Vs axial component of air flow velocity in the stream tube that encloses the propeller disc, far aft of the propeller. If the flow is always subsonic everywhere in the flow field, the complexity of the governing equation can be simplified considerably. The flow over each wing section is assumed to be independent of what happens elsewhere over any of the other sections.
The vortex plate causes a downwash to the incoming flow, thus reducing the effective angle of attack of the flow.
2 BLADE ELEMENT THEORY
The complicated aerodynamic analysis of a 3-D wing can be simplified by assuming that the flow over any small element of the span (an airfoil of elementary width) is basically 2-dimensionally local, so we can obtain the results of Apply 2-D aerodynamic analysis. . This in turn causes a downwash velocity, which varies across the span and is dependent on the geometry of the 3-D wing. Since the axial direction is perpendicular to the plane of the propeller disc, the resultant velocity seen by the blade element, VR, is therefore given by the following expression.
The angle formed by these two velocity components is known as the entry angle, φ, and is given in terms of terminal velocity as follows.
3 MOMENTUM THEORY
Since the flow from the forward free stream section to immediately upstream of the disk is of constant enthalpy, the Bernoulli equation applied to this region gives the following result. Froude (ref.9) in 1878 to take into account that the value of the inflow factor is generally not constant over the entire surface of the actuator disk. The inflow factor is a function of radial distance from the axis of the propeller and is actually determined by the geometry of the propeller.
The theory can be further refined by noting that a swirl blade would be shed at the trailing edge of the blade, similar to the concept of lifting line theory for the finite lifting wing described in the previous section. However, in the propeller housing the “wing” rotates, which means that the influence of the swirl blade on the main flow is much greater. However, the swirl blade would cause an additional velocity upstream of the disk and a velocity reduction downstream of the disk in the azimuthal direction or the direction of rotation of the propeller blade.
The induced azimuthal velocity must be a small fraction of the rotational speed given by b rΩ, where b is a small positive value and Ω is the rotational speed of the blade in radians per second, as described in the previous section. If the flow were truly inviscid, the fluid particles would regain their undisturbed free-stream velocity not far immediately behind the trailing edge of the wing. However, if the flow is viscous, then friction due to viscosity would slow down the speed of the fluid particles moving over the wing surface.
Moreover, this azimuthal velocity reduction also has an effect on the size of the inflow angle, φ. It is clear that momentum theory alone is insufficient to determine the values of the inflow factor a and the eddy factor b.
4 BLADE ELEMENT MOMENTUM THEORY
All values on the right-hand side of the above equation are known, except for the inlet angle, φ. The axial component of the force is the same as the elementary thrust force due to a single blade, and if the propeller has blades B, then the elementary thrust of the propeller is. The azimuthal component of the elementary resultant force when multiplied by the radial distance, r, is the elementary propeller torque.
Some methods for solving transcendental equations will be discussed in the next section. Since the values of all the terms on the right side of the above equation are known, therefore the value of K1 can be calculated. Once the value of φ is found, the values of the other quantities can be easily calculated as follows.
Using the above method, we can predict the thrust rate and torque values at any radial station, provided the propeller blade geometry is known. For any particular value of blade pitch, β, the value of thrust and torque can be calculated using the method described above. Since the values of the thrust and torque gradients are only known at a limited number of locations, each integral must be replaced by a numerical integration such as
If the blade is divided into N elements of equal width, and the aerofoil shape at the edge, rather than at the midsection, of each element is specified, then the following trapezoidal rule must be used instead of the previous approximation. It should be noted that station 1 is at the blade root, whereas station (N+1) is at the blade tip, where both the pressure and torque degrees can be assumed to be zero.
In the simplest solution method, the value of the solution is guessed, say at the midpoint of the given range or φ π= / 4. The secant method does not require the value of the derivative, but not as efficient as Newton's method. Two of the most popular methods with parentheses are the False Rule and the Bisection Methods.
From the results of the previous calculation, it can be concluded that the initial guess should be as follows. In the division method, we start with the known limiting negative and positive bounds of the range of brackets where the solution is, similar to the regula falsi method. A new value or an assumed value is taken as the midpoint of the interval.
For example, after 10 iterations, the width of the interval is where the solution is in this case. The solution can actually be much more accurate than the final size of the interval if it is calculated using the regula falsi method, i.e. Therefore, let's now define that φleft =0 and φright =π/ 2 are two limit values of the area where the solution is located.
An approximate value of the solution is then calculated using the regula falsi method as follows. After calculating the value of φ at radial station r, the values of thrust and torque gradation can be calculated as follows.
6 PROPELLER PERFORMANCE COMPUTATIONAL ALGORITHM
The aerodynamic performance of a blade element is assumed to be the same as that of the infinitely long two-dimensional airfoil of which it is a part. With this restriction, it is reasonable to assume that the lift coefficient of the airfoil varies linearly with the angle of attack,α, as follows. It is assumed that the value of the lift curve slope Cl,α of each wing section is known, either as a result of wind tunnel measurements or from numerical CFD simulation.
It should be noted that the slope of the lift curve of each airfoil is close to but less than 2π. For each value of j, from j = 1 to j = Nbel, the following data are given: SlopeCl(j) is the slope of the lift curve for the jth airfoil in the unit rad−1. Cdrag(j) is the drag coefficient of the j-th airfoil wing. Alpha0(j) is the zero angle of attack of the jth airfoil in rad Theta(j) is the local pitch angle of the jth airfoil in rad Chord(j) is the chord length of the jth airfoil in m.
Nrot is the rotational speed of the propeller in rpm (rotations per minute). Calculate the rotational speed in radians per second. This requires that the value of the inflow angleφ(j) must be calculated using a numerical method. Also check if the same boundary point has been shifted twice in a row, in which case the function G value of the stagnant point should be halved.
Now calculate the propeller thrust and torque using the trapezoidal rule of integration. To calculate thrust, torque and propeller efficiency for a different value of blade pitch, go back to procedure number 1.1 and change the value of Beta.
7 CONCLUSION
Due to the rotating nature of the propeller flow, it is very difficult to directly analyze the effect of the helical vortex sheet on the flow, e.g. However, it is possible to apply the momentum theory to the annular annulus formed by the rotating wing element, and by combining it with the aerofoil theory, the induced axial and azimuthal velocity components can be shown to depend only on a single parameter, the inflow angle. Furthermore, the BEMT method can be used to calculate the inflow angle, provided that the 2-dimensional aerodynamic properties of the blade elements are known from the 2-dimensional aerofoil theory.
In this report, the theoretical basis of the blade element momentum theory is discussed in great detail, so that it is sufficient to be translated into an algorithm for solving this type of problem. The algorithm is suitable for analyzing the propeller performance of any propeller if the propeller blade geometry is specified. The performance of a propeller obviously depends on the operating conditions, such as the forward speed of the aircraft, the rotational speed of the blades and the density of the fluid, as well as the blade pitch setting angle.
Given all these data, the algorithm described in this report can predict the performance of a propeller, provided the aerodynamic properties of each blade element that makes up the overall blade are also given.
8 RECOMMENDATIONS
9 REFERENCES
