*3.2. Method*

The number of inhabitants is a first measure of the degree of exposure to flooding. Demographic data were first mapped at the level of the peripheral municipalities and boroughs of the CUA. This map was used to identify the annual demographic evolution between 1993 and 2018 and the population in 2018. Demographic data were then mapped at the level of the Fokontany in order to obtain the residential population density in 2018 (Scheme 1). Residential population density is defined here as the ratio between the number of inhabitants and the area of built spaces.

Considering that in 2018 the average residential population density of the Fokontany of the agglomeration was estimated at 250 inhab/built ha, the following thresholds were established:


The density map highlights the overall geographic distribution of the population throughout the agglomeration. Crossed with the area of flood-prone zones of each Fokontany in 2018, it allows estimation of percentage of the population living in flood-prone areas (Scheme 1).

**Scheme 1.** Overview of method adopted in the study.

Historic topographic maps and recent aerial data were used to map the extent of urbanization in 1953, 1975, 2006, and 2017. This makes it possible to understand the actual structuring of urbanization and the evaluation of the rate of waterproofing of the soil. The urbanization maps were used to determine the progress of construction in flood-prone zones (Scheme 1).

Two groups of Fokontany from the south plain of the CUA were studied in order to understand the factors underlying urbanization in these areas. These Fokontany, Ampefiloha Ambodirano and Ampandrana-Besarety and Besarety, which were selected based on previous analyses, are located in flood-prone areas and witnessed strong demographic growth over the last years. Site visits were made in order to map current land uses and study the local drainage systems. These site visits further allowed us to observe the living conditions in these areas so as to better understand the interplay between flooding, urbanization, and socioeconomic drivers/challenges (Scheme 1).

#### **4. Results**

#### *4.1. Demographic Change and Residential Population Density in 2018*

Antananarivo experienced considerable demographic growth over the last decades. It was home to more than half of the urban population of the country and around 11.3% of the total population in 2018. Between 1993 and 2018 (25 years), annual growth was estimated at 3.8%.

In 2018, the CUA had approximately 50% of the total population of the agglomeration. However, population growth in the CUA between 1993 and 2018 was rather modest compared to the growth observed in peripheral municipalities (Figure 6). Population

growth in peripheral municipalities was mainly driven by migratory flows from the CUA, defined as proximity migration [48], but also from other regions of the country [37].

The demographic density was much higher in the CUA than in the peripheral municipalities in 2018 (Figure 7). It can be observed, however, that the distribution of the population within each commune varies from one Fokontany to another, with some denser nodes outside the CUA.

The highest density values in 2018 are seen in the Fokontany in the center of the CUA and within a radius of 2.5 km (Figure 7). The density peak reached up 2000 inhabitants/built ha, four times the average density of the Fokontany of the CUA and eight times that of the agglomeration. Most of the denser Fokontany were in the western floodplain.

In the peripheral municipalities, densification was led by national roads toward five main axes and grew with an area of expansion around a 10 km radius of downtown CUA (Figure 7). In these communes, the Fokontany were less dense than those of the CUA. However, some of them were in flood-prone areas.

Based on the residential population density and the size of the built-up areas in floodprone areas in each Fokontany, it is estimated that about 32% of the population of the agglomeration and 43% of the population of the CUA lived in flood-prone areas in 2018.

**Figure 6.** Annual growth rate between 1993 and 2018 and population in 2018 by borough in the CUA and by commune in the outskirts.

**Figure 7.** Residential population density in 2018.

## *4.2. Evolution of Built Spaces*

Antananarivo developed in the center of the historic region of Imerina (Figure 8), an ethnic group in the central highlands of Madagascar. The first settlement was the Rova, a royal palace established during the 17th century, on the highest hill in the city [49]. Other constructions were progressively erected around the palace [50]. Some time later, the 30,000 inhabitants of city [51] settled on this first urbanized terrace of the city, called the "upper city" [52]. The extension continued beyond the limit of the hill and spread on the flanks and ridges of neighboring hills, in the north and west, toward the second half of the 19th century and formed the "medium city" [50]. It then gradually developed into the plain, with the installation of small settlement cores in the middle of rice fields [53].

Under the French regime, the extent of this encroachment became more important, and Antananarivo underwent its first major urban transformations. Some 20 hectares were backfilled in order to form the first Fokontany in the "lower town", and other new neighborhoods were created [54,55]. Between 1896 and 1903, 35 km of paved roadways were opened, tunnels were dug, places were created [56], and work on railway lines started [57]. This was only the beginning of the urbanization that would occupy the entire lower area a few years later.

In 1953, the built space mainly occupied elevated areas and covered around 1806 ha, i.e., 2.4% of the area of the present agglomeration (Table 2). In the CUA, urbanization was mainly oriented toward the east, around the historic center (Figure 8). On the other hand, the development of buildings in the lower town of the west continued. 10% of the CUA area was urbanized, and some 2.6% of the area was then covered by built areas located in floodprone zones (Table 2). Outside the CUA, the development line progressed gradually to the east and northwest. Throughout the agglomeration, spatial development operated through the densification of spaces that were already built and through "fingerprint" urbanization,

where most constructions were arranged along the main national axes (Figures 8 and 9). Among all built areas, 22% were in flood-prone zones, covering an area of 399 ha (Table 2).

**Figure 8.** Urbanized cells: (**a**) 1953–1975; (**b**) 1975–2006; (**c**) 2006–2017; (**d**) 2017.


**Table 2.** Extent and increase of urbanized cells and built areas in flood-prone zones.

**Figure 9.** Urbanized cells in flood-prone zones: (**a**) 1953; (**b**) 1975; (**c**) 2006; (**d**) 2017.

From the 1960s, the period of independence, backfilling continued in the flood plain (Figures 8 and 9). Several new urban areas were built, and this did not protect the city from flooding. Urbanization continued until 1975, with approximately 2330 ha more of built areas than in 1953, and built areas then covered 5.4% of the agglomeration. The increase compared to 1953 is estimated at 129%, an annual increase of 5.9% (Table 2). Inside the CUA, built areas grew considerably and covered 25.5%. These are divided into two distinct parts: to the east toward the elevated areas and to the west in the flood plain (Figure 9). Built areas in flood-prone zones then represented 8.3% of the CUA, a sharp increase compared to the situation in 1953. In peripheral municipalities, urbanization mainly occurred through the filling of voids and the densification of existing areas. In addition to this urban expansion at the agglomeration level, the share of built areas located in flood-prone zones increased very rapidly. Compared to 1953, built areas located in flood-prone zones increased by 176%, an annual increase of 8%, covering 26.5% of the entire built area at the agglomeration level (Table 2).

From 1975 to 2006, urban expansion progressively shifted toward peripheral municipalities, especially in the neighboring municipalities of the CUA (Figure 8). It was guided by national roads and showed five centers of urban growth. In the CUA, urban consolidation continued and densified the area to the east and northwest. Built areas then occupied 48.2% of the CUA. In the whole agglomeration, between 1975 and 2006, the built areas increased by 94%, corresponding to an annual increase rate of 3%. The built areas then occupied 10.5% of the agglomeration, i.e., 8048 ha. It can be seen from Table 2 that 25% of built areas were then located in flood-prone zones. This is equivalent to an 85% increase compared to the situation of 1975, i.e., a 2.7% yearly increase over the period 1975–2006 (Table 2).

Between 2006 and 2017, urban expansion further intensified, especially in peripheral areas. Constructions was dispersed in areas far from the CUA. Approaching the CUA, the urban fabric became denser, especially in the northwest and the south (Figure 8). In the CUA, despite high density, urbanization continued to progress, to a large extent at the expense of rice fields in the lower town (Figure 8), by filling interstices to the west and rising slightly toward empty spaces to the north (Figures 8 and 9). A total of 55.3% of the CUA was occupied by built areas and 19.8% was occupied by built areas located in flood-prone zones. At the agglomeration level, built areas covered 16,250 ha, or 21.2% of the territory. It doubled over the years 2006–2017. The increase of built areas since 2006 was estimated at 102%, i.e., an annual increase of 9.3%. Constructions kept settling in the flood plain, and 23% of the total built areas was located in flood-prone zones in 2017. With an increase rate of 80%, or 7.3% yearly, built areas located in flood-prone zones occupied 4.8% of the agglomeration (Table 2).

Between 1953 and 2017, the annual rate of increase of built areas in the agglomeration was 6.1%. In flood-prone zones, it was estimated at 6% (Table 2). Urban expansion in flood-prone areas developed in parallel with that in other areas of the agglomeration. Practically, it means that building in flood-prone zones in Antananarivo was driven by the general expansion of the city, which was intense.

#### *4.3. Case Studies*

#### 4.3.1. Case Study 1: Fokontany of Ambodirano Ampefiloha

The lower neighborhoods of the west, including the Fokontany of Ambodirano Ampefiloha, are the Fokontany located in the lower town on the left bank of the Andriantany canal (Figure 10). They constitute the extension of backfilled spaces in the plain of Antananarivo during the colonial period. Urbanization in these places accelerated following major subdivision operations in the 1970s. Apart from these constructions and a few service buildings, most houses are made of recycled materials such as wood, plastic, or brick and are in very poor condition [43] and underserved. The area gathers populations with low income, which prevents them from accessing other housing [54]. Homes are cramped and overcrowding prevails. They are exposed to flooding due to river floods and even

overflows of drainage canals, and to health risks. These are not negligible due to the almost permanent accumulation of water during the rainy season.

**Figure 10.** Map of Fokontany of Ambodirano Ampefiloha in CUA.

Ambodirano Ampefiloha covers an area of 63 ha. It is crossed in the east by the GR channel, the irrigation channel of the agricultural plain, and bordered by the Ikopa in the west. It has a flat topography with areas that form basins accumulating large volumes of water.

Between 1993 and 2018, the population almost tripled. With the growing population, this Fokontany is densely populated, with a residential density estimated at 484 inhabitants/built ha. The dynamics of family migration based on family support to overcome the difficulties encountered in the rural world is a main source of this considerable densification [43]. The residential area occupies 30% of the total area and rice fields 60%. The Fokontany does not have a specific drainage network (SAMVA).

This large urban space should contribute to the storage of water during rainy periods [58]. Nevertheless, it is increasingly invaded by constructions.

4.3.2. Case Study 2: Fokontany of Ampandrana-Besarety and Besarety

The Fokontany of Ampandrana-Besarety and Besarety are classified among the working-class neighborhoods at the foot of the upper town (Figure 11). They are located at altitudes between 1251 and 1252 m and extend over 22 ha. As in most of the Fokontany in the lower areas, backfilling allowed a progressive urbanization of the zone [59]. Floods are mainly linked to the overflow of drainage channels.

With a residential population density of 548 inhab/built ha, well above the CUA average, these Fokontany are categorized as densely populated. Population growth was 2.9% per year between 1993 and 2018. This upsurge was due to the high birth rate and the arrival of new migrants. The proximity to the city center and industrial zones explains why new inhabitants have come to settle there.

**Figure 11.** Map of Fokontany of Ampandrana-Besarety and Besarety in CUA.

Mainly residential activities and constructions are developing, to the detriment of rice fields. Two categories of settlements can be identified here: traditional or modern concrete constructions and small constructions made of sheet metal or wood. Currently, dwellings cover 75% of the total area, and 7% is occupied by cultivated areas. However, the whole area has a waterproofing rate of more than 80%, which generates large runoff.

The drainage system works differently on both sides of the area (SAMVA). It is operated through underground networks to the west and provided by gutters in the east. The whole system is subsequently taken up by primary channels. Despite the existence of this system, the primary network remains constrained by the accumulation of water with an important flow upstream of the site and by the weak slopes of the plain. Furthermore, the sections of these channels are heterogeneous. In certain sections, the load is much greater. In addition, the flow of water is hampered by the congestion of gutters and the obstruction of manholes by deposits of sand or waste [43]. Some installed structures also make it more difficult for water to flow during rainy events [60]. All of this, together with the proximity of the constructions to the drainage network and even their encroachment on the network, makes these Fokontany areas with recurrent floods.

#### **5. Discussion**

In SSA, several urban areas have experienced high population growth in recent decades. Gardi has shown that 12 of the top 30 fastest-growing urban agglomerations in the world are in SSA [17]. Antananarivo ranks 20th in the world and 9th in SSA in this ranking, right next to the major cities of Nairobi, with an annual rate of 3.87%, and Kinshasa, with 3.89% [17]. This growth is due to the high rate of natural increase and the rural exodus to and near urban areas [36]. In SSA, internal growth has been a determining factor for many years [16,61,62]. Nevertheless, migration is also part of the driving force behind population growth and urban sprawl [63–65]. For sociocultural, economic, political, and environmental reasons, whether it is a choice or a necessity, this migration seems to be a way to allow better living conditions for rural migrants [63,66]. In South Africa, due to inadequate social services and the lack of employment in rural areas, people migrate to urban areas [66,67]. In the Democratic Republic of Congo, it is because of conflicts and

insecurity [65,68]. In Burkina Faso and Kenya, it is due to climatic disturbances [69]. The expansion of informal activities [66] and the concentration of services and facilities in cities also easily attract the rural population [65,70]. In Madagascar, social and land insecurity in the countryside [45] as well as economic difficulties [71] due to declining agricultural productivity [34] are pushing rural populations to migrate to Antananarivo. Due to the accessibility of services and infrastructure, students, civil servants, and people involved in small trade migrate and increase the size of the city's population [53].

This demographic growth creates unease for urban centers, since the supply of housing and spaces to be built does not meet the growing demand [63,72–74]. The fragility of urban governance, manifested in the lack of support and prioritization of urban infrastructure and land-use management initiatives by governments, is one of the reasons for this [75]. Parnell, Pieterse, and Watson also refer to a lack of good planning [20] due to poorly conceived planning laws and standards for construction as well as insufficient funding [16]. As a result, many informal settlements are developed [18]. The proliferation of these informal settlements is also related to increasing urban poverty and leads to the involvement of many poor people in the informal economy [65,76]. On the other hand, it is produced by the proximity of informal employment, which pushes migrants and poor households to settle nearby [77,78]. It is also accompanied by the development of new housing areas in flood-prone areas [18,19,79]. These settlements are generally precarious, with poor infrastructure, and are heavily impacted by flooding [18,70,80]. For Antananarivo, given the weakness of urban planning and the absence of housing policies, the demand greatly exceeds the supply that the city can offer [81]. This has led to illegal installations and constructions in the west of the city, in the flood plain [82]. In order to overcome this deficit, public authorities indirectly approved and anticipated the extension of urbanization in the plain [83].

In SSA, most of the urban population live in informal settlements in areas at risk [23], such as floodplains, swamps, and riverbanks [84,85]. This population group mainly consists of the urban poor [84]. They are often excluded from the land market due to unaffordable prices and imposed standards and regulations that they are unable to follow, and that force them to occupy these dangerous lands [11,21,86]. In addition to this group are refugees and persons who are displaced due to forced displacement in cities who settle in these areas for various reasons [87–90]. These situations and the increased population density in these areas make them more vulnerable to flooding [84,89,91].

This set of processes is linked to the history of urbanization in SSA. In the region, the pre-colonial period is characterized by a low level of urban development [76] that intensified during the period of colonization [16,65,76]. The occupation of colonial cities by settlers [76,79] favored the development of informal settlements formed by the indigenous population, who were excluded from planning [92]. In Nairobi, this led to the construction of several illegal settlements by homeless Africans who were only allowed to be in the city for work [92]. It also promoted the development of habitats in flood-prone areas, as in the case of Antananarivo. In the 1930s and 1960s, development of the colonial city led to the displacement of several population groups, who took refuge in low areas [93]. In addition, the overcrowded hills of the middle city and the conveniences of the plain attracted inhabitants of the upper city to the lower city [53]. In the 1950s and 1960s, urbanization was mainly fueled by the development of industrial facilities [43], part of the orientation of the 1954 colonial-era urban plan [83]. These works changed the hydraulic regime of the plain and led to densification of the flood plains [59]. Flood-prone zones also became more favorable for speculators and real estate developers. Actually, building is less onerous and investments more profitable in low lands, given the more suitable topography [83]. By contrast, the extension of existing habitats and the widening of service roads were more expensive in the eastern part, given the steep slopes (up to 20%) [40,94]. There are also landslide risks in this part of the city [95].

In the years following independence, around 1960 and 1970, many cities in SSA experienced rapid population growth [65,79,96]. This growth can be linked to policies adopted after independence, which were related to the deployment of jobs, the establishment of several industries in city centers [65,97], and investment in public works [98]. The average annual population growth rate in urban areas was approximately 5% [99]. However, the cities inherited from the colonial era were not designed to accommodate such a massive population [20,76]. This reinforced the proliferation of informal settlements [76] that are more exposed to flooding [74]. In Accra, Ghana, the informal development of some communities in watersheds around rivers and lagoons increased their exposure to flooding [100]. In addition, the economic crisis that hit Africa in the 1970s fueled such difficulties throughout the region [101]. There has been a decline in investment in urban infrastructure and housing [76]. For Antananarivo, the 1970s were characterized by the completion of development and subdivision work to replace the thousands of homes destroyed following the devastating 1959 floods [83,102]. Other districts were then created in the lower town to accommodate the affected populations [94]. As these social housing units were not affordable for vulnerable populations, the construction of precarious housing proliferated. Antananarivo was also plunged into a lasting multifaceted crisis [56]. The construction sector was strongly affected. Materials were more expensive and scarcer. This led to the proliferation of informal habitats in flood-prone zones [43]. This was favored by the absence or inadequacy of urban land management tools [103]. It is difficult to access land due to expensive administrative procedures and a lack of updated information about the legal status of the land [45].

From the 1990s, urban development accelerated, especially in the peripheral areas around cities [62]. This is partly due to the decline in land values in these areas [104]. However, these habitats are often built outside of planning and regulations [65,104,105], and are places where various risks, including flooding, are prevalent [23]. For Antananarivo, the emergence of several peri-urban cores outside the CUA was supported by the establishment of the city's 2004 urban plan, which proposed unclogging the city center [37]. The migratory flow consequently became more important toward peripheral municipalities. This is further related to the limited accommodation capacity of CUA houses linked to their architecture [106] and the lack of available land for development in residential areas [107]. Peripheral areas were more attractive for residential installations in terms of both availability of building land and cost of living [108]. The establishment of industrial buildings [46] and the proliferation of infrastructure projects led to the acceleration of urbanization, which is increasingly taking place in flood-prone areas where land is cheaper and rents moderate [82,106]. The proximity of these settlements to industrial areas attracts the population, as shown in the case studies. These factors were favored by the absence of urban planning for a long period (from 1968 to 2004) [83].

This growing urbanization in SSA is also accompanied by a lack of infrastructure, including drainage systems, which makes cities vulnerable [19,79,91]. In Antananarivo, the case studies reveal this. The proximity of constructions to drainage canals and informal encroachments on these canals progressively reduce their capacity [60]. As the load on the existing network increases, this leads to greater susceptibility to the effects of further flooding [74].

Scheme 2 synthetizes the co-evolution of urbanization and vulnerability of poorly managed urban environments. The demographic pressure leads to urban sprawl around major cities (1). Due to the lack of housing and appropriate urban management (2), the expansion of the city occurs through informal settlements (3). It is more the case that incoming populations, especially from rural areas, are usually associated with low economic resources (4). Flood-prone areas are associated with the development of informal settlements (5): land is cheaper, and constructions are not authorized by planning documents. Furthermore, flood-prone areas located near canals and rivers are well adapted to maintaining subsidence urban agriculture, especially for inhabitants coming from rural areas. The construction of precarious settlements in flood-prone areas (6) by low-income

groups (7) obviously exacerbates the vulnerability of these groups and their habitat; even more as the drainage system in these areas is often insufficient (8), which can partly be explained by the fast urban growth witnessed by SSA cities (9) and the lack of adequate integration of drainage in urban planning policies (10). The lack of drainage combined with urban sprawl and soil-sealing further contribute to increasing floods at the agglomeration level (11).

**Scheme 2.** Flood vulnerability factors, a co-evolutionary perspective.

#### **6. Conclusions**

Flooding is an occurrence that plagues most countries in the world, particularly in urban centers in SSA. Over time, poorly planned urbanization, combined with several other factors, expose people and buildings to flooding and increase their vulnerability. The agglomeration of Antananarivo is a clear example. Like many African urban agglomerations, it faces strong demographic pressure due to the high rate of natural growth and population migration toward urban areas. The city progressively expanded in the lower zone without much control. Our analysis shows that around 32% of the population of the agglomeration lived in flood-prone zones in 2018. The annual growth of built-up areas in flood-prone zones between 1953 and 2017 is estimated at 6%. In 2017, 23% of the buildings of the agglomeration, i.e., almost one out of four buildings, were in flood-prone zones.

The dynamics of urbanization inherited from the colonial era led over time to the proliferation of informal settlements, which are more exposed to flooding. This led to a modification of the hydraulic system of the city and a high degree of vulnerability in the flood plain. Given the inadequacy of the drainage infrastructure, the agglomeration is suffering from increasingly harsh flood events, especially during the rainy period. This is a trend known and present in the literature on urban agglomerations in SSA.

Faced with the rapid growth of the population, which leads to high demand in terms of constructions, urban planning and management are essential to avoid installations in floodprone areas. This concerns dwellings as well as other structures that may attract inhabitants. The inadequacy or even the absence of planning, most of the time accompanied by a lack of provision of adequate services and equipment, leads to concentrated precariousness in flood-prone zones, where land is obviously cheaper. Integrating flood risk management in spatial planning policies is essential to curb this phenomenon. It should be combined

with proper implementation of these policies, targeting both newly developed and existing urban areas. Since many people live in flood-prone areas, it would be more relevant to work on resilience to reduce vulnerability than to relocate this large volume of people.

#### **7. Limitations of the Research**

In the study of flooding in urban areas, it is advisable to consider the different parameters mentioned above to understand the issue and be able to propose solutions to reduce vulnerability. In the framework of our research, we focused on the product of urbanization and its combination with other parameters that promote vulnerability. We did not consider here some factors that may contribute to increase urban flooding, such as climate change. The flood-prone areas considered in our study remain constant over time, because they are based on the calculation of topographic indices (TWI and SPI) relative to the topography. The study does not consider either the amount of precipitation or the flood history. Finally, in our paper, we considered urbanization relative to the whole of the built spaces. It would be relevant to consider built spaces that are only residential areas and integrate land use and land use change.

**Author Contributions:** Conceptualization and methodology: F.N.R. and J.T.; software, validation, and formal analysis: F.N.R.; investigation and resources: F.N.R. and J.T.; data curation: F.N.R.; writing—original draft preparation: F.N.R. and J.T.; writing—review and editing: F.N.R. and J.T.; visualization: F.N.R. and J.T.; supervision: J.T.; project administration: F.N.R. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Data is contained within the article. They are also available on request from the corresponding author.

**Acknowledgments:** We are grateful to M. Faly Rabemanantsoa, M. Jaotiana Rasolomamonjy, and M. Jean Ruffin Ramiaramanana for their help in carrying out this work and providing relevant data.

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

#### **References**


**Jaan H. Pu 1,\* , Joseph T. Wallwork <sup>1</sup> , Md. Amir Khan <sup>2</sup> , Manish Pandey <sup>3</sup> , Hanif Pourshahbaz <sup>4</sup> , Alfrendo Satyanaga <sup>5</sup> , Prashanth R. Hanmaiahgari <sup>6</sup> and Tim Gough <sup>1</sup>**


**Abstract:** During flooding, the suspended sediment transport usually experiences a wide-range of dilute to hyper-concentrated suspended sediment transport depending on the local flow and ground conditions. This paper assesses the distribution of sediment for a variety of hyper-concentrated and dilute flows. Due to the differences between hyper-concentrated and dilute flows, a linear-power coupled model is proposed to integrate these considerations. A parameterised method combining the sediment size, Rouse number, mean concentration, and flow depth parameters has been used for modelling the sediment profile. The accuracy of the proposed model has been verified against the reported laboratory measurements and comparison with other published analytical methods. The proposed method has been shown to effectively compute the concentration profile for a wide range of suspended sediment conditions from hyper-concentrated to dilute flows. Detailed comparisons reveal that the proposed model calculates the dilute profile with good correspondence to the measured data and other modelling results from literature. For the hyper-concentrated profile, a clear division of lower (bed-load) to upper layer (suspended-load) transport can be observed in the measured data. Using the proposed model, the transitional point from this lower to upper layer transport can be calculated precisely.

**Keywords:** parameterised power-linear model; hyper concentration; dilute concentration; suspended sediment transport; flood; sediment size parameter; rouse number; mean concentration; flow depth

## **1. Introduction**

Sediment transport is a common phenomenon during flooding. When sufficient lift force on sediment particles exists to overcome the frictional grips in between them, flow turbulence especially in the upward direction will generate sediment suspension [1,2]. Unlike the bed load, this suspended load is still not well-understood especially for those sediments with highly soluble behaviour in flow [3].

Two-phase flow is usually subjected to complex mixture between the solid and fluid phases. It is complex to mathematically model, in particular when one considers the natural flow in compound or irregular channels such as those studied by Pu [4] and Pu et al. [5]. Some models [6–8] resolve these complexities by neglecting turbulence and forces acting on the sediment particle surfaces, such as the effects of turbulent diffusion in laminar uniform flow or particle–particle collisions within dilute flows. However, applying these assumptions significantly hinders the modelling accuracy. As a result, recent studies

**Citation:** Pu, J.H.; Wallwork, J.T.; Khan, M..A.; Pandey, M.; Pourshahbaz, H.; Satyanaga, A.; Hanmaiahgari, P.R.; Gough, T. Flood Suspended Sediment Transport: Combined Modelling from Dilute to Hyper-Concentrated Flow. *Water* **2021**, *13*, 379. https://doi.org/ 10.3390/w13030379

Academic Editor: Jorge Leandro and James Shucksmith Received: 13 December 2020 Accepted: 28 January 2021 Published: 1 February 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

have attempted to incorporate the resultant lift and drag forces acting on the particle phase [9–11], and this has resulted in diverse formulations for predicting the suspended sediment profile within the flow.

The general consensus when modelling the sediment-laden flow is to assume a representative two-dimensional plane due to the complexity of full 3D modelling [12]. The sediment concentration is normally considered to change with height from the bed [13]; and various flow parameters can be incorporated into mathematical models to determine the full concentration profile. These parameters commonly include: particle fall velocity, particle diameter, Rouse number and mean concentration [14,15]. Within the field of sediment profiling a range of mathematical concepts has been adopted to predict the concentration profile. Goree et al. [16] used continuum theory and incorporated the effect of drift flux due to flow turbulence implemented using large-eddy simulation. However, it was found that the computed results were less accurate in the near-wall region (also agree with [17,18]). Rouse [6] proposed diffusion theory to form one of the simplest mathematical approaches. Despite the apparent simplicity, this diffusion theory-based calculation gave reasonable and efficient prediction of the suspended solid behaviour and has subsequently been utilised as the basis for many further studies.

Another commonly used mathematical concept is that of kinetic theory. This theory is widely regarded as one of the most precise approaches to model sediment concentration distribution as it includes the response of both the solid and liquid phases as well as the interactions between them [11]. Other theories have also been produced and shown to give reasonable results, such as the combination between kinetic and diffusion theories proposed by Ni et al. [19].

In this paper, we are motivated to seek a representative model to analytically calculate the suspended sediment transport profile, since currently there is a lack of such modelling in literature to inclusively represent the diluted, transitional, and dense suspended sediment transport. In the view of this research gap, in this study, the reported models are analysed and prominent flow parameters are assessed. A method of parameterisation is introduced using an analytical regression analysis technique. Consequently, separate parameterised expressions have been proposed for a wide range of flow conditions (i.e., from dilute to hyper-concentrated flow), before being adopted into a coupled power-linear concentration model. Various tests have also been conducted to validate the proposed model with published experimental data to assess the model's accuracy.

#### **2. Models Review**

Diffusion theory has played an important role in mathematical modelling of the suspended solid transport and has been used as the basis of the models by van Rijn [20], Wang and Ni [15], McLean [21], and Zhong et al. [22]. Rouse [6] derived his model from Fick's Law, which defines diffusion theory, and states that diffusion from an area of high concentration to an area of low concentration should be balanced by the product of the settling velocity and concentration as described in Equation (1) [23]:

$$\mathbf{D}\frac{\mathbf{dc}}{\mathbf{dy}} = -\omega\_0 \mathbf{c} \tag{1}$$

where D is sediment diffusivity (m2/s), c is concentration (dimensionless), ω<sup>0</sup> is settling velocity (m/s), and y represents the vertical space across a flow depth (m). Within Fick's formula, assumptions about sediment diffusivity must be made, for which Rouse proposed that the upward diffusion was a result of the vertical flux due to turbulence and assumed that the suspended particles was only associated with fluid turbulence diffusivity [7]. This agrees with the law of wall such that the sediment diffusivity is defined by the shear velocity u∗. Therefore, D can be defined by (Equation (2)):

$$\mathbf{D} = \kappa \mathbf{y} \mathbf{u}\_\* (1 - \varepsilon) \tag{2}$$

where κ is von Karman constant (dimensionless), u∗ is the shear velocity (m/s) and ε is the characteristic height (dimensionless) defined as the vertical distance, y, from the boundary normalised by the flow depth h (Equation (3)):

$$
\varepsilon = \frac{\mathbf{y}}{\mathbf{h}} \tag{3}
$$

Hence ε is limited by 0 < ε ≤ 1.

Inserting Equation (2) into Equation (1) gives Equation (4) as follow:

$$\frac{1}{\mathbf{c}}\mathbf{dc} = -\frac{\omega\_0}{\kappa \mathbf{y} \mathbf{u}\_\*(1-\varepsilon)} \mathbf{dy} \tag{4}$$

Integrating Equation (4) between the boundaries ε and a reference characteristic height ε<sup>a</sup> gives the Rouse formula (Equation (5))

$$\frac{\mathbf{c}}{\mathbf{c}\_{\mathbf{a}}} = \left[\frac{1-\varepsilon}{\varepsilon} . \frac{\varepsilon\_{\mathbf{a}}}{1-\varepsilon\_{\mathbf{a}}}\right]^{\frac{\omega\_{0}}{\kappa u\_{\ast}}} \tag{5}$$

where c<sup>a</sup> is the concentration at the reference height (dimensionless). ε<sup>a</sup> is described as the point where suspended load transport begins to take place and suggested to be 0.005 by Hsu et al. [7]. Under the assumption made by Rouse, the concentration distribution profile becomes more uniform with decreasing Rouse number which can be achieved by using sediment with low settling velocity or by increasing shear velocity, where the Rouse number P can be described by Equation (6):

$$\mathbf{P} = \frac{\omega\_0}{\kappa \mathbf{u}\_\*} \tag{6}$$

A modified model from Rouse has been presented by Kundu and Ghoshal [14] in which they recognised that the sediment concentration distribution can follow more than one profiles, as depicted in Figure 1. The most common profile (Type I) shows a monotonic decrease in concentration with height, and it happens when the flow concentration is dilute. The Type II profile shows an increase in concentration with height to a peak value above the bed, thereafter the concentration decreasing with height (it happens when flow is experiencing transitional concentration between dilute and dense condition). This Type II profile gives rise to a transitional point splitting the distribution into an upper flow region (above maximum concentration) and a lower flow region (below the maximum concentration in the near-bed region). The Type III profile occurs when the flow is subjected to hyper-concentration of sediment and exhibits a steady increase from the bed followed by a decrease in concentration towards the outer region of the flow.

In terms of modelling, Type I allows the most simplistic solution as it can be fitted using the common Rouse approach. However, the heavy sediment-laden flows usually present Type II or III profile. In common with the Rouse model, the dependent variable for the model presented by Kundu and Ghoshal [14] is *ε*, where its functions can be defined as (Equations (7) and (8))

$$
\mathfrak{q}\_1 = \mathfrak{b}\_1 \mathfrak{e}^{\mathfrak{a}\_1} + \mathfrak{q}\_1 \tag{7}
$$

and,

$$\mathfrak{q}\_2 = \mathfrak{b}\_2 \mathfrak{e}^{\mathfrak{\alpha}\_2} + \mathfrak{q}\_2 \tag{8}$$

in which b1, α1, q1, b2, α<sup>2</sup> and q<sup>2</sup> are empirical coefficients to be determined from experimental data.

**Figure 1.** Type I, II and III Concentration Profiles.

Using the asymptotic matching technique by Almedeij [24], the concentration profile for both adjacent sections can be expressed in the Equation (9):

$$\frac{\mathbf{c}}{\overline{\mathbf{c}}} = \frac{1}{\left(\mathbf{b}\_1 \varepsilon^{\alpha\_1} + \mathbf{q}\_1\right)^{-1} + \left(\mathbf{b}\_2 \varepsilon^{\alpha\_2} + \mathbf{q}\_2\right)^{-1}}\tag{9}$$

within which ϕ<sup>1</sup> represents the lower suspension flow region and ϕ<sup>2</sup> represents the upper suspension region. Using this technique, Kundu and Ghoshal [14] produced the empirical coefficients by calibration with previously published experimental data.

Several experimental studies (i.e., Einstein and Qian [25], Bouvard and Petkovic [26], Wang and Ni [15]) have also shown that the sediment profile follows a power law solution within the dilute-concentrated flow regime. This can be described by Equations (7) and (8) through the following simplification in Equation (10):

$$
\mathfrak{q} = \mathfrak{b}\mathfrak{e}^{\mathfrak{a}} \tag{10}
$$

where it is formed when the parameter q in Equations (7) and (8) is set as zero to produce a power law solution. In this formulation, Equation (10) reverts to a similar form as the Rouse formula shown in Equation (5).

However, within extreme flow conditions such as hyper-concentrated flow where the sediment profile has been proven to deviate from the power law distribution. Experimental results yield a linear profile due to an increase in particle–particle interactions. Equation (7) to (8) should thus take a form of the following in Equation (11):

$$
\mathfrak{q} = \mathfrak{b}\mathfrak{c} + \mathfrak{q} \tag{11}
$$

with exponent α equals to unity.

Limitations of this Rouse-type formulation have been evidenced by the measurements of Sumer et al. [27], Greimann and Holly [9], Jha and Bombardelli [10], Kironoto and Yulistiyanto [28], and Goeree et al. [16]. Owing to its derivation from diffusion theory, the Rouse formula provides a single-phase approach focusing on the sediment particles. As a result, the Rouse formula is limited to the representations of flows exhibiting Type I concentration profiles (Figure 1). Due to the boundary assumptions of the Rouse formula, the resultant concentration profile must always revert to zero at the fluid surface and infinity at the bed [29]. Huang et al. [8] further stated that the Rouse formula can lose its accuracy near-bed particularly when dealing with high boundary roughness. One of the attempts to improve the Rouse model is to incorporate an additional factor β into the

Rouse number producing a damping effect, where β is the coefficient of proportionality for the diffusion coefficient for sediment transfer [29].

Greimann and Holly [9] derived a formula using a two-phase approach to the Rouse model. Within their study it is highlighted that, due to Rouse's lack of consideration of particle–particle interactions, the Rouse formula is only valid when *c* < 0.1. As the Rouse formula is derived from Fick's law, it is only applicable to flow when the bulk Stokes number *S<sup>b</sup>* (which is a parameter commonly used to define characteristic of suspended particles in a fluid flow) is very small such that the fluid and solid phases are transported almost in equilibrium. Therefore, it can be concluded that while the Rouse formula gives reasonable calculation to sediment profiling, it is limited by the absence of mechanical forces such as particle–particle interactions and particle inertia, and by its lack of effective sediment parameterisation, i.e., related to sediment size. In comparison, the models proposed by Wang and Ni [15], Ni et al. [19], and Zhong et al. [22] utilised either exponential or power laws to precisely represent suspended sediment profiles across the whole flow depth and with a variety of concentration levels. They adapted kinetic concepts for considering the particle concentration, thus can model two-phase interactions. Additionally, they used empirical fit to determine the profile characteristic, and identified various flow and sediment parameters that can be potentially used to define the concentration profile.

The aim of this study is to investigate the relationship between various flow and sediment parameters to form an improved representation to Equations (7)–(9). This will form a parameterised expression of final suspended particle characteristic model and allow an effective prediction of its concentration profile. The flow parameters to be investigated are Rouse number P, size parameter Sz, and mean concentration c. Additionally, this kind of formulation using the parameterised expressions to improve the suspended sediment transport modelling has so far not been explored in other studies, hence this investigation is crucially needed to study the performance of such modelling.

#### **3. Proposed Modelling**

Many studies have investigated the relevant parameters for considering a concentration profile, including the Rouse number (defined in Equation (6)), particle size, mean concentration, and flow depth [6,27,30,31]. By referring to Equation (9), the variables are related to the coefficients of power-linear law as follows in Equation (12):

$$\mathbf{b}\_{1\prime} \ \mathbf{b}\_{2\prime} \ \mathbf{a}\_{1\prime} \ \mathbf{a}\_{2\prime} \ \mathbf{q}\_{1\prime} \mathbf{q}\_{2} = \mathbf{f}(\mathbf{P}\_{\prime} \mathbf{S}\_{\mathbf{Z}\prime} \ \mathbf{\tilde{c}}) \tag{12}$$

where, S<sup>z</sup> is the dimensionless size parameter (S<sup>z</sup> = d/h), in which d is the sediment particle diameter and h is the flow depth. In this investigation, we collected data from various reported experimental studies (as detailed in Table 1) to inspect the distribution of each power-linear law coefficient toward the physical parameters of Rouse number, particle size, and mean concentration, and to deduce a modified Rouse model for validation tests. It can be observed from Table 1 that the utilised data sources are in a wide range. In particular, the c range in the utilised literature are ranging from 0.00013 to 0.147, which giving a thorough test of concentrations from dilute to hyper-concentrated flow conditions.


#### *3.1. Rouse Number*

Two of the main parameters affecting drag on a sediment particle are the settling and shear velocities. A dimensionless form of these parameters together with von Karman

constant is the Rouse number as defined in Equation (6). By studying each parameter in Equation (9) against P, we can produce Figures 2–5 below.

**Figure 2.** Rouse number regression analysis for coefficient b<sup>1</sup> .

**Figure 3.** Rouse number regression analysis for coefficient α<sup>1</sup> .

**Figure 4.** Rouse number regression analysis for coefficient b2.

**Figure 5.** Rouse number regression analysis for coefficient α2.

Figures 2–5 show that there is a quadratic relationship for the parameters b<sup>1</sup> and α1, and a linear relationship for b<sup>2</sup> and α<sup>2</sup> against P. The regression analysis shows all coefficients have R <sup>2</sup> > 0.5, except for α2. This finding shows P provides reasonable fit to be represented by power-law and its coefficients in wide range of measured data. As analysed by Kundu and Ghoshal [11], the Rouse number function does not provide a good representation to the hyper-concentrated profile, and hence the hyper-concentrated flow data have been omitted in Figures 2–5.

#### *3.2. Size Parameter*

Particle size is another factor that significantly affects sediment drag and lift. The surface area of a particle, determined by the diameter for a spherical particle, can affect the effectiveness of interactive contacts act on the particle. Additionally, particle diameter also influences its settling velocity [34]. In Figures 6–9, the dimensionless S<sup>Z</sup> is plotted against the proposed model's power-law coefficients. In the original Rouse approach [6] or modified Rouse model (as used in Kundu and Ghoshal [14]), the effect of particle size has not been considered, even though it is a crucial factor in determining the suspended sediment behaviour.

In Figures 6–9, it can be observed that a quadratic relationship describes the variation of b<sup>1</sup> and b<sup>2</sup> with S<sup>Z</sup> while a logarithmic relationship for α<sup>1</sup> and α<sup>2</sup> against S<sup>Z</sup> is observed. All the figures show R 2 regression lower than 0.5 with the exception of b1. This low regression shows that α and b are harder to be represented by SZ, which in turns exposes the difficulty of modelling using SZ. Its analysis further suggests that the particle size factor is harder to be fixed. In the analytical modelling studies of Wang and Ni [15] and Ni et al. [19], the particle diameter has been fitted by using a coefficient in the concentration equation extracted from the measured data. The described tests have shown that it is hard to capture the characteristic of concentration profiles when different sediment diameters have been tested. This further affirms the difficulty of finding a representative function for the particle size parameter investigated here.

**Figure 6.** Size parameter regression analysis for coefficient b<sup>1</sup> .

**Figure 8.** Size parameter regression analysis for coefficient b2.

**Figure 9.** Size parameter regression analysis for coefficient α2.

#### *3.3. Mean Concentration*

Michalik [35] and Cellino and Graf [32] investigated the measured sediment concentration profiles for hyper-concentrated flows. Their results showed that for such flows, the sediment profile follows a more linear distribution as opposed to the common power law observed in other dilute flow studies. Another empirical observation by Machalik [35] was that the mean concentration has key dominant impact on the characteristic of concentration distribution compared to Rouse number or particle size. Hence this study will formulate the analytical approach by mean concentration for use in the linear law modelling. Regression analysis was also used to identify the fit between the linear law coefficients and mean concentration, as presented in Figures 10–13.

**Figure 10.** Mean concentration regression analysis for coefficient b<sup>1</sup> .

**Figure 11.** Mean concentration regression analysis for coefficient q<sup>1</sup> .

**Figure 12.** Mean concentration regression analysis for coefficient b2.

The results show that a linear fit describes the variation for the coefficients b<sup>1</sup> and b<sup>2</sup> with mean concentration; while a quadratic relationship describes the variation of the coefficients q<sup>1</sup> and q<sup>2</sup> with c. The fits show an R 2 regression higher than 0.5 without exception. This finding evidences a clear correlation between c and the hyper-concentrated profile.

#### *3.4. Hyper-To-Dilute Boundary*

The findings from the above sections are adapted into the model of Equation (9) to form a parameterised expression for the sediment concentration distribution across the flow depth. The proposed sediment concentration calculative model is governed by a coupled approach. A power law is utilised to represent the dilute sediment concentration (when 0 < c < 0.1), whereas a linear law is used for the dense hyper-concentration (when c ≥ 0.1). This hyper-to-dilute boundary has been set by benchmarking the investigation of Greimann and Holly [9] on dilute flow definition and Rouse model limit. Our proposed coefficients found from the above sections can be represented as:

For the dilute regime, where 0 < c < 0.1 (Equations (13)–(17)):

$$\mathbf{q}\_1 = \mathbf{q}\_2 = 0\tag{13}$$

$$\mathbf{b}\_{1} = 0.047\mathbf{P}^{2} - 0.23\mathbf{P} + 270\mathbf{S}\_{\mathbf{z}}{}^{2} - 17\mathbf{S}\_{\mathbf{z}} + 0.68,\tag{14}$$

$$\alpha\_1 = 0.19 \text{P}^2 - 0.073 \text{P} - 0.15 \ln(\text{S}\_\text{z}) - 2.0 \,, \tag{15}$$

$$\mathbf{b}\_2 = 0.48\mathbf{P} - 1100\mathbf{S}\_\mathbf{z}^2 + 73\mathbf{S}\_\mathbf{z} + 1.6,\tag{16}$$

$$
\alpha\_2 = -0.016 \text{P} - 0.015 \ln(\text{S}\_{\text{Z}}) + 0.025. \tag{17}
$$

For the hyper-concentrated regime where c ≥ 0.1 (Equations (18)–(22)):

$$
\mathfrak{a}\_1 = \mathfrak{a}\_2 = 1.0 \tag{18}
$$

$$\mathbf{b}\_1 = -1.4\,\overline{\mathbf{c}} - 0.39\,,\tag{19}$$

$$\mathbf{q}\_1 = 2.6\overline{\mathbf{c}}^2 + 0.39\overline{\mathbf{c}} + 0.51,\tag{20}$$

$$\mathbf{b}\_2 = \mathbf{1}1 \,\, \overline{\mathbf{c}} - \mathbf{2} \,\mathbf{1},\tag{21}$$

$$\mathbf{q}\_2 = -11\ \overline{\mathbf{c}}^2 + 7.7\ \overline{\mathbf{c}} - 0.24. \tag{22}$$

#### **4. Model Validations**

The model presented within this paper is validated against the experimental data of Wang and Ni [31], Wang and Qian [36], and Michalik [35]. It has also been compared with the previously proposed models by Wang and Ni [31], Ni et al. [19], and Zhong et al. [22]. Wang and Ni [31] presented a theoretical distribution model derived from the kinetic theory. Their model is limited to dilute flow and therefore predicts Type I and limited Type II profiles only. In their assumption, the particle interaction has been neglected, and as a result, they attributed the classification of distribution profile solely to the fluid-induced lift forces. It is also noteworthy within their study that when particle size is small the distribution tends to follow the Type I profile.

The model proposed by Ni et al. [19] used a fusion of kinetic and continuum theories, where kinetic theory using the Boltzmann equation being applied to the solid-phase and continuum theory to the fluid-phase. Within their derivation, the empirically weighted forces have been used to act upon sediment to represent two-phase interactions. The model has been proposed to be applicable to both dilute and dense flows. The model proposed by Zhong et al. [22] is more complex when compared to the other two above-mentioned models. It is based on a tertiary approach where the model can be simplified under various empirically-driven assumptions. Within their research, the experimental testing covers Type I, II, and III profiles; though due to its complexity, their Type III profile required a dynamic value of the empirical damping function to fit for different flow conditions.

#### *4.1. Wang and Ni*

The measured data of Wang and Ni [31] assessed dilute flow within pipes. The concentrations tested were extremely dilute ranging from 0.00042 ≤ c ≤ 0.0033, where these tested conditions are presented in Table 2. This validation exercise will provide a good test to the proposed model capability to capture extremely dilute flow. The sediments tested were grains and coarse sands with particle diameter ranging from 0.58 mm ≤ d ≤ 2.29 mm. The results are presented in Figure 14A–J. The proposed model shows a reasonable correspondence to the experimental data. The measurements of Wang and Ni [31] show that for dilute flow the sediment concentration tends to follow the Type I or II concentration profile with the maximum concentration occurring in the near-bed region.


**Table 2.** Data by Wang and Ni [31].

**Figure 14.** *Cont*.

**Figure 14.** *Cont*.

**Figure 14.** *Cont*.

**Figure 14.** Modelled results and comparisons against experimental data of Wang and Ni [31].

Observation of Figure 14A–J shows that overall there is a better fit by the proposed and other models away from the near-bed region. This is coherent with the suggestions from literature (i.e., Kundu and Ghoshal [11]; Greimann and Holly [9]) stating that the possibility of particle–particle interactions increases at near-bed to produce more challenging conditions for the mathematical modelling.

The experimental data in Figure 14F utilised the largest particles among all the measured data of Wang and Ni [31]. Therefore, larger interaction forces can be expected between the solid–fluid phases due to the larger surface area of each sediment particle. One can observe that the proposed model shows a concentration distribution for Figure 14F which is consistent with the measurements. Compared with the model of Zhong et al. [22], which does not take into account the particle size, the proposed model shows promising computation of big particle measured data.

#### *4.2. Wang and Qian*

Wang and Qian [36] studied the effect of dilute to dense concentrations in open channel flow using an experimental recirculating-tilting flume. Their experiments tested a wide range of sediment diameters 0.15 mm ≤ d ≤ 0.96 mm and concentrations 0.0102 ≤ c ≤ 0.0906 (as shown in Table 3). Their tests are compared against the proposed and other models, with the results presented in Figure 15A–D. Overall, the models show lower accuracy to reproduced measured data throughout the flow depth with increasing mean concentration. The proposed model shows a reasonable fit to the experimental data in the upper flow region. Within this region, the main forces acting on the sediment particles are the lift and drag due to the fluid induced forces and particle inertia. This supports the hypothesis that the proposed model can compute the solid–fluid interactions for particles with reasonable accuracy owing to its inclusion of Rouse and size parameters.

An overview of all the results displayed in Figure 15A–D demonstrates that the accuracy of different models reduces at the lower suspension region, including the proposed model. Within Type II profiles, the local maximum in concentration is observed closer to the bed compared to Type III profiles. Near to the wall boundary, the sediment distribution is governed by the bed-load behaviour as opposed to the suspended load. The plastic particles were used within the flows tested here, and they generate less significant movement than the normal and natural sediment under the acting of particle–particle interactive forces [36]. Due to this, the proposed model that deals with the forces by coupling expressions of Rouse and size parameters unable to simulate the near-bed concentrations reasonably, as compared to the models presented in Ni et al. [19] and Zhong et al. [22] which used imported empirical functions from the respective experiments into their modelling. In addition, the boundary conditions for flow with transitional concentration (from

dilute to dense concentrations) are hard to be fixed, and hence this difficulty may cause discrepancy in the proposed modelling (especially presented by results at Figure 15A,B).

**Table 3.** Data by Wang and Qian [36].


**Figure 15.** *Cont*.

**Figure 15.** Modelled results and comparisons against experimental data of Wang and Qian [36].

#### *4.3. Michalik*

The experimental data of Michalik [35] quantified the sediment profile of hyperconcentrated flows. The sediment material used was sand with a mean diameter of 0.45 mm and with concentration ranging between 0.15 ≤ c ≤ 0.54 (all test conditions are shown in Table 4). In order to model Michalik's tests, this study uses the linear law within the proposed model. The test results are shown in Figure 16A–F, where the modelled results are compared to measurements. Ni et al. [19] and Zhong et al. [22] models are also incorporated into these figures to compare with the measurements and proposed model.

**Table 4.** Data by Michalik [35].


From the tested hyper-concentrated flows, the sediment concentration distribution illustrates a Type III profile. Within hyper-concentrated flow, the maximum in concentration is difficult to model accurately since there exists no distinct boundary where the sediment changes from bed to suspended load but rather the bed load can diffuse into the suspended state through a transitional region. As a result, the suspended load has sometimes been estimated as part of the bed load, which increases the discrepency in suspended solid modelling. To accurately define the transition region and consequently the location of this maximum turning point, a good estimation of *ε<sup>a</sup>* and *c<sup>a</sup>* is required [27]. In this study, the proposed model uses *c* for the mathematical modelling, since it seems acceptable to conclude that *c* is proportional to *c<sup>a</sup>* [37] given that they are both invariant for a given experiment.

The proposed calculated results show reasonable agreement to the experimental data, in which it exhibits better fit to the measurements compared to the rest of models by Ni et al. [19] and Zhong et al. [22] in Figure 16A–F. The figures demonstrate a Type III profiles that fit into the hyper-concentrated distribution. Studies of hyper-concentrated flow show that the sediment concentration distribution becomes more homogenous when *c* increased as proven in Michalik [35]. Hence power law distributions might not be reasonable to model hyper-concentrated flow. Instead a linear model is adopted here and

is evidenced to produce better accuracy. Even though the power-exponential models of Ni et al. [19] and Zhong et al. [22] are suitable to represent densely-concentrated flow, their accuracy are not encouraging compare to the measurements in majority of relatively hyper-concentrated *c* tests (i.e., at Figure 16D–F).

Additionally, one can observe that an increase in the mean concentration shifts the height of maximum concentration upwards through the characteristic height due to the increase of potential bed-load layer. As the mean concentration is increased, the nearbed region becomes more saturated. The tests in Figure 16 show that when c > 0.31 the settling velocity has limited influence and the dominating interaction forces must arise from particle–particle reactions. In these hyper-concentrated tests of c > 0.31, the proposed model represents the concentration distribution well. In particular, the proposed model accurately predicts the height at which the maximum concentration occurs for most cases.

**Figure 16.** *Cont*.

**Figure 16.** *Cont*.

**Figure 16.** Showing modelled results and comparisons for experimental data of Michalik [35].

#### **5. Conclusions**

A parameterised power-linear coupled model has been introduced for inclusively computing the dilute- to hyper-concentrated distribution across the characteristic height within flow. The parameters used for the formulation of this model were size parameter, Rouse number, and mean concentration. As proven, the model is able to accurately compute the suspended sediment profile for a range of flow conditions including various Rouse numbers. The proposed model shows a reasonable accuracy for low and very high concentration tests across the Type I to III profiles. This can be seen from the comparisons with experimental data of Wang and Ni [31] on very dilute flows, Wang and Qian [36] on mixed dilute to dense flows, and Michalik [35] on hyper-concentrated flows. From the tests, the coupling approach of power to linear modelling has been proven to reasonably represent flow with a wide range of concentrations and sediment sizes.

This type of suspended modelling holds key importance to the accurate prediction of various natural flows, such as river, coastal, or flood flow. In flooded condition, the sediment mixture impacts the flow behaviour that can cause modelling failure in reproducing the real-world flood flow. With this analytical modelling study, the flood induced suspended sediment transport with wide range of dilute to dense concentration can be modelled adequately; and hence to provide the vital capability to flood flow modelling. Additionally, to further this work, different analytical modelling besides Rouse-based model can also be investigated.

**Author Contributions:** J.H.P.: writing—original draft preparation, writing—review and editing, funding acquisition, project administration, data curation, supervision; J.T.W.: writing—original draft preparation, writing—review and editing, data curation; M.A.K.: writing—review and editing, data curation; M.P.: writing—review and editing, data curation; H.P.: writing—review and editing, data curation; A.S.: writing—review and editing, data curation; P.R.H.: writing—review and editing, data curation; T.G.: writing—review and editing, supervision. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** The data presented in this study are available on reasonable request from the corresponding author.

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