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Ensemble Averaging for Unsteady Flows ME 410: Microfluidics

Ensemble Averaging for Unsteady Flows ME 410: Microfluidics -Swati Priyam. 1. Steady Flow Steady flow is one in which the time derivatives of the flow field vanish, i.e., the flow variables are constant with respect to time.

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Ensemble Averaging for Unsteady Flows ME 410: Microfluidics

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  1. Ensemble Averaging for Unsteady Flows ME 410: Microfluidics -Swati Priyam 1

  2. Steady Flow Steady flow is one in which the time derivatives of the flow field vanish, i.e., the flow variables are constant with respect to time. Unsteady Flow Fluid flow is called unsteady when the flow variables vary with time as well. Basic Keywords Used: Figure 1a: Stationary time series for steady flow Figure 1b:Non-stationary time series for unsteady flow 2

  3. Basic Keywords Used: Time Averaging • Consider a measured variable x(t) in a turbulent flow. • Assume that the “average characteristics” of x(t) do not vary with time. (Such a behavior is shown in Figure 1a.) [1] • Time averaging is defined as: --- Eq. 1 3

  4. Basic Keywords Used: Ensemble Averaging • Now consider a case in which the “average characteristics” of x(t) do vary with time (Such a behavior is shown in Figure 1b.) • Here, the average itself is a function of time, hence we cannot apply Eq. 1 since we can not be certain about the size of t0 perform the integration. If t0 is very small then the average obtained is unreliable. Similarly, a very large value of t0 will not be able to give a true “local average”. [1] • Hence in this case, an ensemble average is evaluated in the following steps: • Specify the spatial co-ordinates and time at which the average has to be found. • Repeat the experiment N times under identical conditions. • Note the value of the flow variable at the specified location and time in all the N experiments. • Ensemble average: --- Eq. 2 4

  5. Ensemble averaging: An Example Aim of the Experiment : Velocity measurements in a turbulent jet using ensemble averaging. Setup : Consider a turbulent flow in a channel as shown below. The velocity measurement probe is shown as a red dot. Procedure : 1. Take the velocity readings from the probe as a function of time. 2. Repeat the experiment N (here 10) times. 3. Specify the time t0 at which ensemble averaging has to be found. 4. Evaluate ensemble average using Eq. 2. 4

  6. Interactive Animation The user will be asked to specify the number of experiments N, and the time interval t0 at which the values have to be recorded. The user will then generate N graphs by clicking on the “Generate Simulated Graphs” Tab. (Slide 6) As the graph is being generated, the value of u at the specified time interval t0 gets tabulated for each experiment, which can be seen by clicking on “Show Recorded Values” Tab. (Slide 6) “Calculate Ensemble Average” Tab will first present the formula involved and then show the vale calculated. (Slide 7) “Show Convergence” Tab will create the plots of ensemble average vs the number of experiments to show convergence. The user will be able to see the nature of convergence by varying N. (Slide 8) The proposed animation will be something like this: 5

  7. Observations Flow Velocity recorded by the probe as a function of time for 5 experiments. The Ensemble Average has to be evaluated at t0 Here, Data logger window has been shown for 5 experiments only. 6

  8. Calculating Ensemble Average 7

  9. Convergence of Ensemble Average 8

  10. Questionnaire Q. Why does time-averaging fail in the case of unsteady flows? A. This is because we can not define the time interval with certainty, in which the integration in Eq. 1 has to be performed in order to obtain an average. Q. What is the effect of the number of experiments performed on the value of the ensemble average obtained? A. Greater the number of experiments performed, better is the value of ensemble average obtained. Q. What precautions must be taken while performing experiments to evaluate ensemble average? A. The boundary conditions in the experiment and the flow properties of the working substance should not change in between the experiments. 9

  11. Glossary of Terms The glossary might look something like this: Clicking on each of the tabs shown in the left will open a dialog box with the description of the terms (Slides 2 – 4). 10

  12. References and Further Reading [1] P. K. Kundu, I. M. Cohen & H. H. Hu, Fluid Mechanics. Elsevier Academic Press, 3rd Edition. [2] http://en.wikipedia.org/wiki/Ensemble_(fluid_mechanics) [3] www.qis.ex.nii.ac.jp/qis/documents_YY/y3_02chp1_sld.pdf [4] http://www.cfd-online.com/Wiki/Introduction_to_turbulence/Statistical_analysis/Ensemble_average [5] C. T. Crowe, Multiphase Flow Handbook. Taylor and Francis Group.

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