Abstract
In this work, the reconstruction of subgrid scales in large eddy simulation (LES) of turbulent flows in stratocumulus clouds is addressed. The approach is based on the fractality assumption of turbulent velocity field. The fractal model reconstructs subgrid velocity field from known filtered values on LES grid, by means of fractal interpolation, proposed by Scotti and Meneveau (Physica D 127, 198–232 1999). The characteristics of the reconstructed signal depend on the stretching parameter d, which is related to the fractal dimension of the signal. In many previous studies, the stretching parameter values were assumed to be constant in space and time. To improve the fractal interpolation approach, we account for the stretching parameter variability. The local stretching parameter is calculated from direct numerical simulation (DNS) data with an algorithm proposed by Mazel and Hayes (IEEE Trans. Signal Process 40(7), 1724–1734, 1992), and its probability density function (PDF) is determined. It is found that the PDFs of d have a universal form when the velocity field is filtered to wavenumbers within the inertial range. In order to investigate Reynolds number (Re) dependence, we compare the inertialrange PDFs of d in DNS and large eddy simulation (LES) of stratocumulus cloudtop and experimental airborne data from Physics of Stratocumulus Top (POST) research campaign. Next, fractal reconstruction of the subgrid velocity is performed and energy spectra and statistics of velocity increments are compared with DNS data. It is assumed that the stretching parameter d is a random variable with the prescribed PDF. We show that the agreement with the DNS is in such case better and the error in mass conservation is smaller compared to the use of constant values of d. The motivation of this study is to reproduce effect of subgrid scales on a motion of Lagrangian particles (e.g. droplets) in clouds.
Introduction
Lagrangian tracking of particles in turbulent flows seen in nature requires huge computational resources because the flow of a carrier fluid is spatially and temporally complex, with structures in a wide range of scales. The ratio between the largest and smallest scales of turbulence (Re ≈ L/η) observed in such flows can be as high as 10^{9}. The most detailed research tool, which can resolve all turbulent scales and accurately predict particle trajectories, is the DNS [1]. Owing to its unrealistic computational cost at high Reynolds numbers, LES is used as a compromise to calculate the large scale features (resolved) of the flow while subgrid (unresolved) scales are modelled. Lack of subgrid scales or subgrid scale model errors will lead to the progressive divergence of particle trajectories when compared with those obtained in experiment or DNS [2, 3]. As a result, particle statistics such as preferential concentration, average settling velocity and relative velocities are either over or underestimated [4,5,6,7,8,9,10].
Several attempts have been made to develop models for the effects of subgrid scales of turbulence on the prediction of the Lagrangian dispersion of inertial particles. As classified by [3], there are stochastic and structural models. Stochastic models are based on the solutions of the Langevin equations [11,12,13,14] supplemented with a stochastic Wiener process term [15, 16]. Various works on stochastic models show that they perform well at small Stokes number but have strong sentivitity to the filter size [3, 17].
Structural subgrid models are an alternative to the stochastic models and they aim at mimicking (some of) the subgrid scales in a statistical sense. These models allow for the approximate reconstruction of twopoint particle statistics at the subgrid scale [8, 9, 18]. Example of structural models are the fractal interpolation [2, 19, 20], approximate deconvolution (ADM) [21,22,23,24], spectrally optimized interpolation [25], the kinematic simulations based on Fourier modes [26,27,28,29] and loworder dynamical systems [68]. A good structural subgrid model should not only be able to capture the statistics of small scales but also be computationally efficient and easy to use. Fractal interpolation technique (FIT) seems promising (especially for high Re geophysical applications) with respect to these features. FIT was introduced to construct synthetic, fractal subgridscale fields applied to LES of both stationary and freely decaying isotropic turbulence [20]. The underlying assumption of the model is the existence of fractalscale similarity of velocity fields. An attribute of the constructed subgrid velocity depends on the stretching parameter d, which is related to the fractal dimension of the signal. In Scotti and Meneveau [20], it was assumed that d is constant in space and time for homogeneous and isotropic turbulence. Basu et al. [30] proposed an extension of this work by developing a multiaffine fractal interpolation scheme with bivalued stretching parameter and showed that it preserves the higherorder structure functions and the nonGaussianity of the PDF of velocity increments. They performed extensive analyses on atmospheric boundary layer data and argued that the multiaffine closure model should give satisfactory performance in LES. Marchioli et al. [2] used fractal interpolation and approximate deconvolution technique to model subgrid scale turbulence effects on particle dynamics in wallbounded turbulence. They showed that FIT appears to be inefficient in reintroducing the fluid velocity fluctuation when a constant stretching parameter is assumed [31]. They concluded that an accurate subgrid model for particles may require information on the higherorder moments of the velocity fluctuation [2].
Characteristic features of atmospheric turbulence are the inhomogeneity due to buoyancy and the presence of internal and external intermittency. Internal intermittency means that large velocity gradients are present at small scales and the PDF of velocity differences at high Reynolds number are stretchedexponential at small scales [32]. External intermittency refers to the coexistence of both laminar and turbulent regions in the flow. These attributes make it difficult to synthesize subgrid velocity field using fractal interpolation since the local stretching parameters have been shown to change randomly in space and time [2, 33].
The aim of this work is to develop an improvement to FIT, which can be used as a closure model for Lagrangian tracking of particles in moderate or high Reynold number (complex) turbulent flows. We proceed by accounting for the spatial variability of the stretching parameter: first, its local value is computed with a method proposed by Mazel and Hayes [34]. Then, the PDF of local stretching parameter is used to construct subgrid velocity, starting with the filtered DNS or LES. Three examples of complex turbulent flows are investigated, first moderate Reynolds number DNS velocity field of stratocumulustop boundary layer (STBL) [35], second, LES of stratocumulus cloud at Re comparable with atmospheric turbulence [36] and, finally, experimental data from POST airborne measurements in stratocumulus clouds [37,38,39].
The extracted data of PDF of d are used to perform 3D reconstruction of subgrid eddies. Performance of the new approach is compared with FIT with the constant stretching parameter values from Ref. [20] and the multiaffine fractal interpolation scheme [30]. We calculate energy spectra of the reconstructed velocity field and the statistics of velocity increments. The focus on the statistics of velocity increments is motivated by their ability to quantify internal intermittency of small scale turbulence [32, 40, 41], which can influence inertial particle statistics in turbulent flows [40, 42, 43]. We found the new approach with random d the most favourable in terms of the investigated statistics. It reproduces the Kolmogorov’s − 5/3 scaling of turbulent kinetic energy spectra in the inertial range with the smallest error and without spurious modulations. Moreover, PDFs of increments of the reconstructed velocity have nonGaussian, stretchedexponential tails.
This paper is structured as follows: Section 2 describes the FIT and method for stretching parameter estimation from 1D velocity series. Section 3 discusses briefly the DNS, LES and POST airborne datasets of stratocumulus cloud [36, 39, 44]. Section 4 compares the PDF of local stretching parameter for the three datasets. In Section 5, FIT is applied to 1D filtered DNS velocity field. The PDF of velocity increments of the reconstructed field and PDF of DNS velocity increments are compared. In Section 5 the application of the improved FIT to POST airborne measurement and 3D LES velocity field is also presented.
Fractal Interpolation Techniques
Basics
The fractal interpolation technique is an iterative affine mapping procedure to construct the synthetic (unknown) smallscale eddies of the velocity field u(x,t) from the knowledge of a filtered or coarsegrained field \(\tilde {\mathbf {u}}(\mathbf {x},t)\) [20]. For clarity, if \(\bar {\tilde {\mathbf {u}}}(\mathbf {x},t)\) have resolution \({\Delta }^{\prime }\) while \(\tilde {\mathbf {u}}(\mathbf {x},t)\) have Δ such that \({\Delta }^{\prime } > {\Delta }\), the mapping operator W[.] applied to \(\bar {\tilde {\mathbf {u}}}(\mathbf {x},t)\) gives \(\tilde {\mathbf {u}}(\mathbf {x},t)\) = \(W[\bar {\tilde {\mathbf {u}}}(\mathbf {x},t)]\). To generate synthetic small scale velocity fields, the mapping is performed many times i.e. a fractal synthetic field
For example, if we consider three interpolating points \(\{(x_{i}, \tilde {u_{i}}), i=0,1,2\}\), the fractal interpolation reconstructs a signal w_{j}, j = 1,2 at two additional points placed between points 0 and 1 and points 1 and 2, see Fig. 1. Here, w_{j} has the following transformation structure:
with constraints
The parameters a_{j}, c_{j}, e_{j} and f_{j} can be written in terms of d_{j} (called the stretching parameter) and the interpolation points \(\{(x_{i}, \tilde {u_{i}}), i=0,1,2\}\). Values of d_{j} fix the vertical stretching of the left and right segments at each iteration and determine characteristics of the reconstructed signal. Their values are independent of the interpolation points.
The iterative procedure in the limit \(n\to \infty \) creates a continuous function u_{f}(x) provided that the stretching parameter d_{j} obeys 0 ≤d_{j} < 1. Also, if d_{1} + d_{2} > 1 and \((x_{i},\tilde {u_{i}})\), are not collinear, then the fractal dimension D of the reconstructed signal is the unique real solution of \(d_{1} a_{1}^{D1} + d_{2} a_{2}^{D1} = 1\) (for proof, see [45]). Thus, once the stretching parameter is chosen, the remaining parameters a_{j}, c_{j}, e_{j} and f_{j} are given as
Given that
where N + 1 = N_{A} and N_{A} is the number of anchor points (here, N = 2), the stretching parameter d_{j} relates to the fractal dimension D of a velocity field as:
(for proof, see [19, 45]). It is important to note that the relation between the fractal dimension and the inertialrange selfsimilarity of turbulent velocity field is not trivial. Orey [46] proved that almost all realizations of a Gaussian random field with a powerlaw spectrum, \(E(k) \sim k^{\alpha }\) with 1 < α < 3, give fractal dimension D = (5 − α)/2. For Kolmogorov spectrum, α = 5/3, which results in D = 1.67. Although turbulent velocity fluctuations are not Gaussian, the highReynolds experimental results of Praskovsky et al. [47] and Scotti et al. [48] agree with Orey’s theorem. They concluded that turbulent velocity fluctuations gives a fractal dimension \(D \simeq 1.7 \pm 0.05\), which is close to D = 1.67 expected for Gaussian signals.
If a fractal dimension of 5/3 relating to − 5/3 Kolmogorov scaling in the inertialrange of velocity fields is assumed, then d_{j} = ± 2^{− 1/3} (if d_{j} is assumed to be the same for all grid spacings) [20]. Figure 1 shows the 1D construction of w_{j}. We start from a field with three grid points and we successively apply the map w_{j} with stretching parameter d_{j} = ± 2^{− 1/3}. Shown are the initial field, first, second, fourth and the tenth application of the map w_{j}. The energy spectrum after ten reconstruction steps is shown in Fig. 2.
In 3D, FIT is performed separately along x, y and z directions, see [20]. Let us assume that the filtered field has N grid points in all three directions that is, we have \( \tilde {\mathbf {u}}_{ijk}\) for i,j,k = 0,1,2,...,N − 1. First, 1D intersections of the coarsegrained field \( \tilde {\mathbf {u}}_{ijk}\) along x direction are created and FIT is performed for each 1D intersection (see the blue lines on the schematic Fig. 3), that is, for each j,k = 0,1,...,N − 1. We obtain a field which we denote by \(\tilde {\mathbf {u}}_{ijk}^{x}\), where i = 0,1,...,2N − 1, j,k = 0,1,...,N − 1. Then, FIT for each 1D intersection of \(\tilde {\mathbf {u}}_{ijk}^{x}\) along y direction is performed, which gives a field \(\tilde {\mathbf {u}}_{ijk}^{x,y}\), where i,j = 0,1,...,2N − 1, k = 0,1,...,N − 1 (red lines in Fig. 3). Lastly, FIT is performed on \(\tilde {\mathbf {u}}_{ijk}^{x,y}\) along z direction similar to how it was performed in x and y directions. With this, the field u_{ijk}, where i,j,k = 0,1,...,2N − 1 is finally obtained.
Stretching parameter estimation
Mazel and Hayes [34] proposed an algorithm for computing the local stretching parameter d_{i} of any given arbitrary dataset. The algorithm is based on the property that the resulting fractal field is selfsimilar. For illustration, if we consider a dataset with 5 interpolation points {(x_{i}, u_{i}),i = 0,1,2,3,4}, let μ be the vertical distance between the middle interpolation point (x_{2}, u_{2}) and a straight line between the end points (x_{0}, u_{0}) and (x_{4}, u_{4}), see Fig. 4. The value of μ is positive if the interpolation points are above the straight line and negative otherwise. Let ν_{1} be the vertical distance between (x_{1}, u_{1}) and a straight line between (x_{0}, u_{0}) and (x_{2}, u_{2}) while ν_{2} be the vertical distance between (x_{3}, u_{3}) and a straight line between (x_{2}, u_{2}) and (x_{4}, u_{4}). Both ν_{1} and ν_{2} are positive if their respective interpolation points are above their respective straight lines and negative otherwise. Then the stretching parameters d_{1} and d_{2} are ν_{1}/μ and ν_{2}/μ, respectively. An illustration of this calculation is presented in Fig. 4.
To see how the local values of d vary in space (or time), we applied this algorithm on a 1D intersections of DNS velocity field of STBL (see Section 3.1 for details of this simulation) [33]. Figure 5a shows the local values of d. We see a significant variation of d, with some values outside the interval (− 1,1), due to the intermittent nature of turbulent velocity fields. To verify the applied procedure, we reduced the resolution of the signal from Δ to 2Δ and apply FIT (as described in Section 2.1) once with local values of d as given in Fig. 5a. As expected, the FIT constructed signal is identical with the original one, see Fig. 5b.
As described in Section 2.1, to assure continuity of the reconstructed signal in the limit of \(n \to \infty \) reconstruction steps, d should take values within the interval (− 1,1). In order to satisfy this condition, we further place a constraint d≤ 1 in the calculation of local stretching parameters and neglect d values outside this interval.
Test Cases
The stratocumulus clouds have been a major focus of research in the atmospheric turbulence community because of its huge impact on the Earth’s radiative balance. This type of clouds covers about onequarter of the Earth’s surface. The turbulence is driven by convective instabilities and large shear forcings [38, 49, 50]. Below, we provide a brief description of DNS, POST airborne campaign and LES of stratocumulus cloud top.
DNS of stratocumulustop boundary layer
The DNS aim to mimic the top of the stratocumulus cloud (see Fig. 6). This restricted region was considered uniquely so as to avoid the huge computational cost required to simulate the whole boundary layer. The simulation settings aim to reproduce the cloud dynamics, as measured during the second Dynamics and Chemistry of Marine Stratocumulus (DYCOMSII) field study [51], with Re number being the only parameter which does not match the experimental data, due to extensive numerical cost of the DNS. Details of the simulations can be found in [50].
The cloudtop mixing layer has two infinite horizontal layers of moist air: the lower region, which is cool and saturated and the upper region, which is warm and unsaturated. Cloud turbulence is driven by convective instability caused by longwave radiative cooling of the upper region of the cloud. Other sources of forcings are windshear effects induced by a velocity variation of 3 m s^{− 1} along the cloudtop region and the evaporative cooling induced by the mixing of cloudy and environmental air [52]. The upward radiative flux F_{0} which characterizes radiative cooling and the radiative extinction length L_{0} are estimated from the DYCOMSII field study as, respectively F_{0} = 70W/m^{2} and L_{0} = 15m. With this, the reference buoyancy flux equals B_{0} = F_{0}g/(ρc_{p}T_{0}) = 0.002m^{2}/s^{3}, where g is the gravitational acceleration, ρ is the density of the air, c_{p} denotes the air’s heat capacity and T_{0} is the reference temperature. The Reynolds number based on these scales is Re = U_{0}L_{0}/ν = 800 where the reference velocity scale U_{0} = (B_{0}L_{0})^{1/3} = 0.3 m s^{− 1}. Here the kinematic viscosity ν = 0.0056m^{2}/s is almost 400 times larger than the air’s viscosity. For a later comparison with LES study and measurements, we calculate Reynolds number based on the Taylor microscale \(Re_{\lambda } = u^{\prime } \lambda /\nu \), where \(u^{\prime }\) is the turbulence intensity. The Taylor microscale λ is related to the dissipation of the turbulence kinetic energy ε by the formula \(\lambda = [15 \nu u^{\prime 2}/\varepsilon ]^{0.5}\), see [53]. In the incloud region \(u^{\prime }\approx 0.3 m/s\) and ε ≈ 0.6 ⋅ 10^{− 3}m^{2}/s^{3}, see [50], which gives λ ≈ 3.5m and Re_{λ} ≈ 190.
The horizontal size of the computational domain is 54L_{0} which is discretized with 5120 × 5120 × 2048 points in x (streamwise), y (spanwise) and z (vertical) directions. This ensures the fine resolution of the flow down to the smallest dissipative eddies with a characteristic size \(\eta _{0} \simeq 10\) cm. The simulation is statistically homogeneous in the horizontal x and y directions. The statistics at these horizontal planes depend on the vertical coordinate z and time t. Figure 6 shows the vertical crosssection (at y = 405 m) and horizontal crosssection (at z = 550 m) of the logarithm of enstrophy (i.e. Ω = 1/2ω^{2} where ω is the vorticity field). Turbulence statistics, that is, the mean streamwise velocity component, r.m.s. of horizontal (\(u^{\prime }\), \(v^{\prime }\)) and vertical (\(w^{\prime }\)) velocity fluctuations and the budget of turbulence kinetic energy are presented in Fig. 7.
POST airborne data
To investigate further the variability of the stretching parameter in high Reynolds turbulence, we compute d from the airborne data. We use highresolution in situ measurements of wind velocity fluctuations (time signals) from one of the flight segments in the stratocumulus top boundary layer. This segment was a part of flight 13 of the POST research campaign [37, 39] carried out in the vicinity of Monterey Bay in July and August 2008 (the data are available online via https://www.eol.ucar.edu/projects/post/). The signal’s sampling frequency was 40 Hz (corresponding to \(\sim 1.4\) m spatial resolution) and the duration was T = 438.75 s. The magnitude of the vector difference between the wind and aircraft velocity, averaged over the track vector, was 55ms^{− 1} and the turbulence intensity \(u^{\prime }=0.28 ms^{1}\). Turbulence kinetic energy dissipation rate was estimated from the power spectral density and the secondorder structure function in Ref. [38]. For the flight 13 the average value in the wellmixed cloud top layer was ε = 0.55 ⋅ 10^{− 3}m^{2}/s^{3}. The kinematic viscosity at the flight height was ν = 1.46 ⋅ 10^{− 5}m^{2}/s. With this, the Taylor microscale equals, approximately λ ≈ 0.19m and Re_{λ} ≈ 3900 is around 20 times larger than in the DNS described in Section 3.1.
LES of stratocumulustop boundary layer
The large eddy simulation of stratocumulus top boundary layer for POST flight 13 [36] was performed with a simplified version of 3D nonhydrostatic anelastic EuleriansemiLagrangian (EULAG) model [54] without subgrid scale (SGS) model (also called implicit LES). In implicit LES, the truncation terms of the numerical scheme account for the effect of unresolved turbulence. In previous works [55,56,57], it was shown that implicit LES performs comparable to conventional LES (with SGS model). The code solves the three velocity components (\(\vec {u}\) = (u,v,w)) in x −, y − and z − directions with other atmospheric variables such as potential temperature 𝜃 etc. The velocity components were advected using the Multidimensional Positive Definite Advection Transport Algorithm (MPDATA) [58].
The computational domain is 3 km in x and y directions and 1.2 km in z direction with a grid resolution of 5 m in each directions. The flow is periodic across the horizontal boundaries and impermeable free slip condition is applied at the upper boundary. The lower boundary is impermeable with partial slip conditions imposed through nearsurface momentum fluxes. The initial vertical profile of u and v velocity components were set to the geostrophic wind U_{g} = 5.0 m s^{− 1} and V_{g} = − 7.0 m s^{− 1}, respectively. The time step δt was chosen such that the maximum value of the Courant number throughout the simulation period and computational domain is \(\sim 0.5\). For details, readers are referred to Pedersen et al. [36]. Figure 8 shows u velocity component along the horizontal and vertical crosssections of the flow field. The cloud top region is placed at the vertical height \(z \sim 700\) m. Above this region is the free troposphere, which is warm and unsaturated with low turbulence intensity while the region below the cloud top is moist, saturated with high turbulence intensity. Vertical profile of mean velocity field and r.m.s. of horizontal and vertical velocity fluctuations are shown in Fig. 9.
Probability Distribution Function of the Stretching Parameter
In this section, we analyse available data of velocity field from numerical simulations and experiment described in Section 3. We extract stretching parameters, as discussed in Section 2.2 and calculate the PDFs of the module of d, denoted as f(d). Here, d stands for the sample space variable of the stretching parameter d.
DNS of stratocumulustop mixing layer
For the DNS data described in Section 3.1, we use horizontal profiles of the u, v and w components of velocity at a height corresponding to the incloud region (z = 550m). First, we investigate the variability of d as 1D intersections of DNS velocity field are filtered successively to wavenumber in the inertial range. Starting with the fully resolved DNS field with the grid spacing equal to η_{0}, we reduce the resolution to 2η_{0} by using a lowpass filter and calculate the local values of d for the filtered velocity field. Then, the velocity signal is filtered to a grid resolution of 4η_{0} etc., the cutoff wavenumber is decreased, until the resolution matches the inertialrange (at about 16η_{0} to 128η_{0}). After each filtering, the local values of d are extracted. The lowpass filter used is the finite impulse response (FIR) filter of order 30 designed using the Hamming window method [59]. This is done with decimate function in MATLA\(\text {B}^{{\circledR }}\) software. We use the decimation lowpass filter because it is constructed to downsample the signal (i.e reduce the number of grid points) and guard against aliasing. The downsampling feature of the filter was important to have a reconstructed signal with the same number of grid points as the original (DNS, LES or POST airborne) signal. We calculate the local estimate of d with the algorithm explained in Section 2.2. Stretching parameter values outside the interval (− 1,1) were neglected and absolute value of d was used to calculate its PDF. Since the flow is statistically homogeneous over horizontal planes similar results were obtained for 1D intersections of velocity field calculated either in x or y directions.
Figure 10 presents the PDF of d and average fractal dimension of DNS velocity signals at different grid resolutions. In Fig. 10a, the PDFs change significantly for the first four successive filtering steps but seem to be selfsimilar when filtered successively to inertialrange wavenumbers (at steps 4 to 7). All three velocity components give similar profiles of the PDF of the stretching parameter. The average fractal dimension, calculated according to Eq. 8, decreases to 5/3 inertial range scaling, as seen in Fig. 10b. The calculated vertical profile of averaged d across the STBL (see Fig. 11b) is smaller than 0.6 or 0.7 reported in Ref. [31] even in the core cloud region (at z ≈ 550 m). We expect this result follows from the presence of external intermittency (laminar regions will give zero or close to zero local values of d). Even in the incloud region the volume fraction occupied by turbulent flow is smaller than one and equals approximately γ = 0.9 − 0.95, where γ is the intermittency parameter, see [60]. Due to small values of γ in the outercloud regions, the average value of d also decreases therein to, approximately 0.25, see Fig. 11b.
In Ref. [20], it was shown that a fractal signal will only dissipate energy in the limit of small viscosity if d > 0.5. So as to retain dissipative properties, we neglect d < 0.5 in further subgrid scale reconstruction in Section 5.1. If only d > 0.5 is considered, the average stretching parameter will be similar to values obtained in Ref. [31].
No selfsimilarity of PDFs of d is observed in the vertical z direction, see Fig. 11a. Here, velocity field was filtered to a grid resolution of 16η_{0} or 32η_{0}. This is due to the anisotropy of turbulence caused by large scale shear production and buoyancy. Similar conclusion was made by Marchioli et al. [2, 3] where the locally computed stetching parameters varied significantly in the wallnormal direction. Also, in that study, the average stretching parameter was lower than that obtained experimentally from homogeneous isotropic turbulence.
We observe no significant variation in the horizontally calculated average of d in the incloud region  at 450m ≤ z ≤ 550m (see Fig. 11b). Hence, the remaining analysis is based on the horizontal profiles in this region, where turbulence is close to isotropic. We also observe that probabilities of having positive or negative stretching parameter are equal.
Comparison with LES and POST data
To investigate the variability of the stretching parameter in high Reynolds turbulence, we compute d from LES field and from the POST airborne dataset using the Mazel and Hayes’ algorithm described in Section 2.2. We compare PDF of d from LES and POST data with the respective profile from the DNS. DNS velocity field was filtered with the decimate function [59] (described in Section 4.1) to a spatial resolution of 16η_{0} (equivalent to 1.6 m). For LES, we use the horizontal profile of u velocity component at z = 595m (corresponding to the incloud region with the maximum turbulent intensity) to estimate stretching parameters. Numerical scheme of LES introduces a spurious damping of the energy of the smallest resolved scales [24]. In order to remove these effects, we filtered the LES velocity field with the decimate function (described in Section 4.1), to a grid resolution of 20 m (corresponds to inertialrange wavenumber) from its 5 m grid resolution. Another possibility, instead of filtering, would be to recover the energy of the smallest scales using the approximate deconvolution method  ADM (see [24] for details). Combining the ADM and the FIT is, however, left for further work and we will not address this issue here. We did not filter the POST airborne data since its sampling frequency is within the inertialrange. All the three u, v and w components of POST velocity dataset were combined to form a large dataset since their individual PDF of d were similar.
As seen in Fig. 12 comparison between LES and DNS profiles is very good, in spite of different Reynolds numbers of the two simulations. The PDF of d from POST data seems to oscillate around the respective DNS profile. Some differences observed may be due to the effect of large scale tendencies on measured airborne data and small scale measurement noise. Based on the results shown in Fig. 12, we can conclude the PDF of the stretching parameter in the inertial range is a universal function, independent of the Reynolds number.
Results
Fractal interpolation of filtered DNS
First, we filter 1D horizontal intersections of DNS velocity field at z = 550 m to a spatial resolution of 16η_{0} ≈ 1.6m (which is within the inertial range) using the decimation function described in Section 4.1. We next reconstruct subfilter scales back to the resolution η_{0} using FIT. For this, we select the stretching parameter from its PDF calculated from the DNS data, see Fig. 12, using the inverse transform sampling method [61]. The procedure is briefly indicated as following

1.
Calculate cumulative distribution function F(d) from the PDF f(d]), as
$$ F(\mathrm{d}) = {\int}_{0}^{d} f(s) ds. $$ 
2.
Calculate the inverse function F^{− 1}(y).

3.
If y is a random number from uniform distribution [0,1] then d = F^{− 1}(y) is a random number from the investigated PDF.
Apart from the mathematical constraint d≤ 1, which assures the continuity of the reconstructed signal at \(n \to \infty \) reconstruction steps [45], the second constraint, discussed in Section 3.1 was related to the dissipative properties of the signal [48] and reads d > 0.5. Hence, in practice, in the reconstruction process, we only retained the values d that were larger than 0.5. If selected d≤ 0.5, the procedure was repeated and another random value was chosen, till the condition 0.5 < d≤ 1 was satisfied. Next, the sign of d was selected randomly, such that the positive and negative d have equal probabilities. Under such assumptions, the ensemble average 〈d〉 is comparable to values reported in Ref. [31] and the scaling of the reconstructed energy spectra is close to the theoretical k^{− 5/3}. We checked that using only one constraint, d≤ 1, led to underprediction of the reconstructed spectra. Hence, the choice 0.5 < d≤ 1 seems the most favourable as it is supported by the theoretical constraints [19, 48] and gives satisfactory results.
The inertialrange scale invariance is the property that directly relates to the idea of fractality of velocity field. As seen in Fig. 10a, profiles of PDF f(d) collapse into one curve only for cutoff wavenumbers from the inertial range. For this reason, the fractal reconstruction of the dissipative part of the spectrum is not justified, as no selfsimilarity is observed there. Instead, in the FIT reconstruction process, the inertial range was extended down to η_{0} and properties of such an artificial velocity field were investigated. The lack of the dissipative range could be a possible drawback if Lagrangian particles are tracked in the reconstructed field. This would concern especially the motion of smallinertia particles which are correlated with smaller eddies [14]. However, in practice, FIT in numerical simulations of highRe flows will be restricted to a few reconstruction steps, due to increased computational time. With this, reconstruction of the signal down to η_{0} will not be possible and the smallest resolved scales will belong to the inertial range.
Figure 13 shows the longitudinal energy spectra E_{11}(k) (for u velocity component along x direction) of DNS, filtered and FITreconstructed velocity field for the three investigated versions of the model: with d = ± 2^{1/3} as originally proposed by Scotti & Meneveau [20], with d_{1} = − 0.887, d_{2} = − 0.676 as proposed by Basu et al. [30], and with the new proposal with random d. As it is observed, in case of constant values of d, energy spectra exhibit periodic modulations (see also figures 7 and 8 of [20]). Basu et al. [30] avoid this modulation by applying the discrete Haar wavelet transform. FIT energy spectrum with constant values of d = ± 2^{1/3} is much lower in some wavenumbers than the − 5/3 inertialrange scaling (especially at wavenumbers close to the cutoff scale), although the upper envelope qualitatively follows the − 5/3 inertialrange scaling. The FIT energy spectrum reconstructed with constant values of d = − 0.887, − 0.676 has similar properties as the Scotti & Meneveau approach, except that the range between the upper and lower envelope of the spectrum is somewhat smaller. The FIT energy spectrum reconstructed with random values of d follows the inertialrange scaling closer and shows no periodic modulation.
Next, we investigated statistics of velocity increments at two different points u(x + r,t) −u(x,t). The sample space of velocity increment will be denoted by δu and the distance between points by r = r. The tails of the PDFs of velocity increments in the isotropic turbulence f(δu,r,t), can be approximated by the stretched exponentials, where the stretching exponent varies monotonically from 0.5 for r in the dissipation range to 2 (which is a Gaussian distribution) for r in the integral scale range, [62]. The nonGaussianity of the PDFs for small r indicates the presence of the internal intermittency, that is, the probability of extreme events (large velocity differences) at these scales is much higher than predicted by a Gaussian distribution.
Figure 14 presents the PDFs of velocity increment (δu) for DNS, filtered DNS and FIT velocity signals with constant and random values of d for r = 128η_{0} and r = 64η_{0}. To increase the size of dataset used to calculate PDFs we considered both longitudinal (x) and transverse (y) 1D intersections of the velocity field at z = 550 m and, additionally, averaged results over all three velocity components.
All three FIT approaches presented in Fig. 14 compare well with the corresponding DNS profiles. PDFs at r = 128η_{0} are clearly Gaussian, see [32, 41]. Differences are observed when the distance r decreases to r = 8η_{0}, 4η_{0} and 2η_{0}, see Figs. 15, 16 and 17, respectively. The PDFs of velocity increments are far from Gaussian and slightly skewed. The FIT model with random d provides the best FIT with the DNS at smaller r.
We note here that r = 4η_{0} and 2η_{0} correspond to dissipativerange scales. The intermittency at short r seems to be correctly reproduced, especially by the approach with random d, although in the reconstruction process, the dissipative range is not reproduced in any FIT model. We report here this result which seems interesting, although somewhat unanticipated at first sight.
Fractal interpolation of POST airborne data
In order to test the performance of our subgrid model on high Reynolds number airborne data, we use u component of velocity field from flight 13 in POST airborne research campaign [37, 39]. The velocity signal was filtered with the decimation function (described in Section 4.1) from its frequency of 40Hz to 10Hz (corresponding to the spatial resolution of 5.6 m ). The filtering was done to eliminate measurements errors at large frequencies. We use this filtered signal as our reference velocity dataset to validate the performance of the FIT model. This dataset has a larger inertial range compared to DNS data analyzed earlier (see Fig. 18a). The reference signal is filtered to a frequency of 2.5Hz (about 22.4 m spatial resolution) and FIT with the random values of d from PDF (calculated from the reference POST signal) is applied to reconstruct the subfilter part of the dataset. It is important to note that FIT can be carried out in time by replacing the spatial variable x in Eqs. 1 to 6 with time. The frequency spectra for the reference POST, filtered POST and FIT velocity dataset are shown in Fig. 18a. Once again, we observe the approach with random d follows the inertialrange scaling closer than the other two approaches.
Differences in the observed spectra are quantified as follows. Under the local isotropy assumption and within the validity of the Taylor’s hypothesis, the energy spectra in the inertial range can be converted to the frequency spectra, see e.g. [63]
where the constant C_{1} ≈ 0.49, U = 55ms^{− 1} is the true air speed of the aircraft and ε is the turbulence kinetic energy dissipation rate. The value of ε can be estimated from the linearleastsquares fit procedure applied in a certian range of frequencies. In the present case, we used f = 0.1Hz − 2.0Hz and calculated ε = 4.1306 × 10^{− 4} m^{2}/s^{3}. Deviations of the spectra reconstructed with FIT from the inertialrange scaling were quantified by δ_{S} defined as
where S(f) is the FIT frequency spectrum and f_{cut} = 2.5Hz is the cutoff frequency, i.e. the frequency at which the fractal reconstruction is initiated. We obtained the following values of δ_{S}
 δ_{S} = 0.033:

for FIT procedure with d = ± 2^{1/3}
 δ_{S} = 0.049:

for FIT procedure with d = − 0.887,− 0.676
 δ_{S} = 0.026:

for FIT procedure with random d
The FIT model with random values of d reproduces the inertial range scaling with the smallest error. Next, we compare the PDF of velocity increments for POST reference data, filtered POST and FIT data with random values of d in Fig. 18b. The velocity increments were calculated as u(t + τ) − u(t) with τ = 0.1 s. The PDF of velocity increments for POST reference data and FIT data agree reasonably well.
3D fractal interpolation of LES
Next, fractal interpolation of three components of velocity in 3D using the random values of d (from the PDF shown in Fig. 12) is performed. Each velocity component was filtered with the decimate function (described in Section 4.1) to a grid resolution of 20 m from its initial 5 m grid resolution to remove the smallest scales spuriously damped due to numerical diffusion. Then, we perform a 3D fractal reconstruction of inertialrange scales to a grid resolution of 5 m (i.e. two reconstruction steps), as described in Section 2.1) without considering any correlations which might exist between the directions.
Figure 19 presents contour plots of the longitudinal u velocity component at z = 595 m for filtered LES and FITreconstructed field. The addition of inertialrange subgrid structures in FIT velocity is clearly visible.
Energy spectra of this field are presented in Fig. 20 and compared with the two other FIT approaches with constant d. The deviations from − 5/3 scaling were calculated from a formula analogous to Eq. 10
where E_{th} is described by formula
With the fitting range k = 0.019m^{− 1} − 0.044m^{− 1}, the value ε = 5.0299 × 10^{− 5} m^{2}/s^{3} was estimated. The obtained results read
 δ_{E} = 0.40:

for FIT procedure with d = ± 2^{1/3},
 δ_{E} = 0.52:

for FIT procedure with d = − 0.887,− 0.676,
 δ_{E} = 0.28:

for FIT procedure with random d.
Here again, FIT with random values of d shows the best agreement with the − 5/3 inertialrange scaling.
Figure 21 presents the PDFs of velocity increments for u, v and w components, separately, calculated at the distance r = 5m at horizontal plane z = 595m. We compare LES (without filtering) with the FITreconstructed field. As it is seen, LES results are closer to Gaussian, while PDFs calculated from the FITreconstructed field have the exponential tails. This could be explained by the fact that the smallest resolved scales of LES are spuriously damped due to numerical diffusion, hence, the inertialrange intermittency is not reproduced by LES velocity field.
As reported by [20], a limitation of the fractal model is that the subgrid scale velocity field is not divergence free due to the loss of the correlation between directions. To investigate this issue, the error in mass conservation at different reconstruction steps is calculated. For this, the following formula is used
where n,m,p is the number of grid points in x, y and z directions, respectively. The divergence of velocity was calculated with the central difference scheme. Additionally, the error in mass conservation is characterized by the quantity \(max(\nabla \cdot \vec {u})  min(\nabla \cdot \vec {u})\). LES field was filtered to k_{cut} = 0.150, 0.075 or 0.037m^{− 1}, which corresponds to the grid resolution of approximately 10, 20 and 40m, respectively. Next, the signal was reconstructed back to the resolution 5m, which means that 1 FIT reconstruction step was applied to the signal with k_{cut} = 0.150m^{− 1}, 2 steps to the signal with k_{cut} = 0.075m^{− 1} and 3 steps to the signal with 0.037m^{− 1}. Results are presented in Table 1.
In theory, δ_{∇u} should be zero but due to numerical errors, the divergence of LES velocity fields are usually very small (not zero). We first note that, as seen in Table 1, there is a difference in δ_{∇u} of two orders of magnitude between LES and the filtered LES without FIT reconstruction. In spite of this, filtered velocity fields are commonly used to do a priori tests to estimate e.g. an effect of SGS models on particle statistics [14, 29, 64]. Each FIT reconstruction step increases the error in mass conservation, however, after two steps δ_{∇u} is still of the same order of magnitude as the error of filtered LES without reconstruction, at least for the FIT method with random d and for FIT with constant d = − 0.887,− 0.676.
The difference between the maximum and minimum of the divergence of the velocity fields is seen to be one order of magnitude larger if all FIT models are compared with filtered LES. Of all FIT approaches, the new proposal with random values of d gives a smallest value of δ_{∇u} and \(max(\nabla \cdot \vec {u})  min(\nabla \cdot \vec {u})\). Based on this result, we suggest that one or two iteration steps of FIT to LES velocity field should not be exceeded to keep the error in mass conservation at an acceptable level.
Next, we investigated computational cost of the FIT procedure applied to three velocity components in 3D space. We perform one to four reconstruction steps. As presented in Fig. 22, the computational cost (CPU time) of one, two or even three FIT reconstruction steps is small compared to the time needed for LES to resolve large scale features. This shows that the newly proposed fractal model is not only able to reproduce inertialrange eddies but is also computationally efficient.
Conclusions
In this work, a fractal model for the reconstruction of subgrid scales in large eddy simulation is presented. For the reconstruction process, values of the stretching parameter d should be specified. This parameter determines the characteristics of the reconstructed signal which can be derived from its fractal dimension [20]. In previous works, the stretching parameter was chosen to be constant in time and space. To account for its spatial variability, we estimate the PDF of the absolute value of the stretching parameter d from DNS data of stratocumulustop boundary layer, LES data in the same flow configuration and, additionally, measurement data from POST campaign. For this, 1D intersections of velocity field are first lowpass filtered to certain cutoff wavenmbers. Next, an algorithm proposed by [34] is used to compute the local values of d. It was found that if the cutoff wavenumbers were in the inertial range, the PDFs of the stretching parameter collapsed into one curve, independent of the Reynolds number.
Next, 1D intersections of filtered velocity field were reconstructed such that d is a random variable with the prescribed, previously determined PDF. Performance of the new approach was compared with FITs with constant values of d. It was shown the energy spectra follow the − 5/3 scaling more closely and have no spurious modulations if d is random. Moreover, the nonGaussian, stretchedexponential tails of PDFs of velocity increments are reproduced corretly by the improved model. We investigated these statistics as they quantify internal intermittency of small scale turbulence [32, 41].
The random stretching parameter is also used to construct unresolved scales of POST airborne data (flight 13) and 3D LES of stratocumulustop boundary layer. Statistics of velocity increments showed that the improved fractal subgrid model is capable to reproduce some of the subgrid scale features, which might be lost due to finite grid resolution and numerical effects in LES or finite sampling frequency of measurements.
In the case of LES filed, the fractal reconstruction was extended to three dimensions. The divergencefree condition, which can be violated when using fractal interpolation technique, was addressed. We observe that after two reconstruction steps, the error in mass conservation of the reconstructed field is of the same order of magnitude as the error of filtered LES without reconstruction. Moreover, the computational cost required by FIT was compared with the cost of LES without reconstruction. We conclude that, even with three reconstruction steps, CPU time of LES with FIT is of the same order of magnitude as the CPU time of LES.
A perspective for a further study is to use this fractal model to reconstruct subgrid scales of a carrier fluid flow and study the motion of Lagrangian particles in atmospheric physics applications, see e.g. [65,66,67].
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Acknowledgements
This work received funding from the European Union Horizon 2020 Research and Innovation Programme under the Marie SklodowskaCurie Actions, Grant Agreement No. 675675.
MW and SPM acknowledge matching fund from the Polish Ministry of Science and Higher Education No. 341832/PnH/2016. The authors acknowledge Z. Wacławczyk for helping with Fig. 3.
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Akinlabi, E.O., Wacławczyk, M., Malinowski, S.P. et al. Fractal Reconstruction of SubGrid Scales for Large Eddy Simulation. Flow Turbulence Combust 103, 293–322 (2019). https://doi.org/10.1007/s10494019000302
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Keywords
 Fractal interpolation technique
 Turbulence
 Large Eddy simulation
 Subgrid scale model
 Lagrangian particles