- Email: [email protected]
Influence of porous filling on internal flow and heat transport for the gap-cavity structure subjected to high speed airflow
International Journal of Heat and Mass Transfer 93 (2016) 969–979
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Influence of porous filling on internal flow and heat transport for the gap-cavity structure subjected to high speed airflow Chun Shen a,b,c,⇑, Xin-lin Xia c,⇑, Yong-zhen Wang b, Feng Yu b a
State Key Laboratory of Automotive Simulation and Control, Jilin University, Changchun, China Department of Thermal Energy Engineering, Jilin University, Changchun, China c School of Energy Science and Engineering, Harbin Institute of Technology, Harbin, China b
a r t i c l e
i n f o
Article history: Received 13 June 2015 Received in revised form 23 October 2015 Accepted 30 October 2015
Keywords: All-speed Preconditioned Conjugate heat transfer Transient Seal Porous
a b s t r a c t In this paper, using all-speed preconditioned conjugate heat transfer numerical method and wind tunnel experiment, it is investigated the mechanism of the transient process that high speed hot airflow invades into gap-cavity sealing link structure. On the condition of porous material sealing in top gap of the linking structure, via the experimental method, it is found that there is existing increment–decrement–increment phenomenon of variation of temperature magnitude inside the cavity. Meanwhile this variation phenomenon could be reproduced by the all-speed preconditioned conjugate heat transfer numerical method and it is demonstrated that, the effectiveness of the numerical method established for this invading problem in this paper. Then by the numerical method, the more details with respect to the invasion process are shown to analyze the mechanism corresponding to this increment–decrement–increment phenomenon of variation of temperature magnitude. Finally, the symmetry and stability characteristics of the flow status with and without porous sealing are compared and the differences of the invasion process with and without sealing are further summarized. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction Sealing is a significant key thermal protection for turbomachine [1,2], hypersonic flight vehicle, hypersonic ramjet engine [3–5]. In general, the porous material is located in the gap between the adjacent structures, in order to prevent the hot airflow from invading in. However, due to the penetration feature of porous material, it can’t prevent the heat and mass transfer entirely, especially for the condition that there is large pressure difference on both sides of porous material filling in the gap. The objective of this paper is to further investigate this penetration and heat transfer phenomenon for the gap-cavity structure which is equivalent to the real sealing link structure in hypersonic flight vehicle. In general, the external space outside the linking structure on hypersonic vehicle should be the hypersonic compressible flow field, and the internal space inside the linking structure is generally the analogous cavity structure where the internal flow field would be subsonic compressible [6]. If the sealing porous material is filled in the gap which connects the outside hypersonic field and the inner cavity, the velocity of invading hot air into the cavity would ⇑ Corresponding authors at: State Key Laboratory of Automotive Simulation and Control, Jilin University, Changchun, China. Tel.: +86 0431 85095196 (C. Shen). E-mail addresses: [email protected] (C. Shen), [email protected] (X.-l. Xia). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2015.10.066 0017-9310/Ó 2015 Elsevier Ltd. All rights reserved.
be tens or even several meters per second and so the inner flow field should be subsonic incompressible flow. Hence for the link structure with porous sealing in the hypersonic flow, the fields outside and inside the structure are hypersonic/supersonic and subsonic incompressible, respectively, so the entire flow field involved is all-speed flow field which partially consists of supersonic/hypersonic transonic and subsonic flows. Meanwhile, multiple physical processes including conjugate heat transfer among porous, fluid and solid domains, penetration inside the porous media are involved. For sealing structure with porous sealing, a lot of researches were implemented by the experimental approaches [7–10]. However, these researches primarily focused on the penetration through porous material and summarize the range of penetrating rate of mass flow under the conditions of different pressure drop across the porous sealing zone. After the hot air passes through the porous sealing zone, it will continue to invade into the inner cavity, and the thermal shock that is harmful to the inside parts is formed. Therefore, in order to grasp the mechanism of the thermal shock, this subsequent invasion of hot air into the deep gap and inner cavity of the sealing structure needs to be investigated further. So in this paper, the flow and heat transfer process of hot air invading into the inner cavity of the sealing structure is concentrated on.
970
C. Shen et al. / International Journal of Heat and Mass Transfer 93 (2016) 969–979
Nomenclature a c dp E Ec, Fc, Gc Ev, Fv, Gv F Go H Hf HL Hs J K n N p q Q t S T
speed of sound specific heat capacity (J/(kg K)) the particle diameter of porous phase (m) total energy (J/kg) convective flux vectors viscous flux vectors inertia coefficient; view factor external contribution to the irradiation external incident irradiation (W/m2) height of the upper flow field whole width of the solid structure whole height of the solid structure effective radiative power (W/m2) permeability vertical direction of the boundary the total number of the grid cells pressure (Pa) net radiative heat flux (W/m2) vector of primitive variables time (s) source term temperature (K)
In this paper, the experimental test and numerical simulation method are both adopted to study the invasion of external airflow into the gap-cavity sealing structure. For the fluid/porous coupling problem, density-based algorithm [11–13] or pressure–velocity correction algorithm [14–21] could be used to reproduce the conjugate heat transfer process. The density based method is more accurate for the supersonic/hypersonic flow, and the pressure–velocity method is much better for the transonic and subsonic flow. In this paper, the flow field outside the sealing structure is supersonic/hypersonic, so the density based algorithm is more proper for the flow field outside the sealing structure. Meanwhile, the inside of the sealing structure with porous sealing material in the gap is more inclined to the subsonic incompressible flow. Hence, the entire flow field, including external and internal regions outside and inside the sealing structure, especially for the structure with porous material sealing, possesses the all-speed flowing feature, and the density-based preconditioned all-speed algorithm is more appropriate for this all-speed flow [22–24]. In general, the actual linking structure could be simplified to assembly of gasp and cavities [6], which is called gap-cavity structure in this paper. For requirement of sealing, the porous material is filled in the gap. In this paper, this simplified gap-cavity structure with porous sealing in the gap is used as the substitution for actual sealing link structure, to investigate the process of hot airflow invasion. In this paper, the experimental test that high-speed hot airflow flows over the gap-cavity structure is implemented in the wind tunnel. The temperature data on the bottom surface and in the cavity is obtained by the experimental test. Meanwhile, based on the preconditioned all-speed algorithm, the conjugate porous/fluid/ solid domains method is established and is validated via the experimental data. All the numerical programs are developed in OpenFOAM [25], where the density based preconditioned allspeed algorithm for fluid domain has already been established [24] and it mainly includes preconditioning method with timederivative term and all-speed advection discrete scheme, AUSM +(P), AUSM+up, etc., Finally, associating the results produced by the numerical algorithm with the experimental data, it is investigated that the the flow and heat transfer features in the transient
U U ref U1
velocity vector (m/s) reference velocity fixed reference velocity
Greek symbols s pseudo time k conductivity (W/(m K)) e porosity of porous; emissivity r Stefan–Boltzmann constant (W/(m2 K4)) q density (kg/m3) / relative error C preconditioning matrix Subscripts dimless dimensionless i index number ini initial k surface element index f fluid phase r radiation s solid phase x,y,z coordinate axes x, y and z
process that external hot air invades into the inner cavity inside the sealing link structure.
2. Physical and mathematical model 2.1. Experimental and computational geometric models Fig. 1 shows the experimental gap-cavity structure in the wind tunnel. Fig. 2 illustrates the two-dimensional cross section diagram corresponding to actual experimental structure shown in Fig. 1, and it is the numerical computational geometric model in x–y plane. As shown in Figs. 1 and 2, the porous material fills in the top gap (gap 1 in Fig. 2) at the middle of the graphite plane. The high speed airflow aerodynamically heats the graphite plane and the high temperature gas invades into the cavity along gap 1 (full of porous material) and gap 2 of which the length is 2 mm. The permeability and the porosity of the sealing porous material used in the experimental test are 1 107 m2 and 0.9, respectively. The graphite plane is supported by the metal shell which is made of steel. There are two pieces of thermal insulation blocks. The cavity and gap 2 shown in Fig. 2 are cut at the top and middle part of these two blocks, respectively, and these two blocks are put into the metal shell. These two thermal insulation blocks could prevent the heat of the invading hot air from transferring to the internal solid structure through the gap’s and cavity’s walls and hence, it’s much easier to increase the temperature of the air in the cavity. The temperature data at dot 2 and dot 6 (shown in Fig. 2) is measured by the thermocouple wires of which the legs are welded together at one end. The whole structure is surrounded by the thermal insulating material and then put it in the high speed wind tunnel. The whole height and width of the structure Hs and HL is 200 mm and 250 mm, respectively. According to the wind tunnel conditions, the velocity, temperature and pressure of the inlet in the upper flow field (shown in Fig. 2) are approximately equal to 2350 m/s, 1770 K and 1600 Pa, respectively, and the initial pressure and back pressure are both 400 Pa.
C. Shen et al. / International Journal of Heat and Mass Transfer 93 (2016) 969–979
gap
971
sealing
thermocouple wire
cavity inner thermal metal shell insulation material
graphite plate without gap
a) The assembly diagram
b) The bottom of graphite plate with porous material sealing in gap
Fig. 1. Real picture of gap/cavity experimental structure.
porosity of the sealing porous material. Based on the assumptions above, in the pure fluid and porous region, the governing equations for transient dual-time preconditioning computational method could be consolidated as follows:
porous medium (gap 1)
y Upper flow Uin Tin Pin
graphite
Hf gap 2
gap 1 dot 6
gap 2
graphite plate
dot 1
cavity
dot 2
Hs dot 5
dot 4
@Q @Q @Ec @F c @Gc @Ev @F v @Gv þ þ þ þ þ þ þ ¼S @s @t @x @y @z @x @y @z
ð1Þ
where Q is the vector of primitive variables Q ¼ ½p; u; v ; w; TT , the expression of the preconditioning matrix C is shown in the Ref. [24], and s is the pseudo time for transient dual time scheme. In addition, Ec , F c and Gc are the convective flux vectors and Ev , F v and Gv are the viscous flux vectors, where the subscripts ‘‘c” and ‘‘v” denote the convective and viscous fluxes, respectively. The source term S can be written as follows:
x o
C
dot 3 heat insulating layer
S ¼ ð0; Sx ; Sy ; Sz ; 0ÞT Si ¼
1 2
HL Fig. 2. Schematic diagram of the computational model.
2.2. Governing equation 2.2.1. Fluid/porous domain The space of the porous domain occupies a small proportion of the total domain of the whole sealing structure, so the interfacial [26] effect between the porous and fluid domains are ignored. Hence, the single-domain approach that the hybrid porous/fluid region is considered to be a continuous region [11]. The governing equations are only supplemented with a source term that the extra drag in the porous region is taken into account. Besides, for porous zone, the local thermal equilibrium model (LTE) is adopted for the energy equation, and meanwhile the effect of the porous media on other terms in the governing equations is not considered due to the small volume proportion and the high
lDi þ qjui jF i ui
ð3Þ
where the subscript i dictates the coordinate axes x, y and z. K and F are the permeability and the inertia coefficients,
pffiffiffiffiffiffiffiffiffi F ¼ 1:75= 150e3=2
metal shell
ð2Þ
ð4Þ
where dp and e are the particle diameter and the porosity of the porous phase, respectively. The density-based preconditioned all-speed solver is implemented in OpenFOAM and the numerical method of this method is presented in the Ref. [24] Herein, according to the expression U 2ref ¼ minða2 ; maxðjUj2 ; KjU 1 j2 ÞÞ where K is constant here, if the fixed reference velocity is less than the sound speed (U ref 6 a), the preconditioning matrix C is valid for subsonic flow or even low-speed incompressible flow. U 1 is the control reference velocity and in this paper, it is approximately set equal to average velocity in the cavity (the bottom outlet of gap 2). 2.2.2. Solid domain For the solid domain, the Navier–Stokes equation in scalar form is
qs cs
@T s ¼ r ðks rT s Þ @t
ð5Þ
where qs , cs , ks and T s are the density, specific heat capacity, conductivity and temperature for the solid phase.
972
C. Shen et al. / International Journal of Heat and Mass Transfer 93 (2016) 969–979
2.2.3. Radiative heat transfer All the surfaces of the computational geometric zone constitute an enclosure box. The relationships among the external incident irradiation Hk , effective radiative power Jk and net radiative heat flux qk are shown as follows: Ms X J i F ki þ Ho;k
Hk ¼
ð6Þ
i¼1
J k ¼ ek rT 4k þ ð1 ek ÞHk
ð7Þ
qk ¼ J k H k
ð8Þ
where F denotes the view factor, r denotes the Stefan–Boltzmann constant, e denotes emissivity and the subscript ‘‘k” denotes discrete surface element k. 2.2.4. Coupling boundary Accord to the continuity of the heat flux at boundary interface between fluid and solid domains, and provided that if the solid surface heat the outside airflow, the value of outward radiation heat flux from the solid surface is negative. The following relationship is gotten,
kf
@T @T qr ¼ ks @n f @n s
ð9Þ
where the subscript ‘‘f” and ‘‘s” denote fluid and solid respectively, n denote the vertical direction of the boundary, and qr is the radiation heat flux corresponding to qk in Section 2.2.3. 2.3. Numerical methodology The numerical discrete schemes for the transient dual time allspeed preconditioned algorithm are introduced in Ref. [24], and the spatial accuracy is 2nd order. Other discrete methods corresponding to radiation and coupling boundary are shown in Ref. [6]. The numerical model related the topic in this paper is complex coupled model, including all-speed flow model (supersonic/hypersonic compressible flow for external space outside the sealing structure, subsonic compressible/incompressible flow for internal space inside the sealing structure), solid conduction model and surface radiation. Therefore in order to validate this coupled model, the following steps should be employed. Firstly, the allspeed algorithm which is the core of the coupled method should be validated. Secondly, the fluid and solid coupling method should be validated and naturally, it can also demonstrate the accuracy of the conduction sub-model. Finally, based on the accuracy of the sub-models, the comprehensive effect of the entire numerical method should be validated for the topic of invasion process relating to the gap-cavity structure in this paper, and in this part, the experimental data for the real sealing structure as shown in Figs.1 and 2 could be used. Wherein, the all-speed method was already validated in Ref. [24] and the analogous fluid and solid coupling method was already validated in Ref. [6]. In this paper, especially, the last part of validation mentioned above, i.e. the effect of the entire coupled numerical method to reproduce the temperature variation for the internal space inside the cavity, should be discussed in the following part. 3. Results and discussion 3.1. Accuracy analysis of the experimental temperature measurement In the experiment test, pairs of thermocouple wires of which the legs are welded together at one end are employed to measure
temperature, and there is no metal protective sleeve outside the thermocouple wires. The measured points are at dot 6 at the bottom of the graphite plane and dot 2 in the cavity as shown in Fig. 2, wherein the thermocouple wires which measure the airflow temperature at dot 2 is inserted into the cavity from the bottom of the metal shell, and it is fixed by the channel inside the plastic catheter which is inserted into the cavity. In theory, the range of measurement error of the thermocouple wires is ±0.5 °C. Meanwhile, in order to further verify the ability of the thermocouple wires to capture the transient temperature variation, in other analogous experimental condition (the graphite plane was aerodynamic heated by the hot airflow, and after airflow heating, it is naturally cooling.), non-contact infrared thermometers are employed to test the temperature at the point on the upper surface of the graphite plane, and the test point just locates above dot 6 at the bottom of the graphite plane. It is noted that, the thermocouple wires and the infrared thermometer test different points at the upper and bottom surfaces. However, the graphite plane has good thermal conductivity and is thin, so it is considered that the temperature of the entire graphite plane is consistent, and especially the trends of transient temperature variation corresponding to all the points at the entire plane should be consistent. Therefore, although the test point measured by the infrared thermometer is different from the point tested by the thermocouple wires, it could be still used to validate the ability of the thermocouple wires to capture the transient temperature variation. In summary, the values tested by the infrared thermometer are taken as the calibration to validate the effectiveness of the thermocouple wires to capture the transient temperature variation, and to some extent, the difference of the magnitudes corresponding to these two test points, tested by the infrared thermometer and thermocouple wires respectively, can reflect the measure accuracy of the thermocouple wires. Fig. 3 shows the transient variation of dimensionless temperature tested by thermocouple wires and infrared thermometer. The infrared thermometer has temperature response threshold, and that’s the reason why the curve predicted by infrared thermometer has two sudden change regions on both sides. The part of increment of temperature before the peak corresponds to the process that the graphite was aerodynamically heated and the part of decrement of temperature after the peak corresponds to the process that the plane was naturally cooling. As shown in Fig. 3, in the effective section, the range of the magnitude of relative errors / are from 10% to 5%. Actually, at the initial stage (the upper surface of the graphite is just heated and cooled), the transient effect is strong, so the difference of temperature between the upper surface and bottom surface of the graphite is remarkable. With the heating and cooling process proceeding, this difference is disappearing gradually. Especially at the end stage of heating and cooling process, the relative error is nearly close to zero. Overall, the measurement results of the infrared thermometer and thermocouple wires are consistent. Moreover, the variation trends of the test temperature produced by these two methods are the same with each other, it is confirmed that the thermocouple wires adopted in this paper can effectively capture the transient temperature variation. 3.2. Comparison of the experimental data and computational results Fig. 4 shows the temperature transient variation corresponding to different points (dots 1–6 shown in Fig. 1). Fig. 4(a) presents the temperature variation with time corresponding to dot 6 located at the bottom of the graphite plate. For the temperature magnitude at dot 6, there is nearly no difference between sealing and no-sealing conditions, so only the comparison of the temperature variations without sealing is shown in Fig. 4(a). During the 100 s transient process, the values tested by
973
C. Shen et al. / International Journal of Heat and Mass Transfer 93 (2016) 969–979
0.15 1.0 Tdimless
Relative error φ
0.6 φ
φ / Tdimless
0.10
Infrared Thermometers Thermocouple
0.8
0.05
0.4
0.00
0.2 Relative error φ 0.0
-0.05 0.0
0.2
0.4
0.6
0.8
1.0
0.2
tdimless
a) Temperature measurements and relative error
0.4
0.6 tdimless
0.8
b) Amplification of relative error
Fig. 3. Comparison of measurements of thermocouple wires and infrared thermometer.
550
T/K
500 450 400 computational results experimental data
350 300 0
20
40
60
80
100
t/s
a) dot 6 on graphite plane (no-sealing in gap 1) 420
297
dot 1 dot 2 dot 3
400
360
T/K
T/K
380
340
dot 4 dot 5 experimental data
288
320 279
300
dot 1 dot 2 dot 3
280 260
0
20
40
t/s
dot 4 dot 5 experimental data 60
80
100
b) points in the cavity(no-sealing in gap 1)
270
0
20
40
60
80
100
t/s
c) points in the cavity(sealing in gap 1)
Fig. 4. Temperature variation with time corresponding to different points.
the thermocouple at corresponding times are a little greater than those predicted by the numerical method. The difference of magnitude between these two lines shown in Fig. 4(a) increases before the half (0–50 s), and then decreases during the following process (50–100 s). However, the overall increasing trends of these two lines are in good agreement. It is indicated that the numerical method in this paper can predict the aerothermal heating process that the high speed and hot airflow flows over the sealing link structure in this paper, and in a sense, it is confirmed that the temperature value of the hot air just invading into the gap 1 which is
reproduced by the numerical method is close to the experimental data. Fig. 4(b) and (c) show the thermal variation process in the cavity with sealing and without sealing in the gap 1, respectively. In Fig. 4(b) the temperature lines corresponding to the dot 1 and dot 5 predicted by the numerical method are not smooth. However, on the whole, these two lines with the other three lines (dot 2, dot 3 and dot 4) are all ascending all the time and they are almost synchronously increasing with the line regarding experimental data. The difference between the magnitudes of the
974
C. Shen et al. / International Journal of Heat and Mass Transfer 93 (2016) 969–979
temperature predicted by the numerical method (especially the values corresponding to dot 2, dot 3 and dot 4) and the experiment data at different times are always nearly 50 K. Different from the consistent increasing trend in Fig. 4(b), the changing trend of each line in Fig. 4(c) all increases sharply at the beginning, then falls down and lastly increases again in the end. The changing tendency including the values predicted by the numerical method (especially the values corresponding to dot 4 and dot 5) and the experimental data are still the same and the difference of magnitude between them at different times are always nearly 6 K. By comparison of the numerical results and the experimental data, there are existing stable difference of temperature value between them, regardless of sealing or no-sealing. Besides the factors of numerical computational method, it may arise from the following several difference between the real experimental rigs and the numerical computational model: Firstly, the difference between the experimental structure and the simplified computational structure. as shown in Fig. 2, in the direction perpendicular to the paper plane (the xy plane), or in the gap length direction (z axis direction), the actual length of the gap and the cavity is nearly half length of the graphite plane and the thermal insulation material shown in Fig. 1. However, the computational geometric model shown in Fig. 2, is equivalent to the geometric model that has the length equal to the whole length of the sealing structure in the z axis direction. It means that, under the same conditions, the magnitude of the invading mass flow corresponding to the three dimensional structure in the experiment is half as much as the computational value corresponding to the equivalent two dimensional model shown in Fig. 2. Naturally, the actual temperature magnitude of the invading air in the cavity is lower than the computational results at the corresponding moments. Secondly, the unknowing leaking locations in the real experimental rigs. There may be existing gas leakage at some unknown locations, such as the holes which the thermocouples get through, the connection edge between different structures, etc. Thirdly, the difference between real porosity and permeability and the corresponding magnitude in numerical computation. The porosity and the permeability employed in the computing process may be not accurate enough, because the porous sealing strip was squeezed in the actual installation process and it may lead to the changes in the physical properties of the porous sealing strip. Although there is existing stable difference of magnitude between the numerical results and the experimental data, according to consistency of the changing tendency, it is still demonstrated that the numerical method can capture the flow and thermal characteristics regarding the sealing link structure in this paper. Besides the validation of the numerical method, by comparison of the conditions of sealing and no-sealing in gap 1, as shown in Fig. 4(b) and (c), it is also found that the temperature of hot air invading into the cavity decreases substantially, no matter from the perspective of numerical method or experiment. From viewpoint of numerical method, in the whole 100 s transient process, the decrement extent of the temperature due to sealing in gap 1 is close to 100 K. Meanwhile, the increment–decrement–increment tendency of temperature magnitude in Fig. 3(c) is obviously different from the consistent increment tendency in Fig. 3(b). According to this increment-decrement-increment variation tendency, for the sealing condition, the whole invading process could be divided into three parts. The first part is that, in the beginning process (approximate 0–1 s corresponding to the experiment data), the hot air just recently invades into the cavity, and doesn’t have enough time to transfer its thermal energy to solid structure through the cold wall around the cavity. The second part is that, with the invading process proceeding, the process that the cold wall obtains the heat
from the hot air is faster than the heat supplement process with the newly hot air invading into the cavity. The last part is that, the heat supplement rate in the cavity with the hot air invading is more rapid than the heat loss process in the cavity that the heat transfer from hot invading air to the cold wall around the cavity. However, for the no-sealing condition, in the cavity, the heat supplement rate with the hot air newly invading is always quicker than the heat loss rate with cold wall obtaining heat from hot invading air. The phenomenon of this increment-decrementincrement tendency of temperature magnitude in the cavity is further presented that the effect of the sealing in gap1 on the process that the hot air invades into the cavity. 3.3. Qualitative and quantitative analysis of invading process via computational results In order to further comprehend the invasion process of the hot air into the inner cavity inside the gap-cavity sealing structure, the variable distribution along entrance of gap 1 for sealing and nosealing is shown in Fig. 5. For the y-axis velocity uy, there is approximately existing a change in one order of magnitude. For sealing in gap 1, the mean velocity is nearly 50 m/s, but for no-sealing in gap 1, the velocity at peak reaches up to 500 m/s. Moreover, regardless of sealing or no-sealing, the velocity is decreasing with transient process proceeding, but the decreasing proportion is not great, for the 100 s transient process, approximately about 15– 20%. For the temperature and pressure distributions, regardless of sealing or no-sealing, there are both dramatic rises in the region close to the wall of gap 1, and that’s because the invading hot air directly attack the right wall of gap 1 without any barrier corresponding to the no-sealing condition and the hot air directly heats the right wall intensively. For sealing and no-sealing conditions, the mean temperatures at the entrance are close. However, the difference of the mean pressures are significant, and the magnitude of pressure with sealing is approximately twice of the magnitude of pressure without sealing. As shown in Fig. 5d), the shape of the density curves with sealing and no-sealing are nearly the same, and the magnitude of the mean density with sealing is nearly twice of that without sealing. Overall, except the density distribution, along the entrance of gap 1, the shape of curves with sealing is gentle, while the shape of curves without sealing has dramatic variation. The difference of magnitude of velocity between sealing and no-sealing is the most significant, and there is a difference of one order of magnitude. According to the analysis above, for the sealing condition, compared with the no-sealing condition, the mass flow rate through the gap 1 decreases substantially, i.e. it is merely one thirds of the magnitude of the mass flow rate without sealing, as shown in Fig. 5(e), and the average temperature through the top inlet of gap 1 does not have obvious and substantial variation. According to the numerical results, for the average velocity in gap 1, there is nearly existing difference of one order of magnitude between conditions of sealing and no-sealing. Fig. 6 shows the temperature contours at different moments for the cross section of the whole fluid and solid region, as shown in Fig. 2. Obviously, for the condition of sealing in gap 1, at the same moment, the temperature magnitude in the gaps and the cavity is much lower than the temperature value corresponding to the condition of no-sealing. For the condition of no-sealing in gap 1, with the proceeding of the invasion of hot air, there is significantly circular high temperature area in the middle and lower part of the cavity, and the scale of the circular area is almost of the same order of the length of the cavity. Obviously, according to the shape of the high temperature contour, it is speculated that the invading hot air stream circularly flows clockwise.
975
C. Shen et al. / International Journal of Heat and Mass Transfer 93 (2016) 969–979
1400
0
-200 -300
1000 T/K
uy / m·s-1
1200
sealing t/s no-sealing 0.1 1 10 50 100
-100
sealing t/s no-sealing 0.1 1 10 50 100
800 600
-400
400
-500 -600 0.1240
0.1245
0.1250 x/m
0.1255
200 0.1240
0.1260
0.1245
a) y-axis velocity uy
2000 1750
p / Pa
1500
0.1255
0.1260
b) temperature 0.024
sealing t/s no-sealing 0.1 1 10 50 100
0.020
sealing t/s no-sealing 0.1 1 10 50 100
0.016 ρ / kg·m-3
2250
0.1250 x/m
1250 1000
0.012 0.008
750
0.004
500 250 0.1240
0.1245
0.1250 x/m
0.1255
0.1260
0.000 0.1240
0.1245
c) pressure
0.1260
sealing
no-sealing 0.1 1 10 50 100
0.2 m / kg· m - 2· s - 1
0.1255
d) density
0.4
.
0.1250 x/m
0.0 -0.2 -0.4 -0.6 -0.8 -1.0 0.1240
0.1245
0.1250
0.1255
0.1260
x/m
e) Mass flow rate Fig. 5. Comparison of variable distribution along entrance of gap 1 for sealing and no-sealing.
For the condition of sealing in gap 1, at the moment t = 0.1 s and t = 10 s, the temperature contours are asymmetric and disordered. However, with the proceeding of the invasion of hot air, the temperature contours tend to be symmetric along the center line of the sealing structure and to be ordered, as shown in Fig. 6(b), at the moment t = 50 s and t = 100 s. Meanwhile, for the condition of sealing in gap 1, the color of the contour areas in the lower left and right corners at the moment t = 0.1 s are a little brighter than the color at corresponding positions at the moment t = 10 s, and at the moment t = 50 s and t = 100 s, the temperature values at corresponding positions are
gradually getting more and more higher than the former moments. It is indicated that the temperature at the corresponding positions at initial moment, such as t = 0.1 s, reaches a high magnitude, and in the following period the temperature decreases firstly and then increases until the end. This conclusion agrees with the incrementdecrement-increment tendency of the cavity temperature variation obtained in Fig. 4(c) above. In order to further indicate the flow state inside the cavity, Fig. 7 shows the vorticity contour with sealing and without sealing corresponding to the moments in Fig. 6. In Fig. 7, for sealing or nosealing conditions, the vorticity contours at the four moments
976
C. Shen et al. / International Journal of Heat and Mass Transfer 93 (2016) 969–979
T/K
0.1s
0.1s
10s
10s
50s
50s
100s
100s
(a) No-sealing
(b) Sealing
Fig. 6. Temperature contour of entire field at different times.
are not continuous, i.e. the time intervals between them are large. However, it still could be illustrated that, for no-sealing condition, with the proceeding of the hot air invasion, a big clockwise vortex that nearly occupies the entire middle and bottom space of the cavity is formed and other smaller vortexes generated out of gap 2 are developed and crushed around the principal central clockwise vortex.
Corresponding to the sealing condition, at the initial stage (at the moment t = 0.1 s and t = 10 s in Fig. 7(a) and (b)), the vorticity contours are not stable and it is illustrated that there are existing small vertexes along the central line of the structure. These vertexes are generated after the hot air just enters into the cavity from the gap 2. These vortexes are developed along the central line of the sealing structure and finally vanished as they gradually
C. Shen et al. / International Journal of Heat and Mass Transfer 93 (2016) 969–979
977
ω / s-1
0.1s
0.1s
10s
10s
50s
50s
100s
100s
(a) No-sealing
(b) Sealing
Fig. 7. Vorticity contour inside the structure at different times.
approach the bottom wall of the cavity. In the later transient process (at the moment t = 50 s and t = 100 s in Fig. 7(b)), the flow state of the invading hot air is gradually getting more and more stable and smoother, and the vorticity contours are inclined to be symmetric along the central line of the structure. In all, the features of the vorticity contour shown in Fig. 7 are further clarified the flow mechanism inside the cavity which leads to the thermal distribution phenomena presented in Figs. 4 and 6.
In Figs. 6 and 7, via qualitative comparison of contours, the features of invasion of hot air under the conditions of sealing and nosealing in gap 1 are presented, and subsequently, quantitative comparison of magnitude of the variable in the cavity is further implemented to discuss the mechanism of invasion process. The pressure difference inside and outside the sealing link structure is the primarily driven power for the invasion of the airflow, so the pressure inside the cavity is concentrated on. In Fig. 8, the
C. Shen et al. / International Journal of Heat and Mass Transfer 93 (2016) 969–979
580
440
560
435
540
430
520
p / Pa
p / Pa
978
425
500
dot 1 dot 2 dot 3 dot 4 dot 5
420
480
dot 1 dot 2 dot 3 dot 4 dot 5
460 440 420 400
415 410 405 400
0
20
40
60
80
100
0
t/s a) No-sealing
20
40
t/s
60
80
100
b) Sealing
Fig. 8. Variation of pressure in flow field inside structure with time.
transient variations of pressure inside the cavity are shown. From the viewpoint of numerical results, at the initial stage (in this paper, the first moment t = 0.1 s), the magnitude of pressure inside the cavity sharply increases, regardless of sealing or no-sealing. However, the increasing extent are different. On the condition of no-sealing, the increasing extent of the five observed points are almost more than 50 Pa, and the increasing values of the five points with respect to sealing condition are only more than 15 Pa. Obviously, the difference of the ascending pressure value between sealing and no-sealing conditions at the initial stage are caused by the block effect of the sealing material in gap 1 on the hot air invasion. Except the sharp increment at the initial stage, in the next following 100 s transient process, the magnitudes of the pressure inside the cavity always ascend stably, regardless of sealing or no-sealing. At the stable temperature ascending stage, The slope of the pressure lines corresponding to the no-sealing condition is nearly 1, and the slope corresponding to the sealing condition is only nearly 0.2. In Summary, the increasing rate of magnitude of pressure inside the cavity reflects the fact that the sealing material in gap 1 considerably delays the process of the invasion of hot air into the cavity. 4. Conclusion In this paper, by all-speed preconditioned conjugate heat transfer numerical method and wind tunnel experiment, it is investigated the mechanism of invasion of high speed hot airflow into gap-cavity linking seal structure. On the condition of sealing in top gap, by the experimental method, the increment-decrementincrement phenomenon of variation of temperature magnitude inside the cavity is presented, and meanwhile this variation phenomenon is reproduced by the all-speed preconditioned conjugate heat transfer numerical method. Then by the numerical method, the more details with respect to the invasion process are shown to analyze the mechanism corresponding to this increment– decrement–increment phenomenon of variation of temperature magnitude. After validation of numerical method, the difference of invasion process between sealing and no-sealing conditions are further summarized. For the no-sealing condition, there is existing large principal vortex that nearly occupies the entire middle and bottom space of the cavity. For the sealing condition, in the initial stage of the entire transient process, the flow of invading hot air isn’t stable and there are existing small vortexes developing and vanishing along the central line of the gap-cavity structure. However, with the transient process proceeding, the flows are getting more and more stable, the small vertexes along the central line
are vanished, and the flow fields are nearly symmetric along the central line. Finally, by comparison of variation of pressure magnitude inside the cavity with and without sealing, it is found that the decrement of the increasing rate of pressure magnitude inside the cavity can reflect the effect of the sealing material in the top gap on delaying the process of the invasion of hot air into the cavity. Acknowledgement This study was supported by the National Natural Science Foundation of China (Grant No. 51176038 and Grant No. 51309115). References [1] R.E. Chupp, R.C. Hendricks, S.B. Lattime, et al., Sealing in turbomachinery, J. Propul. Power 22 (2) (2006) 313–347. [2] M.F. Aksit, R.E. Chupp, O.S. Dinc, et al., Advanced seals for industrial turbine applications: design approach and static seal development, J. Propul. Power 18 (6) (2002) 1254–1259. [3] J.R. Finkbeiner, P.H. Dunlap, B.M. Steinetz, et al., Review of seal designs on the apollo spacecraft, J. Propul. Power 45 (5) (2008) 900–910. [4] J.J. Demange, P.H. Dunlap, B.M. Steinetz, Improved seals for high temperature airframe applications, AIAA (2006) 2006–4935. [5] J.J. DeMange, P.H. Dunlap, B.M. Steinetz, et al., An evaluation of high temperature airframe seals for advanced hypersonic vehicles, AIAA (2007) 2007–5743. [6] C. Shen, F.X. Sun, X.L. Xia, Analysis on transient conjugate heat transfer in gapcavity-gap structure heated by high speed airflow, Int. J. Heat Mass Transfer 67 (2013) 1030–1038. [7] P.H. Dunlap, B.M. Steinetz, D.M. Curry, Further investigations of control surface seals for the X-38 re-entry vehicle, AIAA (2001) 2001–3628. [8] P.H. Dunlap, J.J. DeMange, B.M. Steinetz, Performance evaluations of ceramic wafer seals, AIAA (2006) 2006–4934. [9] J.R. Finkbeiner, P.H. Dunlap, B.M. Steinetz, et al., On the development of a unique arc jet test apparatus for control surface seal evaluations, AIAA (2004) 2004–3891. [10] B.M. Steinetz, M.L. Adams, P.A. Bartolotta, et al., High-temperature braided rope seals for static sealing applications, J. Propul. Power 13 (1997) 675–682. [11] H.J. Zhang, Z.P. Zou, Y. Li, et al., Preconditioned density-based algorithm for conjugate porous/fluid/solid domains, Numer. Heat Transfer Part A 60 (2011) 129–153. [12] H.J. Zhang, Z.P. Zou, Numerical investigation of laminar-plate transpiration cooling by the preconditioned density-based algorithm, Numer. Heat Transfer Part A 62 (2012) 761–779. [13] I.A. Bedarev, S.G. Mironov, K.M. Serdyuk, et al., Physical and mathematical modeling of a supersonic flow around a cylinder with a porous insert, J. Appl. Mech. Tech. Phys. 52 (2011) 9–17. [14] F.Q. Wang, Y. Shuai, H.P. Tan, C.L. Yu, Thermal performance analysis of porous media receiver with concentrated solar irradiation, Int. J. Heat Mass Transfer 62 (2013) 247–254. [15] F.Q. Wang, Y. Shuai, Z.Q. Wang, Y. Leng, H.P. Tan, Thermal and chemical reaction performance analyses of steam methane reforming in porous media solar thermochemical reactor, Int. J. Hydrogen Energy 39 (2) (2014) 718–730. [16] Y.T. Yang, M.L. Hwang, Numerical simulation of turbulent fluid flow and heat transfer characteristics in heat exchangers fitted with porous media, Int. J. Heat Mass Transfer 52 (13–14) (2009) 2956–2965.
C. Shen et al. / International Journal of Heat and Mass Transfer 93 (2016) 969–979 [17] Z.D. Cheng, Y.L. He, F.Q. Cui, Numerical investigations on coupled heat transfer and synthetical performance of a pressurized volumetric receiver with MCRTFVM method, Appl. Therm. Eng. 50 (1) (2013) 1044–1054. [18] W.W. Yang, Y.L. He, Y.S. Li, Modeling of dynamic operating behaviors in a liquid-feed direct methanol fuel cell, Int. J. Hydrogen Energy 37 (2012) 18412– 18424. [19] Y.Q. Liu, P.X. Jiang, S.S. Jin, et al., Transpiration cooling of a nose cone by various foreign gases, Int. J. Heat Mass Transfer 53 (2010) 5364– 5372. [20] S.S. Feng, T. Kim, T.J. Lu, Numerical investigation of forced convection in pin/plate-fin heat sinks heated by impinging jet using porous medium approach, Int. J. Numer. Methods Heat Fluid Flow 23 (1) (2013) 88–107. [21] S.S. Feng, J.J. Kuang, T. Wen, T.J. Lu, K. Ichimiya, An experimental and numerical study of finned metal foam heat sinks under impinging air jet cooling, Int. J. Heat Mass Transfer 77 (2014) 1063–1074.
979
[22] J. Blazek, Computational Fluid Dynamics: Principles and Applications, second ed., Elsevier, 2005. [23] W.S. Fu, S.H. Huang, W.H. Wang, An investigation of effects of heights and a length on natural convection of three dimensional parallel plates with a heated bottom surface by a CUDA computation platform, Int. J. Heat Mass Transfer 67 (2013) 118–130. [24] C. Shen, F.X. Sun, X.L. Xia, Implementation of density-based solver for all speeds in the framework of OpenFOAM, Comput. Phys. Commun. 185 (2014) 2730–2741. [25] H.G. Weller, G. Tabor, H. Jasak, et al., A tensorial approach to computational continuum mechanics using object-oriented techniques, Comput. Phys. 12 (1998) 620–631. [26] B. Alazmi, K. Vafai, Analysis of fluid flow and heat transfer interfacial conditions between a porous medium and a fluid layer, Int. J. Heat Mass Transfer 44 (9) (2001) 1735–1749.
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Influence of porous filling on internal flow and heat transport for the gap-cavity structure subjected to high speed airflow Chun Shen a,b,c,⇑, Xin-lin Xia c,⇑, Yong-zhen Wang b, Feng Yu b a
State Key Laboratory of Automotive Simulation and Control, Jilin University, Changchun, China Department of Thermal Energy Engineering, Jilin University, Changchun, China c School of Energy Science and Engineering, Harbin Institute of Technology, Harbin, China b
a r t i c l e
i n f o
Article history: Received 13 June 2015 Received in revised form 23 October 2015 Accepted 30 October 2015
Keywords: All-speed Preconditioned Conjugate heat transfer Transient Seal Porous
a b s t r a c t In this paper, using all-speed preconditioned conjugate heat transfer numerical method and wind tunnel experiment, it is investigated the mechanism of the transient process that high speed hot airflow invades into gap-cavity sealing link structure. On the condition of porous material sealing in top gap of the linking structure, via the experimental method, it is found that there is existing increment–decrement–increment phenomenon of variation of temperature magnitude inside the cavity. Meanwhile this variation phenomenon could be reproduced by the all-speed preconditioned conjugate heat transfer numerical method and it is demonstrated that, the effectiveness of the numerical method established for this invading problem in this paper. Then by the numerical method, the more details with respect to the invasion process are shown to analyze the mechanism corresponding to this increment–decrement–increment phenomenon of variation of temperature magnitude. Finally, the symmetry and stability characteristics of the flow status with and without porous sealing are compared and the differences of the invasion process with and without sealing are further summarized. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction Sealing is a significant key thermal protection for turbomachine [1,2], hypersonic flight vehicle, hypersonic ramjet engine [3–5]. In general, the porous material is located in the gap between the adjacent structures, in order to prevent the hot airflow from invading in. However, due to the penetration feature of porous material, it can’t prevent the heat and mass transfer entirely, especially for the condition that there is large pressure difference on both sides of porous material filling in the gap. The objective of this paper is to further investigate this penetration and heat transfer phenomenon for the gap-cavity structure which is equivalent to the real sealing link structure in hypersonic flight vehicle. In general, the external space outside the linking structure on hypersonic vehicle should be the hypersonic compressible flow field, and the internal space inside the linking structure is generally the analogous cavity structure where the internal flow field would be subsonic compressible [6]. If the sealing porous material is filled in the gap which connects the outside hypersonic field and the inner cavity, the velocity of invading hot air into the cavity would ⇑ Corresponding authors at: State Key Laboratory of Automotive Simulation and Control, Jilin University, Changchun, China. Tel.: +86 0431 85095196 (C. Shen). E-mail addresses: [email protected] (C. Shen), [email protected] (X.-l. Xia). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2015.10.066 0017-9310/Ó 2015 Elsevier Ltd. All rights reserved.
be tens or even several meters per second and so the inner flow field should be subsonic incompressible flow. Hence for the link structure with porous sealing in the hypersonic flow, the fields outside and inside the structure are hypersonic/supersonic and subsonic incompressible, respectively, so the entire flow field involved is all-speed flow field which partially consists of supersonic/hypersonic transonic and subsonic flows. Meanwhile, multiple physical processes including conjugate heat transfer among porous, fluid and solid domains, penetration inside the porous media are involved. For sealing structure with porous sealing, a lot of researches were implemented by the experimental approaches [7–10]. However, these researches primarily focused on the penetration through porous material and summarize the range of penetrating rate of mass flow under the conditions of different pressure drop across the porous sealing zone. After the hot air passes through the porous sealing zone, it will continue to invade into the inner cavity, and the thermal shock that is harmful to the inside parts is formed. Therefore, in order to grasp the mechanism of the thermal shock, this subsequent invasion of hot air into the deep gap and inner cavity of the sealing structure needs to be investigated further. So in this paper, the flow and heat transfer process of hot air invading into the inner cavity of the sealing structure is concentrated on.
970
C. Shen et al. / International Journal of Heat and Mass Transfer 93 (2016) 969–979
Nomenclature a c dp E Ec, Fc, Gc Ev, Fv, Gv F Go H Hf HL Hs J K n N p q Q t S T
speed of sound specific heat capacity (J/(kg K)) the particle diameter of porous phase (m) total energy (J/kg) convective flux vectors viscous flux vectors inertia coefficient; view factor external contribution to the irradiation external incident irradiation (W/m2) height of the upper flow field whole width of the solid structure whole height of the solid structure effective radiative power (W/m2) permeability vertical direction of the boundary the total number of the grid cells pressure (Pa) net radiative heat flux (W/m2) vector of primitive variables time (s) source term temperature (K)
In this paper, the experimental test and numerical simulation method are both adopted to study the invasion of external airflow into the gap-cavity sealing structure. For the fluid/porous coupling problem, density-based algorithm [11–13] or pressure–velocity correction algorithm [14–21] could be used to reproduce the conjugate heat transfer process. The density based method is more accurate for the supersonic/hypersonic flow, and the pressure–velocity method is much better for the transonic and subsonic flow. In this paper, the flow field outside the sealing structure is supersonic/hypersonic, so the density based algorithm is more proper for the flow field outside the sealing structure. Meanwhile, the inside of the sealing structure with porous sealing material in the gap is more inclined to the subsonic incompressible flow. Hence, the entire flow field, including external and internal regions outside and inside the sealing structure, especially for the structure with porous material sealing, possesses the all-speed flowing feature, and the density-based preconditioned all-speed algorithm is more appropriate for this all-speed flow [22–24]. In general, the actual linking structure could be simplified to assembly of gasp and cavities [6], which is called gap-cavity structure in this paper. For requirement of sealing, the porous material is filled in the gap. In this paper, this simplified gap-cavity structure with porous sealing in the gap is used as the substitution for actual sealing link structure, to investigate the process of hot airflow invasion. In this paper, the experimental test that high-speed hot airflow flows over the gap-cavity structure is implemented in the wind tunnel. The temperature data on the bottom surface and in the cavity is obtained by the experimental test. Meanwhile, based on the preconditioned all-speed algorithm, the conjugate porous/fluid/ solid domains method is established and is validated via the experimental data. All the numerical programs are developed in OpenFOAM [25], where the density based preconditioned allspeed algorithm for fluid domain has already been established [24] and it mainly includes preconditioning method with timederivative term and all-speed advection discrete scheme, AUSM +(P), AUSM+up, etc., Finally, associating the results produced by the numerical algorithm with the experimental data, it is investigated that the the flow and heat transfer features in the transient
U U ref U1
velocity vector (m/s) reference velocity fixed reference velocity
Greek symbols s pseudo time k conductivity (W/(m K)) e porosity of porous; emissivity r Stefan–Boltzmann constant (W/(m2 K4)) q density (kg/m3) / relative error C preconditioning matrix Subscripts dimless dimensionless i index number ini initial k surface element index f fluid phase r radiation s solid phase x,y,z coordinate axes x, y and z
process that external hot air invades into the inner cavity inside the sealing link structure.
2. Physical and mathematical model 2.1. Experimental and computational geometric models Fig. 1 shows the experimental gap-cavity structure in the wind tunnel. Fig. 2 illustrates the two-dimensional cross section diagram corresponding to actual experimental structure shown in Fig. 1, and it is the numerical computational geometric model in x–y plane. As shown in Figs. 1 and 2, the porous material fills in the top gap (gap 1 in Fig. 2) at the middle of the graphite plane. The high speed airflow aerodynamically heats the graphite plane and the high temperature gas invades into the cavity along gap 1 (full of porous material) and gap 2 of which the length is 2 mm. The permeability and the porosity of the sealing porous material used in the experimental test are 1 107 m2 and 0.9, respectively. The graphite plane is supported by the metal shell which is made of steel. There are two pieces of thermal insulation blocks. The cavity and gap 2 shown in Fig. 2 are cut at the top and middle part of these two blocks, respectively, and these two blocks are put into the metal shell. These two thermal insulation blocks could prevent the heat of the invading hot air from transferring to the internal solid structure through the gap’s and cavity’s walls and hence, it’s much easier to increase the temperature of the air in the cavity. The temperature data at dot 2 and dot 6 (shown in Fig. 2) is measured by the thermocouple wires of which the legs are welded together at one end. The whole structure is surrounded by the thermal insulating material and then put it in the high speed wind tunnel. The whole height and width of the structure Hs and HL is 200 mm and 250 mm, respectively. According to the wind tunnel conditions, the velocity, temperature and pressure of the inlet in the upper flow field (shown in Fig. 2) are approximately equal to 2350 m/s, 1770 K and 1600 Pa, respectively, and the initial pressure and back pressure are both 400 Pa.
C. Shen et al. / International Journal of Heat and Mass Transfer 93 (2016) 969–979
gap
971
sealing
thermocouple wire
cavity inner thermal metal shell insulation material
graphite plate without gap
a) The assembly diagram
b) The bottom of graphite plate with porous material sealing in gap
Fig. 1. Real picture of gap/cavity experimental structure.
porosity of the sealing porous material. Based on the assumptions above, in the pure fluid and porous region, the governing equations for transient dual-time preconditioning computational method could be consolidated as follows:
porous medium (gap 1)
y Upper flow Uin Tin Pin
graphite
Hf gap 2
gap 1 dot 6
gap 2
graphite plate
dot 1
cavity
dot 2
Hs dot 5
dot 4
@Q @Q @Ec @F c @Gc @Ev @F v @Gv þ þ þ þ þ þ þ ¼S @s @t @x @y @z @x @y @z
ð1Þ
where Q is the vector of primitive variables Q ¼ ½p; u; v ; w; TT , the expression of the preconditioning matrix C is shown in the Ref. [24], and s is the pseudo time for transient dual time scheme. In addition, Ec , F c and Gc are the convective flux vectors and Ev , F v and Gv are the viscous flux vectors, where the subscripts ‘‘c” and ‘‘v” denote the convective and viscous fluxes, respectively. The source term S can be written as follows:
x o
C
dot 3 heat insulating layer
S ¼ ð0; Sx ; Sy ; Sz ; 0ÞT Si ¼
1 2
HL Fig. 2. Schematic diagram of the computational model.
2.2. Governing equation 2.2.1. Fluid/porous domain The space of the porous domain occupies a small proportion of the total domain of the whole sealing structure, so the interfacial [26] effect between the porous and fluid domains are ignored. Hence, the single-domain approach that the hybrid porous/fluid region is considered to be a continuous region [11]. The governing equations are only supplemented with a source term that the extra drag in the porous region is taken into account. Besides, for porous zone, the local thermal equilibrium model (LTE) is adopted for the energy equation, and meanwhile the effect of the porous media on other terms in the governing equations is not considered due to the small volume proportion and the high
lDi þ qjui jF i ui
ð3Þ
where the subscript i dictates the coordinate axes x, y and z. K and F are the permeability and the inertia coefficients,
pffiffiffiffiffiffiffiffiffi F ¼ 1:75= 150e3=2
metal shell
ð2Þ
ð4Þ
where dp and e are the particle diameter and the porosity of the porous phase, respectively. The density-based preconditioned all-speed solver is implemented in OpenFOAM and the numerical method of this method is presented in the Ref. [24] Herein, according to the expression U 2ref ¼ minða2 ; maxðjUj2 ; KjU 1 j2 ÞÞ where K is constant here, if the fixed reference velocity is less than the sound speed (U ref 6 a), the preconditioning matrix C is valid for subsonic flow or even low-speed incompressible flow. U 1 is the control reference velocity and in this paper, it is approximately set equal to average velocity in the cavity (the bottom outlet of gap 2). 2.2.2. Solid domain For the solid domain, the Navier–Stokes equation in scalar form is
qs cs
@T s ¼ r ðks rT s Þ @t
ð5Þ
where qs , cs , ks and T s are the density, specific heat capacity, conductivity and temperature for the solid phase.
972
C. Shen et al. / International Journal of Heat and Mass Transfer 93 (2016) 969–979
2.2.3. Radiative heat transfer All the surfaces of the computational geometric zone constitute an enclosure box. The relationships among the external incident irradiation Hk , effective radiative power Jk and net radiative heat flux qk are shown as follows: Ms X J i F ki þ Ho;k
Hk ¼
ð6Þ
i¼1
J k ¼ ek rT 4k þ ð1 ek ÞHk
ð7Þ
qk ¼ J k H k
ð8Þ
where F denotes the view factor, r denotes the Stefan–Boltzmann constant, e denotes emissivity and the subscript ‘‘k” denotes discrete surface element k. 2.2.4. Coupling boundary Accord to the continuity of the heat flux at boundary interface between fluid and solid domains, and provided that if the solid surface heat the outside airflow, the value of outward radiation heat flux from the solid surface is negative. The following relationship is gotten,
kf
@T @T qr ¼ ks @n f @n s
ð9Þ
where the subscript ‘‘f” and ‘‘s” denote fluid and solid respectively, n denote the vertical direction of the boundary, and qr is the radiation heat flux corresponding to qk in Section 2.2.3. 2.3. Numerical methodology The numerical discrete schemes for the transient dual time allspeed preconditioned algorithm are introduced in Ref. [24], and the spatial accuracy is 2nd order. Other discrete methods corresponding to radiation and coupling boundary are shown in Ref. [6]. The numerical model related the topic in this paper is complex coupled model, including all-speed flow model (supersonic/hypersonic compressible flow for external space outside the sealing structure, subsonic compressible/incompressible flow for internal space inside the sealing structure), solid conduction model and surface radiation. Therefore in order to validate this coupled model, the following steps should be employed. Firstly, the allspeed algorithm which is the core of the coupled method should be validated. Secondly, the fluid and solid coupling method should be validated and naturally, it can also demonstrate the accuracy of the conduction sub-model. Finally, based on the accuracy of the sub-models, the comprehensive effect of the entire numerical method should be validated for the topic of invasion process relating to the gap-cavity structure in this paper, and in this part, the experimental data for the real sealing structure as shown in Figs.1 and 2 could be used. Wherein, the all-speed method was already validated in Ref. [24] and the analogous fluid and solid coupling method was already validated in Ref. [6]. In this paper, especially, the last part of validation mentioned above, i.e. the effect of the entire coupled numerical method to reproduce the temperature variation for the internal space inside the cavity, should be discussed in the following part. 3. Results and discussion 3.1. Accuracy analysis of the experimental temperature measurement In the experiment test, pairs of thermocouple wires of which the legs are welded together at one end are employed to measure
temperature, and there is no metal protective sleeve outside the thermocouple wires. The measured points are at dot 6 at the bottom of the graphite plane and dot 2 in the cavity as shown in Fig. 2, wherein the thermocouple wires which measure the airflow temperature at dot 2 is inserted into the cavity from the bottom of the metal shell, and it is fixed by the channel inside the plastic catheter which is inserted into the cavity. In theory, the range of measurement error of the thermocouple wires is ±0.5 °C. Meanwhile, in order to further verify the ability of the thermocouple wires to capture the transient temperature variation, in other analogous experimental condition (the graphite plane was aerodynamic heated by the hot airflow, and after airflow heating, it is naturally cooling.), non-contact infrared thermometers are employed to test the temperature at the point on the upper surface of the graphite plane, and the test point just locates above dot 6 at the bottom of the graphite plane. It is noted that, the thermocouple wires and the infrared thermometer test different points at the upper and bottom surfaces. However, the graphite plane has good thermal conductivity and is thin, so it is considered that the temperature of the entire graphite plane is consistent, and especially the trends of transient temperature variation corresponding to all the points at the entire plane should be consistent. Therefore, although the test point measured by the infrared thermometer is different from the point tested by the thermocouple wires, it could be still used to validate the ability of the thermocouple wires to capture the transient temperature variation. In summary, the values tested by the infrared thermometer are taken as the calibration to validate the effectiveness of the thermocouple wires to capture the transient temperature variation, and to some extent, the difference of the magnitudes corresponding to these two test points, tested by the infrared thermometer and thermocouple wires respectively, can reflect the measure accuracy of the thermocouple wires. Fig. 3 shows the transient variation of dimensionless temperature tested by thermocouple wires and infrared thermometer. The infrared thermometer has temperature response threshold, and that’s the reason why the curve predicted by infrared thermometer has two sudden change regions on both sides. The part of increment of temperature before the peak corresponds to the process that the graphite was aerodynamically heated and the part of decrement of temperature after the peak corresponds to the process that the plane was naturally cooling. As shown in Fig. 3, in the effective section, the range of the magnitude of relative errors / are from 10% to 5%. Actually, at the initial stage (the upper surface of the graphite is just heated and cooled), the transient effect is strong, so the difference of temperature between the upper surface and bottom surface of the graphite is remarkable. With the heating and cooling process proceeding, this difference is disappearing gradually. Especially at the end stage of heating and cooling process, the relative error is nearly close to zero. Overall, the measurement results of the infrared thermometer and thermocouple wires are consistent. Moreover, the variation trends of the test temperature produced by these two methods are the same with each other, it is confirmed that the thermocouple wires adopted in this paper can effectively capture the transient temperature variation. 3.2. Comparison of the experimental data and computational results Fig. 4 shows the temperature transient variation corresponding to different points (dots 1–6 shown in Fig. 1). Fig. 4(a) presents the temperature variation with time corresponding to dot 6 located at the bottom of the graphite plate. For the temperature magnitude at dot 6, there is nearly no difference between sealing and no-sealing conditions, so only the comparison of the temperature variations without sealing is shown in Fig. 4(a). During the 100 s transient process, the values tested by
973
C. Shen et al. / International Journal of Heat and Mass Transfer 93 (2016) 969–979
0.15 1.0 Tdimless
Relative error φ
0.6 φ
φ / Tdimless
0.10
Infrared Thermometers Thermocouple
0.8
0.05
0.4
0.00
0.2 Relative error φ 0.0
-0.05 0.0
0.2
0.4
0.6
0.8
1.0
0.2
tdimless
a) Temperature measurements and relative error
0.4
0.6 tdimless
0.8
b) Amplification of relative error
Fig. 3. Comparison of measurements of thermocouple wires and infrared thermometer.
550
T/K
500 450 400 computational results experimental data
350 300 0
20
40
60
80
100
t/s
a) dot 6 on graphite plane (no-sealing in gap 1) 420
297
dot 1 dot 2 dot 3
400
360
T/K
T/K
380
340
dot 4 dot 5 experimental data
288
320 279
300
dot 1 dot 2 dot 3
280 260
0
20
40
t/s
dot 4 dot 5 experimental data 60
80
100
b) points in the cavity(no-sealing in gap 1)
270
0
20
40
60
80
100
t/s
c) points in the cavity(sealing in gap 1)
Fig. 4. Temperature variation with time corresponding to different points.
the thermocouple at corresponding times are a little greater than those predicted by the numerical method. The difference of magnitude between these two lines shown in Fig. 4(a) increases before the half (0–50 s), and then decreases during the following process (50–100 s). However, the overall increasing trends of these two lines are in good agreement. It is indicated that the numerical method in this paper can predict the aerothermal heating process that the high speed and hot airflow flows over the sealing link structure in this paper, and in a sense, it is confirmed that the temperature value of the hot air just invading into the gap 1 which is
reproduced by the numerical method is close to the experimental data. Fig. 4(b) and (c) show the thermal variation process in the cavity with sealing and without sealing in the gap 1, respectively. In Fig. 4(b) the temperature lines corresponding to the dot 1 and dot 5 predicted by the numerical method are not smooth. However, on the whole, these two lines with the other three lines (dot 2, dot 3 and dot 4) are all ascending all the time and they are almost synchronously increasing with the line regarding experimental data. The difference between the magnitudes of the
974
C. Shen et al. / International Journal of Heat and Mass Transfer 93 (2016) 969–979
temperature predicted by the numerical method (especially the values corresponding to dot 2, dot 3 and dot 4) and the experiment data at different times are always nearly 50 K. Different from the consistent increasing trend in Fig. 4(b), the changing trend of each line in Fig. 4(c) all increases sharply at the beginning, then falls down and lastly increases again in the end. The changing tendency including the values predicted by the numerical method (especially the values corresponding to dot 4 and dot 5) and the experimental data are still the same and the difference of magnitude between them at different times are always nearly 6 K. By comparison of the numerical results and the experimental data, there are existing stable difference of temperature value between them, regardless of sealing or no-sealing. Besides the factors of numerical computational method, it may arise from the following several difference between the real experimental rigs and the numerical computational model: Firstly, the difference between the experimental structure and the simplified computational structure. as shown in Fig. 2, in the direction perpendicular to the paper plane (the xy plane), or in the gap length direction (z axis direction), the actual length of the gap and the cavity is nearly half length of the graphite plane and the thermal insulation material shown in Fig. 1. However, the computational geometric model shown in Fig. 2, is equivalent to the geometric model that has the length equal to the whole length of the sealing structure in the z axis direction. It means that, under the same conditions, the magnitude of the invading mass flow corresponding to the three dimensional structure in the experiment is half as much as the computational value corresponding to the equivalent two dimensional model shown in Fig. 2. Naturally, the actual temperature magnitude of the invading air in the cavity is lower than the computational results at the corresponding moments. Secondly, the unknowing leaking locations in the real experimental rigs. There may be existing gas leakage at some unknown locations, such as the holes which the thermocouples get through, the connection edge between different structures, etc. Thirdly, the difference between real porosity and permeability and the corresponding magnitude in numerical computation. The porosity and the permeability employed in the computing process may be not accurate enough, because the porous sealing strip was squeezed in the actual installation process and it may lead to the changes in the physical properties of the porous sealing strip. Although there is existing stable difference of magnitude between the numerical results and the experimental data, according to consistency of the changing tendency, it is still demonstrated that the numerical method can capture the flow and thermal characteristics regarding the sealing link structure in this paper. Besides the validation of the numerical method, by comparison of the conditions of sealing and no-sealing in gap 1, as shown in Fig. 4(b) and (c), it is also found that the temperature of hot air invading into the cavity decreases substantially, no matter from the perspective of numerical method or experiment. From viewpoint of numerical method, in the whole 100 s transient process, the decrement extent of the temperature due to sealing in gap 1 is close to 100 K. Meanwhile, the increment–decrement–increment tendency of temperature magnitude in Fig. 3(c) is obviously different from the consistent increment tendency in Fig. 3(b). According to this increment-decrement-increment variation tendency, for the sealing condition, the whole invading process could be divided into three parts. The first part is that, in the beginning process (approximate 0–1 s corresponding to the experiment data), the hot air just recently invades into the cavity, and doesn’t have enough time to transfer its thermal energy to solid structure through the cold wall around the cavity. The second part is that, with the invading process proceeding, the process that the cold wall obtains the heat
from the hot air is faster than the heat supplement process with the newly hot air invading into the cavity. The last part is that, the heat supplement rate in the cavity with the hot air invading is more rapid than the heat loss process in the cavity that the heat transfer from hot invading air to the cold wall around the cavity. However, for the no-sealing condition, in the cavity, the heat supplement rate with the hot air newly invading is always quicker than the heat loss rate with cold wall obtaining heat from hot invading air. The phenomenon of this increment-decrementincrement tendency of temperature magnitude in the cavity is further presented that the effect of the sealing in gap1 on the process that the hot air invades into the cavity. 3.3. Qualitative and quantitative analysis of invading process via computational results In order to further comprehend the invasion process of the hot air into the inner cavity inside the gap-cavity sealing structure, the variable distribution along entrance of gap 1 for sealing and nosealing is shown in Fig. 5. For the y-axis velocity uy, there is approximately existing a change in one order of magnitude. For sealing in gap 1, the mean velocity is nearly 50 m/s, but for no-sealing in gap 1, the velocity at peak reaches up to 500 m/s. Moreover, regardless of sealing or no-sealing, the velocity is decreasing with transient process proceeding, but the decreasing proportion is not great, for the 100 s transient process, approximately about 15– 20%. For the temperature and pressure distributions, regardless of sealing or no-sealing, there are both dramatic rises in the region close to the wall of gap 1, and that’s because the invading hot air directly attack the right wall of gap 1 without any barrier corresponding to the no-sealing condition and the hot air directly heats the right wall intensively. For sealing and no-sealing conditions, the mean temperatures at the entrance are close. However, the difference of the mean pressures are significant, and the magnitude of pressure with sealing is approximately twice of the magnitude of pressure without sealing. As shown in Fig. 5d), the shape of the density curves with sealing and no-sealing are nearly the same, and the magnitude of the mean density with sealing is nearly twice of that without sealing. Overall, except the density distribution, along the entrance of gap 1, the shape of curves with sealing is gentle, while the shape of curves without sealing has dramatic variation. The difference of magnitude of velocity between sealing and no-sealing is the most significant, and there is a difference of one order of magnitude. According to the analysis above, for the sealing condition, compared with the no-sealing condition, the mass flow rate through the gap 1 decreases substantially, i.e. it is merely one thirds of the magnitude of the mass flow rate without sealing, as shown in Fig. 5(e), and the average temperature through the top inlet of gap 1 does not have obvious and substantial variation. According to the numerical results, for the average velocity in gap 1, there is nearly existing difference of one order of magnitude between conditions of sealing and no-sealing. Fig. 6 shows the temperature contours at different moments for the cross section of the whole fluid and solid region, as shown in Fig. 2. Obviously, for the condition of sealing in gap 1, at the same moment, the temperature magnitude in the gaps and the cavity is much lower than the temperature value corresponding to the condition of no-sealing. For the condition of no-sealing in gap 1, with the proceeding of the invasion of hot air, there is significantly circular high temperature area in the middle and lower part of the cavity, and the scale of the circular area is almost of the same order of the length of the cavity. Obviously, according to the shape of the high temperature contour, it is speculated that the invading hot air stream circularly flows clockwise.
975
C. Shen et al. / International Journal of Heat and Mass Transfer 93 (2016) 969–979
1400
0
-200 -300
1000 T/K
uy / m·s-1
1200
sealing t/s no-sealing 0.1 1 10 50 100
-100
sealing t/s no-sealing 0.1 1 10 50 100
800 600
-400
400
-500 -600 0.1240
0.1245
0.1250 x/m
0.1255
200 0.1240
0.1260
0.1245
a) y-axis velocity uy
2000 1750
p / Pa
1500
0.1255
0.1260
b) temperature 0.024
sealing t/s no-sealing 0.1 1 10 50 100
0.020
sealing t/s no-sealing 0.1 1 10 50 100
0.016 ρ / kg·m-3
2250
0.1250 x/m
1250 1000
0.012 0.008
750
0.004
500 250 0.1240
0.1245
0.1250 x/m
0.1255
0.1260
0.000 0.1240
0.1245
c) pressure
0.1260
sealing
no-sealing 0.1 1 10 50 100
0.2 m / kg· m - 2· s - 1
0.1255
d) density
0.4
.
0.1250 x/m
0.0 -0.2 -0.4 -0.6 -0.8 -1.0 0.1240
0.1245
0.1250
0.1255
0.1260
x/m
e) Mass flow rate Fig. 5. Comparison of variable distribution along entrance of gap 1 for sealing and no-sealing.
For the condition of sealing in gap 1, at the moment t = 0.1 s and t = 10 s, the temperature contours are asymmetric and disordered. However, with the proceeding of the invasion of hot air, the temperature contours tend to be symmetric along the center line of the sealing structure and to be ordered, as shown in Fig. 6(b), at the moment t = 50 s and t = 100 s. Meanwhile, for the condition of sealing in gap 1, the color of the contour areas in the lower left and right corners at the moment t = 0.1 s are a little brighter than the color at corresponding positions at the moment t = 10 s, and at the moment t = 50 s and t = 100 s, the temperature values at corresponding positions are
gradually getting more and more higher than the former moments. It is indicated that the temperature at the corresponding positions at initial moment, such as t = 0.1 s, reaches a high magnitude, and in the following period the temperature decreases firstly and then increases until the end. This conclusion agrees with the incrementdecrement-increment tendency of the cavity temperature variation obtained in Fig. 4(c) above. In order to further indicate the flow state inside the cavity, Fig. 7 shows the vorticity contour with sealing and without sealing corresponding to the moments in Fig. 6. In Fig. 7, for sealing or nosealing conditions, the vorticity contours at the four moments
976
C. Shen et al. / International Journal of Heat and Mass Transfer 93 (2016) 969–979
T/K
0.1s
0.1s
10s
10s
50s
50s
100s
100s
(a) No-sealing
(b) Sealing
Fig. 6. Temperature contour of entire field at different times.
are not continuous, i.e. the time intervals between them are large. However, it still could be illustrated that, for no-sealing condition, with the proceeding of the hot air invasion, a big clockwise vortex that nearly occupies the entire middle and bottom space of the cavity is formed and other smaller vortexes generated out of gap 2 are developed and crushed around the principal central clockwise vortex.
Corresponding to the sealing condition, at the initial stage (at the moment t = 0.1 s and t = 10 s in Fig. 7(a) and (b)), the vorticity contours are not stable and it is illustrated that there are existing small vertexes along the central line of the structure. These vertexes are generated after the hot air just enters into the cavity from the gap 2. These vortexes are developed along the central line of the sealing structure and finally vanished as they gradually
C. Shen et al. / International Journal of Heat and Mass Transfer 93 (2016) 969–979
977
ω / s-1
0.1s
0.1s
10s
10s
50s
50s
100s
100s
(a) No-sealing
(b) Sealing
Fig. 7. Vorticity contour inside the structure at different times.
approach the bottom wall of the cavity. In the later transient process (at the moment t = 50 s and t = 100 s in Fig. 7(b)), the flow state of the invading hot air is gradually getting more and more stable and smoother, and the vorticity contours are inclined to be symmetric along the central line of the structure. In all, the features of the vorticity contour shown in Fig. 7 are further clarified the flow mechanism inside the cavity which leads to the thermal distribution phenomena presented in Figs. 4 and 6.
In Figs. 6 and 7, via qualitative comparison of contours, the features of invasion of hot air under the conditions of sealing and nosealing in gap 1 are presented, and subsequently, quantitative comparison of magnitude of the variable in the cavity is further implemented to discuss the mechanism of invasion process. The pressure difference inside and outside the sealing link structure is the primarily driven power for the invasion of the airflow, so the pressure inside the cavity is concentrated on. In Fig. 8, the
C. Shen et al. / International Journal of Heat and Mass Transfer 93 (2016) 969–979
580
440
560
435
540
430
520
p / Pa
p / Pa
978
425
500
dot 1 dot 2 dot 3 dot 4 dot 5
420
480
dot 1 dot 2 dot 3 dot 4 dot 5
460 440 420 400
415 410 405 400
0
20
40
60
80
100
0
t/s a) No-sealing
20
40
t/s
60
80
100
b) Sealing
Fig. 8. Variation of pressure in flow field inside structure with time.
transient variations of pressure inside the cavity are shown. From the viewpoint of numerical results, at the initial stage (in this paper, the first moment t = 0.1 s), the magnitude of pressure inside the cavity sharply increases, regardless of sealing or no-sealing. However, the increasing extent are different. On the condition of no-sealing, the increasing extent of the five observed points are almost more than 50 Pa, and the increasing values of the five points with respect to sealing condition are only more than 15 Pa. Obviously, the difference of the ascending pressure value between sealing and no-sealing conditions at the initial stage are caused by the block effect of the sealing material in gap 1 on the hot air invasion. Except the sharp increment at the initial stage, in the next following 100 s transient process, the magnitudes of the pressure inside the cavity always ascend stably, regardless of sealing or no-sealing. At the stable temperature ascending stage, The slope of the pressure lines corresponding to the no-sealing condition is nearly 1, and the slope corresponding to the sealing condition is only nearly 0.2. In Summary, the increasing rate of magnitude of pressure inside the cavity reflects the fact that the sealing material in gap 1 considerably delays the process of the invasion of hot air into the cavity. 4. Conclusion In this paper, by all-speed preconditioned conjugate heat transfer numerical method and wind tunnel experiment, it is investigated the mechanism of invasion of high speed hot airflow into gap-cavity linking seal structure. On the condition of sealing in top gap, by the experimental method, the increment-decrementincrement phenomenon of variation of temperature magnitude inside the cavity is presented, and meanwhile this variation phenomenon is reproduced by the all-speed preconditioned conjugate heat transfer numerical method. Then by the numerical method, the more details with respect to the invasion process are shown to analyze the mechanism corresponding to this increment– decrement–increment phenomenon of variation of temperature magnitude. After validation of numerical method, the difference of invasion process between sealing and no-sealing conditions are further summarized. For the no-sealing condition, there is existing large principal vortex that nearly occupies the entire middle and bottom space of the cavity. For the sealing condition, in the initial stage of the entire transient process, the flow of invading hot air isn’t stable and there are existing small vortexes developing and vanishing along the central line of the gap-cavity structure. However, with the transient process proceeding, the flows are getting more and more stable, the small vertexes along the central line
are vanished, and the flow fields are nearly symmetric along the central line. Finally, by comparison of variation of pressure magnitude inside the cavity with and without sealing, it is found that the decrement of the increasing rate of pressure magnitude inside the cavity can reflect the effect of the sealing material in the top gap on delaying the process of the invasion of hot air into the cavity. Acknowledgement This study was supported by the National Natural Science Foundation of China (Grant No. 51176038 and Grant No. 51309115). References [1] R.E. Chupp, R.C. Hendricks, S.B. Lattime, et al., Sealing in turbomachinery, J. Propul. Power 22 (2) (2006) 313–347. [2] M.F. Aksit, R.E. Chupp, O.S. Dinc, et al., Advanced seals for industrial turbine applications: design approach and static seal development, J. Propul. Power 18 (6) (2002) 1254–1259. [3] J.R. Finkbeiner, P.H. Dunlap, B.M. Steinetz, et al., Review of seal designs on the apollo spacecraft, J. Propul. Power 45 (5) (2008) 900–910. [4] J.J. Demange, P.H. Dunlap, B.M. Steinetz, Improved seals for high temperature airframe applications, AIAA (2006) 2006–4935. [5] J.J. DeMange, P.H. Dunlap, B.M. Steinetz, et al., An evaluation of high temperature airframe seals for advanced hypersonic vehicles, AIAA (2007) 2007–5743. [6] C. Shen, F.X. Sun, X.L. Xia, Analysis on transient conjugate heat transfer in gapcavity-gap structure heated by high speed airflow, Int. J. Heat Mass Transfer 67 (2013) 1030–1038. [7] P.H. Dunlap, B.M. Steinetz, D.M. Curry, Further investigations of control surface seals for the X-38 re-entry vehicle, AIAA (2001) 2001–3628. [8] P.H. Dunlap, J.J. DeMange, B.M. Steinetz, Performance evaluations of ceramic wafer seals, AIAA (2006) 2006–4934. [9] J.R. Finkbeiner, P.H. Dunlap, B.M. Steinetz, et al., On the development of a unique arc jet test apparatus for control surface seal evaluations, AIAA (2004) 2004–3891. [10] B.M. Steinetz, M.L. Adams, P.A. Bartolotta, et al., High-temperature braided rope seals for static sealing applications, J. Propul. Power 13 (1997) 675–682. [11] H.J. Zhang, Z.P. Zou, Y. Li, et al., Preconditioned density-based algorithm for conjugate porous/fluid/solid domains, Numer. Heat Transfer Part A 60 (2011) 129–153. [12] H.J. Zhang, Z.P. Zou, Numerical investigation of laminar-plate transpiration cooling by the preconditioned density-based algorithm, Numer. Heat Transfer Part A 62 (2012) 761–779. [13] I.A. Bedarev, S.G. Mironov, K.M. Serdyuk, et al., Physical and mathematical modeling of a supersonic flow around a cylinder with a porous insert, J. Appl. Mech. Tech. Phys. 52 (2011) 9–17. [14] F.Q. Wang, Y. Shuai, H.P. Tan, C.L. Yu, Thermal performance analysis of porous media receiver with concentrated solar irradiation, Int. J. Heat Mass Transfer 62 (2013) 247–254. [15] F.Q. Wang, Y. Shuai, Z.Q. Wang, Y. Leng, H.P. Tan, Thermal and chemical reaction performance analyses of steam methane reforming in porous media solar thermochemical reactor, Int. J. Hydrogen Energy 39 (2) (2014) 718–730. [16] Y.T. Yang, M.L. Hwang, Numerical simulation of turbulent fluid flow and heat transfer characteristics in heat exchangers fitted with porous media, Int. J. Heat Mass Transfer 52 (13–14) (2009) 2956–2965.
C. Shen et al. / International Journal of Heat and Mass Transfer 93 (2016) 969–979 [17] Z.D. Cheng, Y.L. He, F.Q. Cui, Numerical investigations on coupled heat transfer and synthetical performance of a pressurized volumetric receiver with MCRTFVM method, Appl. Therm. Eng. 50 (1) (2013) 1044–1054. [18] W.W. Yang, Y.L. He, Y.S. Li, Modeling of dynamic operating behaviors in a liquid-feed direct methanol fuel cell, Int. J. Hydrogen Energy 37 (2012) 18412– 18424. [19] Y.Q. Liu, P.X. Jiang, S.S. Jin, et al., Transpiration cooling of a nose cone by various foreign gases, Int. J. Heat Mass Transfer 53 (2010) 5364– 5372. [20] S.S. Feng, T. Kim, T.J. Lu, Numerical investigation of forced convection in pin/plate-fin heat sinks heated by impinging jet using porous medium approach, Int. J. Numer. Methods Heat Fluid Flow 23 (1) (2013) 88–107. [21] S.S. Feng, J.J. Kuang, T. Wen, T.J. Lu, K. Ichimiya, An experimental and numerical study of finned metal foam heat sinks under impinging air jet cooling, Int. J. Heat Mass Transfer 77 (2014) 1063–1074.
979
[22] J. Blazek, Computational Fluid Dynamics: Principles and Applications, second ed., Elsevier, 2005. [23] W.S. Fu, S.H. Huang, W.H. Wang, An investigation of effects of heights and a length on natural convection of three dimensional parallel plates with a heated bottom surface by a CUDA computation platform, Int. J. Heat Mass Transfer 67 (2013) 118–130. [24] C. Shen, F.X. Sun, X.L. Xia, Implementation of density-based solver for all speeds in the framework of OpenFOAM, Comput. Phys. Commun. 185 (2014) 2730–2741. [25] H.G. Weller, G. Tabor, H. Jasak, et al., A tensorial approach to computational continuum mechanics using object-oriented techniques, Comput. Phys. 12 (1998) 620–631. [26] B. Alazmi, K. Vafai, Analysis of fluid flow and heat transfer interfacial conditions between a porous medium and a fluid layer, Int. J. Heat Mass Transfer 44 (9) (2001) 1735–1749.