NUMERICAL RESEARCH OF THE INFLOW INTO DIFFERENT GULLIES OUTLETS

Carvalho Rita F.1, Leandro Jorge1, David Luís M.2, Martins, Ricardo 3, Melo Nuno4

1,3 IMAR-Institute of Marine Research – Marine and Environmental Research Centre, Department of Civil

Engineering, University of Coimbra /1Assistant Professor; 3Student - Coimbra/ Portugal 2LNEC- National Laboratory of Civil Engineering/Research Officer- Lisboa/Portugal

4IPG- Instituto Politécnico da Guarda/Assistant - Guarda/Portugal

1ritalmfc@dec.uc.pt, leandro@dec.uc.pt,

2ldavid@lnec.pt ,

3ricardo.mm@sapo.pt,

4nuno_melo@ipg.pt

Abstract

Gullies serve the purpose of draining the overland flow from roadways into the sewer system. The inflow into a

gully depends on: the flow and water depth on the surface, the inlet area, the dimension, geometry and slope

from the different devices ditch/street, gutter and also on the location of the outlet,. A numerical study was

conducted using an in-house numerical model based on VOF (Volume of Fluid) / FAVOR (Fractional Area

Volume Obstacle Representation) concepts, to study the gully with different location of the outlet, the connection

between the gully and the sewer drainage system. Connections placed in different places along the gully bottom

were studied with a detailed geometry by a FAVOR / VOF in-house numerical model in a 2DV plan (i.e. 2

dimensional plan vertical versus horizontal). Water depths, stream lines, velocity, pressure fields and the inlet

discharge coefficients are compared for the different gullies with different outlet locations and different flow

rates. Data from the numerical model provides high detailed information about the flow features to the

comparison of the different geometry efficiencies and improves our knowledge of the hydraulic processes

involved in transitional regimes during flooding. The Results will be part of larger database that will be created

to validate the applicability of integrated numerical models.

Keywords Drainage systems, gullies, discharge coefficients, numerical simulations, VOF

1. INTRODUCTION

Gullies are extensively used to intercept runoff from roadways, streets and other public and private pavements to

the underground sewer system, but different geometries may be found throughout the world. It seems their

geometry and details follow some regional traditional criteria and that sometimes they are constrained by the

street and sewer system location. Details are also commonly left to chance. Figure 1 presents two gullies where

the connection to the drainage system is vertical and is located in two different positions along the longitudinal

axis: at the center (a) and slightly to downstream (b) (see also Figure 2 b) and c)).

a) b)

Figure 1. View of two gullies with different location of the connection to the pipe system: a) located at the

center; b) located slightly downstream in the longitudinal direction

The inlet efficiency of gullies is not well known and has to be different from gully to gully since its discharge

coefficient depends mainly on the geometry, the hydraulic head, and the velocity field. In [1] a numerical study

for a wide range of discharges through a gully with central outlet concluded that the discharge through the gully

was greatly dependent on (1) the velocity field and (2) the water depth oscillations in the gully (particularly for

larger flow discharges). It was founded that larger flow discharges could be better drained through the gully by

varying the upstream channel Froude number and the water depth. Finding gully discharges relations are most

valuable for characterizing the linking elements found in Dual-Drainage models [2].

Figure 2. Gullies longitudinal profile view with different vertical connection to the pipe systems

This paper aims to study the hydraulic behaviour of gullies featuring different locations of the vertical outlet that

connects the gully to the underground sewer system. A FAVOR/VOF, 2DV in house numerical model was used

to simulate the flow in four different gully configurations, for a wide range of flows. Figure 2 presents the

longitudinal cross sections of the four gully configurations with the outlet located at the entrance corner

(x=1.24), at the center (x=1.5), at ¾ of the length (x=1.65), and at the end of the gully (x=1.76). Numerical

model, simulations and their discussion are presented in section 2 and 3. Final section summarises the main

conclusion.

2. NUMERICAL MODEL

The numerical model is based on the Navier-Stokes equations governing the motion of the 2D incompressible

flows in the plane x z− , in which the free surface is described using a refined Volume-Of-Fluid (VOF) method

[3] and the internal obstacles are described by means of the Fractional Area-Volume Obstacle Representation

(FAVOR) algorithm [4]. According to this description, the VOF-function ( ), ,F F x z t= and porosity

function ( ),x zθ θ= is included in the mass and momentum conservation equations (Navier-Stokes equations).

( ), ,F F x z t= value is 1 for a point occupied by the fluid and zero elsewhere, and ( ),x zθ θ= value is 0 for

a point inside an obstacle and 1 for a point that can be occupied by the fluid. These equations governing the time

evolution as well as details of the numerical model can be found in [5].

The computational domain of the flow was defined by a 2DV rectangle 3 m long (0.0 m < x < 3.0 m ) and 1.0 m

high (0.0 m < z < 1.0 m) representing a 1% sloped piece of the street pavement, using a grid of 160 × 90 cells

with variable x∆ and z∆ (0.01 m minimum). The 2DV domain aims to represent a vertical cross-section

(central section) in a gully with 0.6 m × 0.3 m of dimensions. Since the gully is symmetrical to this plan, there is

no influence of the transversal direction and 2DV is acceptable. The simulations presented were made by

considering the geometry represented by defining the fractional areas and volumes, so as to block the desired

cells to the flow according to the mathematical formula represented the configuration 1 to 4 by the following

equations (1 to 4) :

<<Λ+−<

<<Λ+−<

<<Λ+−<

0.38.1618.001.0

8.128.13012.0001.0

2.10.0612.001.0

xxz

xxz

xxz

(1)

<<Λ+−<

<<Λ+−<

<<Λ+−<

<<Λ+−<

0.38.1618.001.0

8.154.13008.0001.0

46.12.13012.0001.0

2.10.0612.001.0

xxz

xxz

xxz

xxz

(2)

<<Λ+−<

<<Λ+−<

<<Λ+−<

<<Λ+−<

0.38.1618.001.0

8.169.13008.0001.0

61.12.13012.0001.0

2.10.0612.001.0

xxz

xxz

xxz

xxz

(3)

<<Λ+−<

<<Λ+−<

<<Λ+−<

0.38.1618.001.0

72.12.13012.0001.0

2.10.0612.001.0

xxz

xxz

xxz

(4)

The upstream inflow boundary condition, at x=0, was set according to supercritical uniform conditions in a

hypothetical 0.5 m wide channel and 1% slope street, with fixed height, inh , and uniform velocity,

inU = ( , )

in inu w , along the depth. The inflow condition for the different discharges is presented in Table 1. The

left and bottom boundaries were set as free outflow. In order to minimize the development of instabilities, the

computer simulations were done with a special initial condition. The free-surface was defined by considering

upstream the gully, water depth (inh ), and velocity values ( , )

in inu w (0 < x < 1.2) decreasing from x=1.2 to

zero at x=1.8 where a smaller water depth was consider in

h /2; at the bottom outlet the discharge was calculated

according the discharge formula with discharge coefficient equal to 0.5. This condition, although is not

physically possible, allowed for a stable and convergent solution to be obtained after the first cycle. Thus the

physical behaviour of the flow was realistically reproduced.

Table 1. Inflow uniform boundary conditions of numerical investigation programme

Simulation name qin (l/s/m) inh

(m) inu

(m/s) Fr

Q2 40 0.041 0.97 1.52

Q4 80 0.064 1.24 1.56

Q5 100 0.075 1.34 1.57

Q6 120 0.084 1.42 1.57

Q7 140 0.093 1.50 1.57

Q8 160 0.102 1.57 1.56

Q10 200 0.119 1.68 1.55

Q12 240 0.135 1.78 1.54

Q15 300 0.158 1.90 1.52

Q20 400 0.194 2.06 1.49

3. SIMULATIONS RESULTS, FLOW CHARACTERIZATION AND DISCUSSION

The simulations were first done for the outlet located at the centre bottom (Config. 2;x=1.5; Figure 2b, Eq.(2))

for all the discharges, Q2 to Q20 (40 to 400 l/s/m; Table 1). In these simulations the influence of the flow

velocity and water depth at the top entrance of the gully was demonstrated, namely in the direction of the

streamlines around the bottom outlet and in the discharge coefficients [1]. Different locations of the gully outlet

were tested for the same discharges: outlet at the upstream corner bottom (Config. 1; x=1.24; Figure 2a, Eq. (1));

between the centre and the downstream corner bottom (Config. 3; x=1.65; Figure 2c, Eq. (3)) and at the

downstream corner bottom (Config. 4; x=1.74; Figure 2d, Eq. (4)). In Table 2 the results are presented for all

configurations and discharges, for the time steps that better describe the simulation.

It can be observed that the main flow is done trough the outlet for the majority of the discharges. However,

different factors influence the flow discharge through the gully as the flow trajectory and velocity field in the

gully change, depending on the initial water depth in the gully, on the relation between discharge in the channel

and the outlet discharge capacity and on the distance between the location where jet tend to reach and the outlet.

Table 2. Regime definition on the gully

Conf. 1 Config. 2 Config. 3 Config. 4 Reg Q2

t=5.2 s

t=2.5 s

t=4.7 s

t=3 s

R1

Q4

t=3.8 s

t= 9 s

t=5.5 s

t=2.9 s

Q5

t=3.8 s

t=40 s

t=20s

t=4.6 s

Q6

t=3.5 s

t=40 s

t=20 s

t=21.1 s Q7

t=3 s

t=40 s

t=20 s

t=80 s R2

Q8

t=3 s

t=70 s

t=20 s

t=40s

Q10

t=3 s

t=13.6 s

t=20 s

t=20 s

R3

Q12

t=3 s

t=8 s

t=4 s

t=14 s

Q15

t=2.8 s

t=8.5 s

t=13.6 s

t=2.37s

Q20

t=2 s

t=9 s

t=4.5 s

t=1.6 s

We propose a Regime classification based on the one defined for the drop manhole [6] as follows:

• R1 – The main flow direction hits the box floor upstream of the pipe outlet; it occurs for small discharges

from configurations 2 to 4: R1a – the flow streamlines entering the pipe stay glued to the outlet left corner

(e.g. Config. 4, Q4-Q5); R1b – the flow energy is enough to deflect the jet entering the pipe a contraction is

seen in the outlet left corner (e.g. Config.4, Q6 and Config.2, Q2); R1c- the jet entering the box now reaches

the upstream corner of the pipe causing a slight asymmetry in the streamlines entering the pipe (Config.2,

Q5);

• R2 - The main flow direction hits the pipe outlet, the streamlines are almost symmetric and both water depth

and water volume become very stable: R2a – the jet streamlines enter directly into the pipe outlet, guided by

the existence of a vortex created upstream the pipe outlet; R2b - the main flow follows the outlet because of

the natural trend of the jet trajectory (these two transitions are difficult to identify e.g. Config. 2, Q6-Q7);

R2c – the jet streamline is deflected and guided into the pipe outlet due to the vortex created downstream,

consequently the outlet sees an increase of discharge capacity (e.g. Config. 2, Q8);

• R3 – The main flow direction tends to hit the area downstream of the pipe outlet, the streamlines close to the

pipe outlet become non symmetric causing a decrease of the discharge capacity; occurs only for the larger

upstream discharges: R3a - water depth at the gully is maintained, a mirror equilibrium similar to R1c could

be reached (e.g. Config. 1, Q15); R3b the jet hits to far downstream the pipe outlet such that the main flow is

not directed into the outlet and thus the majority of the upstream discharge flows downstream (e.g. Config. 1,

Q20).

These regimes depend on the initial water depth, the natural jet trajectory and the relationship between the

discharge flow in the channel and the outlet discharge capacity: 1) if the discharge in the channel is much lower

than the outlet discharge capacity, the water depth and water volume decrease faster and the regime depends

only on the jet trajectory; 2) if it is approximately one and if there is water at the gully, the velocity field in the

gully may be influenced by upstream or downstream vortices, and the jet forced into the pipe outlet (R2); 3) if

the discharge in the channel is greater than the outlet capacity, the water depth and water volume may be initially

increase, however vortices on the gully could be formed and the jet deflect away from the pipe outlet (R2c)

increasing its capacity. On the other hand, if the jet tends to hit gully wall to far from the pipe outlet, the vortices

upstream become unable to deflect the flow into the pipe outlet and its discharge becomes lower than the

discharge in the channel. This discussion is also supported on the study of Conf.2 in [1] where water volumes

( F x z∆ ∆ ) in all cells (for the total domain) were calculated and plotted throughout the entire time of the

simulation and where it was possible to analyse the variation of the water volume and conclude of its

stabilization.

Velocity and pressure profiles at several locations were considered to quantify water depth and discharges to

outlet and to the downstream channel as well as streamlines were designed in the gully to measure contraction

coefficient at the gully entrance and outlet sections. Figure 3 illustrates the water depth at the upstream (x= 1.2

m) and downstream channel (x= 1.8 m) (Figure 2) for runs that became stable, contraction coefficient in the

gully entrance (Sc1, z = 0.29 m) and pipe outlet section (Sc2, z = 0.59 m).

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

0.3

0.4

0.5

0.6

0.7

0.8

0 100 200 300 400

q2/q

h (m)

qin (l/s/m)

h1 Config.2 h1 Config.3 h1 Config.4

h2 Config.2 h2 Config.3 h2 Config.4

h3 Config.2 h3 Config.3 h3 Config.4

q3 Config.2 q3 Config.3 q3 Config.4

Q

inside the gully

0 2 4 6 8 10 12 14 16 18 20

Fr1>1 Fr1<1

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

-0.4

-0.2

0

0.2

0.4

0.6

0.8

0 100 200 300 400

Cc2Cc1

qin (l/s/m)

Cc1 Config.1 Cc1 Config.2 Cc1 Config.3 Cc1 Config.4

Cc2 Config.1 Cc2 Config.2 Cc2 Config.3 Cc2 Config.4

Q 0 2 4 6 8 10 12 14 16 18 20

Figure 3. Water depth for x=1.2 m (h1); x=1.5 m (h2) and x=1.8 m (h3), flow discharge to downstream channel

(q3/q1); contraction coefficient at the pipe outlet (Sc1/S1) and gully entrance (Sc2/S2)

Figure 3 raises the following discussion:

• The upstream flow for large discharges becomes dependent on the flow conditions at the gully, i.e. for

discharges larger than 160 l/s/m (Q8) in Config. 3 and 4 the water depth at the gully causes the formation of a

smooth hydraulic jump in the upstream channel (backwater effect); Config. 4 presents the highest values of

water depth, the highest water level reached for x=1.2 (h1) and x=1.5 m (h2) corresponds to the larger

discharge (Q20); at the gully, the highest water depths are found for Conf. 3 until Q7 which then shifts to

Conf.4;

• The highest value for x=1.8 m (h3) is found for Q15 in Config 3; for discharges smaller than Q8 and greater

than Q12, the highest water depths are found for Conf. 3 and for intermediate discharges (Q8 to Q12) shifts

to Conf. 4; this is due to placement of the pipe outlet which stands now closer to the gully downstream end,

for higher discharges the water depths above the pipe outlet becomes higher;

• The discharge trough the pipe outlet (q2) is greatly influenced by the pipe outlet configuration which controls

the velocity field inside the gully; while config. 2 is able to drain all tested discharges through the orifice,

conf. 4 is the configuration with less discharge capacity. Config 3 has a discharge capacity similar to Config

2 except for Q20 when R3 regime is reached.

• Contraction coefficient at the pipe outlet varies between 0.3 and 0.7 and at the gully entrance varies between

0.1 and 0.6; largest contraction coefficient is found for the pipe outlet Config. nº3, and lowest for the

Config.1 (the latter was only estimated for discharges larger than Q6); for the gully entrance, largest values

are also found for Config.3, and lowest values for for Config. 2. Conf. 2 and Conf.3 were found to have the

lowest Cc variations, thus presumably more stable throughout the whole discharge range.

4. CONCLUSIONS

A typical urban drainage gully composed of a 0.6 × 0.3 m2 (length × high) and a 0.08 m outlet located at

different distances along the gully bottom located in a channel with a 1% slope was studied for different

discharges, using a 2DV numerical model. The numerical model used was able to predict complex air-water

interface and turbulence flow. The velocity field was the main responsible for altering gully efficiency and

hydraulic behaviour and it depends deeply on the outlet location. This determines the symmetry of the

streamlines at the outlet and thus the contraction and the discharge coefficient. Three parameters were identified

as the main factors: 1) relation between the discharge at the channel and the outlet discharge capacity, 2) the

relative distance from the area where the jet hits and the outlet location and 3) initial water depth at the gully.

Different regimes were identified based on the regimes defined by the drop manhole.

5. ACKNOWLEDGEMENT

The research presented in this paper is funded by the Foundation for Science and Technology through projects

PTDC/AAC-AMB/101197/2008 and PTDC/ECM/105446/2008.

References

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