over a smooth surface are performed using the Boussinesq approximation to account for the buoyancy effect. The Reynolds number based on the geostrophic ...
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, 3.15×10 6, 6.30×10 6, 1.26×10 7 and 3.15×10 7 in order to examine the effect of the stable 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 the mean velocity (φ) 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
Riw : Richardson number at ground Re : Reynolds number =G h /ν Tw : temperature difference between far upper region and ground t : time T : instantaneous temperature Tτ : friction temperature u : velocity vector =(u, v, w) u τ : friction velocity x(x1), y(x2), z(x3) : coordinates β : volume expansion coefficient δt : turbulent depth = u τ / f φ : angle from geostrophic wind direction ν : kinematic viscosity νtx : streamwise eddy diffusivity
Nomenclature a at dv dθ D
: : : : :
thermal diffusivity thermal eddy diffusivity velocity boundary layer thickness thermal boundary layer thickness Ekman layer depth = 2ν / f
f g G Gr h p q total qh
: : : : : :
Coriolis parameter gravity acceleration vector =(0, -g, 0 ) geostrophic wind velocity Grashof number =g β Tw h 3/ν 2 height of computational domain pressure : vertical total heat flux at ground horizontal turbulent heat flux vector
:
* Received: November, 16, Editor: Koichi HISHIDA † Department 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 θ ρ τ Rh τw τθ Ω
: : : : : : :
spanwise eddy diffusivity converted instantaneous temperature density projected Reynolds stress tensor onto x-z plane total shear stress at ground time-scale of temperature decreasing 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
()
: statistically averaged quantity
Fig. 1
Computational domain
Boussinesq approximation to account for the buoyancy effect. The computational domain is doubled in size compared to that of Coleman et al.[9] without the deterioration of resolution. A temperature field is so introduced that its mean profile is quasi-steady with time. The 3-dimensional characteristics of the mean velocity, the Reynolds stress and the turb ulent heat flux are examined. Moreover, the effects of the stable stratification upon the statistical quantities are also discussed based on the o btained DNS data.
1 Introduction The planetary boundary layer (PBL) is affected by the miscellaneous factors such as the system rotation and the buoyancy force. The boundary layer under the effect of the system rotation is called as the Ekman layer. The works on the laminar PBL have been widely performed mainly by the theoretical methods. On the other hand, the turbulent PBL is much more complex because it includes fine and very large-scale motions and because its mean velocity field is three dimensional in nature. Hence, the measurements and experiments have been the major tools in almost all researches on the turbulent PBL. Hunt et al.[1] and Lenschow[2] are the ones who utilized the measurements for the turbulent PBL. However the threedimensional spatial structure of the turbulent velocity field is not easy to be obtained experimentally. Many turbulence models have been proposed and developed to describe the 3-dimensional spatial structure of the velocity field (e.g. André et al.[3], Hunt[4]). Deardorff[5], Mason[6] and Moeng[7] employed the large eddy simulation (LES) to compute the turbulent PBL numerically. The review of the experimental and the numerical results is given by Wyngaard[8] in 1992. Among these studies on PBL, to the authors' knowledge, the DNS of the turbulent Ekman layer have not been performed except the work of Coleman et al.[9][10] They obtained the Ekman spiral to find that it became shallower compared with that of the laminar flow. They also examined the existence of the large-scale longitudinal vortices. But their computational domain was too small to contain the large-scale motion. In addition, they used the temperature field whose mean profile depended on time. As the result, some of the obtained statistical quantities were not averaged in long time enough to be used for the construction of the turbulence model. In the present work, the neutrally and the stably stratified turbulent Ekman layers over a smooth surface are computed through DNS using the
2 Computational condition Calculated flow field is the turbulent Ekman layer of an incompressible viscous fluid over a smooth flat surface in a vertically oriented gravitational field. The system is rotating about a vertical axis with an angler velocity Ω = (0, f/2, 0), where f is the Coriolis parameter. The flow is driven by a horizontal pressure gradient. In this layer, the pressure gradient, the Coriolis and the viscous forces are balanced. At a position so high for the viscosity effect as to be neglected, the pressure gradient balances with the Coriolis force and thus the mean flow direction becomes perpendicular to the pressure gradient. This type of flow is called as the geostrophic wind (G). In the present work, x- and y-axes are set to be parallel and vertical, respectively, to the geostrophic wind direction. Computational configuration is given in Fig. 1. The periodic boundary condition is imposed in x and z directions. The non-slip and the Neumann conditions are adapted at the bottom and top boundaries. The height of the computational domain is so set to be enough large compared to the boundary layer thickness. For the computational grid, the uniform mesh is used in the x and z directions. On the other hand, the non-uniform mesh is adopted in the y direction in order to ensure a high resolution in the vicinity of the ground. The governing equation is the continuity equation ∇⋅ u = 0 , (1) the Navier-Stokes equation
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Thermal Science & Engineering Vol.10 No. 1 (2002)
Table 1 Grid number
Computational conditions 256 × 96 × 256
+
+
+
Spatial resolution (∆ x , ∆ y , ∆ z ) 6.01, 0.219 − 21.4, 6.01 Re = G ・h/ ν
12,000
Ro = G / (fh)
7.0 3
Gr = g β T w h / ν
2
0.0, 3.15 × 10 6 , 6.30 × 10 6 , 1.26 × 10 7 , 3.15 × 10 7
2 y/δ t
Gr=0 3.15 x 107 U/G V/G -W/G Symbol : Coleman et al.(1990) U/G, V/G
1
0
∂θ ∂θ ∂ 2θ 1 +uj = a 2 + (1 − θ ) . ∂t ∂x j ∂x j τ θ
(5)
The optional parameter τθ means the time-scale of the temperature decrease. Here, time constant τθ is assumed to be positive because the stable stratification is usually caused by the decrease of the ground temperature. The heat source term represents the heat-losses caused by both the heat conduction and the radiation heat transfer. In the present computation, the fractional step method is adopted for the coupling between the continuity and the Navier-Stokes equations. The 2nd-order CrankNicolson and the 2nd-order Adams -Bashforth methods are employed as the time-advance algorithm; the former for the vertical viscous term, the latter for the other viscous and the convection terms. The Coriolis term is solved implicitly to avoid the numerical instability. For the spatial discretization, the finite difference method is adopted. The computational conditions are summarized in Table 1.
∂u 1 + (u ⋅∇ )u + 2Ω× u = − ∇p − Θg + ν∇ 2u , (2) ∂t ρ and the energy equation
(3)
The Boussinesq assumption is used in this set of equations. The quantity Θ is defined as Θ = β (θ - Tw). Here, θ is the temperature converted in the following method. The actual stably stratified temperature field is decaying with time because it is resulted from the cooling at the ground. On the other hand, the time-scale of temperature decrease is very large compared with that of the velocity fluctuation. Therefore, a mean temperature field is assumed to be quasi-static with respect to the time advancement. The instantaneous temperature T is defined as (4) T (x ,y , z ,t ) = −Tw (t ) ⋅ 1− θ ( x, y , z ,t )
(
0.5 U/G, V/G, -W/G 1
Fig. 3 Mean velocity profiles for Gr=0 and 3.15×10 7
Fig. 2 Instantaneous velocity field in neutrally stratified turbulent Ekman layer (White: the second invariants of the velocity gradient tensor II’’+
