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Therma l Science & Engineering Vol.10 No. 1 (2002) Direct Numerical Simulation of Turbulent Heat Transfer in the Stably Stratified Ekman Layer Kenji SHINGAI†, Hiroshi KAWAMURA† Abstract The direct numerical simulations (DNSs) of the neutrally and the stably stratified turbulent Ekman layer over a smooth surface are performed using the Boussinesq approximation to account for the buoyancy effect. The Reynolds number based on the geostrophic wind G, the Ekman layer depth D and the kinematic viscosity ν is 410, which is almost equal to that of Coleman et al.(JFM, 1990) The Grashof number is set to be Gr=0, 3.15•~ 6, 6.30•~ 6, 1.26•~ 7 and 3.15•~ 7 in order to examine the effect of the stable 10 10 10 10 stratification. A temperature field is so introduced that its mean profile is quasi-steady with time. As the result, statistical quantities are obtained for both velocity and temperature fields. In this layer, 3-dimensional velocity profile is observed. The angle between the geostrophic wind and φ) the mean velocity ( at the ground decreases compared to the laminar Ekman layer because of the enhanced vertical momentum transfer. Therefore combination of the sweep and ejection has great influence on the flow direction in the vicinity of the ground. The relation between the flow direction and the vertical velocity fluctuation is discussed quantitatively. The horizontal directions of the mean velocity, the Reynolds stress and the turbulent heat flux are compared. It is found that the horizontal turbulent heat flux is not aligned with the mean velocity, and that it represents a similar profile to the Reynolds stress. The similarity and the difference of these double correlation statistics are shown in detail. The effects of the stable stratification upon the direction of the mean velocity, the turbulent heat flux and the Reynolds stress are also discussed based on the obtained DNS data. Key Words : Turbulence, Direct Numerical Simulation (DNS), Ekman layer, Heat Transfer, Stable Stratification Nomenclature Riw : Richardson number at ground Re : Reynolds number =G h /ν a : thermal diffusivity Tw : temperature difference between far upper region at : thermal eddy diffusivity and ground dv : velocity boundary layer thickness t : time dθ : thermal boundary layer thickness T : instantaneous temperature D : Ekman layer depth = 2ν / f Tτ : friction temperature f : Coriolis parameter u : velocity vector =(u, v, w) g : gravity acceleration vector =(0, -g, 0 ) u τ : friction velocity G : geostrophic wind velocity x(x1), y(x2), z(x3) : coordinates Gr : Grashof number =g β Tw h 3/ν 2 β : volume expansion coefficient h : height of computational domain δt : turbulent depth = u τ / f p : pressure φ : angle from geostrophic wind direction q total : vertical total heat flux at ground ν : kinematic viscosity qh : horizontal turbulent heat flux vector νtx : streamwise eddy diffusivity * Received: November, 16, Editor: Koichi HISHIDA •õDepartment of Mechanical Engineering, Tokyo University of Science (2641 Yamazaki, Noda-shi, Chiba 278-8510, JAPAN) -0- Therma l Science & Engineering Vol.10 No. 1 (2002) νtz : spanwise eddy diffusivity θ : converted instantaneous temperature ρ : density τ Rh : projected Reynolds stress tensor onto x-z plane τw : total shear stress at ground τθ : time-scale of temperature decreasing Ω : angler velocity vector of system =(0, f/2, 0 ) Sub/Superscripts () + : normalized by u τ , ν and Tτ ( )′ : fluctuation component : absolute value ( )rms : root mean square Fig. 1 Computational domain () : statistically averaged quantity Boussinesq approximation to account for the buoyancy effect. The computational domain is doubled in size 1 Introduction compared to that of Coleman et al.[9] without the deterioration of resolution. A temperature field is so The planetary boundary layer (PBL) is affected by the introduced that its mean profile is quasi-steady with time. miscellaneous factors such as the system rotation and The 3-dimensional characteristics of the mean velocity, the buoyancy force. The boundary layer under the effect the Reynolds stress and the turb ulent heat flux are of the system rotation is called as the Ekman layer. The examined. Moreover, the effects of the stable works on the laminar PBL have been widely performed stratification upon the statistical quantities are also mainly by the theoretical methods. On the other hand, the discussed based on the o btained DNS data. turbulent PBL is much more complex because it includes fine and very large-scale motions and because its mean 2 Computational condition velocity field is three dimensional in nature. Hence, the measurements and experiments have been the major tools Calculated flow field is the turbulent Ekman layer of an in almost all researches on the turbulent PBL. Hunt et incompressible viscous fluid over a smooth flat surface in al.[1] and Lenschow[2] are the ones who utilized the a vertically oriented gravitational field. The system is measurements for the turbulent PBL. However the three- rotating about a vertical axis with an angler velocity Ω = dimensional spatial structure of the turbulent velocity (0, f/2, 0), where f is the Coriolis parameter. field is not easy to be obtained experimentally. Many The flow is driven by a horizontal pressure gradient. In turbulence models have been proposed and developed to this layer, the pressure gradient, the Coriolis and the describe the 3-dimensional spatial structure of the viscous forces are balanced. At a position so high for the velocity field (e.g. André et al.[3], Hunt[4]). Deardorff[5], viscosity effect as to be neglected, the pressure gradient Mason[6] and Moeng[7] employed the large eddy balances with the Coriolis force and thus the mean flow simulation (LES) to compute the turbulent PBL direction becomes perpendicular to the pressure gradient. numerically. The review of the experimental and the This type of flow is called as the geostrophic wind (G). In numerical results is given by Wyngaard[8] in 1992. the present work, x- and y-axes are set to be parallel and Among these studies on PBL, to the authors' vertical, respectively, to the geostrophic wind direction. knowledge, the DNS of the turbulent Ekman layer have Computational configuration is given in Fig. 1. The not been performed except the work of Coleman et periodic boundary condition is imposed in x and z al.[9][10] They obtained the Ekman spiral to find that it directions. The non-slip and the Neumann conditions are became shallower compared with that of the laminar flow. adapted at the bottom and top boundaries. The height of They also examined the existence of the large-scale the computational domain is so set to be enough large longitudinal vortices. But their computational domain compared to the boundary layer thickness. For the was too small to contain the large-scale motion. In computational grid, the uniform mesh is used in the x and addition, they used the temperature field whose mean z directions. On the other hand, the non-uniform mesh is profile d epended on time. As the result, some of the adopted in the y direction in order to ensure a high obtained statistical quantities were not averaged in long resolution in the vicinity of the ground. time enough to be used for the construction of the The governing equation is the continuity equation turbulence model. In the present work, the neutrally and ∇⋅ u = 0 , (1) the stably stratified turbulent Ekman layers over a the Navier-Stokes equation smooth surface are computed through DNS using the -2- Thermal Science & Engineering Vol.10 No. 1 (2002) Table 1 Computational conditions Grid number 256 × 96 × 256 + + + Spatial resolution (∆ x , ∆ y , ∆ z ) 6.01, 0.219 •| 21.4, 6.01 Re = G •E/ ν h 12,000 Ro = G / (fh) 7.0 3 2 Gr = g β T w h / ν 0.0, 3.15 × 10 6 , 6.30 × 10 6 , 1.26 × 10 7 , 3.15 × 10 7 2 Gr=0 3.15 x 107 y/δ t U/G V/G -W/G Symbol : Coleman et al.(1990) 1 U/G, V/G 0 0 0.5 U/G, V/G, -W/G 1 Fig. 3 Mean velocity profiles for Gr=0 and Fig. 2 Instantaneous velocity field in neutrally 3.15•~ 7 10 stratified turbulent Ekman layer (White: the second invariants of the velocity gradient ∂θ ∂θ ∂ 2θ 1 tensor II’’+<-0.025, Gray: low-speed streaks u’+<-2.5, +uj = a 2 + (1 − θ ) . (5) ∂t ∂x j ∂x j τ θ Dark gray: high-speed streaks u’+>2.5) The optional parameter τθ means the time-scale of the ∂u 1 temperature decrease. Here, time constant τθ is assumed + (u ⋅∇ )u + 2Ω× u = − ∇p − Θg + ν∇ 2u , (2) to be positive because the stable stratification is usually ∂t ρ caused by the decrease of the ground temperature. The and the energy equation heat source term represents the heat-losses caused by ∂T + u ⋅ ∇T = a∇ 2T . (3) both the heat conduction and the radiation heat transfer. ∂t In the present computation, the fractional step method The Boussinesq assumption is used in this set of is adopted for the coupling between the continuity and equations. The quantity Θ is defined as Θ = β (θ - Tw). the Navier-Stokes equations. The 2nd-order Crank- Here, θ is the temperature converted in the following Nicolson and the 2nd-order Adams -Bashforth methods method. are employed as the time-advance algorithm; the former The actual stably stratified temperature field is for the vertical viscous term, the latter for the other decaying with time because it is resulted from the cooling viscous and the convection terms. The Coriolis term is at the ground. On the other hand, the time-scale of solved implicitly to avoid the numerical instability. For temperature decrease is very large compared with that of the spatial discretization, the finite difference method is the velocity fluctuation. Therefore, a mean temperature adopted. The computational conditions are summarized field is assumed to be quasi-static with respect to the in Table 1. time advancement. The instantaneous temperature T is defined as 3. Results and Discussion ( ) T (x ,y , z ,t ) = −Tw (t ) ⋅ 1− θ ( x, y , z ,t ) (4) 3.1 Instantaneous flow field, mean velocity and Here, Tw is the temperature in the far upper region with temperature profiles the temperature at the ground to be zero. In this case, Tw The instantaneous velocity field is visualized to is a function of time because the temperature difference examine the special structure. The x-direction component between the ground and far upper region increase with of the velocity fluctuation and the second invariant of the time advancement. By the substitution of eq. (4) into the velocity gradient tensor are illustrated in Fig. 2. The (3), one may convert the energy equation into streak structures are obtained in the vicinity of the ground. Most structures are not aligned with the -3- Therma l Science & Engineering Vol.10 No. 1 (2002) Table 2 Comparison of the velocity and the 0 thermal boundary layer thickness normalized by δt W/G Gr dv dθ -0.2 0.00 0.767 0.929 3.15×10 6 0.727 0.900 6.30×10 6 0.722 0.882 -0.4 present Gr=0 1.26×10 7 0.637 0.803 present Gr=3.15 x 10 7 3.15×10 7 0.506 0.663 Coleman et al.(1990) -0.6 Laminar Ekman layer Table 3 Relation between the Grashof number and 0 0.2 0.4 0.6 0.8 U/G 1 the mean velocity direction at the ground Fig. 4 Hodograph of the mean velocity for Gr=0 Gr Ri w φ and 3.15•~ 7 10 0.00 0.00 28.6 3.15×10 6 1.00×10 -3 28.9 2 turbulence 6.30×10 6 1.99×10 -3 29.4 y/δt Gr=0 1.26×10 7 3.85×10 -3 30.9 Gr=3.15 x 10 7 3.15×10 7 8.98×10 -3 34.4 laminar - - 45.0 1 2 Gr=0 3.15 x 10 7 u'+rms rms + v' rms u i '+ + w' rms 0 0 0.2 0.4 0.6 0.8 1 1 θ / Tw Fig. 5 Mean temperature profiles for Gr=0 and 3.15•~ 7 10 geostrophic wind direction (x-direction). The vortex 0 structure is complicated compared with the plane 0 40 + y 80 0 1 y/δ t 2 Poiseuille flow because of the system rotation. Fig. 6 Turbulent intensities for Gr=0 and The mean velocity profiles for Gr=0 and Gr=3.15•~ 3.15•~ 7 10 10 7 are given in Fig. 3 and compared with the result boundary layer. We can find that the thermal boundary obtained by Coleman et al.[9] in the neutrally stratified layer thickness in the case of Gr=3.15•~ 7 is thin 10 Ekman layer. The reference length δt is the turbulent comp ared with that in the case of Gr=0. The velocity and depth defined as u τ / f . The mean velocity profile under thermal boundary layer thicknesses (d v and d θ, the neutral stratification shows a good agreement with respectively) for Gr=0, 3.15•~ 6, 6.30•~ 6, 1.26•~ 7 10 10 10 the results of DNS performed by Coleman et al.[9] The boundary layer thickness for Gr=3.15•~ 7 becomes 10 and 3.15 •~ 7 are compared in table 2. The velocity 10 thin compared with the one for Gr=0, because the boundary layer thickness d v is the height where the turbulent mixing is weakened by the buoyancy effect. spanwise mean velocity becomes 0 firstly and d θ is the The hodograph of the mean velocity is shown in Fig. 4 height where the mean temperature θ reaches to 99% of for Gr=0 and 3.15•~ 7. The result by Coleman et al.[9] 10 Tw. The both velocity and thermal boundary layer and the analytical solution for the laminar Ekman layer are thicknesses decrease with increase of the Grashof also plotted. The well-known Ekman spiral is obtained in number. both cases of Gr=0 and 3.15 •~10 7. The spiral for Gr=3.15•~ 7 is larger than that of Gr=0 and more 10 3.2 Relation between mean velocity and sweep- ejection closer to the laminar one. The angles between the geostrophic wind and the Figure 5 shows mean temperature profiles for Gr=0 mean velocity (φ) at the ground for the cases of Gr=0, and 3.15•~ 7. In both cases the stable stratifications are 10 3.15 •~ 6, 6.30 •~ 6, 1.26 •~ 7 and 3.15 •~ 7 are 10 10 10 10 realized in the vicinity of the ground. This type of compared in table 3. The Richardson number at the temperature profile is often observed in the actual earth's ground Riw is defined as -4- Thermal Science & Engineering Vol.10 No. 1 (2002) 0.2 y+ Vector : Mean velocity at y +=5.33 W/G 7 (y/ δ t=0.0153) 300 Gr=0 3.15 x 10 0 u'θ' v 'θ' 200 w'θ' -0.2 q total 100 -0.4 v' + < -0.156 + v' > 0.156 0 -0.6 -1 0 1 2 3 4 0 0.2 0.4 0.6 0.8 U/G 1 ui'+θ '+ Fig. 7 Hodograph of instantaneous horizontal Fig. 9 Profiles of the turbulent heat fluxes for velocity in the neutrally stratified Ekman layer at Gr=0 and 3.15•~ 7 10 y+=5.33 y+ 0.6 30 Gr=0 3.15 x 10 7 300 u'v' φ [deg.] |u|/G v'w' 200 u'w' 0.3 24 7 Gr=0.0 3.15x10 100 |u| φ 0 18 0 -4 -2 0 2 v'/v' 4 -0.5 0 0.5 ui'+u j'+ 1 rms Fig. 8 Relation between the vertical velocity Fig. 10 Profiles of the Reynolds stresses for Gr=0 fluctuation and the velocity direction (φ) at y+=5.33 and 3.15•~ 7 10 g ΓD 2 , depend on the vertical velocity. In other words, the Riw = (6) Tw G sweep and ejection contribute significantly to the mean velocity direction in the vicinity of the ground. where Γ denotes the mean temperature gradient at the For more detailed investigation, the relation between ground and D is the Ekman layer depth D = 2ν f . φ the vertical velocity and the velocity direction ( ) at The angle φ increases with the increase of Grashof y+=5.33 is shown in Fig. 8. The relation between the number and approaches to the 45 degrees, which is the vertical velocity and the magnitude of the horizontal angle in the laminar flow case. This reason is e xpected velocity vector at y+=5.33 is also given. The angle φ and that the flow field is laminarized by the influence of the the magnitude of horizontal velocity |u| are obtained by stable stratification. ensemble-averaging the instantaneous data with the Figure 6 shows the turbulent intensities normalized by same vertical velocity. One can find from this figure that the friction velocity. All components in the case of the angle φ decreases while the magnitude increases Gr=3.15•~ 7 decrease compared with those of Gr=0. 10 v when the sweep motion is enhanced ( ’<0). This is The hodograph of the instantaneous velocity in the because the higher momentum of geostrophic wind with plane of y+=5.33 is shown in Fig. 7. Here, all symbols φ=0 penetrates more deeply into the near wall region. indicate the heads of the horizontal instantaneous On the other hand, when the ejection takes place velocity vectors with the tails of the vectors fixed at (0, 0). (v’>0), φ increases while |u| stays in a lower value. In The solid symbol implies that the vertical components addition, this tendency is more enhanced in the case of normalized by the friction velocity (v’+) are over 0.156 the stable stratification because the momentum of and the open symbol v’+<-0.156. The solid symbols are geostrophic wind hardly reaches to the near ground gathering nearer to (0, 0) compared with the blank region owing to the buoyancy effect. As the result, the symbols. From this figure, we can find out that the directions of the mean velocity at the ground increase magnitude and the direction of instantaneous velocity with the increase of the Grashof number as shown in the -5- Therma l Science & Engineering Vol.10 No. 1 (2002) 1 y+=87.7 y+=17.4 0 y+=11.4 + + w' θ' Gr=0, -1 Gr=3.15 x10 6 , Gr=1.26 x10 7, Gr=6.30 x10 6 , Gr=3.15 x10 7 0 1 2 3 u'+θ' + 4 Fig. 11 Bird’s-eye view of the horizontal turbulent Fig. 12 Hodograph of the horizontal turbulent heat flux (a) and the projected Reynolds stress tensor heat flux vector onto the horizontal plane (b) in the case of Gr=0 0.2 y+=87.7 y+=27.1 table 3. -v'+w'+ 3.2 Turbulent heat flux and Reynolds stress + The turbulent heat flux and the Reynolds stress exhibit y =9.44 the three-dimensional profiles. The turbulent heat flux 0 profiles are obtained as shown in Fig. 9. The total heat Gr=0 flux in the vertical direction is also plotted. All Gr=3.15x106 Gr=1.26x107 Gr=6.30x106 Gr=3.15x107 components are normalized by the vertical total heat flux at the ground (q w). The vertical total heat flux increases 0 0.2 0.4 -u' +v '+ 0.6 as the ground is approached, and takes a maximum value at the ground. In the present temperature field, the Fig. 13 Hodograph of the projected Reynolds horizontal molecular heat flux is not existent because the stress tensor onto the horizontal plane mean temperature remains constant in x and z directions. As the result, only the turbulent heat flux contributes to 0.06 the horizontal heat transfer. Unlike the usual boundary Reynolds stress Turbulent heat flux layer without the system rotation, the spanwise + 0.2 P+ 2 1 P 1θ component of the turbulent heat flux w′θ ′ emerges and + P 23 + P 3θ is even comparable with the vertical one. 0.03 Figure 10 shows the Reynolds stresses for Gr=0 and 0.1 3.15 •~ 7. The spanwise Reynolds stress component 10 P+ 2 , P + 3 P 1θ , P 3θ 2 + v′w′ also appears. These spanwise components of the turbulent heat flux and the Reynolds stress contribute to 0 0 1 + the three-dimensional profiles. In order to grasp the 3-dimensional characteristics of the turbulent heat flux 0 50 100 y+ 150 and the Reynolds stress, the horizontal turbulent heat flux vectors Fig. 14 Production rates of the Reynolds stress ( ) q h = u ′θ ′, w ′θ ′ (7) u ′v′ , v′w′ and the turbulent heat flux u ′θ ′ , w ′θ ′ for the case of Gr=0 and the projected Reynolds stress tensor onto the horizontal plane 6.30•~ 6, 1.26•~ 7 and 3.15•~ 7 are shown in Figs. 10 10 10 ( ) τ Rh = − u ′v ′, − v ′w ′ (8) 12 and 13, respectively. These figures are the projected envelop curves of Figs. 11 (a) and (b). Both of q h and τ Rh are illustrated in Figs. 11 (a) and (b). The directions of show the oval-shaped profiles. One may notice, however, q h and τ Rh change as a function of the height. The a significant difference in the location of the oval profile magnitude of q h is enhanced in the vicinity of the ground between Figs. 12 and 13. That is, the major part of the while it becomes very small in the higher region. On the projected Reynolds stress profiles falls in the first other hand, the projected Reynolds stress is much more quadrant as seen in Fig. 13, while that of the turbulent enhanced in the higher region compared with that of heat flux vector in the fourth one. When we consider the horizontal turbulent heat flux. effect of the buoyancy force, these oval-shaped profiles The hodographs of q h and τ Rh for Gr=0, 3.15•~ 6, 10 become shrunk as Grashof number increases because the -6- Thermal Science & Engineering Vol.10 No. 1 (2002) 1 300 y/δθ 0.8 y+ 200 0.6 Gr=0 0.4 Gr=0 100 Gr=3.15 x 106 6 Gr=3.15 x 10 6 Gr=6.30 x 106 0.2 Gr=6.30 x 10 7 Gr=1.26 x 107 Gr=1.26 x 10 7 Gr=3.15 x 107 Gr=3.15 x 10 0 0 0 π /2 π 3π /2 0 π /2 φ [rad.] π φ [rad.] Fig. 17 Direction of the horizontal turbulent heat Fig. 15 Direction of the horizontal turbulent flux vector versus the height normalized by the heat flux vector versus the height y+ thermal boundary layer thickness d θ 1 300 y/δ v + 0.8 y 200 0.6 Gr=0 0.4 Gr=0 100 Gr=3.15 x 106 Gr=3.15 x 106 Gr=6.30 x 106 0.2 Gr=6.30 x 106 7 7 Gr=1.26 x 107 Gr=1.26 x 107 Gr=3.15 x 10 Gr=3.15 x 10 0 0 0 π /2 π 3π /2 0 π /2 φ [rad.] π φ [rad.] Fig. 18 Direction of the projected Reynolds Fig. 16 Direction of the projected Reynolds stress stress tensor versus the height normalized by the tensor onto the horizontal plane versus the height y+ velocity boundary layer thickness d v velocity and temperature fluctuation are diminished by their sign at y+=20.3, where ∂W ∂ y = 0 . The equations the stable stratification. (9) and (10) indicate that the production terms P12 and P23 The production terms of u ′v′ , v′w′ , u ′θ ′ and w ′θ ′ are directly proportional to the vertical turbulent intensity are compared in Fig. 14. They are given by v′v′ . From Fig. 6, the vertical turbulent intensity v′v′ ∂U has a peak in a rather high region. Therefore the maximum P = −v′ 2 , (9) 12 ∂y of the absolute values of P12 and P23 arises at a higher position than the corresponding turbulent heat flux ∂W P23 = − v′ 2 , (10) production terms. The magnitude of the vectors q h and ∂y τ Rh are mostly governed by their productions terms. ∂θ ∂U ∂θ ∂U Therefore the height of the maximum τ Rh arises at the Pθ = − u′v′ 1 − v′θ ′ : (ν tx + at ) , (11) ∂y ∂y ∂y ∂y higher position than that of q h and takes place in the first ∂θ ∂W ∂θ ∂W quadrant in the hodograph. P θ = − v′w′ 3 − v′θ ′ : (ν tz + at ) , (12) The direction of q h and τ Rh are compared in Figs. 15 ∂y ∂y ∂y ∂y and 16. The angle φ turns faster in the higher Gr cases. where νtx and νtz are streamwise and spanwise eddy This is because both of the velocity and the thermal diffusivities. In the present case, the mean temperature boundary layers become thin in the higher Gr cases. gradient ∂θ ∂y keeps a positive value in the whole field. Thus, the angles are replotted versus the height As the result, the sign of P12 and P1θ depends on the normalized by the velocity or the thermal boundary layer mean velocity gradient ∂U ∂y , while that of P23 and P3θ thicknesses (d v , d θ) in Figs. 17 and 18. From Figs. 17 and is determined by the ∂W ∂y . This is the reason why the 18, we can conclude that the direction of q h and τ Rh become almost independent of Gr in the vicinity of the productions of v′w′ and w′θ ′ (P23 and P3θ) change ground when the heights are normalized by d v and d θ. -7- Therma l Science & Engineering Vol.10 No. 1 (2002) References 4 Conclusion [1] Hunt, J.C.R., Kaimal, J.C. and Gaynor, J.E., “Some The direct numerical simulations of neutrally and observations of turbulence structure in stable stably stratified turbulent Ekman layer over a smooth layers”, Quart. J. R. Met. Soc., 111 (1985), 793-815. surface are performed. The Grashof number is set to be [2] Lenschow, D.H., “Observations of clear and Gr=0, 3.15•~ 6, 6.30•~ 6, 1.26•~ 6 and 3.15•~ 7 10 10 10 10 cloud-capped convective boundary layers, and in order to examine the effect of stable stratification. A techniques for probing them”, E. J. Plate et al. (eds.), quasi-static temperature field is so introduced that its Buoyant convection in geophysical flows, (1998), mean profile is unchanged with time advancement. As 185-206. the result, statistical quantities are obtained for both [3] André, J. C., De Moor, G., Lacarrère, P., Therry, G. velocity and temperature fields. The relation between the and Du Vachat, R., “Modeling the 24-hour evolution flow direction and sweep or ejection is discussed of the mean and turbulent structures of the quantitatively. The effects of stable stratification upon planetary boundary layer”, J. Atmos. Sci., 35 (1978), the direction of the mean velocity, the turbulent heat flux 1861-1883. and the Reynolds stress are discussed based on the [4] Hunt, J.C.R., “Diffusion in the stably stratified obtained DNS data. The conclusions are derived as atmospheric boundary layer”, J. Clim. Appl. Met., follows: 24 (1985), 1187-1195. 1. Instantaneous flow direction in the vicinity of the [5] Deardorff, J.W., “Numerical investigation of neutral ground is turned to the direction of geostrophic and unstable planetary boundary layers”, J. Atmos. wind by the effect of the sweep, while it is turned to Sci., 29 (1972), 91-115. the pressure gradient direction by the effect of the [6] Mason, P.J., “Large-eddy simulation of the ejection. convective atmospheric boundary layer”, J. Atmos. 2. The maximum value of the projected Reynolds stress Sci., 46 (1989), 1492-1516. tensor arises in the first quadrant in the hodograph [7] Moeng, C.-H., “Large-eddy-simulation model for the while that of the horizontal turbulent heat flux in the study of planetary boundary-layer turbulence”, J. fourth quadrant. This difference can be attributed to Atmos. Sci., 41 (1984), 2052-2062. the vertical turbulent intensity profile. [8] Wyngaard, J.C., “Atmospheric turbulence”, Annu. 3. In the stably stratified layer, the directions of τ Rh and Rev. Fluid Mech., 24 (1992), 205-233. q h changes more rapidly with respect to the height. [9] Coleman, G. N., Ferziger, J. H. and Spalart, P. R., “A This is because the velocity and the thermal numerical study of the turbulent Ekman layer”, J. boundary layer become thinner. Therefore, the Fluid Mech., 213 (1990), 313-348. directions can be well scaled if the height is [10] Coleman, G.N., Ferziger, J.H. and Spalart, P.R., normalized by the velocity or the thermal boundary “Direct simulation of the stably stratified turbulent layer thicknesses. Ekman layer”, J. Fluid Mech., 244 (1992) 677-712. Editor’s Comments (Koichi Hishida) followings: 1. The authors found the buoyancy effects on The paper deals with the direct numerical simulation of boundary layer and laminarization by stable rotating stratified turbulent flow using Boussinesq stratification. This effect means that approximation with buoyancy effects. The authors non-similarity might exist in momentum and indicated the effect of the stable stratification and energy transfer process. After that, the authors characteristics of turbulent heat flux and Reynolds shear conclude the similarity of Reynolds stress and stresses. The detailed information on the flow structure turbulent heat fluxes in a boundary layer. Is this is very valuable for analysis of large-scale turbulent discussion in consistency? boundary layer, especially geographical flows. Thus, the 2. What is the future research of the DNS for actual editor has evaluated that the paper gives a significant geographical flow condition? Could you give us contribution to this kind of research area. an appropriate comment for an extension of the However, there still exist unexplored discussions in the DNS for large-scale flow? paper. The authors would be expected to reply the -8- Therma l Science & Engineering Vol.10 No. 1 (2002) Author’s Response subjects in future computations. 2. In the present calculation, much lower Reynolds We would appreciate the editor’s valuable comments. and higher Rossby numbers are employed 1. In this paper, similarity between the Reynolds compared to the actual geographical flow. The stress and the turbulent heat flux has been simulation of the actual geographical flow using discussed only in conjunction with their the DNS is much beyond the capability of the directions. The laminarization attenuates both the present and foreseeable computers. Thus, a velocity and temperature fluctuations. Figures A turbulent model needs to be introduced for the and B indicate the decay of the turbulent energy calculation of the actual geographical flow. The and the temperature variance due to the role of the DNS is to offer the necessary buoyancy effect. A close similarity can be information to understand the basic turbulent observed between these two figures. The present processes and to optimize empirical coefficients. calculations are, however, made for a rather In order to provide better information for closer moderate buoyancy effect. So, it would be of conditions to actual flows, the steady efforts are great interest to find non-similarity in more highly required towards DNS’s with a higher Reynolds stratified cases, which would be one of the and a lower Rossby numbers. 1 1 y/d θ y/d v Gr=0 0.8 Gr=3.15 x 106 0.8 Gr=6.30 x 106 Gr=1.26 x 107 0.6 Gr=3.15 x 107 0.6 0.4 0.4 0.2 0.2 0 0 0 1 2 3 k+ 0 1 2 θ' 2 + 3 Fig. A The effect of Grashof number to the Fig. B The effect of Grashof number to the turbulent energy temperature variance -2-