A_W   Wetted area surface (m²)

C_d    Total resistance coefficient

C_f     Frictional resistance coefficient

C_vp   Viscous pressure resistance coefficient

CFD   Computational Fluid Dynamics

L_M   Total length of model (m)

L_S    Total length of submarine (m)

(Fn)_M  Froude number of model

(Fn)_S   Froude number of submarine

(Re)_M  Reynolds of model

R      Total resistance (N)

R_f     Frictional resistance (N)

R_vp      Viscous pressure resistance (N)

(Re)_S   Reynolds of submarine

V_M   Speed of model (m/s)

V_S    Speed of ship (m/s)

In every experimental test in towing tank, water tunnel and wind tunnel, in the first step, the speed of the model should be developed to the full-scale vessel (ship or submarine). In the second step, the obtained resistance of the model should be developed. For submarine, there are two modes of movement: surface and submerged mode. There is not any problem in surface mode because, according to Froud's law, the ratio of speed of the model to the full-scale vessel is proportional to the square root of lengths (length of the model on the length of the vessel) [1]-[3]. This leads to a reasonable speed and is not so much for the model that is applicable in the laboratory.

For example, for a submarine at surface mode with a speed of 10 m/s and a scale of 1:100, the required speed of the model in towing tank will be 1m/s that is easily possible. The main problem is in submerged mode (fully submerged). At submerged mode, Froude equation cannot be used because of absence of free surface effects and waves. In the depth of water, there is frictional and viscous pressure resistance and there is not wave resistance [4][5]. Furthermore, the use of Reynold's equation is impracticable because model speed will be too large and impossible to provide [6].

For example, for a submarine with a speed of 10 m/s and a scale of 1:100, the required speed of the model in towing tank will be 100 times of main submarine, which means 1000 m/s that is actually impossible. The related dynamic effects are evaluated in [7]-[11]. Classification of resistance in marine applications is shown in Figure 1, which presents the different levels of resistance. A popular and well known classification in marine engineering for total resistance (R ) is the summation of wave resistance, viscous pressure resistance (R_vp) and friction resistance (R_f) [12][13]. There is not wave resistance for fully submerged submarine. Total resistance coefficient (C_d), friction resistance coefficient(C_f), viscous pressure resistance coefficients (C_vp) are defined as:

Figure 1: 
Decomposition and classification of resistance in marine applications [10][11]

Which V is the velocity in (m/s), and Aw is wetted area surface in m².

There are three important notes about critical Reynolds and resistance coefficients that are described below.

Note 1: Reynolds of model and submarines do not have to be exactly equal. Main aid of Reynold’s equation is to ensure from the existence of a turbulent flow on the surface of the model because the flow regime on real submarines is turbulence. Critical Reynolds is different from 300,000 to about 1000,000 that depend on another condition such as roughness of model, initial flow turbulence, vibration and heat transfer. Here is an important note that says, “providing turbulent flow can be done by many parameters not only by the Reynolds”. By providing these parameters that mentioned above, the required critical Reynolds decreases steeply. For example, by setting a wire or pin on the bow of model, turbulence can be happened at critical Reynolds less than 500,000. Thus, we can be sure that the flow on the model is turbulent even in low Reynolds. Apart from that, providing Reynolds equal to one millions is not difficult and is not out of access because the kinematic viscosity coefficient is about 0.000001 that it means, for example, in a model with 1 meter length, and speed of 1 m/s the Reynolds is equal to one millions.

Note 2: Variation of the curves of frictional resistance coefficient and viscous pressure resistance coefficient after critical Reynolds (turbulent current) is almost horizontal, and shows the constant coefficient. Total resistance in fully submerged mode is equal to frictional resistance plus viscous pressure resistance. Schematic curve of variation is shown in Figures 2 and 3. The diagram of variations of frictional resistance coefficients versus Reynold's number for pipes is presented in all fluid dynamic books as "Moody diagram" thus it is an accepted obvious origin. This paper wants to prove that this origin can be extended to be used in fully submerged resistance of submarine. For this purpose C_f and C_vp diagrams versus Reynolds are plotted for three analyses. These diagrams will show that “after a special Reynolds, these coefficients are almost constant”.

Figure 2: 
Schematic variations of the viscous pressure resistance coefficients versus Reynold's number

Figure 3: 
Schematic variations of the frictional resistance coefficients versus Reynold's number

Note 3: The variations of resistance coefficients versus Reynold's number are independent of the geometry and the shape of objects. For proving this concept, the three samples have the different shape from each other.

In the next section, the results of analysis of three studied cases are presented that contains two cases by CFD method and one case by experimental test in towing tank.

Many extensive studies have been done about resistance (drag) in aerospace engineering such as Ref [14], but none of them didn't any suggestion for developing the model test results to main object for submerged vehicles. Critical Reynolds depends on the shape of object, velocity and environment. There are main differences between specifications of marine and aerial vehicles such as sharp nose, sub and super sonic speed and compressibility of air. Our focus in this paper is finding a critical Reynolds for developing the result of the model to the main vehicle for marine crafts.

This analysis is done by Flow Vision software based on CFD method and solving the RANS equations. Generally, the validity of the results of this software has been done by several experimental test cases, and nowadays this software is accepted as a practicable and reliable software in CFD activities. For modeling these cases in this paper, Finite Volume Method (FVM) is used. A structured mesh with cubic cell has been used to map the space around the submarine. For modeling the boundary layer near the solid surfaces, the selected cell near the object is tiny and very small compared to the other parts of domain. For selecting the proper quantity of the cells, for one certain speed, five different amount of meshes were selected and the results were compared insofar as the results remained constant, and it shows that the results are independent of meshing. For the selection of suitable iteration, it was continued until the results were almost constant with variations less than one percent, which shows the convergence of the solution. In this domain, there is inlet (with uniform flow), Free outlet, Symmetry (in the four faces of the box) and Wall (for the body of submarine). The turbulence model is K-Epsilon. The considered flow is incompressible fluid (fresh water) in 20 degrees centigrade.

The dimensions of the submarine are presented in Figure 4, and The modeling in Flow Vision is shown in Figure 5. Wetted area is 29.27 m2 and the specifications of fresh water are considered. According to Iranian Hydrodynamic Series of Submarines (IHSS) the code of this shape is: IHSS.1001565-30108025. Therefore, the foil section of the tower is NACA0025. The specifications of IHSS are described in [13][14]. Architecture, and general arrangements have a very important role in the selection of the hydrodynamic shape [17][21].

Figure 4: 
Dimensions of the model in case 1

Figure 5: 
Modeling of case 1 in the Flow Vision software

Results are presented in Table 1, and the diagram is shown in Figures 6 and 7.

Figure 6: 
The diagram of variations of total resistance coefficients versus Reynold’s numbers in case 1

Figure 7: 
The diagram of variations of viscous pressure resistance coefficients versus Reynold’s numbers in case 1

Study on the results shows that for the total resistance coefficient, there is a millstone in Reynold’s 5 millions because after this point, the variations are less than 5% (in maximum) that meant almost constant resistance coefficient after this Reynold’s. The diagram of variations of viscous pressure resistance coefficients versus Reynold’s, shows a millstone after Reynold’s 1 millions. In both above-mentioned diagrams, there is a local hump around Reynold’s 8 millions.

The specifications of the model are shown in Figure 8, and the modeling in Flow Vision is presented in Figure 9. All modeling conditions are as mentioned in case 1. Wetted area is 7.87 m2 and the specifications of fresh water are considered. According to Iranian Hydrodynamic Series of Submarines (IHSS) [16] the code of this shape is: IHSS.8336058.

Figure 8: 
Dimensions of model in case 2

Figure 9: 
Modeling of case 2 in the Flow Vision software

Study on the results shows that for the total resistance coefficient, there is a millstone in Reynold’s 5 millions because after this point, the variations are less than 4% (in maximum) that meant almost constant resistance coefficient after this Reynold’s. The diagram of variations of viscous pressure resistance coefficients versus Reynold’s, shows a millstone after Reynold’s 1 and 5 millions. Such as mentioned formerly in case 1, here in both diagrams, there is a local hump around Reynold’s 9 millions.

Results are presented in Table 3, and the diagram is shown in Figure 10 and 11.

Figure 10: 
The diagram of variations of total resistance coefficients versus Reynold’s numbers in case 2

Figure 11: 
The diagram of variations of viscous pressure resistance coefficients versus Reynold’s numbers in case 2

Experiments were conducted in the marine laboratory of Isfahan University of Technology (IUT) in Iran. The towing tank has 108(m) length, 3 (m) width and 2.2 (m) depth. The basin is equipped with a trolley that can operate in through 0.05-6 m/s speed that moves by two 7.5 KW electro-motors with ±0.02 m/s accuracy. The system is prepared with a proper frequency encoder, i.e., 500 pulses in a minute, which decreases the uncertainty of measurements. The dynamometer was calibrated by calibration weights [22]. A three degree of freedom dynamometer is used for force measurements. Data are recorded via an accurate data-acquisition system. The dynamometer is equipped with 100 N load cells. An amplifier set is used to raise signals of load cells and to reduce the noise sensitivity of the system. The experiment is conducted with a submarine model that is made by wood materials according to ITTC recommendations [23]. Tango nose submarine is a type of submarine that has been tested in underwater mode. All data are filtered to eliminate the undesirable acceleration, primary and terminative motion of trolley. The trolley was controlled in a wireless system from control room of lab. The data presented in this paper for each point are an average of a lot of towing tank runs according to [24]. For each run, at least 750 samples in 15 seconds were collected and the ensemble averaged. Schematic of the model and the overall test stand is shown in Figure 12. General considerations for submarine model test are described in [25]-[31].

Figure 12: 
Schematic shape of the test stand

Dimensions of studied submarine in this paper are shown in Table 5 with parallel middle body form. Relation L/D is equal to 8.88. Hull bow has Tango shape and stern is conical. Main submarine has a deck with 28 meters of length, 0.4 meters of height and 1 meter of the beam. In addition; it has a conning tower of 3.2 meters length and 3 meters of height on top of the main hull. Maximum submerged speed is 14 knots, and the wetted surface area is 450 square meters. All dimensions of this submarine have been scaled by 1:32.

Dimensions of studied submarine in this paper are shown in Table 5 with parallel middle body form. Relation L/D is equal to 8.88. Hull bow has Tango shape and stern is conical. Main submarine has a deck with 28 meters of length, 0.4 meters of height and 1 meter of the beam. In addition; it has a conning tower of 3.2 meters length and 3 meters of height on top of the main hull. Maximum submerged speed is 14 knots, and the wetted surface area is 450 square meters. All dimensions of this submarine have been scaled by 1:32.

By study on the experimental results, it is shown that for the total resistance coefficient, there is a millstone in Reynolds 5 millions because after this point, the variations are less than 5.1% (in maximum) that meant almost constant resistance coefficient after this Reynolds. In experimental results, such as mentioned for CFD results, there is a local hump around Reynolds 7-8 millions.

Figure 12: 
The diagram of variations of resistance coefficients versus Reynold’s numbers for model test in towing tank

In this paper, a practical solution was presented for solving an old problem about developing the results of the experimental model to the full-scale submarine in fully submerged mode. In every experimental test, in the first step, the speed of the model should be developed to full-scale submarine. In the second step, the obtained resistance of the model should be developed. The main problem is providing the speed of the model in the laboratory, based on Reynold’s similarity. It leads to a very high and impossible speed for model.

Based on the findings of this paper, if the Reynold’s of submarine at submerged test be more than 5,000,000 it can be actually supposed that total resistance coefficient of the model and full-scale submarine is equal (CTS=CTm) for every speed in the region of the mentioned Reynold’s. It means that the both problems for finding “corresponded speed” and “related resistance coefficient” were simultaneously solved. For Reynold’s number 5,000,000, the error of this assumption can be less than 5 percent.

If providing this Reynold’s be difficult, setting some wire or pin on the bow, can be used for providing turbulent flow. Furthermore, many other ways to providing turbulent flow can be used. In every method that we be confident about turbulent flow, the total resistance coefficient is constant in every related speed.

For example, in case 3, for full-scale submarine with length 32 meters, in every speed greater than 0.16 m/s, the Reynold’s number is more than 5,000,000 thus the flow regime is certainly turbulent. According to the model test results, in all speeds larger than 0.5 m/s the Reynolds are more than 5,000,000 with constant total resistance coefficients equal to 0.004. Therefore, for full-scale submarine for every speed more than 0.16 m/s, we can suppose that total resistance coefficient is constant and equal to 0.004. It should be noted that the maximum speed of the model which was tested was only 1.5 meters that are easily possible for doing.

Another interesting subject, is unexpected local hump in the resistance coefficient diagram in Reynold’s number of about 7-9 millions. This phenomenon is seen in both CFD and experimental results but now, there is not any scientific reason for that.

We can summarize the findings of this paper as below:

1. Total resistance coefficient after Reynolds 5,000,000 is almost constant.

2. There is not any need for highs speed for model test in towing tank because “corresponding speed” (such as in ship model and base on Froud’s law) doesn't define here. On the other hand, Reynold’s similarity for finding “corresponding speed” is an unnecessary process.

3. There is a local hump in the resistance coefficient diagram in Reynold’s number of 7-9 millions.