*3.2. Model Replication of Mixing Processes within the Manhole*

– – Following validation of the hydrodynamic processes, the ability of the CFD model to simulate solute mixing within the manhole was tested by comparing measured and simulated concentration profiles within the pipe network (at the location of Cyclopes 2) for all hydraulic conditions. Solute injections for hydraulic conditions in Tests 1–3 were performed once using either a single or double pulse of solute concentration. Injections during hydraulic conditions in Tests 4–6 were repeated three times each, of which the first two had single pulse and the third had two consecutive concentration pulses. Measured and predicted solute concentration time series at manhole D/S (at the location of Cyclopes 2) were extracted compared to those of experimental data. Figure 4 shows comparison of experimentally measured and modelled concentration time series at the manhole downstream measurement point for each test.

• =

• =

• = 1 −

 ̅

1 

• =

 =1

∑ ( −

1  )

√∑ ( −

∑ (− ) 2 =1 ∑ (−̅) 2 =1 

) 2 =1

∑ (−̅)(−̅)

(−̅) 2

̅

 =1 √∑ (−̅) 2 =1

– **Figure 4.** Comparison of experimental and numerical unsteady concentration profiles in the sewer pipe downstream of the manhole (Cyclopes 2) and the measured concentration at the upstream of the manhole (Cyclopes 1). Tests 4–6 are repeated three times (i, ii, iii).

Different statistical parameters were used to check the quality of model performance in predicting the solute concentration at the downstream of the manhole. The parameters used are listed below.


$$\bullet \quad \text{Pearson product-moment correlation coefficient, } r = \frac{\sum\_{i=1}^{n} \left( \mathcal{O}\_{i} - \overline{\mathcal{O}} \right) \left( \mathcal{P}\_{i} - \overline{\mathcal{P}} \right)}{\sqrt{\sum\_{i=1}^{n} \left( \mathcal{O}\_{i} - \overline{\mathcal{O}} \right)^{2} \left( \mathcal{P}\_{i} - \overline{\mathcal{P}} \right)^{2}}}$$

$$\bullet \quad \text{Nash Studifté coefficient, } N\\\text{SC} = 1 - \frac{\sum\_{i=1}^{n} \left(O\_{i} - P\_{i}\right)^{2}}{\sum\_{i=1}^{n} \left(O\_{i} - \overline{O}\right)^{2}}$$

Where *O<sup>i</sup>* is the observation values, *P<sup>i</sup>* is the model predicted values, and *O* is the average of all observed values, *P* is the average of model predicted values and *n* is the number of observations. The calculated values of the mentioned statistical parameters are shown in Table 2. It shows that BIAS of all the comparisons is negligible. The NSC values are greater than 0.995 in all cases. Therefore, the results show that the model accurately replicates the solute mixing processes within the manhole.


**Table 2.** Statistical comparisons between the concentration time series between experimental and numerical models.

#### *3.3. Modelling of Soluble Mass Exchange to Surface Flows*

The solute transport model was then applied to Test 1-A-B-C, 2-A, 3-A and 4-A, as described in Section 2.3 (i.e., with a uniform solute applied directly to the manhole inlet boundary at Section A). Figure 5 shows example plots of concentration evolution inside the manhole for each of these tests at different time intervals. Time *t*<sup>0</sup> = 0 is taken when average solute concentration at Section A exceeds 1% of the peak value. Instantaneous velocity vectors are also displayed to indicate the travel paths of the solute concentration within the manhole volume.

– **Figure 5.** Instantaneous solute concentrations and velocity vectors at the central vertical plane of the manhole from different tests at time (*t*) from 0–20 s. The arrows are not drawn to scale.

Figure 5 shows that as soon as the solute mass enters the manhole, it diffuses from the high velocity flow region into the manhole volume. A part of the concentrated solute mass hits the opposite manhole wall and travels towards the manhole surface. Later, it interacts with the surface flow and recirculates to the manhole. This recirculating flow brings low concentration flow from the surface into the manhole, maintaining a consistent concentration gradient through the manhole height until the upper part becomes completely mixed. The observed flow structures explored in the tests are relativity insensitive to the pipe inflow rate over the partition ratios used in these tests. The results show that until well mixed conditions are achieved, that the concentration field at the manhole/surface interaction point (section C) is highly heterogeneous. Therefore unlike in the pipe network, (where cross-sectionally averaged values can be reasonably assumed at Sections A and B), quantification of mass flow rate to the surface (i.e., over Section C) requires robust understanding of the spatial variation of solute and velocity over the manhole cross section and how this evolves with time.

The evolution of solute mass exchange through each cross section A, B and C is quantified based on the CFD model. For this purpose, CFD model results of test 1-A-B-C, 2-A, 3-A and 4-A were considered. Due to highly heterogeneous conditions at section C, mass flow rate at each time step for each inlet/outlet junctions (section A, B, C) was calculated using the following Equation,

$$
\dot{M}\_{\mathbf{x}} = \int\_{i=0}^{i=A} c\_i \mathbf{u}\_i d\mathbf{A} \tag{10}
$$

where . *M<sup>x</sup>* is the solute mass flow rate though section A, B or C (i.e., . *MPI*, . *M<sup>e</sup>* or . *MPO*); *u<sup>i</sup>* is the mean velocity vector normal to area *i*; *dA* is an incremental cross section area vector (based on a 10mm slice); and *c<sup>i</sup>* is the solute concentration within area *i*. Hence the integral value of the dot multiplication of these components is used to provide the net mass flow rate through sections A, B and C. The model set-up (uniform concentration applied at Section A) results in a constant . *MPI* over each test after the first 0.2 s of simulation (as given in Table 3). Following the calculation of . *MPI*, . *M<sup>e</sup>* and . *MPO*, the rate of change in solute mass within the manhole was calculated using Equation (1). Figure 6 shows resulting outlet solute mass flow rates as a ratio of manhole inlet mass flow rate over each test. The time axis in the figures represents time (in seconds) since the first solute enters the manhole from the sewer inlet. As in Figure 5, this time (*t*<sup>0</sup> = 0) is taken when average solute concentration at Section A exceeds 1% of the peak value. Significant fluctuation can be observed in the . *Me* . *MPI* values due to the complex heterogeneous nature of the flow at the surface/manhole interaction point (section C).


**Table 3.** Characteristic time scales of solute mixing from different model results. Results are arranged in an ascending order of the mean surface flow partition ratios.

Characteristic time scales, as described in Section 2.2, are defined for each test and are presented in Table 3. Similar to the definition of *t*0, *t*<sup>1</sup> and *t*<sup>2</sup> are taken when averaged solute concentration at Sections B and C exceeds 1% of the peak value, respectively. *t*<sup>3</sup> is defined as the time when *dM<sup>m</sup> dt* falls below 2.5% of its peak value for the first time. Timescales in Table 3 are presented non-dimensionally *Water* **2020**, *12*, 2514

 ̇ 

> ̇

 ̇

in terms of the nominal manhole residence time for flow passing to the surface *Tx*, as calculated using Equation (11),

 = ∫ =

=0

 ̇

$$T\_{\chi} = \frac{L\_{\chi}\frac{\pi}{4}(\Phi\_m)^2}{Q\_{\ell}} \tag{11}$$

̇ , ̇   ̇ 

> ̇ ̇ , ̇

where *L<sup>x</sup>* is the vertical distance between the sewer pipe axis and the manhole top and Φ*<sup>m</sup>* is the manhole cross sectional area. ̇ 

 **Figure 6.** (**a**) Mass exchange ratio at the manhole to sewer pipe outlet from different test results. Horizontal lines indicate *QPO*/*QinS* values for each test. (**b**) Mass exchange ratio at the manhole to surface connection with fitted asymptotic trend lines. Horizontal lines indicate *Qe*/*QinS* values for each test. (**c**) Change in solute mass within the manhole.

As can be seen in Figure 6a,b, for each test the proportion of mass flow rate entering the pipe outlet (*MPO*/*MPI* ) and surface flow (*Me*/*MPI* ) grows asymptotically toward the relevant flow partition ratio (as defined in Equations (2) and (3)). Therefore, solute mass flow exchange to the surface can be described using the following function,

$$\frac{M\_{\varepsilon}}{M\_{\text{Pin}}} = \left(\frac{-1}{\mathbb{C}(t - t\_{2}) + 1} + 1\right) \frac{Q\_{\varepsilon}}{Q\_{\text{inS}}} \tag{12}$$

where *C* is an empirical coefficient. The best fit value of C and resulting goodness of fit (r 2 ) value between fitted equation and CFD model results for each test conducted in this work is given in Table 3. The first arrival of mass at the pipe exit (*t*1) and the surface flow (*t*2) occurs relatively quickly in all conditions (0.09 s < *t*<sup>1</sup> < 0.13 s and 0.40 s < *t*<sup>2</sup> < 0.82 s), while the timescale for complete mixing (*t*3) to be achieved (and thus mass flow rate to the surface flow to become approximately steady and equivalent to the surface flow partition ratio) varies significantly over the tests conducted (13.6 s < *t*<sup>3</sup> < 24.0 s). From

Table 3 the value of C and the non-dimensional timescale (*t*1,2,3/*Tx*) to achieve well mixed conditions tend to increase with the flow partition ratio over the range of conditions tested.

#### **4. Discussion**

A comparison of experimentally measured and modelled discharges within a scaled manhole structure shows that, given knowledge of the boundary conditions, the RANS CFD approach accurately simulates flow exchange from piped to surface flows (within 1.7% in all test cases). Therefore, steady-state flow exchange through similar hydraulic structures during flood events is likely to be well described using RANS CFD. These results concur with previous validation studies utilising similar 3D modelling approaches to simulate hydrodynamics in urban drainage structures (see, e.g., in [29,45]), although in this case the complex interaction with surface flows as well as a solute transport is also recreated. Such models are too computationally expensive to be used in direct flood modelling applications; however, there is further potential to utilise such complex models to evaluate simpler semiempirical weir/orifice relationships currently used to describe surface/sewer interaction. Such semiempirical relationships have been found to be sensitive to interaction structure type, inlet characteristics and geometry as well as unsteady hydraulic conditions [12,30,46,47], and thus benefit from case-specific calibration. Similarly, the calibrated model has been shown to accurately reproduce solute concentration profiles (and thus mass flow rates) measured downstream of the manhole structure under a range of flow rates during cases where sewer flow interacts with surface flood water. Taken together with the agreement of modelled and measured flow partition within the manhole, as well as past results comparing CFD velocity vectors against those obtained using PIV measurement in the same facility [29], this result gives confidence that the CFD model can reproduce flow details and resulting solute mass exchange to the surface during flood conditions. A full validation would benefit from having access to measured values of concentration/mass exchange at the interaction point between sewer and surface flows (section C); however, the current results have demonstrated the hydraulic complexity and spatial heterogeneity of concentration at this position. Therefore, such validation measurements would require complex instrumentation such as Laser-Induced Fluorescence (LIF) to provide detailed spatial data over the manhole area. In addition, further validation of CFD approaches would be valuable in more complex hydraulic conditions (e.g., unsteady flow), in systems with different geometrical features or at different scales or in cases involving sediments which are also commonly present in urban drainage networks and may be susceptible to transportation in flood water.

The modelled flow structures illustrate the complexity of the interaction between surcharging manhole flow and surface flood water; however, flow structures within the manhole appear to be relativity insensitive to the pipe inflow rate over the flow partition ratios explored in these tests. The solute transport and resulting mass flow rates within the system are a process of both advection and diffusion. The solute transport from the manhole inlet to the manhole pipe outlet is dominated by the advection process due to the strong local velocities in this zone. Thus, first arrival time to the sewer outlet (*t*1) is dominated by the sewer inlet velocity with little subsequent variation within these tests. In addition, as the flow partition ratio increases (i.e., more flow is transported to the surface) the corresponding timescales for first arrival of mass at the surface (*t*2) and complete mixing within the manhole (*t*3) decrease slightly due to the increasing advection through the manhole structure to the surface. However, a stronger positive relationship is observed between non-dimensional timescales based on the characteristic manhole residence time (*t*2/*Tx*, *t*3/*Tx*) and surface partition ratio (*Qe*/*QinS*), indicating the relative significance of conical flow structures produced by the inlet pipe and subsequent diffusive mixing processes in the tests conducted.

The work has shown that the sharp arrival of a well-mixed solute at an open manhole results in an asymptotic growth of mass exchange to the surface, converging to a value that is defined by the hydraulic flow partition. Parameterisation of an asymptotic growth function (*C*) may be related to the flow partition ratio and/or the characteristic residence time, with more rapid mixing occurring at lower residence times. Approximately well-mixed conditions (and associated equivalence of sewer to surface solute mass exchange s and flow partition ratios) occur at between 5.8 and 9.2 times the manhole residence time over the conditions tested here. Further work is required to explore these relationships over a range of manhole geometries, using different (time varying) solute injection profiles and unsteady hydraulic conditions as well as in other exchange structures such as gullies featuring grills/covers, including at full scale, such that realistic timescales in real situations can be established. A more complete understanding of this problem should also consider the transport of solids, such as fine sediments and entrained material, which are also present in urban drainage networks. In addition, other flood scenarios (e.g., further exploring the influence of surface flow depth and velocity) and cases where the majority of flow transfers to the surface (*Q<sup>e</sup>* /*QinS* > 0.5) could be explored. In such cases where the surface flow partition ratio is significantly larger, the bulk advection of solute by the flow is likely to increasingly dominate diffusivity arising from local flow structures.

## **5. Conclusions**

A 3D CFD model was applied to simulate flows in an exchange structure involving interacting pipe and surface flows to quantify flow and soluble pollutant mass exchange. The model was validated with a laboratory-scale model, achieving differences of less than 1.7% in flow rates and excellent statistical comparisons between observed and modelled concentration time series. This suggests that a RANS CFD approach is an appropriate methodology to evaluate flow partition and to evaluate how soluble pollutants move from sewer networks to surface flood flows.

The model was extended to different conditions to understand the effects of the manhole separately from the pipe network, and used to calculate the evolution of solute mass transport rate through each manhole open boundary cross section under a range of flow conditions including interactions between sewer flows and surface flood water. A sharp arrival of solute into the structure is shown to result in an asymptotic growth of solute mass exchange ratio to the surface converging to a value equal to the surface flow partition ratio. An analysis of the results demonstrates that the timescales to achieve this convergence are dependent on the diffusive transport inside the structure.

The work in this paper describes initial steps to understand the risks of soluble material from sewer networks entering urban flood waters via exchange structures. The transport of pollutants through these structures will also depend on additional factors including, but not limited to, the presence of manhole coverings and change of structure geometry/shape. In order to build a more complete understanding, such that risks to public health can be understood and quantified, requires significantly more work. This includes further consideration of transport and transformations of both contaminated sediments and soluble materials in urban drainage networks as well as datasets from urban drainage networks, floodwaters and urban surfaces such that transport, survival and fate can be modelled within quantifiable uncertainty bounds.

**Author Contributions:** Conceptualization, R.F.C., J.D.S. and M.R.; methodology, M.N.A.B. and M.R.; software, M.N.A.B. and R.F.C.; validation, M.N.A.B.; formal analysis, M.N.A.B.; writing—original draft preparation, M.N.B. and J.S.; writing—review and editing, J.D.S., M.R. and R.F.C.; visualization, M.N.A.B.; supervision, J.D.S. and R.F.C. All authors have read and agreed to the published version of the manuscript.

**Funding:** The research was supported by the UK Engineering and Physical Sciences Research Council (EP/K040405/1). Author Beg MNA worked as part of QUICS (Quantifying Uncertainty in Integrated Catchment Studies) project which received funding from the European Union's Seventh Framework Programme (Grant agreement No. 607000). Cluster computing system used in this study is supported by the Laboratory for Advanced Computing of the University of Coimbra. Authors would also like acknowledge the support of FCT (Portuguese Foundation for Science and Technology) through the Project UIDB/04292/2020-MARE.

**Conflicts of Interest:** The authors declare no conflict of interest.

**Data Available:** Additional datasets associated with this work can be downloaded from https://zenodo.org/ communities/floodinteract/.
