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Wind flow modeling software is widely used for wind resource assessment and mostly based on either the computational fluid dynamics (CFD) approach or the linear WAsP approach.  They focus primarily on modeling of the neutrally stratified atmospheric surface-layer (ASL), which typically covers the bottom 10% of the atmospheric boundary-layer (ABL). In the ASL, the logarithmic wind profile is a justified approximation, and the models account for the effects of roughness and topography changes. Atmospheric stability and Coriolis effects are mostly ignored or, like in WAsP, are treated as small perturbations to the neutral background flow that can be added after solving the model equations. In order to decrease the uncertainty of predictions, especially in complex terrain, stability and Coriolis effects on the whole atmospheric boundary layer should be included in such models.
Turbulence within the ABL covers a wide range of scales (from less than a centimeter up to several kilometers). Since the solution of the full Navier–Stokes equations is not computationally feasible, high Reynolds number flows can be based on the solution of the Reynolds-averaged Navier–Stokes (RANS) equations. Recently, Sogachev et al. developed an atmospheric model for flows over flat terrain that accounts for stability and Coriolis effects: the energy equation in terms of the potential temperature is solved in parallel to the RANS equations, and a consistent two-equation turbulence model is used to close the equations. Using a two-equation closure method to describe the whole ABL allows the flow to be computed at a much lower computational cost than, e.g., using large-eddy simulations (LES).  The aim of the present work is to develop and validate a RANS ABL model framework describing the whole ABL that can be applied for flows over complex terrain. The starting point is the in-house CFD solver EllipSys3D, developed at the Technical University of Denmark's Wind Energy department (Roskilde, Denmark).[5-7] The solver was initially developed for simulating the near-ground surface-layer flow inside a neutrally stratified domain (from now on referred to as the ASL model) and has, under these conditions, been validated against field experiments.[8, 9] To model ABL flows more appropriately, the solver is modified (from now on referred to as the ABL model), and the two-equation turbulence closure from Sogachev et al. is used.
To get a better understanding of the physical processes involved in ABL flows and to validate ABL models, more data sets from atmospheric experiments on full scale are necessary. Various experiments that focus on neutral flow over complex terrain are available, e.g., the Askervein Hill experiment, or more recently the Bolund experiment.[8, 11] Existing benchmark literature for non-neutral ABL flows mostly focuses on flat terrain,[12-15] and test cases for complex terrain are scarce. Ross et al. present a wind-tunnel study of neutral and stably stratified boundary-flow over a steep hill. The wind-tunnel experiment was designed to represent a realistic non-neutral ABL flow and analyses stability effects over terrain under controlled conditions. Although not real ABL flows at full scale, this test case was chosen to test the applicability and performance of the ABL model for flows over well defined but steep terrain.
The central goal of the present study is to examine how well the ABL model performs in representing neutral and non-neutral ABL flows and to set the starting point to apply the ABL model for flows over complex terrain. In Section 2, the modeling approach is presented, followed by Section 3, where implementation aspects (Section 3.1) and the simulation methodology (Section 3.2) are described. In Section 4, the Monin–Obukhov similarity theory (MOST) is briefly described. Section 5 presents results from four test cases that are used to validate the ABL model.
The simulations are divided in three parts. First, Sections 5.1 and 5.2 focus on neutral ABL flow over rough flat ground. Simulation results are compared against experimental data from the Leipzig wind profile and the Cabauw site. Second, non-neutral ABL flow over rough flat ground is considered in Section 5.3. Experimental data from the second GEWEX (Global Energy and Water cycle EXperiment) ABL Study (GABLS2), which analyses a diurnal cycle in the ABL, is used to validate the ABL model for non-neutral conditions. Additionally, MOST is used to compare simulation results against experimental data from several large-scale field campaigns.[12-14] Third, in Section 5.4, the wind-tunnel experiment analyzing stratified boundary-layer flow over a steep hill is used to assess the applicability of the ABL model for non-neutral flows over terrain. Concluding remarks are given in Section 6.
2 MODELING ABL FLOWS
2.1 Governing equations
The high Reynolds number atmospheric flows considered in this study are based on the solution of the RANS equations. The continuity and momentum equations read:
where xi (x1 = x, x2 = y, x3 = z) are the longitudinal, lateral and vertical directions, respectively. Ui is the mean velocity component along xi, is the pressure and μt is the turbulent eddy viscosity.
Since the unsteady term is retained in equation (2), it is possible to simulate transient phenomena with RANS. This is based on the assumption that time averaging of the RANS equations is performed on a time scale similar to the turbulent fluctuations, whereas the low frequency variations of the mean flow (e.g., diurnal simulations of stratified flows in the ABL, as considered in this study) can be properly resolved by the unsteady RANS equations. Transient RANS allows to simulate the unsteady phenomena of a diurnal cycle in the ABL and is basically the only option besides the more computationally expensive LES approach.
2.2 Basic RANS turbulence closure for ASL flows
The RANS equations are used to describe the air flow in the lowest part of the ABL, where the logarithmic wind profile is a justified approximation. To close the given set of equations, the popular k- ε turbulence model is used. The eddy viscosity is obtained by solving the two differential transport equations for the turbulent kinetic energy k and the dissipation rate ε:
where σk and σε are the Schmidt numbers for k and ε, respectively, and Pk is the rate of shear production of k. Cε1 and Cε2 are model coefficients. The resulting mixing length l and the eddy viscosity μt are expressed in terms of k and ε as
2.3 Adaptation of RANS turbulence closure for ABL flows
When modeling the full ABL, thermal stratification and Coriolis effects (caused by the rotation of the earth) should be included. These effects are introduced into the RANS equation system via additional source/sink terms on the right-hand side of the momentum equation (2):
where gi is the gravitational acceleration , ρ is the varying density and ρ0 is a reference density. and fc = 2Ωsinλ are the Coriolis parameters (with the earth's rotation rate Ω and the latitude λ, and are added explicitly to the momentum equations as an external force. The Coriolis force in vertical direction is neglected since it is small compared with the gravitational acceleration.
To include buoyancy effects, an equation for the potential temperature is solved in addition to the RANS equations:
The potential temperature equation couples with the momentum equations via vertical buoyancy forces gi(ρ − ρ0) that act in the direction of the gravitational acceleration. Density variations as a result of pressure variations are assumed to be small so that the flow is treated incompressible. Hence, the density is not a function of pressure, and temperature and density vary linearly as required by the Boussinesq approximation:
where M = 29 g mol − 1 is the molar mass and R = 8.313 J molK − 1 is the universal gas constant.
To close the given set of equations, Sogachev et al. recently developed a consistent closure method for the two-equation turbulence model previously given by considering the behavior of the ε equation in homogeneous turbulent flow when using a source/sink term associated with stability. The potential temperature equation now couples with the turbulence model via an additional source/sink term B that is added to the two turbulent transport equations:[20, 21]
is a modified Cε1 coefficient (Section 2.3.1), and D is an additional diffusion term (Section 2.3.2). The buoyancy source/sink term B is added to the turbulent kinetic energy equation and also appears in the dissipation equation together with the coefficient Cε3 and depends on the eddy viscosity μt, the gravitational acceleration gi and the density gradient:
In unstable conditions, B is positive and supports the generation of turbulent kinetic energy in the k equation, whereas B turns negative in stable conditions and suppresses turbulence. Details about the specification of the coefficient Cε3 are given in Section 2.4.2.
2.3.1 Length-scale limitation
The standard k- ε model, when applied to ABL flows, is known to be too diffusive, leading to a strongly overestimated turbulent length scale l that grows continuously with height and results in a very large ABL height.[22, 23] In real ABL flows, the maximum size of turbulent eddies is limited, e.g., by the finite ABL height or by stratification (see also). Using a length-scale limiter, as initially proposed by Apsley and Castro, allows to reduce the maximum mixing length in the model, and the resulting ABL height is effectively reduced. The modified coefficient in the length scale determining equation (11) is described by
Apart from the maximum global mixing length, le, no additional coefficients are introduced. When the local mixing length l reaches the specified global maximum mixing length le, equals Cε2, and the production and destruction terms in the dissipation equation are in balance, which limits the local length scale l to le. On the other end, when l < < le, equals Cε1, and the modification still satisfies the logarithmic wind profile in the surface-layer close to the ground.
For neutrally stratified ABL flows over a flat rough surface, le is estimated by an expression from Blackadar. To provide a suitable solution for stratified flows, an expression from Mellor and Yamada is used, which depends on the vertical distribution of turbulent kinetic energy k in the ABL and reflects variations in ABL depth induced by thermal stratification:
where G is the geostrophic wind. The coefficient α is chosen so that both length scales are identical for a neutrally stratified ABL flow (lMY = l0). For several test cases, an empirical value of α = 0.075 results in both length scales to agree reasonably well (lMY ≈ l0). To calculate lMY with the aforementioned expression, the turbulent kinetic energy k needs to be integrated over the domain height. For horizontally homogeneous flow, it is sufficient to perform this integration every time step at one location since the flow is horizontally homogeneous. For complex terrain domains with a curvilinear grid, this integration would have to be performed at every location individually. This is computationally not feasible and, instead, a precursor simulation over flat terrain can be used to determine the time varying values of lMY that are then used within a complex terrain domain.
2.3.2 Diffusion term
An additional diffusion type term D is introduced into the dissipation equation (11) of the k- ε model as proposed by Sogachev et al. Numerical experiments have shown some differences in the behavior of the standard k- ε model of Laundner and Spalding and the k- ω model of Wilcox. Sogachev and Panferov reported that, for example, in forest canopies, the k- ω model is performing slightly better than the k- ε model. The ABL model in the present study is developed using the k- ε model, but to obtain consistent results between the two closures, the k- ε model can be transformed to behave similarly to the k- ω model by including an additional diffusion term:
2.4 Model coefficients
2.4.1 Specification of general constants
The model constants Cε1 and Cε2 in equations (4) and (11) are chosen to be consistent with experimental observations for decaying homogeneous, isotropic turbulence. To ensure that the model solution agrees with the constant-stress logarithmic wind profile near the ground, the relation in the succeeding text has to be satisfied, which follows from considering constant-flux flow with ∂k ∕ ∂z = 0:
For neutral flows, the constant Cμ is typically adjusted to set a desired turbulence level, and Cμ = 0.09 is a typical value for industrial flows, whereas in atmospheric research, a value of Cμ = 0.03 is often used.
2.4.2 Specification of stability-related coefficients
The additional coefficient Cε3 in the dissipation equation (11) has to be specified, and an optimal value is unknown. Using the recently developed consistent closure method for two-equation turbulence models from Sogachev et al., Cε3 is modeled using a stability-related coefficient αB (Table 1):
Table 1. Typical values for the coefficients in the k-ε turbulence model for industrial flows (Cμ = 0.09) and for atmospheric flows (Cμ = 0.03).
αB is specified on the basis of the standard coefficients Cε1 and Cε2 of the production and destruction terms in the ε equation, respectively:
αB now depends on the local gradient Richardson number Rig and on the local ratio of l/ lMY, and, hence, is a function of stability.
In the model, the turbulent Prandtl number σθ in equation (8) is approximated as a function of a slightly modified gradient Richardson number RiG:
The additional term in the denominator of the gradient Richardson number RiG is used to stabilize equation (19): RiG and the resulting σθ are now limited during convective cases, where Pk approaches zero.
Sogachev et al. have shown numerically that the developed model framework is suitable for three flow regimes: grid turbulence, wall-bounded flow and homogeneous shear flow. In contrast to an earlier proposed description of Cε3, the form given in equation (18) is universal and need not be specified for each case. Compared with the ASL model, no additional coefficients are introduced into the ABL model that needs to be calibrated. The formulation does not allow for any tuning of the model and only depends on the closure coefficients given in Table 1.
3 SIMULATING ABL FLOWS
3.1 Implementation into EllipSys3D solver
To simulate ABL flows, the EllipSys3D solver is modified using the ABL model equations presented in the previous text.
To solve the convection–diffusion equations, the third-order accurate QUICK differencing scheme is used. All equations are transformed into general curvilinear coordinates. This allows the model to be applied to complex geometries, like ABL flows over terrain, where the use of Cartesian or rectangular coordinates is often not possible.
The buoyancy forces gi(ρ − ρ0) in the momentum equation (2) are added explicitly as external volume forces. When implemented into the solver, this can create numerical problems: the implementation via a discrete body force generates a numerical decoupling between the pressure and the velocities. This problem was identified to cause oscillations in the solution, especially close to boundaries and under strongly stratified conditions. To avoid this, an algorithm for allocating discrete forces is used following Réthoré and Sørensen. This approach solves the problem by spreading the buoyancy forces on the neighboring cells and by applying an equivalent pressure jump at the cell faces.
In order to improve convergence for small mixing lengths, ambient floor values for the turbulence variables are imposed. Especially during strongly stable conditions, the mixing length l and the eddy viscosity μt approach values close to zero or even negative values. This typically occurs within the stable temperature layer (inversion) in the upper part of the ABL. To avoid numerical convergence issues, k and ε are not allowed to drop below a predefined limit. The ambient values kamb and εamb are defined a priori and are set to a minimum turbulent mixing length via equation (5). They are set using a minimum limiter on the turbulence variables: ε = max(ε,εamb) and k = max(k,kamb). Values below the ambient level are simply overwritten. For the present study, the ambient turbulence values are chosen to be kamb = 1 ⋅ 10 − 4 m − 2s − 2 and εamb = 7.208 ⋅ 10 − 8 m − 2s − 3. Together with the model constants in Table 1 and equation (5), this leads to a minimum turbulent mixing length of 1 m. The modification is only active in the upper part of the ABL, where the turbulence variables approach their predefined ambient levels.
3.2 Model setup
All presented calculations use the set of consistent closure coefficients for ABL flows stated in Table 1. Specific simulation parameters such as roughness length z0, geostrophic wind G, Coriolis parameter fc and surface temperatures θ0 are summarized in Table 2.
Table 2. Values for the simulation parameters associated with each model run: geostrophic wind G that is used to specify the pressure gradient to drive the flow, roughness lengthz0, Coriolis parameter fc, maximum global mixing length scale le and potential surface temperature θ0.
G [m s − 1]
fc [1 s − 1]
Four test cases are presented covering neutral ABL flow over flat terrain (Leipzig, Cabauw), non-neutral ABL flow over flat terrain (GABLS2) and neutral and stable wind-tunnel flow over a steep hill (here, le and fc are not applicable).
To simulate ABL flows over flat terrain, initial conditions are typically specified by the logarithmic wind profile over flat terrain according to a specific surface roughness z0. However, since cyclic boundary conditions are used for the present simulations, the final results do not depend on their initial conditions. To drive the flow in the ABL simulations, a pressure gradient is applied that results in the desired geostrophic wind G. The Coriolis force balances the pressure gradient force at the ABL top, where friction by definition is zero: G = − 1 ∕ ρfcδp ∕ δy.
3.2.1 Computational domain
For all flat terrain simulations, the same computational domain is used: it is 6 km high and uses a grid of 384 stretched cells in vertical direction. The bottom cell is equal to the size of the roughness length zo at the wall, and the mesh is stretched hyperbolically toward the top, resulting in cell heights of about 70 m at the top boundary. In horizontal directions, the domain is 1 x 1 km long with a grid of 32 evenly distributed cells. However, as the modeled flow over flat surfaces is horizontally homogeneous, the horizontal grid structure is irrelevant, and the flow variables are therefore functions of the height z alone. Rough wall boundary conditions are used at the bottom of the domain, and a symmetry condition (no-gradient) is used on top. All vertical boundaries are cyclic. Note that grid independent results could already be obtained using a grid of around 100 cells in the vertical direction.
The wind-tunnel experiments of Ross et al. were conducted at the Environmental Flow Research Laboratory (EnFlo), University of Surrey, UK. The wind tunnel has a working section of 20 m length, 3.5 m width and 1.5 m height, and the shape of the two-dimensional steep hill is given by
where H = 0.229 m is the maximum hill height, x is the distance from the center of the hill and l = 1 m is the width of the hill. The computational domain covers 10 m of the wind-tunnel test section and is 1.5 m high and wide, and the hill is placed 2.5 m behind the upstream boundary. The wind-tunnel flow is solved by the EllipSys3D solver and therefore the domain is 3D, although the model hill and the resulting flow is 2D. The grid has 192 grid points in the horizontal, 24 in the lateral and 96 in the vertical direction. Stretched cells are used in the vertical direction with a height of 0.27 mm at the wall and 5 cm at the top of the domain. In horizontal direction, the mesh is refined on top of the hill with cells of 1.5 cm length and is stretched toward the inlet and outlet, where cells are about 10 cm long. In lateral direction, the mesh is equally spaced with the cells being 6 cm wide. At the inlet and upper boundary, inlet conditions are used, and at the downstream boundary, outlet conditions are used.
Monin–Obukhov similarity theory expresses the vertical structure of the horizontally homogeneous ASL as dimensionless universal functions and is often used to validate ABL models. On the basis of dimensional analysis, all non-dimensionalized mean flow properties within the ASL only depend on a reduced set of key scaling parameters: the friction velocity u * , the height above ground z and the vertical turbulent heat flux H:
From these parameters, a universal length scale, the Obukhov length L, can be formed, which describes the exchange processes in the surface-layer:
where θ0 is the potential temperature at the surface and H0 is the near-surface value of the vertical turbulent heat flux H. L is proportional to the vertical potential temperature gradient and describes the height at which buoyant production of turbulence first exceeds mechanical production due to shear. The dimensionless height ζ = z ∕ L is used as a stability parameter and has the same sign as the Richardson number Ri: positive in stable conditions and negative in unstable conditions. On the basis of the assumptions that the flow within the surface-layer is stationary, horizontally homogeneous and that fluxes are independent of height, ζ is now constant throughout the surface-layer (in contrast to Ri), and the normalized wind speed depends on the universal function ζ alone:
For ideal conditions of stationary and horizontally homogeneous flow, MOST is valid in about the lowest 10% of the ABL where the Coriolis effect is negligible. The forms of the stability function φm are not known from dimensional analysis and is obtained empirically from field experiments over flat terrain: measurements for different values of ζ are substituted into equation (23), and curves are fitted to the resulting data. Despite numerous field experiments, there are still discrepancies in the literature for the exact forms of φm. The forms chosen here are widely used and known as the Businger–Dyer relations:[12, 33]
In the literature, the coefficient δm ranges from 4 to 10, and γm ranges from 15 to 28 (see, e.g.,[13, 34]). During ideal conditions, the relation (23) between dimensionless wind shear and dimensionless height ζ is valid for any wind speed, height, roughness and stability condition in the surface-layer. Ideally, vertical profiles will then collapse on one line.
Integration of equation (23) for the velocity gradient yields a modified logarithmic wind profile for the wind speed U:
where Ψm is a stability function. Using the Dyer stability functions Ψm is defined as
5 RESULTS AND DISCUSSION
5.1 Neutral ABL—Leipzig test case
In this case, the Leipzig wind profile is modeled with the ASL and ABL models run steady-state. The necessary simulation parameters are shown in Table 2, and the computational grid is described in Section 3.2.1. Results are shown in Figure 1 together with the measurements. A comparison of the ASL and ABL models results shows the influence of the Coriolis effect: the additional body force in the ABL model induces a velocity component v perpendicular to the direction of the geostrophic wind G and causes the wind to veer with height. In the ABL model, also the height of the ABL is now limited to about 1300 m, because of the applied length-scale limiter (seen in the middle of Figure 1, where the velocity component v approaches zero). With the chosen value of l0 = 41.8 m, the ABL height is however slightly overpredicted. This length scale is slightly larger than the one suggested by Apsley and Castro, who used l0 = 36 m for their simulation. It is generally accepted that the Leipzig experiment was actually conducted in slightly stable conditions, and when using a lower length scale of l0 = 28 m, the measured and simulated profiles agree perfectly (not shown here). However, the goal was not to match the simulation to a single observation, and the ABL model predicts the flow reasonably well and simulated results are significantly improved compared with the ASL model.
5.2 Neutral ABL—Cabauw test case
The neutral ABL over flat terrain at the Cabauw site in the Netherlands is simulated with the ASL and ABL models run steady-state. The necessary simulation parameters are shown in Table 2, and the computational grid is described in Section 3.2.1. The results are shown in Figure 2. The non-dimensional geostrophic wind components (u − ug) ∕ u * and (v − vg) ∕ u * are shown as functions of the non-dimensional height zfc ∕ u * , plotted with a logarithmic scale. Annual averages from the Cabauw site are shown for three classes of the geostrophic wind (G = 5,10,15 m s − 1) at heights 10, 80 and 200 m, together with simulation results using G = 10 m s − 1. When plotted using the aforementioned non-dimensional form, the simulation results for other geostrophic winds collapse on the same line. As for the Leipzig test case, the turning of the wind with height induced by the Coriolis force and the limitation of the ABL height by the length-scale limiter can be seen when comparing the simulated profiles of the ABL model with the ASL model. Results from the ABL model agree well with the measurements, and the chosen modifications prove applicable.
5.3 Non-neutral ABL—GABLS2 test case
This section focuses on assessing how well the ABL model performs in representing non-neutral ABL flows. Observations from the GABLS2 test case held in Kansas, USA with a strong diurnal cycle are chosen to validate the ABL model. In the study of Svensson et al., the observational dataset is compared against simulation results from 30 different models, and simulating the described diurnal cycle has shown to represent a challenging test case for ABL models.
The simulation uses the computational grid from Section 3.2, and the ABL model is run transient, where non-neutral conditions are induced by a prescribed time-varying ground temperature. A time step of 1 s is used. The initial conditions for the simulation of the diurnal cycle are given in Svensson et al.
Results are shown in Figure 3. Figure 3(a) is showing the time varying ground temperature that is used as a model input together with the resulting diurnal evolution of the computed potential temperature field that adopts to the changing surface conditions. The surface stability conditions are influencing the generation of turbulence and the turbulent mixing. Figure 3(b) shows the friction velocity u * at the surface (on the left axis) and the velocity variation at the 10 m level (on the right axis) over one diurnal cycle together with measurements (symbols), and the spread of the different model results from the model intercomparison study of Svensson et al. (shaded region).
In both plots, Figures 3(a) and 3(b), a clear transition between daytime and nighttime is visible after sunrise around 8:00 and after sunset around 18:00. Stable nighttime conditions before 8:00 are characterized by small turbulence levels and a low ABL depth of around 100 m. The stable stratification suppresses the generation of turbulence and results in small values for the friction velocity as shown in Figure 3(b). The air close to the ground is colder then the air above and, because of the small amount of mixing, only penetrates up to heights of about 100 m, where a steep temperature gradient is visible in Figure 3(a).
During daytime between 12:00 and 18:00, unstable conditions are induced by the heating of the ground. Large amounts of convective turbulence lead to a well-mixed ABL with a greater depth. Because of convection, warm air is rising upwards and penetrates the strong stable temperature gradient that is capping the ABL during night. After 12:00, the stable gradient is not existent anymore, and the ABL continues to grow in height. It is during this period that the turbulent length scales and the friction velocity at the surface reach their maximum values. At around 14:00, the maximum temperature is reached, and before returning to the stable nighttime regime, the ABL flow is close to neutral at around 18:00, where the potential temperature is nearly constant with height.
Also shown in Figure 3(b) is the evolution of the wind speed at the 10 m level. Higher wind speeds are observed during daytime, where the increased turbulence is effective at mixing momentum downward close to the ground and vice versa. The shaded areas indicate the model spread of the 30 models that were intercompared within the study of Svensson et al. Computed results generally are within the observed range, and a clear diurnal pattern is visible. Svensson et al. report that all models underestimate the 10 m wind speed after the morning transition and tend to overestimate the wind speed toward the end of the day. One obvious reason for this is that the geostrophic wind during the simulation was kept constant in space and time, whereas both,observations and mesoscale simulations shown in Svensson et al. show a decrease of the geostrophic wind during the observational period. Also note that the measured turbulent kinetic energy shows a sudden increase at about 3:00, which was reported to be a local disturbance not included in the model forcing and therefore not present in the computed results.
Figure 3(c) shows the computed wind profiles in the first 100 m at different times of the day compared against the observations and the standard logarithmic profiles from the ASL model. Also shown are the theoretical profiles from MOST, where the computed surface heat flux H0 together with equations (22) and (26) are used to determine the Obukhov length L and the modified logarithmic wind profile. Stable conditions at night result in smaller wind speeds close to the ground and higher wind speeds above, when compared against the logarithmic solution of the neutral ASL model. During unstable conditions, the wind speed increases rapidly over the first few meters, whereas it is almost constant with height above. The agreement is good, and the developed model captures the observed and theoretical non-neutral behavior.
In Figure 4, MOST (Section 4) is used to assess the performance of the ABL model. Theoretical φm functions from equation (24) are shown together with simulation results and experimental data from several field campaigns.[12-14] To decrease the spread of the experimental φm values, the data needs to be selected carefully. Especially during transitional periods in the morning and evening, the assumptions underlying MOST (stationary and horizontally homogeneous flow with constant ζ over height) are violated in real ABL flows. The shown simulation results are therefore selected accordingly: only cases for fully developed flow away from the transitional periods are shown. MOST was derived for the range | ζ | < 2, and for higher values, it can be seen that experimental and simulation results start to deviate from MOST.
Note that in this case, the ABL model is run for several days, cyclically repeating the surface temperature from the GABLS2 test case, until a cyclical solution is reached. This ensures that the whole flow field is in equilibrium with the model equations and that the solution is independent of the initial conditions. For the GABLS2 model intercomparison, the spin-up time of the models given in Svensson et al. (time before numerical and observational data is compared) is only 8 h. We find that the solution in this case is still significantly dependant on the initial temperature field (also given in Svensson et al.), and when compared against MOST, the agreement is not as good as for the fully converged results. This indicates that the flow field in the ABL after 8 h of spin-up time is not yet in equilibrium with the surface forcing at the ground. In a recent study, Sogachev also found that a spin-up time of several days is needed, depending on the initial conditions. In summary, we find model results to agree best with MOST when allowing the solution to fully converge to a cyclical solution. Results for the GABlS2 test case are best when following the instructions from Svensson et al. This indicates that if information on a large scale (such as the geostrophic wind or the vertical temperature profile) is available from measurements, those conditions should be used in the model to compare numerical and observational data. However, conditions in the real ABL are often non-stationary and horizontally non-homogeneous and are therefore not necessarily in agreement with empirical theories like MOST or with the model equations of numerical models.
5.4 Non-neutral flow over a steep hill
In this test case, neutral and stably stratified boundary-layer flow over a steep hill is simulated and compared against simulations and wind-tunnel measurements.[16, 37] The wind-tunnel experiment was designed to represent realistic ABL flow over a two-dimensional steep hill. Wind-tunnel flows cannot fully resemble real ABL flow at full scale, and the Coriolis effect is negligible. However, this test case allows to study stability effects under controlled conditions and is chosen to test the applicability and performance of the ABL model for flows over well-defined terrain.
The hill is steep enough to induce flow separation and represents a challenging test case for the developed ABL model. The simulation is run in transient mode with a time step of 0.1 s.
Because of the small scale of the wind-tunnel, the Coriolis effect is neglected in the model, hence equation (14) to determine a maximum length scale le cannot be applied, and no length-scale limitation is used.
The necessary input parameters to simulate the wind-tunnel flow are summarized in Table 2. Two cases are simulated: neutral and stably stratified flow with a relatively weak stratification of about 10 K m − 1 in the lowest 0.5 m and a much stronger stratification of about 40 K m − 1 above. The neutral simulations are solved steady-state and the stable simulations are run transient for 150 s of model time (equivalent to 15 tunnel flow-through times) after which the computed flow has reached a quasi-steady state.
The initial conditions for the neutral simulation are specified by the logarithmic wind profile with a surface roughness of z0 = 0.23 mm (Table 2). For the stably stratified flow, the inlet profiles at the upstream boundary are generated by running a precursor simulation: the experimental velocity and temperature profiles from the wind-tunnel given in Ross et al. are run through the ABL model using the wind tunnel specified in the succeeding text in the absence of the hill. This ensures that the inlet profiles are in equilibrium with the model equations.
Computed wind speed and turbulence properties of the neutral and stably stratified flow are compared against experimental results and simulation results[16, 37]. Figure 5 shows contour plots of the streamwise velocity u in a vertical plane perpendicular to the hill, and Figure 6 shows contour plots of the momentum flux . Results for neutral flow are shown in the left column, and results for stable flow are shown on the right. In both cases, flow separation occurs at the lee side of the hill. For the stable case, the depth of the wake region is slightly increased because the stable stratification acts to suppress vertical motion. Above 0.5 m, the strong temperature gradient of about 40 K m − 1 is effectively capping the flow, and turbulence is limited to the lower part of the domain and is significantly reduced when compared with the neutral case where the momentum fluxes are an order of magnitude larger. The ABL model captures the general effects induced by stratification, although the size of the wake is different. For both neutral and stably stratified flows, the recirculation region in the lee side of the hill is significantly overpredicted compared with the wind-tunnel experiment, whereas it agrees well with the LES results. The velocity above of the hill is generally predicted well. The values in Figure 6 are generally predicted well and agree with both wind-tunnel and LES results from Wan and Port-Agel. However, for the neutral case, the values upstream of the hill are found to be too high when compared with the wind-tunnel values. Similar findings were reported by Wan et al.: the model is found to be too dissipative in this region, which leads to an increased upwards deflection of the flow induced by the hill and leads to a slower velocity recovery downstream of the hill and hence an overestimated wake region.
Since the ABL model was developed for ABL flows at full scale, we cannot expect the model to reproduce the wind-tunnel measurements. Because of the small scale of the wind-tunnel, the Coriolis effect is neglected, and the length-scale limiter is not applied. The implemented turbulence closure has been developed for steady ABL flows, and it cannot be expected that the unsteady wake region in the lee of the hill is predicted correctly, and it is not the aim of the ABL model. The wake region has shown to be sensitive to changes in the model constants. No coefficients were adjusted, and all test cases are run with the same set of constants from Table 2. Although this test case is of limited value to verify the developed ABL model, it is shown that the model can be applied on curvilinear grids without any modification and that general effects of stratification on the flow are captured correctly.
The present work presents an ABL model that aims at describing the wind flow within the whole ABL. The model is successfully validated using four test cases. For neutral ABL flow, two test cases over flat terrain are considered, and the implemented Coriolis effect and the length-scale limited k- ε model prove applicable. Computed profiles for the velocity components agree well with measurements from the Leipzig and the Cabauw test case.[17, 18]
For non-neutral ABL flow, a diurnal cycle is simulated, where a time-varying surface temperature reflects different stability conditions that typically occur within the ABL throughout 1 day. The implementation of the k- ε model developed by Sogachev et al. and of the potential temperature equation, proved applicable, and the ABL model that now accounts for stability effects performed well. Finally, a wind-tunnel test case is used to validate the ABL model for stably stratified flow over a steep hill. Although this test case is of limited value to validate the ABL model, the applicability for flows over terrain using curvilinear grids was shown. For all test cases, the computed velocity, potential temperature and turbulence values compare reasonably well.
The advantage of the presented RANS model framework is its general applicability. All implementations in the ABL model are tuning free, and except for the simulation parameters in Table 2, no additional model coefficients need to be specified a priori the simulation. In summary, the test cases show that the developed ABL model is applicable and gives significantly improved predictions for neutral and non-neutral ABL flow compared with the ASL model. It presents a promising approach to also applying the ABL model to complex topography.
This work has been carried out within the WAUDIT project (Grant no. 238576). This Initial Training Network is a Marie Curie action, funded under the Seventh Framework Program (FP7) of the European Commission. Computations were made possible by the use of the PC-cluster Gorm provided by DCSC and the DTU central computing facility. We would also like to thank the Danish energy agency (EFP07-Metoder til kortlægning af vindforhold i komplekst terræn (ENS-33033-0062)), the Center for Computational Wind Turbine Aerodynamics and Atmospheric Turbulence (under the Danish Council for Strategic Research, Grant no. 09-067216).