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CFD data based neural network functions for predicting hydrodynamic performance of a low-pitch marine cycloidal propeller
Applied Ocean Research 94 (2020) 101981
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CFD data based neural network functions for predicting hydrodynamic performance of a low-pitch marine cycloidal propeller Mohammad Bakhtiari, Hassan Ghassemi
T
⁎
Department of Maritime Engineering, Amirkabir University of Technology, Tehran, Iran
ARTICLE INFO
ABSTRACT
Keywords: Marine cycloidal propeller Hydrodynamic performance CFD simulation Neural network Predictive function
Today, various types of propulsion systems are used in different purpose ship types. Marine cycloidal propeller (MCP) is one of these propulsion systems, which has been designed for ships that require high maneuverability. MCP can be considered as an especial type of marine propulsion systems, since it produces the thrust force which is perpendicular to propeller axis of rotation. The magnitude and direction of the thrust force can be adjusted by controlling the pitching angle of the blades, so no separate rudder is needed to manoeuvre the ship. In this study, mathematical functions for predicting the open water hydrodynamic performance of a low-pitch MCP are presented by training a neural network based on computational fluid dynamics (CFD) data. For this purpose, the four nondimensional parameters of blade number (Z), ratio of blade thickness to MCP diameter (t/D), pitch (e) and advance coefficient (λ) are considered as input variables, whereas the hydrodynamic coefficients of thrust (Ks) and torque (Kd) are considered as targets. CFD simulations are performed for different cases of MCP with different combinations of Z, t/D, e and λ. The results showed that a two-layer feedforward network with one hidden layer of sigmoid neurons and at least 4 neurons in the hidden layer can be well trained by CFD data in order to obtain functions with good accuracy in predicting Ks and Kd coefficients of a low-pitch MCP.
1. Introduction With the development of special purpose ship types in recent years, a well-balanced compromise was demanded between the main requirements of ship propulsion, such as high efficiency, low cavitation, noise and erosion, high maneuverability, low maintenance costs, etc. To meet these requirements, various types of marine propulsion systems have been developed. MCP is a special type of these propulsion systems, which is used for ships that require high maneuverability, such as tugs, ferries etc. As shown in Fig. 1a, a MCP consists of a number of hydrofoil blades fitted to a disc usually set flush with the ship hull. The disc revolves about a vertical axis (propeller axis of rotation) while the blades pitches about their own individual axes. Therefore, unlike a screw propeller, the axis of rotation of a MCP is perpendicular to the thrust force. In addition, the thrust is controllable in magnitude and direction by adjusting the pitching angle of the blades, so that no separate rudder is required to manoeuvre the ship. So far, two different configurations of MCP with different applications have been developed, including low-pitch and high-pitch MCPs (see Fig. 1b). The low-pitch one (known as cycloidal propeller) has been designed for advance coefficients of λ < 1. The high-pitch one (referred as trochoidal propeller) has been designed for advance coefficients of λ > 1, and provides higher hydrodynamic efficiency than low-pitch ⁎
ones allowing high-speed operation. Nevertheless, low-speed maneuvering requirements have led to further development of the low-pitch configuration. Here, = VA/R is absolute advance velocity coefficient of a MCP, where VA, ω and R are advance velocity, rotational velocity and radius of MCP, respectively. MCPs have been in use since 1920s [2,3], however serious theoretical and experimental studies on MCP were carried out much later. The research conducted by Henry in 1959 was a foundation for subsequent theoretical analyses and systematic experiments [4]. Early theoretical models developed for analyzing MCP, like Taniguchi's method [5], were in 2D and made simplifying assumptions leading to inaccurate models. In 1961, Haberman & Harley used Taniguchi's method for calculating the hydrodynamic performance of a cycloidal propeller [6]. In 1969, Sparenberg optimized the blade pitch function of a single-blade propeller by minimizing kinetic energy in the wake [7]. James worked on Sparenberg's optimization work with a more detailed model in 1971 [8]. Much later, in 1991, Brockett developed a computational model based on lifting line and actuator disk theories to optimize the blade pitch function [9]. In 1996, Van Manen and Van Terwisga expanded Brockett's computational model to further optimize the blade pitch function [10]. Systematic series of experiments on MCPs were carried out by Nakonechny [11,12], Van Manen [13], Ficken [14], Bjarne [15], and
Corresponding author. E-mail address: [email protected] (H. Ghassemi).
https://doi.org/10.1016/j.apor.2019.101981 Received 22 June 2019; Received in revised form 25 October 2019; Accepted 31 October 2019 Available online 13 November 2019 0141-1187/ © 2019 Elsevier Ltd. All rights reserved.
Applied Ocean Research 94 (2020) 101981
M. Bakhtiari and H. Ghassemi
Fig. 1. a) A ship with twin cycloidal propellers [1], b) two different configurations of MCP.
Bose and Lai [16] in the years between 1961 and 1989. Van Manen's systematic tests demostrated that the hydrodynamic efficiency of highpitch MCPs is lower that low-pitch ones. Also, Van Manen showed that Sparenberg's pitch function is more efficient than a cycloidal pitch function for multi-bladed propellers. In the last two decades, CFD has been implemented to analyze MCP propulsor. In 2005, Jurgens et al. presented a numerical optimization of blade pitch function for MCP (Voith-Schneider propeller) by using CFD method [17]. In 2009, Jurgens et al. investigated the influence of a MCP on stern slamming conditions, the roll damping capabilities of the MCP and the effect of air ventilation. They used experimental and CFD methods and compared the results of the MCP propulsor with azimuth thruster. They demonstrated that a MCP can decreases the impact of stern slamming pressure because of its vertical rotating axis, and can be used to reduce the roll motion of a vessel. They also showed that MCP is less prone to ventilation than azimuth thruster [18]. At the same year, Jurgens and Heinke investigated cavitation behaviour of heavily loaded MCP with different blade profiles under bollard pull condition [19]. In 2014, Esmailian et al. investigated the hydrodynamic performance of a MCP propeller by CFD analysis [20]. Prabhu et al., in 2015, calculated the hydrodynamic coefficients of a MCP by using an unsteady analysis based on numerical panel method and compared their results with steady analysis [21]. In 2017, Prabhu et al. developed a numerical scheme for investigating the fluid structure interaction of a marine cycloidal propeller blade by using 2D inviscid flow theory [22]. In addition, Prabhu et al. investigated hydrodynamic forces on marine cycloidal propeller during maneuvering by numerical panel method in 2019 [23]. In this study, we present mathematical functions based on neural network model to predict open water hydrodynamic performance of a low-pitch MCP. For this purpose, the four nondimentional parameters of blade number (Z), ratio of blade thickness to MCP diameter (t/D), pitch (e) and advance coefficient (λ) of MCP are considered as input variables, whereas the thrust coefficient (Ks) and torque coefficient (Kd) of MCP are considered as targets. A 2.5D CFD method, after validating against experimental data, is used to analyze the turbulent flow around MCP and calculate Ks and Kd for 168 different combinations of Z, t/D, e
and λ. The 168 calculated CFD data are implemented to train neural network and obtain two mathematical functions to predict Ks and Kd of a low-pitch MCP. The obtained mathematical functions are also validated to accurately fit the CFD data. According to the above mentioned, the contributions of this paper, compared with previous works in the field of MCP, include providing mathematical functions for estimating the hydrodynamic coefficients of low-pitch MCPs based on CFD data obtained from 2.5D numerical simulations of MCP. Furthermore, neural network method is implemented to obtain the hydrodynamic coefficient functions of MCP from CFD data. In the following, first the principles governing the hydrodynamic performance of a MCP is described in Section 2. Then, the CFD model used for analyzing the flow around the MCP and its validation against experimental data are presented in Section 3. In Section 4, the neural network model and training algorithm are explained. In Section 5, the CFD data of Ks and Kd (targets) calculated for different combinations of Z, t/D, e and λ (input variables) are provided, and two mathematical functions are presented for predicting open water hydrodynamic performance of a lowpitch MCP. Finally, some conclusions are drawn in Section 6. 2. Hydrodynamic performance of MCP In a MCP, λ and β are two main parameters which define the kinematics of the MCP. Fig. 2 shows the kinematics of a MCP as well as the hydrodynamic forces acting on a single blade. In this figure, VA is advance velocity of MCP, ω is rotational velocity of MCP about its axis (point O), θ is angle of rotation, β is blade pitch angle and αkin is kinematic angle of attack. VR is the resultant velocity of the blade axis (point P) due to VA and ωR. In addition, lift and drag are the hydrodynamic forces acting on single blade, which produce thrust force (Tb), torque about propeller axis (Qb) and side force (Fb) of the blade. Perhaps the most common blade pitch angle for a MCP is pure cycloidal motion (see Fig. 3a), in which the blade pitch angle changes in such a way that the normal to the chord line at blade axis of rotation (NP ) passes through a fixed point (point N, named as steering center). In a pure cycloidal motion, the blade pitch angle is defined as follows: 2
Applied Ocean Research 94 (2020) 101981
M. Bakhtiari and H. Ghassemi
in the fore and aft part of MCP (see Fig. 2), their side forces are in opposite directions and nearly cancel each other out. In the other hands, thrust forces of the blades are mostly in same and positive direction and amplify each other. Therefore, total side force is significantly lower than the total thrust force. Finally, the unsteady total thrust and torque are averaged over one revolution of propeller to obtain the averaged value of thrust (Tave) and torque (Qave), and consequently, the open water hydrodynamic characteristics of a MCP:
Ks =
Kd =
Eff =
Tave 1 2
DLu2
(2)
2Qave 1 2
D2Lu2
(3)
Ks Kd
(4)
where, Ks, Kd and Eff are thrust coefficient, torque coefficient and open water efficiency of a MCP, respectively. D = 2R and L are propeller diameter and blade span length, and u=ωR is circumferential velocity of the blade.
Fig. 2. Kinematics of a MCP and the hydrodynamic forces acting on a single blade.
3. CFD model 3.1. The fluid flow equations In this study, the unsteady Reynolds-averaged Navier–Stokes (URANS) equations with shear-stress transport (SST) k–ω turbulence model are used to numerically solve the turbulent flow around the MCP. The URANS equations can be written as:
ui =0, xi t
( ui ) +
(i = 1, 2, 3)
xj
( ui uj ) =
(5)
p u + µ i xi xj xj
u iu
j
+ gi ,
( j = 1,
2, 3 )
(6) where, u, p, ρand μ are velocity, pressure, density and viscosity of the fluid, respectively, and g is gravity force. Using Boussinesq hypothesis, the Reynolds stresses term in Eq. (5), u i u j , can be stated as:
uj ui + ) xj xi
u i u j = µt (
2 u ( k + µt l ) ij, 3 xl
(l = 1, 2, 3)
(7)
where, μtis turbulent viscosity. In SST k–ω turbulence model, the turbulent viscosity,μt, is computed as follows:
µt =
( , e) = tan
1
1 1 SF , 2 * 1
max
(8)
where, k and ω are turbulence kinetic energy and specific dissipation rate respectively, obtained from the following transport equations:
Fig. 3. Diagrams of blade pitch angle and blade pitch rate for different pitches of a MCP with pure cycloidal motion .
e sin 1 + e cos
k
t (1)
where, e = ON /R is eccentricity or pitch of a MCP. Generally, for e < 1, the MCP is known as a low-pitch propeller, whereas for e > 1, it is referred as a high-pitch propeller. In practice, due to mechanical constraints and high pitch rate of blade near = , as illustrated in Fig. 3b, the pitch of a VSP can be varied within the range of e ≤ 0.8. By summing the thrust and torque forces of all blades of the MCP, the total thrust and torque are calculated at each time instant. It can be stated that once two blades are located at the same transverse locations
t
( k) +
(
xi
)+
xj
( kui) =
(
uj ) =
µ+
xj
µt k
xj
µ+
k xj
µt xj
+ Gk
+G
*k
(9) 2
+D
(10)
where, Gk and Gω are generations of k and ω respectively. σk and σωare turbulent Prandtl numbers for k and ω: k
= =
3
F1/
k ,1
1 + (1
F1)/
F1/
,1
1 + (1
F1)/
k ,2
,2
(11) (12)
Applied Ocean Research 94 (2020) 101981
M. Bakhtiari and H. Ghassemi
In above equations, F1 and F2 are blending functions.
angular velocity of ω (zone Z7 in Fig. 4), and a fixed zone at far field (zone Z8 in Fig. 4). An interface boundary condition is applied to the contact surfaces of the zones. A translational periodic boundary condition is used for the top and bottom boundaries of each zone. As illustrated in Fig. 5 for a 4-bladed MCP, each of the fluid zones was discretized with an unstructured polyhedral mesh in order to create separate cell zones. Unstructured polyhedral mesh have been introduced by some well-known CFD merchant codes as an innovative approach. This type of mesh includes the advantages of structured hexahedral mesh (such as low numerical diffusion resulting in accurate solution) and tetrahedral mesh (such as rapid automatic generation), and overcome the disadvantages of both the above mentioned types of meshes (time-consuming and sensitive to stretching) [26,27]. Here, the sliding mesh technique is implemented, so that each moving cell zone slides relative to one another along the mesh interface, without any mesh deformation. An inflation polyhedral layer mesh including 45 layers with growth rate of 1.1 and Y+<1 (Y+ is nondimensional distance of the first layer from the wall) was used within near-wall region.
3.2. Computational domain and mesh In a MCP, the blades are exposed to high kinematic angles of attack, which are sometimes beyond the dynamic stall angle [9]. Therefore, the flow on the blades is highly separated and unsteady, and the 3D flows in spanwise direction are significant and should be included in simulations [24]. In other words, a 2D simulation does not provide sufficient accuracy to solve the unsteady flow with large separations around airfoils at relatively high angles of attack [25]. In the other hand, the full 3D simulation of MCP is time consuming and computationally costly. Consequently, a 2.5D CFD method is implemented in this study. In this method, a 2D model is extended in a spanwise direction for a considerable length in order to resolve the effects of the flow in the third dimension (spanwise direction), with periodic boundary conditions at two extremities of the domain. Here, the flow equations are solved numerically in a computational domain with one-chord length in spanwise direction by using Ansys Fluent code. The computational domain and boundary conditions are shown in Fig. 4 for a 6-bladed MCP. In this figure, D is disk diameter and C is blade chord length. As mentioned in Section 2, each blade of MCP rotates with constant angular velocity of ω about MCP axis of rotation and simultaneously pitches with angular velocity of ( , , e) about its own axis. Accordingly, we consider a fluid zone near each blade, which moves with the same velocity as the blade (zones of Z1, Z2, Z3, Z4, Z5 and Z6 shown for 6-bladed MCP in Fig. 4), a moving zone with constant
3.3. Validation and verification of CFD model For investigating the accuracy of the CFD model in predicting the open water hydrodynamic performance of a MCP, we compared it against the experimental data provided by Van Manen [13]. Since the present simulations are carried out for MCPs with pure cycloidal motion including rectangular blades with symmetrical NACA 4-digit profile, the Van Manen's test was found as the most appropriate one for
Fig. 4. Computational domain and boundary conditions . 4
Applied Ocean Research 94 (2020) 101981
M. Bakhtiari and H. Ghassemi
Fig. 5. Computational mesh, a) whole solution domain, b) near-field domain, c) the zone around blade, d) inflation polyhedral mesh inside boundary layer and the mesh on blade surface.
validating the CFD model. Table 1 provide specifications of the MCP model tested by Van Manen. It should be noted that the Van Manen's tests were performed at Reynolds numbers in which the scale effect of a MCP can be neglected [28]. Figs. 6 and 7 show the comparison between CFD and experimental data of Ks, Kd and Eff at different λ for two pitches of e = 0.4 and e = 0.6, respectively. In the same figures, the values of relative errors have been provided. The average relative errors of Ks, Kd and Eff are 3.1%, 2.4% and 4.5% for e = 0.4, and 5.5%, 2.6% and 6.6% for e = 0.6, respectively. Some part of these errors may be refer to some existing effects in the test, which are not included in a 2.5D CFD simulation, such as the blade tip effects and the variable blade thickness. However, there is a good agreement between the CFD results and experimental data. CFD simulations were performed in three different meshes with different cell sizes in order to verify the CFD model based on the grid convergence index (GCI) approach proposed in [29]. This approach is an acceptable and a recommended method that have been tested for several hundred CFD cases. According to [29], the mesh refinement was performed uniformly over all cell zones in order to create three meshes with different resolutions; coarse, medium and fine. Table 2 shows the
properties of these three different meshes. In this table, the number of cells relates to the meshes used for validation case in previous section. As an example, the calculated values of Ks and Kd at e = 0.6 and λ=0.3 in the three different meshes, as well as the discretization error and numerical uncertainty are listed in Table 3. In this Table, N1, N2 and N3 are the total number of cells in fine, medium and coarse meshes, respectively. X1, X2 and X3 are performance coefficients of MCP calculated in 1 fine, medium and coarse meshes, respectively. r21= (N1/ N2) 3 and r32= 1 21 (N2/ N3) 3 are mesh refinement factors. P is apparent order. Xext is the extrapolated value of performance coefficients of MCP. 21 21 21 ea21 = |(X1 X2 )/ X1 | is approximate relative error. eext = |(Xext X1)/ Xext | is extrapolated relative error. As seen from Table 3, the numerical uncertainties in fine-mesh solution (GCIfine) for Ks and Kd are very small, indicating that the solution is grid independent. Therefore, the medium mesh including 45 polyhedral inflation layer inside the boundary layer with Y+ value of 1 was selected to perform the numerical simulations of this study. In order to study the independence of solution from temporal discretization, the simulations in medium mesh were performed for two time-steps of T/360 and T/720, where T is the period of one revolution of MCP. Given that the time-step refinement from T/360 to T/720 had a very little effect on the values of Ks, Kd and Eff, the time step of T/360 was selected for performing the simulations.
Table 1 Specifications of the MCP model selected for validation. Parameter
value
MCP diameter (D) Blade number (Z) Ratio of blade span to MCP diameter (L/D) Ratio of blade chord to MCP diameter (c/D) Ratio of maximum blade thickness to MCP diameter (t/ D) Blade platform Blade section profile Blade pitch motion rotational velocity (n) Reynolds number
0.2 m 4 0.6 0.18 0.026
4. Neural network model Neural network models are computational models that have received much attention from researchers in various fields in recent years [30,31]. Neural networks have shown that they are suitable for function fitting [32], and a fairly simple neural network can fit any practical function. Fig. 8 shows the architecture of a single-neuron network with R-element input vector of p. Three different operations occur in a neuron. First, the individual input elements of vector p, (p1, p2,… pR), are multiplied by weights, (w1, 1, w1, 2,… w1, R), and the weighted values are fed to the summing junction. Their sum is simply the dot
Rectangular Symmetrical Pure cycloidal motion 6 RPS 1.4 × 105
5
Applied Ocean Research 94 (2020) 101981
M. Bakhtiari and H. Ghassemi
Fig. 6. Comparison between CFD and experimental data for e = 0.4.
Fig. 7. Comparison between CFD and experimental data for e = 0.6.
product Wp. Then, the Wp is added to the scalar bias b to form the net input n (The bias is like a weight, except that it has a constant input of 1). Finally, the net input is passed through the transfer function f, which produces the scalar output a. In fact, these operations include three functions: the weight function, the net input function and the transfer function. In many types of neural networks, the weight function is a product of a weight times the input. Also, the most common net input function is the summation of the weighted inputs with the bias. Three of the most commonly used transfer functions are shown in Fig. 9. Two or more of the neurons shown in Fig. 8 can be combined in a layer, and a neural network could contain one or more such layers. For example, Fig. 10 illustrates a three-layer network. This network has R inputs, S1neurons in the first layer, S2neurons in the second layer, etc. Each layer has a weight matrix W, a bias vector b, and an output vector a. The outputs of each intermediate layer are the inputs to the following layer. Thus, layer 2 can be analyzed as a one-layer network with S1inputs, S2neurons, and an S2 × S1weight matrix W2. To distinguish between the weight matrices, output vectors, etc., for each of these
Table 2 Properties of the meshes used for grid convergence study. Mesh
Number of cells
Y+ value
Number of inflation layers
Coarse Medium Fine
459,251 846,327 1,685,952
2 1 0.5
35 45 55
layers in the Fig. 10, the number of the layer is appended as a superscript to the variable of interest. In a multilayer network, the layer that produces the network output is called an output layer, whereas all other layers are called hidden layers. The three-layer network shown in Fig. 10 has two hidden layers (layer 1 and layer 2) and one output layer (layer 3) and. Here, it is assumed that the output of the third layer, a3, is the network output. The mathematical functions for outputs (a) in term of inputs (p), weights (W) and biases (b) for each layer can be written as:
6
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M. Bakhtiari and H. Ghassemi
Table 3 Discretization error and numerical uncertainly calculated for Ks and Kd at e = 0.6 and λ = =0.3. Parameter
Ks
Kd
N1 N2 N3 r21 r32 X1 X2 X3 p
1,685,952 846,327 459,251 1.258 1.226 0.234363 0.234296 0.250147 26.817 0.234364
1,685,952 846,327 459,251 1.258 1.226 0.191489 0.191091 0.209570 18.805 0.191502
ea21
0.03%
0.21%
21 eext GCIfine
0.0002%
0.0067%
0.0003%
0.0084%
21 Xext
Fig. 11. A simple schematic of training a neural network with input and target data. Table 4 The variable input parameters and their values for 168 different cases of MCP. Parameter
Values
Number of blade (Z) Ratio of blade thickness to MCP diameter (t/D) MCP pitch (e) Advance velocity coefficient (λ)
4, 5, 6 0.024, 0.042 0.2, 0.4, 0.6, 0.8 0, 0.1, 0.2, …, (e + 0.1)
a1 = f 1 (IW1,1p + b1)
(13)
a2
b 2)
(14)
a3 = f 3 (IW 3,2a2 + b3)
(15)
=
f 2 (IW 2,1a1
+
By substituting Eqs. (13) and (14) in Eq. (15), the following function is obtained for the three-layer neural network:
Fig. 8. A single-neuron neural network with R-element input vector.
a3 = f 3 (IW 3,2f 2 (IW 2,1f 1(IW1,1p + b1) + b2) + b3) 3
(16)
where, a and pare respectively the outputs and inputs of the network. The main idea of neural networks is that the values of weights, w, and biases, b, can be adjusted so that the network exhibits some desired behavior. Thus, one can adjust, or train, a network, so that a particular input leads to a specific target output. Fig. 11 shows a simple schematic of this situation. Here, the network is adjusted based on a comparison of the output and the target, until the network output matches the target. Typically, many such input/target data pairs are needed to train a network.
Fig. 9. Three of the most common transfer function used in neural networks .
Fig. 10. A three-layer neural network .
7
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Fig. 12. Open water hydrodynamic diagrams of Ks, Kd and Eff from CFD data for Z = 4.
It has been proven that a neural network with one (or more) hidden layers of sigmoid neurons followed by an output layer of linear neurons (known as feedforward network) can be trained to approximate any function with a finite number of discontinuities arbitrarily well, given consistent data and enough neurons in its hidden layer. The hidden layers of neurons with nonlinear transfer functions allow the network to learn nonlinear relationships between inputs and targets. In this study, we create feedforward neural networks with one
hidden layer and train them by CFD data to obtain two mathematical functions for open water Ks and Kd of a MCP. Creating and training procedures are performed by using neural network fitting toolbox in MALAB. The networks will be trained with Levenberg-–Marquardt backpropagation algorithm [33,34,35]. The four parameters of Z, t/D, e and λ are considered as inputs, whereas the Ks and Kd are considered as targets of the neural networks.
8
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5. Results and discussions
and λ. The values of Z, t/D, e and λ, considered as variable input parameters to calculate Ksand Kd, are given in the Table 4. Here, all the simulations are performed for a MCP model with diameter of 0.2 m and constant value of c/D = =0.2. The open water hydrodynamic diagrams of Ks and Kd and Eff from CFD data have been presented in Figs. 12–14 for Z = 4, Z = 5 and Z = 6, respectively. It can be observed that the maximum efficiency of MCP increases with increasing MCP pitch parameter. However, at lower
5.1. CFD data for MCP hydrodynamic performance The CFD model described and validated in Section 3 is implemented in Ansys Fluent solver to simulate the turbulent flow around low-pitch MCP. The open water hydrodynamic coefficients of Ksand Kd are calculated for 168 different cases with different combinations of Z, t/D, e
Fig. 13. Open water hydrodynamic diagrams of Ks, Kd and Eff from CFD data for Z = 5. 9
Applied Ocean Research 94 (2020) 101981
M. Bakhtiari and H. Ghassemi
Fig. 14. Open water hydrodynamic diagrams of Ks, Kd and Eff from CFD data for Z = 6.
advance coefficients, the efficiency decreases by increasing the MCP pitch. Also, the diagrams of Eff demonstrate that the blade number have an undesirable effect on hydrodynamic efficiency, so that the hydrodynamic efficiency of MCP reduces by increasing of blade number. According to the diagrams, the average relative reductions in maximum Eff
due to increasing of blade number from 4 to 6 are about −42%, −25%, −13% and −7% for e = 0.2, e = 0.4, e = 0.6 and e = 0.8, respectively. In the case of t/D effects on Eff, it is clear from the figures that the maximum efficiency of MCP decreases with increasing blade thickness. In fact, the average relative reductions of about −6.3%, −7.4% and 10
Applied Ocean Research 94 (2020) 101981
M. Bakhtiari and H. Ghassemi
Fig. 15. Pressure distribution on blade surface, and velocity distribution at mid-span plane for the 6-bladed MCP with t/D = 0.042.
−7.8% are resulted in maximum Eff for Z = 4, Z = 5 and Z = 6, respectively, due to a 75% increase in blade thickness from t/D = 0.024 to t/D = 0.042. As an example of graphical visualization of CFD simulations, the pressure and velocity contours around the 6-bladed MCP have shown in
Fig. 15 for MCP pitches of e = 0.4, e = 0.6 and e = 0.8. The figures illustrate the pressure and velocity distributions on blade surface and mid-span plane of the MCP, respectively. For each MCP pitch, the contours have been extracted in advance velocity coefficient near the maximum efficiency point. 11
Applied Ocean Research 94 (2020) 101981
M. Bakhtiari and H. Ghassemi
Fig. 16. Training performance diagrams for Ks (upper diagram) and Kd (lower diagram).
5.2. Mathematical functions for MCP hydrodynamic performance
outputs (ai) and targets (Ti) of the validation set, defined as follows:
As mentioned in Section 4, we create two-layer feedforward networks with a sigmoid transfer function in the hidden layer and a linear transfer function in the output layer, and train the networks by 168 CFD data presented in previous section. The four parameters of Z, t/D, e and λ are considered as inputs, whereas the Ks and Kd are considered as targets. In this way, two mathematical functions for the two target variables in term of inputs are presented based on the neural network model. The network is created in MALAB software and trained by Levenberg–Marquardt backpropagation algorithm using neural network fitting toolbox in MALAB. The 168 CFD data are randomly divided into three sets; 70% are used for training, 15% are used to validate that the network is generalizing and to stop training before overfitting, and the last 15% are used as a completely independent test of network generalization. The training continues until the validation error fails to decrease for six iterations (validation stop). In fact, the network weights and biases are saved at the minimum value of mean squared error (MSE) between the network
MSE =
1 N
N
(Ti i=1
ai ) 2
(17)
We created and trained various two-layer feedforward networks with different number of hidden neurons from 1 to 10. We observed that by increasing the number of hidden neurons from 1 to 4, the MSE value significantly decreased to less than 1e-4. However, with increasing from 4 to 10 hidden neurons, the reduction in the MSE value was insignificant. So, in order to avoid the complexity of the proposed neural network functions while having enough accuracy, we selected the network with 4 hidden neurons for training. In the other hand, given the fact that each time the neural network is trained, a different solution is achieved due to different initial weight and bias values and different divisions of CFD data into training, validation, and test sets, we trained the network several times for achieving the best solution. In order to improve the process of neural network training, we fed 12
Applied Ocean Research 94 (2020) 101981
M. Bakhtiari and H. Ghassemi
Fig. 17. Regression diagrams for Ks prediction .
the neural network by the normalized values of the input and target CFD data. Therefore, based on the maximum and minimum values of each variable, the inputs and targets variables were normalized to fall in the range of [−1, 1]. Fig. 16 shows the training performance diagrams for Ks and Kd. According to this figure, the best validation MSE errors for Ks and Kd are 2.5153e-5 and 6.8185e-5, respectively at 60th and 222nd iterations. Figs. 17 and 18 illustrate regression diagrams for Ks and Kd predictions, respectively. In fact, the diagrams of these figures present the network outputs with respect to targets for all CFD data, as well as training, validation, and test sets of the data. In these figures, the dashed line (labeled as X = Y) shows a 45° line, where the network outputs are equal to the targets and we have the perfect fit on this line. The solid line represents the best fit linear regression line between outputs and targets. The R value is an indication of the relationship between the outputs and targets. If R = 1, this indicates that there is an exact linear relationship between outputs and targets. As can be seen, for both Ks and Kd, there is high fitness for all data sets, with R values more than 0.997. The neural network mathematical functions achieved for predicting the open water performance of a low-pitch MCP are as follows:
2
Ksn = 1 + exp
2 × 0 . 07364 × Zn
t D n
0 . 10664 ×
1 1 . 00868 × en + 1 . 47633 ×
n
0 . 78548
× ( 0.99585) 2
+ 1 + exp
2×
0 . 05115 × Zn
2 . 27862 ×
t D n
1 0 . 24339 × en + 1 . 18190 ×
n
0 . 24822
× 0.76604 2
+ 1 + exp
2 × 0 . 03349 × Zn + 0 . 60767 ×
t + 0 . 64258 × en D n
1 × 0 . 60457 ×
n
0 . 46072
1 . 50131 × en + 1 . 54741 ×
n
1 . 76776
1.62865 2
+ 1 + exp
2 × 0 . 10187 × Zn
0 . 49728 ×
t D n
1 ×
0.61787 0.11982
(18)
13
Applied Ocean Research 94 (2020) 101981
M. Bakhtiari and H. Ghassemi
Fig. 18. Regression diagrams for Kd prediction.
where, Zn, en, ( D )n , λn, Ksn and Kdn are normalized parameters in the range of [−1, 1], defined as follows: t
K dn 2
= 1 + exp
2×
0 . 10358 × Zn
0 . 28276 ×
t D n
1 0 . 34662 × en
0 . 53122 ×
2 × (Z 2
Zn =
n + 1 . 40075
× ( 14.21384 ) 2
+ 1 + exp
2×
0 . 03299 × Zn + 1 . 09977 ×
t D n
t D
1 0 . 70398 × en
0 . 42949 ×
n + 1 . 71166
× ( 8.78894 ) + 2×
0 . 04837 × Zn + 0 . 91204 ×
t D n
1 0 . 58975 × en
0 . 56987 ×
n + 1 . 51745
n
× 10.19728
1 + exp
2×
0 . 21103 × Zn
0 . 15990 ×
t D n
t D
0.024
4
)
1,
2 × (e 0.2) 0.6 2× 0.9
1,
6
Z
0.024
0.2
1,
0
(20)
t D
e
0.042 (21)
0.8
(22)
(e + 0.1) (23)
2
+
=
(
1,
0.018
n
en = 2
1 + exp
=
2×
4)
1 0 . 41256 × en
1 . 51709 ×
n + 2 . 06870
× 3.55407 + 8.42958
(19) 14
Ksn =
2 × (Ks + 0.1016) 0.7405
K dn =
2 × (K d + 0.0580) 0.6801
1
(24)
1
(25)
Applied Ocean Research 94 (2020) 101981
M. Bakhtiari and H. Ghassemi
Fig. 19. Comparison between neural network functions of Ks and Kd and CFD data for Z = 5, at t/D = 0.042 (upper diagram) and t/D = 0.024 (lower diagram).
• By increasing the propeller pitch (e), the maximum hydrodynamic efficiency of MCP increases. • The blade number (Z) has an undesirable effect on hydrodynamic
Although, the consistency of Ks and Kd outputs with CFD target data was demonstrated in Figs. 17 and 18, a comparison between predictive neural network functions in Eqs. (18) and (19) and CFD data have been presented in Fig. 19 for Z = 5 and t/D = 0.024 and 0.042. This figure shows the Ks and Kd curves predicted by Eqs. (18) and (19) from e = 0.2 to 0.8, as well as CFD data points for e = 0.2, 0.4, 0.6 and 0.8. From the figure, a very good fit can be observed between the neural network functions and CFD data.
• •
6. Conclusions
•
The open water hydrodynamic coefficients of Ks and Kd were calculated for 168 different low-pitch MCPs including different combinations of Z, t/D, e and λ by using a 2.5D CFD method. After validating the method against experimental data, the obtained CFD data were used to train feedforward neural networks in order to present two mathematical functions for Ks and Kd (as output variables) in term of Z, t/D, e and λ (as input variables). According to the results, the following conclusions can be made:
efficiency, so that the hydrodynamic efficiency of MCP reduces by increasing of blade number. In the case of t/D effects on Eff, the maximum efficiency of MCP decreases with increasing blade thickness. A two-layer feedforward network with one hidden layers of sigmoid neurons and at least 4 neurons in the hidden layer can be well trained by CFD data in order to obtain Ks and Kd functions of MCP with high accuracy and good fitness. Using neural networks with high number of neurons (here more than 10) provides Ks and Kd predictive functions with better fitness, but the complexity of the resulted functions increases.
This is our future plan to extend our calculations to high-pitch MCP and also investigate the hydrodynamic performance of MCPs with other blade pitch motions.
• A 2.5D CFD method can be used as an accurate and computation-
Declaration of Competing Interest
ally-efficient CFD method to simulate and calculate the hydrodynamic force coefficients of a low-pitch MCP.
The authors confirm that there is no conflict of interest. 15
Applied Ocean Research 94 (2020) 101981
M. Bakhtiari and H. Ghassemi
Acknowledgment [16]
Numerical computations presented in this study have been performed on the parallel machines of the high performance computing research center (HPCRC) of Amirkabir University of Technology; their supports are gratefully acknowledged.
[17] [18]
Supplementary materials
[19]
Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.apor.2019.101981.
[20]
References
[21]
[1] www.maritimejournal.com/news101/power-and-propulsion/vsps_enter_psv_ market. [2] Http://patents.google.com/patent/US1432700A. [3] Http://patents.google.com/patent/US1681500A. [4] C.J. Henry, A Survey of Cycloidal Propulsion, Davidson Lab, 1959 Report 728. [5] K. Taniguchi, Sea trial analysis of the vertical axis propellers, Fourth Symposium on Naval Hydrodynamics: Propulsion Hydroelasticity, Office of Naval Research and Web Institute of Naval Architecture, Washington, DC, 1962, pp. 27–31. August, No. 6. [6] W.L. Haberman, E.E. Harley, Performance of Vertical Axis (cycloidal) Propellers Calculated by Taniguchi's Method, Hydromechanics laboratory research and development report, Virginia, 1961 Tech report, Report 1564, SR0090101. [7] J.A. Sparenberg, R. de Graaf, On the optimum one-bladed cycloidal propeller, J. Eng. Math. 3 (1969) 1–20. [8] E.C. James, A Small Perturbation Theory for Cycloidal Propellers, PhD Thesis CIT (1971). [9] T. Brockett, Hydrodynamic analysis of cycloidal propulsors, Propellers-Shafting Symposium 2 The Society of Naval Architects and Marine Engineers, Virginia Beach, VI, 1991, pp. 1–28 17–18 September, SNAME. [10] J. Van Manen, T. van Terwisga, A new way of simulating whale tail propulsion, 21st Symposium on Naval Hydrodynamics, 21 National Academies Press, Trondheim, 1996, pp. 24–28 June. [11] B.V. Nakonechny, Experimental Performance of a Six Bladed Vertical Axis Propeller, (1961) DTMB Report 1446 http://dome.mit.edu/bitstream/handle/ 1721.3/48804/DTMB_1961_1446.pdf. [12] B.V. Nakonechny, Design of a 9 inch Cycloidal Propeller Model Unit and Some Experimental Results, (1975) NSRDC Report 3150. [13] J. Van Manen, Results of systematic tests with vertical axis propellers, Int. Shipbuild. Progress 13 (148) (1966) 382–398. [14] N.L. Ficken, Conditions for the Maximum Efficiency Operation of Cycloidal Propellers, SNAME Chesapeake Section Paper, 1966. [15] E. Bjarne, Comparison of a cycloidal propeller with azimuth thrusters with regard to efficiency, cavitation and noise, Proceedings of International Conference on
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Propulsion for Small Craft, London: Royal Institution of Naval Architects, 19829–10 November. N. Bose, P.S.K. Lai, “Experimental performance of a trochoidal propeller with highaspect ratio blades, Marine Technol. 26 (3) (1989) 192–201. D. Jürgens, M. Palm, S. Singer, K. Urban, Numerical optimization of the VoithSchneider propeller, ZAMM J. Appl. Math. Mech. 87 (10) (2007) 698–710. D. Jürgens, M. Palm, Voith Schneider propeller- An efficient propulsion system for dp controlled vessels, Proc. of the Dynamic Positioning Conference, Houston, TX, 2009 13–14 October http://dynamic-positioning.com/proceedings/dp2009/ thrusters_palm.pdf. D. Jürgens, H.J. Heinke, Voith schneider propeller (VSP)-Investigations of the cavitation behaviour, First International Symposium on Marine Propulsors, Trondheim, SMP, 2009June https://pdfs.semanticscholar.org/f764/ 10afafc3007d9370eecf2ef42a6ea4cb20a0.pdf. E. Esmailian, H. Ghassemi, S.A. Heidari, Numerical investigation of the performance of voith schneider propulsion, Am. J. Marine Sci. 2 (3) (2014) 58–62. J. Prabhu, V. Nagarajan, O.P. Sha, Unsteady Flow Analysis of Marine Cycloidal Propeller, Coastal & Inland Shipping, Kharagpur, India, 2015, pp. 73–82 10-11 DecemberThe Royal Institution of Naval Architects. J. Prabhu, V. Nagarajan, M.R. Sunny, O.P. Sha, On the fluid structure interaction of a marine cycloidal propeller, Appl. Ocean Res. 64 (2017) 105–127. J. Prabhu, A.K. Dash, V. Nagarajan, O.P. Sha, On the hydrodynamic loading of marine mycloidal propeller during maneuvering, Appl. Ocean Res. 86 (2019) 87–110. J. Rom, High Angle of Attack Aerodynamics, Springer, New York, 1992. H. Gao, H. Hu, Z.J. Wang, Computational study of unsteady flows around dragonfly and smooth airfoils at low reynolds numbers, 46th AIAA aerospace sciences meeting and exhibit, Reno: Nevada, 2008 7–10 January. M. Peric, S. Ferguson, The advantage of polyhedral meshes, Dynamics 24 (2005) 4–5. M. Peric, Flow simulation using control volumes of arbitrary polyhedral shape, ERCOFTAC Bull. 62 (2004) 25–29. A.W. Ruys, A comparison of some published results of tests on vertical axis propellers, Int. Shipbuild. Progress 13 (1966) 399–401. I.B. Celik, U. Ghia, P.J. Roache, C.J. Freitas, H. Coleman, P.E. Raad, Procedure for estimation and reporting of uncertainty due to discretization in CFD applications, J. Fluids Eng. 130 (7) (2008) 1–4. S. Zhang, Y. Dong, Y. Ouyang, Z. Yin, K. Peng, Adaptive neural control for robotic manipulators with output constraints and uncertainties, IEEE Trans. Neural Netw. Learn. Syst. 29 (11) (2018) 5554–5564. W. He, Y. Dong, Adaptive fuzzy neural network control for a constrained robot using impedance learning, IEEE Trans. Neural Netw. Learn. Syst. 29 (4) (2018) 1174–1186. A. Najafi, H. Nowruzi, H. Ghassemi, Performance prediction of hydrofoil- supported catamarans using experiment and ANNs, Appl. Ocean Res. 75 (2018) 66–84. D. Marquardt, An algorithm for least-squares estimation of nonlinear parameters, SIAM J. Appl. Math. 11 (2) (1963) 431–441. M.T. Hagan, M. Menhaj, Training feed-forward networks with the marquardt algorithm, IEEE Trans. Neural Netw. 5 (6) (1994) 989–993. M.T. Hagan, H.B. Demuth, M.H. Beale, Neural Network Design, PWS Publishing, Boston, MA, 1996.
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Applied Ocean Research journal homepage: www.elsevier.com/locate/apor
CFD data based neural network functions for predicting hydrodynamic performance of a low-pitch marine cycloidal propeller Mohammad Bakhtiari, Hassan Ghassemi
T
⁎
Department of Maritime Engineering, Amirkabir University of Technology, Tehran, Iran
ARTICLE INFO
ABSTRACT
Keywords: Marine cycloidal propeller Hydrodynamic performance CFD simulation Neural network Predictive function
Today, various types of propulsion systems are used in different purpose ship types. Marine cycloidal propeller (MCP) is one of these propulsion systems, which has been designed for ships that require high maneuverability. MCP can be considered as an especial type of marine propulsion systems, since it produces the thrust force which is perpendicular to propeller axis of rotation. The magnitude and direction of the thrust force can be adjusted by controlling the pitching angle of the blades, so no separate rudder is needed to manoeuvre the ship. In this study, mathematical functions for predicting the open water hydrodynamic performance of a low-pitch MCP are presented by training a neural network based on computational fluid dynamics (CFD) data. For this purpose, the four nondimensional parameters of blade number (Z), ratio of blade thickness to MCP diameter (t/D), pitch (e) and advance coefficient (λ) are considered as input variables, whereas the hydrodynamic coefficients of thrust (Ks) and torque (Kd) are considered as targets. CFD simulations are performed for different cases of MCP with different combinations of Z, t/D, e and λ. The results showed that a two-layer feedforward network with one hidden layer of sigmoid neurons and at least 4 neurons in the hidden layer can be well trained by CFD data in order to obtain functions with good accuracy in predicting Ks and Kd coefficients of a low-pitch MCP.
1. Introduction With the development of special purpose ship types in recent years, a well-balanced compromise was demanded between the main requirements of ship propulsion, such as high efficiency, low cavitation, noise and erosion, high maneuverability, low maintenance costs, etc. To meet these requirements, various types of marine propulsion systems have been developed. MCP is a special type of these propulsion systems, which is used for ships that require high maneuverability, such as tugs, ferries etc. As shown in Fig. 1a, a MCP consists of a number of hydrofoil blades fitted to a disc usually set flush with the ship hull. The disc revolves about a vertical axis (propeller axis of rotation) while the blades pitches about their own individual axes. Therefore, unlike a screw propeller, the axis of rotation of a MCP is perpendicular to the thrust force. In addition, the thrust is controllable in magnitude and direction by adjusting the pitching angle of the blades, so that no separate rudder is required to manoeuvre the ship. So far, two different configurations of MCP with different applications have been developed, including low-pitch and high-pitch MCPs (see Fig. 1b). The low-pitch one (known as cycloidal propeller) has been designed for advance coefficients of λ < 1. The high-pitch one (referred as trochoidal propeller) has been designed for advance coefficients of λ > 1, and provides higher hydrodynamic efficiency than low-pitch ⁎
ones allowing high-speed operation. Nevertheless, low-speed maneuvering requirements have led to further development of the low-pitch configuration. Here, = VA/R is absolute advance velocity coefficient of a MCP, where VA, ω and R are advance velocity, rotational velocity and radius of MCP, respectively. MCPs have been in use since 1920s [2,3], however serious theoretical and experimental studies on MCP were carried out much later. The research conducted by Henry in 1959 was a foundation for subsequent theoretical analyses and systematic experiments [4]. Early theoretical models developed for analyzing MCP, like Taniguchi's method [5], were in 2D and made simplifying assumptions leading to inaccurate models. In 1961, Haberman & Harley used Taniguchi's method for calculating the hydrodynamic performance of a cycloidal propeller [6]. In 1969, Sparenberg optimized the blade pitch function of a single-blade propeller by minimizing kinetic energy in the wake [7]. James worked on Sparenberg's optimization work with a more detailed model in 1971 [8]. Much later, in 1991, Brockett developed a computational model based on lifting line and actuator disk theories to optimize the blade pitch function [9]. In 1996, Van Manen and Van Terwisga expanded Brockett's computational model to further optimize the blade pitch function [10]. Systematic series of experiments on MCPs were carried out by Nakonechny [11,12], Van Manen [13], Ficken [14], Bjarne [15], and
Corresponding author. E-mail address: [email protected] (H. Ghassemi).
https://doi.org/10.1016/j.apor.2019.101981 Received 22 June 2019; Received in revised form 25 October 2019; Accepted 31 October 2019 Available online 13 November 2019 0141-1187/ © 2019 Elsevier Ltd. All rights reserved.
Applied Ocean Research 94 (2020) 101981
M. Bakhtiari and H. Ghassemi
Fig. 1. a) A ship with twin cycloidal propellers [1], b) two different configurations of MCP.
Bose and Lai [16] in the years between 1961 and 1989. Van Manen's systematic tests demostrated that the hydrodynamic efficiency of highpitch MCPs is lower that low-pitch ones. Also, Van Manen showed that Sparenberg's pitch function is more efficient than a cycloidal pitch function for multi-bladed propellers. In the last two decades, CFD has been implemented to analyze MCP propulsor. In 2005, Jurgens et al. presented a numerical optimization of blade pitch function for MCP (Voith-Schneider propeller) by using CFD method [17]. In 2009, Jurgens et al. investigated the influence of a MCP on stern slamming conditions, the roll damping capabilities of the MCP and the effect of air ventilation. They used experimental and CFD methods and compared the results of the MCP propulsor with azimuth thruster. They demonstrated that a MCP can decreases the impact of stern slamming pressure because of its vertical rotating axis, and can be used to reduce the roll motion of a vessel. They also showed that MCP is less prone to ventilation than azimuth thruster [18]. At the same year, Jurgens and Heinke investigated cavitation behaviour of heavily loaded MCP with different blade profiles under bollard pull condition [19]. In 2014, Esmailian et al. investigated the hydrodynamic performance of a MCP propeller by CFD analysis [20]. Prabhu et al., in 2015, calculated the hydrodynamic coefficients of a MCP by using an unsteady analysis based on numerical panel method and compared their results with steady analysis [21]. In 2017, Prabhu et al. developed a numerical scheme for investigating the fluid structure interaction of a marine cycloidal propeller blade by using 2D inviscid flow theory [22]. In addition, Prabhu et al. investigated hydrodynamic forces on marine cycloidal propeller during maneuvering by numerical panel method in 2019 [23]. In this study, we present mathematical functions based on neural network model to predict open water hydrodynamic performance of a low-pitch MCP. For this purpose, the four nondimentional parameters of blade number (Z), ratio of blade thickness to MCP diameter (t/D), pitch (e) and advance coefficient (λ) of MCP are considered as input variables, whereas the thrust coefficient (Ks) and torque coefficient (Kd) of MCP are considered as targets. A 2.5D CFD method, after validating against experimental data, is used to analyze the turbulent flow around MCP and calculate Ks and Kd for 168 different combinations of Z, t/D, e
and λ. The 168 calculated CFD data are implemented to train neural network and obtain two mathematical functions to predict Ks and Kd of a low-pitch MCP. The obtained mathematical functions are also validated to accurately fit the CFD data. According to the above mentioned, the contributions of this paper, compared with previous works in the field of MCP, include providing mathematical functions for estimating the hydrodynamic coefficients of low-pitch MCPs based on CFD data obtained from 2.5D numerical simulations of MCP. Furthermore, neural network method is implemented to obtain the hydrodynamic coefficient functions of MCP from CFD data. In the following, first the principles governing the hydrodynamic performance of a MCP is described in Section 2. Then, the CFD model used for analyzing the flow around the MCP and its validation against experimental data are presented in Section 3. In Section 4, the neural network model and training algorithm are explained. In Section 5, the CFD data of Ks and Kd (targets) calculated for different combinations of Z, t/D, e and λ (input variables) are provided, and two mathematical functions are presented for predicting open water hydrodynamic performance of a lowpitch MCP. Finally, some conclusions are drawn in Section 6. 2. Hydrodynamic performance of MCP In a MCP, λ and β are two main parameters which define the kinematics of the MCP. Fig. 2 shows the kinematics of a MCP as well as the hydrodynamic forces acting on a single blade. In this figure, VA is advance velocity of MCP, ω is rotational velocity of MCP about its axis (point O), θ is angle of rotation, β is blade pitch angle and αkin is kinematic angle of attack. VR is the resultant velocity of the blade axis (point P) due to VA and ωR. In addition, lift and drag are the hydrodynamic forces acting on single blade, which produce thrust force (Tb), torque about propeller axis (Qb) and side force (Fb) of the blade. Perhaps the most common blade pitch angle for a MCP is pure cycloidal motion (see Fig. 3a), in which the blade pitch angle changes in such a way that the normal to the chord line at blade axis of rotation (NP ) passes through a fixed point (point N, named as steering center). In a pure cycloidal motion, the blade pitch angle is defined as follows: 2
Applied Ocean Research 94 (2020) 101981
M. Bakhtiari and H. Ghassemi
in the fore and aft part of MCP (see Fig. 2), their side forces are in opposite directions and nearly cancel each other out. In the other hands, thrust forces of the blades are mostly in same and positive direction and amplify each other. Therefore, total side force is significantly lower than the total thrust force. Finally, the unsteady total thrust and torque are averaged over one revolution of propeller to obtain the averaged value of thrust (Tave) and torque (Qave), and consequently, the open water hydrodynamic characteristics of a MCP:
Ks =
Kd =
Eff =
Tave 1 2
DLu2
(2)
2Qave 1 2
D2Lu2
(3)
Ks Kd
(4)
where, Ks, Kd and Eff are thrust coefficient, torque coefficient and open water efficiency of a MCP, respectively. D = 2R and L are propeller diameter and blade span length, and u=ωR is circumferential velocity of the blade.
Fig. 2. Kinematics of a MCP and the hydrodynamic forces acting on a single blade.
3. CFD model 3.1. The fluid flow equations In this study, the unsteady Reynolds-averaged Navier–Stokes (URANS) equations with shear-stress transport (SST) k–ω turbulence model are used to numerically solve the turbulent flow around the MCP. The URANS equations can be written as:
ui =0, xi t
( ui ) +
(i = 1, 2, 3)
xj
( ui uj ) =
(5)
p u + µ i xi xj xj
u iu
j
+ gi ,
( j = 1,
2, 3 )
(6) where, u, p, ρand μ are velocity, pressure, density and viscosity of the fluid, respectively, and g is gravity force. Using Boussinesq hypothesis, the Reynolds stresses term in Eq. (5), u i u j , can be stated as:
uj ui + ) xj xi
u i u j = µt (
2 u ( k + µt l ) ij, 3 xl
(l = 1, 2, 3)
(7)
where, μtis turbulent viscosity. In SST k–ω turbulence model, the turbulent viscosity,μt, is computed as follows:
µt =
( , e) = tan
1
1 1 SF , 2 * 1
max
(8)
where, k and ω are turbulence kinetic energy and specific dissipation rate respectively, obtained from the following transport equations:
Fig. 3. Diagrams of blade pitch angle and blade pitch rate for different pitches of a MCP with pure cycloidal motion .
e sin 1 + e cos
k
t (1)
where, e = ON /R is eccentricity or pitch of a MCP. Generally, for e < 1, the MCP is known as a low-pitch propeller, whereas for e > 1, it is referred as a high-pitch propeller. In practice, due to mechanical constraints and high pitch rate of blade near = , as illustrated in Fig. 3b, the pitch of a VSP can be varied within the range of e ≤ 0.8. By summing the thrust and torque forces of all blades of the MCP, the total thrust and torque are calculated at each time instant. It can be stated that once two blades are located at the same transverse locations
t
( k) +
(
xi
)+
xj
( kui) =
(
uj ) =
µ+
xj
µt k
xj
µ+
k xj
µt xj
+ Gk
+G
*k
(9) 2
+D
(10)
where, Gk and Gω are generations of k and ω respectively. σk and σωare turbulent Prandtl numbers for k and ω: k
= =
3
F1/
k ,1
1 + (1
F1)/
F1/
,1
1 + (1
F1)/
k ,2
,2
(11) (12)
Applied Ocean Research 94 (2020) 101981
M. Bakhtiari and H. Ghassemi
In above equations, F1 and F2 are blending functions.
angular velocity of ω (zone Z7 in Fig. 4), and a fixed zone at far field (zone Z8 in Fig. 4). An interface boundary condition is applied to the contact surfaces of the zones. A translational periodic boundary condition is used for the top and bottom boundaries of each zone. As illustrated in Fig. 5 for a 4-bladed MCP, each of the fluid zones was discretized with an unstructured polyhedral mesh in order to create separate cell zones. Unstructured polyhedral mesh have been introduced by some well-known CFD merchant codes as an innovative approach. This type of mesh includes the advantages of structured hexahedral mesh (such as low numerical diffusion resulting in accurate solution) and tetrahedral mesh (such as rapid automatic generation), and overcome the disadvantages of both the above mentioned types of meshes (time-consuming and sensitive to stretching) [26,27]. Here, the sliding mesh technique is implemented, so that each moving cell zone slides relative to one another along the mesh interface, without any mesh deformation. An inflation polyhedral layer mesh including 45 layers with growth rate of 1.1 and Y+<1 (Y+ is nondimensional distance of the first layer from the wall) was used within near-wall region.
3.2. Computational domain and mesh In a MCP, the blades are exposed to high kinematic angles of attack, which are sometimes beyond the dynamic stall angle [9]. Therefore, the flow on the blades is highly separated and unsteady, and the 3D flows in spanwise direction are significant and should be included in simulations [24]. In other words, a 2D simulation does not provide sufficient accuracy to solve the unsteady flow with large separations around airfoils at relatively high angles of attack [25]. In the other hand, the full 3D simulation of MCP is time consuming and computationally costly. Consequently, a 2.5D CFD method is implemented in this study. In this method, a 2D model is extended in a spanwise direction for a considerable length in order to resolve the effects of the flow in the third dimension (spanwise direction), with periodic boundary conditions at two extremities of the domain. Here, the flow equations are solved numerically in a computational domain with one-chord length in spanwise direction by using Ansys Fluent code. The computational domain and boundary conditions are shown in Fig. 4 for a 6-bladed MCP. In this figure, D is disk diameter and C is blade chord length. As mentioned in Section 2, each blade of MCP rotates with constant angular velocity of ω about MCP axis of rotation and simultaneously pitches with angular velocity of ( , , e) about its own axis. Accordingly, we consider a fluid zone near each blade, which moves with the same velocity as the blade (zones of Z1, Z2, Z3, Z4, Z5 and Z6 shown for 6-bladed MCP in Fig. 4), a moving zone with constant
3.3. Validation and verification of CFD model For investigating the accuracy of the CFD model in predicting the open water hydrodynamic performance of a MCP, we compared it against the experimental data provided by Van Manen [13]. Since the present simulations are carried out for MCPs with pure cycloidal motion including rectangular blades with symmetrical NACA 4-digit profile, the Van Manen's test was found as the most appropriate one for
Fig. 4. Computational domain and boundary conditions . 4
Applied Ocean Research 94 (2020) 101981
M. Bakhtiari and H. Ghassemi
Fig. 5. Computational mesh, a) whole solution domain, b) near-field domain, c) the zone around blade, d) inflation polyhedral mesh inside boundary layer and the mesh on blade surface.
validating the CFD model. Table 1 provide specifications of the MCP model tested by Van Manen. It should be noted that the Van Manen's tests were performed at Reynolds numbers in which the scale effect of a MCP can be neglected [28]. Figs. 6 and 7 show the comparison between CFD and experimental data of Ks, Kd and Eff at different λ for two pitches of e = 0.4 and e = 0.6, respectively. In the same figures, the values of relative errors have been provided. The average relative errors of Ks, Kd and Eff are 3.1%, 2.4% and 4.5% for e = 0.4, and 5.5%, 2.6% and 6.6% for e = 0.6, respectively. Some part of these errors may be refer to some existing effects in the test, which are not included in a 2.5D CFD simulation, such as the blade tip effects and the variable blade thickness. However, there is a good agreement between the CFD results and experimental data. CFD simulations were performed in three different meshes with different cell sizes in order to verify the CFD model based on the grid convergence index (GCI) approach proposed in [29]. This approach is an acceptable and a recommended method that have been tested for several hundred CFD cases. According to [29], the mesh refinement was performed uniformly over all cell zones in order to create three meshes with different resolutions; coarse, medium and fine. Table 2 shows the
properties of these three different meshes. In this table, the number of cells relates to the meshes used for validation case in previous section. As an example, the calculated values of Ks and Kd at e = 0.6 and λ=0.3 in the three different meshes, as well as the discretization error and numerical uncertainty are listed in Table 3. In this Table, N1, N2 and N3 are the total number of cells in fine, medium and coarse meshes, respectively. X1, X2 and X3 are performance coefficients of MCP calculated in 1 fine, medium and coarse meshes, respectively. r21= (N1/ N2) 3 and r32= 1 21 (N2/ N3) 3 are mesh refinement factors. P is apparent order. Xext is the extrapolated value of performance coefficients of MCP. 21 21 21 ea21 = |(X1 X2 )/ X1 | is approximate relative error. eext = |(Xext X1)/ Xext | is extrapolated relative error. As seen from Table 3, the numerical uncertainties in fine-mesh solution (GCIfine) for Ks and Kd are very small, indicating that the solution is grid independent. Therefore, the medium mesh including 45 polyhedral inflation layer inside the boundary layer with Y+ value of 1 was selected to perform the numerical simulations of this study. In order to study the independence of solution from temporal discretization, the simulations in medium mesh were performed for two time-steps of T/360 and T/720, where T is the period of one revolution of MCP. Given that the time-step refinement from T/360 to T/720 had a very little effect on the values of Ks, Kd and Eff, the time step of T/360 was selected for performing the simulations.
Table 1 Specifications of the MCP model selected for validation. Parameter
value
MCP diameter (D) Blade number (Z) Ratio of blade span to MCP diameter (L/D) Ratio of blade chord to MCP diameter (c/D) Ratio of maximum blade thickness to MCP diameter (t/ D) Blade platform Blade section profile Blade pitch motion rotational velocity (n) Reynolds number
0.2 m 4 0.6 0.18 0.026
4. Neural network model Neural network models are computational models that have received much attention from researchers in various fields in recent years [30,31]. Neural networks have shown that they are suitable for function fitting [32], and a fairly simple neural network can fit any practical function. Fig. 8 shows the architecture of a single-neuron network with R-element input vector of p. Three different operations occur in a neuron. First, the individual input elements of vector p, (p1, p2,… pR), are multiplied by weights, (w1, 1, w1, 2,… w1, R), and the weighted values are fed to the summing junction. Their sum is simply the dot
Rectangular Symmetrical Pure cycloidal motion 6 RPS 1.4 × 105
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Fig. 6. Comparison between CFD and experimental data for e = 0.4.
Fig. 7. Comparison between CFD and experimental data for e = 0.6.
product Wp. Then, the Wp is added to the scalar bias b to form the net input n (The bias is like a weight, except that it has a constant input of 1). Finally, the net input is passed through the transfer function f, which produces the scalar output a. In fact, these operations include three functions: the weight function, the net input function and the transfer function. In many types of neural networks, the weight function is a product of a weight times the input. Also, the most common net input function is the summation of the weighted inputs with the bias. Three of the most commonly used transfer functions are shown in Fig. 9. Two or more of the neurons shown in Fig. 8 can be combined in a layer, and a neural network could contain one or more such layers. For example, Fig. 10 illustrates a three-layer network. This network has R inputs, S1neurons in the first layer, S2neurons in the second layer, etc. Each layer has a weight matrix W, a bias vector b, and an output vector a. The outputs of each intermediate layer are the inputs to the following layer. Thus, layer 2 can be analyzed as a one-layer network with S1inputs, S2neurons, and an S2 × S1weight matrix W2. To distinguish between the weight matrices, output vectors, etc., for each of these
Table 2 Properties of the meshes used for grid convergence study. Mesh
Number of cells
Y+ value
Number of inflation layers
Coarse Medium Fine
459,251 846,327 1,685,952
2 1 0.5
35 45 55
layers in the Fig. 10, the number of the layer is appended as a superscript to the variable of interest. In a multilayer network, the layer that produces the network output is called an output layer, whereas all other layers are called hidden layers. The three-layer network shown in Fig. 10 has two hidden layers (layer 1 and layer 2) and one output layer (layer 3) and. Here, it is assumed that the output of the third layer, a3, is the network output. The mathematical functions for outputs (a) in term of inputs (p), weights (W) and biases (b) for each layer can be written as:
6
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Table 3 Discretization error and numerical uncertainly calculated for Ks and Kd at e = 0.6 and λ = =0.3. Parameter
Ks
Kd
N1 N2 N3 r21 r32 X1 X2 X3 p
1,685,952 846,327 459,251 1.258 1.226 0.234363 0.234296 0.250147 26.817 0.234364
1,685,952 846,327 459,251 1.258 1.226 0.191489 0.191091 0.209570 18.805 0.191502
ea21
0.03%
0.21%
21 eext GCIfine
0.0002%
0.0067%
0.0003%
0.0084%
21 Xext
Fig. 11. A simple schematic of training a neural network with input and target data. Table 4 The variable input parameters and their values for 168 different cases of MCP. Parameter
Values
Number of blade (Z) Ratio of blade thickness to MCP diameter (t/D) MCP pitch (e) Advance velocity coefficient (λ)
4, 5, 6 0.024, 0.042 0.2, 0.4, 0.6, 0.8 0, 0.1, 0.2, …, (e + 0.1)
a1 = f 1 (IW1,1p + b1)
(13)
a2
b 2)
(14)
a3 = f 3 (IW 3,2a2 + b3)
(15)
=
f 2 (IW 2,1a1
+
By substituting Eqs. (13) and (14) in Eq. (15), the following function is obtained for the three-layer neural network:
Fig. 8. A single-neuron neural network with R-element input vector.
a3 = f 3 (IW 3,2f 2 (IW 2,1f 1(IW1,1p + b1) + b2) + b3) 3
(16)
where, a and pare respectively the outputs and inputs of the network. The main idea of neural networks is that the values of weights, w, and biases, b, can be adjusted so that the network exhibits some desired behavior. Thus, one can adjust, or train, a network, so that a particular input leads to a specific target output. Fig. 11 shows a simple schematic of this situation. Here, the network is adjusted based on a comparison of the output and the target, until the network output matches the target. Typically, many such input/target data pairs are needed to train a network.
Fig. 9. Three of the most common transfer function used in neural networks .
Fig. 10. A three-layer neural network .
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Fig. 12. Open water hydrodynamic diagrams of Ks, Kd and Eff from CFD data for Z = 4.
It has been proven that a neural network with one (or more) hidden layers of sigmoid neurons followed by an output layer of linear neurons (known as feedforward network) can be trained to approximate any function with a finite number of discontinuities arbitrarily well, given consistent data and enough neurons in its hidden layer. The hidden layers of neurons with nonlinear transfer functions allow the network to learn nonlinear relationships between inputs and targets. In this study, we create feedforward neural networks with one
hidden layer and train them by CFD data to obtain two mathematical functions for open water Ks and Kd of a MCP. Creating and training procedures are performed by using neural network fitting toolbox in MALAB. The networks will be trained with Levenberg-–Marquardt backpropagation algorithm [33,34,35]. The four parameters of Z, t/D, e and λ are considered as inputs, whereas the Ks and Kd are considered as targets of the neural networks.
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5. Results and discussions
and λ. The values of Z, t/D, e and λ, considered as variable input parameters to calculate Ksand Kd, are given in the Table 4. Here, all the simulations are performed for a MCP model with diameter of 0.2 m and constant value of c/D = =0.2. The open water hydrodynamic diagrams of Ks and Kd and Eff from CFD data have been presented in Figs. 12–14 for Z = 4, Z = 5 and Z = 6, respectively. It can be observed that the maximum efficiency of MCP increases with increasing MCP pitch parameter. However, at lower
5.1. CFD data for MCP hydrodynamic performance The CFD model described and validated in Section 3 is implemented in Ansys Fluent solver to simulate the turbulent flow around low-pitch MCP. The open water hydrodynamic coefficients of Ksand Kd are calculated for 168 different cases with different combinations of Z, t/D, e
Fig. 13. Open water hydrodynamic diagrams of Ks, Kd and Eff from CFD data for Z = 5. 9
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M. Bakhtiari and H. Ghassemi
Fig. 14. Open water hydrodynamic diagrams of Ks, Kd and Eff from CFD data for Z = 6.
advance coefficients, the efficiency decreases by increasing the MCP pitch. Also, the diagrams of Eff demonstrate that the blade number have an undesirable effect on hydrodynamic efficiency, so that the hydrodynamic efficiency of MCP reduces by increasing of blade number. According to the diagrams, the average relative reductions in maximum Eff
due to increasing of blade number from 4 to 6 are about −42%, −25%, −13% and −7% for e = 0.2, e = 0.4, e = 0.6 and e = 0.8, respectively. In the case of t/D effects on Eff, it is clear from the figures that the maximum efficiency of MCP decreases with increasing blade thickness. In fact, the average relative reductions of about −6.3%, −7.4% and 10
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M. Bakhtiari and H. Ghassemi
Fig. 15. Pressure distribution on blade surface, and velocity distribution at mid-span plane for the 6-bladed MCP with t/D = 0.042.
−7.8% are resulted in maximum Eff for Z = 4, Z = 5 and Z = 6, respectively, due to a 75% increase in blade thickness from t/D = 0.024 to t/D = 0.042. As an example of graphical visualization of CFD simulations, the pressure and velocity contours around the 6-bladed MCP have shown in
Fig. 15 for MCP pitches of e = 0.4, e = 0.6 and e = 0.8. The figures illustrate the pressure and velocity distributions on blade surface and mid-span plane of the MCP, respectively. For each MCP pitch, the contours have been extracted in advance velocity coefficient near the maximum efficiency point. 11
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M. Bakhtiari and H. Ghassemi
Fig. 16. Training performance diagrams for Ks (upper diagram) and Kd (lower diagram).
5.2. Mathematical functions for MCP hydrodynamic performance
outputs (ai) and targets (Ti) of the validation set, defined as follows:
As mentioned in Section 4, we create two-layer feedforward networks with a sigmoid transfer function in the hidden layer and a linear transfer function in the output layer, and train the networks by 168 CFD data presented in previous section. The four parameters of Z, t/D, e and λ are considered as inputs, whereas the Ks and Kd are considered as targets. In this way, two mathematical functions for the two target variables in term of inputs are presented based on the neural network model. The network is created in MALAB software and trained by Levenberg–Marquardt backpropagation algorithm using neural network fitting toolbox in MALAB. The 168 CFD data are randomly divided into three sets; 70% are used for training, 15% are used to validate that the network is generalizing and to stop training before overfitting, and the last 15% are used as a completely independent test of network generalization. The training continues until the validation error fails to decrease for six iterations (validation stop). In fact, the network weights and biases are saved at the minimum value of mean squared error (MSE) between the network
MSE =
1 N
N
(Ti i=1
ai ) 2
(17)
We created and trained various two-layer feedforward networks with different number of hidden neurons from 1 to 10. We observed that by increasing the number of hidden neurons from 1 to 4, the MSE value significantly decreased to less than 1e-4. However, with increasing from 4 to 10 hidden neurons, the reduction in the MSE value was insignificant. So, in order to avoid the complexity of the proposed neural network functions while having enough accuracy, we selected the network with 4 hidden neurons for training. In the other hand, given the fact that each time the neural network is trained, a different solution is achieved due to different initial weight and bias values and different divisions of CFD data into training, validation, and test sets, we trained the network several times for achieving the best solution. In order to improve the process of neural network training, we fed 12
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Fig. 17. Regression diagrams for Ks prediction .
the neural network by the normalized values of the input and target CFD data. Therefore, based on the maximum and minimum values of each variable, the inputs and targets variables were normalized to fall in the range of [−1, 1]. Fig. 16 shows the training performance diagrams for Ks and Kd. According to this figure, the best validation MSE errors for Ks and Kd are 2.5153e-5 and 6.8185e-5, respectively at 60th and 222nd iterations. Figs. 17 and 18 illustrate regression diagrams for Ks and Kd predictions, respectively. In fact, the diagrams of these figures present the network outputs with respect to targets for all CFD data, as well as training, validation, and test sets of the data. In these figures, the dashed line (labeled as X = Y) shows a 45° line, where the network outputs are equal to the targets and we have the perfect fit on this line. The solid line represents the best fit linear regression line between outputs and targets. The R value is an indication of the relationship between the outputs and targets. If R = 1, this indicates that there is an exact linear relationship between outputs and targets. As can be seen, for both Ks and Kd, there is high fitness for all data sets, with R values more than 0.997. The neural network mathematical functions achieved for predicting the open water performance of a low-pitch MCP are as follows:
2
Ksn = 1 + exp
2 × 0 . 07364 × Zn
t D n
0 . 10664 ×
1 1 . 00868 × en + 1 . 47633 ×
n
0 . 78548
× ( 0.99585) 2
+ 1 + exp
2×
0 . 05115 × Zn
2 . 27862 ×
t D n
1 0 . 24339 × en + 1 . 18190 ×
n
0 . 24822
× 0.76604 2
+ 1 + exp
2 × 0 . 03349 × Zn + 0 . 60767 ×
t + 0 . 64258 × en D n
1 × 0 . 60457 ×
n
0 . 46072
1 . 50131 × en + 1 . 54741 ×
n
1 . 76776
1.62865 2
+ 1 + exp
2 × 0 . 10187 × Zn
0 . 49728 ×
t D n
1 ×
0.61787 0.11982
(18)
13
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M. Bakhtiari and H. Ghassemi
Fig. 18. Regression diagrams for Kd prediction.
where, Zn, en, ( D )n , λn, Ksn and Kdn are normalized parameters in the range of [−1, 1], defined as follows: t
K dn 2
= 1 + exp
2×
0 . 10358 × Zn
0 . 28276 ×
t D n
1 0 . 34662 × en
0 . 53122 ×
2 × (Z 2
Zn =
n + 1 . 40075
× ( 14.21384 ) 2
+ 1 + exp
2×
0 . 03299 × Zn + 1 . 09977 ×
t D n
t D
1 0 . 70398 × en
0 . 42949 ×
n + 1 . 71166
× ( 8.78894 ) + 2×
0 . 04837 × Zn + 0 . 91204 ×
t D n
1 0 . 58975 × en
0 . 56987 ×
n + 1 . 51745
n
× 10.19728
1 + exp
2×
0 . 21103 × Zn
0 . 15990 ×
t D n
t D
0.024
4
)
1,
2 × (e 0.2) 0.6 2× 0.9
1,
6
Z
0.024
0.2
1,
0
(20)
t D
e
0.042 (21)
0.8
(22)
(e + 0.1) (23)
2
+
=
(
1,
0.018
n
en = 2
1 + exp
=
2×
4)
1 0 . 41256 × en
1 . 51709 ×
n + 2 . 06870
× 3.55407 + 8.42958
(19) 14
Ksn =
2 × (Ks + 0.1016) 0.7405
K dn =
2 × (K d + 0.0580) 0.6801
1
(24)
1
(25)
Applied Ocean Research 94 (2020) 101981
M. Bakhtiari and H. Ghassemi
Fig. 19. Comparison between neural network functions of Ks and Kd and CFD data for Z = 5, at t/D = 0.042 (upper diagram) and t/D = 0.024 (lower diagram).
• By increasing the propeller pitch (e), the maximum hydrodynamic efficiency of MCP increases. • The blade number (Z) has an undesirable effect on hydrodynamic
Although, the consistency of Ks and Kd outputs with CFD target data was demonstrated in Figs. 17 and 18, a comparison between predictive neural network functions in Eqs. (18) and (19) and CFD data have been presented in Fig. 19 for Z = 5 and t/D = 0.024 and 0.042. This figure shows the Ks and Kd curves predicted by Eqs. (18) and (19) from e = 0.2 to 0.8, as well as CFD data points for e = 0.2, 0.4, 0.6 and 0.8. From the figure, a very good fit can be observed between the neural network functions and CFD data.
• •
6. Conclusions
•
The open water hydrodynamic coefficients of Ks and Kd were calculated for 168 different low-pitch MCPs including different combinations of Z, t/D, e and λ by using a 2.5D CFD method. After validating the method against experimental data, the obtained CFD data were used to train feedforward neural networks in order to present two mathematical functions for Ks and Kd (as output variables) in term of Z, t/D, e and λ (as input variables). According to the results, the following conclusions can be made:
efficiency, so that the hydrodynamic efficiency of MCP reduces by increasing of blade number. In the case of t/D effects on Eff, the maximum efficiency of MCP decreases with increasing blade thickness. A two-layer feedforward network with one hidden layers of sigmoid neurons and at least 4 neurons in the hidden layer can be well trained by CFD data in order to obtain Ks and Kd functions of MCP with high accuracy and good fitness. Using neural networks with high number of neurons (here more than 10) provides Ks and Kd predictive functions with better fitness, but the complexity of the resulted functions increases.
This is our future plan to extend our calculations to high-pitch MCP and also investigate the hydrodynamic performance of MCPs with other blade pitch motions.
• A 2.5D CFD method can be used as an accurate and computation-
Declaration of Competing Interest
ally-efficient CFD method to simulate and calculate the hydrodynamic force coefficients of a low-pitch MCP.
The authors confirm that there is no conflict of interest. 15
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Acknowledgment [16]
Numerical computations presented in this study have been performed on the parallel machines of the high performance computing research center (HPCRC) of Amirkabir University of Technology; their supports are gratefully acknowledged.
[17] [18]
Supplementary materials
[19]
Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.apor.2019.101981.
[20]
References
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