_{1}

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.

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 [

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 [

Motivated by the issues of topography-induced wind disasters in high wind speed conditions, Uchida and Ohya [

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 [

Three-dimensional numerical simulations are performed for incompressible density stratified flows (stably stratified flows) over a 2-D hill with a constant cross-section in the spanwise (y) direction. The computational domain and the coordinate system used for the simulations are shown in

z ( x ) = h × { 1 + cos ( π x / a ) } / 2 (1)

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

buoyancy term.

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

∂ u i ∂ x i = 0 (2)

∂ u i ∂ t + u j ∂ u i ∂ x j = − ∂ p ∂ x i + 1 R e ∂ 2 u i ∂ x j ∂ x j − ρ F r 2 δ i 3 (3)

∂ ρ ∂ t + u j ∂ ρ ∂ x j = w (4)

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 (ξ - η - ζ). 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 computational technique of the DNS simulations, the finite-difference method (FDM) is adopted. The pressure-velocity coupling algorithm and the time marching method are based on the fractional-step method [

The number of computational grid points in the x, y, and z directions is 221 × 81 × 101, respectively. The cells in the vicinity of the hill are the smallest, and their dimensions are Δx_{min} = 0.04 h and Δz_{min} = 0.003 h. The spanwise (y) dimension of the computational domain is divided equally into Δy = 0.1 h. A number of grid resolutions were tested, and it was confirmed that the results of the present study are nearly independent of the choice of grid resolution. _{0}Uh/μ) of 2000, where h and U represent the hill height and the uniform inflow velocity, respectively. The non-dimensional time step used in the model is Δt = 0.002.

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 [

The 3-D simulation results from the streamwise cross-section at y = 0 are discussed by comparing them to 2-D simulation (221 ´ 101 grid points) results. First, the temporal change of the drag coefficient, Cd, of the hill is compared between the 2-D and 3-D flows. The comparisons are made for the four cases of stability, K. The drag coefficient of the streamwise cross-section (y = 0) of the hill is evaluated from

C d = 2 ∫ S p z ξ d ξ − 2 R e ∫ S ω y x ξ d ξ (5)

where p is the pressure, ω_{y} is the spanwise (y) component of the vorticity (= ∂u/∂z − ∂w/∂x), and x_{ξ} and z_{ξ} are metrics. Integration is performed over the surface of the hill |x| ≤ a.

In both neutral (K = 0) and weakly stable (K = 0.8) conditions, the value of Cd in the 2-D simulations fluctuates periodically in a continuous manner (

In highly stable conditions (K = 1, 1.3), the trends of the fluctuating Cd in the initial period are similar in the 2-D and 3-D simulations (

Cd subsequently becomes approximately constant both in the 2-D and 3-D simulations (

Finally, the relationship between the stability parameter K and the time-averaged values of Cd is examined for the 2-D and 3-D simulations (

The characteristics of the instantaneous flow field from the 2-D and 3-D simulations are investigated. _{y}, of the flow fields in

The characteristics of the 3-D flows at K = 0 and 0.8 differ noticeably from those of the 2-D flows [

those of the 2-D flows. Specifically, in the 3-D flow, the shear layers which separate from the vicinity of the top of the hill, do not reattach on the downstream slope of the hill, and also curl upward some distance downstream of the hill. Subsequently, vortices form that are not connected to the surface downstream of the hill (indicated by arrows in

In the cases of K = 1 and 1.3, that is, in highly stable conditions, lee waves are excited downstream of the hill in both the 2-D and 3-D simulations as suggested by the streamlines (

Subsequently, the characteristics of the time-averaged flow field from the 2-D and 3-D simulations are examined. _{y}, of the time-averaged flows. For both the 2-D and 3-D flows, time-averaging is performed over the non-dimensional time period between t = 100 and t = 200. For the case of the 3-D flow, spanwise spatial averaging is also performed.

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 (

The mechanism responsible for the decreasing values of SR/h with increasing stability for K > 0.8 is likely identical in the 2-D and 3-D simulations. In this high stability regime, lee waves are excited downstream of the hill unlike at the stabilities of K = 0 and K = 0.8. These lee waves induce strong downward flows, which cause the values of SR/h to be significantly smaller than those at K = 0 and K = 0.8. Particularly at K = 1.3, the lee waves are accompanied by strong downward flows immediately downstream of the hill which exceed the uniform inflow velocity, U. These strong downward flows reduce the value of SR/h significantly. The difference in the streamwise length of the separation bubble, SR/h, between the 2-D and 3-D simulations is evident for both K = 0 and K = 0.8 (

The spanwise (y) flow structure is analyzed for neutral (K = 0) and highly stable (K = 1.3) conditions. Streamlines traced by virtual particles released from the top of the hill and the streamwise (x) component of the vorticity, ω_{x}, (=∂w/∂y − ∂v/∂z), of the 3-D flow are shown in

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

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 [

In neutral (K = 0) and weakly stable (K = 0.8) conditions, the characteristics of the simulated 3-D flow over the hill were significantly different from those of the simulated 2-D flow over the hill. In both the 2-D and 3-D flows, vortices were periodically shed from the hill. However, unlike in the 2-D flow, the shear layers separated from the hill top area in the 3-D flow did not become reattached on the downstream side of the hill and curled upward at some distance downstream of the hill. Subsequently, vortices formed that were not connected to the surface downstream of the hill. These vortices were then shed and advected downstream. As a result, the shear layer which separated from the hill top area became reattached significantly farther downstream (significantly larger value of SR/h) in the 3-D flow than in the 2-D flow. In both the 2-D and 3-D flows, the normalized downwind distance from the hill top at which the separated shear layer is reattached, SR/h, was significantly larger at K = 0.8 than at K = 0. The increased value of SR/h at K = 0.8 was attributable to the reduction in vorticity (momentum) caused by the surface of the hill.

In highly stable conditions (K = 1, 1.3), the characteristics of the 3-D flow in the vicinity of the hill were remarkably similar to those of the 2-D flow. The degree of the similarities was higher at K = 1.3. This result is attributable to the spanwise (y) flow structure, that is, immediately downstream of the hill, strong downward flow was induced homogeneously in the y-direction by lee waves. Furthermore, the streamwise (x) component of the vorticity, ω_{x}, was significantly smaller at K = 1 and K = 1.3 than at K = 0, and the isosurfaces of the vorticity component ω_{x} were linear and short in the streamwise (x) direction. In other words, the spanwise (y) component of the flow was strongly suppressed, and the flow became two dimensional in the x and z directions. Thus, the qualitative behaviors of the flows in the vicinity of the hill were similar between the simulated 2-D and 3-D flows in highly stable conditions. In addition, with the influence of the downward flow of the lee waves, the values of SR/h became shorter in highly stable conditions than in neutral (K = 0) and weakly stable (K = 0.8) conditions in both the 2-D and 3-D simulations. Finally, in highly stable conditions, in the region further downstream of the hill at which rotors were induced by the upward flow of the lee waves, strong disturbances were generated locally.

Uchida, T. (2017) Three-Dimensional Numerical Simulation of Stably Stratified Flows over a Two-Di- mensional Hill. Open Journal of Fluid Dynamics, 7, 579-595. https://doi.org/10.4236/ojfd.2017.74039