Simulation of the unsteady vortical flow of freely falling plates

Abstract

An inviscid vortex shedding model is numerically extended to simulate falling flat plates. The body and vortices separated from the edge of the body are described by vortex sheets. The vortex shedding model has computational limitations when the angle of incidence is small and the free vortex sheet approaches the body closely. These problems are overcome by using numerical procedures such as a method for a near-singular integral and the suppression of vortex shedding at the plate edge. The model is applied to a falling plate of flow regimes of various Froude numbers. For $\text {Fr}=0.5$, the plate develops large-scale side-to-side oscillations. In the case of $\text {Fr}=1$, the plate motion is a combination of side-to-side oscillations and tumbling and is identified as a chaotic type. For $\text {Fr}=1.5$, the plate develops to autorotating motion. Comparisons with previous experimental results show good agreement for the falling pattern. The dependence of change in the vortex structure on the Froude number and its relation with the plate motion is also examined.

Graphical abstract

Discretization and time-integration

We discretize the free and bound vortex sheets by N lagrangian point vortices. We here assume that the free vortex sheets are separated from the leading and trailing edges and there is no free vortex sheet detached to the bound vortex sheet. The numerical method can be easily extended to multiple free vortex sheets.

Using the circulation $\Gamma$ as a Lagrangian variable, we denote the location of the free vortex sheet at current time $t^n$ by $z_j^n = z_{\pm } (\Gamma _j,t^n)$ for $1 \le j \le P-1$ and $Q+1 \le j \le N$. The Lagrangian indices of the leading and trailing edges of the bound vortex sheet are denoted by the integer P and Q, respectively. The bound vortex sheet is discretized by using Gauss-Lobatto collocation nodes [18], which are given by

$$\begin{aligned} s_j = \frac{1}{2} \left[ 1 - \cos \left( \frac{\pi (j-P)}{Q-P} \right) \right] , \qquad P \le j \le Q. \end{aligned}$$

(A1)

The edge circulation is expressed as $\Gamma _P^n = \Gamma _+ (t^n)$ and $\Gamma _Q^n = \Gamma _- (t^n)$.

We assume that $z_j^{n}$, $c^{n}$, $\theta ^{n}$, $\dot{c}^{n}$ and $\dot{\theta }^{n}$ are given at time $t^n$. At each time step, we first determine $\Gamma _P^n$, $\Gamma _Q^n$, and $\gamma _j^n$ by solving the linear Eq. (12) and (14), and Eq. (11), and calculate the velocity of the free vortex sheet from Eq. (5). In Eq. (6), the integration over the free vortex sheet is approximated by using the trapezoidal rule, while the integration for the bound vortex sheet is obtained by using the trapezoidal rule, with the numerical method of a near-singular integral if point vortices in the free vortex sheet are close to the plate.

From Eq. (23), $\dot{f}_j^n$ are the function of $\ddot{c}^n$, $\ddot{\theta }^n$, $\dot{\Gamma }_P^n$ and $\dot{\Gamma }_Q^n$, and in turn, $\dot{\gamma }_j^n$ and $\dot{\Gamma }_j^n$ are also functions of these quantities, from Eqs. (21) and (20). Therefore, Eqs. (15), (16), (22) and (24) are a system of linear equations for the unknowns $\ddot{c}^n$, $\ddot{\theta }^n$, $\dot{\Gamma }_P^n$ and $\dot{\Gamma }_Q^n$. Finding these quantities from the linear equation, we are ready for time-integration. We employ the classical fourth-order Runge–Kutta (RK) method for time-integration to obtain $z_j^{n+1}$, $c^{n+1}$, $\theta ^{n+1}$, $\dot{c}^{n+1}$ and $\dot{\theta }^{n+1}$.

To find the time derivatives of $\Gamma _P^n$ and $\Gamma _Q^n$ is the key step in the numerical method for this type of vortex-body interaction. Our method of finding them differs from previous studies that used the Chebyshev series [16], or an implicit scheme for solving a nonlinear system of equations [23, 27]. We use the formulation of the time derivative of the kinematic condition and determine $\dot{\Gamma }_P^n$ and $\dot{\Gamma }_Q^n$ by solving a system of linear equations.

At each time step, two new point vortices are released from both edges of the body. The free vortex sheet lacks resolution at late times, due to the nonuniform distribution of point vortices. To handle this problem, an adaptive point insertion procedure is applied to maintain the resolution of free vortex sheets [17, 18]. The third-order local polynomial interpolation is used to insert points whenever the distance between two consecutive points exceeds a given threshold.

The solution procedure for the equations requires the initial conditions at a time $t_1 > 0$. We use the self-similar solution of the free vortex sheet separated from a plate for the initial conditions [16]. Two starting point vortices with zero circulation $\Gamma _1 = \Gamma _N = 0$ are placed initially and their locations $z_1$ and $z_N$ are given by the small-time asymptotic expansion of the self-similar solution.

The numerical method is summarized as the following time-integration algorithm:

For given $z_j^{n}$, $c^{n}$, $\theta ^{n}$, $\dot{c}^{n}$ and $\dot{\theta }^{n}$,

• (1) At the first step of the RK method,

  • (i) determine $\Gamma _P^n$, $\Gamma _Q^n$, and $\gamma _j^n$.

  • (ii) compute the velocity of the free vortex sheet, $\dot{z}_j^n$.

  • (iii) find $\ddot{c}^n$, $\ddot{\theta }^n$, $\dot{\Gamma }_P^n$ and $\dot{\Gamma }_Q^n$.

  • (iv) compute $z_j^{n+1/2}$, $c^{n+1/2}$, $\theta ^{n+1/2}$, $\dot{c}^{n+1/2}$ and $\dot{\theta }^{n+1/2}$.

• (2) Repeat steps (i)-(iv) in (1) similarly to the other steps of the RK method.

• (3) Find $z_j^{n+1}$, $c^{n+1}$, $\theta ^{n+1}$, $\dot{c}^{n+1}$ and $\dot{\theta }^{n+1}$ by combining the intermediate values.

• (4) Find the location of new point vortices released from the plate edges.

• (5) A point insertion procedure is performed, when necessary.

To validate the numerical method of the present model, we applied it to the falling plate with the uniform $\delta$-regularization with $\delta =0.2$ and checked that the results of the present model agree well with Jones and Shelley [16].