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,
7,
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 have attempted to incorporate the resultant lift and drag forces acting on the particle phase [
9,
10,
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]:
where
is sediment diffusivity (m
2/s),
is concentration (dimensionless),
is settling velocity (m/s), and
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
. Therefore,
can be defined by (Equation (2)):
where
is von Karman constant (dimensionless),
is the shear velocity (m/s) and
is the characteristic height (dimensionless) defined as the vertical distance,
, from the boundary normalised by the flow depth
(Equation (3)):
Hence is limited by .
Inserting Equation (2) into Equation (1) gives Equation (4) as follow:
Integrating Equation (4) between the boundaries
and a reference characteristic height
gives the Rouse formula (Equation (5))
where
is the concentration at the reference height (dimensionless).
is described as the point where suspended load transport begins to take place and suggested to be
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
can be described by Equation (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
, where its functions can be defined as (Equations (7) and (8))
and,
in which
and
are empirical coefficients to be determined from experimental data.
Using the asymptotic matching technique by Almedeij [
24], the concentration profile for both adjacent sections can be expressed in the Equation (9):
within which
represents the lower suspension flow region and
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):
where it is formed when the parameter
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):
with exponent
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
into the Rouse number producing a damping effect, where
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
(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 , size parameter , and mean concentration . 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.
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.