Javad Taghinia *,Md Mizanur Rahman ,Timo Siikonen

1 Aalto University,School of Engineering,Department of Applied Mechanics,FI-00076,Finland

2 Hong Kong University of Science and Technology,Department of Civil and Environmental Engineering,Clear Water Bay,Hong Kong,China

1.Introduction

Impinging jets are one of the essential components in many industrial processes,such as surface coating,electronic component cooling,drying process and paper production.The nature of jet impingement makes it a suitable tool in providing localized cooling and heating.The impingement mechanism has a complex structure due to the presence of stagnation point,walls,high streamline curvature and development of shear layer at the free jet region.The curvature of surface significantly affects the flow topology due to creation of strong flow entrainment and high streamline curvature in the wall jet region.A curved surface also alters the heat transfer rate along the impingement region.Studies showed that the impinging jet on curved surface can increase the heat transfer rate about 20%than that of flat surface in similar conditions[1].

Numerous studies can be found in the literature regarding the impinging jet on flat surfaces with different turbulence modeling strategies such as Reynolds-averaged-Navier-Stokes(RANS)and large eddy simulation(LES)for various configurations[2-9],however,impinging jets on curved surfaces are less investigated in terms of heat transfer and main flow parameters.Most available works in the literature on experiments and measurement of flow parameters show that the impinging characteristics of curved surfaces are different from those of flat plates[10-14].In terms of numerical simulations,there are a few reported studies on this subject,and most of these research applied RANS turbulence models to analyze the flow behavior of impinging jet.Souris and Liakos[15]numerically investigated the flow field from an impinging jet on a semi-cylindrical surface by using the standard k-ε and Reynolds stress transport model(RSM).A good agreement between model predictions and experimental data[16]is found in their study.Kayansayan and Kucuka[17]applied control-volume finite difference method to model a confined jet impingement on concave surface for different Reynolds numbers.They observed a maximum of 28%deviation between their predictions and measurement at the impingement region.Frageau et al.[18]studied the heat transfer rate of a curved surface of airfoil subjected to array of jets with varying nozzle to-surface spacings.They applied the Spalart-Allmaras turbulence model and found satisfying results for higher nozzle-to-surface ratios.The heat transfer rate and flow field from a row of round jets on a concave semi-cylinder were investigated by Craft et al.[19].In their study,linear and non-linear k-ε models along with a wall function were employed.They concluded that using a standard wall function leads to an under-prediction in the Nusselt number values.Sharif and Mothe[1]applied four turbulence models to simulate the impingement on a curved surface,namely RSM,standard k-ε,RNG k-ε and SST(Shear Stress Transport)k-ω.All these models failed to accurately predict the Nusselt number distribution along the impinging surface.However,RSM showed relatively better performance in recreating the flow features.Singh et al.[20]conducted a numerical study of heat transfer of a jet impingement on a concave surface of a circular cylinder with various RANS turbulence models.The effect of different nozzle-to-surface distances and nozzle diameters is considered in their study.They reported that all the applied turbulence models with different configurations over-predicted the Nusselt number at the impinging region.

According to the above literature review,majority of available numerical investigations on this subject are based on RANS methods which could not produce accurate results.The heat transfer and flow field analysis of impinging jet on curved surfaces are less investigated in terms of LES and hybrid RANS-LES methods.Therefore,establishing a numerical study based on these method is essential for such a significant process.

This paper aims at investigating the heat transfer and flow field of an impinging air jet on a curved surface by applying LES with one-equation SGS model[21]and SST based Scale-Adaptive Simulation(SST-SAS),invariant of an hybrid RANS-LES.The one-equation model is developed recently by Taghinia et al.[21]which benefits from a variable eddy-viscosity coefficient.This coefficient allows the model to adapt with a sudden change in flow topology and turbulent structures,especially near the walls.On the other hand,SST-SAS model utilizes the RANS SST kω model in the boundary layer region and close to the walls,while in the rest of flow domain it applies LES(usually with dynamic Smagorinsky SGS model).The study is carried out for different jet-to surface spacings at two Reynolds numbers.The predictions are compared against available measurements in the literature.

2.Mathematical Formulation

2.1.RAST one-equation SGS model

A spatial filter is used in LES to separate the large scales from the small scales that are to be modeled.By applying a spatial filter on incompressible Navier-Stokes equations and using the commutation characteristics,the LES equations yield:

The overbar notation denotes the application of a top-hat filter andν is the kinematic viscosity.Since in the LES formulation the larger length scales are resolved,it denotes the turbulent SGS stresses which is smaller than its counterpart in RANS.The SGS stress tensor is defined as

The constant coefficient models suggest a value of 1-1.5 for the dissipation term coefficient Cε[21].The grid- filter length(or width)Δ is based on the cell volume:

The variable Ckensures the realizability related to the flow deformation rate,where Ttis the hybrid timescale and ?=|W/S|is a dimensionless parameter that is very useful to characterize the flow.For instance,for a pure shear flow ?=1,whereas for a plane strain flow ?=0.Hybrid time scale Ttis formulated as

2.2.SST-SAS model

The SAS is an alternative method to DES(detached eddy simulation)in which the RANS model is not influenced by the grid spacing.It is based on the SST k-ω turbulence model and modifies the SST by adding a source term with the ω-equation to account for unsteadiness.The SST-SAS model solves transport equations for the turbulent kinetic energy k and the specific dissipation rate ω[22,23]:

3.Computational Schemes

The filtered incompressible Navier-Stokes equations are solved using a finite-volume method.The continuity and momentum equations are coupled using the PISO(Pressure Implicit with Splitting of Operators)algorithm.To avoid the generation of a checker-board pressure mode,a modified Rhie-Chow velocity interpolation method at the cell faces is applied[26]Central differencing schemes are applied for diffusion and convective terms.A Crank-Nicolson second-order accurate scheme is employed for time integration.An algebraic multi-grid method is utilized to accelerate the solution convergence.In order to reduce the calculation time,a k-ε model[27]is used to calculate each case with the respective grid.Afterwards,the LES calculations are initiated from a fully converged k-ε simulation.The statistics are obtained when the flow reached to a statistically steady-state.After that,the averaged statistics are obtained for 10 mean flow residence time which is the ratio of characteristic length(D/2)to mean flow velocity(U).The dimensionless time step ofΔ(=tUe/2B)is tuned in a way to ensure the stability criteria in numerical schemes corresponding to the mean Courant-Friedrich-Lewy(CFL)number of 0.75.An in-house code is used for this study.

4.Computational Domain and Settings

The computational domain is shown in Fig.1.The geometry follows the experimental work of Choi et al.[16]consisting an unconfined impinging air jet on a semi-circular curved surface.The jet is inserted from a rectangular slot with a width of B(5 mm)impinging on a curved surface with a diameter of D(150 mm).In order to better identify the directions of measurements,three directions are specified:parameter a indicates the inward radial direction from the impingement surface,r is the direction from the jet-exit toward the targeted surface at the jet centerline and s represents the circumferential direction along the impinging surface.No-slip condition along with a constant heat flux of 5000 W·m-2is applied on the impingement surface.The jet inlet temperature(Ti)is set to 300 K and the inlet condition at the jet-exit is constructed from a separate calculation of turbulent channel flow with the same width as of the inlet slot.For RANS calculations,k and ε at the inlet are calculatedwhere Tiis turbulent intensity,εinis the dissipation rate with Cμ=0.09,and l is turbulent length scale.An atmospheric pressure boundary condition is applied at the outlet.A periodic boundary condition is implemented at the span-wise direction.The air kinematic viscosity(ν)is set as 1.5×10-5at 300 K.

Fig.1.Computational domain with T i=300 K and constant heat flux of 5000 W·m-2 at the concave surface.

The calculations are carried out for three different jet-to-surface(h/B)ratios of 4,6 and 10 at two different Reynolds numbers,namely 2960 and 4740(based on the jet exit velocity(Ue)and the hydraulic diameter of slot(2B)).

The grid distribution for computational domain is illustrated in Fig.2.The grid distribution is denser close to the impingement surface providing the required resolution to resolve all the turbulent structures close to the impingement area and wall jet region.Three different grid resolutions are used for the grid in dependency study.All of these applied grids corresponded to y+＜1 close to the wall which is necessary for an LES to produce correct results at that location.Four sets of grids are tested at Re=2960 and h/B=10 namely,80×150×60,160×250×70,200×300×70 and 240×340×84 in r-,s-and span-wise directions,respectively.It is noticed that the predicted results are similar for 160×250×70,200×300×70 and 240×340×84(less than 2%).Therefore,in order to accelerate the computational time,the grid with 160×250×70 is applied in this study(Fig.3).

5.Results and Discussion

Fig.2.Mesh distribution with y+＜1 at the curved surface.

Fig.3.Nusselt number profiles for different grid sizes obtained by OEM for h/B=10 and Re=4740(corresponds to U e=7.1 m·s-1).

In this section the predicted results for one-equation and SST-SAS models are presented.In order to highlight the capabilities of these two models,the results from RANS RNG k-ε model is also shown,providing a good benchmark for comparisons.The simulation results are compared with the experimental data of Choi et al.[16].The flow field consists of three distinguished structures:(a)free jet region(b)impinging region(c)wall jet region,the free jet length gets larger as the jet-to-surface distance increases.The impinging jet region experiences a high pressure field and flow deformation creating a stagnation point(Fig.4).As the flow moves further away from the stagnation region,the flow field gradually loses the kinetic energy and follows the curvature of surface,resulting in creation of a wall jet.

Fig.4.Averaged velocity contours for h/B=6 for Re=2960(U e=4.4 m·s-1)by OEM.

Fig.5 represents the local Nusselt number distribution for different jet-to-surface ratios at Re=4740 along the impingement surface.The RNG k-ε over-estimated the values at the impinging region while as the flow expands along the surface,these values are under-estimated for all h/B ratios.Both one-equation and SST-SAS models show a better agreement with experimental data,however SST-SAS reproduced slightly higher values(4%-11%).The ability of one-equation model in predicting the correct values at the stagnation region is clearly noticeable.

The mean axial velocity at different h/B ratios is represented in Figs.6 and 7 for Re=2960 and Re=4740.The velocity is normalized with the jet-exit velocity Ueand h/B is normalized as the distance from the jet exit toward the impingement surface at the jet centerline.The SST-SAS over-estimated the velocities close to the impinging zone.A better agreement between predicted results and the experiments has been achieved at a higher jet-to-surface.This behavior can be seen also at Re=4740 in Fig.7.At lower h/B ratios,the RNG k-ε totally failed to predict the correct velocity profile due to isotropic eddy-viscosity formulation,while the one-equation model produced relatively better results compared to other two models.Over-prediction of velocity distribution may be due to an under-estimation of turbulent viscosity by the SST-SAS at the impingement region,resulting in excessive velocity magnitude.Another reason for an over-prediction of spreading rate of the jet can be related to over-prediction of the entrainment of ambient air,change in pressure gradient and kinetic energy at a lower h/B,making it a challenging task for RNG k-ε to adapt with the deformation in the flow field.While the OEM can respond to this through Cμembedded in the eddy-viscosity term,the under-prediction of values by RNG k-ε can be attributed to the standard wall function approach associated with the model.An increase in the jet-to-surface distance(h/B)causes an increase in the boundary and shear-layer thickness at the wall-jet region.At a low jet-to-surface distance,the fluid entrainment occurs close to jet while at higher distances,this entrainment expands in the entire domain.

Fig.5.Local Nusselt number distribution along the concave surface for Re=4740.

The velocity fluctuations for two different Reynolds numbers are illustrated in Figs.8 and 9.The deviation from experimental data is more evident for h/B=4 where SST-SAS over-predicted the velocity fluctuation values(with maximum deviation of 7%from measurements)approaching the curved surface.Both models correctly calculated the rate of decay of velocity fluctuations and also the fluctuations at the potential core region.At a higher h/B,the predictions are more consistent with the experiments(maximum deviation less than 4%)with an increasing rate of mixing of flow with the ambient air.

The predicted mean velocity profiles at different circumferential locations for h/B=6 and 10 are compared with the experimental data at Re=2960 along the radial direction(s)in Figs.10 and 11.As the flow develops radially along the surface,the predicted results show a better agreement with the measured values which can be related to the fact that all models can predict accurately the wall boundary layer.It can be seen that the SST-SAS over-estimated the values at the wall jet region especially for s/B=4 and 6,while away from the surface a better agreement is obtained.Again here,it can be noticed that as the jet-to-surface distance is increased,the predictions are in better consistency with experimental data.The ability of one-equation model in reproducing the correct behavior of flow structure close to the wall is better than the predictions by RNG k-ε and SST-SAS models.The inclusion of sub-grid scale model enables one-equation model to account small fluctuations of the flow field close to the impingement surface.

Fig.6.Mean axial velocity profiles along the jet centerline at Re=2960 with D=150 mm and h=20 mm,30 mm and 50 mm respectively(from left to right).

Fig.7.Mean axial velocity profiles along the jet centerline at Re=4740.

Fig.8.Axial velocity fluctuations along the centerline for Re=2960.

Fig.9.Axial velocity fluctuations along the centerline for Re=4740.

Fig.10.Mean velocity profiles along the radial direction at different circumferential location for h/B=6 at Re=2960.

6.Conclusions

The recently developed one-equation(OEM)LES and SST-SAS models are applied for the first time to investigate the heat transfer and flow field of jet impingement on a curved surface for various jet to-surface distances and Reynolds numbers.The OEM offers a natural near-wall damping which is sensitive to the flow condition close to the solid boundaries.This feature enables the OEM to respond to the flow deformation by the eddy-viscosity coefficient embedded in the model which is suitable for simulating the jet impingement on curved surfaces related to the industrial fluid- flow simulations.Nusselt number comparisons have revealed that the ability of one-equation and SST-SAS models in predicting the heat transfer rate are better than those of RNG k-ε.In terms of velocity distributions,one-equation and SST-SAS models results are considerably more accurate.However,the performance of one-equation model in predicting the thermal and velocity distributions at the impingement region are better(according to Table 1)than the SST-SAS model.These results have been observed in performance of applied models:

·RNG k-ε is unable to correctly predict the peak velocity location at h/B=6 for Re=2960 while one-equation LES and SST-model successfully determine this location.

·At h/B=10,RNG k-ε predicts the location of maximum velocity,which is,relatively good but it obviously under-estimates the velocity profiles in other locations.

·One-equation model performance is better at lower h/B ratios at which the flow field under-goes a high strain rate and rapid deformation.

Calculations showed that the jet-to-surface ratio(h/B)is an influential factor in the accuracy of predictions than the magnitude of Reynolds number in such a way that the predictions are closer to experiment at higher h/B ratios.Therefore,it is recommended to apply OEM model for designing the impinging process at a lower jet-to-surface distance while for designing processes involving higher jet-to-surface spacings(such as h/B=10)both OEM and SST-SAS can be utilized.However,RNG k-ε can yield qualitatively acceptable results for velocity distribution at high jet-to-surface distances according to current calculations,however it is not suitable for heat transfer prediction.It is also worthwhile mentioning that the one-equation model computational time is almost equal to that of the SST-SAS.

Fig.11.Mean velocity profiles along the radial direction at different circumferential location for h/B=10 at Re=2960.

Table 1 Maximum deviation of predictions from experiment for mean and RMS velocity and Nu

[1]M.A.R.Sharif,K.K.Mothe,Parametric study of turbulent slot-jet impingement heat transfer from concave cylindrical surfaces,Int.J.Therm.Sci.49(2010)428-442.

[2]R.S.Amano,H.Brandt,Numerical study of turbulent axisymmetric jets impinging on a flat plate and flowing into an axisymmetric cavity,ASME J.Fluids Eng.106(1984)410-417.

[3]D.Lytle,R.W.Webb,Air jet impingement heat transfer at low nozzle-to-plate spacings,Int.J.Heat Mass Transfer 37(7)(1994)1687-1697.

[4]C.Cornaro,A.S.Fleischer,R.J.Goldstein,Flow visualization of a round jet impinging on cylindrical surfaces,Exp.Therm.Fluid Sci.20(1999)66-78.

[5]S.D.Hwang,H.H.Cho,Effects of acoustic excitation positions on heat transfer and flow in axisymmetric impinging jet:main jet excitation and shear layer excitation,Int.J.Heat Fluid Flow 24(1)(2003)199-209.

[6]L.A.El-Gabray,D.A.Kaminski,Numerical investigation of jet impingement with cross flow:Comparison of Yang-Shih and standard k-turbulence model,Numer.Heat Transfer A 47(2005)441-469.

[7]J.M.Miao,C.Y.Wu,P.H.Chen,Numerical investigation of confined multiple-jet impingement cooling over a flat plate at different cross flow orientations,Numer.Heat Transfer A 55(2009)1019-1050.

[8]H.Y.So,H.G.Yoon,M.K.Chung,Large eddy simulation of flow characteristics in an unconfined slot impinging jet with various nozzle-to-plate distances,Int.J.Mech.Sci.Technol.25(3)(2011)721-729.

[9]J.Taghinia,M.M.Rahman,T.Siikonen,Numerical investigation of twin-jet impingement with hybrid-type turbulence modeling,Appl.Therm.Eng.73(1)(2014)648-657.

[10]R.S.Bunker,D.E.Metzger,Local heattransfer in internally cooled turbine airfoil leading edge regions.Part I:Impingement cooling without film coolant extraction,J.Turbomach.12(1990)451-458.

[11]C.Gau,C.M.Chung,Surface curvature effect on slot-air-jet impingement cooling flow and heat transfer process,J.Heat Transfer 113(1991)858-864.

[12]D.H.Lee,Y.S.Chung,D.S.Kim,Turbulent flow and heat transfer measurements on a curved surface with a fully developed round impinging jet,Int.J.Heat Fluid Flow 18(1997)160-169.

[13]D.H.Lee,Y.S.Chung,M.G.Kim,Turbulent heat transfer from a convex hemispherical surface to a round impinging jet,Int.J.Heat Mass Transfer 42(1999)1147-1156.

[14]J.Taghinia,M.M.Rahman,T.Siikonen,Heat transfer and flow analysis of jet impingement on concave surfaces,Appl.Therm.Eng.84(2015)448-459.

[15]N.Souris,H.Liakos,Impinging jet cooling on concave surfaces,AICHE J.50(2004)1672-1683.

[16]M.Choi,H.S.Yoo,G.Yang,J.S.Lee,D.K.Sohn,Measurement of impinging jet flow and heat transfer on a semi-circular concave surface,Int.J.Heat Mass Transfer 43(2000)1811-1822.

[17]N.Kayansayan,S.Kucuka,Impingement cooling of a semi-cylindrical concave channel by confined slot-air-jet,Exp.Therm.Fluid Sci.25(2001)383-396.

[18]M.Frageau,F.Saeed,I.Paraschivoiu,Numerical heat transfer correlation for array of hot-air jets impinging on a 3-dimensional concave surface,J.Aircr.42(2005)665-670.

[19]T.J.Craft,H.Iacovides,N.A.Mostafa,Modelling of three-dimensional jet array impingement and heat transfer on a concave surface,Int.J.Heat Fluid Flow 29(2008)687-702.

[20]D.Singh,B.Premachandran,S.Kohli,Experimental and numerical investigation of jet impingement cooling of a circular cylinder,Int.J.Heat Mass Transfer 60(2013)672-688.

[21]J.Taghinia,M.M.Rahman,T.Siikonen,One-equation sub-grid scale model with variable eddy-viscosity coefficient,Comput.Fluids 107(2015)155-164.

[22]F.R.Menter,Two-equation eddy-viscosity turbulence model for engineering application,AIAA J.32(8)(1994)1598-1605.

[23]F.R.Menter,Y.Egorov,The scale-adaptive simulation method for unsteady turbulent flow predictions.Part1:theory and model description,Flow Turbul.Combust.85(2010)113-138.

[24]J.C.Rotta,Statistische Theorie Nichthomogener Turbulenz,Z.Phys.(1951)547-572.

[25]C.M.Winkler,A.J.Dorgan,M.Mani,Scale adaptive simulations of turbulent flows on unstructured grids,20th AIAA Computational Fluid Dynamics Conference,AIAA,Honolulu,Hawaii June 27-30 2011,pp.2011-3559.

[26]M.M.Rahman,T.Siikonen,A.Miettinen,A pressure-correction method for solving fluid flow problems on a collocated grid,Numer.Heat Transfer B Fundam.32(1997)63-84.

[27]M.M.Rahman,T.Siikonen,An explicit algebraic Reynolds stress model in turbulence,Int.J.Num.Methods Fluids 52(2006)1135-1157.