Numerical study of flow characteristics of the blade with different structures

In order to study the effect of blade structures on the induced excitation characteristics, two different structures are investigated, which are the blade with special trailing edge and the staggered blade. Firstly, based on the steady numerical simulation, the time-averaged velocity and pressure distribution are obtained at different Re numbers (5×103 and 1×104). Secondly, the unsteady numerical simulation method LES is applied to acquire the transient and complex flow field around the blade. It is found that both two blade structures significantly change the flow characteristics around the blade, especially in the wake flow region. The evident frequency is observed from the frequency spectrums. It is clear that the dominant frequency is originated from the vortex shedding, which is similar to the Karmen vortex street. Finally, the results will be useful for the design of blade with low noise purpose.


Introduction
Flow around the bluff body is an important problem in engineering due to the complex flow induced excitations, and many researches have been carried out. For flow around the single cylinder, Coutanceau [1] and Williamson [2] studied the flow around a circular cylinder at different Reynolds numbers. It is found that the vortexes in the flow wake region are influenced by the Re number and the Re=3×10 5 is defined as critical Re number. As for the double cylinders, Chen [3][4] and Zdravkovich [5][6] studied about its interference model with different arrangements. Zdravkovich [7][8] and Mederios [9] provided the concrete flow field distribution of the flow around the double cylinders.
When it comes to the flow around the flat plate, Parker R [10] and Nakamura Y [11] have carried out the experimental research about it. With the development of the CFD, the numerical simulation is applied to study the flow around the bluff body. Auteri [12] Leclercq [13] and Dadmarzi [14] have carried out the numerical simulation research about the flow around the flat plate. According to the numerical simulation, the vortexes in the flow wake region are all captured, which are consistent with the experimental data.
For now, the structures of the bluff body have been considered, which may be useful when utilized in vane pumps and other fluid machinery. J. Hu [15] and Seung-Jae Lee [16] have studied the NACA0009 hydrofoil with different trailing edge truncations and beveled trailing edges. It is found that modifying the trailing edge of blade can change the vortex structures in the flow wake region remarkably. Yao [17] and Zeng [18] studied the hydrofoil with Donaldson's trailing edge. Results reveal that the amplitude of the vortex shedding is obvious decreased, which is caused by the energy dissipation of vortex  [19] have studied a modified NACA16 hydrofoil via different numerical simulation methods, they compared the simulated results with the experimental data provided by Bourgoyne [20] .
In this paper, two blade structures are designed and analyzed to clarify the corresponding effect on the flow field and induced excitations. The transient numerical simulation method is mainly adopted considering its high calculation accuracy.

Design schemes of the blade structure
In order to study the influence of different structures on the excitation characteristics of the flow around the blade, two typical blade structures are selected for numerical calculation and analysis. The two blade structures are named as Trailing edge and Staggered blade as shown in Fig.1 and Fig.2.

Mesh generation
It is believed that the quality of the grid has a great influence on the accuracy of calculation results. Firstly, to make sure that the grid is enough for the numerical simulation with expected accuracy, the structured grids are used for the models. Compared with the unstructured grids, convergent characteristic of the structured grid is better. And the structured grids are easier to control, especially around the solid wall. Secondly, the ANSYS-ICEM is utilized to generate the structured grids of two models. To obtain better numerical simulation results, the grids are refined around the solid wall, where large pressure gradient and flow separation are generated. Finally, the number of grid is determined to be 1×10 7 approximately. The averaged y + value on the wall is about 0~2 to meet the requirements of the LES. Fig.3 shows the mesh grid of the calculated model.

Numerical scheme
Turbulent flow is quite complex and unsteady. Various physical parameters of fluid, such as velocity, pressure and temperature, change with time and space. Basically, the fluid follows the conservations of energy, mass and momentum. In the present paper, the flow around the blade does not involve heat transfer. So the control equations are as following.
The continuous equation: When the fluid is not compressible, the above equation is : The momentum equation is : The momentum equation is also known as the Navier-Stokes equation, where is the average static pressure, is the Reynolds time-average velocity in the direction, is the liquid density, and is the generalized source term.
The discretization and solution of N-S equations are the core contents of computational fluid dynamics. Nowadays, the numerical calculation methods developed by researchers are mainly divided into two categories: direct numerical simulation and indirect numerical simulation. Direct numerical simulation is defined as the direct solution of the transient Navier-Stokes equation, which has extremely high requirements for the computer, and the calculation period is also very long. Indirect numerical simulation is to simplify the turbulent flow. In the present paper, considering the time spent and constraints of the computing resources, the LES is chosen for the numerical simulation.
As for LES, the large-scale flow structures are considered and calculated. As for the small-scale flow structure, its effect on large-scale motion is solved by establishing an appropriate model. The variables can be separated by the filter function: is the large-scale part of the turbulent flow, and the is the small-scale flow structure in the flow.
The N-S equation is: Where is the subgrid-scale stress(SGS), which reflects the influence of small-scale vortices for the N-S equation, it is defined in Eq.(6): , 1,2,3 (6) SGS the is unknown variable, in order to close the Eq. (5), a suitable calculation model should be used. In the present paper, the SGS model of Smagorinsky-Lilly is applied, which is believed that it has an adequate ability to deal with the turbulent flow.
To acquire the accurate flow field, the commercial CFD software ANSYS-Fluent is applied in the present paper. At first, the standard k-ω model is utilized to calculate the steady numerical simulation. Then, the results from the steady numerical simulation are set as the initial condition in LES. The inlet boundary condition is set as velocity inlet, and the outlet boundary is outflow. The surface of blade is set as no slip wall. The SIMPLEC scheme is used to couple pressure and velocity. The second order upwind scheme is used for pressure spatial discretization, and the bounded central differencing is used for momentum spatial discretization.
In the LES unsteady numerical simulation, the time step is a significant parameter, which has a greater impact on the accuracy of the calculation results. The time step can be calculated as follows: 0.02 Where is the incoming velocity. Finally, time step is selected as 5 10 to ensure the high accuracy of calculation.

Validation of numerical simulation method
The flow field characteristics of single and double flat blades are studied, and the numerical simulation method is verified by the Strouhal number(St): Where is vortex shedding frequency of the cylinder, also known as the Strouhal frequency. The St number is a dimensionless parameter that reflect the relationship between the vortex shedding frequency, the size and flow velocity. It is a function of the structure and the Reynolds number (Re). The St number of the flow around a circular cylinder under different Reynolds numbers has been measured by Lourenco [21] . In the present paper, four cases has been calculated. The comparison between the numerical simulation results and the data from the above literature is listed in Table 1.Where T/D is the characteristic dimension. As a result, it is accepted that the LES can capture complex flow around the blade accurately.

Effect of different structures on flow around the blade
In the present paper, the Re number is applied, which is defined in Eq.(9).
Where is dynamic viscosity coefficient, and are characteristic parameters. The time-averaged velocity and pressure distributions are shown from Fig.4 to Fig.7, and the distribution of the flow field can be obtained preliminarily.   Fig.6 and Fig.7, it is clear that both two blade structures have a high-pressure region at the leading edge. With the increase of Re number, the pressure at the corresponding position increases. However, a low-pressure region is observed at the lower surface of the staggered blade.

Trailing edge
Staggered(downstream blade) Fig.8. The time-averaged velocity in the region of the wake flow at Re=5×10 3 Fig.8 and Fig.9 present the time-averaged velocity in the region of the wake flow at different Re numbers. It is observed that the time-averaged velocity in the wake flow shows semi-parabolic distribution characteristics. Due to the trailing edge structure, the length of the backflow region is extremely short. L b is defined as the length of the backflow region, which is 0.2D approximately for the blade with trailing edge structure. After the backflow region, the time-average flow velocity of the wake flow increases rapidly showing approximately linear growth trend. It is defined as the "linear growth region", and the time-averaged velocity almost linearly increases from 0 to about 0.6 U ∞ . Then, velocity will change into the "slow growth region", and the flow velocity increases from 0.6 U ∞ to 0.8 U ∞ . In this area, the flow velocity changes from linear growth to non-linear growth, and the growth (c)t=3/4T (d)t=1T (b). Staggered Fig.11. Evolution of the wake flow at Re=1×10 4 Having investigated the distributions of time-averaged velocity and pressure, it is necessary to study the evolution of the wake flow. Fig.10 and Fig.11 shows the evolution of the wake flow around the blade with different structures at different moments under two Re numbers. It can be seen that the  Fig.10(a) and Fig.11(a), the large-scale vortex structure at the downstream of the blade with trailing edge is affected by the typical structure. It moves downstream and falls off from the trailing edge. At the same time, only two vertex structures are obvious observed in the wake flow, which form and alternately fall from the upper and lower surfaces of the blade. The Karmen vortex street is clearly observed in the wake flow region. As for the staggered blade shown in Fig.10(b) and Fig.11(b), the vortexes formed in the upstream are impacted and damaged by the downstream staggered blade, which is affected by the upstream blade. The wake flow of the downstream blade is more complicated as characterized by small-scale vortexes and flow separation phenomena.
In order to clarify the evolution of the vortex structure in the region of the flow wake, the vorticity distributions at different Re numbers are investigated.
According to Fig.12 and Fig.13, it can be observed that both blade structures have obvious periodic changes in the flow around the blade. The large-scale vortex structure downstream the blade is affected by the trailing edge. It moves downstream and falls off at the distance from the trailing edge. The vorticity is concentrated at the corners of the upper and lower surfaces. As for the staggered blade, affected by the upstream blade, the turbulence of the incoming flow changes, and the vortex formed in the upstream is impacted by the downstream blade. It causes that the flow in the wake region of the downstream blade is more complicated. Many small vortices in the wake region are generated around the upper and lower surface edges.

Effect of different structures on unsteady load of the blade
As shown in Fig.14 and Fig.15 10 different blade structures, the lift-drag coefficients continue to fluctuate with time. Similar to the flow around a cylinder, the lift coefficient curve fluctuates significantly over time. The drag coefficient fluctuation amplitude is relatively low, and its average value is greater than zero. Two different regions of high and low resistance exist for the lift-drag coefficients. The region of high resistance corresponds to a large fluctuation of the lift coefficient, and the region of low resistance corresponds to a small fluctuation of the lift coefficient.
The values of lift-drag coefficients are given in Table.2 and Table.3 at different Re numbers. From the above results, it is observed that the lift coefficient for the blade with trailing edge is more stable than the lift coefficient in staggered blade, while the staggered blade has the larger lift fluctuation amplitude due to the small-scale vortexes existing in the wake flow.
Trailing edge Staggered Fig.14 Fig.16. Frequency spectrum To obtain the frequency spectrum of the lift-drag coefficients at different Re numbers, the FFT method is applied, and the results are presented in Fig.16. The abscissa represents the frequency in Hz, and the ordinate represents the amplitude, which essentially represents the magnitude of the vortex energy. In all cases, only one obvious characteristic frequency is captured in the velocity spectrum, and it is the dominant frequency in spectrum. It indicates that the flow field structure around the blade shows periodic characteristics, which is characterized by the dominant large-scale vortex structure in the flow field.

Discussions
In the present paper, two different blade structures are concentrated. The utilized numerical simulation method is validated via experiment data provided in the literature. Then, considering the different Re numbers, the time-averaged velocity and pressure distributions are obtained. To acquire the transient flow field, the LES method is applied to capture the complex flow structures and time-frequency signals. The analysis suggests that the trailing edge structure will increase the contacting area of the fluid around the blade. The flow resistance downstream of the separation point is small, and the overall averaged resistance is reduced. As for the staggered blade, the upstream velocity gradient becomes larger due to the hindrance of the downstream blade. The overall averaged resistance increases. For the frequency spectrums, it is clearly observed that only one characteristic frequency caused by the vortex shedding occurs. Besides, the amplitude of the blade with trailing edge is stronger than the staggered blade slightly.

Conclusions
In the present paper, the blade with trailing edge and staggered blade are investigated by the numerical simulation method to clarify the flow pattern and induced excitations at different Re numbers. Some conclusions are as following.