# Stochastic Wake Modelling Based on POD Analysis

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## Abstract

**:**

## 1. Introduction

## 2. LES Simulations

**PA**rallelized

**L**ES

**M**odel PALM [57,58], which has been extensively used for the simulation of the atmospheric boundary layer for the last 15 years. PALM solves the non-hydrostatic, incompressible Navier-Stokes Equations under the Boussinesq Approximation using central differences on a uniformly spaced Cartesian staggered grid. For the time integration a third-order Runge-Kutta scheme and for the advection terms a fifth-order Wicker-Skamarock scheme is used. Subgrid-scale turbulence is filtered implicitly and is parametrized by a modified Smagorinsky approach following Deardorff [59].

## 3. Methods

#### 3.1. Preprocessing

#### 3.2. POD

#### 3.3. Temporal Stochastic Modelling

#### 3.3.1. Uncorrelated Model

#### 3.3.2. OU-Based Model

#### 3.3.3. Spectral Mode

#### 3.3.4. Comments on More Complex Models

#### 3.4. A Spectral Surrogate in Three Dimensions

#### 3.5. Aeroelastic Simulations and Model Verification

## 4. Truncated PODs

#### 4.1. POD Modes and Eigenvalues

#### 4.2. Performance of the Truncated PODs

#### 4.2.1. Results

#### 4.2.2. Discussion

## 5. Stochastic Wake Models

#### 5.1. Modeling the Weighting Coefficients ${a}_{j}(t)$

#### 5.2. Performance of the Stochastic Wake Models

#### 5.2.1. Results

#### 5.2.2. Discussion

## 6. A Stochastic Wake Model with Added Turbulence

#### 6.1. Basic Idea

#### 6.2. Performance

#### 6.2.1. Results

#### 6.2.2. Discussion

## 7. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A. Details of the Spectral Model

#### Appendix A.1. Fitting Procedure

#### Appendix A.2. Estimated Parameters

## Appendix B. Time Series of Loads

**Figure A2.**Sections of the time series of the different loads for the truncated POD (red) with $N=6$ compared to spectral model (blue), OU-based model (blue), uncorrelated model (magenta).

**Figure A3.**Sections of the time series of the different loads for original LES and for the spectral model with added turbulence including $N=6$ modes.

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**Figure 1.**Statistics of the stream-wise velocity $u(y,z,t)$: (

**a**) Mean field $({\mathrm{ms}}^{-1})$ far upstream of the turbine; (

**b**) Mean field $({\mathrm{ms}}^{-1})$, $3.5$ D away from the turbine; (

**c**) Variance $({\mathrm{m}}^{2}{\mathrm{s}}^{-2})$ $3.5$ D away from the turbine. The large black circle marks the rotor area of a turbine in the wake flow, which is modeled later using aeroelastic simulations. The small black circle marks the central region of the wake, which is used to build a spectral surrogate in Section 6.

**Figure 2.**Different profiles at $y=0$ m corresponding to the figures in Figure 1: (

**a**) Mean field far upstream of the turbine; (

**b**) Mean field $3.5$ D away from the turbine; (

**c**) Variance $3.5$ D away from the turbine.

**Figure 3.**Snapshots of the LES showing the the stream-wise velocity u $(\mathrm{m}{\mathrm{s}}^{-1})$ in a yz-plane $3.5$ D away from the turbine. (

**a**) $t=990$ s; (

**b**) $t=1185$ s; (

**c**) $t=1230$ s. The large black circle marks the rotor area of a turbine in the wake flow, which is modeled later using aeroelastic simulations. The small black circle marks the central region of the wake, which is used to build a spectral surrogate in Section 6.

**Figure 4.**Illustration of our modeling approach and its verification: In the left part of the figure, the different steps leading to reduced wake descriptions are presented. These different descriptions are all used as inflows to aeroelastic simulations. All resulting loads are then compared in order to give insight into the strength and weaknesses of the simplified wake flows. The mathematical notation used in the figure will be explained in the rest of this section.

**Figure 5.**Preprocessing of the velocity field u $(\mathrm{m}{\mathrm{s}}^{-1})$ : (

**a**) Instant Snapshot $t=31.8$ s; (

**b**) Velocity deficit; (

**c**) Extracted deficit.

**Figure 6.**(

**a**) POD eigenvalues; (

**b**) Normalised cumulative spectrum of the POD representing the percentage of captured turbulent kinetic energy; (

**c**) Integral time scale of the weighting coefficients versus mode number.

**Figure 8.**Local SKE ${\langle {u}^{\prime}{(y,z)}^{2}\rangle}_{t}$ $({\mathrm{m}}^{2}{\mathrm{s}}^{-2})$ for original LES and truncated PODs including different numbers of modes N.

**Figure 9.**Time series of the different loads for original LES and a truncated POD including $N=6$ modes. Here and in the following figures TB yaw moment is used as an abbreviation for tower base yaw moment.

**Figure 10.**PSDs of the different loads for original LES and truncated PODs including different numbers of modes N.

**Figure 11.**(

**a**) Variance and (

**b**) damage equivalent loads (DELs) versus the number of modes included in the truncated POD. Both are normalised by the values of the original LES.

**Figure 12.**Rainflow counting histograms (RFCs) of different load time series for original LES and truncated PODs including different numbers of modes. To get an impression of the estimation error, the standard error shown for the original LES is estimated by $\frac{\sqrt{{n}_{i}}}{2}$ where ${n}_{i}$ is the number of half-cycles in bin i.

**Figure 13.**(

**a**) PSDs of the weighting coefficients ${a}_{j}(t)$ for $j=1,2,\dots ,50$; (

**b**) PSD for ${a}_{2}(t)$ and the corresponding fit for the spectral model. Additionally, analytical PSDs for uncorrelated and OU-based model are shown corresponding to the estimated model parameters of these models.

**Figure 14.**Time series of ${a}_{2}(t)$ (black) and the different stochastic models ${\tilde{a}}_{2}(t)$. From top to bottom: spectral model (blue), OU-based model (green) and uncorrelated model (magenta).

**Figure 15.**Properties of the weighting coefficients ${a}_{j}(t)$ and the stochastic models ${\tilde{a}}_{j}(t)$: (

**a**) Variance versus mode number j; (

**b**) Integral time scales versus mode number. The results for the uncorrelated model are not shown since the integral time scale is zero for all j; (

**c**) Estimated PSDs of ${a}_{2}(t)$ and realisations of the different stochastic models ${\tilde{a}}_{2}(t)$.

**Figure 16.**Local SKE ${\langle {u}^{\prime}{(y,z)}^{2}\rangle}_{t}$ $({\mathrm{m}}^{2}{\mathrm{s}}^{-2})$ using $N=6$ POD modesfor truncated POD and the different stochastic wake models: (

**a**) truncated POD; (

**b**) spectral model; (

**c**) OU-based model; (

**d**) uncorrelated model.

**Figure 17.**PSDs of the different loads for original LES and the different wake descriptions using $N=6$ POD modes. Note that we aim to capture the behaviour of truncated PODs here, as pointed out in the beginning of this section.

**Figure 18.**RFCs of the different loads for the original LES and the different wake descriptions using $N=6$ POD modes. To get an impression of the estimation error, the standard error is shown for the original LES. It is estimated by $\frac{\sqrt{{n}_{i}}}{2}$ where ${n}_{i}$ is the number of half-cycles in bin i. Note that we aim to capture the behaviour of truncated PODs here, as pointed out in the beginning of this section.

**Figure 19.**(

**a**) Variance and (

**b**) DELs for different loads versus no. of modes N included in the wake descriptions. The solid lines with filled circles represent the results for the spectral model while the dashed lines show the results for the truncated POD. The curves for the different loads are vertically shifted for a better visualisation, as described in the legend. The corresponding shifted values representing $1.0$ are illustrated by the dotted lines. Standard errors are shown for the spectral model for $N=2,4,6,10,30$. They are estimated through the standard deviation from an ensemble of 10 realisations.

**Figure 21.**Snapshot of the spectral model with added small-scale turbulence (Figure 20) in the wake and ambient ABL turbulence outside the wake structure.

**Figure 22.**Local SKE ${\langle {u}^{\prime}{(y,z)}^{2}\rangle}_{t}$ $({\mathrm{m}}^{2}{\mathrm{s}}^{-2})$ for original LES and the spectral model with added turbulence including different numbers of modes N.

**Figure 23.**PSDs of the different loads for original LES and spectral model with added turbulence including different numbers of POD modes.

**Figure 24.**RFCs for different loads and different numbers of modes included in the spectral model with added turbulence.

**Figure 25.**Variance and DELs for different loads versus no. of modes included in the spectral model with added turbulence. Standard errors are shown for the spectral model for $N=2,4,6,10,30$. They are estimated from the standard deviations from an ensemble of 10 realisations.

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## Share and Cite

**MDPI and ACS Style**

Bastine, D.; Vollmer, L.; Wächter, M.; Peinke, J. Stochastic Wake Modelling Based on POD Analysis. *Energies* **2018**, *11*, 612.
https://doi.org/10.3390/en11030612

**AMA Style**

Bastine D, Vollmer L, Wächter M, Peinke J. Stochastic Wake Modelling Based on POD Analysis. *Energies*. 2018; 11(3):612.
https://doi.org/10.3390/en11030612

**Chicago/Turabian Style**

Bastine, David, Lukas Vollmer, Matthias Wächter, and Joachim Peinke. 2018. "Stochastic Wake Modelling Based on POD Analysis" *Energies* 11, no. 3: 612.
https://doi.org/10.3390/en11030612