Page 1 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.Read More

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