TURBULENT STRUCTURE OF OSCILLATORY SHALLOW FLOWS IN OPEN CHANNEL

Chaoquan Chen and Daoyi Chen*

School of Engineering, University of Manchester, UK

Peter Stansby

Department of Civil and Construction Engineering, UMIST, UK

Abstract: The turbulent structure of oscillatory flows over rough and smooth bed was experimentally investigated in a shallow-water tidal flume using a LDV system. The test periods of oscillatory flow were 30s, 60s and 240s, and measurement at each point covered 50 tidal cycles. The velocity and turbulent properties were obtained through phase-averaged analysis.  The profiles of mean velocity, turbulent intensity and Reynolds stress were shown at different phases in details.  It is found the streamwise mean velocity near the bed can be approximated as logarithmic distributions with different integration constants for different oscillatory periods. The phase shift in different heights appeared in the profiles of the streamwise velocity, turbulent intensities and Reynolds stress with the leading phases near the bed. The peak value of turbulent intensity in vertical direction, which occurs in the outer region of the boundary layer, reaches its maximum within decelerating phases as the flow progresses during the course of the cycle. And the peak Reynolds stress occurs at the phase of more than 15 degree behind the peak streamwise velocity.

1    INTRODUCTION

Turbulent oscillatory flows have been studied experimentally by Hino (1983), Sleath (1987), Jensen (1989) and Nielsen (1992) in closed tunnels. For open channel flows that are more complicated due to the effect of the free surface and also be closer to natural flows, Song (1996) and Nezu (1997) presented experimental results of mean velocity and turbulent structure obtained by ADV and LDV but only over smooth beds. In a recent research project at Manchester on pollutant trapping in tidal flows, a combined LDV/LIF system and a PLIF have been used to measure turbulent properties and pollutant transport in oscillatory flows over rough and smooth beds. The turbulent oscillatory boundary flows are of importance to the pollutant-trapping problem. This paper here mainly presents the profile of mean velocity, turbulent intensities and Reynolds stress from such a study, with details for one rough bed case.

2    EXPERIMENTAL METHODS AND DATA ANALYSIS

The experiments were conducted in an 11.0 m´3.3m´0.2m open shallow-water tidal flume (see Lloyd et al 2001). Oscillatory current flows with a nearly sinusoidal velocity profile are considered here, generated by a uni-directional variable speed pump and an arrangement of four valves. Changing the state of the valves reverses the flow direction. Current velocities and tidal periods were controlled by supplying a sinusoidal voltage input to the pump controller using a PC housed D/A board. The rough bed was made from uniform covering of gravel with the median diameter of 5mm on the smooth bed. A TSI 3-D LDV system with fibre probes was used to measure velocity in 2-D with one channel being used for LIF.  During the course of the experiments, the sampling method was set to coincidence with TBD (Time Between Data) to meet the requirement of working out Reynolds Stress. The fibre probes were fixed on a 3-dimensional traverse table that can move the probes to define locations automatically following a computer signal. For the purpose of obtaining the values in the vicinity of the bed, the two probes were tilted at a certain angle for all the tests. This has been calibrated in the data processing.

The existing software, FIND, provided by TSI, is quite useful to analyse and display the result of steady flows. However, for oscillatory flows, the analysis method is different and a new software package VDP 99 for Windows 95/98 has been developed to analyse the experimental data, with easy access and instantaneous view of the results. 

As mentioned above, the sampling method was set to coincidence with TBD instead of fixed sampling interval, which has improved the data density. Therefore, the statistics of the experiments were worked out through ensemble averaging in the following way.

Mean value

           (1)

Standard deviation

            (2)

Reynolds stress

  (3)

Here  is the time interval between two samples. Equation (3) allows for the adoption of coincidence sampling method, as pointed out above. In the absence of data at some points interpolation was applied to work out the statistics. The sinusoidal controlling signal of the oscillatory flow was used to synchronise the data. Therefore, the starting point for averaging over the following cycles is defined by the zero point of control signal, called the phase signal. The time of zero streamwise velocity may be defined as the starting point. For the current experiment system the starting phase defined by the control signal is leading by 30 degree the one determined with the zero velocity. Since the definition by the control signal is more objective, this method was used to synchronise the data collected at different positions. Several averaging methods have been tested to process the data. Compared with others, the method of phase-averaging and then smoothing within a fixed interval can be meaningful result, especially in the analysis of data with concentration. So this method was used to process all the data. For all cases turbulent velocity data were recorded continuously over around 50 cycles.

For oscillatory smooth flows, the amplitude of fluid particles outside the boundary layer, a, is recognised as the characteristic length for normalisation, which is defined as following,

                                      (4)

where U_om is the maximum velocity in the outer region given by

                                     (5)

For oscillatory flows over rough beds, both a and K_s can be taken as the characteristic length. K_s is the Nikuradse's equivalent roughness. In the present research, K_s is taken equal to 2D_g following Sleath (1987), where D_g is the median diameter of the gravel. Both U_* (friction velocity) and U_om can be regarded as characteristic velocity. Since the gravel was not changed over the course of the present experiment, a could be taken as the characteristic length for normalisation, and U_om was adopted as the characteristic velocity for the comparison between different cases, except the logarithmic law analysis.

Table 1 summarises the test conditions for the measurements. d is the so-called oscillatory boundary layer thickness, estimated using  a formula quoted by Letherman et al (2000),

                             (6)

And  is the roughness Reynolds number defined by

                                 (7)

where U_*m is the maximum value of the friction velocity.

Table 1    Test conditions

3    RESULTS

3.1    Mean velocity 

Figure 1 shows streamwise mean velocity profile of test 1 within half cycle. Both values were normalized by the method mentioned before. The first point measured near the rough bed was determined when the valid signal was collected by the LDV system.  Similar to Sleath's work (1987), the theoretical zero bed level was taken to be 0.2D_g below the first point measured just above the bed.  As can be seen from the figure, the maximum streamwise velocity appeared at about 0.02a high from the bed. The obvious characteristics are the phase shift of flows along the height. The layers near the bed changed its direction and reach the maximum velocity earlier than the upper layers near the free surface. Therefore, around the phases when flow reverses, the upper and lower fluids moved in the opposite directions. Hence a stagnant shear layer was formed, and a ¡°s¡± shaped velocity profile was observed. This phenomenon has been visualized by the particles added into the flows. It is worthy to note that such a phase shift is reduced for both flows of a longer cycle period (240s) and flows over smooth bed. And in the stages of accelerating the phase shift increased compared with the decelerating phases.

Fig.1    Streamwise mean velocity profiles at various phase angles from 0 to 165 degrees in a half tidal cycle of Test 1

Figure 2 shows the streamwise velocity contour with phases and heights. The zero velocity appears at different phases of up to 15 degree for different heights. The maximum velocities are located in the phase range between 105 and 120 degree and the height from 0.016a to 0.024a.

Figure 3 shows the vertical velocity profiles of test 1. The following points appear to be noteworthy. First, along with the oscillatory streamwise velocity, the vertical velocity follows the similar trend; however, the maximum value appears 15 degree later than the maximum streamwise velocity. Second, vertical velocities always appear to be negative when streamwise velocities are positive. The similar phenomenon has been revealed and explained with a theoretical relation by Song (1996). Third, along the height from the bed the vertical velocity increases and then drops. The maximum value occurs at a small distance (y/a=0.015) from the bed.

Fig.2  The streamwise velocity contour with phases and heights of Test 1

Fig.3  The vertical velocity profiles at various height of Test 1

3.2    Turbulent intensity

Figure 4 gives the streamwise and vertical turbulent intensity of test 1, namely root-mean-square of u and v. The kinetic energy k can be estimated following Justesen¡¯s research,

                                (8)

Fig.4    Tubulent intensity profiles at various phase angles from 0 to 165 degrees. a). Streamwise direction; b).vertical direction

Figure 5 shows k contour distribution in the phase and height space. It is noted that the turbulent intensity in accelerating phases is less than that in decelerating phases and the maximum value takes place in a small distance from the bed ( ). One of the reasons for such a difference might be: in the decelerating phases the energy resulted from the velocity¡¯s reduction is transformed to turbulence, while in accelerating phases the energy provided by the negative pressure gradient contributes to the speedup of the flows.

Fig.5    Kinetic energy k contour distribution in the phase and height space

In fact if turbulent intensities are plotted against phase, we can see their periodic variation follows the streamwise velocity with a phase lag of 15 degree near the bed, and the phase lag increases with the distance from the bed. This is in accord with the phase shift of streamwise mean velocity described above. In comparison with streamwise turbulent intensity  (=ru in Fig. 4), the vertical turbulent intensity  (=rv in Fig. 4) are only about one half of the streamwise one. It is interesting to note that the maximum  and  do not appear at the same height. In accelerating phases both  and  take place at 0.02a in Figure 4; however, in the decelerating phases maximum streamwise intensities appear at less than 0.01a, much higher than that for vertical intensities, which are still at the height of more than 0.015a.

The change of turbulent intensity with phases and heights depicted above shows how the turbulence evolves as the flow progresses in phases. Generally it builds up near the bed and diffuses constantly into the boundary layer, as the boundary layer develops in time. The near-bed production of turbulence increases from the early stage of increasing phases and reaches the strongest at the phase of more than 15 degree later than the maximum streamwise velocity. This phase range may be characterised by turbulence production since the diffusion within this range is negligible, and the remaining range is characterised by turbulence production and diffusion across the height.

As expected, it is observed that turbulence over smooth bed decreases dramatically in comparison with over rough bed, although the streamwise mean velocity increases as shown in table 1. The flows of shorter period are associated with stronger turbulence because the quick reversal of main stream enhances the production of turbulence.

3.3    Reynolds stress

Figure 6 shows the variation of Reynolds stress  (= RS in Fig. 6) with phase at different depths of test 1. It is noted that the maximum Reynolds stress in the accelerating phases appears at the position about 0.025a , while in the decelerating phases, at the lower position about 0.015a. Also, the magnitude of Reynolds stress during the accelerating phases was less than that during decelerating phases. This is in accord with the result of turbulent intensity, because both are related to turbulent properties. Compared with the result from rough bed tests, Reynolds stress in smooth test was one order smaller for all the periods.

Fig.6    Reynolds stress profiles at various height.

The variations of the Reynolds stress with phase shown in Figure 6 are of importance. Following sinusoidal streamwise velocities, Reynolds stress has two peaks during one cycle, each corresponding to the phase of around 15 degrees later than that of peak horizontal velocity near the bed. The Reynolds stress appears to form a Gaussian-shaped peak of about 60-degree width. However, the phase difference between the peak horizontal velocity and the peak Reynolds stress is smaller for tests of longer periods. It is worthy to point out that this observation may account for the pollutant trapping in the oscillatory flows.

Similar to turbulent intensities the time when the peak Reynolds stress occurs varies along different depths.   The peak Reynolds stress appears earlier near the bed than close to the free surface for all the tests of different periods. It is interesting to note that the mean velocities, the turbulent intensity and the Reynolds stress following the similar phase shift from the bed up to the free surface.

4    DISCUSSION

For steady boundary flows over rough and smooth bed, logarithmic distribution of streamwise velocity has been derived in theory and supported by extensive experiments (Schlichting, 1979). For oscillatory flows studied here, Figure 7 shows the mean streamwise velocity distribution of test 1 in the semi-log co-ordinates. It is found that close to the bed logarithmic laws are valid over all the phases, which can be described as following,

                     (9)

where K is Karman coefficient, and k_s is the Nikuradse equivalent roughness of the gravel, here taken to be 2D_g.  It is in agreement with the results presented by Sleath (1987). The straight lines in Figure 7 are fitted within the range where logarithmic law prevails.

Fig.7    The mean streamwise velocity distribution of test 1 in the semi-log co-ordinate

system

Nezu (1993) summarised 5 methods to determine friction velocity of open channel flows. Streamwise velocity profile and eddy correlation are frequently used in the experimental research (Sanford, 1999). If Karman coefficient ks is taken to be 0.40 in equation (9), friction velocity can be derived from the linear fitting in Figure 7. Figure 8 illustrates the friction velocities by this method.

It is found that the integration constant A is not universal but dependent on bed forms, i.e. , similar to the suggestion by Schlichting (1979) for steady flows. For T = 60s, the A value is about -5.1 and for T=240s about -2.8, although the wall roughness is the same. In the early stage of accelerating and the later stage of decelerating constant A changes dramatically.  Figure 9 shows the range of heights and phases where the logarithmic curves are fitted. Obviously the logarithmic layers extend further into the flows in decelerating phases than in accelerating phases. In the later stages of decelerating this zone can be half of the water depth. In addition, from Figure 8 it is interesting to note that the mean velocity profile near the water surface may be fitted as logarithmic curves as well.

In spite of these, detailed variations simply applying the standard k-e model with logarithmic wall functions gave good mean flow predictions, including bed shear stress (Letherman, et al. 2000).

   Fig.8    Friction velocity obtained by mean velocity profiles and eddy correlation.

   Fig.9    The height (y) where the Log was fitted.

5    CONCLUSION

The main results are as follows for the range of conditions covered in the present experiment of turbulent oscillatory flows in open channel:

• The logarithmic layer exists in turbulent oscillatory flows over rough bed. The integration constant is dependent on the period and keeps reasonably oscillating around a constant except the early stages of accelerating and the late stage of decelerating during the course of the cycle. The height where the logarithmic law is valid can be up to half of the water depth.

• The phase shift exists along the height for mean velocities, turbulent intensities and Reynolds stress. The leading phases for all the quantities occur near the bed and this is probably due to the influence of the free surface.

•  The turbulent characteristics of oscillatory flows in accelerating and decelerating phases are different. Larger turbulent intensities and Reynolds stress appear in the height closer to the bed in decelerating phases compared with in accelerating phases. In addition, different from in accelerating phases, the maximum streamwise and vertical turbulent intensities do not take place in the same height.

• Turbulence is produced near the bed and spread into the boundary. The strongest turbulence takes places in the phase after the maximum streamwise velocity during the course of the cycle.

• Two peaks of Reynolds stress take place during the course of the cycle, each corresponding to the strongest turbulence built-up later than the maximum streamwise mean velocity.

• The introduction of roughness on the bed results in double turbulent intensities but reducing streamwise velocity.

Acknowledgement

This study is supported by a grant (GR/L/34570) from the UK Engineering and Physical Science Research Council (EPSRC). The 3-D LDV was made available from the EPSRC Instrument Pool.

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* Corresponding author (daoyi.chen@man.ac.uk)