20%) the corresponding value for the droplets of d1 ¼ 50 mm (curve 3). These conclusions agree qualitatively with calculation results of [17]. This is explained by a considerable decrease in the interface and by the fact that the inertial droplets are poorly entrained into the pulsation motion of the gas phase and most of them deposit on the impinging surface in the stagnation zone. A change in the value of heat transfer coefficient at stagnation point of the impinging jet depending on the Stokes number is illustrated in Fig. 11. Stokes number Stk ¼ s/sf is the dimensionless parameter and it is the ratio of particles relaxation time s to the appropriate time scale of the carrier phase sf. Here s is the particle relaxation time with consideration of deviation from the Stokes power law and sf ¼ 2R/Um1 is the turbulent time macroscale. The dispersed phase with small Stokes number Stk < 1 is found to be in velocity equilibrium with the gas phase. Particles are sensitive to gas phase fluctuations and will move unaffected through the eddies. The particles with large Stokes number Stk > 1 are found to be no longer in equilibrium with the fluid phase. They are insensitive to gas phase fluctuations and move without interaction with gas eddies. The Stokes number or droplet initial diameter is one of parameters which control the rate of heat transfer. An increase in the Stokes number a drastic rise in heat transfer intensity is observed. For smallest Stk (d1 <10 mm) heat transfer intensity is lower than that for the droplets of large diameter because they evaporate far from the wall and do not come to the near-wall zone; the large droplets demonstrate the less effect on heat transfer between the two-phase jet and the heated wall surface. Then, a reduction of heat transfer rate is shown because the interface is significantly smaller for the large droplets (Stokes number), and droplets evaporation is much slower. This finding relates to the data in Fig. 10b. Therefore, there is some optimum between the droplet size and heat transfer increase. The effect of distance between the pipe edge and plate surface on heat transfer coefficient at the critical point in the gas-droplet jet is presented in Fig. 12, where H0 ¼ H=ð2RÞ is a relative distance from the pipe edge to the impinging surface. Predictions are

M.A. Pakhomov, V.I. Terekhov / International Journal of Thermal Sciences 60 (2012) 1e12

Fig. 12. The effect of distance between the pipe edge and the impinging surface on heat transfer at stagnation point of the impinging jet at Re ¼ 2.66  104, d1 ¼ 50 mm. 1 e ML1 ¼ 0, 2 e 0.2%, 3 e 0.5%, 4 e 1%.

executed with the help of SMC (solid lines). Results of calculation by the ke3 model (dashed line) for the single-phase flow (ML1 ¼ 0) and maximal mass concentration of water droplets at ML1 ¼1% are also shown for comparison. As for the single-phase flow [1e5], the maximum of heat transfer intensity in the gas-droplet impinging jet is at distance H/(2R) ¼ 6e7. Then, heat transfer intensity decreases because the distance from the pipe edge to the impinging surface increases due to attenuation of the axial gas velocity. The difference in computations by the SMC and ke3 models is up to 15e20%. The increase in mass concentration of droplets intensifies heat transfer between the two-phase jet and impinging surface. 6. Comparison with the experimental results in the twophase mist impinging jet Comparison between predictions and data for steam-droplet impinging slot jet of [15] is shown in Fig. 13. Here x is the distance from the stagnation point and B is the slot width. Predictions are performed under the following conditions: B ¼ 7.5 mm; distance between the slot and flat plate H/B ¼ 3; Reynolds number Re ¼ 14,500; heat flux density on the impinging surface qW ¼ 7.54 kW/m2; initial mass concentration of water droplets

Fig. 13. The distribution of heat transfer coefficient in steam-droplets jet impingement. Points are experiments of [15]; solid lines are results by this paper, dashed lines the predictions by the model [21]. 1 e ML1 ¼ 0 (single-phase impinging steam jet); 2 e 1.5%.


Fig. 14. Local heat transfer profiles along the radial coordinate in the gas-droplets impinging flow. Points are experiments [19], lines are predictions: solid lines are simulations of the current work, dashed line is calculation of [21]. Re ¼ 5  104, qW ¼ 1.4 kW/m2, d1 ¼14 mm, H/(2R) ¼ 8. 1 e single-phase flow (ML1 ¼ 0), 2 e ML1 ¼ 0.09%, 3 e 0.15%.

ML1 ¼1.5% and initial average diameter of droplets d1 ¼ 7 mm; temperature of saturated steam T* ¼ 378 K, and steam pressure P ¼ 1.1 bar. The jet impacts the target wall with the length of 250 mm spaced at H ¼ 22.5 mm from the injection plate. A drastic increase in the heat transfer coefficient (over 2.5 times) in the mist steamevapor impinging jet is observed. In general, the agreement between author’s computational results and measurements data of [21] is quite good. The difference between measurement results and predictions is up to 10%, excluding the second experimental point at x/B ¼ 1. Here this difference is up to 30%. The reasons for a significant increase in heat transfer within this zone are unclear for the authors. Distributions of heat transfer at variation of mass fraction of water droplets are shown in the Fig. 14. The use of gas-droplets mist round impinging flow as a coolant causes a significant increase in heat transfer intensity (more than twice in comparison with the single-phase round impinging jet) (curves 2 and 3) because of latent vaporization heat at droplet evaporation. It should be noted that in experiments of [19] and in author’s simulation main intensification of heat transfer occurs in the stagnation zone. Only one maximum in distribution of local heat transfer coefficient is explained by a large distance between pipe exit and cooled surface [1,2]. The difference between measurement results and predictions is up to 10%. The difference between SMC and ke3 model [28] with Durbin’s correction [11] is not significant (see Fig. 14). We still do not know the exact reason for this phenomenon. One of the possible reasons for this relatively small difference between SMC or ke3 model is a very low value of droplets concentration (ML1 < 0.02%) in measurements of [19]. This finding requires the additional investigation and is the potential basis for our future study.Computations by the SMC and ke3 model give qualitatively the same results for single- and two-phase jets impingement for the surface of target. Simulations by model [21] overpredict heat transfer (approximately by 10e25%) in the stagnation region of the impinging single-phase jet, and in the zone of wall jet development this difference is significantly lower. 7. Conclusions The Eulerian model for the two-phase flows laden with small droplets has been presented. This model is based on the SMC by


M.A. Pakhomov, V.I. Terekhov / International Journal of Thermal Sciences 60 (2012) 1e12

[33] for single-phase turbulent impinging jet including two-way coupling of [24]. In the present study, heat transfer and fluid flow characteristics of gas-droplets impinging jets are numerically investigated for varied droplet sizes and under different mass fraction conditions. An increase in mass concentration of droplets intensified heat transfer significantly (almost twice) in comparison with the singlephase impinging air jet because of latent heat of phase change at their evaporation. Heat transfer intensity for the smallest droplets is lower than that for the droplets with a large diameter. The rise in size of dispersed phase leads to a drastic increase in heat transfer intensity. Reduction of heat transfer is typical for large droplets because they have smaller interfacial surface and their heating and evaporation are much slower then for the smaller droplets. Quantitative agreement between the computation results and measurements of [15,19] is obtained. The main increase in heat transfer rate occurs in the stagnation zone. This finding is observed both in experiments of [15,19] and in this study. The use of the SMC model [33] allows correct predictions of anisotropic behavior in gas turbulence near the stagnation point and quantitative distribution of heat transfer coefficient over the impinging surface both for single-phase and gas-droplets impinging jets. Computations by the ke3 model [28] with Durbin’s correction [11] predict the higher value of heat transfer (up to 20%) at the point of impinging jet stagnation in comparison with the SMC. Acknowledgements The work was partially supported by the Russian Foundation for Basic Research (RFBR grant no. 12-08-00504), Foundation of the President of Russian Federation for Young Researchers (grant no. MD 670.2012.8), and Program of Department of Energetic, Mechanics, Mechanical Engineering and Process Control of Russian Academy of Sciences (no. 10-2). References [1] K. Martin, Heat and mass transfer between impinging gas jets and solid surfaces, Advances in Heat Transfer 13 (1977) 1e60. [2] E.P. Dyban, A.I. Mazur, Convective Heat Transfer in Jet Flows Around Bodies, Naukova Dumkova, Kiev, 1982 [in Russian]. [3] K. Jambunathan, E. Lai, M.A. Moss, B.L. Button, A review of heat transfer data for single circular jet impingement, International Journal of Heat and Fluid Flow 13 (1992) 106e115. [4] B.W. Webb, C.F. Ma, Single-phase liquid jet impingement heat transfer, Advances in Heat Transfer 26 (1995) 105e217. [5] A.S. Mujumdar, Impingement drying, in: A.S. Mujumdar (Ed.), Handbook of Industrial Drying, third ed. Taylor & Francis Group, New York, 2007, pp. 385e395. [6] T.J. Craft, L.J.W. Graham, B.E. Launder, Impinging jet studies for turbulence model assessment  II. An examination of the performance of four turbulence models, International Journal of Heat and Mass Transfer 36 (1993) 2685e2697. [7] K. Heyerichs, A. Pollard, Heat transfer in separated and impinging turbulent flows, International Journal of Heat and Mass Transfer 39 (1996) 2385e2400. [8] A. Abdon, B. Sunden, Numerical investigation of impingement heat transfer using linear and nonlinear two-equation turbulence models, Numerical Heat Transfer 40 (Part A) (2001) 563e578. [9] H.M. Hofmann, R. Kaiser, M. Kind, H. Martin, Calculations of steady and pulsating impinging jets e an assessment of 13 widely used turbulence models, Numerical Heat Transfer 51 (Part B) (2007) 565e583. [10] M. Kato, B.E. Launder, The modeling of turbulent flow around stationary and vibrating cylinders, Proceedings of the 9th Symposium on Turbulent Shear Flows, Kyoto, Japan (1993) 10.4.1e10.4.6. [11] P.A. Durbin, On the k3 stagnation point anomaly, International Journal of Heat and Fluid Flow 17 (1996) 89e90. [12] Yu.A. Buyevich, V.N. Mankevich, Cooling of a superheated surface with a jet mist flow, International Journal of Heat and Mass Transfer 39 (1996) 2353e2362. [13] K.M. Graham, S. Ramadhyani, Experimental and theoretical studies mist jet impingement cooling, ASME Journal of Heat Transfer 116 (1996) 161e167.

[14] M. Pasandideh-Fard, S.D. Aziz, S. Chandra, J. Mostaghimi, Cooling effectiveness of a water drop on a hot surface, International Journal of Heat and Fluid Flow 22 (2001) 343e349. [15] X. Li, J.L. Gaddis, T. Wang, Mist/steam heat transfer in confined slot jet impingement, ASME Journal of Turbomachinery 123 (2000) 161e167. [16] T. Wang, J.L. Gaddis, X. Li, Modeling of heat transfer in a mist/steam impinging jet, ASME Journal of Heat Transfer 124 (2001) 1086e1092. [17] M. Garbero, M. Vanni, U. Fritsching, Gas/surface heat transfer in spray deposition processes, International Journal of Heat and Fluid Flow 27 (2006) 105e122. [18] T. Wang, T.S. Dhanasekaran, Calibration of computational model for predict mist/steam impinging jets cooling with an application to gas turbine blades, ASME Journal of Heat Transfer 132 (2010) 1e11. Paper 122201. [19] A. Kanamori, M. Hiwada, J. Minatsu, H. Sugimoto, K. Oyakawa, Control of impingement heat transfer using mist, Journal of Thermal Science and Technology 4 (2009) 202e213. [20] H. Shokouhmand, M.M. Heyhat, A numerical study on heat transfer enhancement in a mist/air impingement jet, Journal of Enhanced Heat Transfer 17 (2010) 231e242. [21] M.A. Pakhomov, V.I. Terekhov, Enhancement of an impingement heat transfer between turbulent mist jet and flat surface, International Journal of Heat and Mass Transfer 53 (2010) 3156e3165. [22] M.A. Pakhomov, V.I. Terekhov, Enhancement of turbulent heat transfer during interaction of an impinging axisymmetric mist jet with a target, International Journal of Heat and Mass Transfer 54 (2011) 4266e4274. [23] I.V. Derevich, L.I. Zaichik, Particle deposition from a turbulent flow, Fluid Dynamics 23 (1988) 722e729. [24] L.I. Zaichik, A statistical model of particle transport and heat transfer in turbulent shear flows, Physics of Fluids A 11 (1999) 1521e1534. [25] M.W. Reeks, On a kinetic equation for the transport of particles in turbulent flows, Physics of Fluids A 3 (1991) 446e456. [26] J. Pozorski, J.-P. Minier, Probability density function modeling of dispersed two-phase turbulent flows, Physical Review E 59 (1999) 855e863. [27] I.V. Derevich, Statistical modelling of mass transfer in turbulent two-phase dispersed flows. 1. Model development, International Journal of Heat and Mass Transfer 43 (2000) 3709e3723. [28] C.B. Hwang, C.A. Lin, Improved low-Reynolds-number ke~3 model based on direct simulation data, AIAA Journal 36 (1998) 38e43. [29] D.A. Drew, Mathematical modeling of two-phase flow, Annual Review of Fluid Mechanics 15 (1983) 261e291. [30] A.A. Mostafa, S.E. Elghobashi, A two-equation turbulence model for jet flow with vaporizing droplets, International Journal of Multiphase Flow 11 (1985) 515e534. [31] N. Beishuizen, B. Naud, D. Roekaerts, Evaluation of a modified Reynolds stress model for turbulent dispersed two-phase flows including two-way coupling, Flow, Turbulence and Combustion 79 (2007) 321e341. [32] A. Kartushinsky, E.E. Michaelides, Y. Rudi, G. Nathan, RANS modeling of a particulate turbulent round jet, Chemical Engineering Sciences 65 (2010) 3384e3393. [33] T.J. Craft, B.E. Launder, New wall-reflection model applied to the turbulent impinging jet, AIAA Journal 30 (1992) 2970e2972. [34] K. Mastanaiah, E.N. Ganic, Heat transfer in two-component dispersed flow, ASME Journal of Heat Transfer 103 (1981) 300e306. [35] W.E. Ranz, W.R. Marshall Jr., Evaporation from drops, part I, Chemical Engineering Progress 48 (1952) 141e146. [36] D. Cooper, D.C. Jackson, B.E. Launder, G.X. Liao, Impinging jet studies for turbulence model assessment e I. Flow field experiments, International Journal of Heat and Mass Transfer 36 (1993) 2675e2684. [37] J.W. Baughn, S. Shimizu, Heat transfer measurements from a surface with uniform heat flux and an impinging jet, ASME Journal of Heat Transfer 111 (1989) 1096e1098. [38] J.W. Baughn, A. Hechanova, X. Yan, An experimental study of entrainment effects on the heat transfer from a flat surface to a heated circular impinging jet, ASME Journal of Heat Transfer 113 (1991) 1023e1025. [39] D. Lytle, B.W. Webb, Air jet impingement heat transfer at low nozzle-plate spacings, International Journal of Heat and Mass Transfer 37 (1994) 1687e1697. [40] D.W. Colucci, R. Viskanta, Effect of nozzle geometry on local convective heat transfer to a confined impinging air jet, Experimental Thermal and Fluid Science 13 (1996) 71e80. [41] K.N. Volkov, Unsteady-state heat transfer in the region of interaction between a turbulent jet and an obstacle, High Temperature 45 (2007) 818e825. [42] M. Behnia, S. Parneix, P.A. Durbin, Prediction of heat transfer in an axisymmetric turbulent jet impinging on a flat plate, International Journal of Heat and Mass Transfer 41 (1998) 1845e1855. [43] P.M. Brdlik, V.K. Savin, Stagnation heat transfer in axisymmetrical jets, streamlined normally oriented flat surfaces, Journal of Engineering Physics and Thermophysics 8 (1965) 423e428 (in Russian). [44] W.P. Jones, B.E. Launder, The calculation of low-Reynolds-number phenomena with a two-equation model of turbulence, International Journal of Heat and Mass Transfer 15 (1973) 1119e1130. [45] G.N. Abramovich, The Theory of Turbulent Jets, The MIT Press Classics, Boston, USA, 1963.