Brahim Berrabah,Miloud Aminallah

Department of Mechanical Engineering,Materials and Reactive Systems Laboratory,Faculty of Technology,Djillali Liabes University,Sidi Bel-Abbes 22000,Algeria

Effect of Coriolis and centrifugal forces on flow and heat transfer at high rotation number and high density ratio in non orthogonally internal cooling channel

Brahim Berrabah*,Miloud Aminallah

Department of Mechanical Engineering,Materials and Reactive Systems Laboratory,Faculty of Technology,Djillali Liabes University,Sidi Bel-Abbes 22000,Algeria

Blade cooling;Computational Fluid Dynamic(CFD);Heat transfer;High buoyancy parameter;High rotation number

Numerical predictions of three-dimensional flow and heat transfer are performed for a two-pass square channel with 45°staggered ribs in non-orthogonally mode-rotation using the second moment closure model.At Reynolds number of 25,000,the rotation numbers studied were 0,0.24,0.35 and 1.00.The density ratios were 0.13,0.23 and 0.50.The results show that at high buoyancy parameter and high rotation number with a low density ratio,the flow in the first passage is governed by the secondary flow induced by the rotation whereas the secondary flow induced by the skewed ribs was almost distorted.As a result the heat transfer rate is enhanced on both co-trailing and co-leading sides compared to low and medium rotation number.In contrast,for the second passage,the rotation slightly reduces the heat transfer rate on co-leading side at high rotation number with a low density ratio and degrades it significantly on both co-trailing and co-leading sides at high buoyancy parameter compared to the stationary,low and medium rotation numbers.The numerical results are in fair agreement with available experimental data in the bend region and the second passage,while in the first passage were overestimated at low and medium rotation numbers.

1.Introduction

Gas turbines are designed to operate at increasingly high,inlet temperatures witha bladecoolingsystemin ordertoobtain better thermal efficiencies.One of the techniques of cooling used is the internal cooling by forced convection which consists of extracting the cooling air of the compressor and makes it circulate in channels with complex geometry arranged inside theblades.The presence of ribs in these channels improves heat transfer.The aerodynamic shape of the blade requires the use ofcoolantchannelsofdifferentconfigurationsanddifferentorientations compared to the axis of rotation.The combination of acomplexgeometryandtherib-androtationinducedsecondary flows develops a complicated flow structure within the serpentine coolant passage inside the turbine blades,which must be examined in detail in order to reduce the effect of Coriolis force and centrifugal buoyancy.Several experimental and numerical works are investigated on the effect of rotation on the flow and heat transfer in coolant channels of the blades turbine.

1.1.Previous experimental studies

For β =0°or β =90 channel orientation,Johnson et al.1investigated the effect of rotation on heat transfer in rotating serpentine passages with staggered r

ibs;their results show that Coriolis forces and buoyancy effects can strongly influence heat transfer.They determine the effect of rotation number,Reynolds number and buoyancy parameter for three configurations(smooth walls,90°and 45°staggered ribs)and they concluded that 45°staggered ribs should be employed for rotating coolant passages.Dutta et al.2studied the effect of ribs configuration on heat transfer in rotating two-pass square channel.Unfortunately they do not find a configuration with better heat transfer in both passages,but they concluded that 60°staggered ribs has high performance of heat transfer in the first pass and 60°parallel ribs enhance heat transfer in the second pass.Liou et al.3explored the field of flow and pressure drop in a rotating two-pass channel with staggered 45°ribs.They found that the flow characteristics in the two passages are a mirror and angled ribs reduce the pressure drop compared to the staggered 90°ribs.

For β =45°or β =135°channel orientation,Han et al.4investigated the effect of rotation on heat transfer in rotating one pass rectangular channel with parallel 45°ribs.They concluded that channel with aspect ratio(AR)of 4:1 has better performance on heat transfer than the channel with AR=1:1 and 2:1,and 135°channel orientation enhances heat transfer almost on all the surfaces.Parsons et al.5reported the effects of the model orientation on the local heat transfer coefficients in a rotating two-pass square channel with ribbed walls.Their result showed that 135°channel orientation reduces rotation effect.Azad et al.6studied the effect of channel orientation with respect to the axis of rotation on heat transfer in rotating two-pass rectangular channels for both smooth and 45°ribbed walls.The results show that the channel orientation effect was less sensitive compared to the case of smooth walls.Johnson et al.7investigated heat transfer in rotating four-pass channel with smooth walls and 45°staggered ribs.They are determined the effect of rotation number and channel orientation on heat transfer ratio for both roughened and smooth walls and the effect of rotation direction only in the case of smooth walls.Their results have provided a database for this study.

1.2.Previous numerical studies

For β =0°or β =90°channel orientation,Jang et al.8simulated the flow and heat transfer in a single-pass rotating square channel with staggered 45°ribs investigated experimentally by Johnson et al.1using the second-moment closure model.Their heat transfer coefficient prediction compared well with the experimental data.Bonhoff et al.9studied the flow and heat transfer in rotating two-pass channels with smooth and ribbed wall investigated experimentally by Johnson et al.1The tested model is second moment closure model,and the results show that the Nusselt number ratio(Nu/Nu0)is qualitatively predicted compared with the experimental data.Al-Qahtani et al.10predicted the flow and heat transfer in one-pass rotating rectangular channels with 45°ribs using the secondmoment closure model.Their heat transfer predictions were in good agreement with the experimental data.Shih et al.11predicted the flow and heat transfer in a rotating two-pass square duct with staggered 45°ribs using a low Reynolds numberk-ω model.However,no comparison with experiment was made.Su et al.12predicted the flow and heat transfer in two-pass rotating rectangular channels(AR=1:1,AR=1:2 and AR=1:4)with 45°angled ribs using second moment closure model.Their results show that the square duct again produced the highest heat transfer enhancement on both the leading and trailing surfaces of the first passage atRe=10000 and rotation numberRo=0.14.Stephens et al.13studied the effect of 45°ribs on the heat transfer rate in a rotating two-pass square channel using a low-Renumberk-ω turbulence model.They examined the effects of Reynolds numbers,rotation numbers,and buoyancy parameters.They found that the skewed ribs create an asymmetric heat transfer coefficient distribution.In a more recent study,Wang et al.14calculated the flow and heat Transfer characteristics in twopass square channels with 45°ribbed walls(staggered ribs with square cross-section)cooled by superheated steam.The results indicate that the superheated steam has better heat transfer performance than air.

For β =45°and β =135°channel orientation,Al-Qahtani et al.15studied heat transfer in rotating rectangular channels with 45°parallel ribs investigated experimentally by Azad et al.6The multi-block Reynolds-averaged Navier-Stokes(RANS)equations with a near-wall second-moment turbulence closure was used for the analysis.The result showed that the rotation induced secondary flow is dominated by the rib induced secondary flow atRe=10000 andRo=0.11.Their heat transfer ratio prediction compared well with the experimental data.Also,Suet al.16predicted the flow and heat transfer in one-pass rotating rectangular channel(AR=4:1),withv-shaped ribs,using a second moment closure model.Their result shows that the change of channel orientation from β =90°to β =135°leads to an increase in the span-wiseaveraged Nusselt number ratios on the bottom surface and a minor decrease of the spanwise-averaged Nusselt number ratios on the top surface,and their results are in good agreement with experimental data.Recently,Chuet al.17studied flow and heat transfer in rotating cooling passage with turning vane in hub region.Two channel orientations are examined(45°and 90°)with aspect ratio(1:2)for Reynolds number varying from 10,000 to 25,000 and rotation number from 0 to 0.2.Their results show that the flow and heat transfer are signi ficantlyaffected after turning vane.

In their experiment,Johnson et al.7varied the rotation number from 0 to0.35 and showedits effect onheat transferrate but they did not study the effect of density ratio(Δρ/ρ),therefore,the present study determines in detail the effect of high rotation number and high density ratio(high buoyancy number)on the flow and heat transfer characteristics in a two-pass channel in non-orthogonal mode-rotation using the second moment closure model.The numerical solutions are performed using commercial software ANSYS CFX 14.0,and the numerical method and the governing equations of the turbulence model were described in detail.18

2.Description of problem

The experimental profile of theu-component velocity to the channel entrance,given by Johnson et al.1is used(Fig.2).At the outlet,with the aim of avoiding the problem of back flow,a condition of the opening type is applied.The density of air is approximated by ρ = ρ0T0/T,determining the effect of the temperature,where ρ0is the density of coolant at inlet,while the effect of temperatureon viscosity and thermal conductivity is given by the Sutherland law.Nusselt numbersNu=hDh/λ were normalized with a smooth tube correlation(Kays and Pekins19)for fully developed,non-rotating,turbulent flowNu0=0.0176Re0.8.Table 1 shows a summary of all cases studied,wherebois the buoyancy parameter.

3.Computational procedure

3.1.Governing equations

For relative velocity formulation,the governing equation of fluid flow for steadily rotating frame can be written as follows.

(1)Conservation of mass

Table 1 Summary of the cases studied(Re=25000 and pref=1×106N/m2).

(2)Conservation of momentum

(3)Conservation of energy

The momentum equation contains two additional acceleration terms:the Coriolis acceleration(2Ω×U)and the centripetal acceleration Ω × (Ω × r).ˉτ is the viscous stress except that relative velocity derivatives are used and the energy equation is written in terms of the relative internal energyH.r is location vector from the origin of rotating frame.In Eq.(2),? is the dyadic operator(or tensor product),(?U)Tthe matrix transposition and δ the identity matrix.In Eq.(3),SEis the energy source.

3.2.Omega-based Reynolds stress model

The Reynolds stress ω turbulence model,or SMC-ω model,is a Reynolds stress model which uses an ω-equation for the length scale equation.The advantage of the ω-equation is that it allows for a more accurate near wall treatment with an automatic switch from a wall function to a low-Reynolds number formulation based on the grid spacing.One adopts the notation used for the Reynolds stresses18τij=-uiuj.

The modeled equations for the Reynolds stresses can be written as

The Omega Reynolds stress model uses the following equation for ω:

whereGij= ρΩk(τjmεikm+ τimεjkm)is the production term by system rotation,with εijkbeing a levi-chivita factor.

The constitutive relation for the pressure-strain correlation is given by

The production tensor of Reynolds stresses is given by

The tensorDijparticipating in the pressure-strain model(Eq.(6))differs from the production tensor in the dotproduct indices:

The turbulent viscosity in the diffusion terms of balance equations(Eqs.(4)and(5))is defined as

The strain rate tensor is given by

The coefficients for the turbulence model are cited in Table 2.

The idea behind the automatic near-wall treatment is that the model shifts gradually between a viscous sub-layer formulation and wall functions,based on the grid density.The ωequation provides analytical solutions,both for the sub-layer and the logarithmic region.A blending function depending onycan be defined.The solutions for ω in the linear and the logarithmic near-wall region are:

whereuτis the friction velocity and κ the Von Karman constant.They can be re-formulated in terms of the dimensionless distance from the wally+and a smooth blending can be performed:

A similar formulation is used for the velocity profile near the wall:

This formulation gives the relation between the velocity at(i=1)andthewallshearstress.Forthetreatmentoftheenergy equation near the wall,an algebraic formulation is required to link the temperature and the heat flux.The formulation of Kader20is used this study

whereqwis the wall heat flux,cpthe heat capacity andPris Prandtl number.This formulation is valid through the entirey+range of the viscous sub-layer and the logarithmic profile.

3.3.Computational grids detail

Fig.2 shows the numerical grid generated using ICEMCFD.14.0.We have used mesh law of half cosinus 1 and 2,close to all walls and full co-sinus in stream-wise direction which gave very small mesh sizes in the vicinity of the walls and ribs better than the mesh law of bi-geometric or uniform.For automatic wall treatment,meany+for both smooth and ribbed walls isˉy+=1 andˉy+=1，8,respectively.To resolve the near wall viscous region 13 grid points were placed in the boundary layer near all walls.The shape of each rib needs 39 grid points to be resolved.The convergence criterion for all quantities error was 10-6.A grid-refinement study was performed using three different grid distributions of 37×37×1123,37×37×1250 and 55×55×1123 points.The grid refinement in the stream-wise and cross-stream direction of the passages and the bend region did not enhance the solution further,because the automatic wall treatment used by the model SMC-ω is less sensitive to the grid spacing.In addition,one modifies the blocks tov-shaped as shown in Fig.2(b),does not affect the results.Thus all results presented in this paper are based on 37×37×1123 grid distribution,which resulted in 1537387 grid points.

4.Results and discussion

The nomenclature used in this study for C-L and C-LS correspond to low pressure surfaces(suction surface)and for C-T and C-TS correspond to high pressure surfaces(pressure surface).The low and high pressure surfaces are defined with respect to Coriolis force direction as illustrates in Fig.1(b).Therefore,C-T surface in the first passage with outward flow is on the high pressure side of the passage.Similarly,co-leading surface in the second passage with inward flow is on the high pressure side.In contrast,the co-trailing sidewall and co-leading sidewall are on high and low pressure sides,respectively in both passages.Rotation generates a Coriolis force:

Table 2 Coefficient for the model SMC-ω.

Assume a Cartesian coordinate system in which i,j and k are unit vectors in the three directions(x,y,z).Fcocan be expressed as follows:

whereuis the stream-wise mean velocity inx-direction,vis the vertical mean velocity iny-direction andwis the transverse mean velocity inz-direction,which equal to zero in this case.

Centrifugal force:

The Coriolis force tends to deviate in the first pass the core flow towards C-T and C-TS,while in the second pass towards C-L and C-TS.In the presence of density gradient(rotating heated channel);the centrifugal force induces centrifugal buoyancy de fined by a buoyancy parameter,which is determined by the variations of the density ratio and the rotation number as follows:

The centrifugal buoyancy has two physical effects.The first effect is the acceleration of the cold air,denser,near the high pressure surfaces and assists to Coriolis effect known in literature as aiding buoyancy.The second effect is the displacement of the hot air,lighter,towards the axis of rotation causing thus a reverse flow which depends strongly on rotation number,known in literature as opposing buoyancy.

This section presents the principal results of temperature distribution and secondary flow streamlines for the seven cases considered as shown in Figs.3–9.In the first passage and bend,the secondary flow patterns are viewed from the upstream towards downstream direction.In the second passage,they are viewed from the downstream towards upstream direction as indicated in Fig.1(d).

4.1.Effect of high rotation number and high density ratio

4.1.1.First passage radially outward flow

At low rotation number(Case 2:Ro=0.24 and DR=0.13 andbo=0.3),a structure of two counter-rotating cells due to the Coriolis effect surmounted by two small cells,also counterrotating caused by the opposing buoyancy is established ats/Dh=2 as shown in Fig.3(b).

As mentioned above,in the stream-wise direction,the Coriolis force deviates the core of the main flow towards the high pressure surfaces(C-T and C-TS).It results that the air near of these sides became colder than that near the low pressure surfaces(C-L and C-LS).Therefore,the aiding buoyancy accelerates the cold air,denser,and assists the Coriolis effect on the one hand and the opposing buoyancy tends to move the hot air,lighter,towards the axis of rotation causing thus a reverse flow on the other hand.This reverse flow meets the cold air at the channel entrance and superimposes to the main flow by forming buoyancy cells,and the effect depends strongly on buoyancy number.

At medium rotation number(Case 3:Ro=0.35 and DR=0.13),the two counter-rotating cells caused by the opposing buoyancy observed in Case 2 are established and are shifted towards the co-leading sidewall(LP)with the counterclockwise cell relatively larger than the clockwise cell.These cells surmount the two counter-rotating Coriolis cells with the clockwise Coriolis cell relatively larger than that observed in Case 2 as shown ats/Dh=2 in Fig.3(c).

At high rotation number and low density ratio(case 4:Ro=1.00,DR=0.13),the counterclockwise buoyancy cell becomes too large compared to the clockwise buoyancy cell and reinforces the counterclockwise Coriolis cell giving rise to a large cell,which compresses the clockwise Coriolis cell.This effect is shown at locations/Dh=2 in Fig.4(a).

In the stationary case(Case 1),the vortices carry the cold air towards the corners,while in the rotating case(Cases 2 and 3),the Coriolis cells transport the cold air towards high pressure sides,which is evident from Fig.3(b)and(c).In contrast at high rotation number(Case 4),the large vortex convect the cold air towards high and low pressure sides as can be seen from Fig.4(a)ats/Dh=2.

At high rotation number and high density ratio(Case 7:Ro=1.00,DR=0.50 andbo=20.5),unlike Case 4,the clockwise buoyancy cell becomes larger than the counterclockwise buoyancy and reinforces the clockwise Coriolis cell on the one hand and compresses the counterclockwise Coriolis cell towards the C-T on the other hand.This effect is shown at locations/Dh=2 in Fig.5.

In the ribbed part(3＜s/Dh＜14),for the stationary case(Case 1),it develops behind each rib;a helical vortex then aligns with the main flow.The combination of these vortices gives rise to two appreciably symmetrical counter-rotating vortices as shown at locationss/Dh=6 ands/Dh=14 in Fig.3(a).As these vortices tend to equalize the temperature field,the mean coolant temperature is increased as can be seen from temperature contours of the same figure.

For rotating case,unlike the stationary case,three types of secondary flows(Coriolis driven cells,rib-induced vortices and buoyancy driven cells)interacted in complex manner,indeed:

At low rotation number(Case 2:Ro=0.24),this interaction isclearly seenatlocations/Dh=6in Fig.3(b),wherethe clockwise Coriolis cell of the smooth part persists and reinforces the clockwise buoyancy cell;this effect prevents the development of the vortices induced by the ribs of the co-leading side(LP).In contrast,the counterclockwise Coriolis cells induced at each inter-ribbed region of co-trailing side(HP)distort partially the vortices induced by the ribs of the co-trailing side on the one hand and reinforce an additional Coriolis cell generated on the co-trailing side near the C-TS,(because the streamwise velocity varies in span-wise direction)on the other hand.Afters/Dh=10,the clockwise Coriolis cell separates from the buoyancy cells generated at each inter-ribbed region of the co-leading side(LP)because it becomes weak but it reinforced by the ribs that are not distorted,while the Coriolis cells generated at each inter-ribbed of the co-trailing side are oscillate in the size and intensity,so that at the exit of passage one finds a structure formed by a clockwise vortex induced by the rib and counterclockwise Coriolis cell surmounted by buoyancy cell as shown ats/Dh=14 in Fig.3(b).

The maximum magnitude of secondary flow(v2+w2)0.5/Ubvaries from 0.7 ats/Dh=3 to 0.35 ats/Dh=14,while the stream-wise velocityu/Ubvaries from 1.5 ats/Dh=3–2 ats/Dh=14 near high pressure surfaces(1.1 rib height),along the diagonal plane.This flow acceleration in stream-wise direction is due to the aiding buoyancy effect(cold air).In contrast near low pressure surface,u/Ubvaries from 0.19 to-0.6 due to the opposing buoyancy(hot air).

At medium rotation number(Case 3:Ro=0.35),the large buoyancy cell persists in the ribbed part and combines with the counterclockwise buoyancy cells induced at each inter-ribbed region of the C-L and the counterclockwise Coriolis cells induced at each inter-ribbed region near the co-trailing side(HP),which gives rise to a large vortex as shown ats/Dh=6 in Fig.3(c).This vortex prevents the development of the vortices induced by the ribs of the co-trailing and coleading sides on the one hand and the clockwise Coriolis cell on the other hand as shown ats/Dh=14 in Fig.3(c).One notes that the Coriolis cells are larger than buoyancy cells and the maximum cross-stream flow(v2+w2)0.5/Ubvaries from 0.8 ats/Dh=3 to 0.5 ats/Dh=14.The stream-wise velocity(u/Ub)kept the value of 1.9 almost constant near high pressure surfaces,while near low pressure surfaces,it varies from 0.17 to-1.

At high rotation number(Case 4:Ro=1.00),the clockwise buoyancy cell is attenuated while the large cell persists.Unlike Case 3,the buoyancy cells are larger than the Coriolis cells,indeed the large buoyancy cell(counterclockwise)combines with the counterclockwise buoyancy cells induced at each inter-ribbed region near low pressures surfaces and the counterclockwise Coriolis cells induced at each inter-ribbed region near high pressure surface(C-T).This effect gives rise to a strong vortex as shown at locationss/Dh=6 ands/Dh=14 in Fig.4(a).This vortex prevents the development of the clockwise Coriolis cells on the one hand and prevents the vortices induced by the ribs of the co-trailing side(HP)to mix with main flow,and weakened the vortices induced by the ribs of the co-leading side(LP)as compared to those observed in Case 1 on the other hand.This vortex convects the cold air towards all four sides and mainly towards high pressure sides,by enhancing heat exchange fluid-surface.

The maximum cross-stream flow(v2+w2)0.5/Ubvaries from 2.3 ats/Dh=3 to 1.17 ats/Dh=14,while the streamwise velocity(u/Ub)varies from 3.3 ats/Dh=3 to 2.1 ats/Dh=14 near high pressure sides,along the diagonal plane.In contrast near low pressure sides,u/Ubvaries from-1.5 to 0.3.Comparison of the temperature contours for Cases 1–4 shown in Figs.3 and 4 indicates that the flow is well mixed in the stationary case(Case 1)and at high rotation number(Case 4)than at low and medium rotation numbers(Cases 2 and 3).

At high rotation number and high density ratio(Case 7:Ro=1.00 and DR=0.50),unlike Case 4(Ro=1.00 and DR=0.13),the large clockwise buoyancy cell persists and combines with the clockwise buoyancy cells generated at each inter-ribbed region near low pressure surfaces on the one hand and reinforces the clockwise Coriolis cells induced at each inter-ribbed region,near high pressure surface(C-TS)on the other hand.This effect gives rise to a large clockwise vortex,which compresses the vortices induced by the ribs of the cotrailing side(HP)and prevents the development of the counterclockwise Coriolis cells,while it distorts the vortices induced by the ribs of the co-leading side(LP)as shown at locations/Dh=6 in Fig.5.When the flow approaches the bend the clockwise Coriolis cells start to attenuate,while the counterclockwise Coriolis cells start to develop.This effect gives rise to two counter-rotating vortices at the exit of passage as shown at locations/Dh=14 in Fig.5.

Comparison of velocity field for Cases 4 and 7 shows that ats/Dh=3,u/Ubincreases from 3.3 in Case 4 to 7 in Case 7 and ats/Dh=14,it increases from 2.1 in Case 4 to 3.5 in Case 7 near high pressure sides(1.1 rib height),along the diagonal plane.For both Case 4 and Case 7,near–wall secondary flow(u2+w2)0.5/Ubincreases significantly on the co-trailing surface(HP)of the first passage compared to Cases 2 and 3 as shown in Fig.10(b),(c),(d)(g).The temperature contours show that the coolant becomes completely hot at the exit of passage in Case 7 as compared to that in Case 4.

4.1.2.Bend region

In the bend region for Case 1(Ro=0),the vortices induced by the ribs of the first passage persist and prevent the development of Dean-type double vortex as shown at the middle of the bend(s/Dh=16.97)in Fig.6(a),while in the rotating cases,the flow is governed by the combined effect of the Coriolis force,centrifugal buoyancy and radial pressure gradient.

At low rotation number(Case 2:Ro=0.24),the cold air near high pressure surfaces(C-T and C-TS)is pushed towards the concave side of the bend and the co-leading side by the combined effect of the aiding buoyancy and Coriolis force due to v-component velocity,leading to development of two vortices of Dean-type;one near the C-T side and the other next to the C-L side,while the hot air near the low pressure surfaces(C-L and C-LS)is convected towards the convex wall when the flow enters in the bend due to the opposing buoyancy,constitutes thus a vortex reinforced by the vortex of Dean-type induced on the co-leading side.This effect gives rise to two counter-rotating vortices:the large vortex nextto the co-leading side and the small one next to the co-trailing side.The large vortex convects the cold air towards the co-leading side and the convex wall of the bend,which results in the fact that the air heats more and due to combined effect of the opposing buoyancy and radial pressure gradient,the flow does not turn towards the concave wall of the bend near the co-trailing side but reverse its direction towards the convex wall of the bend,thus inducing third vortex as shown ats/Dh=16.97 in Fig.6(b).In the second part of the bend,the cold air is pushed towards the concave side by the aiding buoyancy effect inducing thus a vortex which reinforces the third vortex and another vortex is induced on the co-trailing side next to the concave side of the bend.This effect gives rise to a large clockwise vortex shown at locations/Dh=19.5 in Fig.7(b).

At medium rotation number(Case 3:Ro=0.35),almost the same physical effect occurs,whereas the third vortex was suppressed,because the vortex generated by the opposing buoyancy becomes strong.This modifies the structure of two counter-rotating vortices formed at the middle of the bend in which the large vortex is much stronger than the small one and occupies the largest section of the middle of the bend as shown ats/Dh=16.97 in Fig.6(c).In this casev-component velocity is all positive(i.e.varies from 0.12Ubnear C-T to 1.28Ubnear C-L).In the second part of the bend,the cold air is pushed towards the concave side forms thus a clockwise vortex,which develops with the detriment of the large counterclockwise vortex,which gives rise to two counter-rotating vortices with the large vortex next the co-trailing side(LP)and the small one next the co-leading side(HP)as shown at locations/Dh=19.5 in Fig.7(c).

At high rotation number and low density ratio(Case 4:Ro=1.00 and DR=0.13),when the flow enters the bend,the Dean-type double vortex starts to develop with the detriment of the large counterclockwise vortex set up in the first passage,where the vortex of Dean-type induced on the co-leadingsidecombineswiththevortexgeneratedneartheconvex wall by the opposing buoyancy,thus forms a large vortex.This latter compresses the vortex of Dean-type induced on the co-trailing side.This effect gives rise to two counter-rotating vortices with the vortex next to the co-leading side occupying much area,then begins to reverse its direction towards the convex wall to form the third vortex.This effect is shown ats/Dh=16.97 in Fig.6(d).

At high rotation number and high density ratio(Case 7:Ro=1.00 and DR=0.50),unlike Case 4,two vortices are induced by the opposing buoyancy.Indeed the clockwise vortex of the first passage persists in the bend and combines with the clockwise vortex generated by the opposing buoyancy on the convex side next the co-leading side,which suppresses the vortex of Dean-type induced on the co-leading side.Meanwhile,the counterclockwise vortex of the first passage attenuates with the development of the third vortex induced on the convex wall next to the co-trailing side,which reinforces the vortex of Dean-type induced on the co-trailing side.This effect gives rise to a structure of two counter-rotating vortices at the middle of the bend(s/Dh=16.97)as shown in Fig.6(g).

4.1.3.Second passage with radially inward flow

For the stationary case(Case 1),the vortices induced by the ribs of the first passage are attenuated,only the clockwise vortex induced by the last rib of the co-leading side persists and two additional vortices are developed on the convex wall of the second part of the bend due to the vorticity generation,a counterclockwise vortex next to the co-leading side and a clockwise vortex next to the co-trailing side as shown ats/Dh=19.5 in Fig.7(a),therefore the two vortices which circulate in clockwise sense combine with the vortices induced by the ribs of the co-trailing side and the counterclockwise vortex combines with the vortices induced by the ribs of the co-leading side.This effect gives rise to two counter-rotating vortices with the large vortex next to the co-trailing side and the small one next to the co-leading side as shown at locationss/Dh=25.7 ands/Dh=31 in Fig.7(a).

For the rotating cases,the flow in the second passage is radially inward;therefore the Coriolis force due to u-component velocity reverses its direction and the buoyancy force aligns with main flow.

For Case 2 and Case 3(Ro=0.24 andRo=0.35),the structure shown ats/Dh=19.5 in Fig.7(b)and(c)persists in the second passage and interacts with rib-induced vortices.Indeed for Case 2,the large vortex starts to attenuate with profit of the development of the vortices induced by the ribs of the co-trailing(LP)and co-leading(HP)sides.This effect gives rise to two counter-rotating vortices with this near the co-leading side(LP)is relatively larger than that next to the co-trailing side(LP)as shown at locationss/Dh=25.7 ands/Dh=31 in Fig.7(b).In contrast,for Case 3,the large vortex combines with the vortices induced by the ribs of the cotrailing side(LP),which constitute with the vortices induced by the ribs of the co-leading side(HP);two counter-rotating vortices with the vortex next to the co-leading side is relatively larger than the one next to the co-trailing side as shown at locationss/Dh=25.7 ands/Dh=31 in Fig.7(c).For both cases,these vortices tend to equalize more the temperature field.A comparison of the two-vortex structure in Cases 1–3,indicates that the only effect of rotation is expanding the vortex near the co-leading side(HP)and the reduction this near of the co-trailing side(LP).

Athigh rotation number(Case4:Ro=1.00 and DR=0.13)and at exit of the bend,the large vortex persists and combines with a vortex generated near the convex side on the one hand,and interacts with another vortex induced on the co-leading side on the other hand,while the third vortex is attenuated as shown ats/Dh=19.5 in Fig.8(a).In the second passage,the vortex near the convex side combines with the vortices induced by the ribs of the co-leading side(HP),while the other vortex is attenuated.When the flow is redeveloped in this passage,the vortices induced by the ribs of the co-trailing side(LP)are expanded by the Coriolis effect,while the vortices induced by the ribs of the co-trailing side(HP)are considerably reduced.This effect gives rise to two counter-rotating vortices with the clockwise vortex larger than the counterclockwise vortex as shown ats/Dh=25.7 in Fig.8(a).This structure remains almost unchanged until the exit of the passage as shown ats/Dh=31 in Fig.8(a).

At high rotation number and high density ratio(Case 7:Ro=1.00 and DR=0.50),the large vortex established in the bend persists and reinforces the vortices induced by the ribs of the co-leading side(HP),while the clockwise vortex is attenuated.When the flow is redeveloped in this passage,the vortices induced by the ribs of the co-trailing(LP)are expended by the Coriolis effect but the formation of the buoyancy cells at each inter-ribbed region of the co-trailing side(LP)and next to the C-TS reduces these vortices.This effect gives rise to a structure of three vortices as shown ats/Dh=25.7 in Fig.9,then the buoyancy cells formed in inter-ribbed regions are attenuated,while the buoyancy cell near the C-TS develops towards the co-leading side(HP),where it reinforces the vortices induced by the ribs of the co-leading side(HP).This effect gives rise to a structure of two counter-rotating vortices as shown ats/Dh=31 in Fig.9,in which the large vortex occupies much area.The temperature contours show that the higher temperatureoccupiesmuchareainCase7ascomparedtothatinCase4.

4.2.Effect of density ratio

At rotation number of 0.24,three values of the inlet density ratio were studied(i.e.,DR=0.13，0.23 and 0.50).For Case 2,Case 5 and Case 6,the secondary flow streamlines and temperature contours are shown in Figs.3,4 and 6–8.

The effect of varying the density ratio can be clearly seen by the comparison between Figs.3(b)and 4(b)and(c)for the first passage,Fig.6(b),(e)and(f)for the bend region and Figs.7(b)and 8(b)and(c)for the second passage.From this comparison,the following conclusions can be drawn.

In the smooth part of Case 5(DR=0.23)and unlike Case 2(DR=0.13),the clockwise Coriolis cell combines with the clockwise buoyancy cells before the ribbed part and reinforces an additional clockwise buoyancy cell generated near the CLS,while the counterclockwise buoyancy cell is suppressed as shown ats/Dh=2 in Fig.4(b).In Contrast,for Case 6,the two buoyancy cells become large,which surmount the two Coriolis cells and consequently,the two Coriolis cells are relatively reduced as compared to those in Case 2 and Case 5 as shown ats/Dh=2 in Fig.4(c).In the ribbed part of the first passage ats/Dh=6 in Fig.4(b),for Case 5 and unlike Case 2,two buoyancy cells are formed;the large buoyancy cell induced near the co-leading sidewall(LP)reinforces the small buoyancy cell induced at each inter-ribbed region of the coleading side(LP).At high density ratio(Case 6),the buoyancy cells become relatively large while the clockwise Coriolis cell is reduced as shown ats/Dh=6 in Fig.4(c).At the exit of the passage,almost the same structure is established in the three cases;the difference is the development of an additional counterclockwise Coriolis cells on the co-trailing side in Case 5 and the increase of the size of Coriolis cell and the vortex induced by the rib in Case 6 and Case 2,respectively as shown ats/Dh=14 in Figs.3(b)and 4(b)and(c).The temperature contours of the same Figures indicate that the temperature gradient is reduced in case 6 as compared to that in Case 2 and Case 5.In the bend region,for Case 5,Fig.6(e),and for Case 6,Fig.6(f),almost the same physical effect occurs as in Case 2.The vortex of Dean-type induced on the co-trailing side is diminished and increased in Case 5 and Case 6,respectively.In addition,the development of the third vortex near the convex side of the bend of Case 5 and Case 6 is retarded,because the large vortex near the co-trailing side starts to reverse its direction towards the convex wall of the bend constitutes thus a vortex which develops near the co-trailing side with detriment of the large vortex on the one hand and reinforces the third vortex on the other hand.This effect gives rise to the structure shown ats/Dh=16.97 in Fig.6(e)and(f),in which the clockwise vortex occupies a large area.In the second passage,after the attenuation of the bend effect,the flow is dominated by the vortices induced by the ribs of the co-leading side(HP)and the co-trailing side(LP)leading to the formation of two counter-rotating vortices.In Case 6,the vortex near the co-leading side(HP)is relatively larger than that near the co-trailing side(LP)as shown ats/Dh=31 in Fig.8(c),while they become almost appreciably symmetrical in Case 5 as shown ats/Dh=31 in Fig.8(b).The coolant temperature is higher at high density ratio(Case 6)and hot spots are created as can be seen from temperature contours in Fig.8(c).

4.3.Variation of Coriolis and centrifugal forces in stream-wise direction

The variations of Coriolis and centrifugal forces across a vertical line between the fifth and sixth ribs in the diagonal plane of the first and second passage in terms of velocityu/Uband temperature θ between two extremes limits low buoyancy number(Case 2)and high buoyancy number(Case 7)are depicted in Figs.11 and 12.The aiding and opposing buoyancy effects are increased significantly at high buoyancy number(Case 7)as compared to low buoyancy numbers(Case 2).However,in the first passage with radially outward flow at high buoyancy number(Case 7),the aiding buoyancy is increased a factor almost three in terms ofu/Ubcompared to low buoyancy number(Case 2).The opposing buoyancy is increased a factor almost two in terms ofu/Ubcompared to Case 2.In second passage with radially inward flow,at low buoyancy number(Case 2),the Coriolis force is dominated than the buoyancy force,while at high buoyancy number,the opposite hold is valid indeed;the aiding buoyancy accelerates the hot air near the co-leading and co-trailing sidewall(HP),while the opposing buoyancy decelerates the cold air near the co-trailing and co-leading sidewall(LP).The aiding buoyancy increases by approximately 40%in term ofu/Ubin Case 7 compared to Case 2 and the opposing buoyancy increases by approximately 13%compared to Case 2.

4.4.Effect of high rotation number and high density ratio on turbulence

4.4.1.Turbulent kinetic energy

4.4.2.Reynolds stresses

In the first passage ats/Dh=14,Fig.13(d),the major difference is observed at high rotation number and low density ratio(Case 4),low rotation number and high density ratio(Case 6)and high rotation number and high density ratio(Case 7)indeed,the shear stressuvincreases signi ficantly with positive sign in the core region with a maximum value of+0.25U2bobtained in Case 7,+0.14U2bin Case 4 and+0.06U2bin Case 6.In addition,the shear stressuvis increased with negative sign near the co-trailing side(HP)with a maximum value of-0.18U2bobtained in Case 7,-0.12U2bin case 6 and-0.08U2bin Case 4.In the middle of the bend(s/Dh=16.97),the shear stressvwincreases markedly with positive sign in mid-section core area in Case 4 compared to Cases 1,2,3,5 and 6,while it exhibits different trends in Case 7,where it increases with negative sign in the core region and near the co-leading side(HP),while it increases with positive sign near the cotraining side(LP)as can be seen from Fig.13(e).In the second passage,ats/Dh=26.6,the shear stressuvis increased with positive sign in the core region in Case 4,while in Case 7,uvincreases with positive sign at same location as in Case 4.Alsouvincreases with negative sign near the co-trailing side(LP),but it remains lower to that in the first passage.However,no major difference is observed between Cases 1,2,3,5 and 6 as can be seen from Fig.13(f).

4.5.Surfaces heat transfer distributions

The data of local Nusselt number distributions in rotating serpentine coolant passages with skewed ribs for 45°channel ori-entation are not available in open literature until now,while the numerical predictions provide the details that are difficult to obtain by experiments.

Fig.14 shows the local Nusselt number ratio distributions(Nu/Nu0)on the C-T and C-L sides for the cases considered.From the comparison between the distribution of local Nusselt number ratio(Nu/Nu0)in the stationary Case 1 and the rotating cases(Cases 2–7),the following conclusions can be drawn.

In the stationary case(Case 1),the distributions ofNu/Nu0on both co-trailing and co-leading surfaces of the first passage is qualitatively but not quantitatively similar to those of the second passage due to the mean coolant temperature increase in the second passage,while for the rotating cases(Cases 2–4),the distribution ofNu/Nu0is modified due to the rotation.Indeed the destructive effect on the co-trailing surface(HP)for Case 2 is clearly seen in Fig.14(i),which results in the fact that lower heat transfer ratio occurs at the zone of generation of the vortices induced by the ribs and higher heat transfer ratio is located at zone of reattachment of Coriolis cells in the inter-ribbed region.For Case 3,Fig.14(j),the large vortex slightly reduces the zones of lower heat transfer ratio observed in Case 2,while in Case 4 and Case 7,Nu/Nu0is increased markedly whereas the lower zones of heat transfer rate were almost suppressed as shown in Fig.14(k)and(n);this heat transfer enhancement is due to secondary flow induced by rotation.The co-leading surface(LP)is characterized by very low heat transfer rate in most area of inter-ribbed region for Case 2 due to the opposing buoyancy,while this degradation of local heat transfer ratio is reduced in Case 3,in contrast it eliminated in Case 4 and Case 7 as can be seen from Fig.14(d),and 14(g).ThedistributionsofNu/Nu0on both co-trailing and co-leading surfaces of the second passage at low and medium rotation number(Case 2 and Case 3)are almost similar to that for the stationary case as can be seen from Fig.14(a),(b),(c),(h),(i)and(j),because at this level of rotation(Ro=0.24 andRo=0.35),the second passage with inward flow is less sensitive to the rotation,since it was governed by the rib-induced vortices;while this distribution ofNu/Nu0is changed in Case 4 and Case 7 due to the large effect of Coriolis and buoyancy forces,respectively;in the second passage for Case 4,the local heat transfer ratio(Nu/Nu0)is enhanced in most area of inter-ribbed region of the cotrailing surface(LP)on the one hand but it is reduced on the co-leading surface(HP)on the other hand compared to the stationary case as shown in Fig.14(d)and(k).In contrast,in Case 7,Nu/Nu0is considerably reduced on both co-trailing and co-leading sides of the second passage compared to the stationary case as can be seen from Fig.14(g)and(n).This is because of the formation of buoyancy cells due to opposing buoyancy.In the bends region,the local heat transfer ratio is enhanced in Case 4,while it is reduced in Case 7 compared to Case 2 and Case 3.

For Case 5 and Case 6 and on high and low pressure surfaces,almost the same distribution ofNu/Nu0is observed as in Case 2,the difference is the reduction and expanding of the zones with higher and lower values ofNu/Nu0,respectively on both co-leading and co-trailing sides as can be seen from Fig.14(e),(f),(l)and(m).

4.6.Pitch-averaged heat transfer augmentation ratios

4.6.1.Comparison with experimental data

The evolution of averaged Nusselt number ratios on the co-trailing and co-leading sides for the cases studied is shown in Fig.15,in the figure,EXP means the experimental data.Comparisons of this ratios(Nu/Nu0)were made with the experimental data of Johnson et al.7for the stationary Case 1(Ro=0),Case 2(Ro=0.24)and Case 3(Ro=0.35).

In order to validate the numerical results,Fig.15 shows the Nusselt number ratios for Cases 1–3 as well as the measured.Each point(measured and calculated)presents the average result over heater section of 4Dhlong.The experimental data relating to the ribbed walls are evaluated for projected surfaces and numerical for real surfaces(including 45°skewed ribs surface).To be able to compare our numerical results with those experimental,we have divided the latter per 1.15,because according to Johnson et al.,7real surface is 1.15 times of the projected surface.

Johnson et al.7show that the Nusselt number ratio in the first passage is 100–200%higher than that for the fullydeveloped,smooth-wall correlation19;while the calculations(Case 1)show thatNu/Nu0is 100%–300%higher than that for the fully-developed,smooth-wall correlation.19The stationary calculations show that the development of stream-wise flow in the first passage is very significant,as a consequence of the strong near-wall secondary flow(u2+w2)0.5/Ub,which reaches 1.5 as can be seen from Fig.10(a).The 45°staggered ribs generate two high velocity regions(both reaching 1.5Ub),close to the co-leading and co-trailing sides.In addition to a strong impingement on the ribs(2Ub),based upon the correlation,19these flow characteristics lead to an extra 300%increase in heat transfer rate.Unlike the first passage,the predicted and measured Nusselt number ratios are in good agreement on both co-trailing and co-leading sides of the bend region and the second passage as shown in Fig.15.

At low rotation number(Case 2:Ro=0.24),the predicted and the measured Nusselt number ratios on the co-leading side(LP)of the first passage are decreased compared to stationary case as can be seen from Fig.15(b),the measuredNu/Nu0decreases from 2.6 to 1.5,while the predicted decreases from 4 to 2.2.This degradation of heat transfer rate is attributed to the opposing buoyancy effect which prevents the formation of vortices induced by the ribs of the co-leading side(LP).In the bend region and the second passage,the predicted and measured Nusselt number ratios are in good agreement as shown in Fig.15(b).On the co-trailing side(HP)of the first passage,the measured and predictedNu/Nu0are in good agreement qualitatively and show the same increase ofNu/Nu0compared to the stationary case as can be seen from Fig.15(a).In the bend region and the second passage the agreement is fairly good as shown Fig.15(a),and the slight enhancement of heat transfer rate observed on the co-trailing(HP)of the first passage is attributed to the combined effect of Coriolis force and aiding buoyancy.

At medium rotation number(Case 3:Ro=0.35),the Nusselt number ratio(Nu/Nu0)measured and predicted(Case 3)on the co-trailing side(HP)of the first passage are in fairly good agreement as can bee seen from Fig.15(a),and the predictedNu/Nu0increases slightly compared to that in Case 2.In contrast on the co-leading side(LP),the predicted is higher and lower than measured in the first and second passage,respectively as shown in Fig.15(b).Explanations for these anomalous measurements were not presented by Johnson et al.7According to flow field discussed above,the flow is characterized by a large vortex,which reattaches at each interribbed region of both co-trailing and co-leading sides of the first passage;indeed this reattachment enhances the heat exchange on both sides,therefore the difference between C-T and C-L is the enhancement of the heat transfer on the cotrailing side caused by the Coriolis force and the minor reduction of the heat transfer on the co-leading side due to the weakening of vortices induced by the ribs as compared to those induced on the co-trailing side.In addition near-wall secondary flow shown in Fig.10(a),(b)and(c)indicates that the strong secondary flow is obtained in the stationary case compared to Case 2 and Case 3.In the second passage,the difference between predicted and measured is reduced and shows a small difference between the stationary case and low and medium rotation number indicating a small effect of rotation.

4.6.2.Effect of high rotation number and high density ratio on heat transfer ratio

In order to study the combined effect of high rotation number and high density ratio(high buoyancy parameter)on heat transfer rate,Nusselt number is also calculated at 9 locations on the co-leading and co-trailing surfaces forRo=1.00 and DR=0.50.Fig.16 shows the results.

On the co-trailing surface(HP)of the first passage,the heat transfer rate increases slightly as Ro increases except at the locations/Dh=8 as can be seen from Fig.16(a),while in the second part of the bend(s/Dh=18),it decreases at medium rotation number(Ro=0.35).In the second passage,no major difference is observed except at the beginning of the passage,where the heat transfer rate increases slightly compared to the stationary case.On the co-leading side(LP)of the first passage,the heat transfer rate increases as the rotation number increases but remains lower than that atRo=0,while in the second passage remains almost unchanged on the co-leading(HP).On the co-trailing side(LP),no major difference is observed betweenRo=0，Ro=0.24 andRo=0.35 after the fourth rib as shown in Fig.16.

At high rotation number and low density ratio(Case 4:Ro=1.00,DR=0.13),the heat transfer rate increases significantly on the co-trailing side of the first passage and slightly in the bend region,while in the second passage remains the same as the stationary case as can be seen from Fig.16(a).On the co-leading side,Fig.16(b),the heat transfer rate increases in the first passage,while it decreases in the second passage compared to the stationary case.The heat transfer enhancement observed in the first passage on both co-trailing and coleading sides is attributed to reattachment of the large vortex,while the degradation of heat transfer rate on the co-leading side of the second passage is attributed to the reduction of vortices induced by the ribs caused by the rotation.

At high rotation number and high density ratio(Case 7:Ro=1.00,DR=0.50),the heat transfer rate increases on both co-trailing and co-leading sides in the first passage but remains lower than that atRo=1.00 and DR=0.13,in contrast it decreases significantly on both co-trailing and co-leading sides in the second passage compared to all cases as can be seen from Fig.16(a)and(b).The degradation of heat transfer rate observed on low and high pressure surfaces of the second passage contributes to the formation of the buoyancy cells caused by opposing buoyancy effect.

4.6.3.Effect of density ratio on heat transfer ratio

In order to study the effect of increasing density ratio on heat transfer rate,the evolution of averaged Nusselt number ratios on the co-trailing and co-leading sides for Cases 1,2,5 and 6 is shown in Fig.17.No experimental data was available for the effect of density ratio in work of Johnson et al.7

Increasing the density ratio from 0.13(Case 2)to 0.23(Case 5)decreased the heat transfer on the co-trailing side(HP)ats/Dh=8,and increased it in the bend region,while it remains almost unchanged in the second passage.

At high density ratio(Case 6:DR=0.50),the heat transfer rate decreases on low and high pressure surfaces of both passages compared to Case 2 and Case 5 as shown in Fig.17(a)and(b).This reduction of heat transfer rate in the first passage is attributed to the partially destructive interaction between Coriolis cells-rib-induced vortices and the domination of the opposing buoyancy(buoyancy cells)on the one hand and the reduction of near–wall secondary flow as shown in Fig.10(f)on the other hand.In the second passage it is due to the mean coolant temperature increase.

5.Conclusions

The study of effect of high rotation number and high density on flow and heat transfer in two-pass channel with skewed ribs in non orthogonally mode-rotation,the following conclusions are drawn.

(1)At rotation number of 0.24 and density ratio of 0.13(Case 2),the interaction between the three types of secondary flows(rib-induced vortices,Coriolis driven cells and buoyancy cells)is partially destructive in the first passage with radially outward flow leading to a slight enhancement and a reduction of heat transfer rate on the co-trailing(HP)and the co-leading side(LP)respectively,compared to the stationary case.In contrast,in the second passage with radially inward flow,the rotation expands and reduces the vortices induced by the ribs of the co-leading side(HP)and the co-trailing side(LP)respectively,causing a minor effect on heat transfer rate compared to the stationary case.

(2)At rotation number of 0.35 and density ratio of 0.13(Case 3),in the first passage,there is a constructive interaction between the Coriolis and buoyancy cells which gives rise to a large vortex.This effect leads to an enhancement of heat transfer rate of the co-leading side(LP)compared to low rotation number(Case 2).In contrast,in the second passage,almost the same heat transfer characteristics as in Case 2 are observed.

(3)At rotation number of 1 and density ratio of 0.13(Case 4),the counterclockwise Coriolis cells combining constructively with counterclockwise buoyancy cells gives rise to a strong vortex which prevent the vortices induced by the ribs to mix with main flow in the first passage,which leads to an significant increase in heat transfer rate,while in the second passage,the rotation expands and reduces the vortices induced by the ribs of the co-trailing side(LP)and the co-leading side(HP),respectively compared to Case 2 and Case 3,so rate on the co-trailing and co-leading sides of the if rst and second passage compared to the stationary case.

(5)At rotation number of 1 and density ratio of 0.5(Case 7),the interaction between the clockwise Coriolis cells and clockwise buoyancy cells produces a structure of two counter-rotating vortices in the first passage,leads to an increase of heat transfer rate on high and low pressure surfaces compared to low and medium rotation number but it remains lower than that at high rotation number and low density ratio.In contrast,in the second passage,the interaction between rib-induced vortices and buoyancy cells degrades significantly the heat transfer rate compared to all cases and consequently,hot zones are created.

(6)At high rotation number and high density ratio(Case 7),the turbulent kinetic energy and shear stresses increase significantly mainly in the mid-section core area compared to low and medium rotation number.

(7)The predicted heat transfer for stationary case(Case 1)is in good agreement with measurements in the second passage,while in the first passage,the measured is qualitatively predicted.

(8)At low rotation number(Case 2),the predicted and measured heat transfer is in good agreement in the second passage and the bend region,while in the first passage the measured is over-predicted on both co-trailing and co-leading surfaces.

(9)At medium rotation(Case 3),the calculated and measured heat transfer is in fairly agreement on the co-trailing surface of the first and second passage,while the calculated is over-predicted and under-predicted the measured on the co-leading surface of the first and second passage,respectively.

(10)The results from the present study show that the numerical predictions provide characteristics of the flow field,which lead to a better understanding of the physics and turbulent transport mechanisms in the cooling channels of turbine blades at high buoyancy parameter,which are difficult to obtain by experiments.that the heat transfer rate is reduced on the co-leading side(HP)and remains almost unchanged on the cotrailing side compared to the stationary case.

(4)At rotation number of 0.24,one increases the density ratio from 0.13 to 0.50,degrades the heat transfer

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6 February 2016;revised 27 July 2016;accepted 29 August 2016

Available online 22 December 2016

?2016 Chinese Society of Aeronautics and Astronautics.Production and hosting by Elsevier Ltd.This is anopenaccessarticleundertheCCBY-NC-NDlicense(http://creativecommons.org/licenses/by-nc-nd/4.0/).

*Corresponding author.

E-mail addresses: berrabahb70@yahoo.fr (B. Berrabah),aminallahm@yahoo.fr(M.Aminallah).

Peer review under responsibility of Editorial Committee of CJA.