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DNS, LES and RANS of turbulent heat transfer in boundary layer with suddenly changing wall thermal conditions
International Journal of Heat and Fluid Flow 41 (2013) 34–44
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DNS, LES and RANS of turbulent heat transfer in boundary layer with suddenly changing wall thermal conditions Hirofumi Hattori a,⇑, Shohei Yamada a, Masahiro Tanaka a, Tomoya Houra a, Yasutaka Nagano a,b a b
Department of Mechanical Engineering, Nagoya Institute of Technology, Gokiso-cho, Showa-ku, Nagoya 466-8555, Japan Department of Research, Nagoya Industrial Science Research Institute, Yotsuya-dori 1-13, Chikusa-ku, Nagoya 464-0819, Japan
a r t i c l e
i n f o
Article history: Received 25 October 2012 Received in revised form 26 February 2013 Accepted 26 March 2013 Available online 1 May 2013 Keywords: Turbulent boundary layer Heat transfer Turbulent structure DNS Wall thermal condition LES RANS
a b s t r a c t The objectives of this study are to investigate a thermal field in a turbulent boundary layer with suddenly changing wall thermal conditions by means of direct numerical simulation (DNS), and to evaluate predictions of a turbulence model in such a thermal field, in which DNS of spatially developing boundary layers with heat transfer can be conducted using the generation of turbulent inflow data as a method. In this study, two types of wall thermal condition are investigated using DNS and predicted by large eddy simulation (LES) and Reynolds-averaged Navier–Stokes equation simulation (RANS). In the first case, the velocity boundary layer only develops in the entrance of simulation, and the flat plate is heated from the halfway point, i.e., the adiabatic wall condition is adopted in the entrance, and the entrance region of thermal field in turbulence is simulated. Then, the thermal boundary layer develops along a constant temperature wall followed by adiabatic wall. In the second case, velocity and thermal boundary layers simultaneously develop, and the wall thermal condition is changed from a constant temperature to an adiabatic wall in the downstream region. DNS results clearly show the statistics and structure of turbulent heat transfer in a constant temperature wall followed by an adiabatic wall. In the first case, the entrance region of thermal field in turbulence can be also observed. Thus, both the development and the entrance regions in thermal fields can be explored, and the effects upstream of the thermal field on the adiabatic region are investigated. On the other hand, evaluations of predictions by LES and RANS are conducted using DNS results. The predictions of both LES and RANS almost agree with the DNS results in both cases, but the predicted temperature variances near the wall by RANS give different results as compared with DNS. This is because the dissipation rate of temperature variance is difficult to predict by the present RANS, which is found by the evaluation using DNS results. Ó 2013 Elsevier Inc. All rights reserved.
1. Introduction There are a number of experimental, numerical and theoretical investigations of the turbulent boundary layer with heat transfer (Reynolds et al., 1958a; Subramanian and Antonia, 1981a; Kong et al., 2000; Hattori et al., 2007; Houra and Nagano, 2008). In general, wall boundary condition of thermal field can be considered in three cases: a constant wall temperature, a constant heat flux wall and an adiabatic wall. Although the above-mentioned investigations have dealt with a simple thermal wall condition so as to know a fundamental turbulent heat transfer phenomena, the wall condition of a thermal field often changes in practical thermal fields. Thus, the present study deals with the step change of wall thermal conditions in the turbulent boundary layer. Especially, the thermal field of boundary layer whose heated wall condition suddenly vanishes in the downstream region, i.e., the wall heated ⇑ Corresponding author. Tel./fax: +81 52 735 5359. E-mail address: [email protected] (H. Hattori). 0142-727X/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.ijheatfluidflow.2013.03.014
by a constant temperature followed by an adiabatic condition, is investigated by means of DNS. These investigations were experimentally and numerically conducted (Reynolds et al., 1958b; Charnay et al., 1979; Antonia et al., 1977; Subramanian and Antonia, 1981b; Browne and Antonia, 1981; Nagano et al., 1997; Hattori and Nagano, 1998), but the express turbulent heat transfer statistics and structures close to the wall were difficult to observe. The detailed investigations and observations of the near-wall turbulent heat transfer phenomena have relied increasingly on DNS (Hattori et al., 2007; Hattori and Nagano, 2010; Hattori and Nagano, 2004), because DNS can provide the in-depth data of turbulent statistics including high-order moments near the wall, though there are limitations of Reynolds and Prandtl numbers. Hence, the step change of wall temperature is simulated using DNS (Hattori et al., 2012), in which the turbulent boundary layer on the flat plate with step change of wall temperature and turbulent boundary layer over forward-facing step with the heated wall in front of step and the unheated wall on the step are simulated, and the detailed characteristics of turbulent heat transfer in such fields are investigated.
H. Hattori et al. / International Journal of Heat and Fluid Flow 41 (2013) 34–44
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Nomenclature cp k P Pr qj qw Red2;in
specific heat at constant pressure turbulence energy, ¼ ui ui =2 mean static pressure Prandtl number, =m/a SGS heat flux or turbulent heat flux wall heat flux Reynolds number based on free stream velocity and momentum thickness at the inlet of the driver part, ¼ U 0 d2;in =m t time U; V; W mean velocity in x-, y- and z-directions, respectively Ui mean velocity in xi-direction U0 free stream mean velocity u,v,w instantaneous velocity or turbulent fluctuation in x-, yand z-directions, respectively ui instantaneous velocity or turbulent fluctuation in xidirection pffiffiffiffiffiffiffiffiffiffiffiffi us friction velocity, ¼ sw =q xc adiabatic starting point xi Cartesian coordinate in i-direction
These simulations are carried out under the simultaneous developing boundary layer of both velocity and thermal field. On the other hand, the case of the developed boundary layer of velocity field and the entrance region of thermal field is also interesting for the study of turbulent heat transfer problems. Therefore, in order to obtain detailed knowledge of a turbulent thermal field around the point where the thermal boundary condition is changed, i.e., a constant temperature wall followed by an adiabatic wall condition, DNSs are carried out, in which the entrance region of thermal field in turbulence is also simulated. On the other hand, the turbulence models of thermal field in the Reynolds-averaged Navier–Stokes equation simulation (RANS) and the large eddy simulation (LES) have been improved recently using DNS results so as to modify the near-wall predictions (Hattori et al., 2009; Hattori et al., 2006; Hattori and Nagano, 1998; Inagaki et al., 2011), because heat is conducted and transferred usually from and near the wall in engineering problems. Thus, one of the topics in this study is the evaluation of recent turbulent heat transfer models using the present DNS in such a thermal field.
x, y, z
Cartesian coordinate in streamwise, wall-normal and spanwise directions, respectively
Greek symbols a molecular diffusivity for heat aSGS eddy diffusivities of SGS component for heat at eddy diffusivity for heat d2 momentum thickness dij Kronecker delta m kinematic viscosity mSGS eddy diffusivities of SGS component for momentum mt eddy diffusivity for momentum q density sij SGS stress tensor or Reynolds stress sw wall shear stress H mean temperature H0 free stream mean temperature Hw wall temperature h instantaneous temperature or temperature fluctuation h2 temperature variance
d2,in, and the temperature difference between the wall, Hw and the free stream, H0 , DH, at the inlet of the driver part, i.e., the dimensionless temperature is defined as h ¼ ðh Hw Þ=ðH0 Hw Þ, because the constant wall temperature is mainly set for the wall thermal condition. Red2;in is Reynolds number based on the free stream velocity and the momentum thickness at the inlet of the driver part, and Pr(=m/a) is a Prandtl number. In case of DNS, turbulent stress, sij, and turbulent heat flux, qj in Eqs. (2) and (3) are equal to zero, and ðÞ means an instantaneous value in equations. The filtered turbulent quantities are used in equations for the LES (Inagaki et al., 2005; Inagaki, 2011; Inagaki i u j et al., 2011), and ðÞ means a GS component. Thus, sij ¼ ui uj u
2. Numerical procedure The governing equations used in DNS, LES and RANS are the Navier–Stokes equation without buoyancy, the continuity equation for the velocity field, and the energy equation for the thermal field, in which incompressibility is assumed as follows:
i @u ¼0 @xi i i @u @u @p @ 1 @u j i ¼ þu þ sij @xi @xj Red2;in @xj @t @xj ! @ h @ h @ 1 @ h j þu ¼ qj @t @xj @xj Red2;in Pr @xj
ð1Þ ! ð2Þ ð3Þ
where the Einstein summation convention applies to repeated indi i is the dimensionless velocity component in xi direction, ces. u h is is the dimensionless pressure, t the dimensionless temperature, p is the dimensionless time, and xi is the dimensionless spatial coordinate in the i direction, respectively. All equations are non-dimensionalized by the free stream velocity, U 0 , the momentum thickness,
Fig. 1. Flow fields, wall conditions and coordinate system.
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j is an SGS heat flux, which hu is an SGS stress tensor, and qj ¼ huj represent the effect of SGS components, and should be modelled. In case of RANS, since the Reynolds-averaged turbulent quantities are employed in equation, ðÞ means an ensemble-average or a timeaverage value. Therefore, sij ¼ ui uj is Reynolds stress, and qj ¼ uj h is the turbulent heat fluxIn the present study, the gradient diffusion type model is used for the modellings of sij = 2mSGSSij + (2/ 3)dijk (LES), sij = 2mtSij + (2/3)dijk (RANS), qj ¼ aSGS ð@ h=@xj Þ (LES) and qj ¼ at ð@ h=@xj Þ (RANS), where mSGS and aSGS are the eddy diffusivities of SGS component for momentum and heat, respectively, mt and at are the eddy diffusivities for momentum and heat, respectively, k is a turbulence energy or a turbulence energy of SGS i =@xj þ @ u j =@xi Þ=2. In order to solve mSGS component, and Sij ¼ ð@ u and aSGS, the modified mixed-timescale model (MTSn) (Inagaki, 2011) and the mixed-timescale model for thermal field (MTSt) (Inagaki et al., 2011) are adopted for the LES. The characteristic of employed model in LES is the time scale for the eddy diffusivities for momentum and heat, i.e., since the characteristic time scale of SGS is adequately determined by the model in both velocity and
thermal fields, the model gives good predictions of the wall turbulent flows with heat transfer. As for the RANS, mt and at are solved using the two-equation heat transfer model (Hattori and Nagano, 1998). This RANS model can properly predict turbulent quantities including the budget of transport equations in the wall turbulent flows with heat transfer, i.e., the budgets of transport equations for temperature variance and its dissipation rate are adequately predicted (Hattori and Nagano, 1998). Flow fields, wall conditions and coordinate system of the present simulations are shown in Fig. 1. In order to simulate the spatially developing turbulent boundary layer, DNS and LES have the driver and the main simulation parts as shown in Fig. 1, in which the turbulent boundary layer of the velocity is generated in the driver part, and the main simulation part receives turbulent inflow data from the driver part. Thus, the spatially developing boundary
Table 1 Computational method and conditions. Grid system
Staggered grid
Coupling algorithm Time advancement Spatial schemes
Fractional step method Adams–Bashforth method (non-linear terms) Crank–Nicolson method (linear terms) 2nd-order central difference
Parameters Reynolds number
Red2;in ¼ U 0d2;in =m ¼ 300
Prandtl number
Pr = m/a = 0.71
Domain size (x y z) Driver Main
100d2,in 30d2,in 40d2,in 200d2,in 30d2,in 40d2,in
Grid points (x y z) DNS Driver Main
192 128 128 384 128 128
LES
Driver Main
96 64 64 192 64 64
Spatial resolutions (inlet of main simulation part) DNS Dx + 8.3 Dy + 0.043 11 + Dz 5 LES
Dx + Dy + Dz +
16.6 0.086 22 10
Fig. 2. Variations of wall temperature.
Fig. 3. Distributions of mean temperature near the wall.
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layer is achieved without the periodic boundary condition in the streamwise direction (Lund et al., 1998; Kong et al., 2000; Hattori et al., 2007; Hattori and Nagano, 2012). The conditions of thermal field are established in two cases as indicated in Fig. 1. In the first case (hereinafter referred to as case 1), the velocity boundary layer only develops in the driver part, and the flat plate is only heated in the main simulation part, i.e., the adiabatic wall condition is adopted in the driver part. In the main simulation part of case 1, the thermal boundary layer develops along a constant temperature wall followed by an adiabatic wall. Note that the adiabatic wall condition is set in the entrance of the main simulation part in order to avoid the numerical error due to a sudden increase of wall temperature. In the second case (hereinafter referred to as case 2), the velocity and thermal boundary layers simultaneously develop in the driver part, and the wall thermal condition is changed from the constant temperature to adiabatic in the main simulation part. In Fig. 1, the distance, xc, indicates the starting line of the adiabatic wall (the starting line of the adiabatic wall is x/d2,in = 35.4 in both cases). In case 1, since the boundary layer of thermal field develops only in the main simulation part, the boundary layer of thermal field may become very thinner much than the boundary layer of velocity field in the case, i.e., the entrance region of thermal field in turbulence is simulated. Thus, in case 1, it is possible to observe the entrance region of thermal field in the turbulent boundary layer. As for the RANS, the same conditions of DNS and LES are carried out by RANS in the main simulation part of the two-dimensional field (Hattori and Nagano, 1998). The grid numbers DNS used are: Nx Ny Nz = 192 128 128 for the driver part, and Nx Ny Nz = 384 128 128 for the main simulation part. The grid numbers of LES are arranged to be half those of DNS. The boundary conditions for the velocity
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field are the non-slip conditions on the walls, and @u/@y = 0, @w/ @y = 0, @v/@y = (@u/@x + @w/@z) on the upper boundary (free stream). At the outlet of both parts, the convective boundary conditions are applied, and periodic boundary conditions are used in the spanwise direction for both the velocity and thermal fields. Reynolds number, Red2;in ¼ U 0 d2;in =m, is set 300, and Prandtl number, Pr = m/a, is 0.71. Computational method and conditions are summarised in Table 1.
3. Results and discussion 3.1. DNS investigation of turbulent heat transfer in boundary layer with suddenly changing wall thermal conditions In order to demonstrate the variation of temperature difference by the difference of condition of thermal boundary layer in the upstream, the variation of wall temperature which is defined as the difference between the free stream and wall are shown in Fig. 2. It can be observed that the wall temperature of case 1 drops faster than that of case 2, because the undeveloped boundary layer of thermal field encounters the adiabatic wall condition in case 1. In order to observe behaviour of turbulent thermal field on both the isothermal and the adiabatic walls in detail, Fig. 3 shows the distributions of mean temperature near the wall. Developing the thermal boundary layer on the heated wall can be observed in case 1 as shown in Fig. 3a, but the thermal boundary layer of case 2 is developed very little due to the generating boundary layer which is developed simultaneously with velocity field in the driver part as indicated in Fig. 3b. In the adiabatic wall region, the mean temperature rapidly decreases near the wall region in case 1, but the
Fig. 4. Budgets of energy equation (Case 1).
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mean temperature of case 2 is maintained higher than in case 1, which are clearly shown in Fig. 3c. Moreover, the mean temperature diffuses toward the outer region on the adiabatic wall in case 1, but this phenomenon is difficult to observe in case 2 as shown in Fig. 3c. Figs. 4 and 5 show the budget of Reynolds-averaged energy equation in the 2-dimensional field as given below:
! @H @H 1 @2H @2H @ @ 0 ¼ U V þ þ 2 uh v h @x @y @x @y Red2;in Pr @x2 @y
ð4Þ
Comparing the budget of case 1 with that of case 2, the convection term in x-direction obviously affects distribution of mean temperature in case 1 due to the entrance region of thermal field. At the immediate aftermath of the adiabatic starting point (x/d2, in = 40), the convection term in x-direction works in the loss side near the wall in both cases. Conversely, the thermal diffusion term in ydirection works in the gain side near the wall. In case 1, since the convection term in x-direction also contributes in the gain side away from the wall, the mean temperature diffuses toward the outer region as mentioned above. In the region of gradual variation of wall temperature (x/d2,in = 100), the convection term in x-direction clearly affects the distribution of mean temperature of both cases in the outer region as shown in Figs. 4 and 5 and Fig. 5d. Thus, it is also observed in case 2 that the mean temperature distributes toward the outer region at the region of gradual variation of wall temperature as demonstrated in Fig. 3b. Investigating the wall-normal turbulent heat flux as shown in Fig. 6, the wall-normal turbulent heat fluxes of both cases decrease clearly in the near-wall region on the adiabatic wall. Also, the increases in wall-normal turbulent heat fluxes away from the wall in case 1 are clearly found due to the occurrence of a mean temper-
Fig. 6. Near-wall distributions of wall-normal turbulent heat fluxes.
0.002
-0.002
Fig. 5. Budgets of energy equation (Case 2).
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ature gradient by expansion of a high mean temperature region near the outer region as indicated in Fig. 3a, i.e., the turbulent heat transfer in the entrance region of thermal field can be observed. In the entrance region of thermal field, the peak of wall-normal turbulent heat flux which gradually moves over away from the wall, is obviously observed in Fig. 6a. On the adiabatic wall, the peak of wall-normal turbulent heat flux also moves over away from the wall in case 1, but this phenomenon is difficult to observe in
Fig. 9. Thermal streaks in constant wall temperature followed by adiabatic wall; red h > 0.1, blue h < 0.1. Fig. 7. Distributions of wall-normal turbulent heat fluxes near the wall.
Fig. 8. Near-wall distributions of temperature variances.
Fig. 10. Comparisons of turbulence quantities of velocity field predicted with DNS results.
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Fig. 11. Comparisons of variation of wall temperature predicted with DNS results. Fig. 12. Comparisons of mean temperature predicted with DNS results.
case 2. In case 2, the distributions of wall-normal heat flux hardly changes on the heated wall. The peak of wall-normal heat flux moves over away from the wall on the adiabatic wall, and the maintained profiles of wall-normal turbulent heat flux in the outer region are observed, but a slight variation is found as shown in Fig. 6b. On the other hand, the streamwise turbulent heat fluxes of both cases are shown in Fig. 7. It is well-known that the streamwise turbulent heat flux hardly affects the distribution of mean temperature on the heated wall. The turbulent heat flux, however, slightly influences the adiabatic wall as demonstrated in Figs. 4 and 5. Also here, the shift in the peak of streamwise turbulent heat flux is evidently observed in case 1, but the shift of the peak is not found in case 2. On the adiabatic wall, the spreading toward the outer region and decrease of streamwise turbulent heat flux are discovered in case 1, but it is difficult to observe in case 2 as shown in Fig. 7b. On the other hand, the distributions of temperature variances are shown in Fig. 8. Since the temperature variance is allowed on the wall in the adiabatic wall region, it is obvious that the temperature variance increases near the wall region in both cases, in which it seems that the increase of temperature variance near the wall is also caused by the decrease in the dissipation rate of the temperature variance as shown in the budget of temperature variance of Figs. 16 and 17. In the outer region, the temperature variances remarkably diffuse in common with the distributions of wall-normal turbulent heat flux in case 1. Also, the distributions of temperature variance in case 2 are similar to that of the wallnormal turbulent heat flux. On the adiabatic wall, although it is obvious that a temperature variance is hardly produced exists, i.e., the mean temperature gradient and wall-normal turbulent heat flux hardly distribute near the wall; the temperature variance insistently remains consistent for the reason mentioned above.
As for the turbulent structure in the thermal field Fig. 9 shows the thermal streaks of both cases in the constant wall temperature followed by the adiabatic wall. In case 2, a sort of messy streak structure can be observed due to the developed thermal boundary layer on the heated wall. In contrast, a gracile streak structure is found in case 1 due to the entrance region of thermal field. In the downstream region on the adiabatic wall, the streak structures can be observed awhile in both cases. Although the temperature variance remains on the adiabatic wall as shown in Fig. 8a, the streak structures of case 1 cannot be almost detected by the employed criterion (h > 0.1 or h < 0.1) on the adiabatic wall. On the other hand, the structures of case 2 are seen to remain on the adiabatic wall. This is because the turbulent thermal boundary layer is developing so that the temperature variance is also distributed away from the wall. 3.2. Evaluations of turbulence model predictions The predictions of both cases by LES (Inagaki, 2011; Inagaki et al., 2011) and the two-equation heat transfer model of RANS Hattori and Nagano (1998) are demonstrated. In order to evaluate the prediction of velocity field, the streamwise mean velocity, Reynolds shear stress and turbulent energy the LES and RANS predicted are shown in Fig. 10 as compared with DNS results. Both the LES and RANS give adequate predictions for the velocity field, where the prediction of LES indicates the Reynolds-averaged GS component, and since the Reynolds-averaged SGS components are very small, the components are omitted in the predictions as indicated below. Fig. 11 shows comparisons between DNS and predictions for variations of wall temperature. The predictions of both LES and
H. Hattori et al. / International Journal of Heat and Fluid Flow 41 (2013) 34–44
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RANS give good agreement with the DNS result in case 1, but slight disagreements are found in case 2. In case 2, the prediction of LES slightly decreases and the RANS gives overprediction a little as compared with the DNS result. The predictions tend to be almost the same in both cases, i.e., the prediction of LES is smaller than that of RANS. In order to evaluate the predictions of mean temperature near the wall in detail, comparisons of the predicted mean temperature with DNS results are indicated in Fig. 12. The LES properly predicts the distributions of mean temperature DNS obtained on the whole region in case 1, but the RANS slightly gives underprediction. In case 2, both LES and RANS give good predictions of mean temperature near the wall on the heated wall. On the adiabatic wall, the LES slightly under-predicts the DNS results in the downstream region, and the RANS slightly over-predicts the DNS results. Thus, both LES and RANS give slight different prediction for the variation of wall temperature. The prediction of wall-normal turbulent heat flux is important to obtain the adequate profile of mean temperature. Thus, the predicted wall-normal turbulent heat fluxes are shown in Fig. 13 as compared with DNS results. The tendencies in the distributions of wall-normal turbulent heat flux are properly captured by LES and RANS in both cases. In case 1, since the wall-normal turbulent heat fluxes of LES are properly predicted as shown in Fig. 13a, the mean temperature is also captured. However, as shown in Fig. 4a, the convection term also contributes the profile of mean temperature due to the entrance region of thermal field. Thus, it seems that the overprediction of wall-normal turbulent heat fluxes by RANS around the adiabatic starting point does not affect the profile of mean temperature. In case 2, the slight quantitative differences
of predictions by both LES and RANS may give inadequately-predicted mean temperatures near the wall as shown in Fig. 12b. Here we discuss how the streamwise turbulent heat fluxes are the important turbulent quantity for the variation of thermal field in the streamwise direction, where the streamwise turbulent heat flux may slightly influence the distribution of mean temperature as shown in Figs. 4 and 5 in the present cases. Note that the twoequation model cannot adequately predict the streamwise turbulent heat flux due to the principle of its modelling. It is obvious that RANS hardly predicts the turbulent heat flux, but the mean temperature is adequately predicted by RANS as shown in Fig. 12. Thus, in the present cases, the streamwise turbulent heat flux does not seem to very much influence the distribution of mean temperature. However, as mentioned above, it is important to obtain the streamwise turbulent heat flux by the model for the variation of thermal field in the streamwise direction or the heated complex wall shape. Therefore, the evaluation results for streamwise turbulent heat flux are shown in Fig. 14, in which the LES can almost capture the DNS results in both cases and the RANS can hardly predict the streamwise turbulent heat flux. In case 2, however, the overpredictions of LES are found on the adiabatic wall in the downstream region. Fig. 15 shows comparisons of the predicted temperature variances with DNS results. The LES properly predicts temperature variance of DNS results, in which the near-wall distributions of temperature variance are adequately predicted as shown in Fig. 15. On the other hand, the near-wall predictions of RANS are not only larger than the DNS results but also the rapid decrease of temperature variance is predicted near the wall on the adiabatic
Fig. 13. Comparisons of wall-normal turbulent heat fluxes predicted with DNS results.
Fig. 14. Comparisons of streamwise turbulent heat fluxes predicted with DNS results.
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wall in both cases. Therefore, in order to investigate the prediction of RANS in detail, the budgets of transport equation of temperature variance RANS predicted are shown in Figs. 16 and 17. Since the evaluated RANS is modelled to adequately predict the budget of temperature variance and its dissipation-rate, the comparison with the budget of DNS results is worth evaluating for the prediction performance of the turbulence model. The transport equation of temperature variance is given as follows:
Uj
Fig. 15. Comparisons of temperature variances predicted with DNS results.
@h2 @ 2 h2 @uj h2 @H @h @h ¼a 2uj h 2a @xj @xj @xj @xj @xj @xj @xj
ð5Þ
where, the first term of the left-hand side is the convection, the first term of the right-hand side is the molecular diffusion, the second is the turbulent diffusion, the third is the production, and the last term is the dissipation, respectively. In case 1, the predicted budget qualitatively agrees with the DNS results, but the near-wall dissipation and production are over-predicted on the heated wall. On the adiabatic wall, it is found from the predicted budget that the rapid decrease of temperature variance RANS predicted is due to the overprediction of dissipation near the wall. As for case 2, the predicted budget is in good agreement with the budget of DNS result as shown Fig. 17a. In the immediate aftermath of the adiabatic starting point (x/d2,in = 40), the predicted budget gives a good prediction of the budget excluding the streamwise production, 2uhð@ H=@xÞ, which works on the loss side, so the temperature variance is overpredicted as shown in Fig. 15b. Downstream of the adiabatic wall region, the dissipation is also overpredicted. Thus, the rapid decrease of temperature variance near the wall is predicted by RANS. Therefore, these evaluations of turbulence model may suggest an improvement of the modelled equation of dissipation-rate of temperature variance for the prediction of various wall thermal conditions.
Fig. 16. Budgets of temperature variance (Case 1); symbols: DNS, lines: prediction by RANS.
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Fig. 17. Budgets of temperature variance (Case 2); symbols: DNS, lines: prediction by RANS.
4. Conclusions DNSs of spatially developing turbulent boundary layer of both velocity and thermal fields are carried out, in which the wall heating suddenly vanishes in the downstream region, i.e., the wall is heated by a constant temperature condition followed by an adiabatic condition. In this study, two types of thermal field with different wall boundary conditions are investigated via DNS. DNS clearly shows the statistics and structures of thermal field in the present situations. Also, the performances of LES and two-equation heat transfer model of RANS are evaluated using DNS results, in which it is indicated clearly that the prediction gives almost satisfactory results. The improvement of the model prediction, however, is needed in the near-wall region in order to obtain the exact prediction of complex heat transfer phenomena such as in the present cases.
Acknowledgement This research was supported by a Grant-in-Aid for Scientific Research (C), No. 23560225, from the Japan Society for the Promotion of Science (JSPS).
References Antonia, R.A., Danh, H.Q., Prabhu, A., 1977. Response of a turbulent boundary layer to a step change in surface heat flux. Journal of Fluid Mechanics 80, 153–177. Browne, L.W.B., Antonia, R.A., 1981. Calculation of a turbulent boundary layer downstream of a sudden decrease in surface heat flux or wall temperature. In: Proceedings of 3rd Symposium on Turbulent Shear Flows, pp. 10.18–10.22. Charnay, G., Schon, P., Alcaraz, E., Mathieu, J., 1979. Thermal characteristics of a turbulent boundary layer with an inversion of wall heat flux. Turbulent Shear Flows 1, 104–118.
Hattori, H., Nagano, Y., 1998. Rigorous formulation of two-equation heat-transfer model of turbulence using direct simulations. Numerical Heat Transfer, Part B 33, 153–180. Hattori, H., Nagano, Y., 2004. Direct numerical simulation of turbulent heat transfer in plane impinging jet. International Journal of Heat and Fluid Flow 25, 749– 758. Hattori, H., Nagano, Y., 2010. Investigation of turbulent boundary layer over forward-facing step via direct numerical simulation. International Journal of Heat and Fluid Flow 31, 284–294. Hattori, H., Nagano, Y., 2012. Structures and mechanism of heat transfer phenomena in turbulent boundary layer with separation and reattachment via DNS. International Journal of Heat and Fluid Flow 37, 81–92. Hattori, H., Morita, A., Nagano, Y., 2006. Nonlinear eddy diffusivity models reflecting buoyancy effect for wall shear flows and heat transfer. International Journal of Heat and Fluid Flow 27, 671–683. Hattori, H., Houra, T., Nagano, Y., 2007. Direct numerical simulation of stable and unstable turbulent thermal boundary layers. International Journal of Heat and Fluid Flow 28, 1262–1271. Hattori, H., Ohiwa, N., Kozuka, M., Nagano, Y., 2009. Improvement of nonlinear eddy diffusivity model for rotational turbulent heat transfer at various rotating axes. Fluid Dynamics Research 41, 012402 (26pages). Hattori, H., Yamada, S., Houra, T., 2012. DNS study for effects of suddenly-vanishing wall heating in turbulent boundary layer. Journal of Thermal Science and Technology 7 (1), 313–321. Houra, T., Nagano, Y., 2008. Spatio-temporal turbulent structures of thermal boundary layer subjected to non-equilibrium adverse pressure gradient. International Journal of Heat and Fluid Flow 29, 591–601. Inagaki, M., 2011. A new wall-damping function for large eddy simulation employing Kolmogorov velocity scale. International Journal of Heat and Fluid Flow 32 (1), 26–40. Inagaki, M., Kondoh, T., Nagano, Y., 2005. A mixed-time-scale SGS model with fixed model-parameters for practical LES. Journal of Fluids Engineering 127 (1), 1–13. Inagaki, M., Hattori, H., Nagano, Y., 2011. A mixed-timescale SGS model for thermal field at various Prandtl numbers. International Journal of Heat and Fluid Flow 34, 47–61. Kong, H., Choi, H., Lee, J.S., 2000. Direct numerical simulation of turbulent thermal boundary layers. Physics of Fluids 12, 2555–2568. Lund, T., Wu, X., Squires, K., 1998. Generation of turbulent inflow data for spatiallydeveloping boundary layer simulation. Journal of Computational Physics 140, 233–258. Nagano, Y., Hattori, H., Abe, K., 1997. Modeling the turbulent heat and momentum transfer in flows under different thermal conditions. Fluid Dynamics Research 20, 127–142.
44
H. Hattori et al. / International Journal of Heat and Fluid Flow 41 (2013) 34–44
Reynolds, W.C., Kays, W.M., Kline, S.J., 1958a. Heat transfer in the incompressible boundary layer, I – Constant wall temperature. NASA 12-3-58W. Reynolds, W.C., Kays, W.M., Kline, S.J., 1958b. Heat transfer in the incompressible boundary layer, III – Arbitrary wall temperature and NASA MEMO, III-12-3-58W.
turbulent MEMO, Iturbulent heat flux.
Subramanian, C.S., Antonia, R.A., 1981a. Effect of reynolds number on a slightly heated turbulent boundary layer. Journal of Fluid Mechanics 176, 265–281. Subramanian, C.S., Antonia, R.A., 1981b. Response of a turbulent boundary layer to a sudden decrease in wall heat flux. Journal of Heat and Mass Transfer 24, 1641– 1647.
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International Journal of Heat and Fluid Flow journal homepage: www.elsevier.com/locate/ijhff
DNS, LES and RANS of turbulent heat transfer in boundary layer with suddenly changing wall thermal conditions Hirofumi Hattori a,⇑, Shohei Yamada a, Masahiro Tanaka a, Tomoya Houra a, Yasutaka Nagano a,b a b
Department of Mechanical Engineering, Nagoya Institute of Technology, Gokiso-cho, Showa-ku, Nagoya 466-8555, Japan Department of Research, Nagoya Industrial Science Research Institute, Yotsuya-dori 1-13, Chikusa-ku, Nagoya 464-0819, Japan
a r t i c l e
i n f o
Article history: Received 25 October 2012 Received in revised form 26 February 2013 Accepted 26 March 2013 Available online 1 May 2013 Keywords: Turbulent boundary layer Heat transfer Turbulent structure DNS Wall thermal condition LES RANS
a b s t r a c t The objectives of this study are to investigate a thermal field in a turbulent boundary layer with suddenly changing wall thermal conditions by means of direct numerical simulation (DNS), and to evaluate predictions of a turbulence model in such a thermal field, in which DNS of spatially developing boundary layers with heat transfer can be conducted using the generation of turbulent inflow data as a method. In this study, two types of wall thermal condition are investigated using DNS and predicted by large eddy simulation (LES) and Reynolds-averaged Navier–Stokes equation simulation (RANS). In the first case, the velocity boundary layer only develops in the entrance of simulation, and the flat plate is heated from the halfway point, i.e., the adiabatic wall condition is adopted in the entrance, and the entrance region of thermal field in turbulence is simulated. Then, the thermal boundary layer develops along a constant temperature wall followed by adiabatic wall. In the second case, velocity and thermal boundary layers simultaneously develop, and the wall thermal condition is changed from a constant temperature to an adiabatic wall in the downstream region. DNS results clearly show the statistics and structure of turbulent heat transfer in a constant temperature wall followed by an adiabatic wall. In the first case, the entrance region of thermal field in turbulence can be also observed. Thus, both the development and the entrance regions in thermal fields can be explored, and the effects upstream of the thermal field on the adiabatic region are investigated. On the other hand, evaluations of predictions by LES and RANS are conducted using DNS results. The predictions of both LES and RANS almost agree with the DNS results in both cases, but the predicted temperature variances near the wall by RANS give different results as compared with DNS. This is because the dissipation rate of temperature variance is difficult to predict by the present RANS, which is found by the evaluation using DNS results. Ó 2013 Elsevier Inc. All rights reserved.
1. Introduction There are a number of experimental, numerical and theoretical investigations of the turbulent boundary layer with heat transfer (Reynolds et al., 1958a; Subramanian and Antonia, 1981a; Kong et al., 2000; Hattori et al., 2007; Houra and Nagano, 2008). In general, wall boundary condition of thermal field can be considered in three cases: a constant wall temperature, a constant heat flux wall and an adiabatic wall. Although the above-mentioned investigations have dealt with a simple thermal wall condition so as to know a fundamental turbulent heat transfer phenomena, the wall condition of a thermal field often changes in practical thermal fields. Thus, the present study deals with the step change of wall thermal conditions in the turbulent boundary layer. Especially, the thermal field of boundary layer whose heated wall condition suddenly vanishes in the downstream region, i.e., the wall heated ⇑ Corresponding author. Tel./fax: +81 52 735 5359. E-mail address: [email protected] (H. Hattori). 0142-727X/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.ijheatfluidflow.2013.03.014
by a constant temperature followed by an adiabatic condition, is investigated by means of DNS. These investigations were experimentally and numerically conducted (Reynolds et al., 1958b; Charnay et al., 1979; Antonia et al., 1977; Subramanian and Antonia, 1981b; Browne and Antonia, 1981; Nagano et al., 1997; Hattori and Nagano, 1998), but the express turbulent heat transfer statistics and structures close to the wall were difficult to observe. The detailed investigations and observations of the near-wall turbulent heat transfer phenomena have relied increasingly on DNS (Hattori et al., 2007; Hattori and Nagano, 2010; Hattori and Nagano, 2004), because DNS can provide the in-depth data of turbulent statistics including high-order moments near the wall, though there are limitations of Reynolds and Prandtl numbers. Hence, the step change of wall temperature is simulated using DNS (Hattori et al., 2012), in which the turbulent boundary layer on the flat plate with step change of wall temperature and turbulent boundary layer over forward-facing step with the heated wall in front of step and the unheated wall on the step are simulated, and the detailed characteristics of turbulent heat transfer in such fields are investigated.
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Nomenclature cp k P Pr qj qw Red2;in
specific heat at constant pressure turbulence energy, ¼ ui ui =2 mean static pressure Prandtl number, =m/a SGS heat flux or turbulent heat flux wall heat flux Reynolds number based on free stream velocity and momentum thickness at the inlet of the driver part, ¼ U 0 d2;in =m t time U; V; W mean velocity in x-, y- and z-directions, respectively Ui mean velocity in xi-direction U0 free stream mean velocity u,v,w instantaneous velocity or turbulent fluctuation in x-, yand z-directions, respectively ui instantaneous velocity or turbulent fluctuation in xidirection pffiffiffiffiffiffiffiffiffiffiffiffi us friction velocity, ¼ sw =q xc adiabatic starting point xi Cartesian coordinate in i-direction
These simulations are carried out under the simultaneous developing boundary layer of both velocity and thermal field. On the other hand, the case of the developed boundary layer of velocity field and the entrance region of thermal field is also interesting for the study of turbulent heat transfer problems. Therefore, in order to obtain detailed knowledge of a turbulent thermal field around the point where the thermal boundary condition is changed, i.e., a constant temperature wall followed by an adiabatic wall condition, DNSs are carried out, in which the entrance region of thermal field in turbulence is also simulated. On the other hand, the turbulence models of thermal field in the Reynolds-averaged Navier–Stokes equation simulation (RANS) and the large eddy simulation (LES) have been improved recently using DNS results so as to modify the near-wall predictions (Hattori et al., 2009; Hattori et al., 2006; Hattori and Nagano, 1998; Inagaki et al., 2011), because heat is conducted and transferred usually from and near the wall in engineering problems. Thus, one of the topics in this study is the evaluation of recent turbulent heat transfer models using the present DNS in such a thermal field.
x, y, z
Cartesian coordinate in streamwise, wall-normal and spanwise directions, respectively
Greek symbols a molecular diffusivity for heat aSGS eddy diffusivities of SGS component for heat at eddy diffusivity for heat d2 momentum thickness dij Kronecker delta m kinematic viscosity mSGS eddy diffusivities of SGS component for momentum mt eddy diffusivity for momentum q density sij SGS stress tensor or Reynolds stress sw wall shear stress H mean temperature H0 free stream mean temperature Hw wall temperature h instantaneous temperature or temperature fluctuation h2 temperature variance
d2,in, and the temperature difference between the wall, Hw and the free stream, H0 , DH, at the inlet of the driver part, i.e., the dimensionless temperature is defined as h ¼ ðh Hw Þ=ðH0 Hw Þ, because the constant wall temperature is mainly set for the wall thermal condition. Red2;in is Reynolds number based on the free stream velocity and the momentum thickness at the inlet of the driver part, and Pr(=m/a) is a Prandtl number. In case of DNS, turbulent stress, sij, and turbulent heat flux, qj in Eqs. (2) and (3) are equal to zero, and ðÞ means an instantaneous value in equations. The filtered turbulent quantities are used in equations for the LES (Inagaki et al., 2005; Inagaki, 2011; Inagaki i u j et al., 2011), and ðÞ means a GS component. Thus, sij ¼ ui uj u
2. Numerical procedure The governing equations used in DNS, LES and RANS are the Navier–Stokes equation without buoyancy, the continuity equation for the velocity field, and the energy equation for the thermal field, in which incompressibility is assumed as follows:
i @u ¼0 @xi i i @u @u @p @ 1 @u j i ¼ þu þ sij @xi @xj Red2;in @xj @t @xj ! @ h @ h @ 1 @ h j þu ¼ qj @t @xj @xj Red2;in Pr @xj
ð1Þ ! ð2Þ ð3Þ
where the Einstein summation convention applies to repeated indi i is the dimensionless velocity component in xi direction, ces. u h is is the dimensionless pressure, t the dimensionless temperature, p is the dimensionless time, and xi is the dimensionless spatial coordinate in the i direction, respectively. All equations are non-dimensionalized by the free stream velocity, U 0 , the momentum thickness,
Fig. 1. Flow fields, wall conditions and coordinate system.
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j is an SGS heat flux, which hu is an SGS stress tensor, and qj ¼ huj represent the effect of SGS components, and should be modelled. In case of RANS, since the Reynolds-averaged turbulent quantities are employed in equation, ðÞ means an ensemble-average or a timeaverage value. Therefore, sij ¼ ui uj is Reynolds stress, and qj ¼ uj h is the turbulent heat fluxIn the present study, the gradient diffusion type model is used for the modellings of sij = 2mSGSSij + (2/ 3)dijk (LES), sij = 2mtSij + (2/3)dijk (RANS), qj ¼ aSGS ð@ h=@xj Þ (LES) and qj ¼ at ð@ h=@xj Þ (RANS), where mSGS and aSGS are the eddy diffusivities of SGS component for momentum and heat, respectively, mt and at are the eddy diffusivities for momentum and heat, respectively, k is a turbulence energy or a turbulence energy of SGS i =@xj þ @ u j =@xi Þ=2. In order to solve mSGS component, and Sij ¼ ð@ u and aSGS, the modified mixed-timescale model (MTSn) (Inagaki, 2011) and the mixed-timescale model for thermal field (MTSt) (Inagaki et al., 2011) are adopted for the LES. The characteristic of employed model in LES is the time scale for the eddy diffusivities for momentum and heat, i.e., since the characteristic time scale of SGS is adequately determined by the model in both velocity and
thermal fields, the model gives good predictions of the wall turbulent flows with heat transfer. As for the RANS, mt and at are solved using the two-equation heat transfer model (Hattori and Nagano, 1998). This RANS model can properly predict turbulent quantities including the budget of transport equations in the wall turbulent flows with heat transfer, i.e., the budgets of transport equations for temperature variance and its dissipation rate are adequately predicted (Hattori and Nagano, 1998). Flow fields, wall conditions and coordinate system of the present simulations are shown in Fig. 1. In order to simulate the spatially developing turbulent boundary layer, DNS and LES have the driver and the main simulation parts as shown in Fig. 1, in which the turbulent boundary layer of the velocity is generated in the driver part, and the main simulation part receives turbulent inflow data from the driver part. Thus, the spatially developing boundary
Table 1 Computational method and conditions. Grid system
Staggered grid
Coupling algorithm Time advancement Spatial schemes
Fractional step method Adams–Bashforth method (non-linear terms) Crank–Nicolson method (linear terms) 2nd-order central difference
Parameters Reynolds number
Red2;in ¼ U 0d2;in =m ¼ 300
Prandtl number
Pr = m/a = 0.71
Domain size (x y z) Driver Main
100d2,in 30d2,in 40d2,in 200d2,in 30d2,in 40d2,in
Grid points (x y z) DNS Driver Main
192 128 128 384 128 128
LES
Driver Main
96 64 64 192 64 64
Spatial resolutions (inlet of main simulation part) DNS Dx + 8.3 Dy + 0.043 11 + Dz 5 LES
Dx + Dy + Dz +
16.6 0.086 22 10
Fig. 2. Variations of wall temperature.
Fig. 3. Distributions of mean temperature near the wall.
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layer is achieved without the periodic boundary condition in the streamwise direction (Lund et al., 1998; Kong et al., 2000; Hattori et al., 2007; Hattori and Nagano, 2012). The conditions of thermal field are established in two cases as indicated in Fig. 1. In the first case (hereinafter referred to as case 1), the velocity boundary layer only develops in the driver part, and the flat plate is only heated in the main simulation part, i.e., the adiabatic wall condition is adopted in the driver part. In the main simulation part of case 1, the thermal boundary layer develops along a constant temperature wall followed by an adiabatic wall. Note that the adiabatic wall condition is set in the entrance of the main simulation part in order to avoid the numerical error due to a sudden increase of wall temperature. In the second case (hereinafter referred to as case 2), the velocity and thermal boundary layers simultaneously develop in the driver part, and the wall thermal condition is changed from the constant temperature to adiabatic in the main simulation part. In Fig. 1, the distance, xc, indicates the starting line of the adiabatic wall (the starting line of the adiabatic wall is x/d2,in = 35.4 in both cases). In case 1, since the boundary layer of thermal field develops only in the main simulation part, the boundary layer of thermal field may become very thinner much than the boundary layer of velocity field in the case, i.e., the entrance region of thermal field in turbulence is simulated. Thus, in case 1, it is possible to observe the entrance region of thermal field in the turbulent boundary layer. As for the RANS, the same conditions of DNS and LES are carried out by RANS in the main simulation part of the two-dimensional field (Hattori and Nagano, 1998). The grid numbers DNS used are: Nx Ny Nz = 192 128 128 for the driver part, and Nx Ny Nz = 384 128 128 for the main simulation part. The grid numbers of LES are arranged to be half those of DNS. The boundary conditions for the velocity
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field are the non-slip conditions on the walls, and @u/@y = 0, @w/ @y = 0, @v/@y = (@u/@x + @w/@z) on the upper boundary (free stream). At the outlet of both parts, the convective boundary conditions are applied, and periodic boundary conditions are used in the spanwise direction for both the velocity and thermal fields. Reynolds number, Red2;in ¼ U 0 d2;in =m, is set 300, and Prandtl number, Pr = m/a, is 0.71. Computational method and conditions are summarised in Table 1.
3. Results and discussion 3.1. DNS investigation of turbulent heat transfer in boundary layer with suddenly changing wall thermal conditions In order to demonstrate the variation of temperature difference by the difference of condition of thermal boundary layer in the upstream, the variation of wall temperature which is defined as the difference between the free stream and wall are shown in Fig. 2. It can be observed that the wall temperature of case 1 drops faster than that of case 2, because the undeveloped boundary layer of thermal field encounters the adiabatic wall condition in case 1. In order to observe behaviour of turbulent thermal field on both the isothermal and the adiabatic walls in detail, Fig. 3 shows the distributions of mean temperature near the wall. Developing the thermal boundary layer on the heated wall can be observed in case 1 as shown in Fig. 3a, but the thermal boundary layer of case 2 is developed very little due to the generating boundary layer which is developed simultaneously with velocity field in the driver part as indicated in Fig. 3b. In the adiabatic wall region, the mean temperature rapidly decreases near the wall region in case 1, but the
Fig. 4. Budgets of energy equation (Case 1).
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mean temperature of case 2 is maintained higher than in case 1, which are clearly shown in Fig. 3c. Moreover, the mean temperature diffuses toward the outer region on the adiabatic wall in case 1, but this phenomenon is difficult to observe in case 2 as shown in Fig. 3c. Figs. 4 and 5 show the budget of Reynolds-averaged energy equation in the 2-dimensional field as given below:
! @H @H 1 @2H @2H @ @ 0 ¼ U V þ þ 2 uh v h @x @y @x @y Red2;in Pr @x2 @y
ð4Þ
Comparing the budget of case 1 with that of case 2, the convection term in x-direction obviously affects distribution of mean temperature in case 1 due to the entrance region of thermal field. At the immediate aftermath of the adiabatic starting point (x/d2, in = 40), the convection term in x-direction works in the loss side near the wall in both cases. Conversely, the thermal diffusion term in ydirection works in the gain side near the wall. In case 1, since the convection term in x-direction also contributes in the gain side away from the wall, the mean temperature diffuses toward the outer region as mentioned above. In the region of gradual variation of wall temperature (x/d2,in = 100), the convection term in x-direction clearly affects the distribution of mean temperature of both cases in the outer region as shown in Figs. 4 and 5 and Fig. 5d. Thus, it is also observed in case 2 that the mean temperature distributes toward the outer region at the region of gradual variation of wall temperature as demonstrated in Fig. 3b. Investigating the wall-normal turbulent heat flux as shown in Fig. 6, the wall-normal turbulent heat fluxes of both cases decrease clearly in the near-wall region on the adiabatic wall. Also, the increases in wall-normal turbulent heat fluxes away from the wall in case 1 are clearly found due to the occurrence of a mean temper-
Fig. 6. Near-wall distributions of wall-normal turbulent heat fluxes.
0.002
-0.002
Fig. 5. Budgets of energy equation (Case 2).
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ature gradient by expansion of a high mean temperature region near the outer region as indicated in Fig. 3a, i.e., the turbulent heat transfer in the entrance region of thermal field can be observed. In the entrance region of thermal field, the peak of wall-normal turbulent heat flux which gradually moves over away from the wall, is obviously observed in Fig. 6a. On the adiabatic wall, the peak of wall-normal turbulent heat flux also moves over away from the wall in case 1, but this phenomenon is difficult to observe in
Fig. 9. Thermal streaks in constant wall temperature followed by adiabatic wall; red h > 0.1, blue h < 0.1. Fig. 7. Distributions of wall-normal turbulent heat fluxes near the wall.
Fig. 8. Near-wall distributions of temperature variances.
Fig. 10. Comparisons of turbulence quantities of velocity field predicted with DNS results.
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Fig. 11. Comparisons of variation of wall temperature predicted with DNS results. Fig. 12. Comparisons of mean temperature predicted with DNS results.
case 2. In case 2, the distributions of wall-normal heat flux hardly changes on the heated wall. The peak of wall-normal heat flux moves over away from the wall on the adiabatic wall, and the maintained profiles of wall-normal turbulent heat flux in the outer region are observed, but a slight variation is found as shown in Fig. 6b. On the other hand, the streamwise turbulent heat fluxes of both cases are shown in Fig. 7. It is well-known that the streamwise turbulent heat flux hardly affects the distribution of mean temperature on the heated wall. The turbulent heat flux, however, slightly influences the adiabatic wall as demonstrated in Figs. 4 and 5. Also here, the shift in the peak of streamwise turbulent heat flux is evidently observed in case 1, but the shift of the peak is not found in case 2. On the adiabatic wall, the spreading toward the outer region and decrease of streamwise turbulent heat flux are discovered in case 1, but it is difficult to observe in case 2 as shown in Fig. 7b. On the other hand, the distributions of temperature variances are shown in Fig. 8. Since the temperature variance is allowed on the wall in the adiabatic wall region, it is obvious that the temperature variance increases near the wall region in both cases, in which it seems that the increase of temperature variance near the wall is also caused by the decrease in the dissipation rate of the temperature variance as shown in the budget of temperature variance of Figs. 16 and 17. In the outer region, the temperature variances remarkably diffuse in common with the distributions of wall-normal turbulent heat flux in case 1. Also, the distributions of temperature variance in case 2 are similar to that of the wallnormal turbulent heat flux. On the adiabatic wall, although it is obvious that a temperature variance is hardly produced exists, i.e., the mean temperature gradient and wall-normal turbulent heat flux hardly distribute near the wall; the temperature variance insistently remains consistent for the reason mentioned above.
As for the turbulent structure in the thermal field Fig. 9 shows the thermal streaks of both cases in the constant wall temperature followed by the adiabatic wall. In case 2, a sort of messy streak structure can be observed due to the developed thermal boundary layer on the heated wall. In contrast, a gracile streak structure is found in case 1 due to the entrance region of thermal field. In the downstream region on the adiabatic wall, the streak structures can be observed awhile in both cases. Although the temperature variance remains on the adiabatic wall as shown in Fig. 8a, the streak structures of case 1 cannot be almost detected by the employed criterion (h > 0.1 or h < 0.1) on the adiabatic wall. On the other hand, the structures of case 2 are seen to remain on the adiabatic wall. This is because the turbulent thermal boundary layer is developing so that the temperature variance is also distributed away from the wall. 3.2. Evaluations of turbulence model predictions The predictions of both cases by LES (Inagaki, 2011; Inagaki et al., 2011) and the two-equation heat transfer model of RANS Hattori and Nagano (1998) are demonstrated. In order to evaluate the prediction of velocity field, the streamwise mean velocity, Reynolds shear stress and turbulent energy the LES and RANS predicted are shown in Fig. 10 as compared with DNS results. Both the LES and RANS give adequate predictions for the velocity field, where the prediction of LES indicates the Reynolds-averaged GS component, and since the Reynolds-averaged SGS components are very small, the components are omitted in the predictions as indicated below. Fig. 11 shows comparisons between DNS and predictions for variations of wall temperature. The predictions of both LES and
H. Hattori et al. / International Journal of Heat and Fluid Flow 41 (2013) 34–44
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RANS give good agreement with the DNS result in case 1, but slight disagreements are found in case 2. In case 2, the prediction of LES slightly decreases and the RANS gives overprediction a little as compared with the DNS result. The predictions tend to be almost the same in both cases, i.e., the prediction of LES is smaller than that of RANS. In order to evaluate the predictions of mean temperature near the wall in detail, comparisons of the predicted mean temperature with DNS results are indicated in Fig. 12. The LES properly predicts the distributions of mean temperature DNS obtained on the whole region in case 1, but the RANS slightly gives underprediction. In case 2, both LES and RANS give good predictions of mean temperature near the wall on the heated wall. On the adiabatic wall, the LES slightly under-predicts the DNS results in the downstream region, and the RANS slightly over-predicts the DNS results. Thus, both LES and RANS give slight different prediction for the variation of wall temperature. The prediction of wall-normal turbulent heat flux is important to obtain the adequate profile of mean temperature. Thus, the predicted wall-normal turbulent heat fluxes are shown in Fig. 13 as compared with DNS results. The tendencies in the distributions of wall-normal turbulent heat flux are properly captured by LES and RANS in both cases. In case 1, since the wall-normal turbulent heat fluxes of LES are properly predicted as shown in Fig. 13a, the mean temperature is also captured. However, as shown in Fig. 4a, the convection term also contributes the profile of mean temperature due to the entrance region of thermal field. Thus, it seems that the overprediction of wall-normal turbulent heat fluxes by RANS around the adiabatic starting point does not affect the profile of mean temperature. In case 2, the slight quantitative differences
of predictions by both LES and RANS may give inadequately-predicted mean temperatures near the wall as shown in Fig. 12b. Here we discuss how the streamwise turbulent heat fluxes are the important turbulent quantity for the variation of thermal field in the streamwise direction, where the streamwise turbulent heat flux may slightly influence the distribution of mean temperature as shown in Figs. 4 and 5 in the present cases. Note that the twoequation model cannot adequately predict the streamwise turbulent heat flux due to the principle of its modelling. It is obvious that RANS hardly predicts the turbulent heat flux, but the mean temperature is adequately predicted by RANS as shown in Fig. 12. Thus, in the present cases, the streamwise turbulent heat flux does not seem to very much influence the distribution of mean temperature. However, as mentioned above, it is important to obtain the streamwise turbulent heat flux by the model for the variation of thermal field in the streamwise direction or the heated complex wall shape. Therefore, the evaluation results for streamwise turbulent heat flux are shown in Fig. 14, in which the LES can almost capture the DNS results in both cases and the RANS can hardly predict the streamwise turbulent heat flux. In case 2, however, the overpredictions of LES are found on the adiabatic wall in the downstream region. Fig. 15 shows comparisons of the predicted temperature variances with DNS results. The LES properly predicts temperature variance of DNS results, in which the near-wall distributions of temperature variance are adequately predicted as shown in Fig. 15. On the other hand, the near-wall predictions of RANS are not only larger than the DNS results but also the rapid decrease of temperature variance is predicted near the wall on the adiabatic
Fig. 13. Comparisons of wall-normal turbulent heat fluxes predicted with DNS results.
Fig. 14. Comparisons of streamwise turbulent heat fluxes predicted with DNS results.
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wall in both cases. Therefore, in order to investigate the prediction of RANS in detail, the budgets of transport equation of temperature variance RANS predicted are shown in Figs. 16 and 17. Since the evaluated RANS is modelled to adequately predict the budget of temperature variance and its dissipation-rate, the comparison with the budget of DNS results is worth evaluating for the prediction performance of the turbulence model. The transport equation of temperature variance is given as follows:
Uj
Fig. 15. Comparisons of temperature variances predicted with DNS results.
@h2 @ 2 h2 @uj h2 @H @h @h ¼a 2uj h 2a @xj @xj @xj @xj @xj @xj @xj
ð5Þ
where, the first term of the left-hand side is the convection, the first term of the right-hand side is the molecular diffusion, the second is the turbulent diffusion, the third is the production, and the last term is the dissipation, respectively. In case 1, the predicted budget qualitatively agrees with the DNS results, but the near-wall dissipation and production are over-predicted on the heated wall. On the adiabatic wall, it is found from the predicted budget that the rapid decrease of temperature variance RANS predicted is due to the overprediction of dissipation near the wall. As for case 2, the predicted budget is in good agreement with the budget of DNS result as shown Fig. 17a. In the immediate aftermath of the adiabatic starting point (x/d2,in = 40), the predicted budget gives a good prediction of the budget excluding the streamwise production, 2uhð@ H=@xÞ, which works on the loss side, so the temperature variance is overpredicted as shown in Fig. 15b. Downstream of the adiabatic wall region, the dissipation is also overpredicted. Thus, the rapid decrease of temperature variance near the wall is predicted by RANS. Therefore, these evaluations of turbulence model may suggest an improvement of the modelled equation of dissipation-rate of temperature variance for the prediction of various wall thermal conditions.
Fig. 16. Budgets of temperature variance (Case 1); symbols: DNS, lines: prediction by RANS.
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Fig. 17. Budgets of temperature variance (Case 2); symbols: DNS, lines: prediction by RANS.
4. Conclusions DNSs of spatially developing turbulent boundary layer of both velocity and thermal fields are carried out, in which the wall heating suddenly vanishes in the downstream region, i.e., the wall is heated by a constant temperature condition followed by an adiabatic condition. In this study, two types of thermal field with different wall boundary conditions are investigated via DNS. DNS clearly shows the statistics and structures of thermal field in the present situations. Also, the performances of LES and two-equation heat transfer model of RANS are evaluated using DNS results, in which it is indicated clearly that the prediction gives almost satisfactory results. The improvement of the model prediction, however, is needed in the near-wall region in order to obtain the exact prediction of complex heat transfer phenomena such as in the present cases.
Acknowledgement This research was supported by a Grant-in-Aid for Scientific Research (C), No. 23560225, from the Japan Society for the Promotion of Science (JSPS).
References Antonia, R.A., Danh, H.Q., Prabhu, A., 1977. Response of a turbulent boundary layer to a step change in surface heat flux. Journal of Fluid Mechanics 80, 153–177. Browne, L.W.B., Antonia, R.A., 1981. Calculation of a turbulent boundary layer downstream of a sudden decrease in surface heat flux or wall temperature. In: Proceedings of 3rd Symposium on Turbulent Shear Flows, pp. 10.18–10.22. Charnay, G., Schon, P., Alcaraz, E., Mathieu, J., 1979. Thermal characteristics of a turbulent boundary layer with an inversion of wall heat flux. Turbulent Shear Flows 1, 104–118.
Hattori, H., Nagano, Y., 1998. Rigorous formulation of two-equation heat-transfer model of turbulence using direct simulations. Numerical Heat Transfer, Part B 33, 153–180. Hattori, H., Nagano, Y., 2004. Direct numerical simulation of turbulent heat transfer in plane impinging jet. International Journal of Heat and Fluid Flow 25, 749– 758. Hattori, H., Nagano, Y., 2010. Investigation of turbulent boundary layer over forward-facing step via direct numerical simulation. International Journal of Heat and Fluid Flow 31, 284–294. Hattori, H., Nagano, Y., 2012. Structures and mechanism of heat transfer phenomena in turbulent boundary layer with separation and reattachment via DNS. International Journal of Heat and Fluid Flow 37, 81–92. Hattori, H., Morita, A., Nagano, Y., 2006. Nonlinear eddy diffusivity models reflecting buoyancy effect for wall shear flows and heat transfer. International Journal of Heat and Fluid Flow 27, 671–683. Hattori, H., Houra, T., Nagano, Y., 2007. Direct numerical simulation of stable and unstable turbulent thermal boundary layers. International Journal of Heat and Fluid Flow 28, 1262–1271. Hattori, H., Ohiwa, N., Kozuka, M., Nagano, Y., 2009. Improvement of nonlinear eddy diffusivity model for rotational turbulent heat transfer at various rotating axes. Fluid Dynamics Research 41, 012402 (26pages). Hattori, H., Yamada, S., Houra, T., 2012. DNS study for effects of suddenly-vanishing wall heating in turbulent boundary layer. Journal of Thermal Science and Technology 7 (1), 313–321. Houra, T., Nagano, Y., 2008. Spatio-temporal turbulent structures of thermal boundary layer subjected to non-equilibrium adverse pressure gradient. International Journal of Heat and Fluid Flow 29, 591–601. Inagaki, M., 2011. A new wall-damping function for large eddy simulation employing Kolmogorov velocity scale. International Journal of Heat and Fluid Flow 32 (1), 26–40. Inagaki, M., Kondoh, T., Nagano, Y., 2005. A mixed-time-scale SGS model with fixed model-parameters for practical LES. Journal of Fluids Engineering 127 (1), 1–13. Inagaki, M., Hattori, H., Nagano, Y., 2011. A mixed-timescale SGS model for thermal field at various Prandtl numbers. International Journal of Heat and Fluid Flow 34, 47–61. Kong, H., Choi, H., Lee, J.S., 2000. Direct numerical simulation of turbulent thermal boundary layers. Physics of Fluids 12, 2555–2568. Lund, T., Wu, X., Squires, K., 1998. Generation of turbulent inflow data for spatiallydeveloping boundary layer simulation. Journal of Computational Physics 140, 233–258. Nagano, Y., Hattori, H., Abe, K., 1997. Modeling the turbulent heat and momentum transfer in flows under different thermal conditions. Fluid Dynamics Research 20, 127–142.
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H. Hattori et al. / International Journal of Heat and Fluid Flow 41 (2013) 34–44
Reynolds, W.C., Kays, W.M., Kline, S.J., 1958a. Heat transfer in the incompressible boundary layer, I – Constant wall temperature. NASA 12-3-58W. Reynolds, W.C., Kays, W.M., Kline, S.J., 1958b. Heat transfer in the incompressible boundary layer, III – Arbitrary wall temperature and NASA MEMO, III-12-3-58W.
turbulent MEMO, Iturbulent heat flux.
Subramanian, C.S., Antonia, R.A., 1981a. Effect of reynolds number on a slightly heated turbulent boundary layer. Journal of Fluid Mechanics 176, 265–281. Subramanian, C.S., Antonia, R.A., 1981b. Response of a turbulent boundary layer to a sudden decrease in wall heat flux. Journal of Heat and Mass Transfer 24, 1641– 1647.