Heaving Flat Plate Computational Fluid Dynamics (CFD) Simulation (In-House CFD Code)

     The adventure 🏕 🚵 I started a while ago to make my own CFD 🌬 code / software 💻 for my digital CV and to make a shiny new turbulence mode (one-day perhaps) is going along nicely. This post is about a 2D 10% flat plate undergoing forced heaving motion. Heaving motion is achieved by Eqn. 1.

h_y = H_o*sin(2*π*f_h*t)                                              Eqn. 1

     w.r.t. Eqn. 1 reduced frequency is defined as (f_h*H_o/U∞)), f_h is the frequency of oscillations, while t is the instantaneous time. H_o is the heaving amplitude and U∞ is the free stream velocity. h_y is the position of the flat plate. The animation is shown in Fig. 1.

     The Strouhal number is 0.228 and Reynolds number is at 500. As we can see, the in-house CFD code works very well for this complex CFD simulation. Validation of this work will never be completed 😆. As soon, I will move on to the next project without completing this one. Anyway, discretized Navier-Stokes equations are available here, in both C++ and MATLAB formats if you want to validate this non sense yourself! Good Luck!

The animation from in-house CFD simulation

     If you want to hire me as your PhD student in the research projects related to turbo-machinery, aerodynamics, renewable energy, please reach out. Thank you very much for reading.

Aperiodic Aero-foil Kinematics

     This post is about a 2D NACA 0012 aero-foil undergoing forced aperiodic heaving. Heaving motion is achieved by the plot shown in Fig. 1. Plot within Fig. 1 represents position of airfoil at various time steps.

Fig. 1, The position of aero-foil

     The animation of the vorticity contours are shown in Fig. 2. The velocity, pressure and vorticity for aperiodic heaving is shown in Fig. 3. A comparison will be made with heaving later, if ever 😀. As far as aerodynamic forces are concerned, per-cycle C_{l, avg} is at 0.63 as compared to 0.0 for periodic heaving. C_{d, avg} aperiodic heaving is at 0.162 as compared to 0.085 for periodic heaving. Of course, this is done on a coarse mesh. If ever I write a paper about this... 😀

Fig. 2, Top Row, Aperiodic, periodic heaving airfoil 

     

Fig. 3, Top Row, L-R, Vorticity, pressure. Bottom Row, Velocity 

     If you want to collaborate on the research projects related to turbomachinery, aerodynamics, renewable energy, please reach out. Thank you very much for reading. 

Flapping Aerofoil For Propulsion

     This post is about a 2D NACA 0012 aerofoil undergoing forced flapping motion for propulsion purposes. Heaving motion is achieved by applying a vertical velocity on the aerofoil based on the Eqn. 1. Similarly the pitching motion is achieved by applying a rotational velocity, governed by Eqn. 2.

v_y = 2*π*f_h*H_o*sin(2*π*f_h*t)                                              Eqn. 1

ω = -2*π*f_h*ϑ*sin[(2*π*f_h*t) + 1.5708]                               Eqn. 2

     w.r.t. Eqn. 1-2 reduced frequency is defined as (2*π*f_h*H_o/U∞)), f_h is the frequency of oscillations, while ω, t and ϑ_o represent rotational velocity, instantaneous time and maximum pitching angle. H_o is the heaving amplitude and U∞ is the free stream velocity.

     The flapping motion is achieved by a combination of the heaving and pitching. In this particular simulation, the aerofoil is in the propulsion mode, meaning the feathering parameter χ is less in magnitude than 1.0. Feathering parameter is defined by Eqn. 3.

χ = ϑ/arctan(h0*2*π*f_h/U∞)                                  Eqn. 3

     The boundary conditions employed for the simulation are at Re 1,000, K = 1.41, H_o = aerofoil chord length, χ = 0.5489 and f_h = 0.003391 Hz. The animation of the pressure, vorticity and velocity contours is shown in Fig. 1.

Fig. 1, Flow animation, fluid flow direction is from left to right.

     The results of present simulation are compared with [1]. In terms of maximum lift, a maximum deviation of 5% is observed as compared to [1], as shown in Fig. 2. The maximum lift coefficient for available data is ~4.224 while the maximum lift coefficient from the present simulation is ~4.057. The average thrust produced is within 2% of [1]. Average thrust coefficient per cycle from [1] is 0.9957 while the result from present simulation reveals the thrust coefficient to be 1.0098.

Fig. 2, A comparison of coefficient of lift.

If you want to collaborate on the research projects related to turbo-machinery, aerodynamics, renewable energy, please reach out. Thank you very much for reading.

References

[1] https://doi.org/10.1017/jfm.2017.508

The Aerofoil

     Aerofoil, a simple geometric shape that is responsible for heavier than air flight and energy generation from of wind, hydraulic and steam turbines. However, much mystery and confusion exists about how the aerofoil works. Here an explanation is presented about the working of an aerofoil by using computational fluid dynamics and without using any equations.

     The fluid bends and tends to follow the shape of an object placed in its path when the fluid flows around the said object such as an aerofoil. This phenomenon happens due to the Coanda effect. Fig. 1 shows streamlines around an aerofoil at a Mach number of 0.22 and Reynolds number of 5e6. It can be seen from Fig.1 that the fluid starts to bend as soon as it reaches the leading edge of the aerofoil and the fluid follows the shape of the aerofoil.

Fig. 1, The white arrows represent the direction of fluid flow.

     It is well understood that, as a moving fluid bends (changes direction), a pressure difference is created across the flow path. To understand this better, consider a tornado or a typhoon (not the aircrafts). In a tornado, the fluid revolves around a central axis. Consider a point at the center of the tornado. As this point moves towards the circumference of the tornado i.e. away from the center, the pressure increases and vice-versa. This happens due to the curvature of the streamlines inside a tornado. The more the curvature difference, the more the pressure difference across the streamlines.

     In the case of symmetric aerofoils (which have top and bottom half at the same shape), there is no lift generated because the curvature of the streamlines is same on both the suction (top) and pressure (bottom) sides of the aerofoil. The resulting pressure difference between the suction and the pressure sides is zero. This can be seen in the negative coefficient of pressure (-C_p) plot shown in Fig. 2. The coefficient of pressures can be seen to overlap. This plot and subsequent figures and plots are generated using the data obtained from the computational fluid dynamics analysis of the aerofoil. Fig. 3 shows pressure distribution around the aerofoil. It is quite clear that the pressures at the top and bottom surface of the aerofoil are same, hence no lift generation. It is also evident that a pressure difference exists between leading and trailing edge of the aerofoil, hence the presence of the drag force (pressure drag) even at no angle of attack.

Fig. 2, Along the horizontal axis, 0 refers to leading edge.

Fig. 3, Air flow is from left to right.

     But, if the same aerofoil is placed at an angle to the flow, the curvature of the streamlines change, as visible in Fig. 4. Due to the different curvature on the suction and pressure side of the aerofoil, a pressure gradient in created between the suction and pressure side of the aerofoil with lower pressure at the top and higher pressure at the bottom, as shown in Figs. 5. The -C_p plots for the aerofoil at the angle of attack is shown in Fig. 5. The pressure difference is quite clear in both Figs. 5-6.

Fig. 4, The white arrows represent direction of fluid flow.

Fig. 5, Along the horizontal axis, 0 refers to leading edge.

Fig. 6, Air flow is from left to right.

     In the far field, the pressure is uniform, colored by green in Figs. 3, 6. In a case when the fluid is turning, the pressure increases as away from the center of the curvature and vice versa. Looking at the suction side, the pressure will decrease as distance to the center increases. The pressure gradient at the bottom can be explained by the same reason. This difference in pressure is what causes the lift force, as evident from Fig. 5.

     Velocity distribution around the aerofoil at an angle of attack is shown in Fig. 7. It can be seen that the fluid has more velocity at the suction side of the aerofoil as compared to the pressure side. The velocity distribution on the aerofoil without an angle of attack is same on both the pressure and suction sides of the aerofoil and is shown in Fig. 8.

Fig. 7, Air flow is from left to right.

Fig. 8, Air flow is from left to right.

     

     This again, can be explained by the pressure gradient. It can be seen from Figs. 5-6 that the pressure gradient at the suction side of the aerofoil is much more favorable as compared to the pressure side. It can be seen from Figs. 5-6 that the pressure is highest at the leading edge of the aerofoil (stagnation point). The pressure falls to its lowest magnitude past the leading edge of the aerofoil on the suction side. Meanwhile, on the pressure side, the pressure drop is less severe as compared to the suction side. As a result, the fluid faces less resistance on suction side of the aerofoil in comparison with the pressure side. This is the reason why fluid velocity is more at the top as compared to the bottom of the aerofoil, not vice versa. In all the figures, the color red means maximum magnitude and the color blue implies minimum magnitude.

If you want to collaborate on the research projects related to turbomachinery, aerodynamics, renewable energy, please reach out. Thank you very much for reading.