Computational fluid-structure interaction of a restrained ogive-cylindrical body with a blunt elliptical base at a high incidence

The flow around an inclined tangent ogive-cylindrical body of a finite length with a blunt elliptical base, placed in a wind tunnel and allowed to yaw in the x-y plane, is studied computationally. The flow is three-dimensional, compressible, and laminar with a Reynolds number of 30 000 based on the body diameter and a Mach number of 0.2. The study covers the range of angles of attack from 0° to 65°, both for fixed and yaw-restrained configurations where solutions are determined after removal of initial disturbances. The flow around the fixed body, for α > 52° and after disturbance removal, is nonstationary with large amplitude oscillations; for 0°< α < 52° the flow is almost steady with small nonstationary amplitude oscillations possibly due to the body's blunt end nonstationary wake. We identify a transition region between 45° < α < 52° where the attached wake behind the body end becomes detached. The resulting yaw response exhibits an intricate bifurcation structure that includes bistable periodic (finite amplitude) and nonstationary (small amplitude) limit cycles for moderate angles of attack, and nonstationary (finite amplitude) oscillations for high angles of attack. Investigation of the restraint torsion stiffness reveals a resonant periodic lock-in phenomena for low angles of attack which becomes quasiperiodic for a 40° angle of attack.

Figure 1

Body and coordinate system (a) and top-view (b) of a tangent 3.5-diameter forebody, a 6.5-diameter cylindrical afterbody and blunt elliptic base. The slender body is mounted on a hinge-mounted torsion spring placed at 5.2-diameter that restricts motion to yaw and prevents pitch and roll.

Figure 2

Computational grid (a) and zoom-in on the body's blunt end (b) and on the ogive forebody (c). The grid consisted of 200 equispaced circumferential planes extending completely around the body. In each circumferential plane, the grid contained 60 radial points between the body surface and the computational outer boundary and 104 axial points between the nose and the end of the body.

Figure 3

Time histories and power spectra of side-force coefficient $\mathit{Cy}$ of the flow around a fixed body at angles of attack $\alpha =$ 30°,40°,50°, and 60°, (a)–(d) respectively; and three disturbance locations: $4^{\circ }$ (blue), ${94}^{\circ }$ (red), and ${178}^{\circ }$ (orange) circumferentially from the windward plane of symmetry.

Figure 4

Time histories (left) and power spectra (right) of side-force coefficient $\mathit{Cy}$ for a fixed body after disturbance removal with angles of attack $\alpha =$ 30°, 40°, 50°, and 60°, (a)–(d) respectively.

Figure 5

Bifurcation diagrams for the averaged (a) and modulated RMS amplitude (b) side force coefficient as a function of AOA for a fixed body. Full circles (blue) – after disturbance removal, hollow triangles (orange) – with disturbance. We divided the diagram into three regions: A – steady solution, B – transitional and C - nonstationary.

Figure 6

Velocity magnitude contours of a fixed body after disturbance removal for (a) $\alpha =30{}^{\circ }$, (b) 40°, (c) 50° and (d) 60°.

Figure 7

Helicity density contours of a fixed body after disturbance removal for (a) $\alpha =$ 30°, (b) 40°, (c) 50°, (d) 60° and helicity cross section at $x/D=5.2$ for each angle (e)–(h), respectively.

Figure 8

Snapshots of helicity-density contours of a fixed body after disturbance removal for one period $\mathrm{T}=100$ [dimensionless time units] of major frequency (see Fig. 4) at axial location (I) $x/D=3.0$ and, (II) 7.0 and AOA $\alpha ={60}^{\circ }$, with equal time interval of T/4, (i)–(iv), respectively.

Figure 9

Side-force coefficients of the flow around $0.3D$ thick slices along the body.

Figure 10

Time histories (left) and power spectra (right) of yaw angle (ψ) of a restrained body after disturbance removal for (a) $\alpha =$ 10°, (b) 30° (high), (c) 30° (low), (d) 40°, (e) 50°, (f) 60°. Spring stiffness $k_{\psi }=0.1$. Note that cases (a), (b), and (d) are periodic, case (c) has low yaw angle amplitude, (e),(f) exhibit nonstationary behavior. The dashed line shows the frequency ∼0.037, which is the same for all high amplitude cases for regions I–IV.

Figure 11

Time histories (left) and power spectra (right) of side-force coefficient $\mathit{Cy}$ for a restrained body after disturbance removal for (a) $\alpha =$ 10°, (b) 30° (high), (c) 30° (low), (d) 40°, (e) 50°, (f) 60°. Spring stiffness $k_{\psi }=0.1$.

Figure 12

State-space diagrams and overlaid Poincare maps (red circles), defined by sampling the moment every positive zero-crossing, at (a) $\alpha =$ 30°, (b) 40°, (c) 45°, (d) 55°, (e) 60°, (f) 65°. Spring stiffness $k_{\psi }=0.1$. All cases here have a large amplitude yaw angle magnitude, where cases (a) and (b) show periodic behavior (regions II and III, Fig. 14), all the rest are nonstationary. Note that (c) and (d) are localized nontransit whereas (e),(f) are wide-spread transit Poincare maps.

Figure 13

State-space diagrams and overlaid Poincare maps (red circles), for (a) $\alpha = {24}^{\circ }$, (b) ${26}^{\circ }$, (c) ${30}^{\circ }$ and (d) ${34}^{\circ }$ of the lower branch of the region II of the bifurcation diagram (Fig. 14). The lower branch starting as nonstationary ($\alpha = {24}^{\circ }$) and ends with periodic like solution ($\alpha = {34}^{\circ }$). Spring stiffness $k_{\psi }=0.1$.

Figure 14

Bifurcation diagram of the yaw angle amplitude of a restrained body after disturbance removal. Spring stiffness $k_{\psi }=0.1$. The diagram divided to five regions: (I) large amplitude periodic oscillations at low angles of attack, (II) bistable/coexisting large and small amplitudes, (III) large amplitude periodic oscillations at moderate angles of attack, (IV) nonstationary transit oscillations, (V) nonstationary nontransit oscillations.

Figure 15

Bifurcation diagrams for the averaged (a) and modulated RMS amplitude (b) side force coefficient as a function of AOA for a yaw restrained body after disturbance removal. Spring stiffness $k_{\psi }=0.1$.

Figure 16

Helicity density and velocity magnitude contours (I and II, respectively) of a restrained yaw body after disturbance removal for (a) $\alpha = {10}^{\circ }$, (b) 30°, (c) 50°, (d) 60°.

Figure 17

Snapshots of helicity density contours for one period at axial locations $x/D=$ 3.0 and 7.0 (I and II, respectively) at AOA $\alpha = {30}^{\circ }$.

Figure 18

Snapshots of helicity density contours for one period at axial locations $x/D=$ 3.0 and 7.0 (I and II, respectively) at AOA $\alpha = {60}^{\circ }$.

Figure 19

Bifurcation diagrams for the yaw angle amplitude vs torsion spring stiffness coefficient, for AOA $\alpha = {40}^{\circ }$, starting with a soft spring $\left(k_{\psi }=0.002\right)$ to a hard spring $\left(k_{\psi }=0.4\right)$. Maximum yaw amplitude is obtained for a spring with $k_{\psi }=0.04$.

Figure 20

Bifurcation diagrams for the maximum yaw angle amplitude (a), and (b) normalized limit cycle oscillation frequency as a function of a normalized reduced velocity $\left(U^{*}\right)$, for $\alpha = {40}^{\circ }, {30}^{\circ }, {20}^{\circ }$ (blue, gray, and orange circles).

Figure 21

State-space diagrams and overlaid Poincare maps (red circles - left) with corresponding power spectra (right) for $\alpha =$ 30° with (a) $k_{\psi }\left(U^{*}\right)=0.07\left(4.7\right)$, and, (b), $k_{\psi }\left(U^{*}\right)=0.007\left(15.0\right)$.

Figure 22

State-space diagrams and overlaid Poincare maps (red circles) and corresponding power spectra for $\alpha =$ 40° with (a) $k_{\psi }\left(U^{*}\right)=0.04\left(6.28\right)$ and (b) $k_{\psi }\left(U^{*}\right)=0.005\left(17.8\right)$.