A synthetic inflow turbulence generator was implemented
in the idealised Weather Research and Forecasting large eddy simulation
(WRF-LES v3.6.1) model under neutral atmospheric conditions. This method is
based on an exponential correlation function and generates a series of
two-dimensional slices of data which are correlated both in space and in
time. These data satisfy a spectrum with a near “-5/3” inertial subrange,
suggesting its excellent capability for high Reynolds number atmospheric
flows. It is more computationally efficient than other synthetic turbulence
generation approaches, such as three-dimensional digital filter methods. A
WRF-LES simulation with periodic boundary conditions was conducted to
provide prior mean profiles of first and second moments of turbulence for the
synthetic turbulence generation method, and the results of the periodic case
were also used to evaluate the inflow case. The inflow case generated
similar turbulence structures to those of the periodic case after a short
adjustment distance. The inflow case yielded a mean velocity profile and
second-moment profiles that agreed well with those generated using periodic
boundary conditions, after a short adjustment distance. For the range of the
integral length scales of the inflow turbulence (±40 %), its effect on
the mean velocity profiles is negligible, whereas its influence on the
second-moment profiles is more visible, in particular for the smallest
integral length scales, e.g. those with the friction velocity of less than 4 %
error of the reference data at x/H=7. This implementation enables a WRF-LES
simulation of a horizontally inhomogeneous case with non-repeated surface
land-use patterns and can be extended so as to conduct a multi-scale seamless
nesting simulation from a meso-scale domain with a kilometre-scale resolution down to LES
domains with metre-scale resolutions.
Introduction
Atmospheric boundary layer flow involves a wide range of scales of eddies,
from quasi-two-dimensional structures at the mesoscales to three-dimensional
turbulence (normally with higher Reynolds number, i.e. Re∼108–109) at the microscale (Muñoz-Esparza et al.,
2015). The Weather Research and Forecasting (WRF) model (Skamarock
and Klemp, 2008) is capable of simulating atmospheric systems at a variety
of scales. At the mesoscale and synoptic scales, the WRF model allows grid
nesting for downscaling from 10–100 to 1–10 km using a fully compressible
and non-hydrostatic Reynolds-averaged Navier–Stokes (RANS) solver
(Skamarock and Klemp, 2008), which captures the behaviour of mean
flows only. At the microscale, a large eddy simulation (LES) can be
activated in the WRF model (WRF-LES), enabling users to simulate the
characteristics of energy-containing eddies in the atmospheric boundary
layer. Challenges remain in downscaling from the mesoscale
(resolutions down to 1 km, capturing mean information only) to the LES scale
(tens of metres or below, capturing additional turbulence information)
(Doubrawa et al., 2018; Talbot et al., 2012; Chu et al., 2014; Liu et
al., 2011), such as specifying the appropriate inflow conditions for an LES
domain and the sub-grid-scale turbulence schemes for the “grey-zone”
resolution, to which neither planetary boundary layer (PBL) nor LES
parameterisation schemes can apply well. Consequently, microscale and mesoscale
flows are typically studied separately. Most LES models of atmospheric
boundary layer flow at the microscale use periodic boundary conditions and
simplified large-scale geostrophic forcing for idealised simulations.
However, the use of periodic boundary conditions implicitly assumes that atmospheric fields and the underlying land use have repeated
periodic features. This assumption may be unrealistic for real landscapes
where land-use patterns and the atmospheric phenomena coupled to them can be
very heterogeneous. Therefore, such periodic WRF-LES simulations are
restricted to studies of the atmospheric boundary layer flow with a single
domain (e.g. Zhu et al., 2016; Kirkil et al., 2012; Kang and Lenschow,
2014; Ma and Liu, 2017) or the outermost domain of either one-way nested
cases (e.g. Nunalee et al., 2014) or two-way nested cases
(e.g. Moeng et al., 2007). Here we implement a well-tested
synthetic turbulence inflow scheme (Xie and Castro, 2008) in the WRF-LES
model (v.3.6.1), in which the meso-scale model could provide the mean flow
information as the input of the synthetic turbulence inflow scheme. This
scheme provides a step towards enabling WRF's capability of nesting
micro-scale turbulent flows within realistic meso-scale meteorological
fields.
Dhamankar et al. (2018) reviewed three broad classes of methods
to generate the turbulent inflow conditions for LES models, mainly for
engineering applications. The first class is the library-based method, which
relies on an external turbulence library to provide inflow turbulence. The
turbulence library can be based on either (a) the precursor or concurrent
simulation (e.g. Munters et al., 2016) on the same geometry to a
main LES simulation; or (b) a pre-existing database (e.g. Schluter et
al., 2004; Keating et al., 2004) from experiments or computations (on a
different geometry to a main LES simulation). Although this method is
usually limited to specialised applications, it can provide good-quality
inflow turbulence. The second class is the recycling–rescaling-based method
(e.g. Lund et al., 1998; Morgan et al., 2011), in which the velocity
field is recycled from some suitably selected downstream plane back to the
inflow boundary plane. Although this method may be effective in producing
well-established turbulence, there are some limitations, e.g. the
requirements of an equilibrium region near the inlet and a relatively large
domain. The turbulence profile determined by the geometry of the precursor
simulation can be added on the top of any given mean profile, which could be
modified and varied in time for more realistic applications. The third class
is the synthetic turbulence generator, which includes a variety of methods
such as the Fourier transform-based method (e.g. Kraichnan, 1970; Lee et
al., 1992), proper orthogonal-decomposition-based method (e.g. Berkooz et
al., 1993; Kerschen et al., 2005), digital-filter-based method (e.g. Klein et al., 2003; Xie
and Castro, 2008; Kim et al., 2013), diffusion-based
method (e.g. Kempf et al., 2005), vortex method
(e.g. Benhamadouche et al., 2006) and synthetic eddy method
(e.g. Jarrin et al., 2006). The synthetic turbulence
generator has the potential to be used for a wide range of flows. Due to the
imperfection of the synthetic turbulence, which is not directly derived from
generic flow equations, these methods normally require some inputs and a
certain adjustment distance for turbulence to become well-established. For
more information about the above synthetic turbulence generation methods, we
recommend Tabor and Baba-Ahmadi (2010), Wu (2017), and
Bercin et al. (2018).
Several other methods have been developed to generate inflow turbulence for
atmospheric boundary layer flow in nested WRF-LES models. Mirocha
et al. (2014) introduced simple sinusoidal perturbations to the potential
temperature and horizontal momentum equations near the inflow boundaries.
This method can speed up the development of turbulence and generally has a
satisfactory performance in the nested WRF-LES domains, providing promising
results. Muñoz-Esparza et al. (2014) extended the perturbation
method of Mirocha et al. (2014) and proposed four methods, i.e. the
point perturbation method, cell perturbation method, spectral inertial
subrange method, and spectral production range perturbations, to generate
perturbations of potential temperature for a buffer region near the nested
inflow planes. The cell perturbation method was found to have the best
performance regarding the adjustment distance for the turbulence to be
fully developed. It has the advantages of negligible computational cost,
minimal parameter tuning, not requiring a priori turbulent information, and
efficiency to accelerate the development of turbulence.
Muñoz-Esparza et al. (2015) further generalised the cell
perturbation method of Muñoz-Esparza et al. (2014) under a
variety of large-scale forcing conditions for the neutral atmospheric
boundary layer. The perturbation Eckert number (describing the interaction
between the large-scale forcing and the buoyancy contribution due to the
perturbation of potential temperature) was identified as the key parameter
that governs the transition to turbulent flow for nested domains. They found
an optimal Eckert number to establish a developed turbulent state under
neutral atmospheric conditions. These methods impose temperature
perturbations at specific length scales and timescales related to the highest
resolved wave number in the LES. It was demonstrated in
Muñoz-Esparza et al. (2015) that a distance of about 15
boundary layer depths is required to allow the flow to be fully turbulent
when the temperature perturbation method is adopted in the one-way nesting
WRF model. It is to be noted that the temperature perturbation method was
introduced for mesoscale to microscale coupling approach where smooth
mesoscale flow (no resolved turbulence) forces microscale flow by using the
one-way nesting approach in WRF. Muñoz-Esparza et al. (2014)
stated “the perturbation method is to provide a mechanism that accelerates
the transition towards turbulence, rather than to impose a developed
turbulent field at the inflow planes as the synthetic turbulence generation
methods pursue”, and “the use of temperature perturbations presents an
alternative to the classical velocity perturbations commonly used by most of
the techniques.” The optimisation and generalisation of these methods would
require intensive testing. Muñoz-Esparza and Kosovic (2018)
extended the cell perturbation method of the inflow turbulence generation to
non-neutral atmospheric boundary layers. Instead of adopting temperature
perturbations in the original cell perturbation method, Mazzaro
et al. (2019) further explored the random force perturbation method (FCPM) in
the multiscale nested domains.
Due to its accuracy, efficiency, and, in particular, the capability for high
Reynolds number flows, the synthetic inflow turbulence generator
(Xie and Castro, 2008) has been implemented and tested on codes
developed for engineering applications, such as Star-CD (Xie and
Castro, 2009) and OpenFOAM (Kim and Xie, 2016), and the
micro-scale meteorology code PALM (PALM, 2017; Maronga et al., 2020).
This study focuses on an implementation of this synthetic inflow turbulence
generator (Xie and Castro, 2008) in the idealised WRF-LES (v3.6.1)
model under neutral atmospheric conditions. In this paper, Sect. 2
describes the methodology of the WRF-LES model and the technique of the
synthetic inflow turbulence generator, Sect. 3 presents the results of the
WRF-LES model with the use of the synthetic inflow turbulence generator, and
Sect. 4 states the conclusions and future work.
MethodologyWRF-LES model
The atmospheric boundary layer is simulated by the compressible
non-hydrostatic WRF-LES model, which computes large energy-containing eddies
at the resolved scale directly and parameterises the effect of small
unresolved eddies on the resolved field using sub-grid-scale (SGS) turbulence
schemes (Moeng et al., 2007). The Favre-filtered equations are as follows
(Nottrott et al., 2014; Muñoz-Esparza et al., 2015):
1∂ρ̃∂t+∂ρ̃ũj∂xj=0,2∂ũi∂t+∂ũiũj∂xj=υ∂2ũi∂xj∂xj-1ρ̃∂p̃∂xi-∂τij∂xj+F̃i,
where i (or j) =1, 2, 3, represents the component of the spatial
coordinate, ũi is the filtered velocity, xi is the spatial
coordinate, t is the time, p̃ denotes the filtered pressure,
ρ̃ is the filtered density, υ is the fluid kinematic
viscosity, τij are the SGS stresses, and Fĩ represents
external force terms (normally involving the Coriolis force caused by the
rotation of the Earth and the large-scale geostrophic forcing).
For the closure of Eq. (2), τij is parameterised using a SGS
model. In this study, the 1.5-order turbulence kinetic energy (TKE) SGS
model is used,
τij=-2υsgsS̃ij,
where S̃ij is the filtered strain-rate tensor and calculated as
S̃ij=12(∂ũi∂xj+∂ũj∂xi).υsgs denotes the SGS eddy viscosity and is defined as
υsgs=Cklksgs1/2,
where Ck is a model constant, and l is the SGS length scale,
which equals the grid volume of size (Δ) under neutral
conditions (Deardorff, 1970),
Δ=(ΔxΔyΔz)1/3.ksgs is the SGS TKE with the transport equation:
∂ksgs∂t+∂∂xiksgsũi=-υsgsPrgθ0∂θ̃∂z+2υsgsS̃ijS̃ij+υ+υsgs∂2ksgs∂xi∂xi-Cεksgs1.5l,
where θ̃ is the filtered potential temperature, Pr is the
turbulent Prandtl number, and Cε is a dissipation
coefficient (for more details about the parameterisation see Moeng
et al., 2007). Without loss of generality, the “̃” notation for
all filtered variables is omitted hereafter.
Synthetic inflow turbulence generator
The synthetic inflow turbulence generator in Xie and Castro (2008)
adopted the digital filter-based method and is used in this study. For
simplicity, a one-dimensional problem (the streamwise velocity, u, along the
x direction) is used as an illustration to describe this method. The two-point
velocity correlations RuukΔx are assumed
to be represented by an exponential function:
umum+k‾umum‾=RuukΔx=exp-πk2n,
where m, the index that the averaging operator is applied, denotes the
mth element of a vector (one-dimensional data series of, for example, the
digital-filtered velocity, u, in Eq. 9 below), k is the number of elements
for the two-point distance of kΔx, n is related to the
integral length scale L=nΔx with the grid size of
Δx, and um is the digital-filtered velocity,
um=∑k=-NNbjrm+k,
where rm is a sequence of random data with mean rm‾=0 and
variance rmrm‾=1, N is related to the length scale for the
filter (here N≥2n), and bj is the filter coefficient and can be
estimated from
bk=b̃k/∑j=-NNb̃j21/2,
where b̃k≅exp-πkn.
For a two-dimensional filter coefficient, it can be obtained that
bjk=bjbk,
which will then be used to filter the two-dimensional random data at each
time step,
φβt,xj,xk=∑j=-NjNj∑k=-NkNkbjkrm+j,m+k,
where β indicates the velocity component. At the next time step, the
filtered velocity field is calculated as
Ψβt+Δt,xj,xk=Ψβt,xj,xkexp-πΔt2T+φβt,xj,xk1-exp-πΔtT0.5,
where T is the Lagrangian timescale representing the persistence of the
turbulence, and φmt,xj,xk is
calculated based on Eq. (12). Xie and Castro (2008) demonstrated
that Eq. (13) satisfies the correlation functions in an exponential form in
space and in time. The two-dimensional filter in Xie and Castro
(2008) is more computationally efficient than a three-dimensional filter.
Finally, the velocity field is obtained by using the simplified
transformation proposed by Lund et al. (1998),
ũi=u‾i+αiβΨβ,
where
[αiβ]=R̃111/200R̃21/α11R̃22-α2121/20R̃31/α11(R̃32-α21α31)/α22R̃33-α312-α3221/2,
and R̃iβ is the resolved Reynolds stress tensor, which can
be estimated based on measurements or other simulations with periodic
boundary conditions. The calculations of αiβ follow an
iterative order: α11, α21, α22,
α31, α32, and α33.
Integral length scales prescribed at the inlet used in the inflow
BASE case (LS1.0).
Model coupling and configuration
In this study, we firstly configured a WRF-LES model with periodic boundary
conditions in both the streamwise and spanwise directions to obtain prior mean
profiles of first and second moments of turbulence, such as the vertical
profiles of mean velocity and Reynolds stress components, which are required
as inputs by the synthetic inflow turbulence generator. Additional essential
quantities as inputs of the inflow generator are three integral length scales
in the x, y, and z directions, denoted by Lx, Ly, and Lz,
respectively (or Li, i=x,y,z). For the inflow BASE case (denoted by LS1.0),
the vertical profiles of Li are specified as functions of z/H, where
H is the boundary layer height (500 m in this study), shown as Fig. 1,
similar to those in Xie and Castro (2008). The streamwise length
scale (Lx) is specified based on the mean streamwise velocity profile
(〈u〉) and a constant Lagrangian timescale T (prescribed in Eq. 13), i.e. Lx=T〈u〉 using Taylor's hypothesis (turbulence is assumed to be frozen while it is
moving downstream with a mean speed of
〈u〉). The spanwise length scale (Ly) is specified as a constant value.
The vertical length scale (Lz) is specified as a smaller constant value
near the bottom and a larger constant value for the upper domain to be
closer to the measured length scales, as explained in Xie and Castro
(2008) and Veloudis et al. (2007). We conducted a sensitivity
study of integral length scales by varying all three baselines Lx, Ly, and Lz with the ratio of 0.6, 0.8, 1.0, 1.2, or 1.4; these individual
cases are denoted by “LS0.6”, “LS0.8”, “LS1.0”, “LS1.2”, and “LS1.4”,
respectively, in which “LS1.0” is the base case. The size of the
computational domain is 9.98km×2.54km×0.5km (in the
x, y, and z directions), with the resolutions of Δx=Δy=20m and stretched Δz (from about 3 up to 27 m). The grid number is then 499×127×49. In
order to achieve the constant wind direction vertically, the Coriolis force
is not activated in this study. The external driving force is specified as a
constant pressure gradient force in Eq. (2), similar to that used in
Ma and Liu (2017), resulting in a prevailing wind speed of about 10 m s-1 at the domain top. At the top boundary, a rigid lid
(“top_lid” in the “namelist.input” file of the WRF-LES
model) is specified, and a Rayleigh damping layer of 50 m is used to prevent
undesirable reflections (Nottrott et al., 2014; Ma and Liu, 2017) and to
maintain a neutral atmospheric boundary layer.
For the cases with the synthetic turbulence at the inlet and periodic
conditions in the spanwise direction, the constant pressure gradient force
is not necessary anymore. Instead, a pressure-drop between the inlet and
outlet is implicitly derived from the prescribed mean momentum profiles as
part of the synthetic inflow and the outflow boundary conditions in the
solver. The periodic case is used for the validation of the results from the
inflow case. The WRF-LES is solved at a time step of 0.2 s. A spin-up period
of 6 h is adopted for all inflow cases to allow turbulence inside the domain
to reach quasi-equilibrium. The further 1 h outputs with 5 s interval
(approximately the advection timescale of the smallest resolved eddies,
which is equivalently twice the grid resolution of 20 m) were used for the
analysis. We take advantage of the homogeneous turbulence in the spanwise
direction (Ghannam et al., 2015) and calculate all resolved-scale
turbulent quantities by averaging in the spanwise direction (the y direction)
and in time t over the last 1 h period. This averaging is referred to as
“the y-t averaging” hereafter and is denoted by
〈φ〉, for example, for the y-t-averaged φ. For a 4D variable, φ(t,x,y,z), the y-t-averaged φ is a function of xz, i.e. 〈φ〉(x,z); for a variable defined on the x-y plane, e.g. friction velocity u∗(t,x,y), the y-t averaging u∗ is a function of x, i.e. 〈u∗〉(x).
In the synthetic inflow turbulence generator, a uniform mesh is used with
resolutions of Δy=20 m (same as that on the physical
inlet of the WRF-LES domain) and Δz=4.2 m (slightly
larger than the smallest vertical grid spacing of the WRF-LES domain). The
three filtered velocity components at the inlet from the inflow generator
are then interpolated onto the vertically non-uniform mesh in the WRF-LES
domain. It should be noted that the grid resolution can differ between the
inflow patch and the inlet of the WRF-LES domain. The standalone synthetic
turbulence generator code in Xie and Castro (2008) was originally
run on a single processor, whereas the WRF-LES simulation here is run in
parallel mode. It is therefore necessary to ensure that each processor in
the parallel mode has the same information of the two-dimensional slice of
flow field before each processor can extract the corresponding patch from
the same two-dimensional inlet data. In this implementation, the synthetic
turbulence generator code is firstly run on the master processor at each
WRF-LES time step. The generated inlet data are then passed to other
processors. The flow field at the inlet of each corresponding processor was
then be updated at every time step. The additional computational time for
the inflow case is associated with the synthetic inflow turbulence generator
and data passing, i.e. non-parallelisation of the current inflow generator.
Increasing the integral length scale would increase the computation time
since bigger arrays are constructed and computed for the filtered velocity
in the synthetic inflow turbulence generator, as in Eq. (9) for the larger
integral length scale.
Horizontal slice of instantaneous streamwise velocity component,
u (m s-1), at z/H=0.1 after a simulation time of 6 h: (a) the fully periodic
case, (b) the synthetic inflow BASE case (LS1.0), and (c) the inflow case
without perturbations at the inlet.
ResultsBASE case outputHorizontal slices of instantaneous streamwise velocity component
Figure 2 illustrates the horizontal slices of the instantaneous streamwise
velocity component at z/H=0.1 in the periodic case, the synthetic inflow
case (LS1.0 in Fig. 1a), and the inflow case without inlet perturbations
(with mean information only) after a simulation time of 6 h. The synthetic
turbulence structures imposed at the inlet are advected into the domain and
are adjusted by the model dynamics at further downwind distances. After an
adjustment distance (about x/H=5–10), the inflow case (LS1.0) clearly
generates turbulence streaks, which are similar to these in the periodic
case. Other quantities that may further demonstrate this adjustment distance
will be discussed in the following subsections. This suggests that the
synthetic turbulence generated at the inlet can develop into realistic
turbulence with well-configured structures from an adjustment distance
downwind of about x/H=5–10. For the inflow case without inlet velocity
perturbations, there is almost no turbulence generated in the domain even
after several hours of simulation. This is consistent with other similar
tests using engineering CFD codes with no synthetic turbulence added at the
inlet, e.g. Xie and Castro (2008), which confirms that a very long
distance (e.g. 100 times the boundary layer thickness) is needed to allow
turbulence to develop. This indicates the importance of imposing synthetic
turbulence, or at least some form of random perturbations
(e.g. Muñoz-Esparza et al., 2015), at the inlet. The inflow
case without the inlet velocity perturbations is not presented in later
sections.
Spatial variation of
〈u∗〉/u∗ for the periodic case and the inflow
case (LS1.0), where 〈u∗〉 is the y-t-averaged local friction velocity and u∗
is the x-y-t-averaged friction velocity for the periodic case.
Development of local friction velocity
Figure 3 shows the development of the y-t-averaged local friction velocity,
〈u∗〉x, for the periodic case and the inflow BASE case (LS1.0),
normalised by u∗, the x-y-t-averaged friction velocity for the periodic
case. The variation of the local friction velocity is within ±0.5 % of
u∗ along the streamwise direction for the periodic case and is
within 1.5 % of u∗ for the inflow case after a downwind distance of
x/H=7. There is a larger variation close to the inlet region (x/H<7) for
the inflow case. This is because the imposed turbulence on the inflow plane
is “synthetic”, of which only the first-order and second-order moments,
integral length scales, and the spectra aim to match the prescribed data
(Bercin et al., 2018). It must develop over a certain
distance in the WRF-LES domain before it can be fully developed “realistic”
turbulence.
Horizontal profiles (spatially and temporally averaged) of (a)〈u〉/u∗, (b)〈u′2〉/u∗2, (c)〈v′2〉/u∗2, (d)〈w′2〉/u∗2, (e)〈u′w′〉/u∗2, and (f)〈TKE〉/u∗2 at
z/H=0.1 and z/H=0.5 in the periodic case and the inflow case (LS1.0).
Horizontal profiles of mean flow and turbulence quantities
Figure 4 illustrates the y-t-averaged horizontal profiles of the normalised
mean streamwise velocity component, normal and shear turbulent stresses, and
TKE at z/H=0.1 and z/H=0.5 for the periodic case and the inflow case
(LS1.0), respectively. These horizontal profiles show the development of
synthetic turbulence along the streamwise direction. There are only slight
differences in the normalised mean streamwise velocity component
(〈u〉/u∗) between the periodic case and the inflow case. This
suggests that the inflow case reproduces successfully the desired mean wind.
The curves of normalised streamwise velocity variance
(〈u′2〉/u∗2) for both cases match well with each other downstream from
x/H=7–8, although there is a sudden jump close to the inlet and a
subsequent decrease until the location of convergence. The horizontal
profiles of normalised cross-stream velocity variance
(〈v′2〉/u∗2) for the inflow case are in a good agreement after a
developing distance of x/H=10–12, compared with those for the periodic
case. The development convergence of normalised vertical velocity variance
(〈w′2〉/u∗2) is achieved after a distance of about x/H=5–10 from the
inlet. The development distance of turbulent shear stress
(〈u′w′〉/u∗2) is about x/H=5–15. Since the streamwise velocity
variance comprises a large proportion of TKE, the development distance for
TKE is similar to that for the streamwise velocity variance, i.e. about
x/H=7–8. The distance needed for different quantities to reach a converged
state differs from each other, and it is about x/H=5–15.
Spatially and temporally averaged vertical profiles of (a)〈u〉/u∗, (b)〈u′2〉/u∗2, (c)〈v′2〉/u∗2, (d)〈w′2〉/u∗2, (e)〈u′w′〉/u∗2, and (f)〈TKE〉/u∗2 at a
series of downwind locations in the inflow case (LS1.0), and the periodic
case (also averaged in the streamwise direction).
Vertical profiles of mean flow and turbulence quantities
Figure 5 shows the y-t-averaged vertical profiles of the normalised mean
streamwise velocity component, normal and shear turbulent stresses, and TKE
at a series of downwind locations, x/H=0, 4, 6, and 10, for the inflow
case (LS1.0). Inflow cases are not averaged in the streamwise direction so
that the development of turbulence at each downwind location (x/H) can be
investigated. Red lines in Fig. 5 are the spatially (including both in the
streamwise and spanwise directions) and temporally averaged vertical
profiles for the periodic case. It is noted again that these data for the
periodic case are also used as the inputs for a priori turbulence information
required by the synthetic inflow turbulence generator. It is also noted that
the profiles of the mean velocity and second-order moments at the inlet
(x/H=0) are overall in a good agreement with these of the periodic case,
which suggests precise settings of the turbulence generator. The profiles of
the normalised mean streamwise velocity component
(〈u〉/u∗) in the inflow case match closely those of the periodic case.
Although the sampled data are limited, this confirms again that the inflow
case achieves the desired the mean wind. The normalised streamwise velocity
variance (〈u′2〉/u∗2) converges towards the periodic profile after x/H=6 as
shown in Fig. 5b. Although the vertical profiles of
〈v′2〉/u∗2, 〈w′2〉/u∗2 and 〈TKE〉/u∗2 for the inflow case show small variations between
different locations, they are all in a good agreement with the corresponding
data of the periodic case. These are consistent with the results shown in
Fig. 4. The turbulent shear stress
〈u′w′〉, which is the cross-correlation between the streamwise and vertical
velocity fluctuations, usually converges more slowly than the normal
turbulent stresses, e.g. 〈v′2〉. Overall, the synthetic inflow turbulence generator performs well in terms
of the mean flow and the turbulence quantities against the data from the
periodic case, as well as the short development distance.
Spectra of streamwise velocity component for a series of downwind
locations at the height of z/H=0.5, k is the angular wavenumber, with
〈u〉 and 〈u′2〉 the spatially averaged mean and streamwise normal turbulent stress,
respectively.
Spectral analysis
Figure 6 illustrates the spectra of the streamwise velocity component at a
series of downwind locations (x/H=0, 4, 6, and 10) at
z/H=0.5 for the periodic case and the inflow case (LS1.0). For each
x location, e.g. x/H=10, the spectrum for the inflow case was first
calculated from the streamwise velocity component over a time series of 3600 s with an interval of 5 s for five selected sample locations of yn
(y/H=1.76, 2.16, 2.56, 2.96, and 3.36), namely, ũ(t,2H,yn,0.5H). The spectral data were then averaged over
yn to give the spectra plotted in Fig. 6.
The spectrum for the periodic case is calculated using the same method as
that used for the inflow case, with an additional average over the
streamwise direction x. The spectrum at the inlet (x/H=0) possesses the
broadest range of wavenumbers where eddies exhibit inertial sub-range
behaviour, as evidenced by the wavenumber range within which the slope of
each spectrum is approximately -5/3. There is evidence of the tendency
in the profiles from the inlet downstream to recover to that of the periodic
case. The spectrum drops slightly at high wavenumbers from the imposed
spectra at x/H=0 to downwind locations and approaches the spectrum of the
periodic case. The slight drop suggests a decay of small eddies due to the
SGS viscosities and the numerical dissipation originating from the advection
scheme in the WRF-LES model. The spectra in Muñoz-Esparza et
al. (2015) drop at lower wave numbers than those in Fig. 6, mainly due to a
coarser resolution (than the current one). Our resolution of 20 m in the
horizontal direction is much finer than the resolution of 90 m in
Muñoz-Esparza et al. (2015). In other words, the size of the
smallest eddy (twice the grid resolution) that can be resolved by the LES
model is 40 m in our paper vs. 180 m in Muñoz-Esparza et al. (2015). These confirm that synthetic turbulence with an inertial subrange in
the spectrum generated by using the Xie and Castro (2008) method is able
to be mostly sustained in WRF-LES for a high resolution. It is noted that
for a very high resolution, e.g. of the order of magnitude of 1 m, similar
to that used in the simulations of PALM (PALM, 2017; Maronga et al., 2020), the
inertial subrange in the spectrum is much wider. It is to be noted that
Muñoz-Esparza et al. (2015) also tested the Xie and
Castro (2008) method in WRF-LES using the same resolution of 90 m as that
for the temperature perturbation method. Again this is rather a coarse
resolution to test the performance of the Xie and Castro (2008)
method when a spectrum is of interest.
Development of local friction velocity (averaged over spanwise
direction and time) with various integral length scales.
〈u∗〉 is the local friction velocity along the streamwise direction, and
u∗ is the x-y-t-averaged friction velocity for
the periodic case.
Sensitivity tests of integral length scale in the flow cases
It is not trivial to obtain “accurate” integral length scales of the inlet
turbulence generator. Indeed these data are always incomplete. Therefore, it
is necessary to conduct sensitivity tests of the integral length scales.
Figure 7 shows the influence of integral length scale on the development of
local friction velocity. Various integral length scale ratios (ranging from
0.6 to 1.4) to those (Lx, Ly, and Lz respectively) in
the LS1.0 case are tested. Note that these three integral length scales
(Lx, Ly, and Lz) are in the same ratio as those
respectively in the LS1.0 case. For all inflow cases, there is a sudden
change near the inlet due to the imposed “imperfect” inflow turbulence. The
adjustment distance to well-established turbulence (i.e. within 4 %
error) is generally short, i.e. about x/H=2–7 for the studied
cases LS0.6–1.4, but seems shorter for the case with the smaller integral
length scales. This suggests that the imposed integral length scales for the
inflow turbulence slightly affect the convergence to well-developed
turbulence. We conclude that a variation of ±40 % in the integral
length scale in the cases LS0.6–1.4 yields a variation of less than 4 % in
the local friction velocity after about x/H=7, and that the
sensitivity of integral length scale on the local friction velocity is not
significant in the WRF-LES model if the used integral length scale is within
a reasonable range. This is consistent with that in engineering-type CFD
solvers in Xie and Castro (2008).
Horizontal profiles (spatially and temporally averaged) of (a)〈u〉/u∗, (b)〈u′2〉/u∗2, (c)〈v′2〉/u∗2, (d)〈w′2〉/u∗2, (e)〈u′w′〉/u∗2, and (f)〈TKE〉/u∗2 at z/H=0.5 with various
integral length scales.
Figure 8 shows the effects of integral length scale on the horizontal
profiles of the normalised mean streamwise velocity, normal and shear
turbulent stresses, and TKE at z/H=0.5. Figure 8a shows that
〈u〉/u∗ is slightly greater for the length scale ratio less than 1.0.
This is likely due to a slightly smaller u∗, which is common for
smaller integral length scale cases (as shown in Fig. 7). Figure 8b–d
and f show that in general the normal stresses,
〈u′2〉/u∗2, 〈v′2〉/u∗2, 〈w′2〉/u∗2, and 〈TKE〉/u∗2, increase as the length scale ratio increases. This is
because small eddies tend to decay faster than large eddies. It is crucial
to note that for those with the integral length scales close to those of
LS1.0 (the base case) the development distance to converged turbulence is
shorter compared to other cases, indicating that the length scales of the
base case are reasonable estimations.
Vertical profiles (spatially and temporally averaged) of (a)〈u〉/u∗, (b)〈u′2〉/u∗2, (c)〈v′2〉/u∗2, (d)〈w′2〉/u∗2, (e)〈u′w′〉/u∗2, and (f)〈TKE〉/u∗2 at x/H=10 with various
integral length scales.
Figure 9 shows effects of integral length scale on vertical profiles of the
mean velocity, normal and shear turbulent stresses, and TKE at a typical
streamwise location (x/H=10). These profiles are consistent with
those in Fig. 8 and draw the same conclusions as from Fig. 8. For all the
tested integral length scales, downstream from x/H=10 both mean
and turbulent quantities converge to the periodic case. This suggests again
that the mean velocity and the turbulent stresses are not very sensitive to
the integral length scales if they are not too different from the realistic
values. In general, there are slight differences in
〈u〉/u∗ between each case. The magnitudes of turbulent quantities for
smaller integral length scales are generally smaller than those for larger
integral length scales.
Spectra of streamwise velocity component for a series of downwind
locations at x/H=10 and z/H=0.5 with various integral length scales; k is the
angular wavenumber, with 〈u〉 and 〈u′2〉 the spatially averaged mean and streamwise normal turbulent stress,
respectively.
Figure 10 shows the effect of the integral length scale on the spectra of
the streamwise velocity component at x/H=10 and z/H=0.5. For all cases
tested in the current study, the spectra with various integral length scales
generally match those of the periodic case at a distance of x/H=10 from
the inlet albeit with slight changes of the spectrum for small wavenumber
turbulence. A very small variation of the spectra is within margins of
uncertainty in the calculation of the spectra from the raw data. All spectra
show an inertial subrange of -5/3 slope, which are consistent with those
in the references, such as Xie and Castro (2008), indicating the
robustness of the synthetic turbulence generator on the generation of an
inertial subrange.
Discussion and conclusions
A synthetic inflow turbulence generator (Xie and Castro, 2008) was
implemented in an idealised WRF-LES (v3.6.1) model under neutral atmospheric
conditions. A WRF-LES model with periodic boundary conditions was firstly
configured to provide a priori turbulence statistical data for the synthetic inflow
turbulence generator. Previous studies (e.g. Xie and Castro,
2008) suggest that it is important to have an approximation of the integral
length scales, which are the key inputs of the inflow turbulence generator.
The results from the inflow cases were then compared with those from the
periodic case. Sensitivity
tests were conducted for the response of the local friction velocity, the
mean flow, the Reynolds stresses, and the turbulence spectra for the flow
cases for varying integral length scales.
The inflow case with the baseline integral length scales generates similar
turbulence structures to those for the periodic case after an adjustment
distance of x/H=5–15. The WRF-LES model with the inflow generator
reproduces realistic features of turbulence in the neutral atmospheric
boundary layer. The development of local friction velocity suggests that a
downwind distance of about x/H=7 is required to recover the local friction
force for the inflow case, which agrees with the findings in Xie and
Castro (2008) and Kim et al. (2013). Keating et al. (2004)
suggested a development distance of about 20 times the half-channel depth for
modelling a plane channel flow. The difference between this value and our
results can be attributed to the different synthetic turbulence generation
approaches adopted here versus those adopted by Keating et al. (2004). Laraufie et al. (2011) suggested that an increase in the
Reynolds number decreases the adjustment distance when a synthetic inflow
turbulence generator is used. For our simulated atmospheric boundary layer
flow here, the Reynolds number is extremely large. Thus adopting a synthetic
inflow turbulence generator for the atmospheric boundary layer should also
be advantageous in engineering applications. Regarding the minimum
resolution required to generate turbulence synthetically, our presented
results confirm that the tested grid resolution sufficiently resolves the
important features.
Horizontal and vertical profiles of mean velocity and second-moment
statistics further confirm that a short adjustment distance is required for
the development of synthetic turbulence. The mean velocity profiles at all
tested locations were in very good agreement with the reference data, while
the turbulence second-moment statistics profiles were in reasonable
agreement with the reference data about x/H=5–15 downwind of the inlet. An
accurate estimation of the second-order moments is crucial for the
assessment of the synthetic inflow turbulence generator, in particular when
the inflow turbulence information is not completely available. We found
varying the integral length scale within ±40 % of the value in the
base case has a negligible influence on the mean velocity profiles, while
the effects of the variation on the turbulent second-order moment statistics
are visible, for example the local friction velocity was within 4 % error
of the reference data at x/H=7. The synthetic inflow turbulence generator
requires additional computational time compared to periodic boundary
conditions. This will be certainly improved by running the synthetic inflow
generation subroutine in parallel as a future task. This study is focused on
the feasibility of implementing the inflow method (Xie and Castro, 2008) in
the meso-to-micro-scale meteorological code WRF and the impact of the key
variables (i.e. the integral length scales) on the simulated turbulence
development inside the domain. This inflow subroutine has previously been
implemented in both serial and parallel mode in several codes, including
engineering-type codes Star-CD (Xie and Castro, 2009) and OpenFOAM (Kim
and Xie, 2016), and the micro-scale meteorology code PALM (PALM, 2017).
Although the current implementation in WRF is affordable for a
moderate-sized simulation (e.g. resolution of tens of metres), the technical
parallelisation of this inflow subroutine in WRF-LES can be the future work
for very large simulation domains with high resolutions.
In summary, the synthetic inflow turbulence generator is implemented
successfully into the idealised WRF-LES model. The generated synthetic turbulence is correlated both in space and in time in the exponential
form. The spectrum of these data shows an inertial subrange of -5/3
slope, and this again suggests the capability of the method to generate high
Reynolds number flows. The tests on WRF also confirm that this method
yields a satisfactory accuracy, after having compared the local friction
velocity, the mean velocity, the Reynolds stresses, and the turbulence
spectra against the reference data. The WRF-LES model with the synthetic
turbulence generator provides promising results as evaluated against the
periodic case. The limitation of this method is the requirement of a priori
turbulence statistics data and integral length scales, which can be estimated
by the similarity theory of the atmospheric boundary layer or experimental
data. Sensitivity studies have been performed to address this issue, in
particular in terms of the effect of the integral length scale. The
implementation of the synthetic inflow turbulence generator (Xie and
Castro, 2008) can be extended to the WRF-LES simulation of a horizontally
inhomogeneous case with non-repeated surface land-use patterns and be
further developed for the multi-scale seamless nesting case from a
meso-scale domain with a kilometre-scale resolution down to LES domains with metre-scale resolutions. It is also worthwhile to examine the wind spiral case
induced by the Coriolis force in the atmospheric boundary layer.
Code and data availability
The standard version of WRF v3.6.1 tar file is available at https://www2.mmm.ucar.edu/wrf/users/download/get_sources.html (last access: 17 September 2018, UCAR, 2018).
The coupling WRF v3.6.1 code with the synthetic inflow turbulence generator
and case settings are archived on Zenodo (10.5281/zenodo.3668352, Zhong et al., 2019).
Author contributions
The study was conceived by XC and ZTX. JZ implemented the synthetic inflow
turbulence generator code (from ZTX) to the WRF-LES model (v3.6.1) and ran
model simulations. All authors contributed to writing the paper.
Competing interests
The authors declare that they have no conflict of interest.
Acknowledgements
This work used the ARCHER
UK National Supercomputing Service (http://www.archer.ac.uk, last access: 10 January 2020).
Financial support
This research has been supported by the UK Natural Environment Research Council (NERC) (grant no. NE/N003195/1).
Review statement
This paper was edited by James R. Maddison and reviewed by three anonymous referees.
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