Investigation into the selection of viewing configurations for three-component planar Doppler velocimetry measurements

Tom O. H. Charrett, David S. Nobes, and Ralph P. Tatam

Author Affiliations

Tom O. H. Charrett,^{1} David S. Nobes,^{1,}^{2} and Ralph P. Tatam^{1,}^{*}

^{1}Engineering Photonics Group, Centre for Photonics and Optical Engineering, School of Engineering, Cranfield University, Cranfield, Bedfordshire MK43 0AL, United Kingdom

^{2}Currently with the Department of Mechanical Engineering, University of Alberta, Edmonton, Alberta T6G 2V4, Canada

Tom O. H. Charrett, David S. Nobes, and Ralph P. Tatam, "Investigation into the selection of viewing configurations for three-component planar Doppler velocimetry measurements," Appl. Opt. 46, 4102-4116 (2007)

A method for the calculation of three orthogonal velocity components in planar Doppler velocimetry (PDV) using four or more measured velocity components (to the
three typically used) is presented. The advantages and disadvantages are assessed by use of a
Monte Carlo simulation and experimental measurements of the velocity field of a rotating
disk. The addition of a fourth velocity component has been shown to lead to reductions in
the final errors of up to 25%. The selection of viewing configurations for experiments is
discussed by simulation of the level of errors in measured velocity components and
investigation of the final level of errors in the orthogonal velocity components. Experimental
measurements of the velocity field of a rotating disk are presented, demonstrating the effect
of the viewing configuration on the final level of error.

J.-E. Franzkowiak, P. Mercier, G. Prudhomme, and L. Berthe Appl. Opt. 57(11) 2766-2772 (2018)

References

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Condition Numbers for the Viewing Configurations Used in the Investigation (1^{st} Three Views Only)

Configuration

Condition Number

κ_{1}

κ_{2}

κ_{3}

κ_{4}

A

14.75

9.13

10.41

14.55

B

7.11

5.03

6.63

8.98

C

3.47

1.38

3.13

3.23

D

6.15

2.79

4.49

4.68

E

4.50

2.41

3.87

4.00

F

8.45

5.43

7.31

8.95

Table 4

Conditions Used in Simulation

Parameter

Value

Field of view

100 by 100 mm

Image size

400 by 400 pixels

Imaging distance

1.5 m

Illuminating wavelength

514.5 nm

Light sheet

Parallel rays with direction î

Seed particle size range

0.1–0.4 μm^{∗}

CCD A∕D conversion factor

5e^{−}∕count

^{∗}Particle size range selected to match the output of the seeder used in previous experimental investigations at Cranfield [17, 18, 23].

Table 5

Computed Standard Deviation of Orthogonal Component Residuals (Computed Values Minus Original Values) for Three Measured Components (3C) with a Constant Error and Variable Error on Measured Velocity Components for a Velocity Field of (10, 100, 10) m∕s

Condition Number^{(1)}

Standard Deviation of Computed Orthogonal Component Residuals (m∕s)

Constant Error^{(2)}

Variable Error^{(3)}

U

V

W

U

V

W

A

14.75

6.0

2.8

10.1

5.9

3.2

10.4

B

7.11

4.2

2.0

6.1

4.8

3.2

8.4

A^{∗}

14.75

—

—

—

9.2

7.6

19.8

B^{∗}

7.11

—

—

—

7.9

7.1

17.3

C

3.47

2.2

2.3

1.7

3.1

3.4

2.4

D

6.15

1.3

3.5

3.6

1.3

3.3

3.5

E

4.50

2.0

3.5

2.0

2.3

4.3

2.8

F

8.45

6.3

3.5

4.5

7.7

4.8

7.3

^{(1)}Condition number κ1. ^{(2)}Constant error, ERR_{Con}, added to components (standard deviation 2.0 m∕s). ^{(3)}Variable error, ERR_{Var}, added to components-errors calculated using a Doppler shift uncertainty of 2.5 MHz and uncertainty in view angles of 0.1°. ^{∗}Velocity field of (30, 300, 30) m∕s used.

Table 6

Computed Standard Deviation of Orthogonal Component Residuals (Computed Values Minus Original Values) for a Velocity Field of (10, 100, 10) m∕s

Standard Deviation of Computed Orthogonal Component Residuals (m∕s)

3C Method^{(1)}

4C Method^{(2)}

U

V

W

U

V

W

A

5.9

3.2

10.4

4.1

3.2

6.0

B

4.8

3.2

8.4

3.5

3.2

4.7

A^{∗}

9.2

7.6

19.8

4.4

7.6

6.5

B^{∗}

7.9

7.1

17.3

4.0

7.0

5.0

C

3.1

3.4

2.4

2.8

3.5

2.1

D

1.3

3.3

3.5

1.3

3.3

3.2

E

2.3

4.2

2.8

2.0

2.4

1.6

F

7.7

4.8

7.3

4.1

3.6

3.1

^{(1)}3C method. ^{(2)}4C method with equal weighting of all velocity components. ^{∗}Velocity field of (30, 300, 30) m∕s used.

Table 7

Comparison of the Computed Standard Deviation of Orthogonal Component Residuals (Computed Values Minus Original Values) Using the Best 3 of 4 Views and the 4C Method for a Velocity Field of (10, 100, 10) m∕s. Negative Percentage Changes Indicates a Reduction in Uncertainty When Using the 4C Method

Standard Deviation of Computed Orthogonal Component Residuals (m∕s)

‘Best’ 3C Method

4C Method

% Change

U

V

W

U

V

W

U

V

W

A

5.2

3.2

7.1

4.1

3.2

6.0

−21.2

0.0

−15.5

B

3.2

3.2

5.1

3.5

3.2

4.7

9.4

0.0

−7.8

A^{∗}

4.5

7.6

7.3

4.4

7.6

6.5

−2.2

0.0

−11.0

B^{∗}

3.3

7.0

9.6

4.0

7.0

5.0

21.2

0.0

−47.9

C

3.1

3.4

2.3

2.8

3.5

2.1

−9.7

2.9

−8.7

D

1.3

3.3

3.5

1.3

3.3

3.2

0.0

0.0

−8.6

E

2.0

2.4

2.1

2.0

2.4

1.6

0.0

0.0

−23.8

F

4.0

3.7

3.6

4.1

3.6

3.1

2.5

−2.7

−13.9

^{∗}Velocity field of (30, 300, 30) m∕s used.

Table 8

Viewing Angles Used for Experimental Measurements (Previously Defined Value)

Configuration

View Angle (°)

View 1

View 2

View 3

View 4

A

α

180

180

−135

180

β

21 (45)

−21 (45)

0

0

B

α

135

135

−135

135

β

21 (45)

−21 (45)

0

0

Table 9

Standard Deviation of Orthogonal Component Residuals (Experimental Values Minus Theoretical Values) for a Velocity Field of a Rotating Disk

Standard Deviation of the Orthogonal Component Residuals (m∕s)

3C Method

4C Method

U

V

W

U

V

W

A

3.25

5.50

4.90

2.81

5.50

3.69

B

2.40

4.22

3.16

2.18

4.22

2.38

Tables (9)

Table 1

Definitions of the Viewing Configurations Used Showing the Cartesian Components and Viewing Angles for Each Observation Direction

Configuration A

Configuration B

${\widehat{\mathrm{o}}}_{1}$

${\widehat{\mathrm{o}}}_{2}$

${\widehat{\mathrm{o}}}_{3}$

${\widehat{\mathrm{o}}}_{4}$

${\widehat{\mathrm{o}}}_{1}$

${\widehat{\mathrm{o}}}_{2}$

${\widehat{\mathrm{o}}}_{3}$

${\widehat{\mathrm{o}}}_{4}$

X

0.000

0.000

−0.707

0.000

X

0.500

0.500

−0.707

0.707

Y

0.707

−0.707

0.000

0.000

Y

0.707

−0.707

0.000

0.000

Z

0.707

0.707

0.707

1.000

Z

0.500

0.500

0.707

0.707

$\alpha (\xb0)$

90

90

135

90

$\alpha (\xb0)$

45

45

135

45

$\beta (\xb0)$

45

−45

0

0

$\beta (\xb0)$

45

−45

0

0

Configuration C^{8}

Configuration D

${\widehat{\mathrm{o}}}_{1}$

${\widehat{\mathrm{o}}}_{2}$

${\widehat{\mathrm{o}}}_{3}$

${\widehat{\mathrm{o}}}_{4}$

${\widehat{\mathrm{o}}}_{1}$

${\widehat{\mathrm{o}}}_{2}$

${\widehat{\mathrm{o}}}_{3}$

${\widehat{\mathrm{o}}}_{4}$

X

0.460

0.460

0.383

0.383

X

−0.500

−0.500

−0.707

0.707

Y

0.628

−0.628

0.000

0.000

Y

0.707

−0.707

0.000

0.000

Z

0.628

0.628

−0.924

0.924

Z

0.500

0.500

−0.707

0.707

$\alpha (\xb0)$

54

54

−67

67

$\alpha (\xb0)$

135

135

225

45

$\beta (\xb0)$

40

−40

0

0

$\beta (\xb0)$

45

−45

0

0

Configuration E^{7}

Configuration F

${\widehat{\mathrm{o}}}_{1}$

${\widehat{\mathrm{o}}}_{1}$

${\widehat{\mathrm{o}}}_{2}$

X

0.000

Four orthogonal laser sheets in the same plane used sequentially (1, 0, 0), (−1, 0, 0), (0, 1, 0) & (0, −1, 0)

X

0.000

0.500

Two light sheets used sequentially (1, 0, 0) & (0, 1, 0)

Y

0.000

Y

0.000

0.707

Z

1.000

Z

1.000

0.500

$\alpha (\xb0)$

90

$\alpha (\xb0)$

90

45

$\beta (\xb0)$

0

$\beta (\xb0)$

0

45

Table 2

Sensitivity Vectors of the Ideal Case Configuration, and the Computed Matrix Conditioning Numbers for the Different Definitions

Cartesian Components of the Measured Velocity Components∕Sensitivity Vectors

Condition Numbers for the Viewing Configurations Used in the Investigation (1^{st} Three Views Only)

Configuration

Condition Number

κ_{1}

κ_{2}

κ_{3}

κ_{4}

A

14.75

9.13

10.41

14.55

B

7.11

5.03

6.63

8.98

C

3.47

1.38

3.13

3.23

D

6.15

2.79

4.49

4.68

E

4.50

2.41

3.87

4.00

F

8.45

5.43

7.31

8.95

Table 4

Conditions Used in Simulation

Parameter

Value

Field of view

100 by 100 mm

Image size

400 by 400 pixels

Imaging distance

1.5 m

Illuminating wavelength

514.5 nm

Light sheet

Parallel rays with direction î

Seed particle size range

0.1–0.4 μm^{∗}

CCD A∕D conversion factor

5e^{−}∕count

^{∗}Particle size range selected to match the output of the seeder used in previous experimental investigations at Cranfield [17, 18, 23].

Table 5

Computed Standard Deviation of Orthogonal Component Residuals (Computed Values Minus Original Values) for Three Measured Components (3C) with a Constant Error and Variable Error on Measured Velocity Components for a Velocity Field of (10, 100, 10) m∕s

Condition Number^{(1)}

Standard Deviation of Computed Orthogonal Component Residuals (m∕s)

Constant Error^{(2)}

Variable Error^{(3)}

U

V

W

U

V

W

A

14.75

6.0

2.8

10.1

5.9

3.2

10.4

B

7.11

4.2

2.0

6.1

4.8

3.2

8.4

A^{∗}

14.75

—

—

—

9.2

7.6

19.8

B^{∗}

7.11

—

—

—

7.9

7.1

17.3

C

3.47

2.2

2.3

1.7

3.1

3.4

2.4

D

6.15

1.3

3.5

3.6

1.3

3.3

3.5

E

4.50

2.0

3.5

2.0

2.3

4.3

2.8

F

8.45

6.3

3.5

4.5

7.7

4.8

7.3

^{(1)}Condition number κ1. ^{(2)}Constant error, ERR_{Con}, added to components (standard deviation 2.0 m∕s). ^{(3)}Variable error, ERR_{Var}, added to components-errors calculated using a Doppler shift uncertainty of 2.5 MHz and uncertainty in view angles of 0.1°. ^{∗}Velocity field of (30, 300, 30) m∕s used.

Table 6

Computed Standard Deviation of Orthogonal Component Residuals (Computed Values Minus Original Values) for a Velocity Field of (10, 100, 10) m∕s

Standard Deviation of Computed Orthogonal Component Residuals (m∕s)

3C Method^{(1)}

4C Method^{(2)}

U

V

W

U

V

W

A

5.9

3.2

10.4

4.1

3.2

6.0

B

4.8

3.2

8.4

3.5

3.2

4.7

A^{∗}

9.2

7.6

19.8

4.4

7.6

6.5

B^{∗}

7.9

7.1

17.3

4.0

7.0

5.0

C

3.1

3.4

2.4

2.8

3.5

2.1

D

1.3

3.3

3.5

1.3

3.3

3.2

E

2.3

4.2

2.8

2.0

2.4

1.6

F

7.7

4.8

7.3

4.1

3.6

3.1

^{(1)}3C method. ^{(2)}4C method with equal weighting of all velocity components. ^{∗}Velocity field of (30, 300, 30) m∕s used.

Table 7

Comparison of the Computed Standard Deviation of Orthogonal Component Residuals (Computed Values Minus Original Values) Using the Best 3 of 4 Views and the 4C Method for a Velocity Field of (10, 100, 10) m∕s. Negative Percentage Changes Indicates a Reduction in Uncertainty When Using the 4C Method

Standard Deviation of Computed Orthogonal Component Residuals (m∕s)

‘Best’ 3C Method

4C Method

% Change

U

V

W

U

V

W

U

V

W

A

5.2

3.2

7.1

4.1

3.2

6.0

−21.2

0.0

−15.5

B

3.2

3.2

5.1

3.5

3.2

4.7

9.4

0.0

−7.8

A^{∗}

4.5

7.6

7.3

4.4

7.6

6.5

−2.2

0.0

−11.0

B^{∗}

3.3

7.0

9.6

4.0

7.0

5.0

21.2

0.0

−47.9

C

3.1

3.4

2.3

2.8

3.5

2.1

−9.7

2.9

−8.7

D

1.3

3.3

3.5

1.3

3.3

3.2

0.0

0.0

−8.6

E

2.0

2.4

2.1

2.0

2.4

1.6

0.0

0.0

−23.8

F

4.0

3.7

3.6

4.1

3.6

3.1

2.5

−2.7

−13.9

^{∗}Velocity field of (30, 300, 30) m∕s used.

Table 8

Viewing Angles Used for Experimental Measurements (Previously Defined Value)

Configuration

View Angle (°)

View 1

View 2

View 3

View 4

A

α

180

180

−135

180

β

21 (45)

−21 (45)

0

0

B

α

135

135

−135

135

β

21 (45)

−21 (45)

0

0

Table 9

Standard Deviation of Orthogonal Component Residuals (Experimental Values Minus Theoretical Values) for a Velocity Field of a Rotating Disk

Standard Deviation of the Orthogonal Component Residuals (m∕s)