Numerical Study of an External Flow around a Corrugated Wing using Lattice Boltzmann Method

. During the course of recent studies on wings at low Reynold number, it was observed that wing corrugation is often assumed to play an important role as well. However, studies show that corrugation of the wing is intended for structural purposes, and not aerodynamics. Corrugated wings have the advantage of being light and sturdy. Therefore, the main aim of this study is to understand the flow behaviour of the corrugated insect-scale wing; by conducting, a geometric parametric study during a non-oscillatory flight at a particular low Reynolds number and at two different angles of attack. In this computational study, a 3-D section of the corrugated wing along the chord is considered. The lattice Boltzmann method offers an alternative framework compared to the Navier-Stokes simulations. An open-source Parallel Lattice Boltzmann Solver on a high-performance computing platform is used for this computational analysis. The present study shows that the flow-related performance of the corrugated wing in terms of forces and kinetic energy is predominantly governed by the geometric variations that can largely affect the formation of vortices and their mutual interaction. The study reveals that the presence of corrugation does not affect the enhancement of forces and corrugation near the leading edge generally affects the performance due to large flow separation affecting the suction.


Introduction
Researchers working on biomimetic robots are always curious about studying wing corrugation relation with the aerodynamic performance of these bionic references.For example, a simulation on the 50% cross sectional of a corrugated dragonfly forewing at a low Reynolds number clearly shows that with astroke amplitude typically below a certain value (low) usually generates very minimal thrust force.[1] When compared to a noncorrugated simulation, this simulation has no non-realistic fluctuations.[2] From various existing literature, it is revealed that insect wing corrugation only benefits in structural terms by increasing the bending stiffness of the wing and does little to enhance the aerodynamic performance of flapping wings.[3] The thorough study of very accurate flapping oscillatory regimes that shows the corrugated aerofoils and their performance compared to smooth ones, are not yet investigated properly [4] This might not be the case in other flow conditions.Internal corrugations, for example, are responsible for flow instabilities in corrugated channels underflow, which synchronise with the channel's longitudinal acoustic modes, producing powerful pure tones.[5] Tonal sounds can be produced by airflow through corrugated tubing.These sounds, typically called whistling, is an intriguing phenomenon inside the tubing because the corrugations transform this simple tubing structure into an aerodynamic source of toning with different resonant frequencies.These Corrugations bring flexibility to both metallic and non-metallic structures making them useful in a wide range of engineering, industrial, and household applications.[6] Corrugation of insect wings comes with lots of benefits from their flight point of view.For example, it makes the wings stiff, lightweight and with low membrane stress.Corrugations on wings slightly decrease the lift and drag (16% at 15°-25° AoA) and near the leading edge, they are responsible for pushing the leading-edge-separation layer slightly upwards.This decreases the suction pressure or increase pressure near the upper surface with large separation bubble.[7] When it comes to understanding the complex fluid flow characteristics at micro scale, it is important to look into phenomenon that correspond to reactive flows, viscous forces, flow slips, kinetics, capillarity etc.The effect of inertial or gravitational forces are not that important.[8] The superior results of numerical investigations are due to scientific computing power and memory.To accurately model bionic reference locomotion, multiscale simulations are required in the biomimetics domain.This can be accomplished at various levels of computing, ranging from standard computers to petascale supercomputers, as well as combinations of these.[9] or to the most recent emerging exascale highperformance computing environments.[10] There are several models and programmes that can be used to conduct such high-fidelity numerical investigations in order to seek long-term sustainable solutions.The Lattice Boltzmann Method is one such method.For transitional flows of engineering interest, after verifying the accuracy as well as its performance, it was observed that the entropic lattice Boltzmann model is a viable, parameter-free alternative to computational and modelling approaches with similar resolution requirements, such as large-eddy simulations.[11] Researchers have also observed that lattice-Boltzmann method accurately predicts mean aerodynamic parameters like thrust and torque, thereby p[performance when it is compared with numerical method like Reynolds Averaged Navier-Stokes, DNS and experiments provided that mesh resolutions are proper.[12] When lattice Boltzmann (LB) is compared with the Navier Stokes (NS)-based simulations for parallel performance in high-performance computers, both simulations can capture the flow physics with absolute correctness and always remain in good agreement with the S E3S Web of Conferences 477, 00099 (2024) https://doi.org/10.1051/e3sconf/202447700099STAR'2023 experimental results.Numerical codes based on both the frameworks are configured to run on parallel computational platform with similar time and computational resources used.Because of the use of CUDA and other GPU based programming, the numerical approach based on LBM is estimated a lot cheaper (25 times) than the NS based investigation.Thus, it clearly reveals that a nested type LB-Large Eddy Simulation solver can compute extremely well, the calculations of wind flow with parallel/cloud-based computational resources.[13] The lattice Boltzmann method (LBM) appears to be an important computational algorithm for CFD.For simple flow around the two-dimensional (2D) NACA airfoil, at different angles of attack shows promising results with the value of lift and pressure coefficients in good agreement with experimentation from literature.[14] LBM is also advantageous and reliable in terms of its efficiency; near-ideal scalability on high-performance platforms.[15] Commercially also the LB based formulation with highly efficient octree mesh generator reduces mesh complexity computational cost substantially for industrial grade flows.[16] The combination of this method with methods like immersed boundary also produces sensible results for complex problems like biomimetics and enhances the performance.[17,18]   In this paper, the Lattice Boltzmann Method is used to numerically investigate the effect of corrugation on the aerodynamics and flow characteristics of a section of a Mosquitoinspired wing in non-oscillatory conditions.The open-source LBM-based parallel solver is used in this analysis to simulate the condition and compare the results with a plane noncorrugated wing, thereby analysing the aerodynamic forces.The numerical platform is an inhouse high-performance computing facility.

About Wing Modelling
Figure 2 (a -b) shows the three-dimensional section of the wings computationally tested in this work.The LBM based solver is used for the numerical study.The wing sections were modelled in Autodesk Fusion 360 using the morphology of the mosquito wing studied earlier with chord length (c ~ 3.0 mm) and thickness (0.01c) applied for both the wing profiles scaled in dimensions for the LBM computational domain.The smooth wing (noncorrugated wing) was used as a planar one and the corrugated wing is modelled based on the wing design of the actual mosquito aerial vehicle under development at the Department of Aerospace Engineering, Faculty of Engineering, Universiti Putra Malaysia.For aerodynamic performance, the lower hump of the corrugated wing is where the wing membrane is used in the prototype wing and the upper humps represent venation.There are  The wing shapes in Figure 2 are static but can be used as in flapping case also with passive wing twist and rotation as described below.The section of three-dimensional wing kinematics at r = 0.55R (for full-scale wings used in the prototype), which is described in Figure 2 (a -d) was used to investigate the wing corrugation effect on the aerodynamic performance of the 3D wing.If a simple case of flapping is considered, then the wing based trailing edge rotation angles () and AoA are represented by the below given equations (1) for down stroke and (2) for upstroke: where the AoA in the translation phase of the wing at r = 0.55R are around (55 to 63)° and (37 to 47)°), respectively.Table 1 shows the wing morphological details in static (nonoscillatory) conditions.Non-oscillatory 5 0 and 10 0 0.55

Lattice Boltzmann Method and Governing equations
The Lattice Boltzmann method emerged as a numerical alternative to the Navier-Stokes equations at macro and meso level computations.Now it has been extended to the realm of micro and nanofluidics Considering a mesoscopic model of a fluid, which is bounded by the incompressible Navier-Stokes equations ( 3) and ( 4) at the macroscopic level [20] and used for insect and bird flight [21,22], we have: The LBM is quite stable with less dissipation, as such makes it a suitable method to solve NS equations.One such example is the determination of the instantaneous flow field of the MAV rotor concerning the ground as shown in figure 3(b).Due to its use of the Cartesian grids, a substantial amount of grid points are needed to compute the flow, particularly near walls.Suppose at any location  and time , (, ) is the probability to find the particle set with a velocity   , then as per [12], the governing equation ( 5) can be written as: where basically   is the discrete velocity of N number of particle sets and   is the BGK operator as a representation of inner lattice collision for these pairs of particles.
Contrary to this, fluid flow around the solid domain can also be calculated using NSequations with the volume penalization technique in equation ( 6).The negative term is penalization,  is mask function.  .With a collision step and a streaming step, the numerical method looks into relaxation of particle density or population and their transportation to nearby sites with velocity   respectively.For filtering purposes, the available Smagorinsky BGK model is used and implemented in the code.The LBM equations are solved using the open source LBM software [24] on a high-performance computing platform.Solid surface models created in Autodesk Fusion 360 as STL mesh files are treated as bounce-back nodes and in the fluidsolid interface, a bounce-back mechanism is employed.These STL geometries used here are shown in Figures 2(a) and 2(b) for both corrugated and plane wing sections.

Numerical Method, Set Up and Validation
The LBM-based code used here [24] is a high-performance method and has been validated before on several applications one of which is shown in figure 4, with an example of a flow around a sphere, its mesh, iso-surfaces of Q-criterion and comparison with experimental results.So, the code is simply validated with several benchmark cases, with the mesh refinement framework, and the MPI parallelization method.
This study shows the numeric of an external flow around a corrugated insect-inspired aerial vehicle wing in static conditions using large eddy simulation or LES method.At the inlet, an inlet velocity of 0.1 m/s is set as an inlet boundary condition, whereas the wings are defined as a no-slip wall boundary condition.Smagorinsky LES model with constant c = 0.14 is used here.The parallel computation is performed on the "Quanta" high-performance computing facility available here at Universiti Putra Malaysia.The domain consists of a control volume of dimensions 29.5 × 9:0 × 9.0.This additional space was added to allow the wake behind the wing to get stabilize before whacking any boundaries.As previously stated, the time step is determined by the Reynolds number of the system, so it varies with inlet velocity.The wing was imported as an STL mesh file with a simplistic triangular mesh set.The MeshLab was used to resample and simplify the mesh.Lattice velocity was set to 0.04 as per the lattice Boltzmann guidelines.The velocity unit set was chosen as D3Q19.Since the Reynolds number has a strong correlation with the input velocity, the whole case can be controlled by controlling the flow velocity.The grid resolution is 120 in the y-direction with 5.53million allocated cells.The types of boundaries used for the numerical study here are shown in table 2. A simple schematic of the control volume is depicted in figure 5.

Results and Discussion
The analysis demonstrates the Lattice Boltzmann Method's ability to investigate complex phenomena such as flow over a corrugated wing at a positive angle of attack.The LBM technique and high-performance computing were used to compute the instantaneous flow structures, particularly in the vicinity of the wing and in the close wake.Differences in how the outlet boundary conditions were formulated in the two examples can be cited for certain downstream variations.As one would anticipate, improved resolution in the case of the corrugated-shaped obstruction leads to improved simulation accuracy.The instantaneous patterns of the confined flow past the wings, which amplify with the angle of attack, are shown in this section.Figures 6(a -d) show the shape contour maps of the velocity magnitude and instantaneous streamlines that were calculated for two time instants separated by a quarter of the vortex shedding duration.The massive separation bubbles caused by the vorticity shed from the wing and rolling along, may be observed in addition to the large separation regions and associated vortices directly aft of the wing.Additionally significant are the effects of Reynolds number on wing flow.The thin viscous layer of fluid adjacent to the lid contains kinetic energy , which is successively transmitted during wing flow by viscous diffusion.According to the estimate  = Re -1/2 , the total kinetic energy often decreases as the Reynolds number changes, but the average increases over time as seen in Figure 7(a) for both positive angles of attack.The flow field surrounding the corrugated wing is less disturbed by the increase in the positive angle of attack.The asymmetry in the corrugated wing might be the reason for this.Existing literature says that for a negative angle of attack, the corrugation valleys are larger and wider.The vortex shedding, therefore, happens at somewhat lower Reynolds numbers.The current findings can shed light on the hydrodynamics of corrugated wings at positive angles of attack for a variety of Reynolds numbers where insects often fly.Figures 9(a) and 9(b) represents the Iso-surface of the Q-criterion for both the angles of attack depicting wakes produced and their strength.It is important to note the relative dominance of the vortices as like areas where magnitude of vorticity is higher than the magnitude of the rate of strain i.e. near the leading edge.At higher rotation angles or angles of attack, the magnitude decreases particularly near the leading edge corrugation and inside corrugation edges.

Conclusion
In this work, a Lattice Boltzmann Method and technique were employed to simulate flow past a steady obstacle that is a small insect wing section with corrugation.The results show that the corrugation does not affect much in the aerodynamics of the flow during the static flight at a particular angle of attack.The instantaneous three-dimensional velocity and vorticity fields have been computed on an integration time to capture long-time scales of the wing flow.The findings suggest that the effect of corrugation is to affect the aerodynamic forces near the leading edge.The magnitude of vortices is strong near the lower portion of the corrugated wing.A very important observation appears from the front portion of the corrugation edge i.e. near the first leading edge corrugation, resulting in large flow separation and increases in area and magnitude, affecting the suction and the enhancement of aerodynamic force.

Figure 1 (
a) shows actual corrugations in terms of venation on an insect wing and figure 1(b) represents flow around a non-corrugated and corrugated wing.The flow past these insect wings simulated using lattice Boltzmann method at low Reynolds numbers helps designing these small aerial vehicles from micro to pico scale.[19]

S
E3S Web of Conferences 477, 00099 (2024) https://doi.org/10.1051/e3sconf/202447700099STAR'2023 different shapes of the leading edge.The leading edge to trailing for a corrugated has variation in the mean chord length.These wings are designed to produce stronger leadingedge vortices during the translational phase.

Fig. 2 .
Fig. 2. (a) Section of a plane mosquito wing (b) Section of a corrugated mosquito wing (c) Side view (plane wing) (d) Side view of the corrugated wing

S
E3S Web of Conferences 477, 00099 (2024) https://doi.org/10.1051/e3sconf/202447700099STAR'2023 where  is the fluid velocity,  as the kinematic viscosity, and  as the fluid pressure.With incompressibility condition, density has been set to the conventional value  = 1.[17].The fluid based lattice Boltzmann method model for computation is based on a 3D lattice known as D3Q19.The elementary cell structure of the D3Q19 lattice is shown in Figure 3(a).

Fig. 3 .
Fig. 3. (a) D3Q19 unit cell with nearest, next neighbours and so on.Adopted from [23] (b) Instantaneous flow field of the micro aerial vehicle rotor with large eddy simulation based on LBM.Adopted from[12] For the present 3D numerical investigation carried out here, for space and time, the discretization is on a Cartesian grid.The particle velocities use D3Q19 lattice stencils for discretization which has fewer number of DoF compared to the D3Q27 The governing equation is solved 19 times at each discrete velocity   .This complete kinetic method keeps the conservation of mass and momentum that depends upon the number density of particles

Fig. 4 .
Fig.4.Numerical investigation of flow past a sphere with computational mesh, iso-surfaces of Qcriterion in the x-direction, compared to experimental reference data.Adopted from[24]

Fig. 5 .
Fig. 5. 3D Schematics of the control volume with boundaries and the model.

Fig. 6 .
Fig. 6.Contour maps of the velocity magnitude of the steady symmetric flow past a corrugated wing (a) 50 AoA at t = 20 (b) 100 AoA at t = 20 (c) 50 AoA at t = 35 and (d) 100 AoA at t = 35.
Figures6(a -d) show the shape contour maps of the velocity magnitude and instantaneous streamlines that were calculated for two time instants separated by a quarter of the vortex shedding duration.The massive separation bubbles caused by the vorticity shed from the wing and rolling along, may be observed in addition to the large separation regions and associated vortices directly aft of the wing.Additionally significant are the effects of Reynolds number on wing flow.The thin viscous layer of fluid adjacent to the lid contains kinetic energy , which is successively transmitted during wing flow by viscous diffusion.According to the estimate  = Re -1/2 , the total kinetic energy often decreases as the Reynolds number changes, but the average increases over time as seen in Figure7(a) for both positive angles of attack.The flow field surrounding the corrugated wing is less disturbed by the increase in the positive angle of attack.The asymmetry in the corrugated wing might be the reason for this.Existing literature says that for a negative angle of attack, the corrugation valleys are larger and wider.The vortex shedding, therefore, happens at somewhat lower Reynolds numbers.Figure 7(b) shows the streamlines over the wing at a resolution of 500 at time t = 35.

Fig. 7 . 2023 Fig. 8 .
Fig. 7. (a) Average kinetic energy for the corrugated during the analysis (b) Streamlines over the wing showing the magnitude of flow at 50 AoA at t = 35.In addition, the size of the vortices is lower for a positive angle of attack.All these observations are recorded in Figures 8(a) and 8(b) showing 3D vorticity contours in the ydirection at two different time zones.Vortices near the leading edge (first corrugated hump) have higher strength and hence give larger acceleration to the fluid.They are also responsible for large flow separation areas affecting the formation of force.

Table 1 :
Wing Morphological Details

Table 2 :
Types of boundary conditions set for this study