Journal of Applied Fluid Mechanics, Vol. 14, No. 3, pp. 681-690, 2021.
Available online at www.jafmonline.net, ISSN 1735-3572, EISSN 1735-3645.
DOI: 10.47176/jafm.14.03.31815
Stability of Solids in Stepped Flume Nappe Flows:
Subsidies for Human Stability in Flows
H. de B. Ribeiro1, A. L. A. Simões1†, L. D. da Luz1, L. S. G. Mangieri2 and H. E. Schulz3,4
1
Federal University of Bahia, Salvador, Bahia, 40210-630, Brazil
2 Salvador University, Salvador, Bahia, 41770-235, Brazil
3 Hydro-Engineering Solutions (Hydro-LLC), Auburn, Alabama, 36830, USA
4 São Paulo University, São Carlos, São Paulo, 13566-590, Brazil
†Corresponding Author Email: andre.simoes@ufba.br
(Received May 12, 2020; accepted October 8, 2020)
ABSTRACT
Knowing the details of the interaction between people and runoff flows caused by heavy rainfall or by floods
due for example by the rupture of reservoirs or dams is essential to prevent accidents with humans. There are
information in the literature on the equilibrium capacity of individuals partially immersed in flows occurring
in flat-bottomed channels, but there are many gaps regarding the use of urban draining staircases during the
occurrence of rainfalls that generate runoff over their steps, and their impact on people. This study considered
the effect of the flow on the stability of five obstacles positioned on one of the steps of a reduced model of a
draining staircase. The results were used to calculate dimensionless parameters which involve the mass and
height of the obstacle, the water density, critical depth of the flow and step height. These parameters were
justified by a fundamental toppling and drag formulation, and good correlations between the obtained
dimensionless parameters were obtained following adequate power laws. Comparisons between the data
obtained in the present reduced model of staircase and literature data of flat bottom channels showed similar
behaviors. Finally, a scaling procedure to compare results of different scales and situations was also
presented. Excellent correlations using different literature data and those of the present study were obtained.
Keywords: Stepped chute; Draining staircases; Safety in floods; Stability of solids in flows.
NOMENCLATURE
A
b
Cd
e
F
Frt
f, G, J
g
h1
hc
H
H*
H*max
H*min
H*n
Io
l
m
M*
M*max
frontal area
width of the block
drag coefficient
block thickness
force
Froude number
generic functions
acceleration due to gravity
water column
critical depth
block height
dimensionless block height
maximum H*
minimum H*
normalized H*=H*/H*max
slope
step length
mass of the block
dimensionless mass
maximum M*
M*min
M*n
P
q
R
Ret
s
V
minimum M*
normalized M*
weight of the block
unit discharge
correlation coefficient
Reynolds number
step height
mean flow velocity
α
angle of attack
algebraic coefficients (i=1,2,3)
algebraic coefficients (i=1,2)
algebraic coefficients (i=1,2,3)
friction coefficient
kinematic viscosity
density of water
obstacle density
expoent
algebraic coefficients (i=1,2,3)
i
i
i
s
i
H. de B. Ribeiro et al. / JAFM, Vol. 14, No. 3, pp. 681-690, 2021.
1.
force meters, and which was fixed in a 50 m long
and 20 m wide basin. The authors proposed a
computational model for the instability of human
beings in flows that uses human characteristics
normalized with the height. Their results are used in
the literature to compare with data from several
other authors.
INTRODUCTION
Urban flooding has the potential to drag people,
causing loss of life or serious injury. There is a
growing concern across the globe with the
accelerated changes in the built environments
(urban regions), which have waterproofed large
areas of the urban soil surfaces and restricted the
regions of water evacuation, thereby generating
areas of flooding and rapid flow in environments
frequented by humans. Examples of the literature
which emphasize this concern are Wade et al.
(2005), Wallingford (2006), Cox et al. (2010),
Smith and Rahman (2016), Kvočka et al. (2016),
Yao et al. (2017), Martínez-Gomariz et al. (2019),
and Rezende et al. (2019). People partially
immersed in flows may lose their balance for a
number of overlapping reasons: i) reduced friction
of the shoe or foot with the floor, ii) transfer of
momentum from the flow to the human body, which
may cause tipping and dragging and iii) fluctuation
due to the distribution of pressure on the body
(Archimedes buoyancy). There is also the
possibility of collision with floating objects and the
instability of the human balance due to the
generation and release of vortexes and the forces
caused by these detachments (evidenced by Simões
et al. 2016).
In the sense of brining more data and confidence for
the already proposed relevant parameters, Karvonen
et al. (2000) conducted a study in which seven
participants aged between 17 and 60 years, having
heights between 1.60 m and 1.95 m, and masses
between 48 kg and 100 kg, were tested in a channel
130 m long and 11 m width. The authors analyzed
their results using the products hV and Hm,
therefore following the analysis conducted by Abt
el al. (1989), having found lower hV values in
relation to those of Abt et al. (1989) for humans,
and greater in relation to the monolith. The results
showed that more data and analyses are needed.
The relevance of the studies in this field was
pointed by Jonkman (2005), who analyzed the
deaths caused by flood events in the world between
the years of 1975 and 2000, reporting that 1826
events were known, and that they killed more than
175,000 people. The author also mentioned that the
cases of tsunamis, rupture of dams and storm tides
can be even more catastrophic in terms of loss of
life. Later Jonkman and Penning-Rowsell (2008)
described a set of experiments done on full scale in
the River Lea (England), which was possible due to
a floodgate system and a flood relief channel. The
experiments were carried out in a section with about
1% slope and a width of approximately 70 m. A
healthy male person was used for the experiments,
with a height of 1.70 m and a mass of 68.25 kg.
Two experiments with the person standing and four
experiments with the person walking were
performed and the empirical and numerical results
of these two conditions were discussed by Jonkman
and Penning-Rowsell (2008). The authors presented
calculations for dragging (sliding) and tipping
(moment of force) and argued that limit values of
the flow variables (for instability) depend on the
mass of the person.
Considering the direct interaction between runoff
and the human being, Foster and Cox (1973)
studied the instability caused in six male children
aged 9 to 13 years, heights between 1.27 and 1.45 m
and masses between 25 and 37 kg. The tests were
conducted in a channel with a length of 6.0 m,
width of 0.6 m and depth of the cross section of
0.75 m. The authors discussed several aspects that
can lead to instability and commented that even
water depths less than 30 cm can generate
instability for velocities above 1.5 m/s.
Following the pioneering work of Foster and Cox
(1973), the study of Abt et al. (1989) was conducted
to clarify the ability of men, women and a monolith
to resist the aforementioned destabilizing factors.
Twenty volunteers participated in the study, aged
between 19 and 54 years, having heights between
1.52 m and 1.83 m and masses between 40.9 kg and
91.4 kg. The authors used a rectangular channel
2.44 m wide, 61 m long and 1.22 m deep, for
bottom slopes of Io = 0.005 m/m (0.5%) and Io =
0.015 m/m (1.5 %). As a result of their research,
Abt et al. (1989) proposed a relationship between
discharge per unit length q = hV, (the product of the
flow depth and the average velocity of the flow),
and the product between the person's mass, m, and
his height, H. The monolith data resulted in lower
values for hV when compared to those found for
humans.
Intending to link the problem of human security in
flows to physical conceptual basis, Cox et al.
(2010) cited theoretical studies that explored
different aspects of the problem of human
instability in flows, presenting equations based on
physical principles, and also evaluations conducted
with computational models. In this sense, the
authors cited the works of Keller and Mitsch
(1993), Lind et al. (2004), Ramsbottom et al. (2004,
2006) and Ishigaki et al. (2005, 2008a and 2008b,
2009), which gave the basis for their arguments.
Conceptual models were also presented by Milanesi
et al. (2014), who proposed a risk classification,
generating limit curves to be adopted as
vulnerability criteria.
The literature of the area (see, for example, Cox et
al. 2010) mentions the Japanese language study
conducted by Takahashi et al. (1992), who carried
out experiments with 3 male people, with heights
between 1.63 and 1.84 m and masses between 63
and 73 kg. The experimental device consisted of
placing the volunteer on a platform equipped with
Introducing the geometrical complexity of the
human body in laboratory scale, Xia et al. (2014)
used a reduced model of a human being with 0.30 m
in height and mass equal to 0.334 kg, partially
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immersed in a rectangular channel. The authors
showed that the adopted reduced model resulted in
lower hV values when compared to those by Abt et
al. (1989) and Karvonen et al. (2000), and
concluded that the different results are due to the
people's ability to progressively adapt to adverse
flow conditions, a reaction that evidently does not
exist in the employed inanimade reduced model.
use of staircases as a means to simplify and to help
the locomotion on the hillsides. Because of their
location, these staircases also serve as “not designed
drainage channels” during the occurrence of floods.
This condition induced the Brazilian architect João
Filgueiras to study the problem and to conceive the
so-called “draining staircases”, a concept presented
in 1979, whose fundamental objective is to allow
the simultaneous use of the stairs as water drainage
and human locomotion. The water drains under the
stairs, while the people use the upper surfaces. It
allows the transit of people without entering into
contact with potentially dangerous flows. The
application of the concept was immediate, and the
survey of Mangieri (2012) about the existing
drainage stairs in the city of Salvador showed that
four different models are used. However, no design
methodologies were found for the adequate
dimensioning of the drainage characteristics of the
implemented stairs, or for the analyses of the
stability of the users in flow conditions. As a matter
of fact, the consulted literature for the present study
showed that there are no studies directed to the
human vulnerability or instability in the situation of
flows over stairs. However, taking into account that
the use of urban staircases is a historic option
already perpetuated for dislocations in urban
environments (see, for example, Taşke, 2002), and
that flows over staircases in heavy rains are being
more frequent due to the growing of the cities,
involving even stairs of underground subway
stations (see, for example, Compton et al. 2009; or
Yu et al. 2019), this gap must be filled.
Evolving in the conceptual discussion of the
problem, Arrighi (2016) and Arrighi et al. (2017)
presented a perhaps more substantiated study of the
hydrodynamics of pedestrians in flood regions. The
authors used physical principles and dimensional
analysis to present their results as dependent on the
Froude number. The authors defined adequate
mobility parameters for their analyses, which
showed good correlations with the Froude number
defined in terms of the mean velocity and flow
depth.
Considering the previous results and discussions of
the literature, Shu et al. (2016) presented arguments
about the geometric, kinematic and dynamic
similarities in the experimental study conducted
with human models generated by three-dimensional
printing. The authors analyzed their own and
literature data, presenting an equation for tipping
instability that, according to the authors, also best
represents the literature data.
With the exception of studies by Arrighi (2016),
Arrighi et al. (2017) and Shu et al. (2016),
according to the conculted literature, the quantities
used to characterize the problem are presented in
dimensional form (that is, not normalized in the
sense of allowing exploring scale effects). Simões
et al. (2016) applied the Vaschy-Buckingham
theorem in the sense to generalize the formulation
of human instability in flows. The authors selected a
functional involving the Reynolds number, Froude
number, drag coefficients, aspect ratios, relative
roughness, slope of background, and three non
conventional dimensionless parameters: the first
related to the individual's age, the second related to
the individual's psychical interaction with the flood
and the third related to the individual's mass.
Literature data of Abt et al. (1989), Karvonen et al.
(2000) and Xia et al. (2014) were used in the study,
which involved a sensitivity analysis between the
different parameters, enabling Simões et al. (2016)
to obtain equations that correlate the mentioned
dimensionless parameters. Good correlations were
presented between those parameters indicated as the
most relevant in the sensitivity analysis. The study
was thus adequate to indicate the mentioned
representative parameters, in which the Froude
number better represented the flow information, a
conclusion aligned with the indications of Arrighi
(2016) and Arrighi et al. (2017), although the
studies were conducted independently.
This work presents a methodology for studying the
instability of adequate shaped solids in flows over
staircases considering the results of a semiempirical dimensionless formulation for the
assessment of the stability of the obstacles subjected
to these flows. The following specific objectives
were established:
1) to build, test and use an experimental equipment
suitable for this study;
2) to formulate mathematically the problem of
stability and to check it with experimental
results, introducing the empirical information in
the conceptual formulation;
3) to analyze the equations between the
dimensionless parameters for two distinct
situations: considering the obstacle on the steps
(data from the present study), and comparing
the present data with literature results for flat
surfaces.
2.
BASIC
FORMULATION
DIMENSIONLESS
Simões et al. (2016) presented a classification of
different types of instability for human beings in
flows. Among them it is mentioned the direct
impact of the water on the body, which can cause
tipping and dragging of the human being.
Regarding the flow of superficial rainwater (runoff)
in built environments, it is worth mentioning the
solutions for human locomotion by feet between
nearby areas located on hillsides, as they occur for
example in Salvador and Rio de Janeiro, two big
Brazilian cities in Brazil. The usual solution is the
The tipping threshold is quantified by the balance of
moments of force, while the drag threshold is
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H. de B. Ribeiro et al. / JAFM, Vol. 14, No. 3, pp. 681-690, 2021.
quantified by the balance of forces. For the present
study, the scheme of Fig. 1 is considered.
the terms in the equation results in:
𝑏
𝑀* = 𝑓 [4𝐶𝑑 + (2𝐶𝑑 + 1)𝑓 +
𝑒
𝑓2
3
𝑠
3
]( ) ,
ℎ𝑐
(6)
in which M* is the tipping force normalized with
the weight of a cubic volume of water with sides
equal to the critical depth (critical volume for
brevity of nomenclature). Equation (6) relates the
tipping force to the weight (of the block) that tends
to avoid this tipping (via moments of force). The
𝑓2
𝑏
factor 𝐺 = 𝑓 [4𝐶𝑑 + (2𝐶𝑑 + 1)𝑓 + ] is an
𝑒
3
unknown function of the flow (remembering that f
was defined as a generic function of the flow).
Since the independent variable in Eq. (6) is s/hc, it
was assumed that the factor G can also be expressed
in terms of this independent variable, that is, G = G
(s/hc), which produces:
Fig. 1. Scheme of the variables used in the
formulation.
𝐹 = 𝐶𝑑 𝑉 2 𝐴 + 0.5𝑔ℎ1 𝐴 ,
While there is a balance of moments, there is also
balance of forces (equilibrium of moments implies
in equilibrium of forces in this geometry and
condition). In this case, considering the impulsive
forces expressed by Eq. (1) and the resistive
horizontal forces (the weight multiplied by the
friction coefficient ), the equilibrium condition
imposes that:
1
𝐶𝑑 𝑉 2 𝐴 + 𝑔ℎ1 𝐴 = 𝑚𝑔.
Or, rearranging:
𝐻
ℎ𝑐
(2)
1
) + 𝑔ℎ1 𝐴 (𝑠 +
2
ℎ1
3
)=
𝑚𝑔𝑒
2
.
𝑠
= 𝑓(flow) or ℎ1 = 𝑠𝑓 .
(3)
𝐻
ℎ𝑐
𝑏
𝑒
𝑓
𝑠 3 𝑓 (1 + ) +
2
𝑏
𝑒
𝑓
3
𝑚
2
𝑠
(9)
ℎ𝑐
𝑓𝑠
(2𝐶𝑑
𝑠 𝑒
1
+ 𝑓) is also an
2
𝑠
𝑠
= J( ) .
(10)
ℎ𝑐 ℎ𝑐
It is usual in dimensional analysis to express
unknown functions through power laws. The
functions G and J of Eqs. (7) and (10) were then
substituted by power laws, in the form:
(4)
𝑠 3 𝑓 2 (1 + ) = ,
1
+ 𝑓) .
Equations (7) and (10) are adequately aligned with
each other, considering that the mass m of the block
depends on its vertical dimension H. Thus, when
relating m and H to the same independent variable
s/hc, the similar forms obtained for Eqs. (7) and (10)
show the adequacy of the present analysis.
With these considerations, Eq. (3) becomes:
4𝐶𝑑
𝑓𝑠
(2𝐶𝑑
𝑠 𝑒
unknown function of the flow. Following the same
argument of Eq. (7), since the independent variable
in Eq. (9) is s/hc, the factor J was expressed in terms
of this independent variable, that is, J = J (s/hc),
leading to:
The characteristic velocity of the water caused by
falling down a step is √2𝑔𝑠. In this study V was
taken as proportional to the characteristic velocity.
Additionally, the height h1 attained by the water
impinging the block depends on this velocity,
represented here by a generic function f(flow).
Taking the depth h1 normalized with the height of
the step that generates the characteristic velocity, s,
we thus have:
ℎ1
=
The factor 𝐽 =
in which m is the mass of the block. For the balance
of moments, each component of force must be
multiplied by the corresponding arm, providing:
2
(8)
2
The force that produces the counterclockwise
moment is the weight P of the block, given by Eq.
2:
ℎ1
(7)
Equation (7) suggests that the equilibrium condition
of moments of force can be expressed by the
dimensionless parameters M* and s/hc.
Cd is the drag coefficient, is the density of the
water, V is the horizontal velocity of the water, g is
the acceleration of gravity and A is the frontal area
of the block that is into contact with the water. This
area is calculated as A = bh1, with b being the width
of the block and h1 being the depth of the water next
to the block.
𝐶𝑑 𝑉 2 𝐴 (𝑠 +
ℎ𝑐
ℎ𝑐
(1)
𝑃 = 𝑚𝑔,
3
𝑠
𝑠
𝑀* = 𝐺 ( ) ( ) .
The balance of the moments of forces was
performed around the point “o” in Fig. 1 (outer
edge of the step). In this case, the force F that
produces the clockwise moment is due to the water
hitting the block and the pressure of the column of
water indicated by h1, being calculated as:
(5)
𝑠
2
𝑠 3
𝑠
3
𝑀* = 1 ( ) ( ) = 1 ( ) ,
in which is the proportionality constant used for
the velocity. To obtain a dimensionless equation
relating basic characteristics of the flow, Eq. (5)
was divided by hc3, with hc being the critical depth
of the flow. Defining M*=m/(hc3) and rearranging
𝐻
ℎ𝑐
ℎ𝑐
𝑠
= 1 ( )
ℎ𝑐
2 𝑠
ℎ𝑐
ℎ𝑐
𝑠
ℎ𝑐
3
= 1 ( ) ,
ℎ𝑐
in which 3 = 2 + 3 and 3 = 2 + 1.
684
(11)
(12)
H. de B. Ribeiro et al. / JAFM, Vol. 14, No. 3, pp. 681-690, 2021.
were s/hc = 5.5 and s/hc = 2.9, respectively. s/l and
s/hc are used to define the boundaries between the
flow regimes. The flows studied here are classified
as nappe flow regime. Figure 3a shows the
maximum and minimum values of s/hc compared to
the classification curves of Ohtsu et al. (2001) and
Simões et al. (2011). Figure 3b illustrates the
occurrence of nappe flow over the first step in this
study.
As a consequence of Eqs. (11) and (12), the
dimensionless parameter M* (instability force
normalized with the weight of the critical volume of
water) can be expressed in different ways:
𝑠
𝑀* = 1 ( )
ℎ𝑐
3
𝐻 2
𝑀* = 1 ( )
ℎ𝑐
𝐻 2
𝑀* = 1 ( )
ℎ𝑐
(Presented in Eq. 11),
, 1 =
𝑠
3
( ) .
ℎ𝑐
1
/3
13
and 2 =
(13a)
3
3
,
(13b)
(13c)
Eq. (13c) was obtained combining the previous
coefficients, adopting M* for Eq. (13a) and M*(1-)
for Eq. (13b), and multiplying the resulting
equations.
The validity of this dimensionless formulation was
tested with experimental data generated in the
present study and with data obtained from the
literature in this field.
3.
EXPERIMENTAL SETUP
3.1 Material and Methods
The experimental study was conducted at the
Hydraulics Laboratory of the Federal University of
Bahia, in a physical model with the following
characteristics: rectangular stepped chute, with s =
1.99 cm (step height), l = 2.4 cm (floor length), b =
15 cm (channel width). At the inlet a rectangular
weir was used to measure the flow supplied by the
pump, as shown in Fig. 2. A capture of part of the
flow rate through a grid drain was added to control
the condition of the flow over the steps. The flow
rate deviated by the grid was calculated from
measurements of volume and time. The flow over
the stepped chute was calculated through the mass
conservation equation (mass balance).
Fig. 3. (a) Flow regime zones I, II, III, and IV by
Ohtsu et al. (2001) and Simões et al. (2011), and
limit measured points. I: nappe flow; II:
transition flow; III: type A skimming flow; IV:
type B skimming flow; (b) first step flow lengths
to obtain the real regime zones (Ribeiro, 2017).
Prismatic obstacles were constructed using plaster
and wood, with the dimensions shown in Table 1.
Dimensions and masses were chosen to attain an
average density close to 970 kg/m³, similar to the
average density of the human body with lungs filled
with air (Daibert, 2008) and a body mass index
BMI = mass/height², equal to 19 kg/m², a value
between those considered adequate for good health,
18.5 and 25 kg/m², according to Anjos (1992).
Fig. 2. Physical model and experimental
apparatus.
The submersible centrifugal pump used to
recirculate the water in the model furnished a flow
rate per unit length between 0.68 and 1.72 L/(sm),
controlled by a valve. The values of s/hc,
corresponding to these two limits of the flow rates
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H. de B. Ribeiro et al. / JAFM, Vol. 14, No. 3, pp. 681-690, 2021.
coefficients equal to 0.999 and 0.983, respectively.
Table 1 Characteristics of obstacles
H
[m]
e
[m]
b
[m]
m
[kg]
0.09
0.11
0.12
0.14
0.17
0.024
0.024
0.024
0.024
0.024
0.073
0.090
0.098
0.114
0.139
0.1539
0.2299
0.2736
0.3724
0.5491
ρs
BMI
[kgm-3] [kgm-2]
976.0
967.6
969.4
972.2
968.2
𝑀∗ = 0.44𝐻∗ 2.64 ,
𝑀∗ = 7.87(𝑠/ℎ𝑐 )4.29 .
19
19
19
19
19
(15)
The behavior of M* following Eq. (13c) was also
evaluated. In this case Eq. (16) was obtained, with
excellent adherence to the experimental data, as can
be seen in Fig. 5, and correlation coefficient equal
to 0.999.
The experiments were carried out by placing one of
the obstacles on the second step (from bottom to
top) and measuring the flow that caused its tipping.
The dimensionless parameters shown by Simões et
al. (2016), and in section 2, M* = m/(hc³), H* =
H/hc, and s/hc, together with the needed critical
depth hc = (q²/g)1/3, were then calculated. Here q is
the flow rate per unit length, and g is the gravity
acceleration.
12000
(14)
12000
Present study with data from Ribeiro (2017)
M* (Calculated)
Perfect fit
Present study with data from Ribeiro (2017)
6000
0
Equation 14
0
6000
M* (Experimental)
12000
Fig. 5. Comparison between experimental data
and Eq. (15).
M*
8000
𝑀∗ = 0.493𝐻 ∗ 2.4 (𝑠/ℎ𝑐 )0.484 ,
(16)
Returning to dimensional variables from Eq. (16)
produces, for the flow rate per unit length:
4000
𝑚 12.93
𝑞 = 9371.5 ( )
0
0
25
H*
50
(17)
The coefficients have their origin explained in the
derivation of Eq. (13c) and have empirical values
based on the analyzed experimental results. For
tipping or sliding velocity we have then:
(a)
12000
√𝑔
.
𝐻 31.03 𝑠 6.259
Present study with data from Ribeiro (2017)
Equation 15
𝑚 12.93
𝑉 = 9371.5 ( )
8000
√𝑔
.
ℎ1 𝐻 31.03𝑠 6.259
(18)
M*
This procedure exposes the variable q used by Abt
et al. (1989) and the relation with the person's mass,
m, and his height, H. The form of dependence,
however, follows the here proposed relationship.
Note that Eqs. (17) and (18) also define a Reynolds
number for tipping or sliding, in the form:
4000
𝑚 12.93
𝑅𝑒𝑡 = 9371.5 ( )
0
2
4
s/hc
6
(b)
Fig. 4. Relationship between the dimensionless
parameters M*, H* and s/hc, having uncertainty
of 2.2% for M* (error propagation analysis).
4.
𝑅𝑒𝑡 = 9371.5 (
𝑚
𝐻 3
√𝑔
𝐻 31.03 𝑠 6.259
, or
12.93 𝐻 6.259 √𝑔𝐻 3
)
( )
𝑠
(19)
.
(20)
As noted, the Reynolds number of tipping/sliding
Ret is dependent on the characteristics of the
obstacle (human being, as the final objective of
the security studies), expressed by the mass m
and height H, as well as the characteristics of the
flowing fluid, expressed by the density and the
kinematic viscosity . This is the limiting
Reynolds number that corresponds to the
particular case of instability of this study. The
mass and height variables presented in Eqs. (19)
and (20) are related to the average limiting
velocity of Eq. (18) that produced the instability.
RESULTS AND ANALYSIS
The obtained results show that there is a very good
correlation between M*, H* and s/hc, as can be seen
in Fig. 4. Power laws were adjusted for both M*
and H*, as suggested by Eqs. (13a) and (13b),
leading to Eqs. (14) and (15) with good adherence
to the experimental data, and correlation
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H. de B. Ribeiro et al. / JAFM, Vol. 14, No. 3, pp. 681-690, 2021.
This velocity (to attain instability) depends on the
measured variables, and so also the limiting
Reynolds number. In the case of stepped chutes,
as considered in the present study, it also depends
on the height of the step, s. In the present study
the Reynolds number was not considered as an a
priori variable, but emerged from the data
analysis based on the balance equations for the
forces and for the moments of forces. Equations
(16) and (20) have the same information, and Eq.
(20) presents them linked to the immediately
measurable characteristics (not involving the
calculation of the critical depth). The tipping
Reynolds number can also be obtained directly
from the measurements of water speed and depth,
associated with the viscosity of the water (which
depends on the temperature). When using Eq.
(14) the Reynolds number is given by Eq. (21).
𝑅𝑒𝑡 = 30.59 (
𝑚
𝐻 3
)
4.167 √𝑔𝐻 3
Table 2 Minimum and maximum values of the
parameters M* and H*
Author
Abt et al.
(1989),
Io=0.5%
Abt et al.
(1989),
Io=1.5%
Karvonen et al.
(2000)
Xia et al.
(2014)
Ribeiro (2017)
𝑚
𝐻 3
)
4.167
𝐻 3/2
( )
ℎ1
H*max
H*min
2.33
0.20
3.82
0.83
2.57
0.22
4.67
1.12
3.56
0.57
4.56
1.25
4.30
0.99
7.84
5.97
13.52 521.88
47.05 11637.50
Abt et al. (1989): Io=0.5%
Abt et al. (1989): Io=1.5%
Karvonen et al. (2000)
Xia et al. (2014)
Ribeiro (2017)
Equation 22
(21)
.
M*/M*max
0.5
(22)
0.0
0.25
Equations (21) and (22) show that the two
dimensionless parameters Ret and Frt may be
related to the sliding and tipping events in flood
security studies. This coincides with the discussion
of Simões et al. (2016), in which these two
parameters (Ret and Frt) were also considered in the
question of human instability in floods.
0.50
0.75
1.00
H*/H*max
(a)
1.0
Abt et al. (1989): Io=0.5%
Abt et al. (1989): Io=1.5%
Karvonen et al. (2000)
Xia et al. (2014)
Ribeiro (2017)
Perfect fit
M*/M*max (power law)
4.1 Comparison with Literature Studies
Data from Abt et al. (1989), Karvonen et al (2000)
and Xia et al. (2014) were analyzed using the
dimensionless M* and H* as defined in the present
work. To minimize the scale effects between the
different physical models and situations, M* and H*
were normalized with adjusted maximum values
obtained for each set of data. The normalized
parameters thus have a maximum theoretical
variation between 0 and 1, being expressed as M*n =
M*/M*max and H*n = H*/H*max. Because M*max and
H*max are adjusted values, the extremes 0 and 1 may
not be reached in the data analysis. This procedure
proved to be adequate for the analyzed data set, as
can be seen in Fig. 6a. Following the indication of
Eq. (13b), the general behavior of the experimental
data was adjusted to the power law expressed by
Eq. (23). The correlation coefficient between
experimental and calculated data is R = 0.97, and
Fig. 6b shows the good approximation obtained by
using the power law to calculate M*n. The
minimum and maximum values of H* and M* are
shown in Table 2.
𝑀𝑛∗ = 0.97𝐻𝑛∗ 2.78 .
M*min
1.0
.
This equation shows that it is possible to consider
Ret also in cases in which the step height is not
involved (flat bottoms, for example). Further, by
dividing and multiplying the right member of Eq.
(21) by qh11/2, the Reynolds number is simplified
and Eq. (21) may be written in terms of a
tipping/sliding Froude number Frt
𝐹𝑟𝑡 = 30.59 (
H*min
0.5
0.0
0.0
(b)
0.5
M*/M*max (experimental)
1.0
Fig. 6. Relationship between normalized
dimensionless, M*/M*max and H*/H*max and Eq.
(23) (a); comparison between the Eq. (23) and
the experimental data (b).
The dimensionless parameters M* and H* showed a
great capacity to unify data from our own
experiments and from different sources in the
literature. The normalization with maximum values
of M* and H* adjusted for each experiment
emphasized the subjacent similarity between the
profiles obtained for the data of different sources.
This is seen as a very positive characteristic of the
methodology here described.
(23)
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5.
CONCLUSION
REFERENCES
Theoretical analyses were performed about the
phenomenon of tipping/dragging of obstacles in
flows, together with semi-empirical proposals of
power laws, and correlation analyses of
experimental data obtained in a reduced model of
a draining staircase. The correlation analyses
followed the indications of the theoretical study,
and evidenced the dependence between the
dimensionless parameters M*, H* and s/hc for the
tested obstacles. From the theoretical analyses, the
equations were expressed as M* = F(H*), M* =
F(s/hc) and for M* = F(H*, s/hc) using power
laws.
Abt, S. R., R. J. Wittier, A. Taylor and D. J. Love
(1989). Human stability in a high flood
hazard. Water Resources Bulletin 25(4), 881890.
Anjos, L. A. (1992). Índice de massa corporal
como indicador do estado nutricional de
adultos: revisão da literatura. Revista de
Saúde Pública 26(6), 431-436. (In
Portuguese)
Arrighi, C. (2016). Vehicles, pedestrians and
flood risk: a focus on the incipient motion
due to the mean flow. PhD. thesis, University
of Braunschweig and University of Florence.
The three functional forms were checked,
generating excellent correlations and adherence
between experimental data and calculated results.
Arrighi, C., H. Oumeraci and F. Castelli (2017).
Hydrodynamics of pedestrian’s instability in
floodwaters. Hydrology and Earth System
Sciences-Discussions 21, 515–531.
Considering the functional form M* = F(H*), in
which s/hc is not an explicit variable, it was
observed that the experimental data of the present
study, for a drainage staircase, could be analyzed
together with data of the literature, for flat bottom
channels. A procedure was proposed to minimize
the effects of different scales and experimental
conditions between the data of different sources. In
this sense, a normalization of M* and H* was
performed using maximum adjusted values of these
parameters for each independent set of data. When
plotted together, the normlized profiles showed
similar evolution for M*/M*max against H*/H*max
independently of their origin. It is understood that
this is a promising way to conduct joint analyzes to
assess the vulnerability criteria of human beings
exposed to urban floods.
Compton, K. L., R. Faber, T. Y. Ermolieva, J.
Linerooth-Bayer and H. P. Nachtnebel
(2009). Uncertainty and disaster risk
management.
RR-09-02,
International
Institute for Applied Systems Analyses,
Laxenburg, Austria.
Cox, R. J., T. D. Shand and M. J. Blacka (2010).
Australian Rainfall & Runoff Project 10. The
University of New South Wales, School of
Civil and Environmental Engineering.
Daibert, J. B. C. (2008). Os benefícios da natação
para bebês. Monography (Graduation).
Instituto de Biociências de Rio Claro,
Universidade
Estadual
Paulista.
(In
Portuguese)
Considering the Reynolds and Froude numbers,
the present analysis allows relating both
parameters to the formulation of sliding or tipping
in open flows.
Foster, D. N. and R. J. Cox (1973). Stability of
children on roads used as floodways,
Technical Report 73/13, Water Research
Laboratory, The University of New South
Wales, School of Civil and Environmental
Engineering.
In terms of specific results, it was shown that M*
consistently grows with H* and s/hc following
power laws. For the study conducted in the draining
staircase the exponents were 2.64 and 4.29,
respectively, evidencing the strong dependence
between the different parameters. The correlation
coefficients presented the values of 0.999 and
0.983, respectively.
Ishigaki, T., R. Kawanaka, Y. Onishi, H.
Shimada, K. Toda and Y. Baba(2008b).
Assessment of safety on evacuation route
during underground flooding, Proceedings of
the 16th APD-IAHR, Nanjing, China, pp.141146.
For the joint analysis of the present data and
those of the literature using the normalized
parameters Mn* and Hn*, the exponent of the
power law was 2.78, once more showing the
strong dependence between the parameters
defined in this study. The correlation coefficient
presented the value of 0.97.
Ishigaki, T., Y. Asai, Y. Nakahata, H. Shimada,
Y. Baba and K. Toda (2009). Evacuation of
aged persons from inundated underground
space, Proceedings of the 8th International
Conference on Urban Drainage Modelling,
Tokyo.
ACKNOWLEDGEMENTS
Ishigaki, T., Y. Baba, K. Toda and K.
Inoue(2005).
Experimental
study
on
evacuation from underground space in urban
flood, Proceedings of the 31st IAHR
Congress, Seoul, pp.261-262.
To FAPESB, for the support related to the
infrastructure project (PIE0021/2016). The last coauthor thanks CNPq for the Scientific Productivity
Scholarship 307105/2015-6, which guaranteed the
necessary dedication to this study.
Ishigaki, T., Y. Onishi, Y. Asai, K. Toda and H.
Shimada (2008a). Evacuation criteria during
688
H. de B. Ribeiro et al. / JAFM, Vol. 14, No. 3, pp. 681-690, 2021.
urban flooding in underground space,
Proceedings of the 11th ICUD, Scotland,
UK.
2 Draft Inception Report.
Ramsbottom, D., P. Floyd and E. PenningTowsell (2006). Flood risks to people, Phase
2 Project Record, FD2321/PR, Department
for Environment Food and Rural Affairs,
United Kingdom.
Jonkman, S. N. (2005). Global perspectives of
loss of human life caused by floods. Natural
Hazards 34, 151-175.
Rezende, O. M., F. M. Miranda, A. N. Haddad
and M. G. Miguez (2019). A framework to
evaluate urban flood resilience of design
alternatives for flood defence considering
future adverse scenarios. Water 11, 14851513.
Jonkman, S. N. and E. Penning-Rowsell (2008).
Human instability in flood flows. Journal of
the American Water Resources Association
44(4), 11.
Karvonen, R. A., H. K. Hepojoki, H. K. Huhta
and A. Louhio (2000). The use of physical
models in dam-break analysis. Helsinki:
Helsinki University of Technology, 2000.
RESCDAM Final Report.
Ribeiro, H. de B. (2017). Estabilidade de
pedestres sobre escadarias drenantes.
Monography (Graduação). Escola Politécnica
– Universidade Federal da Bahia, SalvadorBA. (In Portuguese)
Keller, R. J. and B. Mitsch (1993). Safety aspects
of the design of roadways as floodways,
Research Report 69, Urban Water Research
Association of Australia.
Kvočka, D., R. A. Falconer and M. Bray (2016).
Flood hazard assessment for extreme flood
events. Nat. Hazards, Springer 84, 1569–
1599.
Shu, C. W., S. S. Han, W. N. Kong and B. L.
Dong (2016). Mechanism of toppling
instability of the human body in floodwaters,
International Conference on Water Resource
and Environment, IOP Conf. Series: Earth
and Environmental Science, 39, 23–26 July
2016, Shanghai, China.
Lind, N., D. Hartford and H. Assaf (2004).
Hydrodynamic models of human stability in
a flood. Journal of the American Water
Resources Association 40(1), 89-96.
Simões, A. L. A., H. E. Schulz and L. D. da Luz
(2016). Dimensionless formulation for
human stability in open flows. RBRH 21(4),
666-673.
Mangieri, L. S. G. (2012). Avaliação dos sistemas
de escadarias e rampas drenantes
implantadas em assentamentos espontâneos
na cidade do Salvador – Bahia. MSc
Dissertation - Escola Politécnica –
Universidade Federal da Bahia, Salvador-BA
(inPortuguese). (In Portuguese)
Simões, A. L. A., H. E. Schulz, R. J. Lobosco and
R. M. Porto (2011). Stepped spillways:
theoretical, experimental and numerical
studies. In: Schulz, H.E., Simões, A.L.A.,
Lobosco, R.J. Hydrodynamics - Natural
Water Bodies, ISBN 978-953-307-893-9,
InTech Open Access Publisher.
Martínez-Gomariz, E., L. Locatelli, M. Guerrero,
B. Russo and M. Martínez (2019). SocioEconomic Potential Impacts Due to Urban
Pluvial Floods in Badalona (Spain) in a
Context of Climate Change. Water 11(12),
2658.
Smith, G. P. and P. F. Rahman (2016).
Approaches for estimating flood fatalities
relevant to floodplain management, WRL
Technical Report 2015/09, The University of
New South Wales, School of Civil and
Environmental Engineering.
Milanesi, L., M. Pilotti and R. Ranzi (2014). A
conceptual model of people’s vulnerability to
floods. Water Resources Research 51(1),
182-197.
Takahashi, S., K. Endoh and Z. I. Muro (1992).
Experimental study on people’s safety
against overtopping waves on breakwaters.
Report of the Port and Harbour Research
Institute 34(4), 4-31. (in Japanese)
Ohtsu, I., Y. Yasuda and M. Takahashi (2001).
Onset of skimming flow on stepped spillways
– Discussion. Journal of Hydraulic
Engineering. 127, 522-524. Discussion on:
Chamani, M. R., Rajaratnam, N. Onset of
skimming flow on stepped spillways. ASCE,
Journal of Hydraulic Engineering. 125(9),
969-971, Sept, 1999.
Taşke, N. (2002). Stairways as spatial elements in
an urban environment, MSc Dissertation,
Izmir|Institute of Technology, Turkey, 214.
Wade, S., D. Ramsbottom, P. Floyd, E. PenningRowsell and S. Surendran (2005). Risks to
people: developing new approaches for flood
hazard
and
vulnerability
mapping,
Proceedings of the 40th Defra Flood and
Coastal Management Conference, HRPP340.
Ohtsu, I., Y. Yasuda and M. Takahashi (2004).
Flows characteristics of skimming flows in
stepped channels. ASCE, Journal of
Hydraulic Engineering 130(9), 860-869.
Wallingford, H. R. (2006). Flood Risks to People,
Phase 2, Defra/Environment Agency,
FD2321/TR1.
Ramsbottom, D., P. Floyd and E. PenningTowsell (2004). Flood risks to people, Phase
689
H. de B. Ribeiro et al. / JAFM, Vol. 14, No. 3, pp. 681-690, 2021.
Xia, J., R. A. Falconer, Y. Wang and X. Xiao
(2014). New criterion for the stability of a
human body in floodwaters. Journal of
Hydraulic Research 52(1), 93-104.
Catchments of Beijing. China, International
Journal of Environmental Research and
Public Health 14, 239-255.
Yu, H., C. Ling, P. Li, K. Niu, F. Du, J. Shao and
Y. Liu (2019). Evaluation of waterlogging
risk in an urban subway station. Advances in
Civil Engineering, Hindawi 19, 1-12.
Yao, L., L. Chen and W. Wei (2017). Exploring
the Linkage between Urban Flood Risk and
Spatial Patterns in Small Urbanized
690