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An investigation of the conjugate heat transfer in an intercooled compressor vane based on a discontinuous Galerkin method
Applied Thermal Engineering 114 (2017) 85–97
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Applied Thermal Engineering journal homepage: www.elsevier.com/locate/apthermeng
Research Paper
An investigation of the conjugate heat transfer in an intercooled compressor vane based on a discontinuous Galerkin method Long-gang Liu, Chun-wei Gu, Xiaodong Ren ⇑ Key Laboratory for Thermal Science and Power Engineering of Ministry of Education, Department of Thermal Engineering, Tsinghua University, Beijing 100084, China
h i g h l i g h t s An intercooled compressor vane has been numerically studied. The cooling influence on the boundary layer is investigated. An empirical equation of Nu and Re has been validated.
a r t i c l e
i n f o
Article history: Received 26 August 2016 Revised 25 November 2016 Accepted 27 November 2016 Available online 29 November 2016 Keywords: Intercooled compressor Heat transfer Boundary layer Adverse pressure Discontinuous Galerkin method
a b s t r a c t In recent years, the convective cooling channel has been applied in the compressor vanes to improve the efficiency of Brayton cycle. In order to investigate the influence of cooling on the aerodynamic and heat transfer performance of an intercooled compressor vane, a discontinuous Galerkin (DG) method has been employed in this paper. The DG method has been verified by using a typical turbine vane problem and the results are in good agreement with the experimental data. A two-dimensional intercooled compressor vane with five cooling channels has been numerically simulated in different conditions. The results are compared with the one without cooling. The effect of cooling on the boundary layer under an adverse pressure gradient has been given, and the relationship between Nusselt number (Nu) and Reynolds number (Re) on the fluid-solid interface has been analyzed. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction The development of modern aero-engines and heavy-duty gas turbines complicates the feature of flow and heat transfer in these machines. In order to improve their performance, the intercooled and recuperated (ICR) method was proposed, which might reduce the fuel consumption. Since 1946, Rolls Royce and Royal Navy designed an ICR marine gas turbine RM60 and used it in the gunboats [1]. However, the ICR gas turbine was expensive and the space of heat exchanger increased. Besides, the ashes in the heat exchanger may decrease the efficiency. In 1965, the Soviet Union developed an ICR marine gas turbine CTY-20 for a cargo ship. Since 1985, GE transformed the aero-engine F404 to an ICR gas turbine LM-1600-ICR [2–4]. Rolls Royce developed Spey SMIC-ICR marine gas turbine. At the same time, MTU in Germany developed an ICR gas turbine SGT12, the power of which was similar to LM1600-ICR and Spey SMIC-ICR. Since 1991, a marine gas turbine ⇑ Corresponding author. E-mail addresses: [email protected] (L.-g. Liu), [email protected]. edu.cn (C.-w. Gu), [email protected] (X. Ren). http://dx.doi.org/10.1016/j.applthermaleng.2016.11.190 1359-4311/Ó 2016 Elsevier Ltd. All rights reserved.
engine WR-21 was developed by Rolls Royce, etc. for US Navy [5,6]. It succeeded in 500-h ICR production engine test program in 1997. In 2000, it was used in the Type 45 Aera Defense destroyer for Royal Navy [7]. In 2006, GE’s ICR gas turbine LMS100 was achieved and the pressure ratio of 42 in it is much higher than other machines with the same size at that time [8]. In the traditional ICR marine gas turbines, the intercooler is behind the low pressure compressor. And the weight and space of the extra heat exchanger and pipe in the ICR engine unacceptably increased. As a result, it is difficult for the ICR method to be applied in the aero-engines. Ito and Nagasaki [9] proposed a new ICR aero-engine method in 2011, as shown in Fig. 1. In order to solve the former problems, they used the vanes in the compressor as the air cooler, and vanes in the bypass duct as the radiator in the intercooling system. Besides, they used vanes in the combustor as the air heater, and vanes in the core nozzle as the heat absorber in the recuperating system. In the compressor vanes, they used five serial cooling channels for the convective heat transfer, and the coolant they chose was supercritical carbon dioxide. The coolant has a larger density and a larger specific heat capacity than the air.
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Nomenclature cp Dj ^j D Fj F^j h L LC LG n Pt Ps q S Tt T U y+ yn
specific heat capacity diffusive flux vector numerical diffusive flux vector convective flux vector numerical convective flux vector convective heat transfer coefficient axial chord length chord length tangential spacing outward normal unit vector of element interfaces total pressure static pressure heat flux source term vector total temperature temperature conservative variable vector non-dimensional wall distance normal wall distance
Greek symbols a angle of attack b installation angle k thermal conductivity m test function n incidence angle
q
X
density bounded connected domain
Subscripts 1 inlet 2 outlet non non-dimensional num numerical result ref reference state Acronyms CHT conjugate heat transfer DG discontinuous Galerkin FEM finite element method FVM finite volume method ICR intercooled and recuperated Ma Mach number NASA National Aeronautics and Space Administration Nu Nusselt number Pr Prandtl number PS pressure side RANS Reynolds-averaged Navier-Stokes Re Reynolds number SS suction side Tu turbulence intensity
Fig. 1. Intercooled and recuperated jet engine [9].
In the analysis of the compressor vanes in the new ICR aeroengine, the boundary layer has large effect on their aerodynamic and heat transfer performance. The separation may occur due to the adverse pressure gradient in the flow direction. Thus, it is necessary to investigate the effect of the adverse pressure gradient, the streamline curvature and the distribution of temperature on the boundary layer. The turbulent boundary layer on the flat plate with a sustained adverse pressure gradient has been studied in the past 60 years. The experimental research was conducted by Clauser [10] in 1954 and Bradshaw [11] in 1967. Rotta [12] investigated the problem theoretically in 1962. Self-similar characteristics have been proposed and were investigated by the experiments of Skare and Krogstad [13] in 1994 and the direct numerical simulation (DNS) of Stoke et al. [14] in 1998. In real applications, the pressure gradient is various in the flow direction. Non-equilibrium pressure gradient boundary layer has been investigated by the experiments of Samuel and Joubert [15] in 1974 and Houra et al. [16] in 2000. The numerical simulations
were conducted by Spalart and Watmuff [17] in 1993 and Coleman et al. [18] in 2003. The thermal fields in the boundary layer was investigated by Bradshaw and Huang [19] in 1995 and Volino and Simon [20] in 1997. They pointed out that the velocity law of the wall is more robust than the thermal law under pressure gradients. Houra and Nagano [21] experimentally studied the boundary layer under non-equilibrium adverse pressure gradient on a uniformly heated plate in 2006. In 2000, Kafoussias and Xenos [22] investigated the effect of adverse pressure gradient on the separation of the compressible turbulent boundary layer with the mass transfer (suction and injection) and heat transfer. In 1998, Khalatov and Shevchuk [23] studied the heat transfer and hydrodynamics over a convexly curved surface with the pressure gradient, and pointed out that the convex curvature decreases the rate of heat transfer and surface friction. In this paper, we investigate the cooling influence on the boundary layer under the adverse pressure gradient and streamline curvature in the new intercooled compressor. There are five
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cooling channels in the compressor vane, and both aerodynamic and heat transfer performances are studied under two different conditions. A discontinuous Galerkin (DG) method is employed to solve the conjugate heat transfer (CHT) problems, which has been verified previously [24,27].
!
Z I Z l1 X ~ kr du @/ks ^j D ^ j Þnj dr þ /kr /ks dX ¼ /ks ðF ðFj Dj ÞdX dt EkF @EkF EkF @xj r¼0 Z þ /ks SdX; 0 6 s 6 l 1; 8k ð5Þ EkF
In order to calculate the gradients of U, auxiliary variants Pj are introduced as follow
2. The DG method for CHT simulation For the compressor vane with cooling channels, the heat transfer occurs between the coolant and the compressor vanes, and between the main stream and vanes. In order to accurately calculate the heat flux on the interface, the CHT method [25,26] involving both the solid domain and the fluid domain should be used generally, because the heat transfer conditions are unknown in most cases. The accuracy of the numerical method and the data exchange on the domain interface has significantly influence on the numerical prediction [27]. Generally, a high fidelity method has lower numerical dissipation and gives us better prediction. The DG method is a suitable choice for us to simulate the CHT problems. DG method is firstly proposed by Reed and Hill in 1973 [28]. Then it was developed by Cockburn and Shu [29,30]. DG method can achieve a high order accuracy without relying on a large reconstruction stencil. A more direct data exchange process can be obtained due to its compactness, and it enhances the accuracy and stability on the domain interface when the problem has a complex geometry and a poor quality mesh. Furthermore, a better result can be obtained by using a DG method rather than the traditional finite element method (FEM) [31]. Authors have focused on DG methods [27,32–35] and applied the method on the CHT problems [27].
@U ¼ 0: @xj
Pj
Z n X EkF
k¼1
v
ð6Þ
v Pj d X
I @EkF
^ j dr þ v Un
Z EkF
! @v UdX ¼ 0: @xj
ð7Þ
Supposing that the numerical solution P h;j and the test function belongs to Ceq h , we can obtain that
l1 X
~ kj;r p
!
Z EkF
r¼0
/kr /ks d
X ¼
I @EkF
^ j dr /ks Un
Z EkF
@/ks Uh dX: @xj
ð8Þ
~ kj;s is one of the freedom of Ph;j jEk as where p F
Ph;j ðx; tÞjEk ¼ F
l1 X
~ kj;s ðtÞ/ks ðxÞ: p
ð9Þ
s¼0
In this paper, the All-Speed Roe-type scheme [35–39], which provide a complete unified solution for both compressible and incompressible flows and thus can ensure the reasonable numerical dissipation in the situation of low Ma, is used to calculate the inviscid numerical flux F^j . The central scheme [40] is used to ^ j . The explicit three-stage calculate the numerical diffusive flux D Runge-Kutta scheme is used for the time discretization.
2.1. Fluid domain 2.2. Solid domain The DG method for compressible flow simulations is briefly described as follows. The compressible Reynolds-averaged Navier-Stokes (RANS) equations are given as
qS cp;S
@U @Fj @Dj þ ¼ S: @t @xj @xj
ð1Þ
Using Galerkin method and the integration by parts, we obtain "Z # I Z Z @U @v ^j D ^ j Þnj dr v v ðF ðFj Dj ÞdX v SdX ¼ 0: dX þ @t EkF @EkF EkF @xj EkF k¼1 ð2Þ
k¼1
is a complete partition of flow domain XF ,
v
is the
^ j is the viscous test function, F^j is the inviscid numerical flux, D numerical flux. Supposing that the numerical solution Uh and the test function v belongs to Ceq h where 2 Ceq h ¼ fv 2 L ðXF Þ : v jEk F
2 Pm EkF ; 8kg:
A local polynomial basis
n
/ks
ol1 s¼0
ð3Þ is adopted to define the
approximate polynomial solution in each cell
U h ðx; tÞjEk ¼ F
l1 X ~ ks ðtÞ/ks ðxÞ; u
EkF
ð10Þ
qj ¼ kS
@T : @xj
ð11Þ
Again use the Galerkin method, we can obtain the following formulation
! Z I Z l1 X dh~kr @/ks k k ^j nj dr þ qS cp;S /r /s dX ¼ /ks q qh;j dX; dt Eks @Eks Eks @xj r¼0
ð12Þ l1 X
g~ kj;r
r¼0
Z EkS
! /kr /ks dX
I ¼
@EkS
^ j dr þ /ks kS Tn
Z EkS
@/ks kS T h dX; @xj
ð13Þ
where ~ hks is the unknown coefficient of the approximation solution ~ kj;s is the unknown freedom of the approximation solution T h jEk , and g s
^ ^j and T. qh;j jEk . The central scheme is applied for the numerical flux q s
ð4Þ
s¼0
~ ks u
@T @qj þ ¼ 0; @t @xj
where
n X
n on where EkF
In the solid domain, the Fourier equation are given by
in which is one of the freedom, and l is the number of basis functions. Substituting Eq. (4) into Eq. (2) and using the basis function as the test function, we have
2.3. Interface of fluid and solid domain On the fluid-solid interface, it needs to ensure the continuity of temperature and normal heat flux. The principles of conjugate heat transfer method are as follows. In each time step, the temperature on the fluid-solid interface is predicted by using the solid solver and used as the boundary
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L.-g. Liu et al. / Applied Thermal Engineering 114 (2017) 85–97 Table 1 Geometry parameters of C3X. Installation angle Air entrance angle Air exit angle Tangential spacing Chord length Axial chord
59.89° 0° 72.38° 117.73 mm 144.93 mm 78.16 mm
condition for the flow computation in the next time step. In contrast to that, the heat flux on the interface is calculated by using the fluid solver and imposed on the solid boundary for the next time step, as shown in Eq. (14). In this way, the data exchange process can be locally and directly achieved without any interpolation since the solution distribution in each element is known in each time step.
TjFluid ¼ TjSolid ; qjSolid nFS ¼ qjFluid nFS :
ð14Þ
The numerical flux on the interface in the fluid solver is given by
8 ^ > > < U ¼ U ðTjSolid ; . . .Þ ^ F j ¼ F j ðTjSolid ; . . .Þ > > :D ^ j ¼ Dj Tj ; q j Solid
j Fluid
ð15Þ
... ;
and the numerical flux on the interface for the solid solver is given by
(
T^ ¼ TjSolid ; ^j ¼ qj jSolid þ ðqk jFluid qk jSolid Þnk nj : q
ð16Þ
Based on the above method, an in-house DG code has been developed by the authors. In the fluid domain, we use a vertexbased hierarchical slope limiter which was proposed by Kuzmin [41] to prevent the numerical oscillation.
Fig. 2. Mesh for the turbine vane C3X.
Table 2 5411 test conditions. Pt1 (bar)
Tt1 (K)
P2 (bar)
Tu
3.119
798
1.553
6.5%
3. Code verification The turbine vane that we use to verify the code is C3X, which was designed and tested by NASA [42,43]. It has 10 internal convective cooling channels. The material of the C3X vane is ASTM310, whose thermal conductivity is k ¼ 0:020176 T þ 6:811 W=ðm KÞ [44], density is 7900 kg/m3 and specific heat capacity is 586.15 J/ (kg K). The geometry parameters of the vane are shown in Table 1. Here an unstructured triangular mesh is used. In order to improve accuracy on the fluid-solid interface, 30 layers of quadrilateral elements are adopted around the vane surface. The distance between the solid wall and the first grid line is set to be y+ < 1. As shown in Fig. 2, the number of elements in the flow domain (red1) is 22,700, and that of the solid domain (blue) is 9600. According to our previous study, the mesh independence is satisfied. Table 2 shows the test condition named 5411. We give the convective heat transfer coefficient and the average temperature of the cooling channels to be the boundary conditions of the solid domain. The convective heat transfer coefficient is calculated by
Nu ¼ Cr ð0:022 Pr0:5 Re0:8 Þ
ð17Þ
where Cr is a coefficient given by [42], Pr is the Prandtl number and Re is the Reynolds number. The SST turbulence model [45] with or without the c Reh transition model [46] is considered to show the model influence. Here the boundary layer transition is studied. The position where Tu rapidly increases can be considered as the onset of transition. 1 For interpretation of color in Fig. 2, the reader is referred to the web version of this article.
Fig. 3. The turbulent intensity (Tu) contour with c Reh transition model.
When the c Reh transition model is employed, the boundary layer transition can be clearly captured, as shown in Fig. 3 and Fig. 4 (left), the transition occurs at about x=L ¼ 0:43 on the suction side and the transition onset momentum thickness Reynolds number is
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320. However, the transition cannot be captured when the c Reh transition model is not considered, as shown in Fig. 4 (right). It indicates that the transition model must be used for the flow simulation with the boundary layer transition. The static pressure and temperature distributions on the vane surface are shown in Fig. 5, and compared with the experimental data. The static pressure is normalized by the inlet total pressure pref ¼ 311; 900 Pa, and the static temperature is normalized by T ref ¼ 811 K. The numerical results are in good agreement with the experiment data, especially for the one with the transition model.
89
convective cooling channels along the camber line of the profile to make the compressor vane as a heat exchanger. The material of the compressor vane is SUS304 whose thermal conductivity is 16 W/(m K). The density of SUS304 is 7930 kg/m3 and the specific heat capacity is 502 J/(kg K). The outer diameter of the heat exchanger tubes is 3.18 mm and the inner diameter is 1.74 mm. Because the material of the tubes is same to the compressor vane, we make it as a whole part in the numerical simulation. The geometry of the vane is shown in Fig. 6 and the geometry parameters of the vane are shown in Table 3. The angle of attack is 10.5°. The solidity r ¼ LC =LG is 1.5, where LC is the chord length and LG is the tangential spacing.
4. Numerical simulation of the intercooled compressor vane 4.2. Computational mesh 4.1. Geometric configuration In this section, an intercooled compressor vane is numerically studied. The vane has the same geometry as that Ito et al. used [47]. The profile is same to the NACA65-(12A2I8b)10 and the original profile is given by NASA in 1956 [48,49]. There are five internal
The unstructured triangular mesh is used here. The domain mesh with the cooling channels are shown in Fig. 7. For the domain without cooling channels, the mesh is similar with it. In order to achieve a high accuracy on the domain interface, 25 layers quadrilateral mesh is used around the vane surface. The distance between
Fig. 4. Transition on the suction side (left: SST + c Reh ; right: SST).
Fig. 5. The static pressure (left) and temperature (right) distributions on the vane surface.
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the solid wall and the first grid line is set to be y+ < 1. Based on the pervious grid-independence study, as shown in Table 4, the mesh used in this paper (The number of elements in the flow domain is 29,094, while the number of elements in the solid domain with cooling channels is 6204, and without cooling channel is 5773) is fine enough to get a mesh convergence result. 4.3. Boundary conditions Two test conditions are considered in this paper, and the inlet Mach number is Ma1 = 0.3 and Ma1 = 0.6, respectively. Table 5 lists the inlet Mach number, total pressure, total temperature and turbulence intensity. For the compressor vane with cooling channels, the convective heat transfer coefficient and the average temperature of the coolant are given. The supercritical carbon dioxide under 10 MPa is used to be the coolant. The properties of the supercritical carbon dioxide can be obtained from [50]. The empirical Eq. (18) proposed by Ito [51] can be applied to calculate the convective heat transfer coefficient. The calculate results are shown in Table 6. SST model with c Reh transition model is adopted.
Nu ¼ 0:0231 Pr0:300 Re0:823
ð18Þ
5. Results and discussions Fig. 6. Geometry of the intercooled compressor vane.
5.1. Comparison between numerical results and experiment data Table 3 Geometry parameters of the vane. Installation angle, b Angle of attack, a Incidence angle, n Chord length, LC Tangential spacing, LG Axial chord length, L
45.0° 10.5° 1.03° 44.003 mm 29.335 mm 31.115 mm
Distributions of the static pressure on the vane surface with or without cooling channels are compared with NASA’s experimental data, as shown in Figs. 8 and 9. The pressure is normalized by the inlet total pressure (pref ¼ 110; 000 Pa). x is the distance from the leading edge and Lc is the chord length. The numerical results are in good agreement with the experimental data. On the pressure side, the pressure decreases near the leading edge and the numer-
Fig. 7. Mesh of fluid domain (red) and solid domain with cooling channels (blue). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
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Table 4 Grid-independence study – comparison of pressure ratio.
Mesh Mesh Mesh Mesh
1 2 3 4
Elements number of fluid domain
Elements number of solid domain with cooling channels
0.6 cooling
0.3 cooling
11,000 29,100 55,000 93,000
2000 5800 9400 16,200
1.08653 1.08670 1.08667 1.08669
1.02016 1.02032 1.02029 1.02031
Table 5 Test conditions.
Ma1 = 0.3 Ma1 = 0.6
Pt1 (Pa)
Tt1 (K)
Tu
110,000 110,000
330.85 333.26
6.5% 6.5%
Table 6 Boundary conditions on cooling channels’ surface.
Channel Channel Channel Channel Channel
1 2 3 4 5
Ave. T (K)
Re
Pr
h (W/m2 K)
312.96 312.75 312.54 312.33 312.12
56,038 55,917 55,795 55,674 55,553
1.6812 1.6819 1.6826 1.6833 1.6841
13,724 13,722 13,719 13,715 13,712
Fig. 9. The distribution of pressure on the vane surface with Ma1 = 0.3.
Fig. 10. Comparison of the temperature at Ma1 = 0.6.
T non ¼ Fig. 8. The distribution of pressure on the vane surface with Ma1 = 0.6.
ical prediction is slightly higher than the experiment. On the suction side, the pressure increases around the middle chord. Due to the separation appears near the trailing edge, the numerical pressure there is lower than the experimental one. It also indicates that the cooling has little influence on the distribution of the static pressure on the vane surface. The numerical result of the static temperature distribution on the surface is compared with Ito’s data. They measured the temperature at four points and calculated the distribution by the inverse heat transfer method at the condition Ma1 = 0.6 [51], as shown in Fig. 10. The static temperature on the vane surface is normalized by the inlet adiabatic temperature of the main stream 330 K and the outlet static temperature of the coolant 311.5 K as
T 311:5 330 311:5
ð19Þ
The prediction of the temperature trend is similar with one given by Ito et al. [51], but the value has a little difference. It can be considered as the deviation due to both the numerical deviation and that of temperature measurement in the experiments. The temperature achieves its lowest value around the middle chord. 5.2. Effect on heat transfer performance The distribution of the temperature on the vane surface with cooling channels is compared with the one without cooling channel, as shown in Figs. 11 and 12. It obviously shows that the convective cooling has large effect on the vane temperature. The cooling air takes away a part of the heat and makes the blade temperature at the middle part lower than that at other parts. The distance between the blade leading edge and the nearest cooling
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Fig. 13. Distribution of heat transfer coefficient.
Fig. 11. The distribution of temperature on the vane surface with Ma1 = 0.6.
Table 7 Comparison of Nu.
Nunum Nuplate NuIto-2012
Ma = 0.6
Ma = 0.3
760.4 1142.2 740.9
466.4 724.5 470.0
Heat flux in the normal direction of the turbulent boundary layer is:
q ¼ k
Fig. 12. The distribution of temperature on the vane surface with Ma1 = 0.3.
channel is larger than the one for the blade middle part, thus, the heat transfer rate near the blade leading edge is lower than the one at the blade middle part, leading to a higher blade temperature at the blade leading edge. Besides, the blade temperature at the rear part is also higher than that at the middle as the similar reason. In general, the cooling effectively reduce the temperature of the middle part of the stator vane. The convective heat transfer coefficient on the surface of compressor vane can be obtained. Fig. 13 shows the convective heat transfer coefficient at the condition Ma1 = 0.3. On both pressure and suction side, the boundary layer becomes thicker along the streamwise direction, and it weakens the heat transfer. Heat flux in the normal direction of the laminar boundary layer is:
q ¼ k
dT : dy
ð20Þ
dT qcp T 0 v 0 : dy
ð21Þ
where T 0 v 0 increases the heat transfer in the turbulent boundary layer. As a result, transition enhances the heat transfer and makes the convective heat transfer coefficient increase at about x=LC ¼ 0:25 on the pressure surface. On the suction surface, laminar boundary layer exists at the front and middle part of the suction side. Transition and separation occur near the trailing edge and enhance the heat transfer. The relationship between the average Nusselt number (Nu) and the average Reynolds number (Re) on the fluid-solid interface has been analyzed. The average convective heat transfer coefficient can be calculated by
P
hair ¼ P
Alocal q ; Alocal ðT air;ad T airfoil Þ
ð22Þ
where T air;ad ¼ 330 K. The average Nu can be calculated by
Nuair ¼
hair LC ; kair
ð23Þ
and the average Re can be calculated by
Reair ¼
uair;in LC
mair
ð24Þ
where chord length LC ¼ 0:044 m. For both conditions Ma1 = 0.6, Nunum ¼ 760:4, Renum ¼ 473; 323 and Ma1 = 0.3, Nunum ¼ 466:4, Renum ¼ 267; 968, the empirical
L.-g. Liu et al. / Applied Thermal Engineering 114 (2017) 85–97
Fig. 14. Contours of Tu.
(a) 0.6 without cooling
(b) 0.6 cooling
Fig. 15. Temperature contours at the middle part of the vane pressure side (Ma1 = 0.6).
equation of well-developed turbulent boundary layer on the flat plate [47] was given by 1=3 Nuplate ¼ 0:037Re0:8 air Pr air
ð25Þ
In 2012, Ito et al. [47] used the compressor vane with five cooling channels and LG ¼ LC =1:0 to do experiments and got an empirical equation at the condition Ma1 = 0.75 with different attack angles as 1=3 NuIto-2012 ¼ 0:024Re0:8 air Pr air
ð26Þ
Nu number predicted by using the DG method are compared with that calculated by the empirical equations, and the results are shown in Table 7. It indicates that the result from Eq. (26) is in good agreement with the numerical one. The relative error is about 2.6% for Ma1 = 0.6 and 0.77% for Ma1 = 0.3. 5.3. Effect on aerodynamic performance The contours of Tu are shown in Fig. 14. For all the conditions, Tu increases at about x=LC ¼ 0:25 on the pressure side of the vane, which can be seen as the onset of transition. The transition position on the suction side is at about x=LC ¼ 0:60. As a result, the cooling channels mainly affect the turbulent boundary layer on the pres-
Fig. 16. Velocity profiles at Point A (Ma1 = 0.6).
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(a) 0.3 without cooling
(b) 0.3 cooling
Fig. 17. Temperature contours at the middle part of the pressure side (Ma1 = 0.3).
Fig. 19. Velocity profiles at Point B (Ma1 = 0.6). Fig. 18. Velocity profiles at Point A (Ma1 = 0.3).
sure side, and have an effect on both the laminar boundary layer and the turbulent boundary layer on the suction side. In the compressible boundary layer, dynamic viscosity and density will be changed. Because of the wall cooling on the compressor vane, the changing temperature leads to a changing viscosity and density, which may change the velocity profile and the temperature profile. In the turbulent boundary layer, decrease of temperature leads to a larger density near the wall, which makes the boundary layer thinner. In the laminar boundary layer, dynamic viscosity reflects the ability of momentum transport normal to the shear layer. The relationship between dynamic viscosity of air and temperature can be obtained by the Sutherland’s formula [52]:
l T ¼ T0 l0 where n ¼ 1:5.
n T0 þ Ts T þ Ts
ð27Þ
l0 ¼ 17:161 106 ðPa sÞ, T 0 ¼ 273:0 ðKÞ, T s ¼ 124 ðKÞ, Fig. 20. Velocity profiles at Point B (Ma1 = 0.3).
L.-g. Liu et al. / Applied Thermal Engineering 114 (2017) 85–97
Fig. 22. Velocity profiles at Point C (Ma1 = 0.3).
Fig. 21. Velocity profiles at Point C (Ma1 = 0.6).
(a) 0.6 without cooling
(b) 0.6 cooling Fig. 23. Pressure contours (Ma1 = 0.6).
(a) 0.3 without cooling
(b) 0.3 cooling Fig. 24. Pressure contours (Ma1 = 0.3).
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It indicates that dynamic viscosity decreases with cooling and increases with heating. As a result, the dynamic viscosity of the boundary layer on the intercooled compressor surface becomes smaller. In the two-dimensional boundary layer, the smaller dynamic viscosity leads to a smaller shear stress (s ¼ lð@u=@yÞ). Smaller shear stress results in a larger velocity near the wall. The displacement thickness d becomes smaller because:
d ¼
Z
h
0
qu dy: 1 qe ue
ð28Þ
where q becomes larger when cooling. It indicates that decrease of the dynamic viscosity leads to a thinner influence region. As a result of the thinner boundary layer, the compressor has a larger throughflow area to have a higher pressure ratio. At Point A in Fig. 14, the effect of cooling channels on the turbulent boundary layer has been analyzed. Because of the cooling channels, the temperature of the turbulent boundary layer decreases, as shown in Figs. 15 and 17. As a result, the boundary layer becomes thinner. In order to see the boundary layer clearly, Figs. 16 and 18 show the velocity profiles in the boundary layer at Point A. On the suction side, two points B and C in Fig. 14 are considered here, and the velocity profiles are shown in Figs. 19–22. Although cooling channels have effect on the boundary layer of suction side, the results show that the effect is not significant. The static pressure contour at Ma1 = 0.6 is shown in Fig. 23, and Fig. 24 is the static pressure contour at Ma1 = 0.3. Because of the effect of cooling channels, the boundary layer becomes thinner, especially on the pressure side. Thus, the cascade has a larger area ratio for the main stream. The compressor vane has a better ability for the pressure rising. The isoline of the pressure contour moves upstream in the rear part of the vane at both test conditions. In a real compressor, the temperature is actually higher than the temperature at these conditions, thus, the effect of cooling channels on the pressure rising may be more significant in the real compressor. 6. Conclusion In this paper, DG method is applied to discuss the conjugate heat transfer performance in a two-dimensional turbine vane and an intercooled compressor vane. Our in-house code shows a good numerical prediction based on the comparison between the numerical results and the experiment data. The conclusions can be obtained as follows: 1. The cooling channels on the compressor vane can reduce the temperature of the vane and protect the vane from the high temperature. 2. At a certain range, the empirical equation of Nu and Re proposed by Ito in 2012 is a suitable equation. It can be applied in the analysis of heat transfer problems. 1=3 Nuair ¼ 0:024Re0:8 air Pr air
3. The cooling channels have an effect on the laminar boundary layer and the turbulent boundary layer. They make the boundary layer thinner and increase the pressure ratio of the stator vane. Cooling in the compressor vanes may be used to improve the aerodynamic performance in a real compressor in the future.
Acknowledgments This work is supported by National Natural Science Foundation of China (Grant No. 51276093). Thanks to Yutaka Ito et al. in Tokyo Institute of Technology for providing the experiment data.
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Contents lists available at ScienceDirect
Applied Thermal Engineering journal homepage: www.elsevier.com/locate/apthermeng
Research Paper
An investigation of the conjugate heat transfer in an intercooled compressor vane based on a discontinuous Galerkin method Long-gang Liu, Chun-wei Gu, Xiaodong Ren ⇑ Key Laboratory for Thermal Science and Power Engineering of Ministry of Education, Department of Thermal Engineering, Tsinghua University, Beijing 100084, China
h i g h l i g h t s An intercooled compressor vane has been numerically studied. The cooling influence on the boundary layer is investigated. An empirical equation of Nu and Re has been validated.
a r t i c l e
i n f o
Article history: Received 26 August 2016 Revised 25 November 2016 Accepted 27 November 2016 Available online 29 November 2016 Keywords: Intercooled compressor Heat transfer Boundary layer Adverse pressure Discontinuous Galerkin method
a b s t r a c t In recent years, the convective cooling channel has been applied in the compressor vanes to improve the efficiency of Brayton cycle. In order to investigate the influence of cooling on the aerodynamic and heat transfer performance of an intercooled compressor vane, a discontinuous Galerkin (DG) method has been employed in this paper. The DG method has been verified by using a typical turbine vane problem and the results are in good agreement with the experimental data. A two-dimensional intercooled compressor vane with five cooling channels has been numerically simulated in different conditions. The results are compared with the one without cooling. The effect of cooling on the boundary layer under an adverse pressure gradient has been given, and the relationship between Nusselt number (Nu) and Reynolds number (Re) on the fluid-solid interface has been analyzed. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction The development of modern aero-engines and heavy-duty gas turbines complicates the feature of flow and heat transfer in these machines. In order to improve their performance, the intercooled and recuperated (ICR) method was proposed, which might reduce the fuel consumption. Since 1946, Rolls Royce and Royal Navy designed an ICR marine gas turbine RM60 and used it in the gunboats [1]. However, the ICR gas turbine was expensive and the space of heat exchanger increased. Besides, the ashes in the heat exchanger may decrease the efficiency. In 1965, the Soviet Union developed an ICR marine gas turbine CTY-20 for a cargo ship. Since 1985, GE transformed the aero-engine F404 to an ICR gas turbine LM-1600-ICR [2–4]. Rolls Royce developed Spey SMIC-ICR marine gas turbine. At the same time, MTU in Germany developed an ICR gas turbine SGT12, the power of which was similar to LM1600-ICR and Spey SMIC-ICR. Since 1991, a marine gas turbine ⇑ Corresponding author. E-mail addresses: [email protected] (L.-g. Liu), [email protected]. edu.cn (C.-w. Gu), [email protected] (X. Ren). http://dx.doi.org/10.1016/j.applthermaleng.2016.11.190 1359-4311/Ó 2016 Elsevier Ltd. All rights reserved.
engine WR-21 was developed by Rolls Royce, etc. for US Navy [5,6]. It succeeded in 500-h ICR production engine test program in 1997. In 2000, it was used in the Type 45 Aera Defense destroyer for Royal Navy [7]. In 2006, GE’s ICR gas turbine LMS100 was achieved and the pressure ratio of 42 in it is much higher than other machines with the same size at that time [8]. In the traditional ICR marine gas turbines, the intercooler is behind the low pressure compressor. And the weight and space of the extra heat exchanger and pipe in the ICR engine unacceptably increased. As a result, it is difficult for the ICR method to be applied in the aero-engines. Ito and Nagasaki [9] proposed a new ICR aero-engine method in 2011, as shown in Fig. 1. In order to solve the former problems, they used the vanes in the compressor as the air cooler, and vanes in the bypass duct as the radiator in the intercooling system. Besides, they used vanes in the combustor as the air heater, and vanes in the core nozzle as the heat absorber in the recuperating system. In the compressor vanes, they used five serial cooling channels for the convective heat transfer, and the coolant they chose was supercritical carbon dioxide. The coolant has a larger density and a larger specific heat capacity than the air.
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Nomenclature cp Dj ^j D Fj F^j h L LC LG n Pt Ps q S Tt T U y+ yn
specific heat capacity diffusive flux vector numerical diffusive flux vector convective flux vector numerical convective flux vector convective heat transfer coefficient axial chord length chord length tangential spacing outward normal unit vector of element interfaces total pressure static pressure heat flux source term vector total temperature temperature conservative variable vector non-dimensional wall distance normal wall distance
Greek symbols a angle of attack b installation angle k thermal conductivity m test function n incidence angle
q
X
density bounded connected domain
Subscripts 1 inlet 2 outlet non non-dimensional num numerical result ref reference state Acronyms CHT conjugate heat transfer DG discontinuous Galerkin FEM finite element method FVM finite volume method ICR intercooled and recuperated Ma Mach number NASA National Aeronautics and Space Administration Nu Nusselt number Pr Prandtl number PS pressure side RANS Reynolds-averaged Navier-Stokes Re Reynolds number SS suction side Tu turbulence intensity
Fig. 1. Intercooled and recuperated jet engine [9].
In the analysis of the compressor vanes in the new ICR aeroengine, the boundary layer has large effect on their aerodynamic and heat transfer performance. The separation may occur due to the adverse pressure gradient in the flow direction. Thus, it is necessary to investigate the effect of the adverse pressure gradient, the streamline curvature and the distribution of temperature on the boundary layer. The turbulent boundary layer on the flat plate with a sustained adverse pressure gradient has been studied in the past 60 years. The experimental research was conducted by Clauser [10] in 1954 and Bradshaw [11] in 1967. Rotta [12] investigated the problem theoretically in 1962. Self-similar characteristics have been proposed and were investigated by the experiments of Skare and Krogstad [13] in 1994 and the direct numerical simulation (DNS) of Stoke et al. [14] in 1998. In real applications, the pressure gradient is various in the flow direction. Non-equilibrium pressure gradient boundary layer has been investigated by the experiments of Samuel and Joubert [15] in 1974 and Houra et al. [16] in 2000. The numerical simulations
were conducted by Spalart and Watmuff [17] in 1993 and Coleman et al. [18] in 2003. The thermal fields in the boundary layer was investigated by Bradshaw and Huang [19] in 1995 and Volino and Simon [20] in 1997. They pointed out that the velocity law of the wall is more robust than the thermal law under pressure gradients. Houra and Nagano [21] experimentally studied the boundary layer under non-equilibrium adverse pressure gradient on a uniformly heated plate in 2006. In 2000, Kafoussias and Xenos [22] investigated the effect of adverse pressure gradient on the separation of the compressible turbulent boundary layer with the mass transfer (suction and injection) and heat transfer. In 1998, Khalatov and Shevchuk [23] studied the heat transfer and hydrodynamics over a convexly curved surface with the pressure gradient, and pointed out that the convex curvature decreases the rate of heat transfer and surface friction. In this paper, we investigate the cooling influence on the boundary layer under the adverse pressure gradient and streamline curvature in the new intercooled compressor. There are five
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cooling channels in the compressor vane, and both aerodynamic and heat transfer performances are studied under two different conditions. A discontinuous Galerkin (DG) method is employed to solve the conjugate heat transfer (CHT) problems, which has been verified previously [24,27].
!
Z I Z l1 X ~ kr du @/ks ^j D ^ j Þnj dr þ /kr /ks dX ¼ /ks ðF ðFj Dj ÞdX dt EkF @EkF EkF @xj r¼0 Z þ /ks SdX; 0 6 s 6 l 1; 8k ð5Þ EkF
In order to calculate the gradients of U, auxiliary variants Pj are introduced as follow
2. The DG method for CHT simulation For the compressor vane with cooling channels, the heat transfer occurs between the coolant and the compressor vanes, and between the main stream and vanes. In order to accurately calculate the heat flux on the interface, the CHT method [25,26] involving both the solid domain and the fluid domain should be used generally, because the heat transfer conditions are unknown in most cases. The accuracy of the numerical method and the data exchange on the domain interface has significantly influence on the numerical prediction [27]. Generally, a high fidelity method has lower numerical dissipation and gives us better prediction. The DG method is a suitable choice for us to simulate the CHT problems. DG method is firstly proposed by Reed and Hill in 1973 [28]. Then it was developed by Cockburn and Shu [29,30]. DG method can achieve a high order accuracy without relying on a large reconstruction stencil. A more direct data exchange process can be obtained due to its compactness, and it enhances the accuracy and stability on the domain interface when the problem has a complex geometry and a poor quality mesh. Furthermore, a better result can be obtained by using a DG method rather than the traditional finite element method (FEM) [31]. Authors have focused on DG methods [27,32–35] and applied the method on the CHT problems [27].
@U ¼ 0: @xj
Pj
Z n X EkF
k¼1
v
ð6Þ
v Pj d X
I @EkF
^ j dr þ v Un
Z EkF
! @v UdX ¼ 0: @xj
ð7Þ
Supposing that the numerical solution P h;j and the test function belongs to Ceq h , we can obtain that
l1 X
~ kj;r p
!
Z EkF
r¼0
/kr /ks d
X ¼
I @EkF
^ j dr /ks Un
Z EkF
@/ks Uh dX: @xj
ð8Þ
~ kj;s is one of the freedom of Ph;j jEk as where p F
Ph;j ðx; tÞjEk ¼ F
l1 X
~ kj;s ðtÞ/ks ðxÞ: p
ð9Þ
s¼0
In this paper, the All-Speed Roe-type scheme [35–39], which provide a complete unified solution for both compressible and incompressible flows and thus can ensure the reasonable numerical dissipation in the situation of low Ma, is used to calculate the inviscid numerical flux F^j . The central scheme [40] is used to ^ j . The explicit three-stage calculate the numerical diffusive flux D Runge-Kutta scheme is used for the time discretization.
2.1. Fluid domain 2.2. Solid domain The DG method for compressible flow simulations is briefly described as follows. The compressible Reynolds-averaged Navier-Stokes (RANS) equations are given as
qS cp;S
@U @Fj @Dj þ ¼ S: @t @xj @xj
ð1Þ
Using Galerkin method and the integration by parts, we obtain "Z # I Z Z @U @v ^j D ^ j Þnj dr v v ðF ðFj Dj ÞdX v SdX ¼ 0: dX þ @t EkF @EkF EkF @xj EkF k¼1 ð2Þ
k¼1
is a complete partition of flow domain XF ,
v
is the
^ j is the viscous test function, F^j is the inviscid numerical flux, D numerical flux. Supposing that the numerical solution Uh and the test function v belongs to Ceq h where 2 Ceq h ¼ fv 2 L ðXF Þ : v jEk F
2 Pm EkF ; 8kg:
A local polynomial basis
n
/ks
ol1 s¼0
ð3Þ is adopted to define the
approximate polynomial solution in each cell
U h ðx; tÞjEk ¼ F
l1 X ~ ks ðtÞ/ks ðxÞ; u
EkF
ð10Þ
qj ¼ kS
@T : @xj
ð11Þ
Again use the Galerkin method, we can obtain the following formulation
! Z I Z l1 X dh~kr @/ks k k ^j nj dr þ qS cp;S /r /s dX ¼ /ks q qh;j dX; dt Eks @Eks Eks @xj r¼0
ð12Þ l1 X
g~ kj;r
r¼0
Z EkS
! /kr /ks dX
I ¼
@EkS
^ j dr þ /ks kS Tn
Z EkS
@/ks kS T h dX; @xj
ð13Þ
where ~ hks is the unknown coefficient of the approximation solution ~ kj;s is the unknown freedom of the approximation solution T h jEk , and g s
^ ^j and T. qh;j jEk . The central scheme is applied for the numerical flux q s
ð4Þ
s¼0
~ ks u
@T @qj þ ¼ 0; @t @xj
where
n X
n on where EkF
In the solid domain, the Fourier equation are given by
in which is one of the freedom, and l is the number of basis functions. Substituting Eq. (4) into Eq. (2) and using the basis function as the test function, we have
2.3. Interface of fluid and solid domain On the fluid-solid interface, it needs to ensure the continuity of temperature and normal heat flux. The principles of conjugate heat transfer method are as follows. In each time step, the temperature on the fluid-solid interface is predicted by using the solid solver and used as the boundary
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L.-g. Liu et al. / Applied Thermal Engineering 114 (2017) 85–97 Table 1 Geometry parameters of C3X. Installation angle Air entrance angle Air exit angle Tangential spacing Chord length Axial chord
59.89° 0° 72.38° 117.73 mm 144.93 mm 78.16 mm
condition for the flow computation in the next time step. In contrast to that, the heat flux on the interface is calculated by using the fluid solver and imposed on the solid boundary for the next time step, as shown in Eq. (14). In this way, the data exchange process can be locally and directly achieved without any interpolation since the solution distribution in each element is known in each time step.
TjFluid ¼ TjSolid ; qjSolid nFS ¼ qjFluid nFS :
ð14Þ
The numerical flux on the interface in the fluid solver is given by
8 ^ > > < U ¼ U ðTjSolid ; . . .Þ ^ F j ¼ F j ðTjSolid ; . . .Þ > > :D ^ j ¼ Dj Tj ; q j Solid
j Fluid
ð15Þ
... ;
and the numerical flux on the interface for the solid solver is given by
(
T^ ¼ TjSolid ; ^j ¼ qj jSolid þ ðqk jFluid qk jSolid Þnk nj : q
ð16Þ
Based on the above method, an in-house DG code has been developed by the authors. In the fluid domain, we use a vertexbased hierarchical slope limiter which was proposed by Kuzmin [41] to prevent the numerical oscillation.
Fig. 2. Mesh for the turbine vane C3X.
Table 2 5411 test conditions. Pt1 (bar)
Tt1 (K)
P2 (bar)
Tu
3.119
798
1.553
6.5%
3. Code verification The turbine vane that we use to verify the code is C3X, which was designed and tested by NASA [42,43]. It has 10 internal convective cooling channels. The material of the C3X vane is ASTM310, whose thermal conductivity is k ¼ 0:020176 T þ 6:811 W=ðm KÞ [44], density is 7900 kg/m3 and specific heat capacity is 586.15 J/ (kg K). The geometry parameters of the vane are shown in Table 1. Here an unstructured triangular mesh is used. In order to improve accuracy on the fluid-solid interface, 30 layers of quadrilateral elements are adopted around the vane surface. The distance between the solid wall and the first grid line is set to be y+ < 1. As shown in Fig. 2, the number of elements in the flow domain (red1) is 22,700, and that of the solid domain (blue) is 9600. According to our previous study, the mesh independence is satisfied. Table 2 shows the test condition named 5411. We give the convective heat transfer coefficient and the average temperature of the cooling channels to be the boundary conditions of the solid domain. The convective heat transfer coefficient is calculated by
Nu ¼ Cr ð0:022 Pr0:5 Re0:8 Þ
ð17Þ
where Cr is a coefficient given by [42], Pr is the Prandtl number and Re is the Reynolds number. The SST turbulence model [45] with or without the c Reh transition model [46] is considered to show the model influence. Here the boundary layer transition is studied. The position where Tu rapidly increases can be considered as the onset of transition. 1 For interpretation of color in Fig. 2, the reader is referred to the web version of this article.
Fig. 3. The turbulent intensity (Tu) contour with c Reh transition model.
When the c Reh transition model is employed, the boundary layer transition can be clearly captured, as shown in Fig. 3 and Fig. 4 (left), the transition occurs at about x=L ¼ 0:43 on the suction side and the transition onset momentum thickness Reynolds number is
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320. However, the transition cannot be captured when the c Reh transition model is not considered, as shown in Fig. 4 (right). It indicates that the transition model must be used for the flow simulation with the boundary layer transition. The static pressure and temperature distributions on the vane surface are shown in Fig. 5, and compared with the experimental data. The static pressure is normalized by the inlet total pressure pref ¼ 311; 900 Pa, and the static temperature is normalized by T ref ¼ 811 K. The numerical results are in good agreement with the experiment data, especially for the one with the transition model.
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convective cooling channels along the camber line of the profile to make the compressor vane as a heat exchanger. The material of the compressor vane is SUS304 whose thermal conductivity is 16 W/(m K). The density of SUS304 is 7930 kg/m3 and the specific heat capacity is 502 J/(kg K). The outer diameter of the heat exchanger tubes is 3.18 mm and the inner diameter is 1.74 mm. Because the material of the tubes is same to the compressor vane, we make it as a whole part in the numerical simulation. The geometry of the vane is shown in Fig. 6 and the geometry parameters of the vane are shown in Table 3. The angle of attack is 10.5°. The solidity r ¼ LC =LG is 1.5, where LC is the chord length and LG is the tangential spacing.
4. Numerical simulation of the intercooled compressor vane 4.2. Computational mesh 4.1. Geometric configuration In this section, an intercooled compressor vane is numerically studied. The vane has the same geometry as that Ito et al. used [47]. The profile is same to the NACA65-(12A2I8b)10 and the original profile is given by NASA in 1956 [48,49]. There are five internal
The unstructured triangular mesh is used here. The domain mesh with the cooling channels are shown in Fig. 7. For the domain without cooling channels, the mesh is similar with it. In order to achieve a high accuracy on the domain interface, 25 layers quadrilateral mesh is used around the vane surface. The distance between
Fig. 4. Transition on the suction side (left: SST + c Reh ; right: SST).
Fig. 5. The static pressure (left) and temperature (right) distributions on the vane surface.
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the solid wall and the first grid line is set to be y+ < 1. Based on the pervious grid-independence study, as shown in Table 4, the mesh used in this paper (The number of elements in the flow domain is 29,094, while the number of elements in the solid domain with cooling channels is 6204, and without cooling channel is 5773) is fine enough to get a mesh convergence result. 4.3. Boundary conditions Two test conditions are considered in this paper, and the inlet Mach number is Ma1 = 0.3 and Ma1 = 0.6, respectively. Table 5 lists the inlet Mach number, total pressure, total temperature and turbulence intensity. For the compressor vane with cooling channels, the convective heat transfer coefficient and the average temperature of the coolant are given. The supercritical carbon dioxide under 10 MPa is used to be the coolant. The properties of the supercritical carbon dioxide can be obtained from [50]. The empirical Eq. (18) proposed by Ito [51] can be applied to calculate the convective heat transfer coefficient. The calculate results are shown in Table 6. SST model with c Reh transition model is adopted.
Nu ¼ 0:0231 Pr0:300 Re0:823
ð18Þ
5. Results and discussions Fig. 6. Geometry of the intercooled compressor vane.
5.1. Comparison between numerical results and experiment data Table 3 Geometry parameters of the vane. Installation angle, b Angle of attack, a Incidence angle, n Chord length, LC Tangential spacing, LG Axial chord length, L
45.0° 10.5° 1.03° 44.003 mm 29.335 mm 31.115 mm
Distributions of the static pressure on the vane surface with or without cooling channels are compared with NASA’s experimental data, as shown in Figs. 8 and 9. The pressure is normalized by the inlet total pressure (pref ¼ 110; 000 Pa). x is the distance from the leading edge and Lc is the chord length. The numerical results are in good agreement with the experimental data. On the pressure side, the pressure decreases near the leading edge and the numer-
Fig. 7. Mesh of fluid domain (red) and solid domain with cooling channels (blue). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
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Table 4 Grid-independence study – comparison of pressure ratio.
Mesh Mesh Mesh Mesh
1 2 3 4
Elements number of fluid domain
Elements number of solid domain with cooling channels
0.6 cooling
0.3 cooling
11,000 29,100 55,000 93,000
2000 5800 9400 16,200
1.08653 1.08670 1.08667 1.08669
1.02016 1.02032 1.02029 1.02031
Table 5 Test conditions.
Ma1 = 0.3 Ma1 = 0.6
Pt1 (Pa)
Tt1 (K)
Tu
110,000 110,000
330.85 333.26
6.5% 6.5%
Table 6 Boundary conditions on cooling channels’ surface.
Channel Channel Channel Channel Channel
1 2 3 4 5
Ave. T (K)
Re
Pr
h (W/m2 K)
312.96 312.75 312.54 312.33 312.12
56,038 55,917 55,795 55,674 55,553
1.6812 1.6819 1.6826 1.6833 1.6841
13,724 13,722 13,719 13,715 13,712
Fig. 9. The distribution of pressure on the vane surface with Ma1 = 0.3.
Fig. 10. Comparison of the temperature at Ma1 = 0.6.
T non ¼ Fig. 8. The distribution of pressure on the vane surface with Ma1 = 0.6.
ical prediction is slightly higher than the experiment. On the suction side, the pressure increases around the middle chord. Due to the separation appears near the trailing edge, the numerical pressure there is lower than the experimental one. It also indicates that the cooling has little influence on the distribution of the static pressure on the vane surface. The numerical result of the static temperature distribution on the surface is compared with Ito’s data. They measured the temperature at four points and calculated the distribution by the inverse heat transfer method at the condition Ma1 = 0.6 [51], as shown in Fig. 10. The static temperature on the vane surface is normalized by the inlet adiabatic temperature of the main stream 330 K and the outlet static temperature of the coolant 311.5 K as
T 311:5 330 311:5
ð19Þ
The prediction of the temperature trend is similar with one given by Ito et al. [51], but the value has a little difference. It can be considered as the deviation due to both the numerical deviation and that of temperature measurement in the experiments. The temperature achieves its lowest value around the middle chord. 5.2. Effect on heat transfer performance The distribution of the temperature on the vane surface with cooling channels is compared with the one without cooling channel, as shown in Figs. 11 and 12. It obviously shows that the convective cooling has large effect on the vane temperature. The cooling air takes away a part of the heat and makes the blade temperature at the middle part lower than that at other parts. The distance between the blade leading edge and the nearest cooling
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Fig. 13. Distribution of heat transfer coefficient.
Fig. 11. The distribution of temperature on the vane surface with Ma1 = 0.6.
Table 7 Comparison of Nu.
Nunum Nuplate NuIto-2012
Ma = 0.6
Ma = 0.3
760.4 1142.2 740.9
466.4 724.5 470.0
Heat flux in the normal direction of the turbulent boundary layer is:
q ¼ k
Fig. 12. The distribution of temperature on the vane surface with Ma1 = 0.3.
channel is larger than the one for the blade middle part, thus, the heat transfer rate near the blade leading edge is lower than the one at the blade middle part, leading to a higher blade temperature at the blade leading edge. Besides, the blade temperature at the rear part is also higher than that at the middle as the similar reason. In general, the cooling effectively reduce the temperature of the middle part of the stator vane. The convective heat transfer coefficient on the surface of compressor vane can be obtained. Fig. 13 shows the convective heat transfer coefficient at the condition Ma1 = 0.3. On both pressure and suction side, the boundary layer becomes thicker along the streamwise direction, and it weakens the heat transfer. Heat flux in the normal direction of the laminar boundary layer is:
q ¼ k
dT : dy
ð20Þ
dT qcp T 0 v 0 : dy
ð21Þ
where T 0 v 0 increases the heat transfer in the turbulent boundary layer. As a result, transition enhances the heat transfer and makes the convective heat transfer coefficient increase at about x=LC ¼ 0:25 on the pressure surface. On the suction surface, laminar boundary layer exists at the front and middle part of the suction side. Transition and separation occur near the trailing edge and enhance the heat transfer. The relationship between the average Nusselt number (Nu) and the average Reynolds number (Re) on the fluid-solid interface has been analyzed. The average convective heat transfer coefficient can be calculated by
P
hair ¼ P
Alocal q ; Alocal ðT air;ad T airfoil Þ
ð22Þ
where T air;ad ¼ 330 K. The average Nu can be calculated by
Nuair ¼
hair LC ; kair
ð23Þ
and the average Re can be calculated by
Reair ¼
uair;in LC
mair
ð24Þ
where chord length LC ¼ 0:044 m. For both conditions Ma1 = 0.6, Nunum ¼ 760:4, Renum ¼ 473; 323 and Ma1 = 0.3, Nunum ¼ 466:4, Renum ¼ 267; 968, the empirical
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Fig. 14. Contours of Tu.
(a) 0.6 without cooling
(b) 0.6 cooling
Fig. 15. Temperature contours at the middle part of the vane pressure side (Ma1 = 0.6).
equation of well-developed turbulent boundary layer on the flat plate [47] was given by 1=3 Nuplate ¼ 0:037Re0:8 air Pr air
ð25Þ
In 2012, Ito et al. [47] used the compressor vane with five cooling channels and LG ¼ LC =1:0 to do experiments and got an empirical equation at the condition Ma1 = 0.75 with different attack angles as 1=3 NuIto-2012 ¼ 0:024Re0:8 air Pr air
ð26Þ
Nu number predicted by using the DG method are compared with that calculated by the empirical equations, and the results are shown in Table 7. It indicates that the result from Eq. (26) is in good agreement with the numerical one. The relative error is about 2.6% for Ma1 = 0.6 and 0.77% for Ma1 = 0.3. 5.3. Effect on aerodynamic performance The contours of Tu are shown in Fig. 14. For all the conditions, Tu increases at about x=LC ¼ 0:25 on the pressure side of the vane, which can be seen as the onset of transition. The transition position on the suction side is at about x=LC ¼ 0:60. As a result, the cooling channels mainly affect the turbulent boundary layer on the pres-
Fig. 16. Velocity profiles at Point A (Ma1 = 0.6).
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(a) 0.3 without cooling
(b) 0.3 cooling
Fig. 17. Temperature contours at the middle part of the pressure side (Ma1 = 0.3).
Fig. 19. Velocity profiles at Point B (Ma1 = 0.6). Fig. 18. Velocity profiles at Point A (Ma1 = 0.3).
sure side, and have an effect on both the laminar boundary layer and the turbulent boundary layer on the suction side. In the compressible boundary layer, dynamic viscosity and density will be changed. Because of the wall cooling on the compressor vane, the changing temperature leads to a changing viscosity and density, which may change the velocity profile and the temperature profile. In the turbulent boundary layer, decrease of temperature leads to a larger density near the wall, which makes the boundary layer thinner. In the laminar boundary layer, dynamic viscosity reflects the ability of momentum transport normal to the shear layer. The relationship between dynamic viscosity of air and temperature can be obtained by the Sutherland’s formula [52]:
l T ¼ T0 l0 where n ¼ 1:5.
n T0 þ Ts T þ Ts
ð27Þ
l0 ¼ 17:161 106 ðPa sÞ, T 0 ¼ 273:0 ðKÞ, T s ¼ 124 ðKÞ, Fig. 20. Velocity profiles at Point B (Ma1 = 0.3).
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Fig. 22. Velocity profiles at Point C (Ma1 = 0.3).
Fig. 21. Velocity profiles at Point C (Ma1 = 0.6).
(a) 0.6 without cooling
(b) 0.6 cooling Fig. 23. Pressure contours (Ma1 = 0.6).
(a) 0.3 without cooling
(b) 0.3 cooling Fig. 24. Pressure contours (Ma1 = 0.3).
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It indicates that dynamic viscosity decreases with cooling and increases with heating. As a result, the dynamic viscosity of the boundary layer on the intercooled compressor surface becomes smaller. In the two-dimensional boundary layer, the smaller dynamic viscosity leads to a smaller shear stress (s ¼ lð@u=@yÞ). Smaller shear stress results in a larger velocity near the wall. The displacement thickness d becomes smaller because:
d ¼
Z
h
0
qu dy: 1 qe ue
ð28Þ
where q becomes larger when cooling. It indicates that decrease of the dynamic viscosity leads to a thinner influence region. As a result of the thinner boundary layer, the compressor has a larger throughflow area to have a higher pressure ratio. At Point A in Fig. 14, the effect of cooling channels on the turbulent boundary layer has been analyzed. Because of the cooling channels, the temperature of the turbulent boundary layer decreases, as shown in Figs. 15 and 17. As a result, the boundary layer becomes thinner. In order to see the boundary layer clearly, Figs. 16 and 18 show the velocity profiles in the boundary layer at Point A. On the suction side, two points B and C in Fig. 14 are considered here, and the velocity profiles are shown in Figs. 19–22. Although cooling channels have effect on the boundary layer of suction side, the results show that the effect is not significant. The static pressure contour at Ma1 = 0.6 is shown in Fig. 23, and Fig. 24 is the static pressure contour at Ma1 = 0.3. Because of the effect of cooling channels, the boundary layer becomes thinner, especially on the pressure side. Thus, the cascade has a larger area ratio for the main stream. The compressor vane has a better ability for the pressure rising. The isoline of the pressure contour moves upstream in the rear part of the vane at both test conditions. In a real compressor, the temperature is actually higher than the temperature at these conditions, thus, the effect of cooling channels on the pressure rising may be more significant in the real compressor. 6. Conclusion In this paper, DG method is applied to discuss the conjugate heat transfer performance in a two-dimensional turbine vane and an intercooled compressor vane. Our in-house code shows a good numerical prediction based on the comparison between the numerical results and the experiment data. The conclusions can be obtained as follows: 1. The cooling channels on the compressor vane can reduce the temperature of the vane and protect the vane from the high temperature. 2. At a certain range, the empirical equation of Nu and Re proposed by Ito in 2012 is a suitable equation. It can be applied in the analysis of heat transfer problems. 1=3 Nuair ¼ 0:024Re0:8 air Pr air
3. The cooling channels have an effect on the laminar boundary layer and the turbulent boundary layer. They make the boundary layer thinner and increase the pressure ratio of the stator vane. Cooling in the compressor vanes may be used to improve the aerodynamic performance in a real compressor in the future.
Acknowledgments This work is supported by National Natural Science Foundation of China (Grant No. 51276093). Thanks to Yutaka Ito et al. in Tokyo Institute of Technology for providing the experiment data.
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