iccsa-21-wind

commit 5307a7e758c7dbafa4d0e8e3fde41b5857091529
parent aa45203011b608cfc43c042ee366023631245988
Author: Ivan Gankevich <i.gankevich@spbu.ru>
Date:   Tue,  4 May 2021 12:25:51 +0300

Final corrections.

- Spell-check.
- Replace anemometer picture.
- Re-read the whole text.

Diffstat:
Makefile
 | 
1
+
gnuplot/turbulence.gnuplot
 | 
6
+++---
inkscape/anemometer.svg
 | 
2064
+++++++++++++++++++++++++++++++++++--------------------------------------------
main.bib
 | 
21
+++++++++++++++------
main.tex
 | 
204
+++++++++++++++++++++++++++++++++++++++++++++++--------------------------------

5 files changed, 1050 insertions(+), 1246 deletions(-)
diff --git a/Makefile b/Makefile
@@ -14,6 +14,7 @@ all: build/Gavrikov-Wind.pdf
 all: build/iccsa-21-wind-slides.pdf
 
 build/main.pdf: main.tex
+build/main.pdf: main.bib
 build/main.pdf: build/inkscape/anemometer.eps
 build/main.pdf: build/daily-stats.eps
 build/main.pdf: build/daily-rmse.eps
diff --git a/gnuplot/turbulence.gnuplot b/gnuplot/turbulence.gnuplot
@@ -25,16 +25,16 @@ set xlabel 'Velocity X, m/s'
 set ylabel 'Turbulence coefficient X'
 plot \
 '< sort -nk1 build/turbulence' using (abs($1)):4 every n with points pt 6 lc '#404040' notitle,\
-'' using (abs($1)):4 smooth bezier with lines lc '#c04040' notitle
+'' using (abs($1)):4 smooth bezier with lines lc '#c04040' lw 2 notitle
 
 set xlabel 'Velocity Y, m/s'
 set ylabel 'Turbulence coefficient Y'
 plot \
 '< sort -nk2 build/turbulence' using (abs($2)):5 every n with points pt 6 lc '#404040' notitle,\
-'' using (abs($2)):5 smooth bezier with lines lc '#c04040' notitle
+'' using (abs($2)):5 smooth bezier with lines lc '#c04040' lw 2 notitle
 
 set xlabel 'Velocity Z, m/s'
 set ylabel 'Turbulence coefficient Z'
 plot \
 '< sort -nk3 build/turbulence' using (abs($3)):6 every n with points pt 6 lc '#404040' notitle,\
-'' using (abs($3)):6 smooth bezier with lines lc '#c04040' notitle
+'' using (abs($3)):6 smooth bezier with lines lc '#c04040' lw 2 notitle
diff --git a/inkscape/anemometer.svg b/inkscape/anemometer.svg
@@ -12,9 +12,9 @@
    xmlns:inkscape="http://www.inkscape.org/namespaces/inkscape"
    version="1.1"
    id="svg941"
-   width="245.59766"
-   height="227.07179"
-   viewBox="0 0 245.59765 227.07179"
+   width="237.8241"
+   height="236.6918"
+   viewBox="0 0 237.82409 236.6918"
    sodipodi:docname="anemometer.svg"
    inkscape:export-filename="/home/igankevich/workspace/iccsa-21-wind/figures/anemometer-annotated.png"
    inkscape:export-xdpi="600"
@@ -47,9 +47,9 @@
      inkscape:window-height="1058"
      id="namedview943"
      showgrid="false"
-     inkscape:zoom="2.9238922"
-     inkscape:cx="81.665547"
-     inkscape:cy="134.36176"
+     inkscape:zoom="1.033752"
+     inkscape:cx="-99.676214"
+     inkscape:cy="144.2514"
      inkscape:window-x="0"
      inkscape:window-y="22"
      inkscape:window-maximized="0"
@@ -58,1190 +58,946 @@
      fit-margin-left="0"
      fit-margin-right="0"
      fit-margin-bottom="0" />
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diff --git a/main.bib b/main.bib
@@ -9,7 +9,6 @@
   year          = {2008},
   issn          = {0196-8904},
   doi           = {10.1016/j.enconman.2007.10.017},
-  url           = {https://www.sciencedirect.com/science/article/pii/S0196890407003767},
   author        = {José A. Carta and Celia Bueno and Penélope Ramírez},
   keywords      = {Mixture distributions, Wind direction, Circular normal
                   distributions, von Mises distributions},
@@ -26,7 +25,6 @@
   year          = {2000},
   issn          = {1352-2310},
   doi           = {10.1016/S1352-2310(99)00414-8},
-  url           = {https://www.sciencedirect.com/science/article/pii/S1352231099004148},
   author        = {John Z Yim and Chun-Ren Chou and Wei-Po Huang},
   keywords      = {Turbulent fluctuations, Probability distribution, Gaussian
                   distribution, Gram–Charlier series expansions},
@@ -45,7 +43,6 @@
   number        = {7},
   doi           = {10.1175/1520-0450(1976)015<0673:NAOPOF>2.0.CO;2},
   pages         = {673--678},
-  url           = {https://journals.ametsoc.org/view/journals/apme/15/7/1520-0450_1976_015_0673_naopof_2_0_co_2.xml},
   comment       = {Wind speed distribution (Weibull).}
 }
 
@@ -74,7 +71,6 @@
   year          = {2015},
   number        = {4},
   pages         = {3075--3092},
-  url           = {https://www.mdpi.com/1996-1073/8/4/3075},
   issn          = {1996-1073},
   doi           = {10.3390/en8043075}
 }
@@ -89,7 +85,6 @@
   year          = {2008},
   issn          = {0196-8904},
   doi           = {10.1016/j.enconman.2008.01.010},
-  url           = {https://www.sciencedirect.com/science/article/pii/S0196890408000228},
   author        = {José A. Carta and Penélope Ramírez and Celia Bueno}
 }
 
@@ -126,7 +121,6 @@
   year          = {2020},
   issn          = {0378-7788},
   doi           = {10.1016/j.enbuild.2020.109805},
-  url           = {https://www.sciencedirect.com/science/article/pii/S0378778819331044},
   author        = {Edward Arens and Ali Ghahramani and Richard Przybyla and
                   Michael Andersen and Syung Min and Therese Peffer and Paul
                   Raftery and Megan Zhu and Vy Luu and Hui Zhang}
@@ -173,3 +167,18 @@
   isbn          = {978-3-030-58817-5},
   doi           = {10.1007/978-3-030-58817-5\_1}
 }
+
+@InProceedings{   gavrikov2020wind,
+  author        = {Gavrikov, Anton and Degtyarev, Alexander and Egorov, Denis
+                  and Gankevich, Ivan and Grigorev, Artemii and Khramushin,
+                  Vasily and Petriakov, Ivan},
+  title         = {Virtual Testbed: Simulation of Air Flow Around Ship Hull and
+                  Its Effect on Ship Motions},
+  booktitle     = {Computational Science and Its Applications -- ICCSA 2020},
+  year          = {2020},
+  publisher     = {Springer International Publishing},
+  address       = {Cham},
+  pages         = {18--28},
+  isbn          = {978-3-030-58817-5},
+  doi           = {10.1007/978-3-030-58817-5\_2},
+}
diff --git a/main.tex b/main.tex
@@ -54,7 +54,7 @@ Wind simulation in the context of ship motion simulation is the topic where
 multiple mathematical models are possible, and the choice of the model depends
 on the purpose of the model. In the course of the research where we apply
 autoregressive model to ocean wave simulation, we decided to investigate
-whether the same model can be used to simulat wind flow round ship hull.
+whether the same model can be used to simulate wind flow round ship hull.
 
 Wind simulation is studied at different scales and the closest scale for a ship
 in the ocean is wind turbine. One of the model that is used to
@@ -84,36 +84,36 @@ coefficients are different~\cite{veers1984modeling}.
 %\end{align*}
 The spectrum describes wind velocity vector in the plane that is perpendicular
 the mean wind direction vector and travels in the same direction
-with mean wind speed. Time series is generated as Fourier series coefficients
-of which are determined from the spectrum and phases are random variables:
+with mean wind speed. Time series is generated as Fourier series, coefficients
+of which are determined from the spectrum, and phases are random variables:
 \begin{align*}
 & V(t) = \overline{V} + \sum\limits_{j=1}^{n}
 	\left( A_j \sin\omega_jt + B_j \cos\omega_jt\right), \\
 & A_j = \sqrt{\frac{1}{2} S_j \Delta\omega} \sin\phi_j, \\
-& B_j = \sqrt{\frac{1}{2} S_j \Delta\omega} \cos\phi_j. \\
+& B_j = \sqrt{\frac{1}{2} S_j \Delta\omega} \cos\phi_j.
 \end{align*}
 Here \(S_j\) is spectrum value at frequency \(\omega_j\),
 \(\phi_j\) is random variable which is uniformly distributed in \([0,2\pi]\).
 The result is one-dimensional vector-valued time series, each element of
 which is velocity vector at a specified point in time and space.
 
-In order to simulate wind velocity vector in multiple points in space,
+In order to simulate wind velocity vector at multiple points in space,
 the authors use the function of coherency~--- the amount of correlation between
 wind speed at two points in space. This function has a form of an exponent and
 depends on frequency~\cite{veers1988wind}:
 \begin{equation*}
 \text{Coh}_{jk}(f) = \exp\left( -\frac {C\Delta r_{jk} f} {U(z)} \right),
 \end{equation*}
-where \(\Delta{}r_{jk}\) is the distance betweein \(i\) and \(j\) points and
+where \(\Delta{}r_{jk}\) is the distance between \(i\) and \(j\) points and
 \(C\) is coherency decrement. Time series for each wind velocity vector
-component are generated independenty and after that their spectra are modified
+component are generated independently and after that their spectra are modified
 in accordance with coherence function (the formulae are not presented here).
 
 In order to simulate wind with autoregressive model it is easier to use
 autocovariance function instead of spectra and coherence function. We can obtain
 autocovariance function from spectra as inverse Fourier transform using
 Wiener---Khinchin theorem. The formula that we obtained this way using
-computer algebra programmes is to complex, but can be approximated by
+various computer algebra programmes is too complex, but can be approximated by
 a decaying exponent:
 \begin{equation*}
 \gamma(t) = \sigma^2 \exp\left( -0.1 \frac{c_2^{3/5}}{c_1} t\right),
@@ -121,15 +121,15 @@ a decaying exponent:
 where \(\sigma^2\) is process variance (area under the spectrum).
 
 Unfortunately, this autocovariance function is one-dimensional and there is
-easy way of obtaining three-dimensional analogue from the spectrums.  In order
+easy way of obtaining three-dimensional analogue from the spectra.  In order
 to solve this problem we looked into datasets of wind speed measurements
 available for the research. However, most of them contain either one or two
-wind velocity vector components (wind speed and direction), they are difficult
-to get and their discretization is very coarse (we found only one paper that
+wind velocity vector components (in a form of wind speed and direction), they are difficult
+to get and their resolution is very small (we found only one paper that
 deals with three components~\cite{yim2000}). For our purposes we need
-descretization of at least one second to simulate gusts and some form of
+resolution of at least one sample per second to simulate gusts and some form of
 turbulence. To summarise, our requirements for the dataset is to provide all
-three wind velocity vector components and has discretization of at least one
+three wind velocity vector components and has resolution of at least one sample per
 second.
 
 We failed to find such datasets and continued our research into anemometers
@@ -139,7 +139,7 @@ anemometer~\cite{cosgrove2007ultra,arens2020anemometer,yakunin20173d}.  As
 commercial anemometers are too expensive for this research, we built our own
 version from the generally available electric components. However, this version
 failed to capture any meaningful data, and incidentally we decided to build an
-anemometer from load cells and strain gauges. This anemometer is
+anemometer from load cells and strain gauges (which were originally intended for different research work). This anemometer is
 straightforward to construct, the electrical components are inexpensive, and it
 is easy to protect them from bad weather. This anemometer is able to record
 all three wind velocity vector components multiple times per second.
@@ -147,29 +147,29 @@ all three wind velocity vector components multiple times per second.
 In this paper we describe how load cell anemometer is built, then we collect
 dataset of all three wind speed components with one-second resolution, verify
 this anemometer using commercial analogue, verify measurements from the dataset
-using well-established distribution for wind speed and direction, and finally
+using well-established distributions for wind speed and direction, and
 estimate autocovariance functions for autoregressive model from our dataset.
-Finally, we present preliminary wind simulation results using this model.
+Finally, we present preliminary wind simulation results using autoregressive model.
 
 \section{Methods}
 
 \subsection{Three-axis wind velocity measurements with load cell anemometer}
 
 In order to generate wind velocity field using four-dimensional (one temporal
-and three spatial dimensions) auoregressive model, we need to use
+and three spatial dimensions) autoregressive model, we need to use
 four-dimensional wind velocity autocovariance function. Using Wiener---Khinchin
 theorem it is easy to compute the function from the spectrum. Unfortunately,
 most of the existing wind velocity historical data contains only wind velocity
 magnitude and direction. We can use them to reconstruct \(x\) and \(y\)
 spectrum, but there is no way to get spectrum for \(z\) coordinate from this
-data. Also, data discretisation in historical data is too coarse for our
-purposes (wind simulation for ship motions). To solve these problems, we decided
+data. Also, resolution of historical data is too small for
+wind simulation for ship motions. To solve these problems, we decided
 to build our own three-axis anemometer that measures wind velocity for all
-three axes at one point in space.
+three axes at one point in space multiple times per second.
 
 To measure wind velocity we used resistive foil strain gauges mounted on the
 arms aligned perpendicular to the axes directions. Inside each arm we placed
-aluminum load cell with two strain gauges: one on the bending side and another
+aluminium load cell with two strain gauges: one on the bending side and another
 one on the lateral side. Load cells use Wheatstone bridge to measure the
 resistance of the gauges and are connected to the circuit that measures the
 resistance and transmits it to the microcontroller in digital format.
@@ -180,18 +180,30 @@ figure~\ref{fig-anemometer}.
 
 \begin{figure}
     \centering
-    \includegraphics{build/inkscape/anemometer.eps}
+    \begin{minipage}{0.55\textwidth}
+        \includegraphics{build/inkscape/anemometer.eps}
+    \end{minipage}
+    \begin{minipage}{0.35\textwidth}
+        Anemometer properties:\phantom{12345}
+        \begin{tabular}{ll}
+            \toprule
+            Load cell capacity & 1 kg \\
+            Load cell amplifier & HX711 \\
+            Microcontroller & ATmega328P \\
+            \bottomrule
+        \end{tabular}
+    \end{minipage}
     %\includegraphics{build/gnuplot/anemometer.eps}
     \caption{Three-axis anemometer. Arms are inserted into the Housing
     and fixed using bolts that go through the circular holes.\label{fig-anemometer}}
 \end{figure}
 
-Each load cell face the direction that is perpendicular to the directions of
-other load cells. When the wind blows in the direction of the load cell, only
+Each load cell faces the direction that is perpendicular to the directions of
+other load cells. When the wind blows in the direction of particular load cell, only
 this cell bends. When the wind blows in an arbitrary direction which is not
 perpendicular to any load cell faces, then all load cells bend, but the
 pressure force is smaller. Pressures of all load cells are recorded
-simultaneously, and we can use Bernoulli's equaion to compute wind velocity
+simultaneously, and we can use Bernoulli's equation to compute wind velocity
 from them.
 
 Bernoulli's equation is written as
@@ -221,13 +233,13 @@ For all load cells we have a system of three equations
 Hence \(\upsilon_{x}=\alpha_{x}\sqrt{F_{x}}\),
 \(\upsilon_{y}=\alpha_{y}\sqrt{F_{y}}\),
 \(\upsilon_{z}=\alpha_{z}\sqrt{F_{z}}\) where \(\alpha_{x,y,z}\) are constants of
-proportionality. Therefore, to obtain wind velocity we take square root of the 
+proportionality. Therefore, to obtain wind velocity we \emph{take square root} of the 
 value measured by the load cell and multiply it by some coefficient determined
 empirically during anemometer calibration.
 
 \subsection{Per-axis probability distribution function for wind velocity}
 
-Scalar wind velocity is described by Weibull disttribution~\cite{justus1976}.
+Scalar wind velocity is described by Weibull distribution~\cite{justus1976}.
 Weibull probability distribution function is written as
 \begin{equation}
     f\left(\upsilon;b,c\right) = b c \left(b \upsilon\right)^{c-1}
@@ -238,8 +250,8 @@ is shape parameter. This function is defined for positive wind velocity, since
 scalar wind velocity is a length of wind velocity vector
 \(\upsilon=\sqrt{\smash[b]{\upsilon_x^2+\upsilon_y^2+\upsilon_z^2}}\). However, projection of
 wind velocity vector on \(x\), \(y\) or \(z\) axis may be negative.
-Our solution to this problem is to use two Weibull distributions: one both positive and
-one for negative projections --- but with different parameters.
+Our solution to this problem is to use two Weibull distributions: one for positive and
+one for negative projection --- but with different parameters.
 \begin{equation}
     f\left(\upsilon_x;b_1,c_1,b_2,c_2\right) = \begin{cases}
         b_1 c_1 \left(b_1 \upsilon_x\right)^{c_1-1} \exp\left(-\left(b_1\upsilon_x\right)^{c_1}\right).
@@ -258,7 +270,7 @@ vector on \(x\) axis. The same formula is used for \(y\) and \(z\) axis.
 %because the maximum value may not be located at \(\upsilon=0\). Our solution to
 %this problem is to make substitution \(\upsilon_x'=\left|\upsilon_x-\Mode\upsilon_x\right|\),
 %where \(\Mode\upsilon_x\) is the mode of wind velocity projection to the corresponding axis,
-%and modulus makes the graph simmetric with respect to \(\upsilon_x'=0\).
+%and modulus makes the graph symmetric with respect to \(\upsilon_x'=0\).
 %The substitution yields
 %\begin{equation}
 %    f\left(\upsilon_x';b,c\right) = b c
@@ -283,7 +295,7 @@ vector on \(x\) axis. The same formula is used for \(y\) and \(z\) axis.
 
 \subsection{Three-dimensional ACF of wind velocity}
 
-Usually, autocovariance is modeled using exponential functions~\cite{box1976time}.
+Usually, autocovariance is modelled using exponential functions~\cite{box1976time}.
 In this paper we use one-dimensional autocovariance function written as
 \begin{equation}
     K\left(t\right) = a_3 \exp\left(-\left(b_3 t\right)^{c_3} \right).
@@ -329,7 +341,7 @@ placed the fan behind and in front of each arm of our anemometer and measured
 values that the corresponding load cell reported for each side of the arm with
 and without the fan. Then we were able to calculate two coefficients for each
 axis: one for negative and another one for positive wind velocities. The
-coefficient equals the sensor value that is equivalent to the wind speed of 1
+coefficient equals the raw sensor value that is equivalent to the wind speed of 1
 m/s (table~\ref{tab-coefficients}).
 
 \begin{table}
@@ -352,10 +364,10 @@ We noticed that ambient temperature affects values reported by our load cells:
 when the load cell heats up (cools down), it reports values that increase
 (decrease) linearly in time due to thermal expansion of the material. We
 removed this linear trend from the measured values using linear regression. The
-code in R~\cite{r-language} that transforms sensor values into wind speed is
+code in R~\cite{r-language} that transforms raw sensor values into wind speed is
 presented in listing~\ref{lst-sample-to-speed}.
 
-\begin{lstlisting}[label={lst-sample-to-speed},caption={The code that transforms raw load cell sensor values into wind speed projections to the corresponding axis.}]
+\begin{lstlisting}[label={lst-sample-to-speed},caption={The code that transforms raw load cell raw sensor values into wind speed projections to the corresponding axis.}]
 sampleToSpeed <- function(x, c1, c2) {
  t <- c(1:length(x))
  reg <- lm(x~t)
@@ -371,14 +383,16 @@ Over a period of one month we collected 3.1M samples and filtered out 12\% of
 them as having too large unnatural values. We attributed these values to
 measurement errors as they spread uniformly across all the time span and are
 surrounded by the values of regular magnitude.
-
-From the remaning data we choose days with wind speeds above the average as
-reported by EMERCOM of Russia\footnote{https://en.mchs.gov.ru/} for Saint
-Petersburg. After that we divided each day into two-hour intervals over which
-we collected the statistics individually. From these intervals we choose the
-ones with data distributions close to unimodal, so we could easily fit them
-into the model.  The statistics for each interval is presented in
-figure~\ref{fig-intervals}.
+%From the remaining data we choose days with wind speeds above the average as
+%reported by EMERCOM of Russia\footnote{https://en.mchs.gov.ru/} for Saint
+%Petersburg.
+After that we divided each day into two-hour intervals over which
+we collected the statistics individually.
+%From these intervals we choose the
+%ones with data distributions close to unimodal, so we could easily fit them
+%into the model.
+The statistics for each interval is presented in
+figure~\ref{fig-intervals}, dataset properties are presented in table~\ref{tab-dataset}.
 
 Unique feature of three-axis anemometer is that it measures both velocity of
 incident air flow towards the arms and the turbulent flow that forms behind the
@@ -395,8 +409,8 @@ for the purpose of the research.
         \(\upsilon_x\), \(\upsilon_y\) and \(\upsilon_z\) are mean speeds for
         each interval for the corresponding axes, \(\upsilon\) is mean scalar wind speed
         for each interval. The horizontal line shows overall average speed.
-        Yellow rectangles denote days when EMERCOM of Russia reported wind
-        speeds above average.
+        Yellow rectangles denote days when EMERCOM of Russia (https://en.mchs.gov.ru/)
+        reported wind speeds above average.
         \label{fig-intervals}}
 \end{figure}
 
@@ -414,20 +428,6 @@ for the purpose of the research.
     \caption{Dataset properties.\label{tab-dataset}}
 \end{table}
 
-\begin{table}
-    \centering
-    \begin{tabular}{ll}
-        \toprule
-        Load cell capacity & 1 kg \\
-        Load cell amplifier & HX711 \\
-        Microcontroller & ATmega328P \\
-        \bottomrule
-    \end{tabular}
-    \caption{Anemometer properties.\label{tab-anemometer}}
-\end{table}
-
-
-
 \section{Results}
 
 \subsection{Anemometer verification}
@@ -471,9 +471,11 @@ distributions for each axis are presented in figure~\ref{fig-velocity-distributi
 Wind direction was approximated by von Mises distribution using least-squares
 fitting. Following the common
 practice~\cite{feng2015sectors} we divided direction axis
-into sectors: from -180\textdegree to -90\textdegree, from from -90\textdegree to
-0\textdegree, from 0 to 90\textdegree and from 90\textdegree to 180\textdegree --- and fitted each sector independently. We chose four sector to have one sector for each side of the anemometer.
-The best-fit and worst-fit distributions for each axis
+into sectors: from -180\textdegree{} to -90\textdegree{}, from from -90\textdegree{} to
+0\textdegree{}, from 0 to 90\textdegree{} and from 90\textdegree{} to
+180\textdegree{} --- and fitted each sector independently. We chose four sector
+to have one sector for each side of the anemometer.
+The best-fit and worst-fit distributions for each sector
 are presented in figure~\ref{fig-direction-distributions}.
 
 \begin{figure}
@@ -495,7 +497,7 @@ are presented in figure~\ref{fig-direction-distributions}.
         Red line shows estimated
         probability density of positive direction angles, blue line shows estimated
         probability density of negative direction angles and circles denote observed
-        probability density of direction angles.
+        probability density of direction angles. 0\textdegree{} is north.
     \label{fig-direction-distributions}}
 \end{figure}
 
@@ -507,7 +509,7 @@ only. The data from the two anemometers was recorded synchronously. From the
 data we selected intervals with wind gusts and compared the measurements of the
 two anemometers. To compare them we scaled load cell anemometer measurements to
 minimise RMSE of the difference between the two time series to compensate for
-the errors in calibration coefficients. The results showed that there is good
+the errors in calibration coefficients. The results showed that there is some
 correspondence between the measurements of the two anemometers
 (figure~\ref{fig-hold-peak}).
 
@@ -544,18 +546,44 @@ are presented in figure~\ref{fig-acf}.
     \label{fig-acf}}
 \end{figure}
 
+\subsection{Turbulence coefficient}
+
+Since our anemometer measures both incident and turbulent flow we took an
+opportunity to study the speed of the turbulent flow in relation to incident
+flow. We calculated the absolute mean values of positive and negative wind
+velocity projections for each time interval of the dataset. We considered the
+flow with the larger absolute mean value incident, and the other one is
+turbulent.  Then we calculated turbulence coefficient as the ratio of absolute
+mean speed of turbulent flow to the absolute mean speed of incident flow. We
+found that the average ratio is close to 60\%. It seems, that the ratio
+decreases as the wind speed increases for wind speeds of 1--5 m/s, but for
+higher wind speeds we do not have the data. Turbulence coefficient
+can be used in wind simulation models to control the magnitude of turbulent
+flow that forms behind the obstacle~\cite{gavrikov2020wind}.
+
+\begin{figure}
+    \centering
+    \includegraphics{build/gnuplot/turbulence.eps}
+    \caption{The ratio of absolute mean speed of turbulent flow to the absolute mean
+    speed of incident flow for each axis.\label{fig-turbulence}}
+\end{figure}
+
 \subsection{Wind simulation using measured ACFs}
 
 We simulated three-dimensional wind velocity using autoregressive model
-implemented in Virtual Testbed~\cite{bogdanov2020vtestbed} and using the data
-obtained with three-axis anemometer on March 28, 2021, 01:00-03:00 UTC. We
-approximated four-dimensional autocovariance function using~\eqref{eq-acf} by
-setting the corresponding parameters from one-dimensional autocovariance
-functions estimated from the data obtained with anemometer, all other
-parameters were set close to nought. Non-nought parameters are listed in
-table~\ref{tab-arma-parameters}.  We found that velocity and direction
-distributions and ACFs of each axis of simulated wind and real wind are close
-to each other (see~figure~\ref{fig-vtestbed}).
+implemented in Virtual Testbed~\cite{bogdanov2020vtestbed}~--- a
+programme for workstations that simulates ship motions in extreme conditions
+and physical phenomena that causes them (ocean waves, wind, compartment
+flooding etc.).  Using the data obtained with three-axis anemometer on March 28,
+2021, 01:00-03:00 UTC we approximated four-dimensional autocovariance function
+using~\eqref{eq-acf} by setting the corresponding parameters from
+one-dimensional autocovariance functions estimated from the data obtained with
+anemometer, all other parameters were set close to nought. Non-nought
+parameters are listed in table~\ref{tab-arma-parameters}.  We found that
+velocity and direction distributions and ACFs of each axis of simulated wind
+and real wind are similar in shape, but are too far away from each other
+(see~figure~\ref{fig-vtestbed}).  We consider these results preliminary and
+will investigate further in future work.
 
 \begin{table}
     \centering
@@ -563,7 +591,7 @@ to each other (see~figure~\ref{fig-vtestbed}).
     \label{tab-arma-parameters}}
     \begin{tabular}{lllll}
         \toprule
-        Axis & ACF \(a\) & \phantom{0}ACF \(b\) & \phantom{0}ACF \(c\) & \phantom{0}Mean velocity \\
+        Axis & ACF \(a\) & \phantom{0}ACF \(b\) & \phantom{0}ACF \(c\) & \phantom{0}Mean velocity, m/s \\
         \midrule
         \(x\) & 1.793 & \phantom{0}0.0214 & \phantom{0}0.2603 & \phantom{0}-2.439 \\
         \(y\) & 1.423 & \phantom{0}0.01429 & \phantom{0}0.2852 & \phantom{0}-2.158 \\
@@ -597,26 +625,36 @@ RMSE of wind direction distribution approximation has negative correlation with
 wind speed: the larger the wind speed, the smaller the error and vice versa.
 This is in agreement with physical laws: the faster the flow is the more
 determinate its mean direction becomes, and the slower the flow is the more
-undeterminate its mean direction is.
-
-One disadvantage of three-axis anemometer is that the arm for the \(z\) axis is
-horizontal, and snow and rain put additional load on this cell distorting the
-measurements. Also, thermal expansion and contraction of the material changes
-the resistance of load cells and distorts the measurements. Both these
-deficiences can be compensated in software by removing linear trend from the
-corresponding interval. 
+indeterminate its mean direction is.
+
+Three-axis anemometer disadvantages are the following.
+\begin{itemize}
+\item 
+The arm for the \(z\) axis is horizontal, and snow and rain put additional load
+on this cell distorting the measurements. 
+\item
+Also, thermal expansion and contraction of the material changes
+the resistance of load cells and distorts the measurements. 
+\item Pressure force on the arm is exerted by individual air particles and
+    is represented by choppy time series, as opposed to real physical signal
+    that is represented by smooth graph.
+\end{itemize}
+The first two defficiences can be compensated in software by removing linear trend
+from the corresponding interval. The last one make anemometer useful only for
+offline studies, i.e.~it is useful to gather statistics, but is unable to measure
+immediate wind speed and direction.
 
 \section{Conclusion}
 
 In this paper we proposed three-axis anemometer that measures wind speed for
 each axis independently. We analysed the data collected by this anemometer and
 verified that per-axis wind speeds fit into Weibull distribution with the
-largest RMSE of 6.7\% and wind directions fit into von Mises distibution with
+largest RMSE of 6.7\% and wind directions fit into von Mises distribution with
 the largest RMSE of 11\%. We estimated autocovariance functions for wind speed
 for each axis of the anemometer and used this approximations to simulate wind
 flow in Virtual Testbed. The parameters of these functions allow to control
 both wind speed and mean direction. The future work is to construct new
-anemometer that is able to measure spatial autocovariances using the proposed
+anemometer that is able to measure spatial autocovariance using the proposed
 anemometer as the base.
 
 \subsubsection*{Acknowledgements.}