Importance of Pressure Measurementsme.jhu.edu/lefd/PPIV/PPIVprinciple.pdf · 2011-04-29 · We all know that pressure measurement is very important. Pressure is a primary concern
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• Importance of Pressure Measurements: – Pressure is a primary concern for many engineering applications , e.g. lift and form drag. – Cavitation : Pressure is of fundamental importance in understanding and modeling cavitation. – Turbulence : Velocity-Pressure-Gradient tensor, which can be decomposed into the pressure diffusion and pressure-strain tensors, is critical for modeling turbulence, especially near boundaries. However, due to the lack of the experimental capability, this tensor has never been measured directly*. Background and Motivation: Importance of Pressure Measurements i j j i x u x u p i j j i x p u x p u 1 Pressure diffusion Pressure-strain Velocity-Pressure-Gradient p u x p u x i j j i *Has been determined based on balance of all other terms in turbulence kinetic energy transport equation, e.g., Liu and Thomas (2004), Gutmark and Wygnanski (1976) and Wygnanski and Fiedler (1969).
– Pressure is a primary concern for many engineering applications , e.g. lift and form drag.
– Cavitation : Pressure is of fundamental importance in understanding and modeling cavitation.
– Turbulence: Velocity-Pressure-Gradient tensor, which can be decomposed into the pressure diffusion and pressure-strain tensors, is critical for modeling turbulence, especially near boundaries. However, due to the lack of the experimental capability, this tensor has never been measured directly*.
Background and Motivation: Importance of Pressure Measurements
*Has been determined based on balance of all other terms in turbulence kinetic energy transport equation, e.g., Liu and Thomas (2004), Gutmark and Wygnanski (1976) and Wygnanski and Fiedler (1969).
Presenter
Presentation Notes
Now the background and the motivation. We all know that pressure measurement is very important. Pressure is a primary concern for many engineering applications. For example, pressure is responsible for both lift and form drag acting on a moving body in fluid, and we know that lift and drag are the two key parameters in aircraft and marine vehicle designs. Pressure is also of fundamental importance for understanding and modeling cavitation. It is well established that cavitation inception occurs when small bubbles or nuclei in liquid grow explosively due to exposure to low pressure. In turbulence research, the velocity-pressure-gradient tensor in the Reynolds stress transport equation, which is typically decomposed into the pressure diffusion and the pressure-strain tensors, is critical for understanding and modeling turbulence. We know that the pressure diffusion, together with the turbulence and the viscous diffusion, serves to transport turbulence away from regions of high mean strain to those locations of low production of turbulence. The pressure-strain tensor serves to re-distribute energy among components of turbulence fluctuations and plays a central role in Reynolds stress transport equation. However, due to the lack of the adequate experimental capability, the velocity-pressure-gradient tensor has never been measured directly.
Background and Motivation: Toolbox Available for Pressure Measurement
• Available Techniques for Pressure Measurement Are Quite Limited:
– Surface Pressure Measurement: – Pressure taps leading to transducers; – Surface flush-mounted pressure transducers– Pressure sensitive paint.
– Pressure Measurement away from Boundaries:– Pitot Tube; Five Hole Probe and Seven Hole Probes.– Drawbacks:
Intrusive;Frequency response is limited;Point measurement, no simultaneous global data.
Presenter
Presentation Notes
We may have to recognize that the available techniques for pressure measurement are quite limited. For surface pressure measurement, we may open pressure taps on the surface of the testing body which may further lead to pressure transducers. We may also place flush mounted pressure transducers on the surface of the testing body. In addition, we may also use the pressure sensitive paint to measure the surface pressure distribution. For spatial pressure measurements, we have only Pitot-tube type of measurement probes such as the five hole and seven hole probes. But these probes are intrusive, not suitable for dynamic measurement, and can only provide point measurement data.
Background and Motivation: Objectives• Objectives and Unique Features of the Present Method:
– To develop a system that can measure the instantaneous global pressure distribution in a non-intrusive manner based on PIV technology.
– The system utilizes four-exposure PIV to measure the distribution of material acceleration and then integrating it to obtain the pressure field.
– The system can measure the instantaneous velocity, material acceleration and pressure field simultaneously.
Presenter
Presentation Notes
In recognition of the lack of sufficient experimental tools for pressure measurement, we would like to develop a system that can measure the instantaneous global pressure distribution in a non-intrusive manner based on PIV technology. This measurement system utilizes four-exposure PIV to measure the distribution of material acceleration and then integrating it to obtain the pressure distribution. With this system, the instantaneous velocity, material acceleration, and pressure field can be measured simultaneously.
Principles of the Technique– Navier-Stokes Equation:
– Obtain the material acceleration based on PIV technology.
– With a measured reference pressure at one point, one can integrate the pressure gradient field to obtain the instantaneous pressure distribution.
DtUD
U
DtUDp
2
Dominant term Negligible for high Re flowand in regions away from the wall
Two 2K2K CCD cameras
Two orthogonally polarized dual-head Nd:Yag lasers
Four consecutively exposed images vector map
Two consecutive velocity vector maps
DtUD
Presenter
Presentation Notes
Now, I would like to talk about the principles of our pressure measurement technique. Theoretically, this technique stems from the Navier-Stokes equation, from which, the pressure gradient can be expressed in this expression. For high Reynolds number flow and in regions away from the wall, where there is no extremely high velocity gradient, the viscous term in the Navier-Stokes equation is negligible. Therefore, the material acceleration is the dominant term, which is balanced by the pressure gradient term on the left hand side of the equation. If we can somehow obtain the material acceleration DU/Dt based on PIV technology, we can then integrate the pressure gradient to obtain the pressure distribution. This is the basic idea of this measurement technique. In particular, this technique utilizes two 2K by 2K CCD cameras and two orthogonally polarized dual-head Nd:Yag lasers to obtain four consecutively exposed images, from which, we can get two consecutive velocity vector maps and then, from these two velocity vector maps, we can obtain the material acceleration. Once the material acceleration is known, the pressure gradient is also known. We can then integrate the pressure gradient field to obtain the instantaneous pressure distribution.
Obtaining the Material Acceleration from Four-Exposure PIV Images
Now let’s look at the details of how we determine the material acceleration. Suppose we have four consecutively exposed PIV images, evenly separated in time with a constant time interval delta t. Here images 1 and 3 are recorded by camera 1 and images 2 and 4 are recorded by camera 2. The cross-correlation between images 1 and 3 will give us velocity vector map U13, and similarly, cross-correlation between images 2 and 4 will give us velocity vector map U24. Obviously, the time interval between these two velocity vector maps is delta t.To measure the material acceleration, we have to compare the velocity of the same group of particles at two different times. Suppose at time t, a group of particles is located at x_a, and the velocity of this particle group at this specific location and at this specific time is represented by U13(evaluated at x_a and time t). After a short time increment delta t, this same group of particles moves to a new locaiton (x_a plus U_a times delta t), where U_a is the Lagrangian velocity with which the particle group moves from here to here. The instantaneous velocity of the particle group at this new place (x_a plus U_a times delta t) and this instant of time (t plus delta t) is represented by U24 (evaluated at this location (x_a plus U_a times delta t) and this instant of time(t plus delta t)). Now the question is: how can we determine the Lagrangian velocity U_a? Well, a natural way to determine U_a is just to take the average of these two velocities U13 and U24. Please note this is both a spatial and temporal average of the velocity of the same group particles at the two different locations and two consecutive instants of time. Moreover, this expression is implicit, because in the expression of U24, we also see the unknown U_a. This requires an iteration process to obtain U_a. Typically, 2 or 3 rounds of iteration will give us a converged value of U_a. Once U_a is found, we can then plug the velocities into this expression to get the estimate of the material acceleration. Please note that the material acceleration is evaluated at this location and at this instant of time. This implies that if we start from a uniform grid at time t then we may end up with a material acceleration map evaluated at a distorted grid, because U_a may vary from place to place.
Virtual Boundary Omni-Directional Integration
– Integrate the measured vector map of material acceleration, starting from a reference point.
– To reduce uncertainty, use Virtual Boundary Omni-Directional Integration over the entire flow field to obtain the instantaneous spatial pressure distribution:
Now, once we get the material acceleration field, we can then integrate it to obtain the pressure distribution. We know that the pressure field is a scalar field, and therefore integration of the pressure gradient must be independent of the integration path. However, we cannot expect to get the correct integration value for pressure only along a single integration path, because we are dealing with experimental data, -- for any experimental data, inevitably there are always experimental errors associated with them. So, to minimize the error of the pressure measurement, we use an integration method featuring Averaged, Shortest-path, Omni-directional integration over the entire flow field to obtain the instantaneous spatial pressure distribution. This figure shows the measured material acceleration and this sketch shows integration paths of the omni-directional integration, and this figure shows the instantaneous pressure distribution obtained by the omni-directional integration.
Overlapped images 1 and 3 Overlapped images 2 and 4
Feasibility Study with Synthetic Images: Pure Rotational Flow
Presenter
Presentation Notes
To study the feasibility of this pressure measurement technique, we applied the aforementioned procedures to synthetic images of solid-body rotation and stagnation point flows to obtain the material acceleration and subsequently, the pressure distribution. These are the four consecutive synthetic images for the pure rotational flow. The simulated seed particles with Gaussian size distribution are distributed homogeneously in a 2K by 2K image using a random number generator available in Matlab. If we overlap the corresponding images on top of each other, we can clearly see the solid-body rotation of the synthetic flow field. Now I would like to show you the results.
Radial Pressure Distribution
Crp 22
21
Measured from the synthetic vortex flow data.
r (pixel)
Pres
sure
(Arb
itrar
y U
nit)
Magnitude of Material Acceleration
Integration path
Averaged, shortest path, omni-directional integration is utilized for pressure calculation in order to minimize the error.
Spatial Pressure Distribution
Standard Deviation of the Relative Error =1.2%
Prob
abili
ty D
ensi
ty F
unct
ion
(100
%)
Relative Error (100%)
Prob
abili
ty D
ensi
ty F
unct
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(100
%)
PDF of the Relative Error of Pressure
Demonstration Using a Synthetic Vortex Flow
Presenter
Presentation Notes
This figure shows the magnitude–contour of the measured material acceleration. If we use the averaged-shortest-path-omni-directional integration over the entire acceleration field, we can then obtain the spatial pressure distribution. This result clearly shows that the averaged-omni-directional integration, which serves as a low-pass filter, can extract the correct pressure information out of a noisy acceleration measurement. We know that for a solid-body rotational flow, the radial distribution of pressure is a quadratic parabola And this figure clearly shows the measured radial pressure distribution agrees with the theoretical curve very well. This figure shows the probability density function of the relative error of the measured pressure, from which we can see the standard deviation of the relative error is only 1.2%.
• Image size: 2048x2048 pixel
• Particle Intensity: 25 per interrogation window of 32x32 pixel
• Interrogation window size: 32x32 pixel
• Strain Rate: S=0.025 (1/sec.)
• Time Interval between images: 0.5sec.
• Particle size: Gaussian distributed with mean diameter of 2.4 pixel, standard deviation of 0.8 pixel.
Exposure 1 Exposure 2 Exposure 3 Exposure 4
Synthetic Image: Constant Strain Rate Flow (Stagnation Point Flow)
Presenter
Presentation Notes
As I said, we also applied the pressure measurement procedures to the synthetic stagnation point flow images. Like the solid-body rotational flow, we generated four consecutive synthetic images. If we overlap the corresponding images on top of each other, the stagnation point flow pattern can easily be recognized.
Results of the Constant Strain Rate Flow (Stagnation Point Flow)
Presenter
Presentation Notes
Now the results of the stagnation point flow. Again, like the pure rotational flow case, the measured pressure value agrees with the theoretical curve very well. And the standard deviation of the relative error in this case is only 1.9%.
