1. Introduction
In-stream structures have been used widely in engineering practice for various purposes, e.g., river training, channel stabilization, erosion control and habitat restoration. In the past several decades, the science and engineering communities have gradually realized that Mother Nature provides great examples of good in-stream structures, which serve multiple purposes. One such natural structure is large woody debris (LWD), which consists of fallen trees, stumps and root-wads, with irregular and complex geometries. Their ecological significance has been recognized since the 1980s [
1,
2,
3]. Studies have found that LWD greatly increases stream habitat complexity [
4]. Organic matter can easily accumulate in-between the debris of snags, leaves and root-wads, which provides nutrients and food for living organisms. The debris structures were reported to contain more than half of the organic matter in lower-order streams [
5], which provides abundant food sources for invertebrates, fishes and macro-invertebrate biomass. The structures can also function as shelters for fishes and prevent them from being attacked by predators [
6]. The slow flow region in the wake of in-stream structures provides a resting area and helps fish regain energy [
7]. The ecological benefits of in-stream structures highly rely on the hydrodynamics within and around them. Recognizing their importance to rivers, in-stream structures have been widely used in stream restoration projects [
8]. Examples of these structures include single log vane, log cross vane, mud sill, multi-log vane deflector, water jack and engineered log jams (ELJ) [
9], among which ELJ is a very attractive option due to its abundant eco-hydraulic benefits [
10].
The proper design of in-stream structures depends on the understanding of the associated hydraulics. These in-stream structures can be regarded as roughness elements in rivers. They alter the flow conditions by slowing down surrounding flow and changing local water depths [
11]. In the measurements of Cultus River, Manga et al. [
12] found that LWD, which only covered 2% of flow area, contributed half of the total flow resistance in the stream. A recent flume study also showed that if these structures are fixed, they can delay the flooding time and significantly lower the runoff coefficient, which helps reduce erosion and suppress flooding damage [
13]. Thus, one important design parameter is the flow resistance due to LWDs. On the other hand, the drag force experienced by the structures is an important factor for their stability.
There are two major methods to quantify the flow resistance due to in-stream structures. The first one is to use the concept of stress partitioning, which was first introduced for the study of fluvial dynamics [
14]. This method separates the flow resistance due to debris from the total. For example, some researchers created new Darcy–Weisbach friction factors for the prediction of flow resistance in the stream, which is constituted of bed shear, bend curvature and drag from debris [
12,
15,
16]. In this method, it is critical yet difficult to estimate the debris drag coefficient. It is often observed that LWDs contain much sediment and many leaves inside the gaps between snags and root-wads. Therefore, these structures are treated as solid barriers, and as a result, simple theoretical models can be applied to calculate the drag force exerted on them [
15]. In early studies, a very common approximation is to use a single cylinder as a proxy for a log in the streams. Flume experiments were conducted to investigate how the side-walls, bed bottom and orientation would affect the drag and lift coefficients for a single log [
17,
18,
19].
Another method is to use the one-dimensional (1D) momentum equation to compute the drag force exerted on in-stream structures. For example, Turcotte et al. [
20] found that the drag forces on large cylinders calculated from the momentum equation matched well with the measured values. They also found that the classic drag force formulas underestimate the force if the cylinder diameter is much larger than critical water depth. Compared with the stress partitioning method, the momentum analysis method is more applicable to complex and porous accumulations, like debris jams. For example, Manners et al. [
21] measured detailed velocity around natural woody debris jams and calculated drag coefficients through the moment equation.
Numerous studies have shown the complexity of in-stream structures and the deviation of their hydraulic behavior from the predictions made by the two methods mentioned above. For example, Manners et al.’s [
21] results show the drag coefficient for the debris jams ranges from 2.6 to 9.0. In contrast, the drag coefficient is about 0.8 for a cylinder and one for a cube. Manners et al.’s [
22] field study indicates that for debris jams, the drag decreases as the porosity increases. A more recent study of Shields and Alonso [
23] shows that the roughness on the surface of cylinder increases lift, but reduces wave force (which is the difference between actual drag force and theoretical drag force). They also found that the effect of surface roughness is not as significant as the bed proximity, blockage rate and submergence. However, under certain conditions, the increase of complexity may reduce the drag coefficient. Gippel et al.’s [
19] flume study showed that branches added to a single trunk would lower the drag coefficient because the increase of drag force was less than the percentage increase of the frontal area.
For the solid barrier model in hydraulics, a very classic application of CFD is in the groyne fields [
24,
25]. The use of CFD to study the porous in-stream structures is rare in the fluvial hydraulics community. As an infrequent case in this point, in order to evaluate the geometric complexity of woody debris, computational simulations were performed by Allen et al. [
26]. They developed a shape complexity factor to quantify the geometric complexity of LWD. Their simulations showed that both the drag coefficient and the integrated turbulence kinetic energy decrease with the complexity factor. However, for the complex and porous structures, it is very difficult to compute the shape complexity factor. There are more CFD studies in coastal engineering for coastal structures, which often consist of amour rocks [
27,
28]. In order to simplify the simulations, these coastal engineering studies usually adopt a porous media flow model by adding resistance source terms, which are calculated with an extended Darcy–Forchheimer equation, into the Navier–Stokes equations. Jensen et al. [
29] calibrated the empirical resistance coefficients for coastal structures. Their simulation results had a good match with the experimental data. A similar porosity model will be evaluated in this study.
This paper presents the findings on the effects of different in-stream structure representations in 3D computational models. In particular, the implications for flow resistance, wake length scale and turbulence characteristics have been analyzed. Three different representations of the selected in-stream structure, i.e., ELJ, were used and compared. The three geometric representations include the fully-resolved model (no simplification in geometry), the porosity model and solid barrier (i.e., treating the ELJ as an impervious box). The simulation case was based on the flume experiments for ELJ in [
10,
30]. In the following, the computational models will be introduced first. Calibrations of all models are then presented, followed by detailed analysis on the simulation results and comparisons. This paper then ends with discussions and conclusions.
3. Results and Discussion
In the following, the fully-resolved ELJ model is first validated with experimental data. Then, the results from simplified representations, i.e., the porosity model and solid barrier, are presented and analyzed.
3.1. Validation of the Fully-Resolved Configuration
For the fully-resolved case, which serves as the baseline for the evaluation of the other two simplifications, we found that the turbulent inlet condition was critical for better comparison with the experiments in Bennett et al. [
10]. In this study, a background case without ELJ was first run to get a fully-developed open channel flow, which was then used at the inlet.
Figure 4 compares the cross-sectional distributions of mean streamwise velocity
u and turbulent kinetic energy (TKE)
k between the numerical and experimental results for the background case. The simulated mean fields were based on averages over 140 s, while in Bennett et al.’s [
10] experiment, the averaging time was 180 s. The results from the 140 s averaging time have little difference from those with 100 s averaging time. Therefore, the 140 s averaging time was used in this work. From the comparison in the figure, a similar pattern and comparable magnitude can be observed for both velocity and TKE. The velocity core is slightly stretched in the simulation than in the experiment.
Bennett et al. [
10] measured the flow on two cross-sections located at the immediate upstream and downstream sides of ELJ at
mm and
mm, respectively.
Figure 5 presents the comparison of the upstream and downstream cross-sectional distribution of the mean streamwise velocity. ELJ was located adjacent to the right sidewall (looking downstream). The red lines indicate the edge of ELJ. On the upstream cross-section, reduced velocity can be observed in the front of ELJ. The pattern and magnitude are quite similar in both experiment and the simulation. However, the high flow region from computational model slightly deviates from the experiment, which might be due to the flattened high velocity core in the simulated incoming flow. For the downstream cross-section, a large velocity gradient can be observed on the eddy of ELJ in both cases. The velocity in the upper main channel flow adjacent to ELJ is slightly larger in LES than in the experiment. However, they are very comparable in general.
The fully-resolved simulation case also shows good overall agreement on TKE with the experiment (
Figure 6). In the computational model, the TKE
k was calculated as:
where
denotes the resolved velocity fluctuation in three directions and
denotes the unresolved sub-grid scale kinetic energy. The simulated upstream distribution of TKE has a noticeable difference from the experiment because of the overall small magnitude. For the downstream cross-section, the shear layer induced by ELJ created a high TKE zone, which was captured well by the computational model.
In the fully-resolved case, the drag force on ELJ can be directly calculated with the integration of pressure and shear stress on the structure surface, namely the first method, which gives
. The second method of using the momentum equation with the simulated flow field gives
, which is very close to the result from the first method. For comparison, the measured drag force in Bennett et al. [
10] was
, which indicates that the simulation underestimated the drag force. This underestimation might be attributed to several factors, including the incoming flow condition, the mismatch between the fabricated ELJ in experiment and the design, etc. In particular, the incoming flow cannot be exactly reproduced as seen in
Figure 4, which leads to slower impingement onto ELJ and lower drag force. In the experiment, it will be ideal, but very time consuming, to measure in great detail the inflow conditions to be used at the inlet in the simulations. In this way, the discrepancy could be greatly reduced.
From the above comparison on mean velocity, TKE and drag force, the fully-resolved case reproduced the experimental condition well considering the complexity of the problem. In the following, the fully-resolved case will be used as the base for comparison with the porous media and solid barrier simplifications.
3.2. Calibration of Porous Media Model
The calibration of the porous media parameters (
α,
β and
) is based on the fully-resolved case, which itself has been validated with experiments in previous section. As summarized in Burcharth et al. [
37] and Jensen et al. [
29], the selection of
α and
β varies for different porous materials and packing modes. Six combinations of
α,
β and
were tested for the ELJ structure.
Table 1 lists the parameters, including the intrinsic and inertial permeability, of the six different combinations.
To quantify the difference between the porosity cases and the fully-resolved case, the cross-sectional error distributions for mean velocity and mean TKE are plotted in
Figure 7. The error is defined as the porous media model results minus the fully-resolved simulation values.
Figure 8 shows the transverse distribution of mean streamwise velocity and TKE on the upstream and downstream cross-sections at
. To reduce the length of this paper, we only plot the results from the case of
and
, which gives better results than the case of
and
. The red dashed rectangle represents the location of ELJ. As shown in
Figure 7a and
Figure 8, for the upstream cross-section, large velocity errors mainly occur near the bed for small
. When
is small (1 mm), the porosity model underestimates the streamwise velocity at the upstream side of ELJ, which is caused by the lower permeability suppressing the flow. The velocity suppression is more significant near the bank and free surface. In contrast, larger
leads to higher permeability and thus overestimates the streamwise velocity in the upstream side of ELJ. For the downstream cross-sections, error occurs mainly in the shear layer. All porosity model cases underestimate the streamwise velocity in the shear layer, with slight overestimation in the near bed region. This near-bed overestimation is less significant when
has the value of the key element diameter. For the TKE distribution shown in
Figure 7b and
Figure 8, the error distribution is different from velocity. With
32 mm and 19 mm, TKE was under-predicted in the shear layer. However, if
is very small (1 mm), TKE was dramatically over-predicted in the shear layer. This is caused by the fact that lower permeability of the porous media region forces more flow to the main channel and generates a larger velocity gradient in the shear layer.
In addition to the overall difference of velocity and TKE, an integral error over a cross-sectional plane can be defined as:
where
denotes the total area of the surface,
denotes the area for the
i-th sub-area and
denotes the flow variable on the
i-th sub-area. As shown in
Table 1, the general trend is that smaller
resulted in smaller permeability, and the error
ε became larger for both upstream and downstream velocity and TKE. However, the drag force error does not always follow this trend. For all six calibration cases, the results also show that the downstream error
ε is always larger than the upstream due to the rich turbulent dynamics behind the ELJ. As seen in
Table 1, the solid barrier case also over-predicted the drag force, and the error is much larger than that of the porosity case. The solid barrier case is closer to the
1 mm porosity case in terms of drag force and spacial integral error.
Overall, based on the results for drag force and error ε, the combination of , and mm gave the best results and, thus, was selected for further analysis.
3.3. Evaluation of Different Representations in the CFD Model
The effects of ELJ representation were investigated with three cases, including the validated fully-resolved case, the calibrated porous media case and the solid barrier case. The aspects of the ELJ hydraulics evaluated included flow adjustment, near-surface and near-bottom flow and turbulence and flow structure.
3.3.1. Flow Adjustment
To the best of our knowledge, there is no quantitative study on how flow will adjust when it passes an in-stream structure, which is usually bounded by stream banks. However, some recent studies investigated unbounded flow around finite and semi-finite (porous) vegetation patches [
38,
39,
40]. They revealed the relationship between the porous media properties and the flow adjustment in terms of streamwise velocity and wake length. We used similar analysis with the aim to find comparable relationships. As in previous studies [
39,
40], the flow domain can be partitioned into three regions as shown in
Figure 9, namely the upstream adjustment region, the interior adjustment region and the wake region. For unbounded wake flows, two wake length scales (
and
) can be defined.
denotes the length scale of the steady wake region where the velocity remains almost constant.
denotes the length scale of the distance between the structure and the von Karmen vortex street. However, Kim et al. [
41] showed that the von Karmen vortex street cannot form in semi-bounded flows, which is consistent with the numerical results in this paper. These two lengths also manifest themselves in the signature of turbulence.
is approximately located at the first peak of turbulence intensity in the wake region, while
is located at the second peak. Chen et al. [
40] proposed the following empirical wake length equations:
where
can be calculated with Equation (
13).
, where
m is the element number per unit planar area and
d is the diameter of the cylindrical element used in Chen et al. [
40].
denotes the flow blockage due to the obstruction. For ELJ, we use the key elements to approximate
n and
d. As a result,
m and
m
m).
Chen et al. [
40] measured the centerline velocity at half depth in flume experiments to represent the depth-averaged velocity for their unbounded flow around a porous vegetation patch. In this study, since we have the simulated velocity distribution around ELJ, local cross-sectional averaging was performed within the adjustment and wake regions. Local regional averaging is more suitable for this semi-bounded flow.
Figure 10 shows the longitudinal profiles of cross-sectionally-averaged streamwise velocity and TKE within the adjustment and wake regions. For the velocity profile, due to the semi-boundedness (wall effect), the incoming normalized streamwise velocity is only about 0.43. The trend of the velocity profiles is in accordance with that in Chen et al. [
40]. In the steady wake region, the larger pore size (
) case tends to match better with the fully-resolved case. The velocity profiles of solid barrier and small pore size case are similar, which makes sense because the small pore size reduces the permeability. They both show negative streamwise velocity inside the steady wake region. This is induced by the strong boundary layer separation and vortex generated at the edge of the object. With the increase of porosity, more passing-through flow reduces the negative pressure gradient, pushes the wake flow further downstream and thus enlarges the wake. For TKE, it has a relatively large value at the upstream side of ELJ edge and a small value at the immediate downstream side where the area is shielded. After that, TKE increases rapidly in the steady wake region. We also found that the growth rate and asymptotic TKE value are affected by the permeability and the resulted flow passing through ELJ. Reduced pass-through flow will induce higher shear, more eddies and, thus, higher TKE in the steady wake region. Neither the porosity model cases, nor the solid barrier case reproduced a good match on TKE in the whole of the wake zones. The porosity model with a large pore size matched with the fully-resolved case in the steady wake zone, while the solid barrier case matched better further downstream. For all porosity model cases, they tend to maintain high turbulence even after the steady wake region.
Table 2 shows the simulated
and
from the cases shown in
Figure 10. The length scales were determined by the peak of turbulent intensity. However, we found that for the porosity cases, it was very difficult to determine the exact value of
due to the lack of clear peaks, as shown in
Figure 10. Thus, only the range of
is presented. The results indicate that the porosity model with
= 19 mm produced a better match on
with the fully-resolved case. Again, the increase of
, i.e., porosity, will increase the wake length
. According to Equations (
13), (
18) and (
19), if we use the channel mean velocity in the equations, we obtain
,
and
, which are much larger than the values in
Table 2. However, if the mean velocity of the whole channel was replaced by the local mean value experienced by the ELJ, i.e.,
, we obtain
,
and
. Obviously, the latter is closer to the simulated values. The replacement of incoming flow velocity with the local mean is reasonable and necessary because of the side wall effects.
3.3.2. Near-Surface Velocity
Another important metric is the near-surface velocity around the in-stream structure, which impacts a host of processes, such as aeration and heat transfer.
Figure 11 shows the time-averaged streamwise velocity distribution at 5 mm below the free surface for different cases. For the porosity model, only the case with
,
and
mm is shown. The comparison shows that the surface velocity is not so sensitive to how ELJ is modeled. The only noticeable difference is in the steady wake region. For the fully-resolved case and the porosity case, there is only a small region showing negative streamwise velocity. However, for the solid barrier case, a much larger region has negative streamwise velocity, which is due to the fact that no passing-through flow causes much stronger recirculation behind the solid barrier. This result is consistent with
Figure 10.
3.3.3. Near-Bed Turbulence
In-stream structures also affect river morphology by changing sediment transport. Thus, near-bed turbulence, the driver of sediment motion, is evaluated. The time-averaged TKE near the bed is used as a proxy for sediment entrainment capacity [
42]. The near-bed TKE distributions are plotted in
Figure 12. The data were sampled at a height of 15 mm above the bed. From the comparison, it is observed that the porosity model gave a better match with the fully-resolved case. The porosity model underestimated TKE in the shear layer, and the shear layer itself was delayed. The solid barrier case greatly overestimated TKE in the shear layer. Thus, from the sediment transport point of view, the solid barrier approximation to the ELJ will over-predict the scour hole around the structure.
3.3.4. Flow Structure
The effects of ELJ representation in models can be evaluated at the simulated flow features.
Figure 13 shows an instantaneous flow field (streamlines) in the wake zone, which is colored by the magnitude of instantaneous velocity. These streamlines were sourced from a horizontal line parallel to the downstream side of ELJ. The source line was located at 50 mm above the bed. As can be seen in the figure, significant discrepancy exists in the steady wake zone in different model representations. For the fully-resolved case, small and irregular flow curvatures can be observed in the steady wake zone, most of which stem from the element surface in ELJ. The incoming flow has to go through the irregular pores within ELJ, and it results in very rich flow dynamics. In the porous media case, such dynamics are totally lost, and the streamlines are much smoother than the other two cases in the steady wake zone. Low permeability in the porous media region only slows down the flow through ELJ. The only similarity between the porous media case and the fully-resolved case is that the passing-through flow causes the delay of the shear layer. In the solid barrier case, a larger recirculation zone and a strong shear layer can be seen at the lee side of the block.
Figure 14 shows the instantaneous isosurfaces of
= 3.
is a commonly-used flow quantity to visualize the vortex structures [
43]. Substantial differences can be observed among the three cases. The fully-resolved case shows strong coherent flow structures inside the ELJ, also shown in
Figure 15. In the shear layer, it takes a long distance for the vortex to shed behind the ELJ to re-attach to the right bank. However, in the porous media case, no such coherent structure was observed inside the porous media domain at the same value of
. The vortex structure is also extremely weak in the near-field compared with the fully-resolved case. However, it shows similar vortex structure in the far-field as the fully-resolved case. In the solid barrier case, the contour of
= 3 shows comparable strong coherent structures as the fully-resolved case in the near-field shear layer. However, the shear layer of the solid barrier case is more diffusive in the far field than the other two cases. In addition, the reattachment distance is much smaller in the solid barrier case. This reduction of reattachment distance is caused by the lack of passing-through flow in the solid barrier case. Overall, in terms of flow structure, the porous media case models well in the far field, while the solid barrier case predicts better in the near field. Nevertheless, neither of them are able to resolve the flow structure inside the ELJ.
4. Conclusions
Using three-dimensional computational fluid dynamics modeling, we evaluated the effects of different modeling representations of ELJ, a popular in-stream structure for river restoration and erosion control. Three different modeling cases, i.e., a fully-resolved case, a porosity model case and a solid barrier case, were simulated. The computational model was validated with experimental data in a flume where a scaled model of ELJ was placed. The simulated results were examined and analyzed on various aspects related to the functionality and stability of ELJ. The applicabilities and limitations of each simplified representation of complex ELJ geometry were discussed.
Full resolution of the ELJ geometry is not feasible for wide use in practice because of the prohibitive computational cost. In this work, it served as the baseline for comparison with other methods. The fully-resolved case reproduced the velocity and TKE distributions well at the upstream and downstream cross-sections near the ELJ. When the porosity model was used, the porosity coefficients were first calibrated to get , , mm. We found that the equivalent grain size should scale as the key element diameter in this particular ELJ. The calibrated porosity model only showed satisfactory performance on drag force, velocity and TKE. The solid barrier case generally gave the least accurate results.
Further comparison showed that the modeling of the passing-through flow has a significant impact on the wake and the shear layer, which is summarized in
Table 3. The porosity model approximates the near-surface velocity and near-bed TKE in the steady wake region well, while the solid barrier model performed slightly better further downstream. According to the near-bed TKE, the shear layer is under-predicted in the porosity model and over-predicted in the solid barrier model. The solid barrier model can predict well the strong flow structure in the near-field shear layer, while the porosity model performs better in the far-field shear layer. However, neither the porosity model, nor the solid barrier model can capture the flow structure inside the ELJ. The flow adjustment analysis shows that patches with higher flow blockage (
) have larger wake length. The wake lengths
and
from Equations (
18) and (
19) agree well with simulated results only if the local velocity, instead of channel mean velocity, is used.
In conclusion, when the porosity model or solid barrier representation is used for ELJ, the benefits of the significantly reduced computational cost come with the price of missing physical information that might be important for the intended purpose of in-stream structures. The porosity model, after careful calibration, seems to perform better than the solid barrier model, especially the drag force. However, this work also identified substantial loss of accuracy in terms of velocity, turbulent shear, sediment transport capacity and coherent structure in the vicinity and the wake zone of ELJ. Thus, care needs to be taken to interpret and use the porosity model results in practice. If the emphasis is in the far field, then the simulation results using both the porosity model and the solid barrier representation did not show dramatic difference from the fully-resolved case. Therefore, the use of such simplified models may be justified.