3.2.1
Background and Definitions
In aircraft design applications, use of lower-scale models of the designed aircraft for certain experimental work is a common procedure due to costs and safety reasons
[121, 122]. However, in order to apply the experiment results of the model aircraft to the full-scale aircraft, certain similarity relations must exist between the two aircraft. Depending on the type of similarity demanded, these relations may include similarity in Reynolds number, Froude number, Mach number or further similarity measures, in addition to the most basic geometric and angle of attack similarities [123]. Similarity in any of these parameters means that, the parameter has the same value for both the lower-scale and the full-scale aircraft. Nevertheless, in most cases, it is not possible to accomplish similarity in all these similarity measures simultaneously [124]. Hence, similarity is realized only in certain parameters [123], which are the most relevant to the phenomena under investigation. For instance, if the compressibility effects are the main focus of the investigation, the lower-scale model and its test conditions are adjusted to match the Mach number of the full-scale aircraft [125].
The dynamic similarity is the type of similarity, under which the forces and the motion of the dynamically-scaled lower-scale aircraft simulate those of the full-scale aircraft. In order to achieve this similarity, the lower-scale aircraft is produced so that, its geometry and mass distribution is the same as those of the full-scale aircraft. Additionally, it is required that the following force ratios are the same for both aircraft [124].
F roude N umber = V ehicle inertial f orce V ehicle weight = mV2/l W = mV2/l mg = V2 gl (3.1)
M ass ratio= V ehicle inertial f orce Aerodynamic f orce = mV2/l ρV2l2 = m ρl3 = W ρgl3 (3.2)
In eqs. (3.1) and (3.2) the terms m, W , and V are the mass, weight, and the speed of the flight vehicle respectively. The terms ρ and g are the air density and the gravitational acceleration at the altitude of interest. The term l is the characteristic length of the flight vehicle.
Based on the definitions given above, the term dynamical scaling is used for re- ferring to obtaining a scaled aircraft from another aircraft by complying with the dynamical similarity requirements stated above.
3.2.2
Dynamic Scaling Coefficients
The scale factors for the quantities, which are used in this work, are derived below. Assuming that:
1. An aircraft, aircraft-2, is the scaled version of an existing aircraft, aircraft-1, with the same geometry,
2. The aircraft dimensions, as well as the densities and gravitational accelerations belonging to each aircraft’s flight conditions are known,
then the following relations are readily available.
Rl= l2 l1 (3.3) Rρ = ρ2 ρ1 (3.4) Rg = g2 g1 (3.5)
Adhering to the notation of Gainer and Hoffman [124], the terms Rl, Rρ and Rg are the length, density and gravity scale factors, respectively. The term scale factor in general is defined as the ratio of any quantity belonging to aircraft-2 to the same quantity of aircraft-1.
Starting from the assumptions given above, using the dynamical scaling require- ments stated in previous section, as well as the scaling factors given in eqs. (3.3) to (3.5), the scaling factors for further quantities can be written as shown below.
Since both aircraft are of the same geometry, from eq. (3.3) the area and volume
scale factors of the aircraft can be written as R2l and R3l, respectively.
Since the Froude number must have the same value for both dynamically-scaled aircraft, eq. (3.6) can be written.
V12 g1l1 =
V22
g2l2 (3.6)
Inserting eqs. (3.3) and (3.5) and rearranging the terms, the velocity scale factor can be written as given in eq. (3.7).
V2 V1 = q Rg q Rl (3.7)
Equating the mass ratios (eq. (3.2)) of both aircraft, eqs. (3.8) and (3.9) are obtained. m1 ρ1l31 = m2 ρ2l32 (3.8) W1 ρ1g1l31 = W2 ρ2g2l23 (3.9)
Inserting the scale factors given in eqs. (3.3) to (3.5) and rearranging, the mass
weight is a force quantity, the weight scale factor is also the force scale factor. m2 m1 = RρR 3 l (3.10) W2 W1 = RρRgR 3 l (3.11)
Looking at the moment of inertia formulas of rigid bodies [126], it can be seen that the moment of inertia I is proportional to the length and mass of the object, such that I ∝ ml2. Based on this, the moment of inertia ratio of the two aircraft
about same axes can be written as given in eq. (3.12).
I2 I1 = m2 m1 l2 l1 !2 (3.12) Placing the mass and length scale factors into eq. (3.12), the moment of inertia
scale factor can be written as stated in eq. (3.13). I2
I1 = RρR 5
l (3.13)
The angular velocity scale factor can be derived based on the relation between the translational and angular velocity. Considering an aerodynamic control surface of the aircraft of chord l as an example, the translational velocity of the surface’s tip under the angular velocity ω can be written as V = ωl. Based on this relation, the angular velocity ratios of both aircraft can be written as given in eq. (3.14).
ω2 ω1 = V2 V1 l1 l2 (3.14)
Inserting the velocity and length scale factors into eq. (3.14), the angular velocity scale factor can be written as eq. (3.15).
ω2 ω1 = q Rg √ Rl (3.15) Since angle is a dimensionless quantity, the dimension of the angular velocity is the reciprocal of the time. Therefore, using eq. (3.15), the time scale factor can be written as given in eq. (3.16).
t2 t1 = √ Rl q Rg (3.16)
3.2.3
Using Dynamic Scaling
This work uses simulation in order to answer the research question stated in Sec- tion 1.3.3. Keeping the leader and follower aircraft identical, it is also investigated in this work, how the answer to the question varies with different sizes of aircraft. Therefore, two separate simulation sets made up of different scales of aircraft are used. Dynamical scaling is used for generating an aircraft of different scale from one existing aircraft flight dynamical model. Then the simulations are performed with each scale of dynamically-scaled aircraft.
In order to obtain a dynamically-scaled aircraft model from an existing aircraft model, the scaling factors derived in the previous section are used. For calculating a quantity of the derived aircraft from the same quantity of an existing aircraft, the quantity is multiplied with the corresponding scale factor. For instance, eqs. (3.17) and (3.18) show the derivation of the flight velocity and mass of the derived aircraft, aircraft-2 from an existing aircraft, aircraft-1.
V2 = q Rg q RlV1 (3.17) m2 = RρR3lm1 (3.18)