Shear stress statistics in a turbulent, open-channel flow over a rough bed

D. HURTHER and U. LEMMIN

Laboratoire de Recherches Hydrauliques

Ecole Polytechnique Fédérale de Lausanne

CH-1015 Lausanne, Switzerland

tel: +41 21 693 2400 fax: + 41 21 693 6767 e-mail: david.hurther@epfl.ch

ABSTRACT

3D quasi-instantaneous velocity profiles are taken at the center of a uniform, turbulent, open-channel flow over a rough bed. A cumulant discard method is applied to describe the statistical properties of the covariance terms and relative to their time means. Conditional statistics and conditional sampling are used to compare the theoretical and experimental relative shear stress contributions from quadrants in the longitudinal and transverse planes. The results in the section show the presence of outward interactions (quadrant I), ejections (quadrant II), inward interactions (quadrant III) and sweeps (quadrant IV) with dominance in quadrant II and IV. The transverse quadrant distribution of fractional events is characterized by a quasi uniform repartition over the transverse plane. A simple linear relation between the third order cumulants of the two probability densities leads to an expression for the turbulent kinetic energy flux indicating its strong dependence on the bursting process. Finally the positive and negative contributions to the stress production are compared with the transport term evaluated from the third order moment of .

Keywords: shear stress dynamics, conditional statistics, conditional sampling, coherent structures, turbulent energy budget.

INTRODUCTION

Coherent structures in a uniform open-channel flow over smooth or rough bed have been studied by many authors . The importance of these near wall burst events in the generation of shear stress over the whole water depth has been demonstrated quantitatively (Grass 1971). Recently, experimental studies of Séchet (1996), Cellino (1998) and Rashidi et al. (1990) have provided information about the correlation between the occurrence of coherent structures and the sediment transport mechanism in the cases of bedload transport and suspension flow. Few studies exist in the field of mixing of matter which consider quantitatively the influence of these events on the turbulent diffusivity depending only on the local statistical properties of the turbulent flow. Therefore, in order to improve estimates of these turbulence depending quantities more experimental studies are needed to determine their statistical characteristics made possible by new instrument. This paper reports on an experimental investigation of the conditionally sampled covariance events and in open-channel flow. The statistical parameters evaluated from the experimental results will be compared to those estimated by a cumulant discard Gram-Charlier distribution of third order also used by Nakagawa and Nezu (1977; hereinafter NN77). Also, a relationship between the turbulent kinetic energy diffusion term and an ejection-sweep parameter will be given without making additional assumptions for the term of the 2D mean flow turbulent energy equation.

EXPERIMENTAL DETAILS AND INSTRUMENTATION

Experiments were carried out in a laboratory open-channel (29m long, 2.45m wide) under uniform flow conditions with a sand bed (mean grain size 1.7mm). The measurement section is placed 13m downstream from the entrance where turbulent flow is well developed. All velocity data presented here were taken at the center of the channel. The hydraulic parameters are given in table 1. These indicate a subcritical turbulent flow. The variables u_* u_*,S and u_*,log represent the friction velocities obtained from linear extrapolation of the mean Reynolds stress at the wall, from the energy line slope formula for uniform flow and from the log law, respectively. The dimensionless roughness value shows an incompletely rough bed. The P parameter in table 1 represents the P factor of Coles wake function.

An Acoustic Doppler Velocity Profiler (ADVP) is used to evaluate the three instantaneous velocity components over the entire water depth. In order to maintain a constant acoustical beamwidth over the whole water depth we use a concentric annular phase array emitter which reduces the sample volume size to p4²´3mm³. Details of this technique are described in Hurther et al. (1998). The temporal resolution is fixed at 31.25 velocity profiles per second. The experiments were conducted in clear water conditions in which the ultrasonic targets are concentration microstructures originating from fine air bubbles following the turbulent motion without inertial lag over the investigated frequency band (Shen et al. 1997).

THEORETICAL ASPECTS

We define the variables as follows: are the zero mean fluctuating longitudinal, transverse and vertical velocity components respectively. are equal to . We shall quantify the contributions from the different planes of the relative shear stress for the covariance terms and . Fig. 1 shows the orientations of the quadrants in the longitudinal and transverse planes. Two characteristic functions and , expressed as the Fourier transforms of the joint probability density functions and respectively, can be expressed as function of the moment and cumulant generating functions in which and denote the moments of (j+k)^th order and and correspond to the cumulants of (j+k)^th order. NN77 expressed the conditionally sampled probability densities over the four quadrants of covariance event . The mathematical manipulation is completely described in NN77. The following equations are given:

(1)

where the index q in p_i,q denotes the quadrant index in the i^th plane with planes 1 and 2 corresponding to the longitudinal and transverse planes respectively. The probability density is directly developed from the corresponding bivariate normal distribution. The non-conditionally sampled probability function of shear stress is with and:

(2)

where, are the corresponding correlation coefficients and is the 0-th order modified Bessel function of the second kind. The physical meaning of the coefficients and will be discussed later. From equations (1) we will calculate the first order moment of each conditional probability density distribution as function of the threshold levels . Increasing the level allows the selection of strong fractional Reynolds stress events. Their directional distribution over the different quadrants in the longitudinal and transverse sections can be investigated using the following expressions with and with . This method is known as the u-w quadrant threshold technique. The parameters , evaluated from the probability densities will be compared to the ones from experimental results in order to give new information about the quadrant distribution of the relative covariance term in the transverse section of the flow.

Statistical analysis and quadrant contributions to shear stress

Fig. 2 represents all third order moments needed to evaluate the factors and of the quadrant probability density functions . In our case a linear combination of these third order moments yields . From this relation we can express the factors in eq. (2) as function of one third order moment and calculate the quadrant probability densities. The last relation shows the importance of the third order moment (or cumulant) when covariance dynamics of events is investigated.

Fig. 3 presents the theoretical and experimental quadrant fractional contributions of the relative covariance term calculated from equations (2) and from the -quadrant technique, respectively. Comparison of the results at four different depths are shown denoting the well-known ejection-sweep dominance in contribution. In the outer flow region (z/h>0.2) the ejection contribution is higher than the one originating from sweeps but in the inner region the sweep contribution increases and reaches the same level as the quadrant II contribution. That phenomena was also found by Grass (1971) who stated that sweep events become larger in the vicinity of the wall. In Fig. 4 we can observe the same quantities as in Fig. 3 but for the shear stress events , i.e. the distribution in the transverse section of the flow. It can be seen that the curves fall off more slowly stretching to much larger values of the threshold level denoting larger variations from their temporal mean and low mean covariance . It is particularly striking that the contribution to the shear stress generation is nearly the same for all four quadrants, independent of the threshold level and depth z/h.

Turbulent energy equation

The dimensionless turbulent kinetic energy flux is written as , which is extracted from the simplified form of the turbulent energy equation for a uniform, incompressible 2D mean flow. NN77 derived this term as: with . This equation indicates the direct relation between the turbulent energy flow budget and the bursting process through the third order moments and . If we assume that a linear combination between the four third order moments is valid (see above), F_k can be written as with . The c_i,jk coefficients are extracted from the linear combination and the coefficients C_i are extracted from the universal semi-empirical relations of the turbulence intensities given by Nezu et al. (1986). In Fig. 5, the F_k profiles estimated by the three last relations are plotted. All curves agree well among themselves except in the inner flow region. The last equation of F_k has the advantage that it requires the knowledge of only one instantaneous velocity component in comparison with the relation proposed by NN77 in which the crossed third order moments are estimated from 2D instantaneous velocities measurements. We found S@ -0.15 from our experimental investigations.

DISCUSSION and CONCLUSION

As mentioned above the equation for the F_k term clearly shows the connection of the turbulent energy budget to the bursting phenomena. The effect of roughness can be identified if the data are compared to those measured by NN77. It is characterized by a rapid decrease of F_k towards the wall, becoming negative for z/h<0.2. In that case the inner region is marked by a downward turbulent energy flux. NN77 showed that this tendency increases with roughness which implies an increase of the turbulent energy flux gradient with roughness and thus an increase of the local energy loss with roughness size. In Fig. 6 the dimensionless turbulent diffusion term (one plot) and production terms (three plots) are represented. The observed distribution over depth of the diffusion term corresponds well with the tendency of the curves in Fig. 5. The three remaining curves in Fig. 6 show the positive, negative and total stress production terms of the turbulent energy equation represented respectively by the contributions of the quadrant two and quadrant four events, the quadrant one and three events and the summation of the last two events. It is noted that a positive production term implies an energy transfer from the mean to the turbulent flow and vice versa. In our case, the total production is positive over the entire flow depth with a maximum value of 14 at z/h=0.09. For 0.1<z/h<0.6 the positive term remains considerably higher than the negative one but in the free surface flow region the two terms are almost of the same magnitude. It should be remembered that the last two points of the profile near the surface were omitted due to the presence of the transducer chamber. In that region an energy transfer to the mean flow could be envisioned.

References

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Nakagawa, H. and Nezu, I. (1977). "Prediction of the contribution to the Reynolds stress from bursting events in open-channel flows." J. Fluid Mech. , 80(1), 99-128.

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Shen, C. and Lemmin, U. (1997). "Ultrasonic scattering in highly turbulent clear water flow." Ultrasonics , 35, 57-64.