Boundary layer theory and thermal convection

Course code: 5C1207 (PhD-students: 5C5118)

Lab No. 3:

Tollmien-Schlichting waves in a flat plate boundary layer

 

1 Background and purpose

Laminar boundary layers are subject to a convective wave instability called Tollmien-Schlichting (TS) waves. TS-waves with a given frequency amplify within a range of Reynolds numbers which can be assessed by linear theory (see Appendix 1). When the wave reaches a large enough amplitude, nonlinear phenomena initiate the transition to turbulnce. This is the dominating scenario in a low-disturbance environment, e.g. on an airfoil or a space craft under atmospheric conditions. In noisy environments, e.g. on turbine blades, the TS-wave instability competes with other transition mechanisms, which may lead to transition at lower Reynolds numbers. This lab will give some practical experience in how to measure the chacacterstics of the TS wave instability. It will also illustrate some different kinds of disturbances which may initaite transition in boundary layers, such as excitation of wave packets and turbulent spots, and free stream turbulence.

2 Experimental equipment

The experiments will be performed on a flat plate mounted in the MTL wind tunnel, which is a wind tunnel with exceptionally low level of background distrubances. The working section of the plate is 2.2 m long. It is mounted horizontally in the lower part of the test section, and the stagnation line flow is controlled by menas of a flap at the downstream end of the plate. The leading edge is designed to give a good approximation to the Blasius boundary layer flow all along the plate.

2.1 Introducing disturbances into the boundary layer

Different kinds of disturbances will be introduced into the boundary layer:

Wave generation. TS-waves are generated by a loudspeaker which are fed with a signal from a function generator amplified by a hifi-system. The sound waves are introduced into the boundary layer either from the free stream or through a slit in the plate, and they excite TS-waves with the same frequency.

Localized transient disturbances. These are generated by for instance by a short air pulse introduced through a whole from beneath the plate, or from the free stream. This may excite either a TS wave packet or or a turbulent spot, depending on the intensity of the disturbance and the Reynolds number.

Free stream turbulence (FST). A grid placed at the entrance of the test section will cause free stream turbulence, which will greatly affect laminar-to-turbulent transition in the boundary layer. The grid used in the lab will give with free stream rms fluctuations of about 2% of the free stream velocity.

 

 

2.2 Hot wire anemometry and traversing system

The streamwise velocity and its fluctuations are measured with a hot-wire. The time signal is logged on a computer where the signal is processed and its power spectrum computed. This gives the dominating wave frequency and the wave amplitude at this frequency. The wave can also be observed qualitatively by watching an oscilloscope connected to the hot wire probe.

The hot wire is calibrated against a Prandtl tube placed in the free stream. Note that the anemometer output is a strongly nonlinear funtion of the velocity, which means that the amplitude of velocity fluctuations observed in the oscilloscope will depend on the local mean velocity at different y-positions.

The hot-wire probe can be traversed normal to the plate as well as in the downstream direction. This makes it possible to measure the amlitude profile of the wave aross the boundary layer as well as its downstream development.

3 Detailed lab instructions

ã Get acquainted with the equipment:

  1. how to run the wind tunnel
  2. how to control the wave frequency and amplitude
  3. how to position the hot wire probe.
    Make sure you know at what coordiante the probe would hit the wall!
  4. Connect one channel of the oscilloscope to the signal generator, and the other channel to the hotwire anemometer. Position the hotwire inside the boundary layer close to the wall and watch the waves excited by the generator on the oscilloscope. Watch the effect of changing amplitude and freqeuency. Move the probe in the downstream direction, while watching the change in phase lag between the wave generator and the hot wire signal.
  5. Connect both signals to the computer. In this lab, the calibration is already done beforehand. Make sure the processing gives you the correct frequency of the wave.

    6. ã Quick check of the free stream pressure gradient

  6. Position the hot wire probe well above the boundary layer at the leading edge, and measure the velocity there with the computer. Traverse the probe downstream to about x=2000, while watching the output voltage of the anemometer. Does ot vary monotonously? Measure the free stream velocity at the downstream position. Make a rough estimate of the pressure gradient from these two measurement points. Is it positive or negative? How would this influence the TS-wave instability?

    7. ã Determine measurement parameters for the TS-wave

  7. Determine a convenient free stream velocity (e.g. 8 m/s) and generator frequency to excite waves with the desired frequency (e.g. F = 86). Determine with the help of the stability diagram at which x-positions branch I and II are located. Estimate the wave length, assuming the wave speed to be approximately 0.3 Uƒ.
  8. Position the probe at an x-position which corresponds to branch II according to linear theory. Watch the oscilloscope and the computed spectrum. What happens when the generator amplitude is increased? Can you identify fundamental or subharmonic wave interactions? Watch the time signal on the ocilloscope, and draw a sketch of it for your lab report.
  9. Move the probe beyond branch II. Where does the flow become fully turbulent? How long is the transition region?
  10. Choose a generator amplitude which excites a purely sinusoidal TS-wave at branch II. Record the wave amplitude with the computer - its maximum should not exceed 0.5% of the free stream velocity (Uƒ).
  11. Check whether the wave is two-dimensional by moving the probe in the spanwise (z-) direction and watching if the wave amplitude varies. What effect would a 3D component have on the amplification rate?

    12. ã Measure the y-profile of the TS-wave amplitude and phase near branch II.

  12. Traverse the probe through the boundary layer and take measurements of the wave amplitude and phase, as well as the mean velocity. Watch the oscilloscope while you traverse and try to detect the phase shift which occurs appoximately in the middle of the boundary layer.
  13. Plot the amplitude and phase as a function of y/d , where d = (nx/Uƒ)1/2, x being the downstream distance from the leading edge and n the kindmatic viscosity. Compare to results from linear theory.

    14. ã Measure the amplification curve A(x) of the TS-wave

  14. Position the probe at a position upstream of branch I. Locate the near wall y-maximum of the wave amplitude and record the wave amplitude A there.
  15. Move the probe downstream by about 100 mm and repeat the same procedure there. Continue until you reach past brach II.
  16. Plot the wave amplitude as N = ln (A/A0), where A0 is hte amplitude at branch I, as a function of Reynolds number based on d. Compare to predictions from linear theory.

    17. ã Wave packets and turbulent spots

  17. Trigger transient localized disturbances from the computer. Position the probe near the plate at an x-position near branch II of the the wave you studied above. Draw a sketch of the ocilloscope trace for your lab report.
  18. Watch how varying disturbance intensities give rise to TS-wave packets and turbulent spots. Try to estimate the dominating frequency of the wave packet by looking at the oscilloscope trace.
  19. Move the probe in the x-direnction and try to determine where the flow becomes turbulent. How long is the transition region?

    20. ã Free stream turbulence

  20. Insert the grid at the entrance of the test section. Readjust the free stream velocity to the same value as before. Position the proble near the plate at an x-position 350 mm downstream of the leading edge. Traverse the probe through the boundary layer taking measurements of the mean and rms velocity. Plot them as a function of d.
  21. Whatch the spectrum oscilloscope time signal and its computed spectrum as you traverse the probe. What can you say about the frequency content in the boundary layer compared to that in the free stream? Record the power spectrum of u near the maximum of urms.
  22. Move the probe in the x-direnction and try to determine where the flow becomes turbulent? How long is the transition region?
  23. Position the probe about 1 mm from the plate at an x-position where the there are no signs of turbulence. Traverse downstream at steps of about 100 mm, and estimate the intermittency factor by watching the oscilloscope. Plot these values as a function of x.

    24. ã Turbulent boundary layer

  24. Position the probe at position where the flow is fully turbulent, and traverse through the boundary layer taking measurements of the mean and rms velocity. Plot them as a function of d. Record the power spectrum of u near the maximum of urms.

4 LAb Report

The lab report should contain at least the following:

  1. TS-wave at fix frequency:
    - plots of the TS-wave amlitude and phase profiles across the boundary layer together with the mean flow velocity profile.
    - a plot of the spectrum of u near the TS-wave maximum
    - a plot of the TS-wave amplification curve compared to linear theory.
    - sketches of the oscilliscope signal in the nonliear region and near transition
  2. Boundary layer subjected to free stream turbulence:
    Plots of the urms- and U-profiles, power spectra near the maximum of urms, and sketches of the oscilloscope signal
    (a) upstream of the transition region (no turbulent spots),
    (b) when the flow is fully turbulent.
  3. Answers to the questions above.
  4. Comments on the different transition scenario's observed.

 

Appendix 1. Tollmien-schlichting waves in plane boundary layers - theoretical background

Appendix 2. Brief introduction to CTA hot wire anemometry