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# Velocity-Measurement: Processing-velocity-vectors Mechanical Engineering Notes | EduRev

## Mechanical Engineering : Velocity-Measurement: Processing-velocity-vectors Mechanical Engineering Notes | EduRev

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Objectives_template
file:///G|/optical_measurement/lecture15/15_1.htm[5/7/2012 12:23:28 PM]
Module 3: Velocity Measurement
Lecture 15: Processing velocity vectors

The Lecture Contains:
Data Analysis from Velocity Vectors
Velocity Differentials
Vorticity and Circulation
RMS Velocity
Drag Coefficient
Streamlines
Turbulent Kinetic Energy Budget

Page 2

Objectives_template
file:///G|/optical_measurement/lecture15/15_1.htm[5/7/2012 12:23:28 PM]
Module 3: Velocity Measurement
Lecture 15: Processing velocity vectors

The Lecture Contains:
Data Analysis from Velocity Vectors
Velocity Differentials
Vorticity and Circulation
RMS Velocity
Drag Coefficient
Streamlines
Turbulent Kinetic Energy Budget

Objectives_template
file:///G|/optical_measurement/lecture15/15_2.htm[5/7/2012 12:23:29 PM]
Module 3: Velocity Measurement
Lecture 15: Processing velocity vectors

Data Analysis from Velocity Vectors
In applications, the velocity information is often necessary but not sufficient and other quantities will
be of interest as well. The velocity field obtained from PIV measurements can be used to estimate
relevant quantities by means of differentiation and integration. The vorticity field is of special interest
because, unlike the velocity field it is independent of the frame of reference. In particular, if it is
resolved temporally, the vorticity field can be much more useful in the study of flow phenomena than
the velocity field. This is particularly true in highly vortical flow such as turbulent shear layers, wake
vortices and complex vortical flows. Integral quantities can also be obtained from the velocity data.
The instantaneous velocity field obtained by PIV can be integrated, yielding either a single path
integrated value or another field such as the stream function. Analogous to the vorticity field, the
circulation obtained through path integration is also of special interest in the study of vortex dynamics,
mainly because it is also independent of the reference frame. In the following section, data analysis
for calculation of various derived quantities from PIV measurements are presented.
Velocity differentials
The differential terms are estimated from the velocity vectors obtained from PIV. Since PIV provides
the velocity vector field sampled on a two dimensional evenly spaced grid specified as  finite
differencing can be employed to get the spatial derivatives. There are a number of finite difference
schemes that can be used to obtain the derivatives. The truncation error associated with each
operator is estimated by means of a Taylor series expansion. The actual uncertainty in differentiation
is due to that in the uncertainty of the velocity estimate  It can be obtained using standard error
propagation methods assuming individual data to be independent of the other. There are two
schemes that reduce the error associated with differentiation: Richardson extrapolation and least
squares approach. The former minimizes the truncation error while the least squares approach
reduces the effect of random error, i.e. the measurement uncertainty,  These approaches are
briefly discussed bellow.
Least squares estimate of the first derivative:
Here, the accuracy is of the order of  and the associated uncertainty is
The derivative using Richardson extrapolation is calculated as:
The accuracy of the above approximation is of order  and the uncertainty associated with the
expression is

Page 3

Objectives_template
file:///G|/optical_measurement/lecture15/15_1.htm[5/7/2012 12:23:28 PM]
Module 3: Velocity Measurement
Lecture 15: Processing velocity vectors

The Lecture Contains:
Data Analysis from Velocity Vectors
Velocity Differentials
Vorticity and Circulation
RMS Velocity
Drag Coefficient
Streamlines
Turbulent Kinetic Energy Budget

Objectives_template
file:///G|/optical_measurement/lecture15/15_2.htm[5/7/2012 12:23:29 PM]
Module 3: Velocity Measurement
Lecture 15: Processing velocity vectors

Data Analysis from Velocity Vectors
In applications, the velocity information is often necessary but not sufficient and other quantities will
be of interest as well. The velocity field obtained from PIV measurements can be used to estimate
relevant quantities by means of differentiation and integration. The vorticity field is of special interest
because, unlike the velocity field it is independent of the frame of reference. In particular, if it is
resolved temporally, the vorticity field can be much more useful in the study of flow phenomena than
the velocity field. This is particularly true in highly vortical flow such as turbulent shear layers, wake
vortices and complex vortical flows. Integral quantities can also be obtained from the velocity data.
The instantaneous velocity field obtained by PIV can be integrated, yielding either a single path
integrated value or another field such as the stream function. Analogous to the vorticity field, the
circulation obtained through path integration is also of special interest in the study of vortex dynamics,
mainly because it is also independent of the reference frame. In the following section, data analysis
for calculation of various derived quantities from PIV measurements are presented.
Velocity differentials
The differential terms are estimated from the velocity vectors obtained from PIV. Since PIV provides
the velocity vector field sampled on a two dimensional evenly spaced grid specified as  finite
differencing can be employed to get the spatial derivatives. There are a number of finite difference
schemes that can be used to obtain the derivatives. The truncation error associated with each
operator is estimated by means of a Taylor series expansion. The actual uncertainty in differentiation
is due to that in the uncertainty of the velocity estimate  It can be obtained using standard error
propagation methods assuming individual data to be independent of the other. There are two
schemes that reduce the error associated with differentiation: Richardson extrapolation and least
squares approach. The former minimizes the truncation error while the least squares approach
reduces the effect of random error, i.e. the measurement uncertainty,  These approaches are
briefly discussed bellow.
Least squares estimate of the first derivative:
Here, the accuracy is of the order of  and the associated uncertainty is
The derivative using Richardson extrapolation is calculated as:
The accuracy of the above approximation is of order  and the uncertainty associated with the
expression is

Objectives_template
file:///G|/optical_measurement/lecture15/15_3.htm[5/7/2012 12:23:29 PM]
Module 3: Velocity Measurement
Lecture 15: Processing velocity vectors

Vorticity and circulation
The vorticity components in the  directions can be calculated from the partial derivatives of
velocity using:
Figure 3.33: Contour for the calculation of circulation for the estimation of
vorticity at a point (i; j).

Page 4

Objectives_template
file:///G|/optical_measurement/lecture15/15_1.htm[5/7/2012 12:23:28 PM]
Module 3: Velocity Measurement
Lecture 15: Processing velocity vectors

The Lecture Contains:
Data Analysis from Velocity Vectors
Velocity Differentials
Vorticity and Circulation
RMS Velocity
Drag Coefficient
Streamlines
Turbulent Kinetic Energy Budget

Objectives_template
file:///G|/optical_measurement/lecture15/15_2.htm[5/7/2012 12:23:29 PM]
Module 3: Velocity Measurement
Lecture 15: Processing velocity vectors

Data Analysis from Velocity Vectors
In applications, the velocity information is often necessary but not sufficient and other quantities will
be of interest as well. The velocity field obtained from PIV measurements can be used to estimate
relevant quantities by means of differentiation and integration. The vorticity field is of special interest
because, unlike the velocity field it is independent of the frame of reference. In particular, if it is
resolved temporally, the vorticity field can be much more useful in the study of flow phenomena than
the velocity field. This is particularly true in highly vortical flow such as turbulent shear layers, wake
vortices and complex vortical flows. Integral quantities can also be obtained from the velocity data.
The instantaneous velocity field obtained by PIV can be integrated, yielding either a single path
integrated value or another field such as the stream function. Analogous to the vorticity field, the
circulation obtained through path integration is also of special interest in the study of vortex dynamics,
mainly because it is also independent of the reference frame. In the following section, data analysis
for calculation of various derived quantities from PIV measurements are presented.
Velocity differentials
The differential terms are estimated from the velocity vectors obtained from PIV. Since PIV provides
the velocity vector field sampled on a two dimensional evenly spaced grid specified as  finite
differencing can be employed to get the spatial derivatives. There are a number of finite difference
schemes that can be used to obtain the derivatives. The truncation error associated with each
operator is estimated by means of a Taylor series expansion. The actual uncertainty in differentiation
is due to that in the uncertainty of the velocity estimate  It can be obtained using standard error
propagation methods assuming individual data to be independent of the other. There are two
schemes that reduce the error associated with differentiation: Richardson extrapolation and least
squares approach. The former minimizes the truncation error while the least squares approach
reduces the effect of random error, i.e. the measurement uncertainty,  These approaches are
briefly discussed bellow.
Least squares estimate of the first derivative:
Here, the accuracy is of the order of  and the associated uncertainty is
The derivative using Richardson extrapolation is calculated as:
The accuracy of the above approximation is of order  and the uncertainty associated with the
expression is

Objectives_template
file:///G|/optical_measurement/lecture15/15_3.htm[5/7/2012 12:23:29 PM]
Module 3: Velocity Measurement
Lecture 15: Processing velocity vectors

Vorticity and circulation
The vorticity components in the  directions can be calculated from the partial derivatives of
velocity using:
Figure 3.33: Contour for the calculation of circulation for the estimation of
vorticity at a point (i; j).

Objectives_template
file:///G|/optical_measurement/lecture15/15_4.htm[5/7/2012 12:23:29 PM]
Module 3: Velocity Measurement
Lecture 15: Processing velocity vectors

The expression of mass continuity for an interrogation spot on the  plane can be written as
Circulation can be computed from either velocity using
or vorticity as

Page 5

Objectives_template
file:///G|/optical_measurement/lecture15/15_1.htm[5/7/2012 12:23:28 PM]
Module 3: Velocity Measurement
Lecture 15: Processing velocity vectors

The Lecture Contains:
Data Analysis from Velocity Vectors
Velocity Differentials
Vorticity and Circulation
RMS Velocity
Drag Coefficient
Streamlines
Turbulent Kinetic Energy Budget

Objectives_template
file:///G|/optical_measurement/lecture15/15_2.htm[5/7/2012 12:23:29 PM]
Module 3: Velocity Measurement
Lecture 15: Processing velocity vectors

Data Analysis from Velocity Vectors
In applications, the velocity information is often necessary but not sufficient and other quantities will
be of interest as well. The velocity field obtained from PIV measurements can be used to estimate
relevant quantities by means of differentiation and integration. The vorticity field is of special interest
because, unlike the velocity field it is independent of the frame of reference. In particular, if it is
resolved temporally, the vorticity field can be much more useful in the study of flow phenomena than
the velocity field. This is particularly true in highly vortical flow such as turbulent shear layers, wake
vortices and complex vortical flows. Integral quantities can also be obtained from the velocity data.
The instantaneous velocity field obtained by PIV can be integrated, yielding either a single path
integrated value or another field such as the stream function. Analogous to the vorticity field, the
circulation obtained through path integration is also of special interest in the study of vortex dynamics,
mainly because it is also independent of the reference frame. In the following section, data analysis
for calculation of various derived quantities from PIV measurements are presented.
Velocity differentials
The differential terms are estimated from the velocity vectors obtained from PIV. Since PIV provides
the velocity vector field sampled on a two dimensional evenly spaced grid specified as  finite
differencing can be employed to get the spatial derivatives. There are a number of finite difference
schemes that can be used to obtain the derivatives. The truncation error associated with each
operator is estimated by means of a Taylor series expansion. The actual uncertainty in differentiation
is due to that in the uncertainty of the velocity estimate  It can be obtained using standard error
propagation methods assuming individual data to be independent of the other. There are two
schemes that reduce the error associated with differentiation: Richardson extrapolation and least
squares approach. The former minimizes the truncation error while the least squares approach
reduces the effect of random error, i.e. the measurement uncertainty,  These approaches are
briefly discussed bellow.
Least squares estimate of the first derivative:
Here, the accuracy is of the order of  and the associated uncertainty is
The derivative using Richardson extrapolation is calculated as:
The accuracy of the above approximation is of order  and the uncertainty associated with the
expression is

Objectives_template
file:///G|/optical_measurement/lecture15/15_3.htm[5/7/2012 12:23:29 PM]
Module 3: Velocity Measurement
Lecture 15: Processing velocity vectors

Vorticity and circulation
The vorticity components in the  directions can be calculated from the partial derivatives of
velocity using:
Figure 3.33: Contour for the calculation of circulation for the estimation of
vorticity at a point (i; j).

Objectives_template
file:///G|/optical_measurement/lecture15/15_4.htm[5/7/2012 12:23:29 PM]
Module 3: Velocity Measurement
Lecture 15: Processing velocity vectors

The expression of mass continuity for an interrogation spot on the  plane can be written as
Circulation can be computed from either velocity using
or vorticity as

Objectives_template
file:///G|/optical_measurement/lecture15/15_5.htm[5/7/2012 12:23:30 PM]
Module 3: Velocity Measurement
Lecture 15: Processing velocity vectors

In the present work, vorticity has been calculated by choosing a small rectangular contour around
which the circulation is calculated from the velocity field using a numerical integration scheme, such
as trapezoidal rule. The local circulation is then divided by the enclosed area to arrive at an average
vorticity for the sub-domain. The following formula provides a vorticity estimate at a point (i; j) based
on circulation using eight neighboring points (see Figure 3.32):
with
It has been observed from experiments that a circulation calculation via the velocity field yields better
estimates of vorticity, and in particular, the peak vorticity that is otherwise under-predicted. At other
locations, the vorticity field determined by the two approaches are practically identical. The accuracy
of the vorticity measurement from PIV data depends on the spatial resolution of the velocity sampling
and the accuracy of the velocity measurements. Therefore, the vorticity error can be associated with
calculation scheme and the grid size used for velocity sampling. Another source of uncertainty is that
propagated from the velocity measurements. PIV velocity measurements are the local averages of the
actual velocity in the sense that it represents a low pass filtered version of the actual velocity field.
Thus, vorticity from PIV data is only a local average of an already averaged velocity field and not a
point measurement.

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