Aerofoil Kinematics Computational Fluid Dynamics (Update: 01)

This post is about a 2D NACA 0010 aerofoil undergoing various forms of forced kinematics i.e. pure heaving and pitching and a combination of two known as flapping.

Heaving motion is achieved by changing the angle of attack on the aerofoil based on the Eqn. 1.

α_e = arctan[2*π*St_a*cos(2*π*f_h*t)] + α_i               Eqn. 1

The pitching motion is achieved by employing the sliding mesh with the rotational velocity governed by Eqn. 2.

ω = 2*π*f_h*ϑ*cos(2*π*f_h*t)                                 Eqn. 2

w.r.t. Eqn. 1-2 α_e is the effective angle of attack, St_a is Strouhal number (defined as (f_h*h0/U∞)), f_h is the frequency of oscillations, while ω, t and ϑ represent rotational velocity, instantaneous time and pitching angle. h0 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 power extraction mode, meaning the feathering parameter χ is greater 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 set of simulations are at Re 50,000, St_a 0.0149, h0 = aerofoil chord length, χ = 1.1 and f_h = 0.5 Hz. The animation of the velocity contours superimposed with streamlines is shown in Fig. 1. The velocity scale ranges from 0 to 7 m/s. Pressure distribution around the aerofoils in various forms of motion, after five complete cycles is shown in Fig. 2.

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

Fig. 2, Fluid flow is from left to right

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Heaving Airfoil Simulation

This post is about a 2D NACA 0012 heaving aerofoil. Heaving motion is achieved by changing the angle of attack on the aerofoil based on the Eqn. 1.

α_e = arctan[2*π*St_a*cos(2*π*f_h*t)]+ α_i               Eqn. 1

w.r.t. Eqn. 1, α_e is the effective angle of attack, St_a is Strouhal number, f_h is the heaving frequency.

The case S1 and H6 from [1] are compared in the animations below.

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References

[1] https://doi.org/10.1121/10.0001419

Hypersonic Flow over a Two Dimensional Heated Cylinder

     This post is about the simulation of hypersonic flow over a heated circular cylinder, in two dimensions.

     Equation 1 is used as a relationship between Mach and the Reynold number.

M= Re*μ*√(R*T) ÷ d*P*√γ     (1)

     w.r.t. equation 1, the parameters represent the following quantities.

     M     Freestream Mach number at 17.6
     Re    Reynolds number at 376,000
     μ     Dynamic viscosity at 1.329045e^-5 Ns.m^-2
     R     Specific gas constant at 286.9 J.(kg.K)^-1
     T     Freestream temperature 200 K
     d     Cylinder diameter at 5.6730225e^-4 m
     P     Freestream pressure at 101325 Pa
     γ     Specific heat ratio at 1.4
     Tw  Wall temperature of cylinder at 500 K
     Pr    Prandtl number at 0.736

     The boundary conditions were taken from [1]. A comparison with [1] is shown in Fig. 1. Inside Fig. 1, the red dotted line with circles represents the data from [1]. The black solid line represents the data from the present simulation. Within Fig. 1, 0° represents the stagnation point. The velocity, pressure, Mach number and temperature contours are shown in Fig. 2.

Fig. 1 A comparison with previous research [1].

Fig. 2, Top Row, L-R: Velocity and pressure contours. Bottom Row, L-R: Mach number and temperature contours.

The computational mesh and the computational domain with boundary conditions visible are shown in Fig. 3-4, respectively. The computational domain had a size of 20D x 20D. The mesh had 836,580 total cells and 944 cells were located at the solid fluid boundary. Several local mesh controls were employed to capture the shockwave properly.

Fig. 3, The computational mesh.

Fig. 4, The computational domain.

     The solution method is Finite Volume method. SIMPLE-R is the solver employed. Implicit central difference scheme for diffusion terms, second-order Upwind scheme for convective terms and first-order implicit for temporal terms are used. The mesh created uses the Cartesian mesh with Immersed Boundary method.

     Reference:

     [1] https://arc.aiaa.org/doi/10.2514/6.2004-2371 

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Flying Wing Design using Computational Fluid Dynamics (Verification and Validation)

     This post is about a transient simulation of the ONERA M-6 flying-wing aircraft with a cross-section of ONERA D airfoil, as shown in Fig. 1.

Fig. 1, The simulated geometry. 

     The mesh had 2,474,614 cells in total with 300,892 cells on the wing surface, as shown in Fig. 2. The computational domain is shown in Fig. 4. The computational domain walls are at a distance equal to ten times the wingspan.

Fig. 2, The computational mesh.

Fig. 3, The computational domain.

     The simulated conditions are taken from [1-4] i.e. a freestream Mach number of 0.8395 at 101,325 Pa and 293.2 K. The angle of attack is 3.06°.

     The lift force from the present numerical simulation is at 19,551.65 N as compared to the lift force of 20,438.53 N as determined by [1-4]. The numerically determined drag force from present simulation is 1,370.88 N as compared to 1,313.24 N, as determined by [1-4].

     The results of lift and drag force from the present simulation are within 4.35% and 4.2% of the results calculated by NASA [4] and [1-3]. Streamlines and pressure surface plot around the aircraft surface are shown in Fig. 4.

Fig. 4, Results.

References

[1] Le Moigne, "A Discrete Navier-Stokes Adjoint Method for Aerodynamic Optimization of Blended Wing-Body, Configurations", PhD thesis, Cranfield University, United Kingdom, 2002.

[2] J. Lee, C. S. Kim, C. Kim, O. H. Rho, and K. D. Lee, "Parallelized Design Optimization for Transonic Wings using Aerodynamic Sensitivity Analysis", AIAA Paper 2002-0264, 2002.

[3] E. J. Neilsen and W. K. Anderson, "Recent Improvements in Aerodynamic Design Optimization on Unstructured Meshes", AIAA Paper 2001-0596, 2001.

[4] 3D ONERA M6 Wing Validation, https://turbmodels.larc.nasa.gov/onerawingnumerics_val_sa.html.

     Thank you for reading. If you would like to collaborate on research projects, please reach out.