Direct measurements of controlled aerodynamic forces on a wire-suspended axisymmetric body

Abstract

A novel in-line miniature force transducer is developed for direct measurements of the net aerodynamic forces and moments on a bluff body. The force transducers are integrated into each of the eight mounting wires that are utilized for suspension of an axisymmetric model in a wind tunnel having minimal wake interference. The aerodynamic forces and moments on the model are altered by induced active local attachment of the separated base flow. Fluidic control is effected by an array of four integrated aft-facing synthetic jet actuators that emanate from narrow, azimuthally segmented slots, equally distributed around the perimeter of the circular tail end. The jet orifices are embedded within a small backward-facing step that extends into a Coanda surface. The altered flow dynamics associated with both quasi-steady and transitory asymmetric activation of the flow control effect is characterized by direct force and PIV measurements.

Appendix: model’s natural frequencies

The axisymmetric body suspended in the hoop frame, as described in Sect. 2, can be modeled as a spring-mass system, having a single central mass supported by eight springs, when only a longitudinal vibration component of the support wires is considered. Consequently, such a simplified model does not incorporate support-wire tension as a parameter, since only transverse component of vibration depends on tension. Each of the mounting wires has a longitudinal and a transverse vibration component. If the transverse mode is secondary due to the dominant central mass, the longitudinal vibration is utilized to model each wire as a spring, and that vibration frequency is proportional to the cross-sectional area of the wire and inversely proportional to the attached mass and the wire length (Rao 1990)

$$ f \approx \sqrt {{\frac{A \times E}{m \times l}}} = \sqrt {\frac{k}{m}} , $$

(1)

where m is the mass, k is the equivalent spring constant, A is the cross-sectional wire area, l is the wire length, and E is Young Modulus of wire. In the present system, each wire is braided bronze that has a cross-sectional area of 1.11 × 10⁻⁶ m², a Young Modulus of 110 Gpa, and a length of 0.33 m. Therefore, the equivalent spring constant is 3.7 × 10⁵ N/m.

The suspended model has a configuration of

$$ \left( {X + x,Y + y,Z + z} \right), $$

(2)

under assumption x, y, z ≪ X, Y, Z, where X, Y, Z are the coordinates of the model about the cg, and x, y, z are perturbations in the three axial directions. The end points of the wires, connected to the hoop frame, are assumed to be rigid. During the small motions of suspended model, each of the springs (mounting wires) provides a restoring force (Rao 1990)

$$ \sum\limits_{1}^{8} {\bar{F}_{i} = m\bar{a}} = \sum\limits_{1}^{8} {k\delta_{i} } , $$

(3)

where $\bar{a} = \ddot{x}\,\bar{i} + \ddot{y}\,\bar{j} + \ddot{z}\,\bar{k},$ and $\delta_{i} = \delta_{i} \left( {x,y,z,X,Y,Z,l} \right),$ $\bar{F}$ is the restoring force, $\bar{a}$ is the model acceleration, m is the mass of the body, and δ is the wire displacement. All the wires are identical (k ₁ = k ₂ = k _i = k) and symmetrical around the central body, having the same angle with respect to the three coordinate axes. Decomposing the forces into the three coordinate axes using geometry yields the following equations,

$$ \begin{aligned} m\ddot{x} & = & 4k \times u_{1} \times \bar{i} \times \left[ {\left( {\sqrt {\left( {Y + y} \right)^{2} + \left( {X + x} \right)^{2} + \left( {Z + z} \right)^{2} } - l} \right) - \left( {\sqrt {\left( {Y - y} \right)^{2} + \left( {X - x} \right)^{2} + \left( {Z - z} \right)^{2} } - l} \right)} \right] \\ m\left( {\ddot{y} + \bar{g}} \right) & = & 4k \times u_{2} \times \bar{j} \times \left[ {\left( {\sqrt {\left( {Y + y} \right)^{2} + \left( {X + x} \right)^{2} + \left( {Z + z} \right)^{2} } - l} \right) - \left( {\sqrt {\left( {Y - y} \right)^{2} + \left( {X - x} \right)^{2} + \left( {Z - z} \right)^{2} } - l} \right)} \right], \\ m\ddot{z} & = & 4k \times u_{3} \times \bar{k} \times \left[ {\left( {\sqrt {\left( {Y + y} \right)^{2} + \left( {X + x} \right)^{2} + \left( {Z + z} \right)^{2} } - l} \right) - \left( {\sqrt {\left( {Y - y} \right)^{2} + \left( {X - x} \right)^{2} + \left( {Z - z} \right)^{2} } - l} \right)} \right] \\ \end{aligned} $$

(4)

where $l = \sqrt {X^{2} + Y^{2} + Z^{2} }$ and u ₁, u ₂, u ₃ are the components of the unit vector, u, along the length of each wire.

Applying a first-order Taylor expansion about (x ₀, y ₀, z ₀),

$$ f(x,y,z) \cong f(x_{0} ,y_{0} ,z_{0} ) + f_{x} (x_{0} ,y_{0} ,z_{0} ) \cdot (x - x_{0} ) + f_{y} (x_{0} ,y_{0} ,z_{0} ) \cdot (y - y_{0} ) + f_{z} (x_{0} ,y_{0} ,z_{0} ) \cdot (z - z_{0} ) $$

where $(x_{0} ,y_{0} ,z_{0} ) = (0,0,0)$, yields the following three equations:

$$ \begin{gathered} \ddot{x} = {\frac{{8 \times k \times u{}_{1}}}{m}}\left( {xu_{1} + yu_{2} + zu_{3} } \right)\bar{i} \hfill \\ \ddot{y} = {\frac{{8 \times k \times u_{2} }}{m}}\left( {xu_{1} + yu_{2} + zu_{3} } \right)\bar{j} - \bar{g}. \hfill \\ \ddot{z} = {\frac{{8 \times k \times u_{3} }}{m}}\left( {xu_{1} + yu_{2} + zu_{3} } \right)\bar{k} \hfill \\ \end{gathered} $$

(5)

The wires in the assembly of the model (Fig. 1) each have a unit vector of $\overline{u} = \pm 0.712\overline{i} + \pm 0.496\overline{j} + \pm 0.496\overline{k}$. Using Matlab, the above equations are solved numerically. The natural frequencies in the (x, y, z) directions for the model are (295, 205, and 205 Hz).