Large Eddy Simulation Study of Moving Objects in ... - AIAA ARC

Large Eddy Simulation Study of Moving Objects in Thermally-Stratified Boundary Layer Flows John Haywood∗1 and Adrian Sescu†1 1

Department of Aerospace Engineering, Mississippi State University, MS 39762

Numerous experimental and numerical studies have shown that the dynamics of the atmospheric boundary layer is affected by various surface non-uniformities that may include canopies, trees or buildings, among others. In this work, a large eddy simulation study of large box moving through a thermally-stratified atmospheric boundary layer is conducted. The focus is on characterizing the wakes generated by the interaction between these moving boxes and the stably stratified atmospheric boundary layer. A concurrent precursor simulation method is implemented in a pseudo-spectral code to impose turbulent inflow conditions upstream of the object. An analysis of the wake generated by the moving object is performed using a three-dimensional proper orthogonal decomposition (POD) method. Various results in terms of time-averaged streamwise velocity and potential temperature, as well as turbulence intensity, vertical turbulent momentum and heat fluxes profiles are reported. Two types of POD modes of different streamwise wavenumbers - coined here as ’sinuous’ and ’varicose’, respectively - are identified in the wake.

I.

Introduction

The atmospheric boundary layer (ABL) is the layer of air in the vicinity of the earth’s surface in which various effects from the surface (friction, heating and cooling) are felt directly on time scales in the order of less than a day, and significant fluxes of momentum and scalars are carried out by turbulent motions (Garratt20 ). The turbulence in the ABL is an important key feature in the dynamics of the air in the atmosphere, and it includes some specific characteristics when compared to flow at lower scales. Turbulence generated by thermal convection (from buoyancy) coexists with the mechanical turbulence which is usually generated by wind shear. In addition, the rotation of the Earth induces a new dynamics into the mean flow which ultimately interacts with turbulence. There is a large range of scales sizing from small eddies in the order of 1 mm to large coherent structures in the order of the height of the boundary layer (order of 1 km). The flow in the ABL may interact with various objects that are either at rest or in motion (a relevant example of large moving objects is a fleet of ships). If these objects exceed a certain size, their effect on the dynamics of the ABL may be significant. In the past, most of the fundamental knowledge of boundary layer turbulence in the atmosphere has been achieved as a result of extensive experimental work (e.g., Augstein3 or Holland and Rasmusson21 ). Large Eddy Simulation(LES) has been widely used in the last decades to predict the flow in ABL, starting with the pioneering work of Deardorff16, 17 who applied LES to neutral and unstable boundary layers. The early LES following Deardorff’s approach focused on the cloud-free, convective boundary layers since reasonable solutions can be obtained with low resolution in the grid. Moeng30 continued the work of Deardorff and developed an algorithm that uses a mixed pseudospectral finite-difference method. Schmidt and Schumann38 used LES to investigate a convective boundary layer uniformly heated from below and topped by a layer of uniformly stratified fluid. Mason28 studied the free convection in ABL with an overlying capping inversion, using different subgrid-scale models, varying the domain size and the mesh resolution, and found out that the gross features of the ABL are not sensitive to the details of the simulations. Sorbjan42, 43 used LES to generate and compare statistics of turbulence fluctuations during penetrative and nonpenetrative dry convention, and examined the effects caused by variation of the potential temperature lapse rate in the ∗ Research † Assistant

Assistant, Department of Aerospace Engineering, Mississippi State University, MS 39762; member AIAA. Professor, Department of Aerospace Engineering, Mississippi State University, MS 39762; Senior member AIAA.

1 of 26 American Institute of Aeronautics and Astronautics

free atmosphere on a shear free convective ABL. Examples of more recent analyses associated with LES applied to convective ABL include Fedorovich et al.18 who studied the relationships between parameters of convective entrainment of a shear-free stratified atmosphere, Conzemius and Fedorovich15 who focused on the dynamics of entrainment and the effects on the evolution of the dry convective ABL under the presence of wind shear, or Botnick and Fedorovich9 who investigated the effects of various initializations for LES. In the area of stable ABL, Mason and Derbyshire28 used an LES framework to study the ABL over uniform, flat terrain. Andren1 studied the dry stably-stratified planetary boundary layer using LES, using an improved subgrid-scale model version, and showed improvements in the near-surface region compared to previous studies. Examples of recent articles related to the application of LES to stable ABL include an inter-comparison of results from different LES codes (Beare et al.7 ), Stoll and Porté-Agel46 who performed simulations using a new dynamic subgrid-scale model generating results that were in better agreement with boundary-layer similarity theory and previous LES studies, or Zhou and Chow50 who used an explicit filtering and reconstruction approach. There are several recent LES studies of the interaction of the ABL with large arrays of wind turbines, since it was found that such large wind farms may affect the local weather. Previous relevant numerical studies involving large-scale interactions between ABL and wind farms include Keith et al. 25 in which a series of simulations using community Global Circulation Models (GCMs) are presented, evaluating possible effects of massive implementations of wind farms at a global scale. Barrie and Kirk-Davidof6 explored possible impacts of large wind farms on weather, and Baidya-Roy et al.4 raised the question whether wind turbine arrays may have an effect on the local meteorological conditions of a particular region. A recent LES study of the stratified ABL interacting with a wind farm was performed by Lee et al.24 Their analysis targeted the impact of the effective roughness length and the stratification on structural loadings of wind turbines, and the effects that the rotor wakes can have on the downstream turbines. Another recent LES study of off-shore wind turbine performance was presented in Ref.2 Calaf et al.12 performed a suite of LES of a fully-developed wind turbine array boundary layer using an actuator disk model under neutral stability conditions (their work was extended to thermally-stratified conditions by Sescu and Meneveau? ). In Porte-Agel et al.,36 a LES framework was validated against experimental data, with three types of models used to parameterize the rotor-induced forces: a standard actuator-disk model without rotation; an actuator-disk model with rotation; and an actuator-line model. In this work, an LES framework is used to study the effect of large moving objects on the dynamics of the ABL. The focus is on the dynamics of the wakes generated by the interaction between these moving objects and the atmospheric flow. The tool is a pseudo-spectral LES code solving for the incompressible Navier-Stokes equations with Boussinesq approximation used to take into account the effect from the thermal stratification in the momentum equations, and with the Coriolis effect included. The box is modeled using an immersed boundary method for both the momentum and temperature. Originally, the code was designed to work with periodic conditions along the horizontal directions; in this work, a concurrent precursor simulation method is implemented to turn off the periodicity along the streamwise direction. The wakes generated by the box are analyzed using a fully three-dimensional POD technique. Various results in terms of timeaveraged streamwise velocity and potential temperature, as well as turbulence intensity, vertical turbulent momentum and heat fluxes profiles are reported. Using the POD method, two types of modes - coined here as sinuous and varicose - are identified in the wake; in addition, different wavenumber sinuous modes are revealed in the wake. Section II briefly introduces the LES framework, discussing the governing equations, boundary conditions, and the specific subgrid-scale model used. In section III, the immersed boundary method used to model the moving box is described. In section IV, the concurrent precursor simulation technique is introduced and discussed in the present framework. Section V discusses the POD technique that is applied to analyze the wakes. In section VI, results consisting of various contour plots of instantaneous or mean streamwise velocity and potential temperature, profiles of streamwise turbulence intensity, turbulent momentum and her fluxes, and contour plots of streamwise velocity POD modes are presented and discussed. Concluding remarks are included in the last section VII.

II.

Large Eddy Simulation Framework

In this study, the LES filtered momentum conservation equations with the Boussinesq approximation, transport equation for potential temperature, and continuity equation are used,


˜ ∂u ˜i ∂u ˜i ∂ p˜∗ ∂ τ˜ij θ˜ − hθi +u ˜j =− − + δi3 g + fc ij3 (˜ uj − ugj ) + Fi ∂t ∂xj ∂xi ∂xj θ0 ∂ θ˜ ∂ θ˜ ∂πj +u ˜j =− , ∂t ∂xj ∂xi

∂u ˜i =0 ∂xi

(1)

(2)

˜ is represented by tilde, u respectively, where the spatial filtering at scale ∆ ˜i , i = 1, 2, 3 are the components of the velocity field corresponding to the axial x1 -direction, lateral x2 -direction, and vertical x3 -direction, respectively, θ˜ is the resolved potential temperature, θ0 is the reference temperature, the angle brackets represent a horizontal average, g is the gravitational acceleration, fc is the Coriolis parameter, δij is the Kronecker delta, ijk is the alternating unit tensor, p˜∗ is the effective pressure divided by reference density, and Fi is a forcing term (here modeling the effect of the wind turbines). The SGS stress and the SGS heat g ˜ They are modeled using a Lagrangian scale-dependent flux are given as τij = ug ˜i u ˜j and πj = u ˜j θ. i uj − u jθ −u model as developed by Bou-Zeid et al.,8 and extended to scalar transport by Porte-Agel.37 Because the Reynolds number is very large in the ABL and the flow in the vicinity of the ground surface is modeled using Monin-Obukhov similarity theory (Monin and Obukhov,32 Obukhov,34 Moeng30 ), the molecular viscous diffusion term in the momentum equation is neglected. This study is concerned with overall effects of a large moving objects on the ABL, with a relatively coarse grid resolution across the objects; as result, the objects are modeled using the immersed boundary method (Peskin,35 Iaccarino,22 Mittal29 ). Using the hypothesis of horizontal homogeneity of turbulence in ABL, periodic boundary conditions can be employed along both horizontal directions. The vertical gradients of velocity and scalars and the vertical component of velocity must vanish at the top boundary which is located well above the boundary layer top. In addition, for the stable case, a sponge layer is added above the boundary layer, to damp out the gravity waves reflecting from the top boundary. A generic equation with damping term is ∂t φ = L(φ)+r(z)(φ−hφi)H(z−zb ) where L() is a spatial differential operator, φ is a prognostic variable, hφi is an average (usually, over horizontal planes), H(x) is Heaviside step function, and zb is the elevation where the sponge layer starts. The relaxation term r(z)(φ − hφi) dampens fluctuations at time scales larger than a prescribed relaxation time scale τ = 1/r (Sorbjan44 ). In this work, the relaxation function is given as r(z) = r0 /2 [1 − cos (π(z − zb )(zt − zb ))] where r0 is a given relaxation constant (of the order of 0.01s−1 ), and zb indicates the top of the computational domain. At the bottom surface, the instantaneous wall stress is related to the velocity at the first vertical node through the application of the Monin-Obukhov similarity theory (Businger et al.10 ). 2

 τi3 |z=0 = −u2∗

κVf u˜i  u˜i ; = − z Vf Vf ln z0 − ΨM

i = 1, 2

(3)

where τ13 |z=0 and τ23 |z=0 are the instantaneous local wall stress components, u∗ is the friction velocity, z0 is the effective roughness length, κ is the von Kármán constant (taken to be κ = 0.4), ΨM is the stability 1/2 correction function for momentum, and Vf = u˜1 (∆z/2)2 + u˜2 (∆z/2)2 is the local filtered horizontal velocity at the first vertical level, in the grid. The surface heat flux is computed as

hw0 θ0 iz=0

u∗ κ θs − θ˜ = ln zz0s − ΨH

(4)

where θs is the imposed surface (ground level) potential temperature, θ˜ denotes the resolved potential temperature at the first vertical level, z0s is the roughness length for scalar (its value is 0.1z0 ), ΨH (ζ) = ´ζ [1 − φH (ζ 0 )dζ 0 /ζ 0 ] is the stability correction function for heat flux (also, ζ = z/L), and φH is given as 0


φH (ζ) = P rt + βζ,

(5)

where the Prandtl number P rt = 0.74, β ≈ 5, for stably stratified conditions, and −1/2

φH (ζ) = P rt (1 − γζ)

,

(6)

where γ ≈ 16, for unstable conditions. In stratified conditions, the profiles of horizontally averaged velocity, temperature and turbulent fluxes are not stationary, since the surface heat flux injects/extracts heat into/from the surface layer. To keep the horizontally averaged temperature profile stationary, while the temperature at the ground is maintained constant, a source or sink of heat, depending on whether the ABL is stable or unstable, respectively, must be applied above the ABL. In reality such exchanges occur through entrainment at the top of the ABL. The numerical fringe region continues upward to the top boundary of the domain (see Sescu and Meneveau40 for more details on the control method). Within the ABL, the wind direction is a function of elevation according to the Ekman spiral, and it is difficult to set the geostrophic velocity components that provide the desired flow direction at the hub height. A gradual adjustment of the geostrophic wind direction is applied here by using a source term (in the form of a Coriolis force) in the momentum equations (see Sescu and Meneveau40 ). The numerical tool is an LES code that employs a pseudo-spectral method (Canuto13 ) in the horizontal directions and a second order accurate centered difference scheme in the vertical direction. The corresponding aliasing errors in the nonlinear terms, that arise from the pseudo-spectral method, are corrected according to the 3/2 rule (Canuto13 ). Continuity equation is enforced through the solution of the Poisson equation resulting from taking the divergence of the momentum equation. The centered difference scheme requires the use of a staggered grid in the vertical direction, with the vertical component of velocity stored halfway between the other variables. The time marching is performed using a fully-explicit second-order accurate Adams-Bashforth scheme (Butcher11 ).

III.

Immersed boundary method

The presence of the moving box in the ABL is modeled using the direct forcing immersed boundary method (IBM) first introduced by Mohd-Yusof.31 In the direct forcing approach, the velocity boundary conditions are imposed by a force added to the discretized momentum equations, un+1 − uni i = RHSi + fi ∆t

(7)

The imposed force can then be calculated by specifying the velocity at the boundary, (ui )box   (ui )box −uni − RHS ; box points i ∆t fi = 0 ; f luid points By substituting the imposed force back into the discretized momentum equations, it is shown that the boundary condition is satisfied at each discrete point. un+1 = (ui )box i

(8)

Since the continuity equation is satisfied by solving the Poisson equation for the pressure, the application of the imposed force needs to be handled carefully. If the imposed force is applied before the Poisson equation is solved, the resulting velocity field after the solution of the Poisson equation will not satisfy the desired boundary conditions; if the imposed force is applied after the Poisson equation is solved, the resulting velocity field will not be divergence free. It is because of these issues that the pressure and the imposed force are calculated using a coupled method (Chester et al.,14 Munters33 ). First, an intermediate velocity u∗i is found by solving the momentum equations without the pressure gradient or the imposed force. In terms of the Adams-Bashforth scheme, the intermediate velocity is 3 1 u∗i = uni + ∆t RHSin − RHSin−1 (9) 2 2 4 of 26 American Institute of Aeronautics and Astronautics

Then, the velocity at the next time step is calculated by adding the pressure and the imposed force terms to the intermediate velocity, 3 n n+1 ∗ n ui = ui + ∆t fi − ∇p (10) 2 The imposed force is calculated by applying the boundary condition   3 ∇pn + (ui )box −u∗i ; box points ∆t fi = 2 0 ; f luid points The Poisson equation for the pressure is found by taking the divergence of equation for the velocity at the n + 1 timestep and requiring that the flow be divergence free at the next time step. ∇2 pn =

2 u∗ ∇( i + fin ) 3 ∆t

(11)

The coupled pressure and imposed force equations are then solved interatively so that the velocity field at the n + 1 timestep satisfies both the continuity equation and the desired boundary conditions. The same direct forcing approach indroduced by Mohd-Yusof can be applied to the energy equation to model the effect of a box that has a different temperature than the surrounding atmosphere (Kang et al.23 ). As with the momentum equations, a temperature forcing term is added to the discretized energy equation to impose a Dirichlet boundary condition. θn+1 − θn = RHSθ + fθ ∆t

(12)

The temperature forcing term can then be calculated by specifing the temperature at the boundary, θb   θbox −θn − RHS ; box points θ ∆t fθ = 0 ; f luid points By substituting the temperature forcing term back into the discretized energy equation, it is shown that the temperature boundary condition is satisfied at each discrete point. θn+1 = θbox

(13)

Because the energy equation is uncoupled from the continuity and momentum in incompressible flow and an explict time-advancement scheme is used, the temperature forcing term does not need to be explicitly calculated (Balaras5 ). By using the direct forcing IBM to model a box that coincides with the grid, the boundary of the object is exactly reproduced. This creates a sharp discontinuity at the surface of the box. High frequency oscillations are introduced at the discontinuity by representing a discontinuous function using finite Fourier series, as is required for a pseudo-spectral method. This behavior of a finite Fourier series at a sharp discontinuity is known as Gibbs phenomenom. The effect of Gibbs phenomenom is minimized by applying a Laplacian smoothing operator to the velocity and temperature fields on and inside the boundary of the box for each horizontal plane (Tseng et al.48 ). old old old old unew (i, j) = (1 − 4λ)uold (14) i i (i, j) + λ ui (i + 1, j) + ui (i − 1, j) + ui (i, j + 1) + ui (i, j − 1) old new old old old old θ (i, j) = (1 − 4λ)θ (i, j) + λ θ (i + 1, j) + θ (i − 1, j) + θ (i, j + 1) + θ (i, j − 1) (15) The horizontal indices (i, j) represent grid points located on or inside the box. The smoothing operator is applied interatively before the spectral derivatives are calculated. Figure 1(a) shows a potential temperature countour before the application of the Laplacian smoothing operator. Oscillations can be clearly seen emanating from the upstream face of the box. Weaker osciallations can also be seen on the spanwise sides of the box. The effect of the smoothing operation is shown in figure 1(b). The smoothing has inhibited the growth of the spurious oscillations and allowed for a crisper resolution of the box.


a)

b)

Figure 1. Contour of the non-dimensional potential temperature on an xy plane through the center of the box: a) without the Laplacian smoothing; b) with the Laplacian smoothing.

IV.

Precursor simulation

Realistic turbulent inflow conditions are required in order to study the effect of an isolated box on the ABL. Otherwise, with the periodic horizontal boundary conditions, the wake of the box would be recycled through the domain. This turns a single box, in the streamwise direction, into an infinite series of boxes. Using previous precursor methods, a separate simulation is carried out prior to the main simulation to generate a library of turbulent data (Tabor47 ). The turbulent library is created by simulating a periodic domain, for example a box of fluid or pipe flow, and, at every time step, writing a slice normal to the streamwise direction of the velocity field to disk. These slices are then rescaled, if necessary, and loaded as inflow conditions for the main simulation. The idea is to provide the main simulation with realistic turbulent scales at the inlet that then develop into the appropriate turbulent structures for the flow. The method of running the precursor simulation concurrently with the main simulation was first introduced in the context of flat plate boundary layers by Lund et al.27 and improved upon by Ferrante and Elghobashi.19 In it, a slice of the velocity field is taken near the inflow of the precursor simulation. The data is rescaled and, along with the appropriate imposed turbulent kinetic energy spectrum and Reynolds stress, used as the inlet conditions for the main simulation. In the concurrent precursor method introduced by Stevens et al.,45 the precursor simulation and the main simulation are run simulatneously, advancing in time together. The inflow conditions are transfered to the main simulation directly from the precursor simulation. Two identical domains are considered: the precursor and the main. The difference between the two domains is that the IBM is included in the main domain to simulate the presence of the box and is excluded from the precursor domain. The simulations in both domains are synchronized in time. At the end of each time step, the velocity in a region at the outflow of the precursor domain is blended on to the end of the main domain. The blending allows the velocity in the main simulation to smoothly transition to the copied precursor simulation velocity, which minimizes the generation of spurious oscillations. Shown in figure 2, the blending region has a length Lblend and is defined from x = Ls to x = Lx , where Lx is the length of either domain and Ls = Lx − Lblend . The length of the blending region required to smoothly transition the 1 velocities is case dependent, with Lblend = 10 Lx being used here. The velocity in the blending region is defined as main ublend (x, y, z, t) = w(x)upre (Ls , y, z, t) (16) i i (x, y, z, t) + [1 − w(x)] ui where upre is the velocity field from the precursor domain and umain is the velocity field from the main i i domain. As x increases, the blending function w(x) diminishes the effect of the main domain velocities and


increases the effect of the precursor domain velocities. The blending function used here is defined as  h i  1 1 − cos π x−Ls ; Ls ≤ x ≤ Lpl Lpl −Ls w(x) = 2 1 ; L 0.99 (27) ∞ k k=1 λ This says that the number of snapshots collected should be large enough to span enough time so that 99% of the turbulent kinetic energy is resolved.


VI. VI.A.

Results and Discussions

Description of the Test Cases

A large rectangular box with dimensions 79×66×79m (in the streamwise, transverse and spanwise directions, respectively) is considered. The box is moving at velocities 10 and 20 m/s in the opposite direction to a geostrophic wind of 8 m/s. In addition, two thermal stratifications are considered: ∆θ = 0.2 and ∆θ = 0.4. The size of both the precursor and main domains is 2513 × 628 × 600 m in the streamwise, transverse and spanwise directions, respectively, and both simulations are performed on 192×64×128 grids. A constant grid spacing from top to bottom, along the vertical direction, since non-uniform, clustered grids near the ground are not needed due to the wall model based on Monin-Obukhov similarity theory. Constant grid spacing is also used along both horizontal directions, discretized here using a pseudo-spectral method. Thermal stratification is set by specifying the temperature difference, ∆θ = θt − θs , between the temperature at the ground surface, θs , and the temperature at the top of the ABL, θt . VI.B.

Contour Plots

Sample contour plots of the instantaneous streamwise velocity and potential temperature are shown in figures 3, 4 and 5. Figure 3 shows qualitative contour plots of the streamwise velocity component (top) and potential temperature (bottom) on xz planes cut through the center of the box (the stratification for this case is ∆θ = 0.2, while the velocity of the box is 10 m/s). The top part shows the flow reversal in the close proximity to the box in the downstream region (dark blue color is associated with zero or even negative velocities), and that there is an acceleration of the upper layers far away from the box (fluid particles with high momentum are driven downward from upper layer). Figure 4 shows the same case (as in figure 3), but in xy planes cutting the domain through the center of the box. The top part show that the momentum wake persists for a long distance, and that it spreads in the spanwise direction. There is also a significantly long thermal wake as shown in the bottom part of the figure (light red is associated with larger temperature); the thermal wake seems to reach the blending region, which means that longer domains have to be employed (subject to future work). It should be mentioned that the blending region where the data is taken from the precursor simulation also acts as a sponge layer, so one would not expect spurious reflections to occur. Figure 5 show similar contour plots, except the thermal stratification is increased from ∆θ = 0.2 to ∆θ = 0.4. Upon comparing figures 4 and 5, one can notice that there are some differences in the size and ’strength’ of the wake: by increasing the stratification, the spreading of the wake increases for both velocity and potential temperature.

Figure 3. Contours of the instantaneous streamwise velocity (top) and potential temperature (bottom) on an xz plane through the center of the box (∆θ = 0.2; ubox = 10 m/s).


Figure 4. Contours of the instantaneous streamwise velocity (top) and potential temperature (bottom) on an xy plane through the center of the box (∆θ = 0.2; ubox = 10 m/s).

Figure 5. Contours of the instantaneous streamwise velocity (top) and potential temperature (bottom) on an xy plane through the center of the box (∆θ = 0.4; ubox = 10 m/s).

Contours of time average streamwise velocity are presented in figure 6 for a stratification of ∆θ = 0.2, and the box velocity of ubox = 10 m/s (top part) and ubox = 20 m/s (bottom part). One can notice that as the velocity of the box is increased the size of the wake increases (not surprisingly). Figure 7 shows contour plots of time average potential temperature for the same cases: ∆θ = 0.2, and ubox = 10 m/s (top part) and ubox = 20 m/s (bottom part). As the velocity of the box is increased the thermal wake shortens, which is expected since higher velocities tend to cool the air more rapidly.


Figure 6. Contours of the time averaged streamwise velocity on an xz plane through the center of the box (∆θ = 0.2): top) ubox = 10 m/s; bottom) ubox = 20 m/s.

Figure 7. Contours of the time averaged potential temperature on an xz plane through the center of the box (∆θ = 0.2): top) ubox = 10 m/s; bottom) ubox = 20 m/s.

VI.C.

Averaged Profiles

The following profiles are of time averaged data sampled every box height downstream of the center of the box (starting at half box streamwise location) with the vertical height normalized with respect to the height of the box, the velocities normalized by the geostrophic wind relative to the box (for ubox = 10 m/s, U = 18 m/s), and the potential temperature normalized by the ground temperature. The streamwise turbulence intensity is shown in figures 8, 9, 10 11 for various cases. It can be noticed in figures 8 and 9 that by increasing the velocity of the box, the variance is also increased starting at the ground up to approximately 1.5 from the box height. This is more prevalent in the vicinity of the box, as one can notice that the increases is only visible in the close proximity to the ground as the distance from the box is increased. Interestingly, the turbulence intensity at the ground (in the downstream) is higher as the stratification is increased (see figure 9). The increase in the stratification has a much weaker effect on the turbulence intensity in the downstream region of the box. At a box velocity of 10 m/s, the increase in stratification reduces the maximum turbulence intensity, but has little effect on the profile below one box height. This can be seen in figure 10. Looking at figure 11, the increase in stratification showed a smaller decrease (almost none) for the 20 m/s box velocity case. The profiles for the two stratification cases at 20 m/s collapsed down on to one profile by 2.5 box heights downstream of the box, whereas for the 10 m/s case, there is a small but noticeable difference between the two profiles even after 4 box heights.


5

x/hbox =0.5

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p hu2 i/U

Figure 8. Streamwise turbulence intensity distribution along the vertical direction sampled every hbox /2 downstream of the box (∆θ = 0.2): blue) ubox = 10 m/s; red) ubox = 20 m/s.

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Figure 9. Streamwise turbulence intensity distribution along the vertical direction sampled every hbox /2 downstream of the box (∆θ = 0.4): blue) ubox = 10 m/s; red) ubox = 20 m/s.


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Figure 10. Streamwise turbulence intensity distribution along the vertical direction sampled every hbox /2 downstream of the box (ubox = 10 m/s): blue) ∆θ = 0.2; red) ∆θ = 0.4.

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Figure 11. Streamwise turbulence intensity distribution along the vertical direction sampled every hbox /2 downstream of the box (ubox = 20 m/s): blue) ∆θ = 0.2; red) ∆θ = 0.4.

Profiles of the vertical turbulent momentum flux are shown in figures 12, 13, 14 and 15 for the same cases as in the previous 4 figures. Figures 12 and 13 show the effect of varying the velocity at the same stratification, either ∆θ = 0.2 or ∆θ = 0.4, respectively. For both stratification cases, increasing the velocity of the box slightly reduces the magnitude of the vertical momentum flux around the elevation of one box height; but it does slightly increase the magnitude of the momentum flux below 3/4 of a box height. From figures 14 and 15, the increase in stratification reduces the maximum magnitude of the momentum flux within 1.5 box heights from the rear of the box. Differences in the stratification case profiles exist farther downstream for the box moving 20 m/s when comparing with the box moving 10 m/s.


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huwi/U 2

Figure 12. Vertical turbulent momentum flux distribution along the vertical direction sampled every hbox /2 downstream of the box (∆θ = 0.2): blue) ubox = 10 m/s; red) ubox = 20 m/s.

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Figure 13. Vertical turbulent momentum flux distribution along the vertical direction sampled every hbox /2 downstream of the box (∆θ = 0.4): blue) ubox = 10 m/s; red) ubox = 20 m/s.


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0

huwi/U 2

Figure 14. Vertical turbulent momentum flux distribution along the vertical direction sampled every hbox /2 downstream of the box (ubox = 10 m/s): blue) ∆θ = 0.2; red) ∆θ = 0.4.

5

x/hbox =0.5

x/hbox =2

x/hbox =3

x/hbox =4

0.2 0.4

4

z/hbox

x/hbox =1

3 2 1 0 -0.01

0

-0.01

0

-0.01

0

-0.01

0

-0.01

0

huwi/U 2

Figure 15. Vertical turbulent momentum flux distribution along the vertical direction sampled every hbox /2 downstream of the box (ubox = 20 m/s): blue) ∆θ = 0.2; red) ∆θ = 0.4.

Figure 16 shows the vertical turbulent heat flux for the two box velocity cases at a stratification of ∆θ = 0.2. Increasing the box velocity decreases the magnitude of the vertical turbulent heat flux immediately behind the box; but by one box height downstream, there is no significant effect. The effects of increasing stratification for a box velocity of 10 m/s on the vertical turbulent heat flux exist for a longer distance downstream of the box. As seen in figure 17, there are noticeable differences in the profiles up until three box heights downstream. The results corresponding to the higher stratification (∆θ = 0.4) are not included here since it was found that they are similar to the ∆θ = 0.2 case.


5

x/hbox =0.5

x/hbox =1

x/hbox =2

x/hbox =3

x/hbox =4

z/hbox

4 3 2 1

10 20

0 -1

0

1 ×10−5

-1

0

1 ×10−5

-1

0

1 ×10−5

-1

0

1 ×10−5

-1

0

1 ×10−5

hwθi/U

Figure 16. Vertical turbulent heat flux distribution along the vertical direction sampled every hbox /2 downstream of the box (∆θ = 0.2): blue) ubox = 10 m/s; red) ubox = 20 m/s.

5

x/hbox =0.5

x/hbox =1

x/hbox =2

x/hbox =3

x/hbox =4

z/hbox

4 3 2 1

0.2 0.4

0 -1

0

1 ×10−5

-1

0

1 ×10−5

-1

0

1 ×10−5

-1

0

1 ×10−5

-1

0

1 ×10−5

hwθi/U

Figure 17. Vertical turbulent heat flux distribution along the vertical direction sampled every hbox /2 downstream of the box (ubox = 10 m/s): blue) ∆θ = 0.2; red) ∆θ = 0.4.

VI.D.

POD Analysis of the Wakes

For each case, 4000 snapshots of the 3D velocity field were saved for the POD analysis with each snapshot being taken 0.2 seconds apart. Initially, a smaller number of snapshots were saved and analyzed, but we realized that a larger number of snapshot are needed to obtained more accurate POD modes (4000 is still small, but enough to capture the important physics of the wakes; computer processing time and storage limitations were the main issues). Next, several important results from the POD analysis are presented. The mode energies normalized by the strongest mode for each case are plotted in figure 18. From figure 18(a), the strength of the modes quickly decreases over the first twenty modes, with the twentieth mode being less than 30% as strong as the first mode. In comparing the different stratification levels for each velocity, the relative energies contained in the first twenty modes is greater for the ∆θ = 0.2 case than for the ∆θ = 0.4 case. For the same stratification level, the modes associated with the 10 m/s case are stronger than for 20 m/s case. Figure 18(b) plots the log of mode energies, again normalized by the strongest mode for each case, versus the log of the mode number. The relative strength of the modes for all cases decreases, roughly following a


λk λ1

∼ k −0.65 trend. When examining the effect of varying stratification levels at constant box velocities, the behavior seen in the first twenty modes with the ∆θ = 0.2 cases being stronger than the ∆θ = 0.4 cases is continued. At constant stratification levels, the 10 m/s cases are greater in strength than 20 m/s cases up until mode 104 for ∆θ = 0.2 and mode 139 for ∆θ = 0.4. At that those respective points, the 20 m/s modes contain more relative energy than the 10 m/s modes. Each subsequent mode after k ≈ 660 and k ≈ 1200, for 10 m/s and 20 m/s cases repsectively, is less than 1% as strong as the strongest mode. The significant portion of the turbulent kinetic energy is spread over nearly twice as many modes for the 20 m/s cases than for the 10 m/s cases. 100

1 0.2 0.2 0.4 0.4

0.9 0.8

10 20 10 20

0.7

λk λ1

λk λ1

0.6 0.5

10−1

0.4 0.3 0.2 0.1 5

10

15

10−2 100

20

101

102

103

k

k

a)

b) Figure 18. POD mode energies

One more observation about figure 18 is that, for all cases, the instances of consecutive modes having almost the same mode energies (for example, k = 2 and k = 3 for ∆θ = 0.2 and ubox = 20 m/s and k = 1 and k = 2 for ∆θ = 0.4 and ubox = 20 m/s). This is the result of the formation of a pair of nearly identical sinuous modes that are shifted by a quarter wavelength to preserve the orthogonality of the modes (see figures 19(a) and 19(b)). This offset of a fourth of a wavelength can clearly be seen in figures 19(a) and 19(b). In each pair of contours, it can be seen that the modes have the same wavenumbers, but are just shifted by a quarter of a wavelength.


a)

b)

c)

d) Figure 19. Contours of the streamwise velocity POD modes on an xy plane through the center of the box, highlighting sinuous modes with different phases and wavenumbers.

In figure 20, the types of POD streamwise velocity modes that were identified are shown: 20(a) of the figure shows a sinuous mode; 20(b) shows a varicose mode; while 20(c) and 20(d) show combinations of sinuous and varicose modes, with the sinuous mode dominating in part 20(c) and the varicose mode dominating in 20(d). Parts 20(e) and 20(f) show a cross-section through the wake, highlighting a sinuous and a varicose mode. Next, we compare the contour plots for the streamwise velocity POD mode from the cases with the moving box (left column) and the corresponding cases without a box (right column). Figure 21 shows POD modes associated with thermal stratification ∆θ = 0.2 and box velocity ubox = 10 m/s. It can be noticed that the first two modes in this case are barely affected by the presence of the moving box, and that the mode third is of varicose type. The next two modes (the fourth and the fifth) are sinuous, while the sixth and seventh (not shown) modes are again unaffected. The eight and the ninth modes are larger wavenumber sinuous modes; the next varicose mode is the sixteenth (while all others that are in between are unaffected - not shown). The next case is the one corresponding to the thermal stratification ∆θ = 0.2 and box velocity ubox = 20 m/s. Here, the first all four modes are affected: the first is a varicose mode, while the other three are sinuous, with the fourth having a very small wavenumber. The fifth mode seems unaffected, while the sixth is a varicose mode. The seventh is a smaller wavenumber sinuous mode (when compared to the second and the third). Then, the following modes were found to be either ’weak’ sinuous or varicose modes.


a)

b)

c)

d)

e)

f)

Figure 20. Contours of typical streamwise velocity POD modes from various cases, highlighting different types: a) sinuous mode; b) varicose mode; c) combined sinuous-varicose mode, where the sinuous mode is dominant; d) combined sinuous-varicose mode, where the varicose mode is dominant; e) cross-section through the wake revealing a sinuous mode; c) cross-section through the wake ervealing a varicose mode.


box

no box

Figure 21. Contours of the streamwise velocity POD modes on an xy plane through the center of the box (∆θ = 0.2, ubox = 10 m/s): top to bottom, k = 1, 2, 3, 4, 5, 6, 8, 9, 16. The left column corresponds to results with the effect of the box included, while in the right column there are results without the effect from the box.

When the stratification is increased to ∆θ = 0.4, while the box velocity is ubox = 10 m/s (figure 23), the first two modes seem unaffected again (as in figure 21). The third and the fifth are combined sinuous-varicose modes, while the fourth and the sixth are sinuous (the sixth has a smaller wavenumber). Most of these modes were found to be unaffected by the box. 20 of 26 American Institute of Aeronautics and Astronautics

box

no box


The last case is shown in figure 24 (thermal stratification ∆θ = 0.4 and box velocity ubox = 20 m/s). It was found that almost all first 15 modes are affected by the presence of the box (nine of them are shown in figure 24). The first two are sinuous, phase-shifted modes, while the next three are varicose. Then several higher wavenumber sinuous modes follow. 21 of 26 American Institute of Aeronautics and Astronautics

box

no box



box

no box



VII.

Conclusions

The dynamics of the wakes generated by a large rectangular box moving through a stable thermallystratified atmospheric boundary layer were studied using Large Eddy Simulations. A direct forcing immersed boundary method was used to model both the physical and thermal existence of the box. A concurrent precursor method generated realistic turbulent inflow conditions to allow for the simulation of an isolated box in the streamwise direction. The structure of the wake of the box was studied using the three-dimensional snapshot proper orthogonal decomposition (POD) method. Two different thermal stratifications were considered along with two different velocities of the moving box. Various results consisting of time-averaged streamwise velocity and potential temperature, as well as turbulence intensity, vertical turbulent momentum and heat fluxes profiles were presented and discussed. The presence of the box showed slight effect on the streamwise turbulence intensity and vertical turbulent momentum and heat flux; the effect was mostly restricted to the region within a height of 2 box heights from the ground. Using the POD method, two types of modes with different streamwise wavenumbers were identified in the wake (they were named sinuous and varicose).

Acknowledgments The authors would like to thank Claire VerHulst from Johns Hopkins University for kindly providing the 3D POD code, and for helping with the code settings. This effort is sponsored by the U.S. Government under Other Transaction number W15QKN-13-9-0001 between the Consortium for Energy, Environment and Demilitarization, and the Government. The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright notation thereon. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the U.S. Government.

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