Description of the Alternative Model

As introduced above, the alternative motion model which separates the parachute and dropwindsonde body is expected to be more appropriate in describing the dropwindsonde motion since the wind tunnel tests showed that the variation of the drag coefficient with angle of attack is not negligible. According to Cockrell (1987), in this alternative model, the parachute and its payload are modelled as two rigid bodies. However, the wind tunnel test results showed that the parachute is able to adjust its orientation to provide a constant drag coefficient. As a result, it is not necessary to model the parachute as a rigid body but it can be simplified as an external deceleration force. Moreover, Cockrell (1987) described both two degrees of freedom motion, i.e. translations in a plane, and three degrees of freedom motion, i.e. adding the rotation to the planar motion. Using the three degrees of freedom motion model, the variation of the dropwindsonde aerodynamics with angle of attack can be taken into account since the rotation explicitly produces the

orientation of the dropwindsonde body. The sketch of this alternative motion model is shown in Fig. 3.23.

Figure 3.23: Sketch of the alternative model and forces acting on the model.

In this alternative motion model, the dropwindsonde body drag needs to be calculated based on the angle of attack, and can be expressed as,

FDx = 1 2ρACD(α)M(u− x˙) FDy = 1 2ρACD(α)M(v− y˙) FDz = 1 2ρACD(α)M(w− z˙) +mg (3.32)

where M is the magnitude of the relative wind vector, (u− x, v˙ − y, w˙ − z˙), and can be expressed as

M =p(u− x˙)2 _{+ (v− y˙)}2_{+ (w− z˙)}2 _(3.33)

In expressions (3.32) and (3.33), the drag coefficient CD(α) is consists of two parts: one

is provided by the dropwindsonde body CDbody(α) which depends on angles of attack,

α, and the other is provided by the parachute CDparachute and is constant regardless of

angle of attack. FD represents the drag force while the subscript indicates the force direction. A is the area of dropwindsonde body cross section. ( ˙x,y,˙ z˙) is a vector giving the velocity of the dropwindsonde at the position (x, y, z) while the vector (u, v, w) gives the velocity of the driving wind. Moreover, the upward direction is defined as positive in all equations, and therefore the gravity should take the value of − 9.8m/s2_.

In contrast to treating the dropwindsonde as a point object experiencing only drag, the alternative motion model allows an explicit expression for the lift, such as

FLx = 1 2ρACLbody(α)M(c(v− y˙)− b(w− z˙)) FLy = 1 2ρACLbody(α)M(− c(u− x˙) +a(w− z˙)) FLz = 1 2ρACLbody(α)M(b(u− x˙)− a(v− y˙)) +mg (3.34) All symbols have similar meanings as in expression (3.32) except the lift coefficient CL

reflects only the contribution made by the dropwindsonde body. ~q = (a, b, c) is a para- metric vector with unit magnitude, i.e. a2_+b2_+c2 _{= 1. This parametric vector is used to}

determine the direction of the lift. In reality, the total aerodynamic force experienced by the dropwindsonde is decomposed into three spatial components. However, if the force is described in the coordinate system defined by the dropwindsonde relative motion and the coordinate system follows the change of the dropwindsonde body orientation, the lift force calculated according to the lift coefficient measured in the wind tunnel test is actu- ally a combined force whose components are perpendicular to the relative motion vector of the dropwindsonde in two orthogonal directions. As a result, the aerodynamic force

experienced by the dropwindsonde needs only be decomposed into the drag and lift, and the parametric vector q, giving the direction of this combined lift force, is calculated as the cross product of the dropwindsonde relative motion vector,m~ = (u− x, v˙ − y, z˙ − z˙), and the orientation vector of the dropwindsonde body, ~p= (px, py, pz), as

~q = m~ ×~p

= (u− x, v˙ − y, w˙ − z˙)×(px, py, pz) (3.35)

In a component fashion, the parametric vector q reads,

a = pz(v− y˙)− py(w− z˙)

b = px(w− z˙)− pz(u− x˙)

c = px(u− x˙)− py(v− y˙) (3.36)

Considering the rigid body motion of the dropwindsonde body is driven by a combi- nation of aerodynamic forces and the weight force as m~a=vecFD+FL~ , it is determined as, mx¨ = 1 2ρAM(CD(α)(u− x˙) +CLbody(α)[c(v− y˙)− b(w− z˙)]) my¨ = 1 2ρAM(CD(α)(v− y˙) +CLbody(α)[− c(u− x˙) +a(w− z˙)]) mz¨ = 1 2ρAM(CD(α)(w− z˙) +CLbody(α)[b(u− x˙)− a(v− z˙)]) +mg (3.37) Equation (3.37) governs the translation of the dropwindsonde.

Considering that the rotation is driven by the torque, its governing equation can be expressed as,

Iβ¨=T(α) (3.38) whereIis the moment of inertia of the dropwindsonde body for rotating around its center,

β is the angle between the dropwindsonde body orientation and a reference direction, and ¨β represents the angular acceleration. One uncertainty in equation (3.38) is the moment of inertia value, I. Since the exact calculation of I is extremely difficult, a sensitivity analysis of the I value on the dropwindsonde motion simulation is performed. In this sensitivity analysis, a single dropwindsonde drop is simulated based on different moment of inertia values, including I1, which is calculated assuming the mass is evenly distributed in the cylinder defined by the outer geometry of the dropwindsonde body,

I2, which is calculated assuming the mass is concentrated in a slender cylinder with a

diameter only 10% of the value found in a real dropwindsonde body and I3, which is

calculated assuming the mass is concentrated in a tube with an outer diameter the same as the cylinder describing the real dropwindsonde body and a thickness of 10% of that diameter. Figure 3.24 shows the simulation results. It is obvious that the influence of using different values of the moment of inertia is negligible, and therefore the value I1 is used in the following simulations and discussions.

100 200 300 400 500 10 20 30 40 50 60 Height (m) Motion Velocity (m/s) Momet of Inertia Comparison I1

I2 I3

Figure 3.24: Comparison of the velocity profiles from different dropsonde drops simulated based on different momentum of inertia values, the meanings of I1, I2, I3 can be found in the text.

In summary, equations (3.37) and (3.38) govern the motion of the dropwindsonde, as a rigid body, and formulate the alternative dropwindsonde motion model.