(1)
Sung-Uk Choi and Hyeongsik Kang
Department of Civil Engineering, Yonsei University, Seoul 120-749, Korea
Tel +82-2-2123-2797; Fax +82-2-364-5300; E-mail schoi@yonsei.ac.kr
Abstract: The Reynolds stress model is applied to open-channel flows with submerged vegetation. The SSG model is used for the pressure strain correlation term in the transport equation of the Reynolds stresses. The mean flow and turbulence structures are reproduced numerically, and they are compared with measurement data and with the results from the k-e model. Comparisons between the numerical and theobserved profiles reveal that the Reynolds stress model predicts the flow better than the k-e model. The difference in the flow structures computed by the non-isotropic and isotropic turbulence models seems to be small, but it may cause a significant difference in predicting sediment or pollutant load by vegetated open-channel flows.
Keywords: Reynolds stress model, SSG model, vegetated open-channel flows
Stream flows over vegetated bottom boundary are quite common in nature. Specifically, most floodplains are vegetative, and they are designed to deliver large discharge during floods. So it is important to reflect vegetation’s impact on the carrying capacity of a channel with floodplains. However, the traditional way to consider resistance by vegetation was just to increase roughness coefficients. This is because open-channel flows with either submerged or emergent vegetation are turbulent flows characterized by 3-dimensional complicated mechanisms. Thus reflecting vegetation in the main channel or in the floodplain has been a difficult task in fluvial hydraulics.
Vegetation in the watercourse raises the water level during floods significantly. This process is achieved by converting mean kinetic energy into turbulent kinetic energy at the scale of plant stems (Nepf, 1999). On the other hand, vegetation mitigates suspended solids (Lopez and Garcia, 1998), and thus improves turbidity level. Vegetation also brings improvement of habitats for wildlife in the river and of water quality by filtering the pollutants.
Mean flow and turbulence characteristics of open-channel flows over rough boundary are studied by either laboratory measurements or numerical computations. If one has to resort to numerical computations, he needs a proper turbulence model. In many engineering problems in fluid mechanics and hydraulics, the k-e model has been most widely used perhaps due to well-established empirical coefficients of the model. In fact, the k-e model has been successfully applied to variety of problems such as simple shear flows, a buoyant jet in a cross flow, and complex wall shear flows. However, to date, none of the existing turbulence models are truly universal, consequently, each model needs to be tuned to specific flows.
The k-e model and similar models based upon the eddy viscosity concept make a basic assumption that Reynolds stress is aligned with the velocity gradient. This assumption is valid in simple shear flows but is not for complicated flows. For example, secondary flows in a compound open-channel, although the magnitude of which is extremely small compared to the mean flow, affect conveyance seriously, and thus should not be neglected. This flow motion in plane normal to the main flow direction is driven by the anisotropy of turbulence.
Shimizu and Tsujimoto (1994) calculated vertical distributions of mean and turbulent flow structures by using the k-e model. Naot et al. (1996) carried out numerical simulation with the algebraic stress model to see the impact of vegetation on the floodplain in the compound open-channel. Lopez (1997) simulated turbulence structures of vegetated open-channel flows by using the k-e model and compared the computed solutions with their experimental results. With the same numerical model, Lopez and Garcia (1998) estimated the amount of suspended sediment load reduced by vegetation.
The present study is an application of the Reynolds stress model to vegetated open-channel flows. The SSG model proposed by Speziale et al. (1991) is used for the pressure strain correlation term. The computed flow structures are compared with numerical solutions from the k-e model as well as with the profiles measured in the laboratory. Improvements of accuracy in predicting vertical structures will contribute to better assessment of sediment and pollutant loads transported by vegetated open-channel flows.
Consider the open-channel flow with submerged vegetation in Figure 1, where the boundary layer coordinates, (x, y, z), represent the streamwise, spanwise, and vertical directions, respectively. In the figure, cylindrical vegetation is uniformly distributed at distances with lx and ly in the x- and y-directions, respectively. Assuming that the flow is at a high Reynolds number in a wide open-channel, the x -momentum equation is expressed as
(1)
where
u is the mean velocity component in
the x-direction, p is the mean pressure, r is the fluid density, n is the laminar viscosity,
is the drag force per unit volume
of water,
is the gravitational acceleration, and
is the channel slope. In eq.(1),
the drag force
due to vegetation is given by
(2)
where CD is the drag coefficient and
is the vegetation density defined
by
(3)
in which D is the cylinder diameter and Ah is the bed area composed of lx and ly. Regarding the drag coefficient for submerged vegetation, Dunn (1996) observed through laboratory experiments that the value of CD for vertical cylinders varies with distance from depth, and it reaches a maximum at a distance close to one third of the cylinder height. Dunn found a mean value of CD = 1.13±15%, which is used in the present computation.
The transport equation of Reynolds stresses takes the following form:
(4)
where Rij is the Reynolds stress, Pij is the rate of production of Rij , Dij is the transport of Rij by diffusion, eij is the rate of dissipation of Rij, and Pij is the transport of Rij due to turbulent pressure-strain correlation. In eq.(3), the rate of production of Rij is given by
(5)
which does not require modeling while the remaining terms need modeling. For Dij, the following model proposed by Hanjalic and Launder (1972) is used:
(6)
with Cs = 0.11. In eq.(6), k is the turbulence kinetic energy and e is the turbulence kinetic energy dissipation rate. Most relationships suggested for eij are isotropic models which are recognized to be incorrect near the solid wall. Thus, herein, the following Rotta's (1951) model is used for the rate of dissipation of Rij:
(7)
One of obstacles in using the complete Reynolds stress model in practical engineering problems lies in the use of the wall damping functions. If the fluid is bounded by the complicated domain, the normal distance from the walls must be evaluated from geometric considerations, which is not readily available in some cases. Moreover, since the wall damps the fluctuations acting only normal to the boundary, a complicate coordinate transformations would be required to determine a component normal to the wall for each Reynolds stress (Basara and Cokljat, 1995). In order to overcome this problem, recently, many researchers proposed the models for pressure-strain term without a wall reflection term. The present RSM employs the SSG model proposed by Speziale, Sarkar, and Gatski (1991) for the pressure-strain term in eq.(4):
(8)
where
bij is the anisotropy
tensor, Sij is the rate of
strain tensor, Wij is the
rotation tensor, and
–
are empirical coefficients. The values of these parameters used herein are
= –3.4,
= 4.2,
= 0.8–1.3(bmnbnm)1/2,
= –1.8,
= 1.25, and
= 0.4. In eq.(8), the bij, Sij, and Wij
are given, respectively, by
;
;
(9
a,b,c)
In
the right hand side of eq.(8), the first term represents the slow contribution
to the pressure-strain correlation. This is the usual Rotta term for the return
to isotropy, which is included in most Reynolds stress models. The remaining
terms represent the rapid contribution to the pressure-strain correlation, where
the second term including
is a non-linear contribution to
the return to isotropy. The third term is linear and the fourth term is
quadratic in bij. The fifth
and the last terms are also linear in bij, and they are known to make a major contribution in
the rapid part.
The damping effects of the wall and the free surface are very similar and they act to increase the level of anisotropy. However, in the vicinity of the free surface, the mean velocity gradients are negligible. This may be a main reason why the SSG model is found not to be very suitable near the free surface (Basara and Cokljat, 1995). Therefore, in order to reflect the damping effect of the free surface, the following model proposed by Shir (1973) is added to the pressure-strain term:
(10)
where
is an empirical constant (=0.05)
and fs is the free surface
damping function given by
(11)
where
is a model parameter (=0.01), L is the dissipation length which is
given by
, and h is the water depth.
Wall functions are used at the bottom boundary to take into account the bottom roughness. The free surface is treated as a symmetry plane for all variables except for the dissipation rate of turbulent kinetic energy (e). The relationship for e from Rodi (1984) is employed to increase turbulence kinetic energy dissipation level at the free surface.
Numerical implementations of the RSM introduced so far were performed based upon the finite volume method. We have applied the numerical models to open-channel flows with submerged vegetation. In this study, it is assumed that vegetation behaves like a rigid cylinder, thus no bending of vegetation occurs. In the simulation of open-channel flows with submerged vegetation, the experimental conditions in Lopez (1997) such as H = 0.335 m, S0 = 0.0036, a = 1.09 m-1, and hp = 0.1175 m are imposed. Measurement data from Lopez (1997) are used for comparisons with computed results.
The mean velocity is given in Figure 2 for the open channel with H/hp = 2.85. Velocity defect due to vegetation is obviously seen below the height of vegetation. In this region, numerical solutions from the k-e model and the RSM are almost the same, and they agree well with measured data by Lopez (1997). However, above the vegetation height, it is seen that the velocity from the RSM agrees well with the measured profile, which is smaller than the solution from the k-e model.
The simulated turbulence intensities together with measurements are given in Figure 3. Turbulence intensity, increasing from the bottom, peaks around the height of vegetation, and it decays. Note that both computed and measured results reflect this trend. The numerical result from the RSM is found to agree well with the measured data. The turbulence intensity computed by the k-e model appears slightly larger and smaller than observed data below and above the vegetation height, respectively.
Figure 4 shows the Reynolds stress profile. The Reynolds stress, which is zero at the bottom, increases up to the height of vegetation, then it decreases. Both numerical solutions from the RSM and the k-e model agree well with experimental data below the height of vegetation. However, above the height of vegetation, the Reynolds stress profile by the k-e model is greater than the measured data as well as the profile predicted by the RSM. The peak value by the k-e model is also the largest among three profiles.
The Reynolds stress model was applied to open-channel flows with submerged vegetation. The SSG model was used for the pressure strain correlation term. Mean flow and turbulence characteristics such as Reynolds stress and turbulence intensity were simulated. The results were compared with not only measurement data but also numerical solution from the k-e model. The profiles computed by the RSM match observed data better than those by the k-e model. The improvement in the prediction may affect significantly the assessment of sediment or pollutant loads in the vegetated open-channel flows.
Acknowledgements
This work was supported by the Brain Korea 21 Project.
References
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Fig. 1 Vegetated open-channel flow simulated
Fig. 2 Mean velocity
Fig. 3 Turbulence intensity
Fig. 4 Reynolds stress