Author Contributions
Conceptualization, D.D. and V.Y.; methodology, V.U.; software, D.D.; validation, D.D.; formal analysis, V.U.; investigation, D.D.; resources, D.D.; data curation, D.D.; writing—original draft preparation, D.D.; writing—review and editing, V.U. and V.Y.; visualization, D.D.; supervision, V.U. All authors have read and agreed to the published version of the manuscript.
Figure 1.
(
a) Sketch of the “small” open wind tunnel used at the University of West Bohemia in Pilsen. The test section is 400 mm long, it has square cross-section of the side of 125 mm. (
b) The localization of the measured airfoil of chord is 80 mm within the test section; its maximum thickness is 14.5 mm, thus the blockage ratio is 11.6%. The area studied by PIV is just behind the trailing edge with dimensions of 33 × 33 mm, i.e., 0.4 × 0.4 chord length. The PIV plane is 45 mm from the bottom wall. (
c) Example of few instantaneous velocity fields. The grayscale corresponds to stream-wise velocity component,
, which means the reference velocity measured in empty wind tunnel, for presented data, is equal to 11.5 m/s. Only every fourth velocity vector is shown. The entire ensemble of 1208 snapshots is animated in
http://home.zcu.cz/~dudad/Ct300a00ab.gif (40 MB, accessed on 29 July 2021).
Figure 1.
(
a) Sketch of the “small” open wind tunnel used at the University of West Bohemia in Pilsen. The test section is 400 mm long, it has square cross-section of the side of 125 mm. (
b) The localization of the measured airfoil of chord is 80 mm within the test section; its maximum thickness is 14.5 mm, thus the blockage ratio is 11.6%. The area studied by PIV is just behind the trailing edge with dimensions of 33 × 33 mm, i.e., 0.4 × 0.4 chord length. The PIV plane is 45 mm from the bottom wall. (
c) Example of few instantaneous velocity fields. The grayscale corresponds to stream-wise velocity component,
, which means the reference velocity measured in empty wind tunnel, for presented data, is equal to 11.5 m/s. Only every fourth velocity vector is shown. The entire ensemble of 1208 snapshots is animated in
http://home.zcu.cz/~dudad/Ct300a00ab.gif (40 MB, accessed on 29 July 2021).
Figure 2.
Ensemble average stream-wise velocity. Panel (a) shows the map of average u (grayscale), the white isotach separates the area of back flow, the last black isotach separates the area of average velocity larger than reference velocity measured in empty wind tunnel. Panel (b) contains the velocity profiles at several distances from the trailing edge denoted as dashed lines in panel (a) distinguished via color: we apologize to readers with grayscale printer.
Figure 2.
Ensemble average stream-wise velocity. Panel (a) shows the map of average u (grayscale), the white isotach separates the area of back flow, the last black isotach separates the area of average velocity larger than reference velocity measured in empty wind tunnel. Panel (b) contains the velocity profiles at several distances from the trailing edge denoted as dashed lines in panel (a) distinguished via color: we apologize to readers with grayscale printer.
Figure 3.
(a) Wake width calculated according to the formulas for boundary layer thickness. The dash-dotted lines represent the calculation with constant , while the dashed line plays for usage of , which is the background of Gaussian fit and it develops with distance x. Dependence of these two velocities on x is displayed in panel (b).
Figure 3.
(a) Wake width calculated according to the formulas for boundary layer thickness. The dash-dotted lines represent the calculation with constant , while the dashed line plays for usage of , which is the background of Gaussian fit and it develops with distance x. Dependence of these two velocities on x is displayed in panel (b).
Figure 4.
Panel (
a) shows data at one of velocity profiles (
Figure 2b) with corresponding Gaussian fit. There is shown the constant value of
Uref, which is velocity measured in empty wind tunnel, and
U0, which is the background of Gaussian fit. (
b) The comparison of mentioned methods based on threshold (first four) or on the Gaussian fit (last two).
Figure 4.
Panel (
a) shows data at one of velocity profiles (
Figure 2b) with corresponding Gaussian fit. There is shown the constant value of
Uref, which is velocity measured in empty wind tunnel, and
U0, which is the background of Gaussian fit. (
b) The comparison of mentioned methods based on threshold (first four) or on the Gaussian fit (last two).
Figure 5.
Instantaneous stream-wise velocity. Panel (a) shows the map of u (grayscale) normalized by reference velocity. The white line represents the centers of Gaussian fits in transverse profiles, the pair of black lines are the centers plus/minus the σ-parameter of the fit. Panel (b) shows a single particular Gaussian fit at the stream-wise distance denoted as dashed line in panel (a).
Figure 5.
Instantaneous stream-wise velocity. Panel (a) shows the map of u (grayscale) normalized by reference velocity. The white line represents the centers of Gaussian fits in transverse profiles, the pair of black lines are the centers plus/minus the σ-parameter of the fit. Panel (b) shows a single particular Gaussian fit at the stream-wise distance denoted as dashed line in panel (a).
Figure 6.
Panel (
a) compares the wake width calculated as average of the instantaneous wake widths (
Figure 5) with the wake width of ensemble-averaged stream-wise velocity by using the Gauss fit (black short-dashed line) and by using the threshold method (gray dashed line). The error bars represent the standard deviation of the instantaneous wake widths in each position. Panel (
b) shows the position of wake centerline calculated on instantaneous (black solid line) or ensemble-averaged (gray dashed line) velocity field. Error bars are the standard deviation of the instantaneous wake positions.
Figure 6.
Panel (
a) compares the wake width calculated as average of the instantaneous wake widths (
Figure 5) with the wake width of ensemble-averaged stream-wise velocity by using the Gauss fit (black short-dashed line) and by using the threshold method (gray dashed line). The error bars represent the standard deviation of the instantaneous wake widths in each position. Panel (
b) shows the position of wake centerline calculated on instantaneous (black solid line) or ensemble-averaged (gray dashed line) velocity field. Error bars are the standard deviation of the instantaneous wake positions.
Figure 7.
Standard deviation of stream-wise velocity. Panel (a) shows the map of σ(u) (grayscale) normalized by reference velocity. Panel (b) contains the standard deviation profiles at several distances from the trailing edge denoted as dashed lines in panel (a); we apologize to readers with grayscale printer.
Figure 7.
Standard deviation of stream-wise velocity. Panel (a) shows the map of σ(u) (grayscale) normalized by reference velocity. Panel (b) contains the standard deviation profiles at several distances from the trailing edge denoted as dashed lines in panel (a); we apologize to readers with grayscale printer.
Figure 8.
Standard deviation of stream-wise velocity. Panel (a) shows the map of σ(u) (grayscale) normalized by reference velocity. Panel (b) contains the standard deviation profiles at several distances from the trailing edge denoted as dashed lines in panel (a).
Figure 8.
Standard deviation of stream-wise velocity. Panel (a) shows the map of σ(u) (grayscale) normalized by reference velocity. Panel (b) contains the standard deviation profiles at several distances from the trailing edge denoted as dashed lines in panel (a).
Figure 9.
(a) Spatial distribution of stream-wise velocity flatness; the thick isoline refers to value of 3, the thin black isoline to 6, the white isoline to values greater than 12. (b) The instantaneous velocity field with value most different from the average in the point denoted by rectangle and a pair of arrows in bottom left corner. Note that this snapshot is a regular case, it is not noisier than the others, however there is some fluid of lower velocity ejected from the wake into the surroundings hitting the denoted point. (c) Spatial distribution of flatness of the ensemble without the snapshot (b). Note that removal of single case of 1208 significantly decreases the flatness signal in the denoted point.
Figure 9.
(a) Spatial distribution of stream-wise velocity flatness; the thick isoline refers to value of 3, the thin black isoline to 6, the white isoline to values greater than 12. (b) The instantaneous velocity field with value most different from the average in the point denoted by rectangle and a pair of arrows in bottom left corner. Note that this snapshot is a regular case, it is not noisier than the others, however there is some fluid of lower velocity ejected from the wake into the surroundings hitting the denoted point. (c) Spatial distribution of flatness of the ensemble without the snapshot (b). Note that removal of single case of 1208 significantly decreases the flatness signal in the denoted point.
Figure 10.
(a) Profiles of flatness coefficient of the stream-wise velocity component. (b) Profiles of the skewness of stream-wise velocity component at depicted distance past the trailing edge distinguished via colors: we apologize to readers with grayscale printer.
Figure 10.
(a) Profiles of flatness coefficient of the stream-wise velocity component. (b) Profiles of the skewness of stream-wise velocity component at depicted distance past the trailing edge distinguished via colors: we apologize to readers with grayscale printer.
Figure 11.
(a) Spatial map of the skewness of stream-wise velocity component. (b) Probability density function (PDF) of the instantaneous stream-wise velocities within the four denoted rectangles, solid line in the wake central region, dashed line in the inner shear layer (SL), dash-dotted line in the outer shear layer and the short, dashed line in the surrounding region. The u values are normalized by the local average and standard deviation. The inset shows the PDF as the function of velocity and it shows different widths and positions of the distributions (note the logarithmic scale in inset).
Figure 11.
(a) Spatial map of the skewness of stream-wise velocity component. (b) Probability density function (PDF) of the instantaneous stream-wise velocities within the four denoted rectangles, solid line in the wake central region, dashed line in the inner shear layer (SL), dash-dotted line in the outer shear layer and the short, dashed line in the surrounding region. The u values are normalized by the local average and standard deviation. The inset shows the PDF as the function of velocity and it shows different widths and positions of the distributions (note the logarithmic scale in inset).
Figure 12.
Comparison of the discussed definitions of wake width past the airfoil NACA 64-618 at zero angle of attack. Horizontal axis is the distance past trailing edge; all lengths are normalized by the airfoil chord. Golden circles: the distance of points, where average velocity deficit crosses half (threshold θ = 0.5) the maximum velocity deficit. Maroon diamonds: twice the thickness σ of the Gauss function fitted to the average velocity profile. Orange diamonds: the average of wake widths fitted to each snapshot. Cyan Xs: the distance of double-Gauss peaks fitted to the profile of standard deviation of stream-wise velocity. Green crosses: distance of mentioned peaks plus the thicknesses of those peaks in the profile of standard deviation of stream-wise velocity. Violet triangles: distance of double-Gauss peaks fitted to the profile of stream-wise velocity skewness (third statistical moment). Blue rectangles: distance of points, where the stream-wise velocity flatness reaches the threshold θ = 6, i.e., 2 × 3, the Gauss reference value.
Figure 12.
Comparison of the discussed definitions of wake width past the airfoil NACA 64-618 at zero angle of attack. Horizontal axis is the distance past trailing edge; all lengths are normalized by the airfoil chord. Golden circles: the distance of points, where average velocity deficit crosses half (threshold θ = 0.5) the maximum velocity deficit. Maroon diamonds: twice the thickness σ of the Gauss function fitted to the average velocity profile. Orange diamonds: the average of wake widths fitted to each snapshot. Cyan Xs: the distance of double-Gauss peaks fitted to the profile of standard deviation of stream-wise velocity. Green crosses: distance of mentioned peaks plus the thicknesses of those peaks in the profile of standard deviation of stream-wise velocity. Violet triangles: distance of double-Gauss peaks fitted to the profile of stream-wise velocity skewness (third statistical moment). Blue rectangles: distance of points, where the stream-wise velocity flatness reaches the threshold θ = 6, i.e., 2 × 3, the Gauss reference value.
Figure 13.
Wake widths past airfoil at different Reynolds numbers (“k” denotes “·103”). Panel (a) shows the wake width calculated as the σ-parameter of Gaussian fit of the ensemble-averaged stream-wise velocity. Panel (b) shows the wake widths calculated by using standard deviation of stream-wise velocity fitted by double-Gauss function, the peak distance plus the width of bottom σB and top σT peak. Note the Re explored up to here is 6.1·104, it is not plotted here.
Figure 13.
Wake widths past airfoil at different Reynolds numbers (“k” denotes “·103”). Panel (a) shows the wake width calculated as the σ-parameter of Gaussian fit of the ensemble-averaged stream-wise velocity. Panel (b) shows the wake widths calculated by using standard deviation of stream-wise velocity fitted by double-Gauss function, the peak distance plus the width of bottom σB and top σT peak. Note the Re explored up to here is 6.1·104, it is not plotted here.
Figure 14.
(a) Comparison of possible wake width definitions at distance x/c = 0.4 past the airfoil NACA 64-618 trailing edge as a function of chord-based Reynolds number. Empty maroon diamonds denote the Gaussian fitting of average velocity profile, filled orange diamond the Gauss fitting of instantaneous velocity fields, the green crosses represent wake width based on peaks of stream-wise velocity standard deviation, cyan Xs represent the fluctuation peak distance only, and the blue squares represent the threshold of stream-wise velocity flatness, which fails for laminar flow, therefore, it is not plotted there. (b), Standard deviation of the instantaneous wake width calculated by using Gaussian fit (orange filled diamonds) and the standard deviation of the wake centerline position (violet empty diamonds), which can be interpreted as wake meandering.
Figure 14.
(a) Comparison of possible wake width definitions at distance x/c = 0.4 past the airfoil NACA 64-618 trailing edge as a function of chord-based Reynolds number. Empty maroon diamonds denote the Gaussian fitting of average velocity profile, filled orange diamond the Gauss fitting of instantaneous velocity fields, the green crosses represent wake width based on peaks of stream-wise velocity standard deviation, cyan Xs represent the fluctuation peak distance only, and the blue squares represent the threshold of stream-wise velocity flatness, which fails for laminar flow, therefore, it is not plotted there. (b), Standard deviation of the instantaneous wake width calculated by using Gaussian fit (orange filled diamonds) and the standard deviation of the wake centerline position (violet empty diamonds), which can be interpreted as wake meandering.
Figure 15.
(
a) Virtual origin
x0 of the wake width, if its stream-wise development is approximated by linear function (8). The uncertainty of flatness-based wake width is so high that it is not plotted. (
b) the grow rate
a of the wake as a function of Reynolds number. The symbols are the same as in
Figure 14.
Figure 15.
(
a) Virtual origin
x0 of the wake width, if its stream-wise development is approximated by linear function (8). The uncertainty of flatness-based wake width is so high that it is not plotted. (
b) the grow rate
a of the wake as a function of Reynolds number. The symbols are the same as in
Figure 14.
Figure 16.
Wake widths at constant distance x/c = 0.4 as a function of number of snapshots used for averaging.
Figure 16.
Wake widths at constant distance x/c = 0.4 as a function of number of snapshots used for averaging.
Table 1.
The possible mathematical methods of extracting parameters of some data profile.
Table 1.
The possible mathematical methods of extracting parameters of some data profile.
Method | Advantages | Disadvantages |
---|
Threshold | - -
Simple and robust. - -
Easy to implement. - -
The physical interpretation is straight forward.
| - -
Results depend on a single point or its surrounding, therefore more sensitive to noise. - -
Threshold choice needs some justification. - -
The thresholded point has to be visible (inside field of view or traverser range).
|
Fitting | - -
Result depends on all data points of the profile. - -
No need to see the entire profile. - -
More stable on noisy data.
| - -
Need for an a priori knowledge of the functional dependence of the profile. - -
Danger of fake local optima, especially in the cases of too many fitting parameters, or when the real functional dependence does not follow the expected one.
|
Table 2.
The momenta of stream-wise velocity, which have been examined in this contribution.
Table 2.
The momenta of stream-wise velocity, which have been examined in this contribution.
Moment of u | Advantages | Disadvantages |
---|
Average | - -
Easiest to obtain by most measuring and computational techniques. - -
A simple functional profile in later stages, therefore only a part of the wake has to be measured. - -
Good repeatability.
| - -
The average velocity does not draw the wake sharply. - -
The resulting width is a product of convention rather than of some physical effect.
|
Standard deviation | - -
Refers to fluctuations, which is the key feature of any flow structure. - -
Reasonable (but not excellent) repeatability.
| - -
It is not yet clear how and when the double-peak character disappears.
|
Skewness | - -
Refers to the entrainment effect of the turbulent flow.
| - -
Need for high statistical quality. - -
The outer edge of the wake boundary is too irregular; thus, the fitting method does not give repeatable results, but any suitable threshold value is not discovered yet. - -
It fails for the laminar flow.
|
Flatness | - -
Refers to rare events at the wake boundary, thus, it is the most physical method to find the wake edge. - -
The wake is drawn quite sharply.
| - -
Need for high statistical quality. - -
It is not yet explored how it behaves nor when the background flow is turbulent. - -
It fails for the laminar flow.
|