4.2 Derivation of Model Scaling Factors

  4.2.1 Before discussing methods to determine the scaling laws, it is necessary to differentiate between fundamental dimensions and derived dimensions. Fundamental dimensions are the basic quantities upon which all other dimensions are based. In free-fall lifeboat studies, fundamental dimensions commonly used are length (L), mass (M), and time (T). The dimensions of all other parameters are derived from these fundamental dimensions. For example, the units of velocity are derived from length and time as LIT (meters per second, for instance).

  4.2.2 Scaling laws are commonly derived from the Buckingham Pi Theorem which states that "any complete physical relationship can be expressed in terms of a set of independent dimensionless products composed of the relevant physical parameters" (Baker, 1973). Bridgman (1931) expressed this concept mathematically in the following manner. If a system can be represented by some function:

where α, β, γ, et cetera are parameters of the system, then the same system can be represented by the function:

 In Equation 4.2 the term π_i are dimensionless products of the independent parameters in Equation 4.1. These dimensionless products are the scaling from which the ratio of full-scale parameters to model parameters can be determined. As will be seen later, there are fewer π-terms in Equation 4.2 than there are parameters in Equation 4.1.

  4.2.3 When conducting model studies to evaluate the behavior of free-fall lifeboats, adequate models are generally used. During such studies, the rigid body kinematic behavior of the lifeboat is of primary concern. As such, only those parameters that affect the kinematic behavior of the lifeboat during the launch need to be properly scaled. These parameters, which can be determined from the differential equations of motion of the lifeboat, are presented in Table 4.1. The units presented with each parameter are its dimensions in the MLT system of fundamental dimensions.

 Because the π_i terms in Equation 4.2 are products of the original parameters, and because these products are of zero dimension, an equation of dimensional homogeneity can be written as:

 The quantity on the left side of Equation 4.3 is said to be dimensionally equal to the quantity on the right side. As such, Equation 4.3 can be expressed in terms of its dimensions, namely:

 The exponents for mass, length, and time then can be equated which yields the three equations:

 Equations 4.5 and 4.7 can be solved for the a₂ and a₅, respectively. The resulting expression for a₅ can be substituted into Equation 4.6 which then can be solved for a₁. After these expressions for a₁, a₂, and a₅ are substituted into Equation 4.3 and like exponents are grouped, Equation 4.3 becomes:

 The terms in parentheses in Equation 4.8 are the π-terms in Equation 4.2, namely:

 It should be noted that π₆ is the inverse of Froude's Number squared and that π₂ is the inverse of Reynolds' Number for the lifeboat.

  4.2.4 These π-terms form the basis of the model laws for the system. The model laws, simply stated, require that the π-terms in the prototype are equal to the corresponding π-term in the model. From the complete set of model laws for the system, and the physical constraints resulting from the environment in which the tests are conducted, the scale factors can be determined. The physical constraints encountered are:

• 1. The tests are conducted on earth so that gravitational acceleration is the same for both the model and the prototype;

• 2. tests will be conducted in normal atmosphere so that the properties of air are the same in both the model and the prototype; and

• 3. The tests will be conducted in sea water so the properties of water will be the same for both the model and the prototype.

  4.2.5 In free-fall lifeboat models, it is desirable to achieve a condition of geometric similarity, i.e. the model and the full-scale lifeboat have similar shape and proportions. Such bodies of geometrically similar shape are commonly called geosims (Comstock, 1967). For this condition to be satisfied, there can be only one length scale factor. In this discussion, the length scale factor, K₈, is defined to be:

 Then, by equating π₁ for the model and prototype, and substituting the physical constraint that the model is tested in air and the length scale factor, it can be shown that the mass scale factor, K_m, is:

 The scale factor for the second moment of mass, K_I can be determined using π₇ and the length and mass scaling factors. The second moment of mass scales as:

 Similarly, the velocity scale factor, K_v can be found from π₆ to be:

 The length and velocity scale factors can be used with π₃ to show that acceleration is the same in the model and prototype. Using π₄ and π₅ for the model and prototype, the scaling factors for angular velocity and angular acceleration are found to be the same as those for velocity and acceleration, respectively. Lastly, the time scale factor, K_t , can be found using π₈to be:

  4.2.6 When using the scaling factors that have been developed thus far, the condition that Froude's Number in the model should be equal to that in the prototype has been satisfied. However, these same scale factors preclude the ability for Reynolds' Number in the model to be the same as that for the prototype. Using π₂ it can be seen that velocity should scale linearly with length for Reynolds' Number to be the same in the model and prototype. As such, it is not possible for Froude's Number and Reynolds' Number to be the sarne, respectively, in the model and prototype. It should be noted that if Reynolds' Number is to be equal in the model and prototype, the other scaling 'factors developed also would have to change accordingly.

  4.2.7 A question arises, then, as to whether Froude scaling or Reynolds' scaling should be used. The question can be answered by examining the launch phenomenon. The launch of a free-fall lifeboat is a gravity dominated event. This is true both during the free fall and during entry into the water. During the launch, the lifeboat begins at rest and accelerates until it impacts the water. The drag forces exerted. on the boat by the air during the free fall are small compared to gravitational forces. Likewise, the frictional drag forces exerted on the boat during water entry are small when compared with the inertia forces caused by impacting the water. That is to say, the viscosity of air and water have a significantly lesser effect on the behavior of the lifeboat than does gravity. Because the effect of gravity is considered in Froude's Number, and is not considered in Reynolds' Number, Froude scaling should be used in the preparation of free-fall lifeboat models.