Vortex dynamics of axisymmetric cones at high angles of attack

Abstract

Vortex asymmetry, dynamics, and breakdown in the wake of an axisymmetric cone have been investigated using direct numerical simulation for a wide range of angles of attack. The immersed boundary method is employed with pseudo-body-conformal grids to ensure the accuracy and resolution requirements near the body while being able to account for topology changes near the cone tip. The separated shear layer originated from the surface of the cone swirls into a strong primary vortex. Beneath the primary vortex on the leeward surface of the cone, a well-coherent counter-rotating secondary vorticity is generated. Beyond a particular threshold of swirl, the attached vortex structure breaks and the flow undergoes a chaotic transformation. Depending on the angle of attack, the flow shows different levels of instabilities and the topology of the vortices changes in the wake. In addition to swirl, spiral vortices that revolve around the primary vortex core often merge with the core and play a role in developing the double-helix mode of instability at the onset of the vortex breakdown. At the angle of attack of 60$^\circ$, the time-averaged side force becomes asymmetric at the stage where the drag overcomes the lift. At the angle of attack of 75$^\circ$, the primary vortex governs the flow asymmetry and the side force. Flow asymmetry is independent of the vortex breakdown. Finally, the contribution of primary vortices to the total forces is quantified using a force partitioning method.

Graphical abstract

Appendices

Appendix A Identification procedure of the primary and secondary vorticity

The selected region and the location of the core of the vortex have been evaluated from the average flow field data. The velocity gradient eigenmodes are found to be insufficient for identifying the vortex core once the vortex breakdown happens. First, an anisotropic Gaussian filter is applied to remove any small-scale structures. Based on the spatial characteristics of the small-scale vortical structures in the flow, the size and strength of the filter are determined [59]. This filtered field allows systematic identification of the primary vortex core. The selected points were obtained by using user-defined threshold values (changes along the axis of the primary vortex core) for helicity and vorticity. A procedure has also been applied for the identification of the zone for the secondary vorticity. If a point on the left of the domain has vorticity opposite the primary vorticity, the radial distance is less than the center of the primary vortex, and situated on the left side of the domain (i.e., $x<0$), that point is considered in the region of the left secondary vorticity. The zones that are identified by the algorithm discussed above is shown in Fig. 18.

Fig. 18

Zone identified for primary and secondary vorticity for different AoAs

Appendix B Partitioning of the surface force

According to Ref. [57, 58, 60], for fluid of volume (V) around an immersed fixed object with the surface ($S_0$), multiplying the Navier–Stokes equations by $\nabla \phi$ yields

$$\begin{aligned} - \int _{S_0} P {\varvec{n}} \cdot \nabla \phi \,\textrm{d}A = -\rho \int _V \left( {\varvec{u}} \times \varvec{\omega } \right) \cdot \nabla \phi \,\textrm{d}V + \mu \int _{S_0} \left( {\varvec{n}} \times \varvec{\omega } \right) \cdot \nabla \phi \,\textrm{d}A \end{aligned}$$

(B1)

where, ${\varvec{n}},\, \phi ,\, \varvec{\omega }$ and $\mu$ are the unit normal pointing outward on the surface ($S_0$), auxiliary potential, vorticity, and dynamic viscosity, respectively. Isolating the second term on the right-hand side, the vortex force on a stationary immersed body with a selected fluid volume V bounded by the surface S can be represented as

$$\begin{aligned} F_i = \int _{V} (\varvec{u} \times \varvec{\omega }) \cdot \nabla \phi _i\,\, \textrm{d}V \end{aligned}$$

(B2)

where $\phi _i$ is the auxiliary potential governed by the following equations and boundary conditions.

$$\begin{aligned} \nabla ^2\phi _i = 0 \end{aligned}$$

(B3)

and boundary conditions for $\phi$ is

$$\begin{aligned} \nabla \phi _i \cdot {\varvec{n}} = {\left\{ \begin{array}{ll} n_i &{} \quad \text {(on body surface }S)\\ 0 &{} \quad \text {(on outer boundary }\Sigma )\\ \end{array}\right. } \end{aligned}$$

(B4)

where $n_i$ is the i component of the unit normal pointing outward on the surface S, and $\Sigma$ is the far-field boundary.

Fig. 19

Auxiliary potential function ($\phi$) on the surface of a sphere a BEM and b analytical solution

1.1 Boundary element method (BEM) for solving auxiliary potential function

The governing Laplace’s equation for auxiliary potential function can be solved for any arbitrary surface by boundary element method (BEM). The Green’s functions of Laplace’s equation constitute a special class of harmonic functions that are singular at an arbitrary point $x_0$. By definition, Green’s function satisfies the singularly forced Laplace’s equation

$$\begin{aligned} \nabla ^2 G({\varvec{x}},{\varvec{x}}_0) + \delta ({\varvec{x}},{\varvec{x}}_0) = 0 \end{aligned}$$

(B5)

By applying Green’s second identity $\nabla \cdot (g\nabla \phi - \phi \nabla g) = 0$ for a non-singular harmonic function $\phi (x)$ and using a Green’s function $G({\varvec{x}},{\varvec{x}}_0)$ in place of g(x), and using the above definition, we obtain

$$\begin{aligned} \phi (x) \, \delta ({\varvec{x}},{\varvec{x}}_0) = \nabla \cdot \left[ \, G({\varvec{x}},{\varvec{x}}_0)\nabla \phi ({\varvec{x}}) - \phi ({\varvec{x}}) \nabla G({\varvec{x}},{\varvec{x}}_0) \, \right] = 0 \end{aligned}$$

(B6)

We can select a control volume V bounded by a closed surface or a collection of surfaces denoted by C. When the pole of Green’s function $x_0$ is placed outside V, the left-hand side of the above equation is non-singular throughout V. Integrating both sides of the equation over V, and using the divergence theorem, we find

$$\begin{aligned} \int _C \left[ \, G({\varvec{x}},{\varvec{x}}_0) \nabla \phi ({\varvec{x}}) - \phi ({\varvec{x}}) \nabla G({\varvec{x}},{\varvec{x}}_0) \, \right] \cdot {\varvec{n}} \, \textrm{d}S = 0 \end{aligned}$$

(B7)

where dS is the differential area on C. In contrast, when $x_0$ is placed inside V, the left-hand side exhibits a singularity at the point $x_0$. Using the distinctive properties of the delta function to perform the integration, we find

$$\begin{aligned} \phi ({\varvec{x}}_0) = -\int _C G({\varvec{x}},{\varvec{x}}_0)\left[ \, {\varvec{n}}\cdot \nabla \phi ({\varvec{x}})\right] \textrm{d}S + \int _C \phi ({\varvec{x}})\left[ \, {\varvec{n}}\cdot \nabla G({\varvec{x}},{\varvec{x}}_0)\right] \textrm{d}S \end{aligned}$$

(B8)

where the unit normal vector ${\varvec{n}}$ points into the control area enclosed by C. Above equation provides us with a boundary-integral representation of a harmonic function in terms of the boundary values and the boundary distribution of the normal derivative of the harmonic function. A similar approach can be made if the point $x_0$ approaches the surface C. To compute the value of $\phi (x)$ at a particular point $x_0$ located on a selected control area, we can evaluate the two boundary integrals on the right-hand side of the last equation and solve the first type kernel equation for $\phi (x_0)$.

Fig. 20

Comparison of auxiliary potential function ($\phi$) in the domain for BEM and analytical solution

1.1.1 Validation of BEM

To verify the solution convergence of the BEM method described above, the results obtained in the BEM method are compared with the analytical solutions (obtained by the equation below in the spherical coordinate system) to Laplace’s equation for potential flow over a sphere in the z-direction.

$$\begin{aligned} \phi = - \frac{U_\infty a^3\cos \theta }{2r^2}, r\ge a \end{aligned}$$

(B9)

where a is the radius of the sphere. Figure 19 shows both the analytical and BEM solution of the auxiliary function on the surface of the sphere.

The contour lines of the analytical solution and the BEM solution of $\phi$ inside the domain are shown in Fig. 20.