九州大学学術情報リポジトリKyushu Three-Dimensional Numerical Simulation of Stably Stratified Flows over a Two-Dimensional Hill

Stably stratified flows over a two-dimensional hill are investigated in a channel of finite depth using a three-dimensional direct numerical simulation (DNS). The present study follows onto our previous two-dimensional DNS studies of stably stratified flows over a hill in a channel of finite depth and provides a more realistic simulation of atmospheric flows than our previous studies. A hill with a constant cross-section in the spanwise (y) direction is placed in a 3-D computational domain. As in the previous 2-D simulations, to avoid the effect of the ground boundary layer that develops upstream of the hill, no-slip conditions are imposed only on the hill surface and the surface downstream of the hill; slip conditions are imposed on the surface upstream of the hill. The simulated 3-D flows are discussed by comparing them to the simulated 2-D flows with a focus on the effect of the stable stratification on the non-periodic separation and reattachment of the flow behind the hill. In neutral (K = 0, where K is a non-dimensional stability parameter) and weakly stable (K = 0.8) conditions, 3-D flows over a hill differ clearly from 2-D flows over a hill mainly because of the three-dimensionality of the flow, that is the development of a spanwise flow component in the 3-D flows. In highly stable conditions (K = 1, 1.3), long-wavelength lee waves develop downstream of the hill in both 2-D and 3-D flows, and the behaviors of the 2-D and 3-D flows are similar in the vicinity of the hill. In other words, the spanwise component of the 3-D flows is strongly suppressed in highly stable conditions, and the flow in the vicinity of the hill becomes approximately two-dimensional in the x and z directions.


Introduction
The atmospheric boundary layer is often characterized by vertical variations of air density. In density stratified layers such as the surface inversion layer that is frequently observed at nighttime, the air density decreases with height. In stably stratified conditions, negative buoyancy forces are present. As a result, when air flows over simple or complex topographies in stably stratified conditions, waves and other fluid phenomena emerge that are not observed in neutrally stratified conditions [1]- [14]. Internal gravity waves are one such phenomenon [1]. When an atmospheric layer (lid) that can reflect internal gravity waves forms at the top of the surface inversion layer, the energy of internal gravity waves (lee waves) excited by the topography is trapped between the ground surface and the atmospheric layer (lid). The trapped internal gravity waves have a significant influence on the flow over the topography. Understanding these wave and other fluid phenomena is necessary to predict wind characteristics in the vicinity of both simple and complex topographies for numerous practical applications. These applications include effective use of wind energy, prediction of air pollutant advection and dispersion, and issues related to high wind speeds such as topography-induced wind disasters (disasters due to high local wind) [2].
In the past, the authors have conducted numerical studies on stably stratified flows over a two-dimensional hill with a model based on the finite differencing method [3]- [11]. In these studies, in order to simulate flow over a hill with an upper atmospheric lid aloft, the hill was set in a channel of finite depth. Uchida and Ohya [10] clarified the relationship between the unsteady flow that emerges over the hill and the behavior of columnar disturbances that propagate upstream of the hill. The same work showed that the characteristics of unsteady flow over a hill were highly influenced by variables related to the non-linearity of the flow such as the Reynolds number, the hill height, the channel depth, the blockage ratio, and the topography configuration parameters. Uchida and Ohya [10] also elucidated the effect of stable stratification on the non-periodic separation and reattachment of the flow behind the hill. Uchida and Ohya [8] investigated the influence of the streamwise grid resolution on computational results and proposed an effective numerical computational method for analyses of flow over a hill in a stably stratified fluid.
Motivated by the issues of topography-induced wind disasters in high wind speed conditions, Uchida and Ohya [7] [9] simulated flows over a hill with uniform inflow speed and evaluated the percentage increase of the wind velocity in areas on the hill with locally increased wind speed. In highly stable conditions, breaking lee waves emerged in the vicinity of the hill. A region of extremely high wind velocity formed downstream of the breaking lee waves, and the wind velocity in the region exceeded the inflow velocity [9]. The series of studies introduced above was conducted with a 2-D direct numerical simulation (DNS), and some of the effects of stable stratification on the spanwise flow structure (three-dimensionality of the flow along the spanwise component) remain unknown. The present study investigates the characteristics of 3-D flows over a hill, which are more realistic than the 2-D flows studied previously. The effect of the ground boundary layer that develops over the upstream side of the hill is not considered as in our previous studies [5], that is, slip conditions are imposed on the surface upstream of the hill, and no-slip conditions are imposed only on the surface of the hill and the surface downstream of the hill. With this set of boundary conditions, numerical simulations are conducted with DNS for a Reynolds number of 2000 as in our previous studies [5]. The simulated 3-D flows will be discussed in terms of the differences and similarities with the simulated 2-D flows, particularly in terms of the effect of the stable stratification on the non-periodic separation and reattachment of the flow behind the hill.

Governing Equations for Stratified Flows over a Hill and Numerical Computational Method
for |x| ≤ a, where the parameter a specifies the steepness of the hill. In the present study, the parameter a is set to a = h (=1) to model a steep 2-D hill. The density and pressure fields are decomposed into base and perturbation components. Far upstream of the hill, the base density field ρ B is defined so that the density decreases linearly in the vertical direction (z) as dρ B /dz = −1. The base pressure field that is required for hydrostatic equilibrium with the base density field is referred to as p B . A vertically uniform profile of wind velocity, U, is imposed on the flow approaching the hill. Furthermore, the Boussinesq approximation is applied so that the density is assumed to be constant except in the  The five unknown variables to be solved for are the velocity components, u i (= u, v, w), the deviation of pressure from the base pressure field, p' (= p − p B ), and the deviation of the density from the base density field, p' (= ρ − ρ B ). The governing equations for density stratified flows over a hill consist of the continuity equation, the Navier-Stokes equation, and the density equation. The non-dimensionalized forms of these governing equations can be expressed as where the prime notations (') indicating the deviation of the density and pressure from their base values have been omitted, and the subscripts represent Einstein notation. In Equation (3), the non-dimensional numbers Re and Fr indicate the Reynolds number (=ρ 0 Uh/μ) and the Froude number (=U/Nh), respectively, where ρ 0 is the reference density, μ is the coefficient of viscosity, N is the buoyancy frequency defined as N 2 = −(g/ρ 0 ) (dρ B /dz), and g is the acceleration of gravity. Because a hill of height h is set within a channel of finite depth, H, in the present study, the non-dimensional stability index K (=NH/πU) is adopted to express the stability of the flow. Fr and K are related by Fr·K = H/πh, thus, the smaller the value of Fr or the larger the value of K, the more stable the stratification of the flow.
In order to avoid numerical instability and to achieve highly accurate predictions of the flow over the hill in stably stratified conditions, the DNS simulations are performed using collocated grids in a general curvilinear coordinate system (ξηζ). In the collocated grid system, staggered allocation is used: velocity components, u i , pressure, p, and density, ρ, are defined at the cell center, and variables that result from the contravariant velocity components, U i (=U, V, W), multiplied by the Jacobian, J, are defined at the cell faces. As for the computa- Four cases are considered for the simulations: 1) neutral stratification at K = 0 (Fr = ∞), 2) weakly stable stratification at K = 0.8 (Fr ≈ 2.39), 3) highly stable stratification at K = 1 (Fr ≈ 1.91), and 4) highly stable stratification at K = 1.3 (Fr ≈ 1.47). The readers are advised to refer to Uchida and Ohya [10] for a discussion of the ways in which the simulated flow is influenced by variables and parameters related to the non-linearity of the flow such as the Reynolds number, the blockage ratio, and the topography configuration parameters.

Flow over the Hill (0 ≤ K ≤ 1.3), Instantaneous Field
The characteristics of the instantaneous flow field from the 2-D and 3-D simulations are investigated. Figure 5 illustrates the streamlines of released virtual fluid particles, and Figure 6 shows the spanwise (y) vorticity, ω y , of the flow fields in        performed. Figure 9 illustrates the relationship between the stability and the streamwise (x) length of the separation bubble formed behind the hill, SR, that is the streamwise distance from the hill center (x = 0) at which the shear layer that separated from the vicinity of the hill top becomes reattached. In Figure 9, SR is normalized by the hill height, h. The trends of the variation of SR/h with respect to the stability, K, are similar between the 2-D and 3-D simulations: the value of SR/h increases with increasing stability between K = 0 and K = 0.8 while it deceases with increasing stability for K > 0.8. From the distribution of the vorticity (Figure 8), it can be speculated that the following mechanism accounts for the value of SR/h at K = 0.8 being larger than that at K = 0, in both the 2-D and 3-D simulations. Due to the effect of the stable stratification, a large velocity deficit occurs in the vicinity of the surface of the hill, and the vorticity (momentum) caused by the hill surface decreases as illustrated by the decreased number of the isolines for vorticity. As a result, the downstream distance at which the separated shear layer becomes reattached increases with increasing stability between K = 0 and K = 0.8.
The mechanism responsible for the decreasing values of SR/h with increasing stability for K > 0. 8

Spanwise (y) Flow Structure in Neutral Conditions (K = 0) and Highly Stable Conditions (K = 1.3)
The spanwise (y) flow structure is analyzed for neutral (K = 0) and highly stable (K Open Journal of Fluid Dynamics  Figure 10 and Figure 11, respectively. At K = 0, the flow varies in the spanwise (y) direction in a complex manner, suggesting that the flow has become three-dimensional (Figure 10(a)). The three-dimensionality of the complex flow has led to the emergence of longitudinal vortices that entwine with one another over a broad area downstream (Figure 11(a)). Furthermore, variation in the strength of vortex roll-up (separation of streamlines) is evident (arrows in Figure 10(a)) as was also observed in numerically simulated 3-D flows around a cylinder [19] [20].
At K = 1.3, lee-waves causes strong downward flows immediately downstream of the hill. These downward flows occur almost homogeneously in the y-direction. Consequently, the values of ω x are smaller at K = 1.3 compared to K = 0, and the structure of the isosurfaces of ω x are linear and short in the streamwise (x) direction (area A in Figure 11(b)). In other words, the spanwise (y) flow component is strongly inhibited, and the flow becomes approximately two-dimensional in the x and z directions. However, in the vicinity of the rotors induced by the upward flow of the lee-waves further downstream of the hill, the flow becomes heterogeneous in the y-direction (indicated by arrows in Figure 10(b) and area B in Figure 11(b)). Therefore, strong disturbances are locally generated within and near the rotors even in highly stably stratified flows.

Summary
Three-dimensional direct numerical simulations (DNS) were conducted for flows over a hill in neutral and stably stratified conditions. The present study followed onto our previous two-dimensional DNS studies on flows over a hill in stably stratified conditions and provided a more realistic simulation of atmospheric flows than our previous studies. In the 3-D simulation, a hill with a constant cross-section in the spanwise (y) direction was placed in the computational domain. The spanwise dimension of the computational domain was eight times the height of the hill. As in our previous research [5], no-slip conditions were imposed only on the hill surface and the surface downstream of the hill; on the surface upstream of the hill, slip conditions were imposed. This set of surface conditions was adopted to investigate the flow over the hill without considering the effect of the ground boundary layer that would develop upstream of the hill.