- Email: [email protected]
Direct numerical simulation of thermally-stratified turbulent boundary layer subjected to adverse pressure gradient
ARTICLE IN PRESS
JID: HFF
[m5G;June 27, 2016;22:30]
International Journal of Heat and Fluid Flow 0 0 0 (2016) 1–16
Contents lists available at ScienceDirect
International Journal of Heat and Fluid Flow journal homepage: www.elsevier.com/locate/ijheatfluidflow
Direct numerical simulation of thermally-stratified turbulent boundary layer subjected to adverse pressure gradient Hirofumi Hattori∗, Amane Kono, Tomoya Houra Department of Mechanical Engineering, Nagoya Institute of Technology, Gokiso-cho, Showa-ku, Nagoya 466-8555, Japan
a r t i c l e
i n f o
Article history: Available online xxx Keywords: Turbulent boundary layer Adverse pressure gradient Thermal stratification Heat transfer DNS
a b s t r a c t The objective of this study is to investigate and observe turbulent heat transfer structures and statistics in thermally-stratified turbulent boundary layers subjected to a non-equilibrium adverse pressure gradient (APG) by means of direct numerical simulation (DNS). DNSs are carried out under conditions of neutral, stable and unstable thermal stratifications with a non-equilibrium APG, in which DNS results reveal heat transfer characteristics of thermally-stratified non-equilibrium APG turbulent boundary layers. In cases of thermally-stratified turbulent boundary layers affected by APG, heat transfer performances increase in comparison with a turbulent boundary layer with neutral thermal stratification and zero pressure gradient (ZPG). Especially, it is found that the friction coefficient and Stanton number decrease along the streamwise direction due to the effects of stable thermal stratification and APG, but those again increase due to the APG effect in the case of weak stable thermal stratification (WSBL). Thus, the analysis for both the friction coefficient and Stanton number in the case of WSBL with/without APG is conducted using the FIK identity in order to investigate contributions from the transport equations, in which it is found that both Reynolds-shear-stress and the mean convection terms of the friction coefficient and both the wallnormal turbulent heat flux and the spatial development terms of the Stanton number contribute to again increase those values in the case of WSBL with APG. The characteristic turbulent statistics of both the velocity and the thermal fields along streamwise direction are clearly indicated, in which the decrease of log-law profile of streamwise mean velocity which was found by experimental study is also observed in the neutral boundary layer of our DNS. DNS results reveal that the turbulent characteristics of both cases of stable and unstable thermal stratification boundary layers differ with the turbulent characteristics of the neutral boundary layer having APG. © 2016 Elsevier Inc. All rights reserved.
1. Introduction Since a turbulent boundary layer subjected to an adverse pressure gradient (APG) yielded by an increase of pressure in the streamwise direction and also causing flow separation, affects the turbulent structure and heat transfer strongly, a turbulent heat transfer phenomenon of APG flow should be explored for the controls of flow and heat transfer. APG turbulent boundary layer with heat transfer has been revealed by experimental (Houra and Nagano, 2006; Nagano et al., 1993; 1998) and DNS (Araya and Castillo, 2013; Lee and Sung, 2008) studies. On the other hand, a thermally-stratified turbulent boundary layer which can be observed in the atmosphere has been also investigated by experimental (Ohya, 2001) and DNS (Hattori et al., 2014; 2007)
∗
Corresponding author. Tel./fax: +81 52 735 5359. E-mail address: [email protected] (H. Hattori).
studies, which expand the understanding of turbulent structure in a thermally-stratified turbulent boundary layer. Although those studies have led to detailed research on structures of turbulent heat transfer, a study on turbulent boundary layer with overlapping of several influencing factors in turbulent structures such as thermal stratifications and pressure gradients is needed in order to determine the mechanism of more complex turbulent heat transfer. Thus, a turbulent heat transfer subjected to thermal stratification and pressure gradient should be explored as the typical situation. In the present study, in order to investigate and observe effects of adverse pressure gradient for a thermally-stratified turbulent boundary layer, DNS of thermally-stratified turbulent boundary layers subjected to adverse pressure gradient is carried out, where a non-equilibrium APG turbulence flow is assumed and various thermally-stratified turbulent boundary layers with a nonequilibrium APG are simulated in order to reveal statistics and structures in such fields.
http://dx.doi.org/10.1016/j.ijheatfluidflow.2016.05.020 0142-727X/© 2016 Elsevier Inc. All rights reserved.
Please cite this article as: H. Hattori et al., Direct numerical simulation of thermally-stratified turbulent boundary layer subjected to adverse pressure gradient, International Journal of Heat and Fluid Flow (2016), http://dx.doi.org/10.1016/j.ijheatfluidflow.2016.05.020
ARTICLE IN PRESS
JID: HFF 2
[m5G;June 27, 2016;22:30]
H. Hattori et al. / International Journal of Heat and Fluid Flow 000 (2016) 1–16
Nomenclature Cf cp gi Lx , Ly , Lz Nx , Ny , Nz P¯ Pr Reδ2 Reδ2,in
Re2 Re2,in
Riδ2,in
Reδ99 St t U¯ , V¯ u, v, w U¯ i U¯ 0 ui uτ xi x, y, z y+
local friction coefficient = 2τw /(ρU¯ 02 ) specific heat at constant pressure gravitational acceleration nondimensional domain size in streamwise, wallnormal and spanwise directions, respectively grid numbers in streamwise, wall-normal and spanwise directions, respectively mean static pressure Prandtl number, = ν /α Reynolds number based on free stream velocity and local momentum thickness, = U¯ 0 δ2 /ν
wall temperature at the inlet of the driver part wall and free-stream temperature temperature fluctuation local friction temperature, = qw /(ρ c p uτ ) ensemble- or time-averaged value normalization by inner variables (uτ , θ τ , ν )
2. DNS of thermally-stratified boundary layer with APG Assuming that the Boussinesq approximation is approved for the Navier-Stokes equation, the governing equations used in the present DNS are indicated as follows (Hattori et al., 2007):
Reynolds number based on free stream velocity and momentum thickness at the inlet of the driver part, = U¯ 0 δ2,in /ν Reynolds number based on free stream velocity and local enthalpy thickness, = U¯ 0 2 /ν
∂ ui ∂ ui ∂p 1 ∂ 2 ui + uj =− + + δi2 Riδ2,in θ ∂t ∂xj ∂ xi Reδ2,in ∂ x j ∂ x j
(1)
∂ ui =0 ∂ xi
(2)
Reynolds number based on free stream velocity and enthalpy thickness at the inlet of the driver part, = U¯ 0 2,in /ν bulk Richardson number based on free stream velocity, momentum thickness at the inlet of the driver part, = gβδ2,in /U¯ 2
∂θ ∂θ 1 ∂ 2θ + uj = ∂t ∂ x j Pr Reδ2,in ∂ x j ∂ x j
(3)
0
Reynolds number based on free stream velocity and local boundary layer thickness, = U¯ 0 δ99 /ν local Stanton number = qw /[ρ c pU¯ 0 ( w − ∞ )] time mean velocity in x- and y-directions, respectively turbulent fluctuation in x-, y- and z-directions, respectively mean velocity in xi -direction free stream mean velocity turbulent fluctuation in xi -direction √ local friction velocity, = τw /ρ Cartesian coordinate in i-direction Cartesian coordinate in streamwise, wall-normal and spanwise directions, respectively nondimensional distance from wall surface, = uτ y/ν
Greek symbols α molecular diffusivity for heat β coefficient of volume expansion δ 99 local boundary layer thickness defined at location where mean velocity is equal to 99% of free-stream velocity δ1 displacement thickness δ2 momentum thickness δ ij Kronecker delta δt thermal boundary layer thickness defined at location where nondimensional mean temperature [= ¯ − ¯ 0 / )] is equal to 99% of nondimensional ( ¯ ∞− ¯ 0 / )] free-stream temperature [= ( thermal boundary layer thickness defined ¯ = at dimensionless mean temperature, ¯ 0 )/ , is equal to 99%. (θ − ¯ ∞− ¯0 temperature difference, =
2 ν ρ τw ¯
¯0 ¯ w, ¯∞ θ θτ ( ) ( )+
enthalpy thickness kinematic viscosity density wall shear stress mean temperature
where the Einstein summation convention applies to repeated indices, and a comma followed by an index indicates differentiation with respect to the indexed spatial coordinate. ui is the dimensionless velocity component in xi direction, θ is the dimensionless temperature difference, p is the dimensionless pressure, t is the dimensionless time, and xi is the dimensionless spatial coordinate in the i direction, respectively. Reδ2,in = U¯ 0 δ2,in /ν is the Reynolds number based on the free stream velocity and the momentum thickness at the inlet of the driver part, δ 2, in . Note that “the driver part” means the inflow data generator for the inlet boundary of the main simulation part (Hattori et al., 2007). P r = ν /α is the Prandtl number, and Riδ2,in = gβδ2,in /U¯ 02 is the bulk Richardson number based on the free stream velocity, the momentum thickness at the inlet of the driver part, and the temperature difference between a free ¯ 0− ¯ w ). In the governing equations, stream and a wall ( = the dimensionless variables are given using the free stream velocity, U¯ 0 , and the free stream temperature, 0 , at the inlet of the driver part, and the wall temperature, w . The Prandtl number is set to 0.71, assuming the working fluid to be air. The Reynolds number is set to 300, and the Richardson numbers are set to −0.005 (unstable thermal stratification boundary layer: USBL), 0 (neutral thermal stratification boundary layer: NBL), 0.005 (weak stable thermal stratification boundary layer: WSBL) and 0.04 (strong stable thermal stratification boundary layer: SSBL). For efficiently conducting the DNS of thermally-stratified turbulent boundary layers subjected to APG, the computational domain is composed of two parts; one is the driver part where a zero-pressure-gradient (ZPG) flow with an isothermal wall is generated and used as the inflow boundary condition for the main simulation, and the other is the main part where a thermallystratified turbulent boundary layers subjected to adverse pressure gradient are simulated (Hattori et al., 2007; Hattori and Nagano, 2004). A central finite-difference method of second-order accuracy is used to solve the equations of continuity, momentum and energy (Hattori et al., 2007; Hattori and Nagano, 2004), where the computational domain, grid numbers and grid resolutions are listed in Table 1. Since it is considered that the boundary layer thickness remarkably increases due to the effects of APG and unstable thermal stratification in the case of USBL, the domain size of a wall-normal direction is given by double the size of the NBL. In the case of WSBL, the domain size in the streamwise direction is expanded due to the influence of the boundary condition at the
Please cite this article as: H. Hattori et al., Direct numerical simulation of thermally-stratified turbulent boundary layer subjected to adverse pressure gradient, International Journal of Heat and Fluid Flow (2016), http://dx.doi.org/10.1016/j.ijheatfluidflow.2016.05.020
ARTICLE IN PRESS
JID: HFF
[m5G;June 27, 2016;22:30]
H. Hattori et al. / International Journal of Heat and Fluid Flow 000 (2016) 1–16
3
Table 1 Computational conditions of main part. Ri
−0.005 (USBL)
0 (NBL) ∗
dCp /dx Lx × Ly × Lz Nx × Ny × Nz x+in y+in + zin x+out y+out + zout t (U¯ 0 /δ2,in ) Total steps
3.3 × 10 600 × 30 × 40 1152 × 128 × 128
0
7.95 0.042 ∼ 11.3 4.77
−4
6.40 0.34 ∼ 9.12 3.84
0
0.005 (WSBL)
3.3 × 10 600 × 60 × 40 1152 × 160 × 128
13.5 0.048 ∼ 18.2 5.4
−4
0
0.04 (SSBL) −4
3.3 × 10 900 × 30 × 40 1728 × 128 × 128
8.28 0.044 ∼ 11.8 5 12.3 3.50 0.043 ∼ 16.5 0.018 ∼ 5 4.9 2.1 0.01 380,0 0 0
3.12 0.016 ∼ 4.45 1.87
0
3.3 × 10−4 600 × 30 × 40 1152 × 128 × 128
3.12 0.016 ∼ 4.44 1.87
1.47 0.0077 ∼ 2.1 0.88
outlet, where the influence of the boundary condition is mentioned later. The dimensionless time step is 0.01 in each case, and in order to adequately obtain turbulent statistics, 380,0 0 0 calculation steps are needed at least for each case. In order to achieve the APG condition in the free stream, the boundary conditions of the free stream are given as follows:
∂u ∂w ∂v ∂u ∂w =0, = 0, =− − ∂y ∂y ∂y ∂x ∂z U¯ e (x ) = U¯ 0
1 − C p (x )
(4)
(5)
where U¯ e (x ) is a local mean free stream velocity, C p = [P (x ) − P (0 )]/(ρU¯ 0 /2 ) is the local pressure coefficient, P(x) is a local mean pressure in the free stream, and P(0) is the free stream pressure at the inlet. Cp is set to 0 at the inlet of main part of DNS, and Cp is increased along the streamwise direction. Finally, Cp increases with dC p /dx∗ [= dC p /(dx/δ2,in )] = 3.3 × 10−4 in order to achieve the APG flow. For the ZPG flow, Cp (x) is set to 0, i.e., dC p /dx∗ = 0. Note that our DNS deals with the non-equilibrium APG turbulent boundary layer, because the equilibrium condition (Clauser, 1954) does not satisfy the relation, β (x ) = (δ ∗ /τw )(dP¯ /dx )=constant, in DNS with APG condition, where δ ∗ is the displacement thickness. The non-slip condition for velocity field and the isothermal condition for thermal field at the wall, and the convective boundary condition (Hattori et al., 2007) at the outlet and the periodic condition for the spanwise direction are adopted for both velocity and thermal fields. On the other hand, the Reynolds-averaged Navier-Stokes equations of Eq. (1) in the two-dimensional field are given as follows:
U¯
∂ U¯ ¯ ∂ U¯ +V ∂x ∂y ∂ P¯ ∂ 1 ∂ U¯ ∂ 1 ∂ U¯ 2 =− + −u + − uv ∂ x ∂ x Reδ2,in ∂ x ∂ y Reδ2,in ∂ y
∂ V¯ ¯ ∂ V¯ ∂ P¯ ∂ 1 ∂ V¯ ¯ +V =− + − uv U ∂x ∂y ∂ y ∂ x Reδ2,in ∂ x ∂ 1 ∂ V¯ + − v2 + Riδ2,in θ ∂ y Reδ2,in ∂ y
Fig. 1. Local friction coefficients.
(6) 3. Results and discussion
3.1. Fundamental Parameters
(7)
where U¯ and V¯ are mean velocities in the streamwise and wallnormal directions, respectively. The spanwise mean velocity vanishes due to the assumption of 2-dimensional mean flow field. Considering these equations, note that the APG effect mainly affects the momentum equation of streamwise direction and the effect of thermal stratification remarkably acts on the momentum equation of wall-normal direction.
Fig. 1 shows the local friction coefficients, Cf . The local friction coefficients of adverse pressure gradient flow (APG) generally decrease in comparison with those of zero pressure gradient flow (ZPG) in all cases, in which the remarkable decreases of Cf along the streamwise direction are found in cases of weak stable thermal stratification boundary layer (WSBL) and strong stable thermal stratification boundary layer (SSBL). In the case of SSBL, decrease of the momentum thickness (Reδ2 ) is observed due to stable thermal stratification (Hattori et al., 2014), and the APG effect overlapping the effect of stable thermal stratification causes the flow separation. Since the APG effect develops the momentum thickness,
Please cite this article as: H. Hattori et al., Direct numerical simulation of thermally-stratified turbulent boundary layer subjected to adverse pressure gradient, International Journal of Heat and Fluid Flow (2016), http://dx.doi.org/10.1016/j.ijheatfluidflow.2016.05.020
JID: HFF 4
ARTICLE IN PRESS
[m5G;June 27, 2016;22:30]
H. Hattori et al. / International Journal of Heat and Fluid Flow 000 (2016) 1–16
Fig. 2. Local friction coefficients in WSBL.
the increase of the momentum thickness can be observed in the downstream region, i.e., the APG effect mainly influences this region. Although Cf of the SSBL case indicates a negative value in the downstream region, Cf comes close to zero near x/δ2,in = 500 in the case of WSBL with APG, as shown in Fig. 1. However, Cf does not become zero, and then Cf again increases in the streamwise direction as x = 600δ2,in . Since it is considered that the re-increase may be caused by the domain size effect, DNS of the expanding domain in the streamwise direction as x = 900δ2,in for the case of WSBL with APG the is carried out for killing the domain size effect. As the result, the domain size effect does not affect the behavior of Cf as shown in Fig. 1. Thus, the separation of flow is not observed in the case of WSBL with APG, but the flow separation is found to occur slightly in the instantaneous field as shown in Fig. 2. The domain size in the streamwise direction of the case of WSBL with ZPG is also set as x = 900δ2,in as shown in Fig 1(b). Again, observing the momentum thickness in Fig. 1(a), the growth of momentum thickness is clearly disturbed by only effect of the stable thermal stratification, but it can be seen that the APG effect overlapping the effect of stable thermal stratification destroys the only effect of the stable thermal stratification. Fig. 2(b) shows distributions of the instantaneous value of Cf on the wall, in which the region gray colored indicates the instantaneous separation region, i.e., the instantaneous value of Cf is negative. Since the large fluctuation of Cf appears in the region between 450 < x/δ 2, in < 700 as shown in Fig. 1, it is considered that this region suggests that the turbulent structure is re-changed by both stable thermal stratification and APG. Fig. 3 shows the local Stanton number, St. The influence of domain size does not also affect in the thermal field as shown in
Fig. 3(b). Stanton numbers of APG are larger than those of ZPG in cases of NBL and USBL. In cases of stable thermal stratification, the Stanton numbers of APG are almost the same as those of ZPG, but a region where the Stanton numbers of APG are smaller than those of ZPG is also found. In the distribution of St, the development of enthalpy thickness (Re2 ) is remarkably disturbed in the case of SSBL due to stable thermal stratification as shown in Fig. 3(a), but decrease of the enthalpy thickness is not found. It can be seen from cases of USBL and NBL that the APG effect hardly influences the development of enthalpy thickness, but the APG effect is observed in the case of WSBL as shown in Fig. 3(a), where St again increases. Thus, it can be considered that the APG effect works increases of both Cf and St. Figs. 4 and 5 are shown the contributions of Reynolds stress and other forces distributions to the friction coefficient and of turbulent heat flux and other effects distributions to Stanton number in order to further investigation unique variations of Cf and St in the case of WSBL with APG. For the velocity field, the following relation concerning the turbulent boundary layer which is based on the FIK identity (Fukagata et al., 2002) was derived (Hou et al., 2006).
C f = cδ + cT + cC + cD + cP
(8)
where cδ = 4(1 − δ1 )/Reδ99 is the laminar drag term, cT = 1 2 0 2(1 − y )(−uv )dy is the contribution of Reynolds shear stress 1 term, cC = 2 0 2(1 − y )(−U¯ V¯ )dy is the mean convection term, 1 cD = 2 0 2(1 − y )2 [∂ U¯ U¯ /∂ x + ∂ uu/∂ x − (1/Reδ )∂ 2U /∂ x∂ x]dy is the 1 spatial development term, and cP = (1/2 ) 0 y2 (−∂ P¯/∂ x )dy is the pressure gradient term, respectively. On the other hand, the
Please cite this article as: H. Hattori et al., Direct numerical simulation of thermally-stratified turbulent boundary layer subjected to adverse pressure gradient, International Journal of Heat and Fluid Flow (2016), http://dx.doi.org/10.1016/j.ijheatfluidflow.2016.05.020
ARTICLE IN PRESS
JID: HFF
[m5G;June 27, 2016;22:30]
H. Hattori et al. / International Journal of Heat and Fluid Flow 000 (2016) 1–16
Fig. 3. Local Stanton numbers.
Fig. 4. Distributions of FIK identity for friction coefficient.
following relation concerning the turbulent thermal boundary layer with a constant wall temperature condition is derived using the identity for the thermal field (Kasagi et al., 2012).
St = cδ + cT + cC + cD
5
(9)
where cδ = 1/Reδ99 P r is the laminar contribution term, cT = 1 0 (−vθ )dy is the contribution of wall-normal turbulent heat flux 1 ¯ dy is the mean convection term, and cD = term, cC = 0 (−V¯ ) 1 2 ¯ ¯ 0 (1 − y ) ∂ U /∂ x + ∂ uθ /∂ x − (1/Reδ P r )∂ /∂ x∂ x dy is the spatial development term, respectively. In the case of WSBL, the contribution of pressure gradient term for Cf hardly varies as shown in Figs. 4(a) and (b), where the spatial development term, cD , and the mean convection term, cC , balance in the most domain. In the re-increase region of Cf in the case of WSBL with APG, however, it can be observed that the mean convection term and the contribution of Reynolds shear stress term, cT , increase so that the balance between cD and cC does not maintain in the region. Thus, the contribution of Reynolds shear stress and the mean convection terms contribute to the increase of Cf . On the other hand, the spatial development term, cD , and the mean convection term, cC , for Stanton number also balance in the most domain as shown in Figs. 5(a) and (b). In the re-increase region of St, however, it can be seen that the mean convection term does not contribute to increase of St, where slight increase of the contribution of wall-normal turbulent heat flux term and slight remain of the spatial development term are observed. Thus, these terms contribute to re-increase of St.
The ratios between Cf /2 and St are shown in Fig. 6 in order to investigate efficiency of heat transfer, where the value of the case of SSBL indicates a negative value due to existing negative value of Cf in the separation region. The ratios between Cf /2 and St of APG in all cases become smaller than those of ZPG, i.e., the efficiency of heat transfer is enhanced due to the APG effect, although the effect of stable thermal stratification also enhances the efficiency of heat transfer. 3.2. Turbulent statistics for velocity field The profiles of turbulent statistics normalized by both the inner and outer scales for velocity field are shown in Figs. 7–11 (Case of SSBL is not shown here). The typical profile of streamwise mean velocity normalized by the inner scale in the case of NBL with APG (Houra and Nagano, 2006; Nagano et al., 1993; 1998) is shown in Fig. 7(a), in which the standard log-law profile is not maintained due to the APG effect and the wake region is raised upward. Since the inner scale in the case of WSBL with APG significantly varies, the remarkably changing profiles of mean velocity are observed as shown in Fig. 7(b). Here, the normalization using inner scale employs the local scale on both axes, i.e., the local friction velocity, uτ , and friction temperature, θ τ , are used, because variations of turbulent statistics at the local location along streamwise direction are shown. On the other hand, the normalization using outer scale is adopted to describe the local boundary layer thickness for the x-axis and the free stream velocity at the inlet of main simulation part. Thus, the absolute variations of statistics from the inlet can be observed. Observing outer scaling profiles of the mean velocity, the
Please cite this article as: H. Hattori et al., Direct numerical simulation of thermally-stratified turbulent boundary layer subjected to adverse pressure gradient, International Journal of Heat and Fluid Flow (2016), http://dx.doi.org/10.1016/j.ijheatfluidflow.2016.05.020
ARTICLE IN PRESS
JID: HFF 6
[m5G;June 27, 2016;22:30]
H. Hattori et al. / International Journal of Heat and Fluid Flow 000 (2016) 1–16
Fig. 5. Distributions of FIK identity for Stanton number.
Fig. 6. Ratios between Cf and St.
remarkable decrease of mean velocity can be seen, and then recovering the mean velocity near the wall in the downstream region is clearly observed as indicated in Fig. 7(b). On the other hand, the decreases of mean velocity in the log-law region are found in the case of USBL, but the wake region is not raised upward as indicated in Fig. 7(c). In all cases, the deceleration of mean velocity normalized by the outer scale due to APG is found in the outer and free stream regions, but the different behaviors of mean velocity can be clearly seen in the inner region. The Reynolds shear stress is demonstrated in Fig. 8, in which the case of ZPG is also included. In the case of NBL, the remarkable increase of Reynolds shear stress normalized by the inner
scale in the case of APG is observed as compared with the case of ZPG along the streamwise direction, but the decrease of the net Reynolds shear stress in the case of APG can be found by the outer scale normalization, though no variations of Reynolds shear stress in the case of ZPG can be seen over x/δ 2, in ∼ 200, in which note that the turbulence spatially develops. In the case of WSBL, although the Reynolds shear stress remarkably decreases in the streamwise direction in both cases of APG and ZPG, it can be seen that the Reynolds shear stress is dramatically re-generated in the downstream region in the case of APG. In the case of ZPG, it can be seen that Reynolds shear stress normalized by the outer scale almost vanishes in the downstream region. Thus, it is considered that the flow becomes laminar in the downstream region, but Reynolds shear stress insistently remains which can be observed in the inner scale normalization. Also, it is revealed by the analysis of identify that the flow does not become laminar as mentioned in the previous section. Therefore, the effect of stable thermal stratification with ZPG of this case cannot laminarize the turbulent flow. Also, since the effect of pure APG does not decrease the turbulence, the flow does not become laminar. Consequently, the local friction coefficient again increases in the downstream region as shown in Fig. 1. Also, the counter gradient diffusion phenomenon (CDP) (Hattori et al., 2014) is found in the outer region near x/δ 2, in ∼ 400 in the case of APG. Since the CDP is also observed in the case of WSBL without APG, the CDP is caused by the stable thermal stratification. In the case of USBL, the increase of Reynolds shear stress normalized by the inner scale can be also seen in both cases of APG and ZPG, but the Reynolds shear stress normalized by the outer scale also increases in the case of APG, though the increase cannot be observed in the case of ZPG. Thus, the effect of unstable thermal stratification with APG enhances the net Reynolds shear stress, which is different phenomenon in comparison with the case of NBL with APG. Thus, the different profiles of mean velocity can be observed as shown in Fig. 7(c). The rms velocity fluctuations in streamwise, wall-normal and spanwise directions are shown in Figs. 9–14, where the case of ZPG is also included in figures. Fig. 9 shows rms velocity fluctu+ + ations normalized by the inner scale, u+ rms , vrms and wrms , in the case of NBL with or without APG. The streamwise rms velocity hardly changes near the wall, but significant increases of the case of APG in the outer region as compared with the case of ZPG can be observed as shown in Fig. 9(a). As for the wall-normal rms velocity fluctuation, remarkable increases of the case of APG in all region can be seen, but v+ rms of the case of ZPG does not vary near the wall as indicated in Fig. 9(b). It can be seen that the distribution of spanwise rms velocity fluctuations indicates the similar tendency of distribution of v+ rms in both cases as shown in Fig. 9(c). Thus, the pure effect of APG increases velocity fluctuations in the outer region, and enhances the wall-normal and spanwise velocity fluctuations as viewed from the perspective of normalization using the inner scale. On the other hand, the rms velocity fluctuations normalized by the outer scale, u∗rms (= urms /U¯ 0 ), v∗rms (= vrms /U¯ 0 ) and w∗rms (= wrms /U¯ 0 ), are shown in Fig. 10, in which v∗rms and w∗rms hardly change in both cases of APG and ZPG, but decrease of u∗rms near the wall can be observed in the case of APG. In view of distributions of Reynolds shear stress as shown in Fig. 7(a), it is considered that the energy transfer from mean streamwise velocity to streamwise fluctuation mainly decreases near the wall by the effect of APG. Figs. 11 and 12 show the distributions of rms velocity fluctuations in the case of WSBL. In view of the inner scale normalization as indicated in Fig. 11, rms velocity fluctuations of all directions remarkably vary in both cases due to the variations of friction velocity caused by the stable thermal stratification, but decreases of rms velocity fluctuations can be observed in the outer region. As for rms velocity fluctuations normalized by the outer scale in the case of WSBL, it can be found that rms velocity
Please cite this article as: H. Hattori et al., Direct numerical simulation of thermally-stratified turbulent boundary layer subjected to adverse pressure gradient, International Journal of Heat and Fluid Flow (2016), http://dx.doi.org/10.1016/j.ijheatfluidflow.2016.05.020
JID: HFF
ARTICLE IN PRESS
[m5G;June 27, 2016;22:30]
H. Hattori et al. / International Journal of Heat and Fluid Flow 000 (2016) 1–16
Fig. 7. Profiles of streamwise mean velocity.
Please cite this article as: H. Hattori et al., Direct numerical simulation of thermally-stratified turbulent boundary layer subjected to adverse pressure gradient, International Journal of Heat and Fluid Flow (2016), http://dx.doi.org/10.1016/j.ijheatfluidflow.2016.05.020
7
JID: HFF 8
ARTICLE IN PRESS
[m5G;June 27, 2016;22:30]
H. Hattori et al. / International Journal of Heat and Fluid Flow 000 (2016) 1–16
(b) Stable stratification
Fig. 8. Profiles of Reynolds shear stress.
Please cite this article as: H. Hattori et al., Direct numerical simulation of thermally-stratified turbulent boundary layer subjected to adverse pressure gradient, International Journal of Heat and Fluid Flow (2016), http://dx.doi.org/10.1016/j.ijheatfluidflow.2016.05.020
JID: HFF
ARTICLE IN PRESS
[m5G;June 27, 2016;22:30]
H. Hattori et al. / International Journal of Heat and Fluid Flow 000 (2016) 1–16
9
Fig. 9. Profiles of rms velocity fluctuation normalized by inner scale in the case of NBL.
Fig. 10. Profiles of rms velocity fluctuation normalized by outer scale in the case of NBL.
fluctuations of all directions decrease in the case of ZPG as shown in Fig. 12. On the other hand, decreases of rms velocity fluctuations of all directions due to thermal stratification can be also observed in the case of APG, but these again increase over x/δ2,in = 400. In particular, increases of rms velocity fluctuations of all directions in the outer region can be observed at x/δ2,in = 800. Thus, the effect of APG in the stable thermal stratification is of increase of velocity fluctuations in the outer region. The case of USBL is demonstrated in Figs. 13 and 14. In view of the inner scale normalization, it can be found that rms velocity fluctuations of all
directions increase in the outer region in both cases of APG and ZPG, but remarkable increases of the near-wall rms velocity fluctuation of spanwise direction can be observed in the case of APG as indicated in Fig. 13. As for the profiles of rms velocity fluctuations normalized by the outer scale, remarkable increases of rms velocity fluctuation of wall-normal direction are obviously found in the case of APG as shown in Fig. 14. The rms velocity fluctuation of spanwise direction also increases near the wall in the case of APG, but rms velocity fluctuation of streamwise direction hardly varies. Thus, the effect of APG in the unstable thermal
Please cite this article as: H. Hattori et al., Direct numerical simulation of thermally-stratified turbulent boundary layer subjected to adverse pressure gradient, International Journal of Heat and Fluid Flow (2016), http://dx.doi.org/10.1016/j.ijheatfluidflow.2016.05.020
JID: HFF 10
ARTICLE IN PRESS
[m5G;June 27, 2016;22:30]
H. Hattori et al. / International Journal of Heat and Fluid Flow 000 (2016) 1–16
Fig. 11. Profiles of rms velocity fluctuation normalized by inner scale in the case of WSBL.
stratification especially is of an enhancement of wall-normal rms velocity fluctuation, which clearly influences an increase of Reynolds shear stress as shown in Fig. 8(c). 3.3. Turbulent statistics for thermal field Turbulent statistics in thermal field are shown in Fig. 15–18 (Also, case of SSBL is not shown here). Profiles of mean temperature are demonstrated in Fig. 15. As for the profiles normalized by ¯ + , only the decrease of mean temperature in the the inner scale,
Fig. 12. Profiles of rms velocity fluctuation normalized by outer scale in the case of WSBL.
log-law region can be seen in the case of NBL, but the maximum ¯ + does not change, because the ratio between Cf and St value of ¯+ hardly varies in the case of NBL where the maximum value of is proportional to 1/θ τ , that is C f /2/St. Hence, the maximum ¯ + decreases with decreasing the ratio in the case of value of USBL as shown in Fig. 15(c). It can be seen that mean temperature normalizer by the outer scale is slow to increase in the case of NBL near the wall as shown in Fig. 15(a), but rapid increases of temperature can be observed in the case of USBL as shown in
Please cite this article as: H. Hattori et al., Direct numerical simulation of thermally-stratified turbulent boundary layer subjected to adverse pressure gradient, International Journal of Heat and Fluid Flow (2016), http://dx.doi.org/10.1016/j.ijheatfluidflow.2016.05.020
JID: HFF
ARTICLE IN PRESS
[m5G;June 27, 2016;22:30]
H. Hattori et al. / International Journal of Heat and Fluid Flow 000 (2016) 1–16
11
Fig. 13. Profiles of rms velocity fluctuation normalized by inner scale in the case of USBL.
Fig. 15(c). As for mean temperature profile normalized by the inner scale in the case of WSBL as indicated in Fig. 15(b), it can be seen that the conductive sublayer is maintained in all region, but the varying profiles in the log-law region are observed. Observing outer scaling profiles of the mean temperature, the remarkable decrease of mean temperature can be seen, and then recovering the mean temperature near the wall in the downstream region is clearly observed as indicated in Fig. 15(b). Turbulent heat fluxes in streamwise and wall-normal directions are shown in Figs. 16 and 17 including the case of ZPG, where the streamwise turbulent heat flux, uθ , clearly affects the Reynolds shear stress in the thermally-stratified turbulent boundary layer, because uθ is contained in the buoyancy term of the transport equation of Reynolds shear stress as Guv = Riδ2,in uθ . The
Fig. 14. Profiles of rms velocity fluctuation normalized by outer scale in the case of USBL.
wall-normal turbulent heat flux normalized by the inner scale, +
vθ , increases in the downstream region in both cases of NBL and USBL. The wall-normal turbulent heat flux normalized by the outer ∗ scale, vθ = vθ /U¯ , in the case of NBL with APG decreases near ∗ the wall in the downstream region, but vθ in the case of USBL with APG increases in the downstream region. Thus, the different profiles of mean temperature between cases of NBL and USBL with APG are obtained as shown in Fig. 15, because vθ is closely associated with a formation of mean temperature profile. On the other hand, in the case of WSBL with or without APG, vθ
+
decreases in
Please cite this article as: H. Hattori et al., Direct numerical simulation of thermally-stratified turbulent boundary layer subjected to adverse pressure gradient, International Journal of Heat and Fluid Flow (2016), http://dx.doi.org/10.1016/j.ijheatfluidflow.2016.05.020
ARTICLE IN PRESS
JID: HFF 12
[m5G;June 27, 2016;22:30]
H. Hattori et al. / International Journal of Heat and Fluid Flow 000 (2016) 1–16
Fig. 15. Profiles of mean temperature.
the downstream region, and then vθ of ZPG, two peak values of vθ
+
+
again increases. In the case
appear at x/δ2,in = 800. Thus, the +
effect of APG in the case of WSBL is of enhancement of vθ at the ∗ ∗ log and outer regions. As for vθ in the case of WSBL, vθ consider∗ ably becomes small at x/δ2,in = 400, but it can be seen that vθ is
∗
re-generated as shown in Fig. 16(b). However, the case of ZPG, vθ almost vanishes at the downstream region. Thus, the wall-normal turbulent heat flux in the case of ZPG does not contribute the enhancement of heat transfer coefficient as revealed by the analysis of identify of Fig. 5(b). On the other hand, because the effect of
Please cite this article as: H. Hattori et al., Direct numerical simulation of thermally-stratified turbulent boundary layer subjected to adverse pressure gradient, International Journal of Heat and Fluid Flow (2016), http://dx.doi.org/10.1016/j.ijheatfluidflow.2016.05.020
JID: HFF
ARTICLE IN PRESS
[m5G;June 27, 2016;22:30]
H. Hattori et al. / International Journal of Heat and Fluid Flow 000 (2016) 1–16
13
Fig. 16. Profiles of wall-normal turbulent heat flux.
APG does not suppress the wall-normal turbulent heat flux, the wall-normal turbulent heat flux contributes the enhancement of heat transfer coefficient as shown in Fig. 5(a). The streamwise turbulent heat flux is demonstrated in Fig. 17. It can be found that the streamwise turbulent heat flux normalized by the inner scale, +
uθ , in cases when both NBL and USBL with APG slightly decrease, ∗ and uθ which is the streamwise turbulent heat flux normalized
by the outer scale, also decreases in cases of both NBL and USBL ∗ with or without APG, but the near-wall uθ obviously increases in the case of USBL with APG. Since uθ affects the Reynolds shear stress in the thermally-stratified turbulent boundary layer as mentioned above, the increase of near-wall uθ works on the enhancement of the Reynolds shear stress in the case of USBL as shown in Fig. 8(c). In the case of WSBL, the behavior of uθ is similar to the
Please cite this article as: H. Hattori et al., Direct numerical simulation of thermally-stratified turbulent boundary layer subjected to adverse pressure gradient, International Journal of Heat and Fluid Flow (2016), http://dx.doi.org/10.1016/j.ijheatfluidflow.2016.05.020
JID: HFF 14
ARTICLE IN PRESS
[m5G;June 27, 2016;22:30]
H. Hattori et al. / International Journal of Heat and Fluid Flow 000 (2016) 1–16
Fig. 17. Profiles of streamwise turbulent heat flux.
behavior of vθ as shown in Fig. 16(b), but it can be observed in the ∗ ∗ case of APG that uθ is not very small in comparison with vθ at ∗ x/δ2,in = 400. Since uθ does not vanish in the case of WSBL with APG, it is considered that the turbulence is maintained in the case of WSBL with APG as mentioned in the discussion of wall-normal turbulent heat flux.
Finally, the rms temperature fluctuation is shown in Fig. 18. In cases of both NBL and USBL with APG, it can be seen that the peak value of the rms temperature fluctuation normalized by +
+
the inner scale, θ 2 , decreases, and θ 2 increases in the outer ∗ region. Although θ 2 which is the rms temperature fluctuation normalized by the outer scale only decreases in the case of NBL,
Please cite this article as: H. Hattori et al., Direct numerical simulation of thermally-stratified turbulent boundary layer subjected to adverse pressure gradient, International Journal of Heat and Fluid Flow (2016), http://dx.doi.org/10.1016/j.ijheatfluidflow.2016.05.020
ARTICLE IN PRESS
JID: HFF
[m5G;June 27, 2016;22:30]
H. Hattori et al. / International Journal of Heat and Fluid Flow 000 (2016) 1–16
15
Fig. 18. Profiles of rms temperature fluctuation.
∗
∗
the increase of θ 2 near the wall and the decrease of θ 2 in the region around 0.1y/δ t ∼ 0.5y/δ t can be observed in the case of USBL with APG. This different profile strongly relates the profiles of the wall-normal turbulent heat flux as shown in Fig. 16, because the wall-normal turbulent heat is included in the production term of transport equation of θ 2 . In the case of WSBL with or without
APG, it can be observed that θ 2 of both the normalizations remains in all regions, although both the wall-normal turbulent heat flux and the gradient of mean temperature which compose the production term of θ 2 considerably become small at x/δ2,in = 400. Thus, it is considered that terms of transport equation except for the production term act on the sustentation of θ 2 .
Please cite this article as: H. Hattori et al., Direct numerical simulation of thermally-stratified turbulent boundary layer subjected to adverse pressure gradient, International Journal of Heat and Fluid Flow (2016), http://dx.doi.org/10.1016/j.ijheatfluidflow.2016.05.020
JID: HFF 16
ARTICLE IN PRESS
[m5G;June 27, 2016;22:30]
H. Hattori et al. / International Journal of Heat and Fluid Flow 000 (2016) 1–16
4. Conclusions
Acknowledgement
DNSs of thermally-stratified turbulent boundary layers subjected to a non-equilibrium APG are carried out. DNS results clearly show the characteristic of thermally-stratified nonequilibrium APG turbulence flow. Especially, it is found that the friction coefficient and Stanton number decrease along the streamwise direction due to the effects of stable thermal stratification and APG, but those again increase due to the APG effect in the case of WSBL. Thus, the analysis for both the friction coefficient and Stanton number in the case of WSBL with/without APG is conducted using the FIK identity (Fukagata et al., 2002; Kasagi et al., 2012) in order to investigate contributions from the transport equations, in which it is found that both Reynolds-shear-stress and the mean convection terms of the friction coefficient and both the wallnormal turbulent heat flux and the spatial development terms of the Stanton number contribute to again increase those values in the case of WSBL with APG. Also, it is found that the mean velocity profiles normalized by the inner scale of APG flow increase in the case of WSBL and decrease in the case of USBL in the outer region due to the variation of friction coefficients, but decreases of free stream due to APG can be observed in both cases. Also, characteristic profiles of turbulence are detected in the thermal field, in which decreases of mean temperature normalized by the inner scale are observed in cases of NBL and USBL, although increases of mean temperature can be seen in the case of WSBL. Consequently, DNS results reveal that the turbulent characteristics of both stable and unstable thermal stratifications with APG are different from those of the NBL affected by APG.
This research was supported by a Grant-in-Aid for Scientific Research (C), 26420144, from the Japan Society for the Promotion of Science (JSPS). References Araya, G., Castillo, L., 2013. Direct numerical simulations of turbulent thermal boundary layers subjected to adverse streamwise pressure gradients. (1994-present). Phys. Fluids 25 (9). Clauser, F.H., 1954. Turbulent boundary layers in adverse pressure gradients. J. Aeronaut. Sci. 21 (2), 91–108. Fukagata, K., Iwamoto, K., Kasagi, N., 2002. Contribution of reynolds stress distribution to the skin friction in wall-bounded flows. Phys. Fluids 14 (11). Hattori, H., Hotta, K., Houra, T., 2014. Characteristics and structures in thermally-stratified turbulent boundary layer with counter diffusion gradient phenomenon. Int. J. Heat Fluid Flow 49, 53–61. Hattori, H., Houra, T., Nagano, Y., 2007. Direct numerical simulation of stable and unstable turbulent thermal boundary layers. Int. J. Heat Fluid Flow 28, 1262–1271. Hattori, H., Nagano, Y., 2004. Direct numerical simulation of turbulent heat transfer in plane impinging jet. Int. J. Heat Fluid Flow 25, 749–758. Hou, Y., Somandepalli, V.S.R., Mungal, M.G., 2006. A technique to determine total shear stress and polymer stress profiles in drag reduced boundary layer flows. Exp. Fluids 40 (4), 589–600. Houra, T., Nagano, Y., 2006. Effects of adverse pressure gradient on heat transfer mechanism in thermal boundary layer. Int. J. Heat Fluid Flow 27 (5), 967– 976. Kasagi, N., Hasegawa, Y., Fukagata, K., Iwamoto, K., 2012. Control of turbulent transport: less friction and more heat transfer. ASME. J. Heat Transfer 134 (3). Lee, J.H., Sung, H.J., 2008. Effects of an adverse pressure gradient on a turbulent boundary layer. Int. J. Heat Fluid Flow 29 (3), 568–578. Nagano, Y., Tagawa, M., Tsuji, T., 1993. Effects of adverse pressure gradients on mean flows and turbulence statistics in a boundary layer. Turbulent Shear Flows 8, 7–21. Nagano, Y., Tsuji, T., Houra, T., 1998. Structure of turbulent boundary layer subjected to adverse pressure gradient. Int. J. Heat Fluid Flow 19 (5), 563–572. Ohya, Y., 2001. Wind-tunnel study of atmospheric stable boundary layers over a rough surface. Boundary-Layer Meteorol. 98, 57–82.
Please cite this article as: H. Hattori et al., Direct numerical simulation of thermally-stratified turbulent boundary layer subjected to adverse pressure gradient, International Journal of Heat and Fluid Flow (2016), http://dx.doi.org/10.1016/j.ijheatfluidflow.2016.05.020
JID: HFF
[m5G;June 27, 2016;22:30]
International Journal of Heat and Fluid Flow 0 0 0 (2016) 1–16
Contents lists available at ScienceDirect
International Journal of Heat and Fluid Flow journal homepage: www.elsevier.com/locate/ijheatfluidflow
Direct numerical simulation of thermally-stratified turbulent boundary layer subjected to adverse pressure gradient Hirofumi Hattori∗, Amane Kono, Tomoya Houra Department of Mechanical Engineering, Nagoya Institute of Technology, Gokiso-cho, Showa-ku, Nagoya 466-8555, Japan
a r t i c l e
i n f o
Article history: Available online xxx Keywords: Turbulent boundary layer Adverse pressure gradient Thermal stratification Heat transfer DNS
a b s t r a c t The objective of this study is to investigate and observe turbulent heat transfer structures and statistics in thermally-stratified turbulent boundary layers subjected to a non-equilibrium adverse pressure gradient (APG) by means of direct numerical simulation (DNS). DNSs are carried out under conditions of neutral, stable and unstable thermal stratifications with a non-equilibrium APG, in which DNS results reveal heat transfer characteristics of thermally-stratified non-equilibrium APG turbulent boundary layers. In cases of thermally-stratified turbulent boundary layers affected by APG, heat transfer performances increase in comparison with a turbulent boundary layer with neutral thermal stratification and zero pressure gradient (ZPG). Especially, it is found that the friction coefficient and Stanton number decrease along the streamwise direction due to the effects of stable thermal stratification and APG, but those again increase due to the APG effect in the case of weak stable thermal stratification (WSBL). Thus, the analysis for both the friction coefficient and Stanton number in the case of WSBL with/without APG is conducted using the FIK identity in order to investigate contributions from the transport equations, in which it is found that both Reynolds-shear-stress and the mean convection terms of the friction coefficient and both the wallnormal turbulent heat flux and the spatial development terms of the Stanton number contribute to again increase those values in the case of WSBL with APG. The characteristic turbulent statistics of both the velocity and the thermal fields along streamwise direction are clearly indicated, in which the decrease of log-law profile of streamwise mean velocity which was found by experimental study is also observed in the neutral boundary layer of our DNS. DNS results reveal that the turbulent characteristics of both cases of stable and unstable thermal stratification boundary layers differ with the turbulent characteristics of the neutral boundary layer having APG. © 2016 Elsevier Inc. All rights reserved.
1. Introduction Since a turbulent boundary layer subjected to an adverse pressure gradient (APG) yielded by an increase of pressure in the streamwise direction and also causing flow separation, affects the turbulent structure and heat transfer strongly, a turbulent heat transfer phenomenon of APG flow should be explored for the controls of flow and heat transfer. APG turbulent boundary layer with heat transfer has been revealed by experimental (Houra and Nagano, 2006; Nagano et al., 1993; 1998) and DNS (Araya and Castillo, 2013; Lee and Sung, 2008) studies. On the other hand, a thermally-stratified turbulent boundary layer which can be observed in the atmosphere has been also investigated by experimental (Ohya, 2001) and DNS (Hattori et al., 2014; 2007)
∗
Corresponding author. Tel./fax: +81 52 735 5359. E-mail address: [email protected] (H. Hattori).
studies, which expand the understanding of turbulent structure in a thermally-stratified turbulent boundary layer. Although those studies have led to detailed research on structures of turbulent heat transfer, a study on turbulent boundary layer with overlapping of several influencing factors in turbulent structures such as thermal stratifications and pressure gradients is needed in order to determine the mechanism of more complex turbulent heat transfer. Thus, a turbulent heat transfer subjected to thermal stratification and pressure gradient should be explored as the typical situation. In the present study, in order to investigate and observe effects of adverse pressure gradient for a thermally-stratified turbulent boundary layer, DNS of thermally-stratified turbulent boundary layers subjected to adverse pressure gradient is carried out, where a non-equilibrium APG turbulence flow is assumed and various thermally-stratified turbulent boundary layers with a nonequilibrium APG are simulated in order to reveal statistics and structures in such fields.
http://dx.doi.org/10.1016/j.ijheatfluidflow.2016.05.020 0142-727X/© 2016 Elsevier Inc. All rights reserved.
Please cite this article as: H. Hattori et al., Direct numerical simulation of thermally-stratified turbulent boundary layer subjected to adverse pressure gradient, International Journal of Heat and Fluid Flow (2016), http://dx.doi.org/10.1016/j.ijheatfluidflow.2016.05.020
ARTICLE IN PRESS
JID: HFF 2
[m5G;June 27, 2016;22:30]
H. Hattori et al. / International Journal of Heat and Fluid Flow 000 (2016) 1–16
Nomenclature Cf cp gi Lx , Ly , Lz Nx , Ny , Nz P¯ Pr Reδ2 Reδ2,in
Re2 Re2,in
Riδ2,in
Reδ99 St t U¯ , V¯ u, v, w U¯ i U¯ 0 ui uτ xi x, y, z y+
local friction coefficient = 2τw /(ρU¯ 02 ) specific heat at constant pressure gravitational acceleration nondimensional domain size in streamwise, wallnormal and spanwise directions, respectively grid numbers in streamwise, wall-normal and spanwise directions, respectively mean static pressure Prandtl number, = ν /α Reynolds number based on free stream velocity and local momentum thickness, = U¯ 0 δ2 /ν
wall temperature at the inlet of the driver part wall and free-stream temperature temperature fluctuation local friction temperature, = qw /(ρ c p uτ ) ensemble- or time-averaged value normalization by inner variables (uτ , θ τ , ν )
2. DNS of thermally-stratified boundary layer with APG Assuming that the Boussinesq approximation is approved for the Navier-Stokes equation, the governing equations used in the present DNS are indicated as follows (Hattori et al., 2007):
Reynolds number based on free stream velocity and momentum thickness at the inlet of the driver part, = U¯ 0 δ2,in /ν Reynolds number based on free stream velocity and local enthalpy thickness, = U¯ 0 2 /ν
∂ ui ∂ ui ∂p 1 ∂ 2 ui + uj =− + + δi2 Riδ2,in θ ∂t ∂xj ∂ xi Reδ2,in ∂ x j ∂ x j
(1)
∂ ui =0 ∂ xi
(2)
Reynolds number based on free stream velocity and enthalpy thickness at the inlet of the driver part, = U¯ 0 2,in /ν bulk Richardson number based on free stream velocity, momentum thickness at the inlet of the driver part, = gβδ2,in /U¯ 2
∂θ ∂θ 1 ∂ 2θ + uj = ∂t ∂ x j Pr Reδ2,in ∂ x j ∂ x j
(3)
0
Reynolds number based on free stream velocity and local boundary layer thickness, = U¯ 0 δ99 /ν local Stanton number = qw /[ρ c pU¯ 0 ( w − ∞ )] time mean velocity in x- and y-directions, respectively turbulent fluctuation in x-, y- and z-directions, respectively mean velocity in xi -direction free stream mean velocity turbulent fluctuation in xi -direction √ local friction velocity, = τw /ρ Cartesian coordinate in i-direction Cartesian coordinate in streamwise, wall-normal and spanwise directions, respectively nondimensional distance from wall surface, = uτ y/ν
Greek symbols α molecular diffusivity for heat β coefficient of volume expansion δ 99 local boundary layer thickness defined at location where mean velocity is equal to 99% of free-stream velocity δ1 displacement thickness δ2 momentum thickness δ ij Kronecker delta δt thermal boundary layer thickness defined at location where nondimensional mean temperature [= ¯ − ¯ 0 / )] is equal to 99% of nondimensional ( ¯ ∞− ¯ 0 / )] free-stream temperature [= ( thermal boundary layer thickness defined ¯ = at dimensionless mean temperature, ¯ 0 )/ , is equal to 99%. (θ − ¯ ∞− ¯0 temperature difference, =
2 ν ρ τw ¯
¯0 ¯ w, ¯∞ θ θτ ( ) ( )+
enthalpy thickness kinematic viscosity density wall shear stress mean temperature
where the Einstein summation convention applies to repeated indices, and a comma followed by an index indicates differentiation with respect to the indexed spatial coordinate. ui is the dimensionless velocity component in xi direction, θ is the dimensionless temperature difference, p is the dimensionless pressure, t is the dimensionless time, and xi is the dimensionless spatial coordinate in the i direction, respectively. Reδ2,in = U¯ 0 δ2,in /ν is the Reynolds number based on the free stream velocity and the momentum thickness at the inlet of the driver part, δ 2, in . Note that “the driver part” means the inflow data generator for the inlet boundary of the main simulation part (Hattori et al., 2007). P r = ν /α is the Prandtl number, and Riδ2,in = gβδ2,in /U¯ 02 is the bulk Richardson number based on the free stream velocity, the momentum thickness at the inlet of the driver part, and the temperature difference between a free ¯ 0− ¯ w ). In the governing equations, stream and a wall ( = the dimensionless variables are given using the free stream velocity, U¯ 0 , and the free stream temperature, 0 , at the inlet of the driver part, and the wall temperature, w . The Prandtl number is set to 0.71, assuming the working fluid to be air. The Reynolds number is set to 300, and the Richardson numbers are set to −0.005 (unstable thermal stratification boundary layer: USBL), 0 (neutral thermal stratification boundary layer: NBL), 0.005 (weak stable thermal stratification boundary layer: WSBL) and 0.04 (strong stable thermal stratification boundary layer: SSBL). For efficiently conducting the DNS of thermally-stratified turbulent boundary layers subjected to APG, the computational domain is composed of two parts; one is the driver part where a zero-pressure-gradient (ZPG) flow with an isothermal wall is generated and used as the inflow boundary condition for the main simulation, and the other is the main part where a thermallystratified turbulent boundary layers subjected to adverse pressure gradient are simulated (Hattori et al., 2007; Hattori and Nagano, 2004). A central finite-difference method of second-order accuracy is used to solve the equations of continuity, momentum and energy (Hattori et al., 2007; Hattori and Nagano, 2004), where the computational domain, grid numbers and grid resolutions are listed in Table 1. Since it is considered that the boundary layer thickness remarkably increases due to the effects of APG and unstable thermal stratification in the case of USBL, the domain size of a wall-normal direction is given by double the size of the NBL. In the case of WSBL, the domain size in the streamwise direction is expanded due to the influence of the boundary condition at the
Please cite this article as: H. Hattori et al., Direct numerical simulation of thermally-stratified turbulent boundary layer subjected to adverse pressure gradient, International Journal of Heat and Fluid Flow (2016), http://dx.doi.org/10.1016/j.ijheatfluidflow.2016.05.020
ARTICLE IN PRESS
JID: HFF
[m5G;June 27, 2016;22:30]
H. Hattori et al. / International Journal of Heat and Fluid Flow 000 (2016) 1–16
3
Table 1 Computational conditions of main part. Ri
−0.005 (USBL)
0 (NBL) ∗
dCp /dx Lx × Ly × Lz Nx × Ny × Nz x+in y+in + zin x+out y+out + zout t (U¯ 0 /δ2,in ) Total steps
3.3 × 10 600 × 30 × 40 1152 × 128 × 128
0
7.95 0.042 ∼ 11.3 4.77
−4
6.40 0.34 ∼ 9.12 3.84
0
0.005 (WSBL)
3.3 × 10 600 × 60 × 40 1152 × 160 × 128
13.5 0.048 ∼ 18.2 5.4
−4
0
0.04 (SSBL) −4
3.3 × 10 900 × 30 × 40 1728 × 128 × 128
8.28 0.044 ∼ 11.8 5 12.3 3.50 0.043 ∼ 16.5 0.018 ∼ 5 4.9 2.1 0.01 380,0 0 0
3.12 0.016 ∼ 4.45 1.87
0
3.3 × 10−4 600 × 30 × 40 1152 × 128 × 128
3.12 0.016 ∼ 4.44 1.87
1.47 0.0077 ∼ 2.1 0.88
outlet, where the influence of the boundary condition is mentioned later. The dimensionless time step is 0.01 in each case, and in order to adequately obtain turbulent statistics, 380,0 0 0 calculation steps are needed at least for each case. In order to achieve the APG condition in the free stream, the boundary conditions of the free stream are given as follows:
∂u ∂w ∂v ∂u ∂w =0, = 0, =− − ∂y ∂y ∂y ∂x ∂z U¯ e (x ) = U¯ 0
1 − C p (x )
(4)
(5)
where U¯ e (x ) is a local mean free stream velocity, C p = [P (x ) − P (0 )]/(ρU¯ 0 /2 ) is the local pressure coefficient, P(x) is a local mean pressure in the free stream, and P(0) is the free stream pressure at the inlet. Cp is set to 0 at the inlet of main part of DNS, and Cp is increased along the streamwise direction. Finally, Cp increases with dC p /dx∗ [= dC p /(dx/δ2,in )] = 3.3 × 10−4 in order to achieve the APG flow. For the ZPG flow, Cp (x) is set to 0, i.e., dC p /dx∗ = 0. Note that our DNS deals with the non-equilibrium APG turbulent boundary layer, because the equilibrium condition (Clauser, 1954) does not satisfy the relation, β (x ) = (δ ∗ /τw )(dP¯ /dx )=constant, in DNS with APG condition, where δ ∗ is the displacement thickness. The non-slip condition for velocity field and the isothermal condition for thermal field at the wall, and the convective boundary condition (Hattori et al., 2007) at the outlet and the periodic condition for the spanwise direction are adopted for both velocity and thermal fields. On the other hand, the Reynolds-averaged Navier-Stokes equations of Eq. (1) in the two-dimensional field are given as follows:
U¯
∂ U¯ ¯ ∂ U¯ +V ∂x ∂y ∂ P¯ ∂ 1 ∂ U¯ ∂ 1 ∂ U¯ 2 =− + −u + − uv ∂ x ∂ x Reδ2,in ∂ x ∂ y Reδ2,in ∂ y
∂ V¯ ¯ ∂ V¯ ∂ P¯ ∂ 1 ∂ V¯ ¯ +V =− + − uv U ∂x ∂y ∂ y ∂ x Reδ2,in ∂ x ∂ 1 ∂ V¯ + − v2 + Riδ2,in θ ∂ y Reδ2,in ∂ y
Fig. 1. Local friction coefficients.
(6) 3. Results and discussion
3.1. Fundamental Parameters
(7)
where U¯ and V¯ are mean velocities in the streamwise and wallnormal directions, respectively. The spanwise mean velocity vanishes due to the assumption of 2-dimensional mean flow field. Considering these equations, note that the APG effect mainly affects the momentum equation of streamwise direction and the effect of thermal stratification remarkably acts on the momentum equation of wall-normal direction.
Fig. 1 shows the local friction coefficients, Cf . The local friction coefficients of adverse pressure gradient flow (APG) generally decrease in comparison with those of zero pressure gradient flow (ZPG) in all cases, in which the remarkable decreases of Cf along the streamwise direction are found in cases of weak stable thermal stratification boundary layer (WSBL) and strong stable thermal stratification boundary layer (SSBL). In the case of SSBL, decrease of the momentum thickness (Reδ2 ) is observed due to stable thermal stratification (Hattori et al., 2014), and the APG effect overlapping the effect of stable thermal stratification causes the flow separation. Since the APG effect develops the momentum thickness,
Please cite this article as: H. Hattori et al., Direct numerical simulation of thermally-stratified turbulent boundary layer subjected to adverse pressure gradient, International Journal of Heat and Fluid Flow (2016), http://dx.doi.org/10.1016/j.ijheatfluidflow.2016.05.020
JID: HFF 4
ARTICLE IN PRESS
[m5G;June 27, 2016;22:30]
H. Hattori et al. / International Journal of Heat and Fluid Flow 000 (2016) 1–16
Fig. 2. Local friction coefficients in WSBL.
the increase of the momentum thickness can be observed in the downstream region, i.e., the APG effect mainly influences this region. Although Cf of the SSBL case indicates a negative value in the downstream region, Cf comes close to zero near x/δ2,in = 500 in the case of WSBL with APG, as shown in Fig. 1. However, Cf does not become zero, and then Cf again increases in the streamwise direction as x = 600δ2,in . Since it is considered that the re-increase may be caused by the domain size effect, DNS of the expanding domain in the streamwise direction as x = 900δ2,in for the case of WSBL with APG the is carried out for killing the domain size effect. As the result, the domain size effect does not affect the behavior of Cf as shown in Fig. 1. Thus, the separation of flow is not observed in the case of WSBL with APG, but the flow separation is found to occur slightly in the instantaneous field as shown in Fig. 2. The domain size in the streamwise direction of the case of WSBL with ZPG is also set as x = 900δ2,in as shown in Fig 1(b). Again, observing the momentum thickness in Fig. 1(a), the growth of momentum thickness is clearly disturbed by only effect of the stable thermal stratification, but it can be seen that the APG effect overlapping the effect of stable thermal stratification destroys the only effect of the stable thermal stratification. Fig. 2(b) shows distributions of the instantaneous value of Cf on the wall, in which the region gray colored indicates the instantaneous separation region, i.e., the instantaneous value of Cf is negative. Since the large fluctuation of Cf appears in the region between 450 < x/δ 2, in < 700 as shown in Fig. 1, it is considered that this region suggests that the turbulent structure is re-changed by both stable thermal stratification and APG. Fig. 3 shows the local Stanton number, St. The influence of domain size does not also affect in the thermal field as shown in
Fig. 3(b). Stanton numbers of APG are larger than those of ZPG in cases of NBL and USBL. In cases of stable thermal stratification, the Stanton numbers of APG are almost the same as those of ZPG, but a region where the Stanton numbers of APG are smaller than those of ZPG is also found. In the distribution of St, the development of enthalpy thickness (Re2 ) is remarkably disturbed in the case of SSBL due to stable thermal stratification as shown in Fig. 3(a), but decrease of the enthalpy thickness is not found. It can be seen from cases of USBL and NBL that the APG effect hardly influences the development of enthalpy thickness, but the APG effect is observed in the case of WSBL as shown in Fig. 3(a), where St again increases. Thus, it can be considered that the APG effect works increases of both Cf and St. Figs. 4 and 5 are shown the contributions of Reynolds stress and other forces distributions to the friction coefficient and of turbulent heat flux and other effects distributions to Stanton number in order to further investigation unique variations of Cf and St in the case of WSBL with APG. For the velocity field, the following relation concerning the turbulent boundary layer which is based on the FIK identity (Fukagata et al., 2002) was derived (Hou et al., 2006).
C f = cδ + cT + cC + cD + cP
(8)
where cδ = 4(1 − δ1 )/Reδ99 is the laminar drag term, cT = 1 2 0 2(1 − y )(−uv )dy is the contribution of Reynolds shear stress 1 term, cC = 2 0 2(1 − y )(−U¯ V¯ )dy is the mean convection term, 1 cD = 2 0 2(1 − y )2 [∂ U¯ U¯ /∂ x + ∂ uu/∂ x − (1/Reδ )∂ 2U /∂ x∂ x]dy is the 1 spatial development term, and cP = (1/2 ) 0 y2 (−∂ P¯/∂ x )dy is the pressure gradient term, respectively. On the other hand, the
Please cite this article as: H. Hattori et al., Direct numerical simulation of thermally-stratified turbulent boundary layer subjected to adverse pressure gradient, International Journal of Heat and Fluid Flow (2016), http://dx.doi.org/10.1016/j.ijheatfluidflow.2016.05.020
ARTICLE IN PRESS
JID: HFF
[m5G;June 27, 2016;22:30]
H. Hattori et al. / International Journal of Heat and Fluid Flow 000 (2016) 1–16
Fig. 3. Local Stanton numbers.
Fig. 4. Distributions of FIK identity for friction coefficient.
following relation concerning the turbulent thermal boundary layer with a constant wall temperature condition is derived using the identity for the thermal field (Kasagi et al., 2012).
St = cδ + cT + cC + cD
5
(9)
where cδ = 1/Reδ99 P r is the laminar contribution term, cT = 1 0 (−vθ )dy is the contribution of wall-normal turbulent heat flux 1 ¯ dy is the mean convection term, and cD = term, cC = 0 (−V¯ ) 1 2 ¯ ¯ 0 (1 − y ) ∂ U /∂ x + ∂ uθ /∂ x − (1/Reδ P r )∂ /∂ x∂ x dy is the spatial development term, respectively. In the case of WSBL, the contribution of pressure gradient term for Cf hardly varies as shown in Figs. 4(a) and (b), where the spatial development term, cD , and the mean convection term, cC , balance in the most domain. In the re-increase region of Cf in the case of WSBL with APG, however, it can be observed that the mean convection term and the contribution of Reynolds shear stress term, cT , increase so that the balance between cD and cC does not maintain in the region. Thus, the contribution of Reynolds shear stress and the mean convection terms contribute to the increase of Cf . On the other hand, the spatial development term, cD , and the mean convection term, cC , for Stanton number also balance in the most domain as shown in Figs. 5(a) and (b). In the re-increase region of St, however, it can be seen that the mean convection term does not contribute to increase of St, where slight increase of the contribution of wall-normal turbulent heat flux term and slight remain of the spatial development term are observed. Thus, these terms contribute to re-increase of St.
The ratios between Cf /2 and St are shown in Fig. 6 in order to investigate efficiency of heat transfer, where the value of the case of SSBL indicates a negative value due to existing negative value of Cf in the separation region. The ratios between Cf /2 and St of APG in all cases become smaller than those of ZPG, i.e., the efficiency of heat transfer is enhanced due to the APG effect, although the effect of stable thermal stratification also enhances the efficiency of heat transfer. 3.2. Turbulent statistics for velocity field The profiles of turbulent statistics normalized by both the inner and outer scales for velocity field are shown in Figs. 7–11 (Case of SSBL is not shown here). The typical profile of streamwise mean velocity normalized by the inner scale in the case of NBL with APG (Houra and Nagano, 2006; Nagano et al., 1993; 1998) is shown in Fig. 7(a), in which the standard log-law profile is not maintained due to the APG effect and the wake region is raised upward. Since the inner scale in the case of WSBL with APG significantly varies, the remarkably changing profiles of mean velocity are observed as shown in Fig. 7(b). Here, the normalization using inner scale employs the local scale on both axes, i.e., the local friction velocity, uτ , and friction temperature, θ τ , are used, because variations of turbulent statistics at the local location along streamwise direction are shown. On the other hand, the normalization using outer scale is adopted to describe the local boundary layer thickness for the x-axis and the free stream velocity at the inlet of main simulation part. Thus, the absolute variations of statistics from the inlet can be observed. Observing outer scaling profiles of the mean velocity, the
Please cite this article as: H. Hattori et al., Direct numerical simulation of thermally-stratified turbulent boundary layer subjected to adverse pressure gradient, International Journal of Heat and Fluid Flow (2016), http://dx.doi.org/10.1016/j.ijheatfluidflow.2016.05.020
ARTICLE IN PRESS
JID: HFF 6
[m5G;June 27, 2016;22:30]
H. Hattori et al. / International Journal of Heat and Fluid Flow 000 (2016) 1–16
Fig. 5. Distributions of FIK identity for Stanton number.
Fig. 6. Ratios between Cf and St.
remarkable decrease of mean velocity can be seen, and then recovering the mean velocity near the wall in the downstream region is clearly observed as indicated in Fig. 7(b). On the other hand, the decreases of mean velocity in the log-law region are found in the case of USBL, but the wake region is not raised upward as indicated in Fig. 7(c). In all cases, the deceleration of mean velocity normalized by the outer scale due to APG is found in the outer and free stream regions, but the different behaviors of mean velocity can be clearly seen in the inner region. The Reynolds shear stress is demonstrated in Fig. 8, in which the case of ZPG is also included. In the case of NBL, the remarkable increase of Reynolds shear stress normalized by the inner
scale in the case of APG is observed as compared with the case of ZPG along the streamwise direction, but the decrease of the net Reynolds shear stress in the case of APG can be found by the outer scale normalization, though no variations of Reynolds shear stress in the case of ZPG can be seen over x/δ 2, in ∼ 200, in which note that the turbulence spatially develops. In the case of WSBL, although the Reynolds shear stress remarkably decreases in the streamwise direction in both cases of APG and ZPG, it can be seen that the Reynolds shear stress is dramatically re-generated in the downstream region in the case of APG. In the case of ZPG, it can be seen that Reynolds shear stress normalized by the outer scale almost vanishes in the downstream region. Thus, it is considered that the flow becomes laminar in the downstream region, but Reynolds shear stress insistently remains which can be observed in the inner scale normalization. Also, it is revealed by the analysis of identify that the flow does not become laminar as mentioned in the previous section. Therefore, the effect of stable thermal stratification with ZPG of this case cannot laminarize the turbulent flow. Also, since the effect of pure APG does not decrease the turbulence, the flow does not become laminar. Consequently, the local friction coefficient again increases in the downstream region as shown in Fig. 1. Also, the counter gradient diffusion phenomenon (CDP) (Hattori et al., 2014) is found in the outer region near x/δ 2, in ∼ 400 in the case of APG. Since the CDP is also observed in the case of WSBL without APG, the CDP is caused by the stable thermal stratification. In the case of USBL, the increase of Reynolds shear stress normalized by the inner scale can be also seen in both cases of APG and ZPG, but the Reynolds shear stress normalized by the outer scale also increases in the case of APG, though the increase cannot be observed in the case of ZPG. Thus, the effect of unstable thermal stratification with APG enhances the net Reynolds shear stress, which is different phenomenon in comparison with the case of NBL with APG. Thus, the different profiles of mean velocity can be observed as shown in Fig. 7(c). The rms velocity fluctuations in streamwise, wall-normal and spanwise directions are shown in Figs. 9–14, where the case of ZPG is also included in figures. Fig. 9 shows rms velocity fluctu+ + ations normalized by the inner scale, u+ rms , vrms and wrms , in the case of NBL with or without APG. The streamwise rms velocity hardly changes near the wall, but significant increases of the case of APG in the outer region as compared with the case of ZPG can be observed as shown in Fig. 9(a). As for the wall-normal rms velocity fluctuation, remarkable increases of the case of APG in all region can be seen, but v+ rms of the case of ZPG does not vary near the wall as indicated in Fig. 9(b). It can be seen that the distribution of spanwise rms velocity fluctuations indicates the similar tendency of distribution of v+ rms in both cases as shown in Fig. 9(c). Thus, the pure effect of APG increases velocity fluctuations in the outer region, and enhances the wall-normal and spanwise velocity fluctuations as viewed from the perspective of normalization using the inner scale. On the other hand, the rms velocity fluctuations normalized by the outer scale, u∗rms (= urms /U¯ 0 ), v∗rms (= vrms /U¯ 0 ) and w∗rms (= wrms /U¯ 0 ), are shown in Fig. 10, in which v∗rms and w∗rms hardly change in both cases of APG and ZPG, but decrease of u∗rms near the wall can be observed in the case of APG. In view of distributions of Reynolds shear stress as shown in Fig. 7(a), it is considered that the energy transfer from mean streamwise velocity to streamwise fluctuation mainly decreases near the wall by the effect of APG. Figs. 11 and 12 show the distributions of rms velocity fluctuations in the case of WSBL. In view of the inner scale normalization as indicated in Fig. 11, rms velocity fluctuations of all directions remarkably vary in both cases due to the variations of friction velocity caused by the stable thermal stratification, but decreases of rms velocity fluctuations can be observed in the outer region. As for rms velocity fluctuations normalized by the outer scale in the case of WSBL, it can be found that rms velocity
Please cite this article as: H. Hattori et al., Direct numerical simulation of thermally-stratified turbulent boundary layer subjected to adverse pressure gradient, International Journal of Heat and Fluid Flow (2016), http://dx.doi.org/10.1016/j.ijheatfluidflow.2016.05.020
JID: HFF
ARTICLE IN PRESS
[m5G;June 27, 2016;22:30]
H. Hattori et al. / International Journal of Heat and Fluid Flow 000 (2016) 1–16
Fig. 7. Profiles of streamwise mean velocity.
Please cite this article as: H. Hattori et al., Direct numerical simulation of thermally-stratified turbulent boundary layer subjected to adverse pressure gradient, International Journal of Heat and Fluid Flow (2016), http://dx.doi.org/10.1016/j.ijheatfluidflow.2016.05.020
7
JID: HFF 8
ARTICLE IN PRESS
[m5G;June 27, 2016;22:30]
H. Hattori et al. / International Journal of Heat and Fluid Flow 000 (2016) 1–16
(b) Stable stratification
Fig. 8. Profiles of Reynolds shear stress.
Please cite this article as: H. Hattori et al., Direct numerical simulation of thermally-stratified turbulent boundary layer subjected to adverse pressure gradient, International Journal of Heat and Fluid Flow (2016), http://dx.doi.org/10.1016/j.ijheatfluidflow.2016.05.020
JID: HFF
ARTICLE IN PRESS
[m5G;June 27, 2016;22:30]
H. Hattori et al. / International Journal of Heat and Fluid Flow 000 (2016) 1–16
9
Fig. 9. Profiles of rms velocity fluctuation normalized by inner scale in the case of NBL.
Fig. 10. Profiles of rms velocity fluctuation normalized by outer scale in the case of NBL.
fluctuations of all directions decrease in the case of ZPG as shown in Fig. 12. On the other hand, decreases of rms velocity fluctuations of all directions due to thermal stratification can be also observed in the case of APG, but these again increase over x/δ2,in = 400. In particular, increases of rms velocity fluctuations of all directions in the outer region can be observed at x/δ2,in = 800. Thus, the effect of APG in the stable thermal stratification is of increase of velocity fluctuations in the outer region. The case of USBL is demonstrated in Figs. 13 and 14. In view of the inner scale normalization, it can be found that rms velocity fluctuations of all
directions increase in the outer region in both cases of APG and ZPG, but remarkable increases of the near-wall rms velocity fluctuation of spanwise direction can be observed in the case of APG as indicated in Fig. 13. As for the profiles of rms velocity fluctuations normalized by the outer scale, remarkable increases of rms velocity fluctuation of wall-normal direction are obviously found in the case of APG as shown in Fig. 14. The rms velocity fluctuation of spanwise direction also increases near the wall in the case of APG, but rms velocity fluctuation of streamwise direction hardly varies. Thus, the effect of APG in the unstable thermal
Please cite this article as: H. Hattori et al., Direct numerical simulation of thermally-stratified turbulent boundary layer subjected to adverse pressure gradient, International Journal of Heat and Fluid Flow (2016), http://dx.doi.org/10.1016/j.ijheatfluidflow.2016.05.020
JID: HFF 10
ARTICLE IN PRESS
[m5G;June 27, 2016;22:30]
H. Hattori et al. / International Journal of Heat and Fluid Flow 000 (2016) 1–16
Fig. 11. Profiles of rms velocity fluctuation normalized by inner scale in the case of WSBL.
stratification especially is of an enhancement of wall-normal rms velocity fluctuation, which clearly influences an increase of Reynolds shear stress as shown in Fig. 8(c). 3.3. Turbulent statistics for thermal field Turbulent statistics in thermal field are shown in Fig. 15–18 (Also, case of SSBL is not shown here). Profiles of mean temperature are demonstrated in Fig. 15. As for the profiles normalized by ¯ + , only the decrease of mean temperature in the the inner scale,
Fig. 12. Profiles of rms velocity fluctuation normalized by outer scale in the case of WSBL.
log-law region can be seen in the case of NBL, but the maximum ¯ + does not change, because the ratio between Cf and St value of ¯+ hardly varies in the case of NBL where the maximum value of is proportional to 1/θ τ , that is C f /2/St. Hence, the maximum ¯ + decreases with decreasing the ratio in the case of value of USBL as shown in Fig. 15(c). It can be seen that mean temperature normalizer by the outer scale is slow to increase in the case of NBL near the wall as shown in Fig. 15(a), but rapid increases of temperature can be observed in the case of USBL as shown in
Please cite this article as: H. Hattori et al., Direct numerical simulation of thermally-stratified turbulent boundary layer subjected to adverse pressure gradient, International Journal of Heat and Fluid Flow (2016), http://dx.doi.org/10.1016/j.ijheatfluidflow.2016.05.020
JID: HFF
ARTICLE IN PRESS
[m5G;June 27, 2016;22:30]
H. Hattori et al. / International Journal of Heat and Fluid Flow 000 (2016) 1–16
11
Fig. 13. Profiles of rms velocity fluctuation normalized by inner scale in the case of USBL.
Fig. 15(c). As for mean temperature profile normalized by the inner scale in the case of WSBL as indicated in Fig. 15(b), it can be seen that the conductive sublayer is maintained in all region, but the varying profiles in the log-law region are observed. Observing outer scaling profiles of the mean temperature, the remarkable decrease of mean temperature can be seen, and then recovering the mean temperature near the wall in the downstream region is clearly observed as indicated in Fig. 15(b). Turbulent heat fluxes in streamwise and wall-normal directions are shown in Figs. 16 and 17 including the case of ZPG, where the streamwise turbulent heat flux, uθ , clearly affects the Reynolds shear stress in the thermally-stratified turbulent boundary layer, because uθ is contained in the buoyancy term of the transport equation of Reynolds shear stress as Guv = Riδ2,in uθ . The
Fig. 14. Profiles of rms velocity fluctuation normalized by outer scale in the case of USBL.
wall-normal turbulent heat flux normalized by the inner scale, +
vθ , increases in the downstream region in both cases of NBL and USBL. The wall-normal turbulent heat flux normalized by the outer ∗ scale, vθ = vθ /U¯ , in the case of NBL with APG decreases near ∗ the wall in the downstream region, but vθ in the case of USBL with APG increases in the downstream region. Thus, the different profiles of mean temperature between cases of NBL and USBL with APG are obtained as shown in Fig. 15, because vθ is closely associated with a formation of mean temperature profile. On the other hand, in the case of WSBL with or without APG, vθ
+
decreases in
Please cite this article as: H. Hattori et al., Direct numerical simulation of thermally-stratified turbulent boundary layer subjected to adverse pressure gradient, International Journal of Heat and Fluid Flow (2016), http://dx.doi.org/10.1016/j.ijheatfluidflow.2016.05.020
ARTICLE IN PRESS
JID: HFF 12
[m5G;June 27, 2016;22:30]
H. Hattori et al. / International Journal of Heat and Fluid Flow 000 (2016) 1–16
Fig. 15. Profiles of mean temperature.
the downstream region, and then vθ of ZPG, two peak values of vθ
+
+
again increases. In the case
appear at x/δ2,in = 800. Thus, the +
effect of APG in the case of WSBL is of enhancement of vθ at the ∗ ∗ log and outer regions. As for vθ in the case of WSBL, vθ consider∗ ably becomes small at x/δ2,in = 400, but it can be seen that vθ is
∗
re-generated as shown in Fig. 16(b). However, the case of ZPG, vθ almost vanishes at the downstream region. Thus, the wall-normal turbulent heat flux in the case of ZPG does not contribute the enhancement of heat transfer coefficient as revealed by the analysis of identify of Fig. 5(b). On the other hand, because the effect of
Please cite this article as: H. Hattori et al., Direct numerical simulation of thermally-stratified turbulent boundary layer subjected to adverse pressure gradient, International Journal of Heat and Fluid Flow (2016), http://dx.doi.org/10.1016/j.ijheatfluidflow.2016.05.020
JID: HFF
ARTICLE IN PRESS
[m5G;June 27, 2016;22:30]
H. Hattori et al. / International Journal of Heat and Fluid Flow 000 (2016) 1–16
13
Fig. 16. Profiles of wall-normal turbulent heat flux.
APG does not suppress the wall-normal turbulent heat flux, the wall-normal turbulent heat flux contributes the enhancement of heat transfer coefficient as shown in Fig. 5(a). The streamwise turbulent heat flux is demonstrated in Fig. 17. It can be found that the streamwise turbulent heat flux normalized by the inner scale, +
uθ , in cases when both NBL and USBL with APG slightly decrease, ∗ and uθ which is the streamwise turbulent heat flux normalized
by the outer scale, also decreases in cases of both NBL and USBL ∗ with or without APG, but the near-wall uθ obviously increases in the case of USBL with APG. Since uθ affects the Reynolds shear stress in the thermally-stratified turbulent boundary layer as mentioned above, the increase of near-wall uθ works on the enhancement of the Reynolds shear stress in the case of USBL as shown in Fig. 8(c). In the case of WSBL, the behavior of uθ is similar to the
Please cite this article as: H. Hattori et al., Direct numerical simulation of thermally-stratified turbulent boundary layer subjected to adverse pressure gradient, International Journal of Heat and Fluid Flow (2016), http://dx.doi.org/10.1016/j.ijheatfluidflow.2016.05.020
JID: HFF 14
ARTICLE IN PRESS
[m5G;June 27, 2016;22:30]
H. Hattori et al. / International Journal of Heat and Fluid Flow 000 (2016) 1–16
Fig. 17. Profiles of streamwise turbulent heat flux.
behavior of vθ as shown in Fig. 16(b), but it can be observed in the ∗ ∗ case of APG that uθ is not very small in comparison with vθ at ∗ x/δ2,in = 400. Since uθ does not vanish in the case of WSBL with APG, it is considered that the turbulence is maintained in the case of WSBL with APG as mentioned in the discussion of wall-normal turbulent heat flux.
Finally, the rms temperature fluctuation is shown in Fig. 18. In cases of both NBL and USBL with APG, it can be seen that the peak value of the rms temperature fluctuation normalized by +
+
the inner scale, θ 2 , decreases, and θ 2 increases in the outer ∗ region. Although θ 2 which is the rms temperature fluctuation normalized by the outer scale only decreases in the case of NBL,
Please cite this article as: H. Hattori et al., Direct numerical simulation of thermally-stratified turbulent boundary layer subjected to adverse pressure gradient, International Journal of Heat and Fluid Flow (2016), http://dx.doi.org/10.1016/j.ijheatfluidflow.2016.05.020
ARTICLE IN PRESS
JID: HFF
[m5G;June 27, 2016;22:30]
H. Hattori et al. / International Journal of Heat and Fluid Flow 000 (2016) 1–16
15
Fig. 18. Profiles of rms temperature fluctuation.
∗
∗
the increase of θ 2 near the wall and the decrease of θ 2 in the region around 0.1y/δ t ∼ 0.5y/δ t can be observed in the case of USBL with APG. This different profile strongly relates the profiles of the wall-normal turbulent heat flux as shown in Fig. 16, because the wall-normal turbulent heat is included in the production term of transport equation of θ 2 . In the case of WSBL with or without
APG, it can be observed that θ 2 of both the normalizations remains in all regions, although both the wall-normal turbulent heat flux and the gradient of mean temperature which compose the production term of θ 2 considerably become small at x/δ2,in = 400. Thus, it is considered that terms of transport equation except for the production term act on the sustentation of θ 2 .
Please cite this article as: H. Hattori et al., Direct numerical simulation of thermally-stratified turbulent boundary layer subjected to adverse pressure gradient, International Journal of Heat and Fluid Flow (2016), http://dx.doi.org/10.1016/j.ijheatfluidflow.2016.05.020
JID: HFF 16
ARTICLE IN PRESS
[m5G;June 27, 2016;22:30]
H. Hattori et al. / International Journal of Heat and Fluid Flow 000 (2016) 1–16
4. Conclusions
Acknowledgement
DNSs of thermally-stratified turbulent boundary layers subjected to a non-equilibrium APG are carried out. DNS results clearly show the characteristic of thermally-stratified nonequilibrium APG turbulence flow. Especially, it is found that the friction coefficient and Stanton number decrease along the streamwise direction due to the effects of stable thermal stratification and APG, but those again increase due to the APG effect in the case of WSBL. Thus, the analysis for both the friction coefficient and Stanton number in the case of WSBL with/without APG is conducted using the FIK identity (Fukagata et al., 2002; Kasagi et al., 2012) in order to investigate contributions from the transport equations, in which it is found that both Reynolds-shear-stress and the mean convection terms of the friction coefficient and both the wallnormal turbulent heat flux and the spatial development terms of the Stanton number contribute to again increase those values in the case of WSBL with APG. Also, it is found that the mean velocity profiles normalized by the inner scale of APG flow increase in the case of WSBL and decrease in the case of USBL in the outer region due to the variation of friction coefficients, but decreases of free stream due to APG can be observed in both cases. Also, characteristic profiles of turbulence are detected in the thermal field, in which decreases of mean temperature normalized by the inner scale are observed in cases of NBL and USBL, although increases of mean temperature can be seen in the case of WSBL. Consequently, DNS results reveal that the turbulent characteristics of both stable and unstable thermal stratifications with APG are different from those of the NBL affected by APG.
This research was supported by a Grant-in-Aid for Scientific Research (C), 26420144, from the Japan Society for the Promotion of Science (JSPS). References Araya, G., Castillo, L., 2013. Direct numerical simulations of turbulent thermal boundary layers subjected to adverse streamwise pressure gradients. (1994-present). Phys. Fluids 25 (9). Clauser, F.H., 1954. Turbulent boundary layers in adverse pressure gradients. J. Aeronaut. Sci. 21 (2), 91–108. Fukagata, K., Iwamoto, K., Kasagi, N., 2002. Contribution of reynolds stress distribution to the skin friction in wall-bounded flows. Phys. Fluids 14 (11). Hattori, H., Hotta, K., Houra, T., 2014. Characteristics and structures in thermally-stratified turbulent boundary layer with counter diffusion gradient phenomenon. Int. J. Heat Fluid Flow 49, 53–61. Hattori, H., Houra, T., Nagano, Y., 2007. Direct numerical simulation of stable and unstable turbulent thermal boundary layers. Int. J. Heat Fluid Flow 28, 1262–1271. Hattori, H., Nagano, Y., 2004. Direct numerical simulation of turbulent heat transfer in plane impinging jet. Int. J. Heat Fluid Flow 25, 749–758. Hou, Y., Somandepalli, V.S.R., Mungal, M.G., 2006. A technique to determine total shear stress and polymer stress profiles in drag reduced boundary layer flows. Exp. Fluids 40 (4), 589–600. Houra, T., Nagano, Y., 2006. Effects of adverse pressure gradient on heat transfer mechanism in thermal boundary layer. Int. J. Heat Fluid Flow 27 (5), 967– 976. Kasagi, N., Hasegawa, Y., Fukagata, K., Iwamoto, K., 2012. Control of turbulent transport: less friction and more heat transfer. ASME. J. Heat Transfer 134 (3). Lee, J.H., Sung, H.J., 2008. Effects of an adverse pressure gradient on a turbulent boundary layer. Int. J. Heat Fluid Flow 29 (3), 568–578. Nagano, Y., Tagawa, M., Tsuji, T., 1993. Effects of adverse pressure gradients on mean flows and turbulence statistics in a boundary layer. Turbulent Shear Flows 8, 7–21. Nagano, Y., Tsuji, T., Houra, T., 1998. Structure of turbulent boundary layer subjected to adverse pressure gradient. Int. J. Heat Fluid Flow 19 (5), 563–572. Ohya, Y., 2001. Wind-tunnel study of atmospheric stable boundary layers over a rough surface. Boundary-Layer Meteorol. 98, 57–82.
Please cite this article as: H. Hattori et al., Direct numerical simulation of thermally-stratified turbulent boundary layer subjected to adverse pressure gradient, International Journal of Heat and Fluid Flow (2016), http://dx.doi.org/10.1016/j.ijheatfluidflow.2016.05.020