Propeller-Induced Jet Impact on Vegetated Flow Fields: Complex Coupled Effect towards the Velocity Profile

Abstract

The failure of swirling ship propellers in marine environments can lead to huge repair costs. One of the main causes of such failure is when propellers tangle with vegetation, especially in shallow flow environments like ports, harbours, or shipyards. In order to understand the above-mentioned issue, this study proposes an analytical approach to explore efficient predictions and provide a flow guideline with respect to the co-existence of vegetation and propeller swirling effects. More specifically, we intend to investigate the full-depth theoretical velocity profile to represent propeller-induced flow under submerged vegetation conditions. This paper first investigates the modified logarithmic law approach to determine its suitability to represent the regional vegetated flow zone before implementing it into a three-layer analytical model. It was found, using the benchmark of literature measurements, that the modified log law improved the near-bed velocity calculation by nearly 70% when compared to an existing model. A propeller jet impact computation coupled into the vegetation analytical model was then investigated in different locations within the vegetated flow, i.e., at free-flow, water–vegetation interface, and vegetation-hindered zones, to study their complex velocity distribution patterns. The results demonstrate adequate validation with the vegetated flow and measured propeller jet data from the literature. This proves the potential of the proposed analytical approach in representing a real-world propeller jet event submerged in water flow with the existence of vegetation. The proposed novel method allows costless computation, i.e., as compared to conventional numerical models, in representing the complex interaction of the propeller jet and vegetated flow.

1. Introduction

Over past decades, studies on velocity distribution and its accuracy in open-channel flow have been conducted [1], which are useful for numerous research fields, including river restoration, sediment transport, river morphology, power plant design, and hydrometry [1,2]. The logarithmic trend has been followed by the vertical velocity profile above aquatic vegetation, as shown by various experimental and theoretical studies on open-channel vegetated flows [3,4,5]. According to Pu et al. [4], a vegetated layer submerged within the flow increases flow resistance and can be treated as exhibiting a similar effect to a roughness layer. In addition, Pu et al. [6] and Pu [7] proposed water-worked bedforms, which usually occur in natural channels, to be another factor that alters velocity distribution in near-bed flow regions. Overall, the non-vegetated flow velocity profile is determined by the bed’s shear stress and physical flow parameters, while drag and friction parameters with respect to vegetation are the key determinants for vegetated flow, i.e., for submerged and emerged cases.

Excessive sediment transport induced by propeller-equipped big ships is common in harbours and waterways [8,9]. Studies by Hong et al. [10] and Hamil et al. [11] discussed the impact of propeller-induced jets on the velocity profile and bed scouring. According to Albertson et al. [12], the characteristics of velocity downstream from jet diffusion are similar to a submerged 3D jet. Besides imitating real-world cases, the bed scouring caused by the 3D propeller-induced flow has also been studied experimentally by Penna et al. [13], and their findings further emphasised the impact of propeller flow, particularly in shallow flow fields. As identified by Hong et al. [10] and Hamil et al. [11], the axial velocity along the propeller axis of rotation is the most important component for altering the flow’s characteristic as it releases forward thrust to push the ship in the desired direction; hence, more research is required to understand its behaviour in various real-world flow conditions to achieve a functional guideline that can maximise ship operations.

Up until now, there has been a lack of analytical modelling investigations of the combined flow field from propeller-induced jet and vegetated flow (the combined effect is shown in the schematic diagram of Figure 1). The existing method to investigate this complex engineering problem is by using numerical models, which can be computationally costly. In view of this research field’s importance to reduce ship malfunction in shallow flow conditions, this study researched an analytical approach to understand and, hence, mitigate this problem. To the author’s knowledge, this study is the first to theoretically combine and study the propeller-induced jet and vegetated flow coherently. To systematically study the mentioned complex flow, this paper will first investigate the flow model with submerged vegetation to obtain a suitable representation of the vegetated velocity profile. After that, the impact of the ship’s propeller will be studied and included into the vegetated flow model to achieve a calculation of the combined velocity characteristic of propeller jet-induced vegetated flow. This study will also validate the presented modelling approach through separate vegetated and propeller-induced flow experiments reported in the literature.

2. Vegetation Model

Logarithmic law (log law) and its modified forms have presented great potential to represent velocity profiles for vegetated flow [14]. To represent velocity distribution across full flow depth, researchers have proposed various two-layer models where velocity has been characterised separately at free water and submerged vegetation layers [4,15,16]. For the vegetation-laden layer, the height of vegetation, including the deflection effect, was calculated using the bending moment caused by the vegetation’s flexibility [17]. To revise the log law, the roughness factor generated by vegetation has also been considered [4,15].

Huai et al. [3] further proposed a three-layer model to more precisely represent submerged rigid vegetation flow. In their proposal, velocity profile modelling was separated into a non-vegetation layer, an inner layer, and an outer layer within vegetation. The model starts with the formulation of the momentum equation for steady flow as follows:

$$\frac{\mathit{\partial }\tau }{\mathit{\partial }\mathrm{z}}+{\mathsf{\rho }}_{\mathrm{o}}\mathrm{g}{\mathrm{S}}_{\mathrm{o}}-\frac{1}{2}{\mathsf{\rho }}_{\mathrm{o}}{\mathrm{C}}_{\mathrm{D}}\mathrm{m}\mathrm{D}{\mathrm{u}}^2=0$$

(1)

where $\tau$ is the shear stress, ${\mathsf{\rho }}_{\mathrm{o}}$ is the water density in the unit of kg/m³, ${\mathrm{S}}_{\mathrm{o}}$ is the bed slope, C_D is the drag coefficient, m is the density of vegetation in the unit of stems/m², D is the vegetation steam diameter, and u is streamwise flow velocity. The mixing length hypothesis can be utilised to find the shear stress for the inner vegetation region close to the bed [18].

$$\tau ={\mathsf{\rho }}_{\mathrm{o}}{\mathrm{l}}_{\mathrm{m}}^2\left|\frac{\mathit{\partial }\mathrm{u}}{\mathit{\partial }\mathrm{z}}\right|\frac{\mathit{\partial }\mathrm{u}}{\mathit{\partial }\mathrm{z}}$$

(2)

Here, the value of ${\mathrm{l}}_{\mathrm{m}}^2$ is the roughness height of the channel due to the boundary layer, and Equation (3) can be obtained by substituting Equation (2) into Equation (1).

$${\mathrm{l}}_{\mathrm{m}}^2\frac{\mathit{\partial }}{\mathit{\partial }\mathrm{z}}\left(\left|\frac{\mathit{\partial }\mathrm{u}}{\mathit{\partial }\mathrm{z}}\right|\frac{\mathit{\partial }\mathrm{u}}{\mathit{\partial }\mathrm{z}}\right)+\mathrm{g}{\mathrm{S}}_{\mathrm{o}}-\frac{1}{2}{\mathsf{\rho }}_{\mathrm{o}}{\mathrm{C}}_{\mathrm{D}}\mathrm{m}\mathrm{D}{\mathrm{u}}^2=0$$

(3)

The partial derivative in Equation (3) can be resolved using various numerical or analytical approaches to calculate velocity.

At the top layer of vegetation, the velocity profile can be represented by

$$\mathrm{u}=\sqrt{2\mathrm{g}{\mathrm{S}}_{\mathrm{o}}\left\{\mathsf{\alpha }{\mathrm{h}}_{\mathrm{u}}\mathrm{e}\mathrm{x}\mathrm{p}\left(\mathsf{\alpha }\left(\mathrm{z}-{\mathrm{h}}_{\mathrm{v}}\right)\right)+1\right\}/\left({\mathrm{C}}_{\mathrm{D}}\mathrm{m}\mathrm{D}\right)}$$

(4)

where h_u is the flow depth from the top of vegetation to the free surface, while h_v is the flow depth from the bed to the top of vegetation. Some Reynolds shear stress assumptions, i.e., ignoring viscosity shear stress and assuming momentum balance in the flow above the vegetated region, have been used to obtain Equation (5).

$$\tau ={\mathsf{\rho }}_{\mathrm{o}}\mathrm{g}{\mathrm{h}}_{\mathrm{u}}{\mathrm{S}}_{\mathrm{o}}\mathrm{e}\mathrm{x}\mathrm{p}\left(\mathsf{\alpha }\left(\mathrm{z}-{\mathrm{h}}_{\mathrm{v}}\right)\right)$$

(5)

In the search for a non-vegetated layer velocity profile, the combination of mixing length and vegetation boundary layer was introduced by Huai et al. [3]. Due to the roughness effect of the vegetated zone, the flow flux through vegetation must be considered in this layer. By omitting viscosity shear stress, one can obtain the below equation for Reynolds shear stress.

$$\overline{{\mathrm{u}}^{\prime }}\overline{{\mathrm{w}}^{\prime }}={-\mathrm{l}}^2({\frac{\mathit{\partial }\mathrm{u}}{\mathit{\partial }\mathrm{z}})}^2$$

(6)

Finally, the equation for the velocity profile can be formulated by solving Equation (6) using the mixing length assumption from [18], i.e., Equation (8)

$$\mathrm{u}={\mathrm{u}}^{\mathrm{*}}\left\{\frac{1}{\mathrm{k}}\ln (1+\mathrm{k}(\mathrm{z}-{\mathrm{h}}_{\mathrm{v}})/{\mathrm{l}}_{\mathrm{o}}\right\}+\mathrm{C}$$

(7)

$$\mathrm{l}=\left[{\mathrm{l}}_{\mathrm{o}}+\mathrm{k}(\mathrm{z}-{\mathrm{h}}_{\mathrm{v}})\right]\sqrt{1-(\mathrm{z}-{\mathrm{h}}_{\mathrm{v}})/{\mathrm{h}}_{\mathrm{u}}}$$

(8)

where l₀ equals the mixing length at the interface of vegetation and flow, and k is the von Karman constant, which is taken as 0.41 [2,3]. In Equation (7), u* is the shear velocity, and C is the continuum coefficient at the vegetation and flow interface, which is suggested by [3] to be

$$\mathrm{C}=\sqrt{2(\mathsf{\alpha }{\mathrm{h}}_{\mathrm{u}}+1)/\left({\mathrm{C}}_{\mathrm{D}}\mathrm{m}\mathrm{D}{\mathrm{h}}_{\mathrm{u}}\right)}$$

(9)

2.1. Modified Log Law Implementation at the Vegetated Bed Layer

At the near-bed region of a submerged vegetation flow, the velocity profile’s estimation is complicated, and finite numerical models (i.e., finite volume, finite element, or finite difference models) have usually been employed for its calculation [3,4]. However, the numerical calculation is usually time-consuming as compared to the analytical model and can defeat the purpose of timely vegetated flow prediction. Since Keulegan’s investigation, the log law and its various modified approaches have been proven to represent a wide range of flow conditions well, including those over smooth and rough beds [19].

In this study, the treatment of the near-bed region of vegetated flow using the modified log law has been proposed to mimic the flow via a representative roughness depth caused by vegetation. This representation method is rarely performed for the vegetation case in the literature, but it can provide a cost-effective method to represent the complex flow condition at the vegetated bed. In summary, to simplify the computation consideration, this study modifies the “rough-bed” representation to the inner layer of vegetation, where the basic model is presented for a fixed rough-bed flow as follows [20]:

$$\mathrm{U}=\frac{{\mathrm{u}}_{\mathrm{*}}}{\mathrm{k}}\ln \left(\frac{\mathrm{z}-{\mathrm{z}}_{\mathrm{o}}}{{\mathrm{k}}_{\mathrm{s}}}\right)+\mathrm{Z}$$

(10)

where k_s is the Nikuradse roughness, z is the depth up to velocity dip position, z_o is the physical zero-velocity position (which is regarded at channel fixed bed for vegetated flow), and Z is the logarithmic integration constant. The vegetation boundary condition is found in [21]:

$$\mathrm{U}\left(\mathsf{\xi }\right)=\left[\frac{{\mathsf{\xi }}_1^2/2+{\mathsf{\xi }}_1+{\mathrm{C}}_{\mathrm{A}\mathrm{r}}}{{\mathsf{\xi }}^2/2+\mathsf{\xi }+{\mathrm{C}}_{\mathrm{A}\mathrm{r}}}\right]\left[\frac{{\mathrm{g}\mathrm{S}}_{\mathrm{o}}\left[\left({\mathsf{\xi }}^2/4+\mathsf{\xi }\right)-\left({\mathsf{\xi }}_1^2/4+{\mathsf{\xi }}_1\right)+{\mathrm{C}}_{\mathrm{A}\mathrm{r}}\ln \left(\mathsf{\xi }/{\mathsf{\xi }}_1\right)\right]}{{\mathrm{u}}_{\mathrm{*}}\mathrm{k}\left({\mathsf{\xi }}_1^2/2+{\mathsf{\xi }}_1+{\mathrm{C}}_{\mathrm{A}\mathrm{r}}\right)}+\mathrm{U}\left({\mathsf{\xi }}_1\right)\right]$$

(11)

in which $\mathsf{\xi }=\mathrm{z}/\mathrm{h}$ (h is full water depth), and it is regarded as the characteristic depth. ${\mathsf{\xi }}_1$ is the characteristic depth for the inner flow region where the vegetated flow velocity profile changes from being log-based to exhibiting linear characteristics. ${\mathrm{C}}_{\mathrm{A}\mathrm{r}}$ is the constant calibrated by [21] based on the aspect ratio of flows.

Similarly to all proposed analytical models in the literature, this study’s model also needs assumptions to function. One such assumption is steady-state flow. A lot of previous studies have employed the same assumption since a non-steady, time-varying approach is too difficult to model using an analytical approach. The examples of the analytical model that used the steady flow assumption include those presented by Huai et al. [3], Huai et al. [15], Rowinski and Kubrak [18], Lassabatere et al. [19], and Pu [21]. In the coming sections, this paper will fully validate the proposed model by comparing it with actual measurements in the literature, in order to prove its functionality and accuracy.

2.2. Validation of the Combined Vegetated Model

To validate the discussed three-layer vegetated model, various reported measurements in the literature have been used in this study. In the first validation, Huai et al. [3] obtained measured data from a vegetated field (created using vegetation measuring 0.6 cm in diameter and 19 cm in height), and calculations were made using the presented model. The utilised experimental data were obtained from two pieces of vegetation situated parallelly, i.e., one after another longitudinally, as this location has been identified as a zone that is significantly affected by vegetation-induced flow turbulence. Also, in this first test, Huai et al. [3] proposed a model for the inner vegetated region that has been used to compare with the proposed modified log law model, benchmarking against the measured data. Zhao and Fan [22] measurements were chosen as the second validation test. Different from Huai et al. [3], Zhao and Fan [22] used truncated cone-shaped vegetation with 0.4 cm lower and 0.2 cm upper diameters at 24 cm height. Their measured data can further examine the capability of the proposed model to represent the unique resulting flow field. Finally, Kubrak et al.’s [23] laboratory data have also been used for validation, as it presented a rough sand bed where the vegetation was seated on. A total of two substantially varying k_s of the rough bed from their study have been employed for our testing, 0.0001 m (for Case 1.1.3) and 0.0158 m (for Case 4.1.1), where the utilised vegetations had elliptical cross-sections with a longer diameter of 0.095 cm and a shorter diameter of 0.07 cm at a height of 16.5 cm. All parameters from the literature-reported data are presented in

Table 1. Input data from each study.

	S0 (-)	m (m−2)	h (m)	Cd (-)	Q (l/s)
Huai et al. [3]—Case 1	0.004	2000	0.29	1.00	21.8
Huai et al. [3]—Case 2	0.004	2000	0.38	1.00	35.8
Zhao and Fan [22]—Run F1	0.0007	9820–9960	0.36	0.50	37.0
Zhao and Fan [22]—Run F2	0.0007	9820–9960	0.36	0.42	43.3
Zhao and Fan [22]—Run F3	0.0007	9820–9960	0.36	0.40	52.0
Zhao and Fan [22]—Run B	0.0001	8000	0.467	0.55	1.7
Kubrak et al. [23]—Case 1.1.3	0.005	10000	0.2475	1.00	33.3
Kubrak et al. [23]—Case 4.1.1	0.009	2500	0.2421	1.40	60.9

, in which they are used as the input conditions for the presented model’s calculation.

Figure 2 shows the comparison of the theoretical models with measurements from Huai et al.’s [3] experimental study. Away from the bed at the free-water zone, the presented model presents some discrepancies relative to the measurements, which can be a result of the error range in measurements in this zone by Huai et al. [3]—as observed from the oscillating measured data. Apart from this discrepancy, the proposed model captures the rest of the measured regions reasonably. At the near-bed region (i.e., at the inner vegetation region), both Huai et al. [3] and the modified log law models have been presented and compared. Using the benchmark of measurements, the modified log law model calculated the velocity profile with better accuracy compared to Huai et al.’s [3] model in the inner vegetation region, where the modified log law model improved velocity estimation accuracy by nearly 70% compared to the latter. This closer correspondence has been consistently shown for both Cases 1 and 2, which demonstrated the capability of the modified log law model. More specifically, Huai et al.’s [3] three-layer model underestimated the point of vegetation’s inner–outer layer interface, which has been improved by the employment of ${\mathsf{\xi }}_1$ in the modified log law approach.

To further explore the capability of the proposed model with the modified log law, the experimental runs by Zhao and Fan [22] have been utilised in this study (refer to the comparison in Figure 3). Their test conditions spread across 1.7 to 52.0 l/s of flow, and this can further investigate the model’s adaptability to fast and slow flows. Additionally, the utilised vegetation exhibited truncated cone shapes, which can assess the competence of the bespoke model to represent the flow field via different vegetation shapes. As observed in Figure 3, the calculation presents good correspondence to the measured data in all runs. It can be further observed that the computed result shows a higher discrepancy relative to the laboratory measurement at Run B’s free-water layer. In this run, the extremely small discharge was used (1.7 L/s), and it presented higher uncertainty to modelling due to drag effect (C_D). This uncertainty would carry on to impact the free-water layer, even above the vegetation. However, in other stages of the velocity profile, the model has performed well to mimic the experimental velocity distribution, and this demonstrates its potential, especially in the extremely slow flow’s near-bed governed by the proposed modified log law.

In Figure 4, the tests by Kubrak et al. [23] have been further investigated using the presented model. The tests used the rough sand bed beside the vegetation to create a more realistic testing criterion to represent natural real-world flow. The tests were chosen for investigating the modelling capability in order to represent the rough-bed flow with high k_s. The results show that the model well-represented both measured data with varying discharges and vegetation densities. Due to the presence of high k_s in Kubrak et al.’s [23] tests, the inner vegetation layer was extended to a high z/h ratio. This characteristic has been captured well by the model, which proved that the employed modified log law has high adaptability and flexibility in representing various smooth and rough bed vegetated flows while maintaining healthy accuracy.

3. Propeller-Induced Jet Model

The propeller-induced jet represents a complex flow condition that is difficult to simulate using analytical or numerical models. In working circumstances, the blades of the propellers will generate tremendous backward pushing force to drive the momentum of the ship forward. To represent this complicated flow phenomenon, Hong et al. [24] and Hamill and Kee [9] have proposed the use of their respective analytical approaches. Their theoretical models have been proven to represent the propeller-induced flow well while allowing fast computation (i.e., as compared to a numerical approach).

The efflux velocity, V_ef, is one of the key products of the jet flow caused by the propeller’s rotation. To model it, Hamill et al. [11] proposed

$${\mathrm{V}}_{\mathrm{e}\mathrm{f}}=1.22{\mathrm{n}}^{1.01}{\mathrm{D}}_{\mathrm{p}}^{0.84}{\mathrm{C}}_{\mathrm{t}}^{0.62}$$

(12)

where D_p is the propeller diameter, n is the rotational velocity by the propeller, and C_t is the thrust coefficient. However, the maximum axial velocities while the propeller is operational, V_max, should be reduced at a location further behind the propeller. Hamill [8] suggested that it should be determined via the ratio of blade area to the total circular area covered by propeller rotation, β; the distance from the propeller, X; and D_p as shown below.

$$\frac{{\mathrm{V}}_{\mathrm{m}\mathrm{a}\mathrm{x}}}{{\mathrm{V}}_{\mathrm{e}\mathrm{f}}}=0.87{\left(\frac{\mathrm{X}}{{\mathrm{D}}_{\mathrm{p}}}\right)}^{-\frac{\mathsf{\beta }}{4}}$$

(13)

According to a theoretical suggestion by Albertson et al. [12], and later by the practical model of McGarvey [25], the axial velocity distribution, V_r, measured from the centre of the propeller, can be modelled using the Gaussian probability relationship. Judging the complex flow movement, overlapping probability functions can also occur, which leads to the following suggested model [9]:

$$\frac{{\mathrm{V}}_{\mathrm{r}}}{{\mathrm{V}}_{\mathrm{m}\mathrm{a}\mathrm{x}}}=\exp \left[-\frac{1}{2}\frac{{\left(\mathrm{r}-\mathrm{R}\right)}^2}{{\mathsf{\sigma }}^2}\right]$$

(14)

where r is the radial distance from the centre of propeller to the end of the blade (or the blade length), and R is the radial distance from the centre of the propeller to the location of V_max. The standard deviation within the probability distribution model, $\mathsf{\sigma }$, can be represented according to the following relationship (provided that X/D_p > 0.5):

$$\mathsf{\sigma }=\frac{1}{2}\mathrm{R}+0.075\left(\mathrm{X}-\frac{{\mathrm{D}}_{\mathrm{p}}}{2}\right)$$

(15)

In Figure 5a, the axial velocity model in Equations (14) and (15) is used to calculate and compare with the laboratory measurements by Hamill [8]. The test conditions for that experiment were as follows: D_p = 7.5 cm, n = 1000 RPM (revolutions per minute), and X/D_p = 2.67. From Figure 5a, there is a clear discrepancy between the measured data and the model. The in-balance bell shape presented by the measured data can be observed on two sides separated by the propeller axis of r = 0; however, the theoretical model presents well-balanced calculated points on the positive and negative sides from r = 0. In the experiment, the thrust spinning momentum was clearly magnified sideward (i.e., due to the installed angle of the propeller), where this effect has not been reproduced in the modelled result. However, in comparison, both experimental and calculated data exhibit a similar curve shape and characteristic, which verifies that the model can represent propeller-induced jets reasonably. Further, the laboratory test conducted by Hamill and Kee [9] was also used to compare with the model. The physical test conditions were as follows: D_p = 7.6 cm and X/D_p = 1.00, where n was tested on multiple RPMs. Figure 5b shows that the model provides consistent results when compared to the measured data at 1000 RPM, which further evidences the accuracy of the propeller jet modelling approach. Also, it was observed that different n values produce a very insignificant variation in the axial velocity profile since the tests were conducted in stagnant water. However, this should change in flowing conditions where each RPM will cause significant alterations to the velocity distribution.

4. Complex Coupled Effect of Propeller-Induced Wave Impact on Vegetated Flow

In this section, the coupled effect of vegetated flow with propeller-induced dynamics is investigated. To study the impact of propeller operation on vegetation-induced flow patterns, the afore-discussed three-layer vegetated flow model (i.e., with modified log law at the inner vegetation zone) was coupled with the propeller model in Equations (14) and (15) to perform various theoretical tests on submerged vegetated flow fields. For the tested vegetation channel, the water depth was set at 35 cm with a vegetation height of 20 cm. A total of four theoretical tests with different propellers were conducted, in which their test conditions are summarised in

Table 2. Input data from theoretical tests on vegetated flow.

	Dp (cm)	Blade Area Ratio, β	Number of Blades, N	x/Dp
Propeller 1	7.6	0.4700	3	4.605
Propeller 2	9.2	0.4525	4	3.804
Propeller 3	10.3	0.6417	4	3.398
Propeller 4	13.1	0.9220	6	2.672

. Each propeller test was executed at a range of speeds, n: 500, 750, 1000, 1250, and 1500 RPM; and the propeller has been placed at three different vertical locations for testing: 26.0 cm above the bed (at free-water layer), 20.0 cm (at interface zone between vegetation and free-water layers), and 10.6 cm (at vegetation layer).

In Figure 6, Figure 7, Figure 8 and Figure 9, one can observe that all calculated coupled velocity profiles for different propellers have reasonable features, such as vegetated flow profile shape and propeller-driven flow turbulence. In terms of distribution across different RPMs of speed, the propeller with a higher n incited higher velocity (consistently featured for all tests), which is sensible. Detailed analysis revealed that there is a mix of flow characteristics when the propeller was placed at vegetated, interfacial, and free-water zones. When it is located at the free-water layer, one can observe that the flow propagation presents a balanced “M” shape. When free from the vegetated field, the free-water flow introduces no resistance to the original propeller-induced profile; hence, the ‘M’ shape remains almost undisturbed. However, at the interface and vegetated zones, the ‘M’ shape was distorted due to vegetation’s obstructive effect. For both of those zones, it was found that the profile was enhanced at the upper portion of the propeller’s hub, expanding its profile as compared to the lower part. The upper part of the interface zone was located at an unobstructed zone and, hence, permitted less restriction to the propagation of flow and caused higher profiles. Moreover, the flow’s momentum from the lower portion of the propeller was transferred to the upper half in order to cause shrinkage in the lower profile. In the vegetated layer, the velocity profile’s calculation was heavily influenced by the coupled model’s k_s at the lower half from the propeller hub, and this caused low velocities.

Due to the importance of this coupled flow phenomenon, it is crucial that further studies will be carried out after this research study. This phenomenon will also be interesting to verify through laboratory approach, which, to the author’s knowledge, has currently not been investigated in the literature. However, it can be seen from the co-examination between the model’s calculation and hydrodynamic laws that the proposed coupled approach represents the propeller-induced vegetated flow appropriately and provides a good initial start to this stream of research.

5. Conclusions

This paper studied a coupled analytical model to estimate the co-effect of propeller jets and vegetated flows on velocity distribution. It successfully proposed an efficient modified log law representation of the flow at the vegetated bed, which was used in a three-layer vegetation flow model. The proposed model was tested with a wide range of vegetated flow measured data reported in the literature, including those with normal vegetation, truncated cone shape vegetation, and vegetation with sand beds under different flow conditions and vegetation densities. All these validations evidenced the competence of the proposed analytical approach with respect to representing various vegetated flows with reasonable accuracy. A validated propeller jet flow model was then adapted to couple with the suggested vegetated flow model to propose a theoretical approach to investigate the influence of propeller-induced jet turbulence in channels with submerged vegetations. Theoretical tests with different propeller configurations and rotational speeds were performed, and their analyses showed practicable solutions to various cases. Lastly, as a recommendation, it is crucial that the proposed coupled model is tested using actual laboratory measurements to further identify and ensure its modelling capability.