Direct Numerical Simulation of Turbulent Heat Transfer in the by ntn18128

VIEWS: 25 PAGES: 9

									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-

								
To top