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Extended synthetic eddy method to generate inflow data for turbulent thermal boundary layer
International Journal of Heat and Mass Transfer 134 (2019) 1261–1267
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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Extended synthetic eddy method to generate inflow data for turbulent thermal boundary layer Geunwoo Oh, Kyung Min Noh, Hyunwook Park, Jung-Il Choi ⇑ Department of Computational Science and Engineering, Yonsei University, Seoul 03722, Republic of Korea
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Article history: Received 30 June 2018 Received in revised form 7 February 2019 Accepted 18 February 2019
Keywords: Synthetic eddy method Inflow boundary condition Turbulent thermal boundary layer Large-eddy simulation
a b s t r a c t We propose an extended synthetic eddy method (XSEM) based on the synthetic eddy method, which includes the temperature fluctuation component in the turbulent flux tensors, to generate timedependent turbulent thermal inflow data for a spatially-developing boundary layer. The proposed XSEM is applied to large eddy simulations of a spatially-developing turbulent thermal boundary layer on a flat plate with an isothermal wall condition. Time-varying turbulent thermal inflow fields are reconstructed by composing the prescribed mean and reproduced fluctuations fields, along with Cholesky decomposition to the turbulent flux tensor with thermal flux. The obtained results indicate that the inflow generated by the proposed XSEM provides self-sustaining turbulence with fully recovered turbulent statistics behind a re-developing boundary layer region with the recovery distance roughly seven times of the inlet boundary layer thickness. Finally, we demonstrate the robustness of the XSEM for providing appropriate inflow data for simulations of turbulent thermal boundary layer flows for different Prandtl numbers. Ó 2019 Elsevier Ltd. All rights reserved.
1. Introduction Understanding the mechanism of turbulent thermal boundary layer flows is one of the primary issues in various fields of engineering wherein surface heat transfer is related to turbulent flows. To address this problem, high-fidelity numerical simulations, such as large-eddy simulations (LES) or direct numerical simulations (DNS), have been conducted to resolve turbulent structures near the surface in order to predict the characteristics of such surface heat transfer. However, performing these simulations requires time-varying realistic thermal turbulent data at an inlet boundary, wherein the generated inflow data satisfies the governing equations and varies stochastically [1,2]. A recent review [2] summarizes the vigorous research activities on inflow turbulence generation methods for eddy-resolving computations of spatiallydeveloping turbulent flows, such as the synthetic Fourier method [3], recycling/rescaling method [4], synthetic eddy method [5], and digital filtering method [6], among others. For more information, please refer to [1,2]. Although significant improvements have been achieved in inflow generation methods, relatively only a few inflow studies on turbulent thermal flows have been conducted.
⇑ Corresponding author. E-mail address: [email protected] (J.-I. Choi). https://doi.org/10.1016/j.ijheatmasstransfer.2019.02.061 0017-9310/Ó 2019 Elsevier Ltd. All rights reserved.
Several numerical studies have been conducted for a spatiallydeveloping turbulent thermal boundary layer utilizing an inflow generation method to generate time-dependent inlet data. Bell and Ferziger [7] performed numerical simulations of turbulent thermal flow on an isothermal wall for a momentum thickness Reynolds number RedM ranging from 300 to 700 and a molecular Prandtl number of 0.1, 0.71, and 2.0. In particular, they generated a thermal boundary layer using the fringe concept, transforming the non-periodic inlet and outlet condition to periodic conditions. Li et al. [8] performed DNS of a turbulent thermal boundary layer for RedM = 830 and Pr = 0.2, 0.71, and 2.0 by employing the fringe method as well [7]; they reported the differences in the scalar behavior at the outer region compared with that in the channel flow. Kong et al. [9] extended the recycling/rescaling method [4] to turbulent thermal boundary layer flows, reintroducing a spatially-developed flow from a downstream location using an appropriate rescaling process for the inlet in order to generate time-dependent thermal inflow data for RedM ¼ 300 and Pr ¼ 0:71; they performed DNS of turbulent boundary layer flows on an iso-thermal wall or iso-flux wall. Furthermore, Araya and Castillo [10] modified the recycling/rescaling method [4,9] for an adverse-pressure gradient turbulent boundary layer; in particular, they performed DNS of the turbulent thermal boundary layer over a rough isothermal wall with weak stratification for RedM ¼ 2300 and Pr ¼ 0:71. Li et al. [11] obtained the inlet flow through an
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auxiliary computation; in their study, instantaneous fluid velocity and temperature fields were extracted in the form of a time series at the location of RedM ¼ 1100. In particular, they performed DNS of turbulent thermal boundary layer for RedM ¼ 1100—1940 with Pr ¼ 0:71 by introducing the precursor data at the inlet; they observed a strong correlation between streamwise velocity and temperature fluctuations near the wall. Wu and Moin [12] conducted a DNS study for spatially-developing turbulent thermal boundary layers at zero-pressure gradient; they observed a transition of the boundary layer for RedM ¼ 80—1950 and Pr ¼ 1:0. In their study, pre-computed isotropic turbulence is superimposed to mean flows at the inlet to generate turbulent boundary layer. Recently, Okaze and Mochida [13] extended the digital-filter
2 pffiffiffiffiffiffiffiffi H11 6 6 H =L 6 21 11 6 Lij ¼ 6 6 H31 =L11 6 4 H41 =L11
0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi H22 L221
0
ðH32 L21 L31 Þ=L22
0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi H33 L231 L232
ðH42 L21 L41 Þ=L22
ðH43 L31 L41 L32 L42 Þ=L33
subscripts i; j ¼ 1; 2; 3 indicate the velocity fluctuations u0 ; v 0 ; w0 in streamwise, wall-normal, and spanwise velocities, respectively, while i; j ¼ 4 denotes temperature fluctuations h0 . It should be noted that / represents an average quantity of / in time and in the spanwise direction because the spanwise direction is homogeneous in a spatially-evolving turbulent boundary layer on a flat plate. In general, / can be defined as an ensemble (or spatiotemporal) average of /. As previously mentioned, in a manner similar to the study by Okaze and Mochida [13], Cholesky decomposition is applied to the turbulent flux tensor Hij to obtain the lower-triangular matrix Lij ;
0
3
7 7 7 7 7: 7 0 7 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5 2 2 2 H44 L41 L42 L41
method [6] to generate the turbulent fluctuations for fluid velocity and contaminant concentration; they applied the digital-filter method to LES of a half-channel flow for contaminant dispersion with no buoyancy effect. In this study, we propose an extended synthetic eddy method (XSEM) that can generate turbulent thermal inflow data based on the synthetic eddy method (SEM) [5] by including temperature fluctuations. In a manner similar to the study by Okaze and Mochida [13], Cholesky decomposition is applied to the turbulent flux tensor involving Reynolds stress and thermal flux. In addition, velocity and temperature fluctuations are generated using reproduced fluctuations, along with an eddy-convecting process in a virtual box. Our proposed XSEM is applied to LES of a spatiallydeveloping turbulent thermal boundary layer with an isothermal wall. Furthermore, we investigate the recovery of turbulent statistics for turbulent thermal boundary layer flows generated by the proposed XSEM and compare the statistics with those obtained using other inflow methods. Finally, we demonstrate the robustness of XSEM for various Prandtl numbers by visualizing turbulent coherent structures and temperature fluctuation fields near the wall; we also present a re-developing distance for the recovery of turbulent statistics based on Prandtl number.
2. Numerical method 2.1. Extended synthetic eddy method We propose the XSEM to generate turbulent thermal inflow data by the superposition of reproduced coherent structures to the prescribed mean fields. The coherent structures can be obtained using an eddy-convecting process in a virtual box, where the fluctuations of velocity components and temperature satisfy the second-order statistics of the turbulent thermal boundary layer based on the prescribed statistics. Detailed descriptions of the synthetic eddy method (SEM) can be found in [5]. An extended turbulent flux tensor Hij ¼ u0i u0j is comprised of Reynolds stresses and thermal fluxes related to temperature fluctuations, where the
0
ð1Þ
Similar to the SEM [5], a real-time varying turbulent thermal inflow field /i ðx; t Þ can be reconstructed by a composition of the prescribed mean field Ui and reproduced fluctuations /0i as follows: N 1 X /i ðx; tÞ ¼ Ui ðxÞ þ /0i ðx; tÞ ¼ Ui ðxÞ þ pffiffiffiffi Lij kj f rðxÞ x xk ; N k¼1
ð2Þ
where /i denotes ½u; v ; w; hT and kj is an arbitrary intensity in steps th
th
of 1 or 1 indicating the j component of the k synthetic eddy. The number of synthetic eddies N can be approximated using the ratio of the area of the inlet plane to the area covered by a synthetic eddy; in our study, we set N ¼ 1184. The virtual box containing the synthetic eddies is defined as ½r; r r; Ly þ r ½r; Lz þ r with parts corresponding to the x; y, and z directions, respectively, where Ly and Lz are the sizes of the computational domain in wall-normal and spanwise directions, respectively. Here, the inlet plane is located at x ¼ 0. Furthermore, it should be noted that r is the size of the synthetic eddies, which will be determined later. The shape function f rðxÞ has a compact support on k pffiffiffiffiffiffi k yy zzk with a normal½r; r; f rðxÞ x xk ¼ V B r3 f xx r f r f r R1 2 ization condition 1 f ðxÞdx ¼ 1; this function determines the velocity distribution of the synthetic eddy at location xk . Here, V B is the volume of the virtual box. The truncated Gaussian function [14], f ðxÞ ¼ C exp 9x2 =2 if jxj < 1, and 0 otherwise, is used in this study for the shape function, where the constant C is set based on the normalization condition. The synthetic eddies convect with a convection velocity U c ; xk ðt þ DtÞ ¼ xk ðt Þ þ U c Dt, where U c is the averaged streamwise mean velocity and Dt is the computational time step. As illustrated in Fig. 1, the synthetic eddies exiting out of the virtual box are re-introduced at the inlet of the box. The generated thermal turbulent fields at x ¼ 0 in the virtual box are imposed to the main simulation domain as the inlet boundary condition. Further information on the general SEM procedure can be found in [5,14]. Considering the influence of thermal fluctuations on the coherent turbulent structures, the size of an eddy is defined as
G. Oh et al. / International Journal of Heat and Mass Transfer 134 (2019) 1261–1267
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Fig. 1. Schematic diagram of the XSEM procedure. The scattered spheres represent synthetic eddies, which are colored based on the instantaneous temperature h.
n
n
o
o
r ¼ max min k3=2 =; jd; jh dT ; Dx , where k is the turbulent kinetic energy, is the dissipation, j is the von Karman constant for fluid flow, d is the boundary layer thickness, jh is the von Karman constant for temperature, and dT is thermal boundary layer thickness. It should be noted that jh dT represents the length scale of the thermal effects on the eddy, where jh is set to 0.47 considering the turbulent Prandtl number Prt ¼ mt =at and the mixing length theory for momentum and heat transfer [15]; furthermore, Pr t ð jus yÞ=ð jh us yÞ ¼ j=jh , where us is the friction velocity. 2.2. Numerical simulations For numerical simulations of a turbulent thermal boundary layer, we consider the non-dimensionalized incompressible Navier-Stokes equations and energy equation for forced convection flow [9]. The governing equations are solved using an efficient monolithic projection method in a staggered Cartesian grid, preserving the second-order accuracy both in time and space [16– 18]. The Crank–Nicolson scheme is adopted to discretize all terms in the governing equations in time along with a linearization technique for the nonlinear convection terms, while second-order finite difference schemes are applied in space. Approximate block lowerupper (LU) decompositions, along with the approximate factorization technique for linear systems are employed to solve the momentum equations as well as energy equation without any iterations. Moreover, the Poisson equation in the projection method is solved using a multi-grid method with a Fourier diagonalization technique for a fast computation [18]. LES of the turbulent thermal boundary layer for RedM ¼ 1440 and Pr ¼ 0:71 are performed to investigate the effects of inflow conditions on the evolving turbulent profiles. A dynamic Vreman model is used as the sub-grid scale turbulence model in our simulations, which is a variant of the Vreman model [19,20]. The computational domain is defined as 0 6 x=dM;0 6 200; 0 6 y=dM;0 6 50 and 0 6 z=dM;0 6 22:5 with 269 61 97 grid points in streamwise, wall-normal, and spanwise directions, respectively, where dM;0 is the momentum thickness at the inlet. In particular, uniform grids are used in streamwise and spanwise directions, while a nonuniform grid clustered to the bottom wall using a hyperbolic tangent function is applied in the wall-normal direction. The corresponding grid spacings in the wall unit are Dxþ 45; Dyþ min 0:5 and Dzþ 22:5 in the respective directions, which provides sufficient resolution for wall-resolved LES [21]. The convective outflow boundary condition is considered at the exit for both velocity and temperature, while the periodic boundary conditions are used in the spanwise direction [22]. Furthermore, the no-slip boundary and zero isothermal boundary conditions are applied to the wall,
while the freestream conditions for the velocity fields are applied to the far-field boundary [22] with temperature set to unity. For the inlet conditions, first, we obtained the realistic velocity and temperature fluctuations at the inlet based on the recycling/ rescaling (R/R) method [9] using an auxiliary simulation of the spatially developing turbulent thermal boundary layer flow. Then, we consider three kinds of inflow generation methods for comparison: (1) Precursor Method: the stored instantaneous plane data of turbulent fluctuations are introduced at the inlet of the main simulation for each time step; (2) XSEM: real-time varying turbulent inflow fields at each time step are described using Eq. (2) with the mean inlet velocity and temperature profiles and the secondorder statistics obtained using the R/R method; (3) Random Fluctuations Method [3]: the resulting fluctuations /0i are scaled such that the calculated fluctuations conform to the turbulent flux tensor Hij , associated with the inlet turbulent profiles obtained using the R/R method.
3. Results and discussion 3.1. Assessment of the XSEM on the recovery of turbulent statistics We performed LES of spatially-evolving turbulent thermal boundary layer flows on a flat plate to evaluate the performance of our proposed XSEM in providing self-sustaining inflow turbulence and downstream recovery of turbulent statistics. The precursor and random fluctuation methods for generating inflow data were also used for numerical simulations to compare their recovery statistics with those of our method. As a baseline flow condition, a turbulent thermal boundary layer with RedM ¼ 1440 at the inlet and Pr ¼ 0:71 is selected for our study. Fig. 2 shows the influence of the inflow turbulence generated by the XSEM on downstream turbulent statistics. The turbulent statistics at the inlet ðx=d0 ¼ 0Þ indicate that XSEM generates inflow data that satisfies the prescribed first- and second-order statistics, compared with those obtained using the precursor method as well as the previous DNS results reported in [10,23]; here, d0 represents the boundary layer thickness at the inlet. However, XSEM slightly over-predicts the mean streamwise velocity U þ , temperature Hþ and thermal flux u0 h0 near the inlet region ðx=d0 ¼ 1Þ, while significantly under-predicting the Reynolds stress u0 v 0 , compared with other results specified above. As the boundary layer develops further downstream, U þ and Hþ , and u0 h0 obtained using the XSEM are similar to other results at x=d0 P 5, while u0 v 0 becomes similar at x=d0 P 7. This confirms that the inflow generated by the XSEM provides self-sustaining turbulence behind the redeveloping boundary layer region ð0 6 x=d0 6 7Þ with reproduc-
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δ
δ
δ
δ
δ
δ
θ
Θ
×
×
Fig. 2. Downstream development of turbulent statistics: (a) mean streamwise velocity, U þ ; (b) Reynolds stress, u0 v 0 ; (c) mean temperature, Hþ ; (d) thermal flux, u0 h0 ; XSEM ( ) and precursor method ( ) at RedM;0 ¼ 1440 with Pr ¼ 0:71; Spalart (j) at RedM ¼ 1410; Araya and Castillo (N) at RedM ¼ 2290.
δ (a)
δ (b)
Fig. 3. Downstream development of (a) C f and (b) St based on the different inflow generation methods.
tion of turbulent kinetic energy, where the corresponding turbulent statistics are fully recovered. It is important to note that the recovery feature of temperature follows that of the fluid flow because the fluid-thermal interaction is one-way coupled for the forced convection dominant flow. Fig. 3 indicates the effect of inflow generation methods on the recovery of the skin friction coefficient C f ¼ sw =ð2qU 1 Þ and Stan ton number St ¼ h= qcp U 1 , where sw is the wall shear stress, q is the density of the fluid, U 1 is the free-stream velocity, h is the convective heat transfer coefficient, and cp is the specific heat of the fluid at constant pressure. Compared with the precursor method, C f and St that are obtained using the XSEM and random fluctuation method sharply decrease near the inlet region. However, the XSEM provides similar predictions for C f and St at further downstream locations ðx=d0 P 7Þ to those obtained using the precursor method; in contrast, the results for the random fluctuation method indicate that the near-wall turbulence severely weakens,
and the flow tends to laminarize, which is consistent with the observation in [5]. To further investigate the recovery of near-wall turbulence based on the inflow generation methods, C f and St are shown in Fig. 4 in terms of boundary layer characteristics such as RedM and RedE representing the Reynolds number based on momentum ðdM Þ and enthalpy thickness ðdE Þ, respectively. For comparison, we consider the semi-empirical correlations derived by Kays and Crawford [24]; 1=4
C f ¼ 0:025RedM ; St ¼ 0:0125Pr
0:5
Re1=4 : dE
ð3Þ ð4Þ
where Eq. (4) is valid in the range of 0:5 6 Pr 6 1:0 with a high Reynolds number. Furthermore, Fig. 4 indicates that the precursor method provides monotonically decreasing C f and St with the development of the boundary layer, which is similar to the feature
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×
×
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δ
δ
Fig. 4. Comparisons of (a) C f and (b) St obtained using different inflow generation methods with respect to boundary layer characteristics.
δ
Θ
θ
δ
(a)
(b)
Fig. 5. Profiles of (a) Hþ and (b) u0 h0 at x=d0 ¼ 7 obtained using the XSEM for different Pr numbers, compared with results from other studies.
of the semi-correlations. The results of the XSEM indicate sharp decreases in C f and St near the inlet region (see Fig. 3), leading to the reduction in the momentum and enthalpy thicknesses as shown in Fig. 4. However, the XSEM shows fairly good predictions of C f and St after the recovery of the turbulent statistics at further downstream locations ðx=d0 P 7:0Þ compared with those obtained using the semi-correlations and precursor method. It should be noted that the corresponding RedM and RedE at x=d0 ¼ 7:0 are roughly 1600 and 1800, respectively. In contrast, the inflow turbulence generated by the random fluctuation method could not maintain characteristics of the turbulent boundary layer because of significant momentum and enthalpy losses near the inlet region.
at x=d0 7, where the location is roughly behind the re-developing region for the recovery of statistics as discussed in Section 3.1. Thus, both predicted results show good agreement with the existing DNS results for the boundary layer flows at RedM ¼ 2290 [10] and channel flows at Res ¼ 180 [25] as well as Kader’s correlation [26]. This confirms that the XSEM is capable of generating reliable inflow data that provides a self-sustaining turbulent thermal boundary layer for different Pr. Fig. 6 shows the downstream development of St=St 0 based on Pr; these results indicate that heat transfer characteristics are recovered at x=d0 10 for the three Pr cases, while the recovery region becomes slightly longer for a
3.2. Applicability of the XSEM for different Prandtl numbers LES of the developing turbulent thermal boundary layer at RedM ¼ 1440 were also performed with Pr ¼ 0:2 and 5:0 to investigate the robustness of XSEM based on different Pr. To impose turbulent inflow data using the XSEM, we obtained the prescribed statistics for the XSEM by performing an auxiliary simulation using the R/R method [9] for each Pr case. Because the thermal boundary layer for Pr ¼ 5:0 is relatively thinner than those for other cases, half of the grid spacings used in the baseline simulation are applied in each direction for a sufficient resolution: Dxþ 22:5; Dyþmin 0:32, and Dzþ 11:3 with 537 71 193 grid points. Fig. 5 indicates that the XSEM provides considerably similar predictions of Hþ and u0 h0 compared with the precursor method regardless of Pr. It should be noted that the statistics are obtained
δ Fig. 6. Downstream development of St=St 0 for different Pr numbers.
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Fig. 7. Vortical structures (a,b,c) and temperature fluctuations h0þ (d,e,f) near the wall for the turbulent thermal boundary layer using the XSEM for different Pr numbers. The vortical structures are identified by k2 method [27] with a criterion; k2 ¼ 0:013. The iso-surfaces are colored based on the value of H. Contour plots of h0þ are shown in a horizontal plane at yþ ¼ 12; 10:5, and 8 for Pr ¼ 0:2; 0:71, and 5.0, respectively.
low Pr fluid. This can be explained by noting that the thermal boundary layer is thicker than the boundary layer for a low Pr fluid, thus requiring a longer re-developing region because of low thermal fluxes, leading to slower recovery. Finally, the vortical structures in the developing turbulent thermal boundary layer flows obtained using the XSEM are visualized in Fig. 7 (a)–(c) using the k2 criterion [27] with k2 ¼ 0:013; these are colored based on the value of H, for different Pr cases. In addition, contour plots of h0þ are shown in Fig. 7 (d)–(f) in the horizontal plane, where the maximum value of u0 h0 is observed at yþ ¼ 12; 10:5, and 8 for Pr ¼ 0:2; 0:71, and 5:0, respectively. Except for near the inlet region, it can be clearly observed that coherent vortical structures are well maintained at downstream locations for all Pr cases. For a high Pr fluid, the vortical structures carrying hot fluid introduce higher thermal fluxes near the wall, resulting in more elongated temperature fluctuation fields in the streamwise direction, while relatively isotropic temperature fluctuations are observed for a low Pr fluid. Similar to the DNS results in the study by Redjem-Saad et al. [28] for turbulent heat transfer in pipe flows, the our obtained results indicate that the thermal fields tend to be more isotropic for a lower Pr fluid owing to the thicker conductive sublayer. However, for a higher Pr fluid, thinner conductive regions reduce the molecular heat flux but enhance the turbulent heat flux normal to the wall, which, in turn, increases the correlation between the thermal and velocity fluctuations, leading to thermal streaky structures. The instantaneous flow and thermal fields in Fig. 7 indicate that the proposed XSEM provides reliable inflow data to simulate a developing turbulent thermal boundary layer flow with a short recovery region.
4. Conclusions In this study, we proposed the XSEM to provide time-varying inflow turbulence data for simulating spatially-developing turbulent thermal boundary layer flows. The thermal inflow fields were reconstructed using the prescribed mean and reproduced fluctua-
tion fields with Cholesky decomposition of the turbulent flux tensor including thermal flux, based on the original SEM [5]. We performed LES of the developing turbulent thermal boundary layer flows for RedM ¼ 1440 at the inlet in order to investigate the influence of the inflow turbulence generated by the XSEM on downstream turbulent statistics. Downstream development of mean velocity and temperature, Reynolds stress, and thermal flux confirms that the XSEM provides self-sustaining turbulence with fully recovered turbulent statistics behind the downstream location x=d0 7. Moreover, the XSEM provides similar predictions of C f and St after the turbulent thermal boundary layer is fully redeveloped behind the location, compared with those obtained using the precursor method. Additional LES results for the boundary layer at different Pr numbers show that the XSEM is capable of generating reliable inflow data that provides self-sustaining turbulent thermal boundary layer behind a short recovery region. Finally, the instantaneous vortical structures as well as temperature fluctuation fields for the turbulent thermal boundary layer obtained using the XSEM demonstrate its robustness at different Prandtl numbers. Conflict of interest The authors declare that there is no conflict of interest regarding the publication of this paper. Acknowledgement This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (Ministry of Science and ICT) (Nos. NRF-2014R1A2A1A11053140 and NRF20151009350). References [1] G.R. Tabor, M. Baba-Ahmadi, Inlet conditions for large eddy simulation: a review, Comput. Fluids 39 (4) (2010) 553–567.
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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Extended synthetic eddy method to generate inflow data for turbulent thermal boundary layer Geunwoo Oh, Kyung Min Noh, Hyunwook Park, Jung-Il Choi ⇑ Department of Computational Science and Engineering, Yonsei University, Seoul 03722, Republic of Korea
a r t i c l e
i n f o
Article history: Received 30 June 2018 Received in revised form 7 February 2019 Accepted 18 February 2019
Keywords: Synthetic eddy method Inflow boundary condition Turbulent thermal boundary layer Large-eddy simulation
a b s t r a c t We propose an extended synthetic eddy method (XSEM) based on the synthetic eddy method, which includes the temperature fluctuation component in the turbulent flux tensors, to generate timedependent turbulent thermal inflow data for a spatially-developing boundary layer. The proposed XSEM is applied to large eddy simulations of a spatially-developing turbulent thermal boundary layer on a flat plate with an isothermal wall condition. Time-varying turbulent thermal inflow fields are reconstructed by composing the prescribed mean and reproduced fluctuations fields, along with Cholesky decomposition to the turbulent flux tensor with thermal flux. The obtained results indicate that the inflow generated by the proposed XSEM provides self-sustaining turbulence with fully recovered turbulent statistics behind a re-developing boundary layer region with the recovery distance roughly seven times of the inlet boundary layer thickness. Finally, we demonstrate the robustness of the XSEM for providing appropriate inflow data for simulations of turbulent thermal boundary layer flows for different Prandtl numbers. Ó 2019 Elsevier Ltd. All rights reserved.
1. Introduction Understanding the mechanism of turbulent thermal boundary layer flows is one of the primary issues in various fields of engineering wherein surface heat transfer is related to turbulent flows. To address this problem, high-fidelity numerical simulations, such as large-eddy simulations (LES) or direct numerical simulations (DNS), have been conducted to resolve turbulent structures near the surface in order to predict the characteristics of such surface heat transfer. However, performing these simulations requires time-varying realistic thermal turbulent data at an inlet boundary, wherein the generated inflow data satisfies the governing equations and varies stochastically [1,2]. A recent review [2] summarizes the vigorous research activities on inflow turbulence generation methods for eddy-resolving computations of spatiallydeveloping turbulent flows, such as the synthetic Fourier method [3], recycling/rescaling method [4], synthetic eddy method [5], and digital filtering method [6], among others. For more information, please refer to [1,2]. Although significant improvements have been achieved in inflow generation methods, relatively only a few inflow studies on turbulent thermal flows have been conducted.
⇑ Corresponding author. E-mail address: [email protected] (J.-I. Choi). https://doi.org/10.1016/j.ijheatmasstransfer.2019.02.061 0017-9310/Ó 2019 Elsevier Ltd. All rights reserved.
Several numerical studies have been conducted for a spatiallydeveloping turbulent thermal boundary layer utilizing an inflow generation method to generate time-dependent inlet data. Bell and Ferziger [7] performed numerical simulations of turbulent thermal flow on an isothermal wall for a momentum thickness Reynolds number RedM ranging from 300 to 700 and a molecular Prandtl number of 0.1, 0.71, and 2.0. In particular, they generated a thermal boundary layer using the fringe concept, transforming the non-periodic inlet and outlet condition to periodic conditions. Li et al. [8] performed DNS of a turbulent thermal boundary layer for RedM = 830 and Pr = 0.2, 0.71, and 2.0 by employing the fringe method as well [7]; they reported the differences in the scalar behavior at the outer region compared with that in the channel flow. Kong et al. [9] extended the recycling/rescaling method [4] to turbulent thermal boundary layer flows, reintroducing a spatially-developed flow from a downstream location using an appropriate rescaling process for the inlet in order to generate time-dependent thermal inflow data for RedM ¼ 300 and Pr ¼ 0:71; they performed DNS of turbulent boundary layer flows on an iso-thermal wall or iso-flux wall. Furthermore, Araya and Castillo [10] modified the recycling/rescaling method [4,9] for an adverse-pressure gradient turbulent boundary layer; in particular, they performed DNS of the turbulent thermal boundary layer over a rough isothermal wall with weak stratification for RedM ¼ 2300 and Pr ¼ 0:71. Li et al. [11] obtained the inlet flow through an
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auxiliary computation; in their study, instantaneous fluid velocity and temperature fields were extracted in the form of a time series at the location of RedM ¼ 1100. In particular, they performed DNS of turbulent thermal boundary layer for RedM ¼ 1100—1940 with Pr ¼ 0:71 by introducing the precursor data at the inlet; they observed a strong correlation between streamwise velocity and temperature fluctuations near the wall. Wu and Moin [12] conducted a DNS study for spatially-developing turbulent thermal boundary layers at zero-pressure gradient; they observed a transition of the boundary layer for RedM ¼ 80—1950 and Pr ¼ 1:0. In their study, pre-computed isotropic turbulence is superimposed to mean flows at the inlet to generate turbulent boundary layer. Recently, Okaze and Mochida [13] extended the digital-filter
2 pffiffiffiffiffiffiffiffi H11 6 6 H =L 6 21 11 6 Lij ¼ 6 6 H31 =L11 6 4 H41 =L11
0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi H22 L221
0
ðH32 L21 L31 Þ=L22
0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi H33 L231 L232
ðH42 L21 L41 Þ=L22
ðH43 L31 L41 L32 L42 Þ=L33
subscripts i; j ¼ 1; 2; 3 indicate the velocity fluctuations u0 ; v 0 ; w0 in streamwise, wall-normal, and spanwise velocities, respectively, while i; j ¼ 4 denotes temperature fluctuations h0 . It should be noted that / represents an average quantity of / in time and in the spanwise direction because the spanwise direction is homogeneous in a spatially-evolving turbulent boundary layer on a flat plate. In general, / can be defined as an ensemble (or spatiotemporal) average of /. As previously mentioned, in a manner similar to the study by Okaze and Mochida [13], Cholesky decomposition is applied to the turbulent flux tensor Hij to obtain the lower-triangular matrix Lij ;
0
3
7 7 7 7 7: 7 0 7 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5 2 2 2 H44 L41 L42 L41
method [6] to generate the turbulent fluctuations for fluid velocity and contaminant concentration; they applied the digital-filter method to LES of a half-channel flow for contaminant dispersion with no buoyancy effect. In this study, we propose an extended synthetic eddy method (XSEM) that can generate turbulent thermal inflow data based on the synthetic eddy method (SEM) [5] by including temperature fluctuations. In a manner similar to the study by Okaze and Mochida [13], Cholesky decomposition is applied to the turbulent flux tensor involving Reynolds stress and thermal flux. In addition, velocity and temperature fluctuations are generated using reproduced fluctuations, along with an eddy-convecting process in a virtual box. Our proposed XSEM is applied to LES of a spatiallydeveloping turbulent thermal boundary layer with an isothermal wall. Furthermore, we investigate the recovery of turbulent statistics for turbulent thermal boundary layer flows generated by the proposed XSEM and compare the statistics with those obtained using other inflow methods. Finally, we demonstrate the robustness of XSEM for various Prandtl numbers by visualizing turbulent coherent structures and temperature fluctuation fields near the wall; we also present a re-developing distance for the recovery of turbulent statistics based on Prandtl number.
2. Numerical method 2.1. Extended synthetic eddy method We propose the XSEM to generate turbulent thermal inflow data by the superposition of reproduced coherent structures to the prescribed mean fields. The coherent structures can be obtained using an eddy-convecting process in a virtual box, where the fluctuations of velocity components and temperature satisfy the second-order statistics of the turbulent thermal boundary layer based on the prescribed statistics. Detailed descriptions of the synthetic eddy method (SEM) can be found in [5]. An extended turbulent flux tensor Hij ¼ u0i u0j is comprised of Reynolds stresses and thermal fluxes related to temperature fluctuations, where the
0
ð1Þ
Similar to the SEM [5], a real-time varying turbulent thermal inflow field /i ðx; t Þ can be reconstructed by a composition of the prescribed mean field Ui and reproduced fluctuations /0i as follows: N 1 X /i ðx; tÞ ¼ Ui ðxÞ þ /0i ðx; tÞ ¼ Ui ðxÞ þ pffiffiffiffi Lij kj f rðxÞ x xk ; N k¼1
ð2Þ
where /i denotes ½u; v ; w; hT and kj is an arbitrary intensity in steps th
th
of 1 or 1 indicating the j component of the k synthetic eddy. The number of synthetic eddies N can be approximated using the ratio of the area of the inlet plane to the area covered by a synthetic eddy; in our study, we set N ¼ 1184. The virtual box containing the synthetic eddies is defined as ½r; r r; Ly þ r ½r; Lz þ r with parts corresponding to the x; y, and z directions, respectively, where Ly and Lz are the sizes of the computational domain in wall-normal and spanwise directions, respectively. Here, the inlet plane is located at x ¼ 0. Furthermore, it should be noted that r is the size of the synthetic eddies, which will be determined later. The shape function f rðxÞ has a compact support on k pffiffiffiffiffiffi k yy zzk with a normal½r; r; f rðxÞ x xk ¼ V B r3 f xx r f r f r R1 2 ization condition 1 f ðxÞdx ¼ 1; this function determines the velocity distribution of the synthetic eddy at location xk . Here, V B is the volume of the virtual box. The truncated Gaussian function [14], f ðxÞ ¼ C exp 9x2 =2 if jxj < 1, and 0 otherwise, is used in this study for the shape function, where the constant C is set based on the normalization condition. The synthetic eddies convect with a convection velocity U c ; xk ðt þ DtÞ ¼ xk ðt Þ þ U c Dt, where U c is the averaged streamwise mean velocity and Dt is the computational time step. As illustrated in Fig. 1, the synthetic eddies exiting out of the virtual box are re-introduced at the inlet of the box. The generated thermal turbulent fields at x ¼ 0 in the virtual box are imposed to the main simulation domain as the inlet boundary condition. Further information on the general SEM procedure can be found in [5,14]. Considering the influence of thermal fluctuations on the coherent turbulent structures, the size of an eddy is defined as
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Fig. 1. Schematic diagram of the XSEM procedure. The scattered spheres represent synthetic eddies, which are colored based on the instantaneous temperature h.
n
n
o
o
r ¼ max min k3=2 =; jd; jh dT ; Dx , where k is the turbulent kinetic energy, is the dissipation, j is the von Karman constant for fluid flow, d is the boundary layer thickness, jh is the von Karman constant for temperature, and dT is thermal boundary layer thickness. It should be noted that jh dT represents the length scale of the thermal effects on the eddy, where jh is set to 0.47 considering the turbulent Prandtl number Prt ¼ mt =at and the mixing length theory for momentum and heat transfer [15]; furthermore, Pr t ð jus yÞ=ð jh us yÞ ¼ j=jh , where us is the friction velocity. 2.2. Numerical simulations For numerical simulations of a turbulent thermal boundary layer, we consider the non-dimensionalized incompressible Navier-Stokes equations and energy equation for forced convection flow [9]. The governing equations are solved using an efficient monolithic projection method in a staggered Cartesian grid, preserving the second-order accuracy both in time and space [16– 18]. The Crank–Nicolson scheme is adopted to discretize all terms in the governing equations in time along with a linearization technique for the nonlinear convection terms, while second-order finite difference schemes are applied in space. Approximate block lowerupper (LU) decompositions, along with the approximate factorization technique for linear systems are employed to solve the momentum equations as well as energy equation without any iterations. Moreover, the Poisson equation in the projection method is solved using a multi-grid method with a Fourier diagonalization technique for a fast computation [18]. LES of the turbulent thermal boundary layer for RedM ¼ 1440 and Pr ¼ 0:71 are performed to investigate the effects of inflow conditions on the evolving turbulent profiles. A dynamic Vreman model is used as the sub-grid scale turbulence model in our simulations, which is a variant of the Vreman model [19,20]. The computational domain is defined as 0 6 x=dM;0 6 200; 0 6 y=dM;0 6 50 and 0 6 z=dM;0 6 22:5 with 269 61 97 grid points in streamwise, wall-normal, and spanwise directions, respectively, where dM;0 is the momentum thickness at the inlet. In particular, uniform grids are used in streamwise and spanwise directions, while a nonuniform grid clustered to the bottom wall using a hyperbolic tangent function is applied in the wall-normal direction. The corresponding grid spacings in the wall unit are Dxþ 45; Dyþ min 0:5 and Dzþ 22:5 in the respective directions, which provides sufficient resolution for wall-resolved LES [21]. The convective outflow boundary condition is considered at the exit for both velocity and temperature, while the periodic boundary conditions are used in the spanwise direction [22]. Furthermore, the no-slip boundary and zero isothermal boundary conditions are applied to the wall,
while the freestream conditions for the velocity fields are applied to the far-field boundary [22] with temperature set to unity. For the inlet conditions, first, we obtained the realistic velocity and temperature fluctuations at the inlet based on the recycling/ rescaling (R/R) method [9] using an auxiliary simulation of the spatially developing turbulent thermal boundary layer flow. Then, we consider three kinds of inflow generation methods for comparison: (1) Precursor Method: the stored instantaneous plane data of turbulent fluctuations are introduced at the inlet of the main simulation for each time step; (2) XSEM: real-time varying turbulent inflow fields at each time step are described using Eq. (2) with the mean inlet velocity and temperature profiles and the secondorder statistics obtained using the R/R method; (3) Random Fluctuations Method [3]: the resulting fluctuations /0i are scaled such that the calculated fluctuations conform to the turbulent flux tensor Hij , associated with the inlet turbulent profiles obtained using the R/R method.
3. Results and discussion 3.1. Assessment of the XSEM on the recovery of turbulent statistics We performed LES of spatially-evolving turbulent thermal boundary layer flows on a flat plate to evaluate the performance of our proposed XSEM in providing self-sustaining inflow turbulence and downstream recovery of turbulent statistics. The precursor and random fluctuation methods for generating inflow data were also used for numerical simulations to compare their recovery statistics with those of our method. As a baseline flow condition, a turbulent thermal boundary layer with RedM ¼ 1440 at the inlet and Pr ¼ 0:71 is selected for our study. Fig. 2 shows the influence of the inflow turbulence generated by the XSEM on downstream turbulent statistics. The turbulent statistics at the inlet ðx=d0 ¼ 0Þ indicate that XSEM generates inflow data that satisfies the prescribed first- and second-order statistics, compared with those obtained using the precursor method as well as the previous DNS results reported in [10,23]; here, d0 represents the boundary layer thickness at the inlet. However, XSEM slightly over-predicts the mean streamwise velocity U þ , temperature Hþ and thermal flux u0 h0 near the inlet region ðx=d0 ¼ 1Þ, while significantly under-predicting the Reynolds stress u0 v 0 , compared with other results specified above. As the boundary layer develops further downstream, U þ and Hþ , and u0 h0 obtained using the XSEM are similar to other results at x=d0 P 5, while u0 v 0 becomes similar at x=d0 P 7. This confirms that the inflow generated by the XSEM provides self-sustaining turbulence behind the redeveloping boundary layer region ð0 6 x=d0 6 7Þ with reproduc-
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δ
δ
δ
δ
δ
δ
θ
Θ
×
×
Fig. 2. Downstream development of turbulent statistics: (a) mean streamwise velocity, U þ ; (b) Reynolds stress, u0 v 0 ; (c) mean temperature, Hþ ; (d) thermal flux, u0 h0 ; XSEM ( ) and precursor method ( ) at RedM;0 ¼ 1440 with Pr ¼ 0:71; Spalart (j) at RedM ¼ 1410; Araya and Castillo (N) at RedM ¼ 2290.
δ (a)
δ (b)
Fig. 3. Downstream development of (a) C f and (b) St based on the different inflow generation methods.
tion of turbulent kinetic energy, where the corresponding turbulent statistics are fully recovered. It is important to note that the recovery feature of temperature follows that of the fluid flow because the fluid-thermal interaction is one-way coupled for the forced convection dominant flow. Fig. 3 indicates the effect of inflow generation methods on the recovery of the skin friction coefficient C f ¼ sw =ð2qU 1 Þ and Stan ton number St ¼ h= qcp U 1 , where sw is the wall shear stress, q is the density of the fluid, U 1 is the free-stream velocity, h is the convective heat transfer coefficient, and cp is the specific heat of the fluid at constant pressure. Compared with the precursor method, C f and St that are obtained using the XSEM and random fluctuation method sharply decrease near the inlet region. However, the XSEM provides similar predictions for C f and St at further downstream locations ðx=d0 P 7Þ to those obtained using the precursor method; in contrast, the results for the random fluctuation method indicate that the near-wall turbulence severely weakens,
and the flow tends to laminarize, which is consistent with the observation in [5]. To further investigate the recovery of near-wall turbulence based on the inflow generation methods, C f and St are shown in Fig. 4 in terms of boundary layer characteristics such as RedM and RedE representing the Reynolds number based on momentum ðdM Þ and enthalpy thickness ðdE Þ, respectively. For comparison, we consider the semi-empirical correlations derived by Kays and Crawford [24]; 1=4
C f ¼ 0:025RedM ; St ¼ 0:0125Pr
0:5
Re1=4 : dE
ð3Þ ð4Þ
where Eq. (4) is valid in the range of 0:5 6 Pr 6 1:0 with a high Reynolds number. Furthermore, Fig. 4 indicates that the precursor method provides monotonically decreasing C f and St with the development of the boundary layer, which is similar to the feature
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×
×
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δ
δ
Fig. 4. Comparisons of (a) C f and (b) St obtained using different inflow generation methods with respect to boundary layer characteristics.
δ
Θ
θ
δ
(a)
(b)
Fig. 5. Profiles of (a) Hþ and (b) u0 h0 at x=d0 ¼ 7 obtained using the XSEM for different Pr numbers, compared with results from other studies.
of the semi-correlations. The results of the XSEM indicate sharp decreases in C f and St near the inlet region (see Fig. 3), leading to the reduction in the momentum and enthalpy thicknesses as shown in Fig. 4. However, the XSEM shows fairly good predictions of C f and St after the recovery of the turbulent statistics at further downstream locations ðx=d0 P 7:0Þ compared with those obtained using the semi-correlations and precursor method. It should be noted that the corresponding RedM and RedE at x=d0 ¼ 7:0 are roughly 1600 and 1800, respectively. In contrast, the inflow turbulence generated by the random fluctuation method could not maintain characteristics of the turbulent boundary layer because of significant momentum and enthalpy losses near the inlet region.
at x=d0 7, where the location is roughly behind the re-developing region for the recovery of statistics as discussed in Section 3.1. Thus, both predicted results show good agreement with the existing DNS results for the boundary layer flows at RedM ¼ 2290 [10] and channel flows at Res ¼ 180 [25] as well as Kader’s correlation [26]. This confirms that the XSEM is capable of generating reliable inflow data that provides a self-sustaining turbulent thermal boundary layer for different Pr. Fig. 6 shows the downstream development of St=St 0 based on Pr; these results indicate that heat transfer characteristics are recovered at x=d0 10 for the three Pr cases, while the recovery region becomes slightly longer for a
3.2. Applicability of the XSEM for different Prandtl numbers LES of the developing turbulent thermal boundary layer at RedM ¼ 1440 were also performed with Pr ¼ 0:2 and 5:0 to investigate the robustness of XSEM based on different Pr. To impose turbulent inflow data using the XSEM, we obtained the prescribed statistics for the XSEM by performing an auxiliary simulation using the R/R method [9] for each Pr case. Because the thermal boundary layer for Pr ¼ 5:0 is relatively thinner than those for other cases, half of the grid spacings used in the baseline simulation are applied in each direction for a sufficient resolution: Dxþ 22:5; Dyþmin 0:32, and Dzþ 11:3 with 537 71 193 grid points. Fig. 5 indicates that the XSEM provides considerably similar predictions of Hþ and u0 h0 compared with the precursor method regardless of Pr. It should be noted that the statistics are obtained
δ Fig. 6. Downstream development of St=St 0 for different Pr numbers.
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Fig. 7. Vortical structures (a,b,c) and temperature fluctuations h0þ (d,e,f) near the wall for the turbulent thermal boundary layer using the XSEM for different Pr numbers. The vortical structures are identified by k2 method [27] with a criterion; k2 ¼ 0:013. The iso-surfaces are colored based on the value of H. Contour plots of h0þ are shown in a horizontal plane at yþ ¼ 12; 10:5, and 8 for Pr ¼ 0:2; 0:71, and 5.0, respectively.
low Pr fluid. This can be explained by noting that the thermal boundary layer is thicker than the boundary layer for a low Pr fluid, thus requiring a longer re-developing region because of low thermal fluxes, leading to slower recovery. Finally, the vortical structures in the developing turbulent thermal boundary layer flows obtained using the XSEM are visualized in Fig. 7 (a)–(c) using the k2 criterion [27] with k2 ¼ 0:013; these are colored based on the value of H, for different Pr cases. In addition, contour plots of h0þ are shown in Fig. 7 (d)–(f) in the horizontal plane, where the maximum value of u0 h0 is observed at yþ ¼ 12; 10:5, and 8 for Pr ¼ 0:2; 0:71, and 5:0, respectively. Except for near the inlet region, it can be clearly observed that coherent vortical structures are well maintained at downstream locations for all Pr cases. For a high Pr fluid, the vortical structures carrying hot fluid introduce higher thermal fluxes near the wall, resulting in more elongated temperature fluctuation fields in the streamwise direction, while relatively isotropic temperature fluctuations are observed for a low Pr fluid. Similar to the DNS results in the study by Redjem-Saad et al. [28] for turbulent heat transfer in pipe flows, the our obtained results indicate that the thermal fields tend to be more isotropic for a lower Pr fluid owing to the thicker conductive sublayer. However, for a higher Pr fluid, thinner conductive regions reduce the molecular heat flux but enhance the turbulent heat flux normal to the wall, which, in turn, increases the correlation between the thermal and velocity fluctuations, leading to thermal streaky structures. The instantaneous flow and thermal fields in Fig. 7 indicate that the proposed XSEM provides reliable inflow data to simulate a developing turbulent thermal boundary layer flow with a short recovery region.
4. Conclusions In this study, we proposed the XSEM to provide time-varying inflow turbulence data for simulating spatially-developing turbulent thermal boundary layer flows. The thermal inflow fields were reconstructed using the prescribed mean and reproduced fluctua-
tion fields with Cholesky decomposition of the turbulent flux tensor including thermal flux, based on the original SEM [5]. We performed LES of the developing turbulent thermal boundary layer flows for RedM ¼ 1440 at the inlet in order to investigate the influence of the inflow turbulence generated by the XSEM on downstream turbulent statistics. Downstream development of mean velocity and temperature, Reynolds stress, and thermal flux confirms that the XSEM provides self-sustaining turbulence with fully recovered turbulent statistics behind the downstream location x=d0 7. Moreover, the XSEM provides similar predictions of C f and St after the turbulent thermal boundary layer is fully redeveloped behind the location, compared with those obtained using the precursor method. Additional LES results for the boundary layer at different Pr numbers show that the XSEM is capable of generating reliable inflow data that provides self-sustaining turbulent thermal boundary layer behind a short recovery region. Finally, the instantaneous vortical structures as well as temperature fluctuation fields for the turbulent thermal boundary layer obtained using the XSEM demonstrate its robustness at different Prandtl numbers. Conflict of interest The authors declare that there is no conflict of interest regarding the publication of this paper. Acknowledgement This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (Ministry of Science and ICT) (Nos. NRF-2014R1A2A1A11053140 and NRF20151009350). References [1] G.R. Tabor, M. Baba-Ahmadi, Inlet conditions for large eddy simulation: a review, Comput. Fluids 39 (4) (2010) 553–567.
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