Analytical, Computational Fluid Dynamics and Flight Dynamics of Coandă MAV

The paper establishes the basic working relationships among various relevant variables and parameters governing the aerodynamics forces and performance measures of Coandă MAV in hover and translatory motion. With such motivation, capitalizing on the basic fundamental principles, the Fluid Dynamics and Flight Mechanics of semi-spherical Coandă MAV configurations are revisited and analyzed as a baseline. To gain better understanding on the principle of Coandă MAV lift generation, a mathematical model for a spherical Coandă MAV is developed and analyzed from first physical principles. To gain further insight into the prevailing flow field around a Coandă MAV, as well as to verify the theoretical prediction presented in the work, a computational fluid dynamic CFD simulation for a Coandă MAV generic model are elaborated using commercial software FLUENT®. In addition, the equation of motion for translatory motion of Coandă MAV is elaborated. The mathematical model and derived performance measures are shown to be capable in describing the physical phenomena of the flow field of the semi-spherical Coandă MAV. The relationships between the relevant parameters of the mathematical model of the Coandă MAV to the forces acting on it are elaborated subsequently.


Introduction
Coandă effect and Coandă jet have found many applications in engineering, among others in aircraft and vehicle technology [1][2][3][4][5], as well as wind-turbines [6][7][8][9]. In particular, Coandă MAV' share in active developments, and various configurations have been proposed and developed. For the purpose of designing a Coandă MAV which could meet the desired mission and design requirements, it is mandatory to establish the basic working relationships among various relevant variables and parameters governing the aerodynamics forces. To assist the analysis, design and developments of Coandă MAV's, several tools can be resorted to. The first is the analytical tool, which capitalizes on the basic fundamental principles. The second is the utilization of Computational Fluid Dynamics, (CFD) which has the advantage of providing visualization for significant insight and identification to the problem at hand, which then can be utilized in enhancing the analysis and identifying specific details. Experimental tools can benefit from the insight gained by analysis, CFD computational and visualization studies, in the preliminary study stage by designing specific experiments as well as in the conceptual and prototype design stages. Various CFD studies have also indicated that there are still significant discrepancies between CFD and experimental studies [6], which necessitate the existence of specific baseline configurations for validation purposes. Coandă MAV is expected to be capable of manoeuvring, and the analysis carried out here is associated with hovering and cruising conditions, which should give further insight on, and can be further elaborated into its manoeuvring performance.
In the study of Coandă MAV, it is desired that the Coandă propulsion system will be able to provide both an efficient cruise phase as well as hover capability, by using Coandă jet blanket for semi-spherical Coandă MAV, which is deflected downward using a curved surface. Thus lift will be generated, first to hover and later for propulsion as well. Such Coandă jet has been utilized for circulation enhancement [3][4][5][6][7][8][9] for fixed wing aircraft and turbine blades in forward flight and movement. The principle of Coandă MAV lift generation as well as the equation of motion for its translatory motion will be derived and elaborated.
The present analysis is carried out on Coandă MAV spherical configuration as a baseline, which utilizes some results from the authors' previous analysis. The equations can be easily modified for other configurations and can be compared to other investigators' results for further assessments. CFD visualization studies will also be utilized to gain further understanding in developing the physical model in the analysis.
Some basic results drawn from the physical and mathematical model has been obtained to describe the physical phenomena of the flow field related to the relevant surfaces influenced by the Coandă effect jet sheets, and rewritten in a more convenient form as necessary to gain better insight on the relationship among the relevant parameters.

System of Coordinates
For the derivation of the equation of motion of the Coandă MAV, the system coordinates that are required for setting the equation will be defined. Following the convention in flight dynamics, four main reference frames are identified, namely the inertial coordinate system, the local horizon reference frame, the body reference frame, and the wind reference frame, as required in the analysis, and are depicted in Figure1a. Without loss of geerality, for simplicity and instructiveness, only twodimensional coordinate systems in the plane of symmetry of the Coandă MAV will be elaborated. These four coordinates are: a) Inertial coordinate system, which is used for defining and the application of the Newton's law of motion. In two dimensions, this coordinate system is depicted by the OExyz. b) The local horizon coordinate system Oxhyhzh, which is fixed at the center of mass of the vehicle, and is parallel with the earth Inertial Coordinate system. c) The body coordinate system Oxbybzb, which is fixed to the vehicle, and follows a conveniently chosen axis of the vehicle. d) The wind axes system Oxwywzw, which moves with the vehicle and the xw axis is coincident with the velocity vector (flight path of the vehicle). In the figure, the wind axes are oriented to the flight path angle γ relative the local horizon axes and by the angle of attack α relative to the body axes. The two dimensional configuration in the vertical plane perpendicular to the earth surface is shown in Figure1 (b).  The mathematical model of Coandă MAV elaborated here follows closely that of Ahmed et al [10][11] and Djojodihardjo and Ahmed [12][13], which was based on first principles and are summarized and reproduced here as a baseline reference for other development. For Coandă MAV with the configuration depicted in Figure2, an actuator rotor can be visualized to be located at the center of the body. For conceptual development, the dimension of the rotor can be set to be small. Part of the flow being drawn by the actuator can be utilized for lift (like in a helicopter), and part of it will be utilized for establishing radial flow for Coandă jet blanket on the surface of the body. Momentum analysis is carried out in the analysis of Coandă effect related to finding the relevant aerodynamic forces and defining the Coandă MAV performance parameters. The latter is developed for a baseline and simplified semi-spherical Coandă MAV as depicted in Figure 3. In the following, a rigorous analysis is carried out to elaborate how Coandă effect contributes to the generation of lift capability of Coandă MAV.

Mathematical Model and Fluid Dynamic Analysis for hovering Semi-spherical Coandă MAV as a Baseline Configuration
where Vj-R is the jet flow velocity, R is the vehicle body radius, h is the jet slot thickness and ṁ is the jet mass flow rate. The momentum equation applied to the control volume in the y (vertical) direction can be differentiated into the contribution due to the Coandă Blanket momentum and the pressure difference on the body due to Coandă Blanket: Control Volume The contribution of lift from the momentum flux through the control volume CV in the y (vertical) direction is given by: Since the Momentum in the radial direction does not contribute to lift, then the momentum equation in the y-direction for the control volume CV depicted in Figure4 is: The contribution of lift from the pressure difference between the upper part (curved MAV body covered by Coandă Blanket) and the lower part of the MAV body can be found by considering various fundamental physical approach. The detail of the elaboration can be found in Ahmed and Djojodihardjo [10][11] and yields: where θi and θ0 are the turning angles at which the jet flow injected and separated from the Coandă surface respectively, as depicted in Figure4. The jet flow assumed to be uniform outflow separating at the sharp edge of the Coandă MAV curved surface. Then the contribution for lift due to the pressure difference across the surfaces of the MAV body, for the latter approach is given by (as elaborated by Djojodihardjo and Ahmed [12][13]), for the significant value of θi): Hence the total lift due to Coandă jet blanket momentum and Coandă jet blanket induced pressure difference is given by Hence, due to the presence of the Coandă blanket, the Coandă MAV has additional lift given in (8). It should be noted that a more exact approach requires additional information, i.e. the energy conservation equation, which is then utilized as the fourth equation. This is elaborated subsequently.
Applying the energy conservation equation on the same control volume, which has been redrawn for convenience in Figure5, assuming uniform properties across the sectional areas at the input and output of the Coandă jet blanket and ignoring the entrainment energy exchange between the ambient air and the Coandă jet blanket, the energy equation can be written as: which is essentially the Bernoulli equation along any streamline between the inlet and the outlet sections. The contribution of the entrainment energy exchange along the Coandă jet blanket can later be incorporated, such as by adopting certain assumptions as a heuristic approach or, applying the complete viscous equation by numerical simulation or CFD approach. The latter should be carried out for more meticulous approach, using the present simplified analytical approach as guidelines. At the same time, the CFD results, by appropriate considerations of the relevant parameters utilized there can be used in developing the heuristic approach as well as for validating and assessing the merit of the analytical approach. Noting that from the outset, the flow is considered to be incompressible, equation (9) Hence, by the application of the conservation energy principle for incompressible flow, and assuming ambient pressure at the inlet and outlet of the Coandă jet blanket and neglecting the energy exchange between the ambient air and the Coandă jet blanket, the following velocity relationship between the inlet and outlet sections of the Coandă jet blanket is given by This should simplify the solution given in equation (8) and reduce the use of idealization or the number of assumption since fewer unknowns are included in the equations. Combining equations (8) and (9) The results will be assessed a posteriori by using more structured CFD simulations, although the latter also contain uncertainties and inaccuracies.

Mathematical Model and Flight Dynamic Analysis for Semi-spherical Coandă MAV in Translatory Motion
In the development of the flight dynamics of Coandă MAV in Translatory Motion, without loss of generalities, the equation of motion is developed in the vertical plane perpendicular to the earth motion, rendering two-dimensional planar motion. The hovering state, as depicted in Figure6, will be utilized as a reference.
Balance of forces in the free-body diagram exhibited in Figure6 and the use of the Coandă lift from equation (12) leads to: In the development of the equation of motion during translatory motion, a further simplifying assumptions will be made, which could be refined at later stage to incorporate more realistic ones. These are: 1. The thrust of the Coandă MAV will not be affected by the attitude of the Coandă MAV configuration during hover. This assumption implies that Coandă jet flow relative to the Coandă MAV does not change during maneuver. More elaborate analytical model should be developed assisted by CFD simulation. 2. During the translatory motion, the thrust of the Coandă MAV is assumed to be:   Weight-Fuel Consumption: It should be noted, that the above relationships are derived for balance of forces and Newton's equation for semi-spherical Coandă MAV treated as a point mass moving in a plane perpendicular to the planar Earth.

Performance Measure during Hover
The feasibility of using Coandă techniques to enhance aerodynamic performance of Coandă MAV can be assessed using some non-dimensional quantities of performance measure. The most logical measure that has commonly been utilized is the Coandă jet momentum coefficient, C (Mamou and Khalid [14], 2007; Djojodihardjo [9]),, which are defined as: While (23) may be appropriate for any Coandă activated vehicle during hover and lift-off, one can define other performance measures to evaluate the aerodynamic performance of the spherical Coandă MAV considered here, based on the rate of the energy of the Coandă jet ejected by Coandă MAV at its ideal inlet velocity Vj-in at the Coandă jet injection (inlet) or Vj-out at its peripheral outlet compared to the rate of momentum influx of the Coandă jet times its inlet velocity Vj-in (non-dimensional): Further realistic assumption can be drawn from CFD visualization, which also added further insight to the problem. Similar performance measures can be defined for Cylindrically Shaped Coandă MAV.
Taking note that Mirkov and Rasuo [15][16] considered the inlet velocity to thrust ratio for a given inlet size could be the most important output value resulting either from experiment on UAV's or by numerical simulation, another performance measure could be defined to be the lift to inlet velocity, which is a dimensional quantity. Substituting equation (13)

Some Results and Examples
To assess the merit and plausibility of the theoretical analysis using first principles carried out here, a comparison of the theoretical prediction for the performance measures PM1, PM2 and PM3 given by (25) -(26) are compared with CFD simulation carried out using ANSYS FLUENT for Spherically Shaped Coandă MAV. The results are depicted in Figs. 9 and 10. The simulation was performed using steady RANS with two equations k-ω Shear Stress Transport (SST) turbulence model. Simple pressure-velocity coupling scheme with the least squares cell base as discretization gradient was applied in the solution method together with second order upwind for the momentum equation and the turbulent kinetic energy.
The performance PM2 as depicted in Figure10 shows the influence of the ratio of the jet slot thickness h to the reference radius Ro. The influence of the jet inlet velocity on the air vehicle performance "lift" may also be of interest from designer's point of view, which enables them to befittingly select the right propulsion system for such MAV. The jet inlet velocity influence on the performance of the Spherical Coandă MAV at constant jet thickness (h=50mm) is depicted in Figure11. For comparative and validation purposes, Figs. 12 (a) and (b) are produced to assess the present theoretical prediction with CFD simulation results for the case considered by Schroijen and van Tooren [15] for cylindrical Coandă MAV, for h/R=0.075 at two different jet inlet velocities.    been introduced, while in the CFD simulation, the full Navier-Stokes equation for incompressible fluid was used. show the velocity magnitude contours for the given configuration and inlet conditions for cases considered by Mirkov and Rasuo [16][17] and Ghozali [18]. Meticulous attempts and grid sensitivity studies on the size of the mesh cells have been performed to enable plausible visualizations of the flow around the whole body with best details.  The numerical study was performed with a moderately small Reynolds number, Re= 68458, based on mean velocity and jet inlet height. The velocity across the inlet, which has a thickness of 0.05 m, is uniform (Vj-in = 20 m/s). The number of the mesh cells was 52830, and the mesh quality was found to be acceptable. The orthogonality quality was 4.96943e-01, which is acceptable from the ANSYS orthogonality quality requirements. Results obtained on the influence of the inlet jet radius on the air vehicle performance (lift) investigated using CFD simulation for two inlet jet radius, Ri=5 mm and Ri=50mm are presented as velocity magnitude contours shown in Figure13 (29) and (31) and CFD simulation for level and climbing flights, respectively.

Conclusions
A comprehensive effort has been made to analyze and describe the governing equations applicable to Coandă MAV in hover and translatory motion utilizing first principles as articulated in fluid dynamics and flight mechanic, using conservation principles for a control volume and free-body diagram for the application of Newton's law of motion. The paper establishes the basic working relationships among various relevant variables and parameters governing the aerodynamics forces and performance measures of Coandă MAV in hover and translatory motion. CFD simulations for a Coandă MAV generic model are carried out to gain insight into the flow situation, establishing the analytical model and validating it subject to the theoretical assumptions. The mathematical model and derived performance measures are shown to be capable in describing the physical phenomena of the flow field of the semi-spherical Coandă MAV. Comparison with CFD simulation serves also to assess the uncertainties and accuracy of the theoretical approach. In addition, the CFD computational and visualization studies, can provide further insight in revealing other characteristics of the flow field, and can assist further in-depth analysis, as well as in the design of experiments to that end. For example, to find out, under what conditions the effect of viscosity may cause the Coandă jet to separate. CFD visualization has certainly assisted the analytical work in assessing the application of fundamental conservation principle in continuum mechanics as well as elaborating the equation of motion for the flight dynamics of the Coandă MAV.