2.1. Governing equations

Consider a fluid flow around an elliptic cylinder with semi-axes $R_x$ and $R_y$ ($R_x \geqslant R_y$) performing high-frequency low-amplitude periodic oscillations along a straight line at an angle $\alpha$ to the major semi-axis (see figure 1) with velocity

(2.1a,b)\begin{equation} \boldsymbol{u}(t)=(u_x(t),u_y(t))= U_0 \tilde u(t) (\cos\alpha,\sin\alpha), \quad \tilde u (t) = \textrm{Re}\left(\sum_{k=1}^\infty\tilde u_k \,\textrm{e}^{\textrm{i} k \omega \tilde{t}}\right). \end{equation}

Here, $\omega$ determines the oscillation period $T = 2 {\rm \pi}/\omega$, whereas $U_{0}$ determines the characteristic amplitude of velocity oscillations. We choose $U_0$ so that the dimensionless velocities $\tilde {u}_k$ satisfy the condition

(2.2)\begin{equation} \sum_{k=1}^\infty k^{{-}1}\left|\tilde{u}_k\right|^2=1. \end{equation}

Figure 1.

Scheme of the motion.

It is convenient to study the flow in the moving elliptic coordinate system $(r, \theta )$ associated with the cylinder. The coordinates $r$ and $\theta$ are related to the Cartesian coordinates $x$ and $y$ as follows:

(2.3a,b)\begin{equation} x=\lambda \cosh r \cos \theta, \quad y=\lambda \sinh r \sin \theta. \end{equation}

Normalizing the spatial coordinates, time and velocity by $R_x$, $\omega ^{-1}$, $U_0$ respectively, we can write the dimensionless Navier–Stokes equation (assuming incompressible flow and neglecting body forces) in terms of streamfunction $\psi$ as

(2.4)\begin{gather} \frac{\partial}{\partial t}\Delta \psi + \frac{\gamma \varepsilon}{\lambda^2 h^2} \frac{\partial (\psi,\Delta \psi)}{\partial (r, \theta)} - \varepsilon ^2 \varDelta ^2 \psi =0, \end{gather}

(2.5a–c)\begin{gather} \frac{\partial \left(\psi , \Delta \psi \right)}{\partial \left(r,\theta \right)} = \left[\frac{\partial \psi }{\partial r}\frac{\partial \Delta \psi}{\partial \theta }- \frac{\partial \psi }{\partial \theta }\frac{\partial \Delta \psi}{\partial r}\right],\quad h=\sqrt{\cosh^2 r - \cos^2 \theta}, \quad \lambda=\sqrt{1-b^2}, \end{gather}

where $\varDelta$ is the Laplace operator. The dimensionless parameters of the problem are defined as follows:

(2.6a–c)\begin{equation} \varepsilon = \frac{d_{st}}{R_x}, \quad \gamma=\frac{U_0 d_{st}}{\nu}, \quad d_{st}=\sqrt{\frac{\nu}{\omega}}, \quad b=R_y/R_x, \end{equation}

where $\nu$ is the kinematic viscosity of the fluid. The dimensionless parameter $\varepsilon$ determines the ratio of the thickness of the Stokes unsteady boundary layer $d_{st}$ (see Landau & Lifshitz Reference Landau and Lifshitz1987) to the length of the semi-major axis. For the case of high-frequency oscillations $d_{st}$ is small so $\varepsilon \ll 1$. The dimensionless parameter $\gamma$ is the analogue of the Reynolds number based on the thickness of the boundary layer $d_{st}$. Further, we assume that it takes values of the order of unity.

In a moving coordinate system on the surface of the cylinder we set no-slip boundary conditions and at infinity we set the boundary condition corresponding to an oscillating flow

(2.7)\begin{gather} r=r_0=\ln\left(\frac{\sqrt{1+b}}{\sqrt{1-b}}\right):\ \psi =\frac{\partial \psi }{\partial r} =0, \end{gather}

(2.8a,b)\begin{gather} r\to \infty :\ \psi \sim \lambda \,\textrm{e}^r \sin (\theta-\alpha) \tilde u,\quad \tilde{u} = \textrm{Re}\left(\sum_{k=1}^\infty\tilde u_k \,\textrm{e}^{\textrm{i} k \tilde{t}}\right). \end{gather}

2.2. Force calculation

One of the most important flow characteristics is the hydrodynamic force exerted on the cylinder by the fluid (per unit length). We will calculate it through the characteristics of the flow at infinity (see, e.g. Noca, Shiels & Jeon Reference Noca, Shiels and Jeon1999). For the considered case, the formulas for calculation of the components of the force $\boldsymbol {F} = (F_x, F_y)$ can be represented in the following form:

(2.9)\begin{gather} F_{x,y} = L_{x,y}(\psi)+\varepsilon N_{x,y}(\psi,\psi) + \varepsilon^2 I_{x,y}(\psi), \end{gather}

(2.10)\begin{gather}L_x(\psi) = \lambda \frac{\partial }{\partial t} \int _{0}^{2{\rm \pi} }\left. \left(\frac{\partial \psi }{\partial r} \sinh r -\psi \cosh r\right)\right| _{r \to \infty } \sin \theta \,\textrm{d}\theta, \end{gather}

(2.11)\begin{gather}L_y(\psi) ={-}\lambda \frac{\partial }{\partial t} \int _{0}^{2{\rm \pi} } \left. \left(\frac{\partial \psi }{\partial r} \cosh r -\psi \sinh r\right)\right| _{r \to \infty } \cos \theta \,\textrm{d}\theta, \end{gather}

(2.12)\begin{align} N_x(\psi,\psi)&={-} \gamma \lambda \left. \int _{0}^{2{\rm \pi} }\left(\frac{\partial \psi}{\partial \theta } \sinh r \Delta \psi \right)\right|_{r \to \infty } \sin \theta \,\textrm{d}\theta \nonumber\\ &\quad - \gamma \int _{0}^{2{\rm \pi} }\left(\frac{1}{2} \frac{\sinh r \cos \theta} {\lambda h^2} \left(\left(\frac{\partial \psi }{\partial r} \frac{\partial \psi }{\partial r}\right) - \left(\frac{\partial \psi }{\partial \theta } \frac{\partial \psi }{\partial \theta } \right) \right)\right.\nonumber\\ &\quad \left.\left. -\frac{\sin \theta \cosh r}{\lambda h^2} \frac{\partial \psi}{\partial \theta } \frac{\partial \psi}{\partial r}\right)\right|_{r \to \infty } \, \textrm{d}\theta, \end{align}

(2.13)\begin{align} N_y(\psi,\psi)&= \gamma \lambda \left. \int _{0}^{2{\rm \pi} }\left(\frac{\partial \psi}{\partial \theta } \cosh r \Delta \psi \right)\right|_{r \to \infty } \cos \theta \,\textrm{d}\theta \nonumber\\ &\quad -\gamma \int _{0}^{2{\rm \pi} }\left(\frac{1}{2} \frac{\cosh r \sin \theta} {\lambda h^2} \left(\left(\frac{\partial \psi}{\partial r} \frac{\partial \psi}{\partial r}\right) - \left(\frac{\partial \psi}{\partial \theta } \frac{\partial \psi}{\partial \theta } \right) \right)\right.\nonumber\\ &\quad \left.\left. +\frac{\cos \theta \sinh r}{\lambda h^2} \frac{\partial \psi}{\partial \theta } \frac{\partial \psi}{\partial r}\right)\right|_{r \to \infty } \,\textrm{d}\theta. \end{align}

The last term in (2.9) vanishes at infinity.

The force $\boldsymbol {F}$ acting on the cylinder in the oscillatory flow differs from the force $\boldsymbol {F}^*$ acting on the oscillating cylinder in the still fluid by the Froude–Krylov force $\boldsymbol {F}^f$

(2.14)\begin{equation} \boldsymbol{F} = \boldsymbol{F}^* + \boldsymbol{F}^f. \end{equation}

For the present case, $\boldsymbol {F}^f=(F^f_x,F^f_y)$ can be calculated as

(2.15)\begin{gather} F^f_x=\frac{\partial \tilde u}{\partial t} \cos \alpha \int _{0}^{2{\rm \pi}} \cos^2 \theta \sqrt{\frac{1- \lambda^2\cos^2 \theta}{1+\lambda^2\cos^2 \theta}} \,\textrm{d}\theta, \end{gather}

(2.16)\begin{gather}F^f_y= b^2 \frac{\partial \tilde u}{\partial t} \sin \alpha \int _{0}^{2{\rm \pi}} \sin^2 \theta \sqrt{\frac{1- \lambda^2\cos^2 \theta}{1+\lambda^2\cos^2 \theta}} \,\textrm{d}\theta. \end{gather}

Note that, in accordance with the normalization used, all the determined components of the force are non-dimensionalized by $F_0=\rho R_x^2 U_0 \omega$.

3.1. Asymptotic expansions

The solution of the problem (2.4)–(2.8a,b) consists of oscillatory $\tilde {\psi }$ and steady $\bar {\psi }$ parts

(3.1)\begin{equation} \psi =\tilde{\psi } +\bar{\psi }. \end{equation}

We separate equation (2.4) into a pair of equations describing oscillatory and stationary flows

(3.2)\begin{gather} \frac{\partial }{\partial t}\Delta \tilde{\psi } + \frac{\gamma \varepsilon}{\lambda^2 h^2}\left[\frac{\partial \left(\bar{\psi },\Delta \tilde{\psi }\right)}{\partial \left(r,\theta \right)} +\frac{\partial \left(\tilde{\psi },\Delta \bar{\psi }\right)}{\partial \left(r,\theta \right)}\right] +\frac{\gamma \varepsilon}{\lambda^2 h^2}{\left[\frac{\partial \left(\tilde{\psi },\Delta \tilde{\psi }\right)}{\partial \left(r,\theta \right)}\right]}_u ={\varepsilon }^{{2}}{\varDelta }^{{2}}\tilde{\psi }, \end{gather}

(3.3)\begin{gather}\frac{\gamma \varepsilon}{\lambda^2 h^2}\left[\frac{\partial \left(\bar{\psi },\Delta \bar{\psi }\right)}{\partial \left(r,\theta \right)}\right] +\frac{\gamma \varepsilon}{\lambda^2 h^2}{\left[\frac{\partial \left(\tilde{\psi },\Delta \tilde{\psi }\right)}{\partial \left(r,\theta \right)}\right]}_s ={\varepsilon }^{{2}}{\varDelta }^{{2}}\bar{\psi }. \end{gather}

Here, subscripts ${u}$ and ${s}$ denote the oscillatory and the steady parts of the expression, respectively. To solve the problem we use the method of matched asymptotic expansions.

As is easy to see, the solution of the problem in the leading term ($\varepsilon ^0$) is a harmonic function that determines the potential flow around the cylinder. Such a solution satisfies only one boundary condition on the body. From this we get that there must be a boundary layer near the cylinder. So we represent the oscillatory and steady components in the form of decompositions in the external and internal (boundary-layer) regions.

In the external region we expand the streamfunction as a power series in parameter $\varepsilon$

(3.4)\begin{equation} \left.\begin{gathered} \tilde{\psi } ={\tilde{\varPsi }}_0\left(r,\theta ,t\right)+ \varepsilon {\tilde{\varPsi }}_{{1}}\left(r,\theta ,t\right) + {\varepsilon }^{{2}}{\tilde{\varPsi }}_{{2}}\left(r,\theta ,t\right) +\cdots , \\ \bar{\psi } =\varepsilon {\bar{\varPsi }}_{{1}}\left(r,\theta \right)+{\varepsilon }^{{2}} {\bar{\varPsi }}_{{2}}\left(r,\theta \right)+\cdots. \end{gathered}\right\} \end{equation}

The expansion in $\varepsilon$ is effectively an expansion in a dimensionless frequency; the effects of finite amplitude are represented by the parameter $\gamma$.

Substituting (3.4) into (3.2) and collecting terms involving the same powers of $\varepsilon$, we get the following sub-problems for the first three unsteady terms:

(3.5)\begin{gather} \Delta {\tilde{\varPsi }}_j =0, \quad j=0,1. \end{gather}

(3.6)\begin{gather}\frac{\partial }{\partial t}\Delta {\tilde{\varPsi }}_{{2}} +\frac{\gamma \varepsilon}{\lambda^2 h^2}\left[\frac{\partial \left({\tilde{\varPsi }}_0,\Delta {\bar{\varPsi }}_{{ 1}}\right)}{\partial \left(r,\theta \right)}\right]=0. \end{gather}

Internal expansion is constructed in the unsteady boundary layer $r-r_0 \sim \varepsilon$. Introducing the boundary-layer coordinate

(3.7)\begin{equation} \eta =\frac{r-r_0}{\varepsilon } \end{equation}

we can write the formal power series expansion of the boundary-layer solution in the form

(3.8)\begin{equation} \left.\begin{gathered} \tilde{\psi } =\varepsilon {\tilde{\psi }}_0\left(\eta ,\theta ,t\right) + {\varepsilon }^{{2}}{\tilde{\psi }}_{{1}}\left(\eta ,\theta ,t\right) + {\varepsilon }^{{3}}{\tilde{\psi }}_{{2}}\left(\eta ,\theta ,t\right) +\cdots ,\\ \bar{\psi } ={\varepsilon }^{{2}}{\bar{\psi }}_{{1}}\left(\eta ,\theta \right)+{\varepsilon }^{{3}} {\bar{\psi }}_{{2}}\left(\eta ,\theta \right) +\cdots. \end{gathered}\right\} \end{equation}

We rewrite (3.2) in the boundary-layer coordinate system $(\eta ,\theta )$, taking into account that

(3.9)\begin{equation} \left.\begin{gathered} \lambda^{{-}2} h^{{-}2} = a_1-a_2\varepsilon+\cdots=h_0^{{-}2}-\lambda^{2}h_0^{{-}4}{2\eta \cosh r_0 \sinh r_0}\varepsilon+\cdots, \\ h_0=\lambda\sqrt{(\cosh^2 r_0-\cos^2 \theta)}. \end{gathered}\right\} \end{equation}

Substituting (3.8) into the resulting equation and collecting terms involving the same powers of $\varepsilon$, we get a number of boundary-layer sub-problems. For the first two unsteady terms we have

(3.10)\begin{gather} \frac{\partial}{\partial t}\frac{\partial^2 \tilde{\psi}_0 }{\partial {\eta }^2} -a_1\dfrac{\partial ^4 \tilde{\psi}_0 }{\partial {\eta }^4}=0, \end{gather}

(3.11)\begin{gather}\frac{\partial}{\partial t}\frac{\partial^2 \tilde{\psi}_1 }{\partial {\eta }^2} -a_1\frac{\partial ^4 \tilde{\psi}_1 }{\partial {\eta }^4}={-}\gamma {\left[\frac{\partial \left({\tilde{\psi }}_0,a_1\dfrac{\partial^2 \tilde{\psi}_0 }{\partial {\eta }^2}\right)}{\partial \left(\eta ,\theta \right)}\right]}_u- \frac{\partial}{\partial {\eta }^2}a_2\frac{\partial^2 \tilde{\psi}_0 }{\partial {\eta }^2}. \end{gather}

The priority in solving the sub-problems is determined by the matching conditions. For functions $\tilde {\psi }_{j}$ they can be written as follows:

(3.12)\begin{equation} \left.\begin{gathered} \left. \tilde{\psi }_{0} \right|_{\eta \to \infty } \sim \left. \tilde{\varPsi }_{1} \right|_{r \to r_0} +\left. \eta \frac{\partial \tilde{\varPsi }_{0} }{\partial r} \right|_{r \to r_0},\\ \left. \tilde{\psi }_{1} \right|_{\eta \to \infty } \sim \left. \tilde{\varPsi }_{2} \right|_{r \to r_0} +\left. \eta \frac{\partial \tilde{\varPsi }_{1} }{\partial r} \right|_{r \to r_0} \left. +\frac{\eta ^{2} }{2} \frac{\partial ^{2} \tilde{\varPsi }_{0} }{\partial r^{2} } \right|_{r \to r_0}. \end{gathered}\right\} \end{equation}

The resulting asymptotic expansion scheme is shown in figure 2.

Figure 2.

Asymptotic expansion scheme.

Hydrodynamic forces acting on a cylinder can be represented as a power series in parameter $\varepsilon$

(3.13)\begin{equation} \boldsymbol{F} =\boldsymbol{f}_0 +\varepsilon \boldsymbol{f}_{{1}} +{\varepsilon }^{{2}} \boldsymbol{f}_{{2}}+\cdots, \end{equation}

where coefficients $\boldsymbol {f}_j$ are calculated by the formulas

(3.14a–c)\begin{equation} \left.\begin{gathered} \boldsymbol{f}_{0} = \boldsymbol{L}\left({\tilde{\varPsi }}_0\right),\quad \boldsymbol{f}_{1} = \boldsymbol{L}\left({\tilde{\varPsi }}_1\right) + \boldsymbol{N} \left({\tilde{\varPsi }}_0,{\tilde{\varPsi }}_0\right) \\ \boldsymbol{f}_{2} = \boldsymbol{L}\left({\tilde{\varPsi }}_2\right) \!+ \boldsymbol{N}\left({\tilde{\varPsi }}_1,{\tilde{\varPsi }}_0\right)\!+ \boldsymbol{N}\left({\tilde{\varPsi }}_0,{\tilde{\varPsi }}_1\right)\!+ \boldsymbol{N}\left({\bar{\varPsi }}_1,{\tilde{\varPsi}}_0\right)+ \boldsymbol{N}\left({\tilde{\varPsi }}_0,{\bar{\varPsi}}_1\right)+\boldsymbol{I}({\tilde{\varPsi}}_0). \end{gathered}\right\} \end{equation}

3.2. Determination of ${\tilde {{ \varPsi }}}_0$

The external solution (3.15a,b) in the leading term is determined from the equation (3.5) with the condition (2.8a,b) at infinity and the no-penetration condition for $r = r_0$ (see, e.g. Milne-Thomson Reference Milne-Thomson1960)

(3.15a,b)\begin{equation} {\tilde{\varPsi }}_0 = \tilde{u}(t) S(r,\theta), \quad S(r,\theta) = \lambda\,\textrm{e}^{r_0} \sinh(r-r_0) \sin (\theta-\alpha).\end{equation}

It describes the potential flow of an ideal fluid around a cylinder.

The force component $\boldsymbol {f}_0 =(\,f_{x,0},\,f_{y,0})$ corresponding to the leading term

(3.16)\begin{gather} f_{x,0}= \frac{\textrm{d}\tilde{u}(t)}{\textrm{d}t} \lambda^2 {\rm \pi}\,\textrm{e}^{r_0} \sinh r_0 \cos \alpha = \frac{\textrm{d}\tilde{u}(t)}{\textrm{d}t} b(1+b) {\rm \pi}\cos \alpha, \end{gather}

(3.17)\begin{gather}f_{y,0}= \frac{\textrm{d}\tilde{u}(t)}{\textrm{d}t} \lambda^2 {\rm \pi}\,\textrm{e}^{r_0} \cosh r_0 \sin \alpha = \frac{\textrm{d}\tilde{u}(t)}{\textrm{d}t} (1+b) {\rm \pi}\sin \alpha \end{gather}

describes added mass force.

3.3. Determination of ${\tilde {\psi }}_0$

The solution of the internal sub-problem (3.10) (for determination of ${\tilde { \psi }}_0$) with boundary conditions

(3.18a–c)\begin{equation} {\left.{\tilde{\psi }}_0\right|}_{\eta =0} =0,\quad {\left.\frac{\partial {\tilde{\psi }}_0}{\partial \eta }\right|}_{\eta =0} =0,\quad {\left.{\tilde{\psi }}_0\right|}_{\eta \to \infty }\sim \left. \eta \frac{\partial \tilde{\varPsi }_{0} }{\partial r} \right|_{r \to r_0} = \eta \tilde{u}(t) \lambda \,\textrm{e}^{r_0} \sin (\theta-\alpha), \end{equation}

can be written as

(3.19)\begin{equation} {\tilde{\psi }}_0 = \lambda \,\textrm{e}^{r_0} \frac{\sin (\theta-\alpha)}{h_0} \,\textrm{Re} \left(\sum_{k=1}^\infty \tilde{u}_k \,\textrm{e}^{\textrm{i}kt}\left(\eta h_0 +\frac{1}{\sqrt{\textrm{i}k}}\left(\exp({-\sqrt{\textrm{i}k}\eta h_0})-1\right)\right)\right). \end{equation}

For $\eta \to \infty$ we get

(3.20)\begin{equation} \eta\to\infty: \ {\tilde{\psi }}_0 =\eta \tilde{u}(t) \lambda \,\textrm{e}^{r_0} \sin (\theta-\alpha)-\lambda \,\textrm{e}^{r_0} \frac{\sin (\theta-\alpha)}{h_0} \frac{\textrm{d}^{1/2}\tilde{x}}{\textrm{d}t^{1/2}} + \exp. \end{equation}

Here and below, ‘exp’ denotes exponentially small terms. The fractional derivative of the function $\tilde {x}$ describing the movement of the cylinder $(\tilde {u} = \textrm {d}\tilde {x}/\textrm {d}t)$ is defined as

(3.21)\begin{equation} \frac{\textrm{d}^{1/2}\tilde{x}}{\textrm{d}t^{1/2}} = \textrm{Re} \left(\sum_{k=1}^\infty {\tilde{x}_k} \sqrt{\textrm{i}k}\,\textrm{e}^{\textrm{i}kt}\right) = \textrm{Re} \left(\sum_{k=1}^\infty \frac{\tilde{u}_k}{\sqrt{\textrm{i}k}} \,\textrm{e}^{\textrm{i}kt}\right). \end{equation}

3.4. Determination of ${\tilde {\varPsi }}_{1}$

We build the solution ${\tilde {\varPsi }}_{1}$ as a harmonic function satisfying equation (3.5) with the boundary conditions

(3.22a,b)\begin{equation} r=1:\ {\tilde{\varPsi }}_{{1}}={-}\lambda \,\textrm{e}^{r_0} \frac{\sin (\theta-\alpha)}{h_0(\theta)}\frac{\textrm{d}^{1/2}\tilde{x}}{\textrm{d}t^{1/2}}; \quad r\to \infty :\ \tilde{\varPsi}_{{1}} =O (1). \end{equation}

The first of them is the matching condition following from (3.12), the second is the boundary condition at infinity. The solution to the corresponding problem has the form

(3.23)\begin{gather} \tilde{\varPsi }_{1} ={-} \lambda \,\textrm{e}^{r_0} \frac{\textrm{d}^{1/2}\tilde{x}}{\textrm{d}t^{1/2}}\left(\sum_{j=1}^\infty\left(A_j\cos \alpha \sin (\kern1.5pt j\theta) - B_j\sin \alpha \cos (\kern1.5pt j\theta) \right) \textrm{e}^{{-}j(r-r_0)} \right), \end{gather}

(3.24a,b)\begin{gather} A_j = \frac{1}{\rm \pi}\int_{-{\rm \pi}}^{\rm \pi}\frac{\sin \theta \sin (\kern1.5pt j\theta)}{\lambda \sqrt{\cosh^2 r_0-\cos^2 \theta}}\,\textrm{d}\theta, \quad B_j = \frac{1}{\rm \pi}\int_{-{\rm \pi}}^{\rm \pi}\frac{\cos \theta \cos (\kern1.5pt j\theta)}{\lambda \sqrt{\cosh^2 r_0-\cos^2 \theta}}\,\textrm{d}\theta. \end{gather}

This solution allows us to calculate the next component of the force using formula (3.14a–c). Since $N_ {x, y} ({\tilde {\varPsi }}_0, {\tilde {\varPsi }}_0 ) = 0$, we have

(3.25)\begin{gather} f_{x,1} = L_x({\tilde{\varPsi }}_1) = \lambda^2\,\textrm{e}^{2r_0} {\rm \pi}\cos \alpha A_1 \frac{\textrm{d}}{\textrm{d}t} \frac{\textrm{d}^{1/2}\tilde{x}}{\textrm{d}t^{1/2}} = (1+b)^2 {\rm \pi}\cos \alpha A_1 \frac{\textrm{d}^{1/2}\tilde{u}}{\textrm{d}t^{1/2}}\nonumber\\ \hspace{-5.5pc}= (1+b)^2 \sqrt{\rm \pi} \cos \alpha A_1 \int_{-\infty}^{t}\frac{\textrm{d}\tilde{u}}{\textrm{d}\tau} \frac{\textrm{d}\tau}{\sqrt{t-\tau}}, \end{gather}

(3.26)\begin{gather} f_{y,1} = (1+b)^2 \sqrt{\rm \pi} \sin \alpha B_1 \int_{-\infty}^{t}\frac{\textrm{d}\tilde{u}}{\textrm{d}\tau} \frac{\textrm{d}\tau}{\sqrt{t-\tau}}. \end{gather}

Thus, $\boldsymbol {f}_{1}$ represents the history force (Basset force) in its classical form.

The integrals $A_1, B_1$ can be calculated analytically in two limiting cases:

1. (i) for the case of a circular cylinder ($b = 1$)

   (3.27)\begin{equation} A_1 = B_1 = 1; \end{equation}

2. (ii) for the case of an infinite thin plate ($b = 0$)

   (3.28)\begin{equation} A_1 = 4 / {\rm \pi}\end{equation}
   and the integral $B_1$ diverges, i.e. it is possible to estimate the hydrodynamic force only for the angles of attack $\alpha = 0, {\rm \pi}$.

For the remaining cases, the values of the integrals can be found numerically. The numerical data in the range $0.05<b<1$ are well approximated by the following formulas:

(3.29a,b)\begin{equation} A_1\approx \frac{3.811b+0.8749}{b^2+3b+0.6874}, \quad B_1\approx \frac{1.357b+0.1067}{b^2+0.4472b+0.01562}. \end{equation}

The maximum difference between these approximations and the exact values of the integrals does not exceed 0.04 %.

3.5. Determination of ${\tilde {\psi }}_{1}$

The component ${\tilde {\psi }}_{1}$ can be found from the solution of the problem (3.11) with boundary conditions

(3.30)\begin{equation} \left.\begin{gathered}{\left.{\tilde{\psi }}_{{1}}\right|}_{\eta =0}={\left.\frac{\partial {\tilde{\psi }}_{{1}}}{\partial \eta }\right|}_{\eta =0}=0,\\ {\left.{\tilde{\psi }}_{{1}}\right|}_{\eta \to \infty }\sim\left. \eta \frac{\partial \tilde{\varPsi }_{1} }{\partial r} \right|_{r \to r_0} + \left.\frac{\eta ^{2} }{2} \frac{\partial ^{2} \tilde{\varPsi }_{0} }{\partial r^{2} } \right|_{r \to r_0}\\ =\eta \lambda \,\textrm{e}^{r_0} \frac{\textrm{d}^{1/2}\tilde{x}}{\textrm{d}t^{1/2}}\left(\sum_{j=1}^\infty j \left(A_j\cos \alpha \sin (\kern1.5pt j\theta) - B_j\sin \alpha \cos (\kern1.5pt j\theta) \right) \right). \end{gathered}\right\} \end{equation}

The last condition here is a matching condition (see (3.12)).

Due to the linearity of the problem (3.11), (3.30), its solution can be represented as the sum of two terms

(3.31)\begin{equation} {\tilde{\psi }}_{{1}} ={\tilde{\psi }}^{(1)}_{{1}} +{\tilde{\psi }}^{(2)}_{{ 1}}. \end{equation}

The first term is the solution of (3.11) with the first term (proportional to $\gamma$) on the right-hand side and zero boundary conditions. The second term is the solution of (3.11) with the second term on the right-hand side and boundary conditions (3.30).

For ${\tilde {\psi }}^{(2)}_{{1}}$ we get

(3.32)\begin{align} {\tilde{\psi }}^{{(2)}}_{{1}} &= \textrm{Re} \left( \sum_{k=1}^\infty \tilde{u}_k k^{{-}1} \,\textrm{e}^{\textrm{i}kt} \tilde{g}_k \left(\eta, \theta \right)\right), \end{align}

(3.33)\begin{align} \tilde{g}_k(\eta,\theta)&=\lambda \,\textrm{e}^{r_0}\lambda^{2} \cosh r_0 \sinh r_0 \frac{\sin(\theta-\alpha)}{2 h_0^4}\nonumber\\ &\quad \times \left(\left(\textrm{i}-\frac{\eta h_0 \sqrt{k}}{\sqrt{i}}-(\eta h_0 \sqrt{k})^2 \right)\exp({-\sqrt{\textrm{i}k}\eta h_0}) - \textrm{i}\right)\nonumber\\ &\quad +\lambda \,\textrm{e}^{r_0} h_0^{{-}1} \left(\sum_{j=1}^\infty j\left(A_j\cos \alpha \sin (\kern1.5pt j\theta) - B_j\sin \alpha \cos (\kern1.5pt j\theta) \right) \right)\nonumber\\ &\quad \times\left(\textrm{i}+\frac{\eta h_0 \sqrt{k}}{\sqrt{\textrm{i}}}-\textrm{i}\exp({-\sqrt{\textrm{i}k}\eta h_0}) \right). \end{align}

For ${\tilde {\psi }}^{(1)}_{\textrm {1}}$ we obtain the following formula:

(3.34)\begin{equation} {\tilde{\psi }}^{{(1)}}_{{1}} =\gamma \frac{1}{h_0}\frac{\partial}{\partial \theta}\left(\frac{ \sin(\theta- \alpha)}{h_0}\right) \frac{\sin(\theta- \alpha)}{h_0} \cdot \sum_{k=1}^\infty \textrm{Re}\left\{\tilde{p}_k(\eta h_0)\,\textrm{e}^{\textrm{i}kt}\right\}. \end{equation}

We will not present here relations for the calculation of the complex-valued functions $\tilde {p}_k (\eta )$. They are very cumbersome and uninformative, since $\tilde {p}_k$ contains an infinite number of functions of $\eta$ proportional to the product of the amplitudes of the time harmonics $\tilde {u}_n, \tilde {u}_m$, $n \pm m = k$. It should be noted that all functions $\tilde {p}_k (h_0 \eta )$ and their first derivatives vanish at $\eta = 0$ and reach (exponentially) a constant value $\tilde {p}^0_k$ at infinity.

The behaviour of inner solution ${\tilde {\psi }}_1$ at infinity gives the following boundary condition for ${\tilde {\varPsi }}_{{2}}$:

(3.35)\begin{align} r =r_0:\ {\tilde{\varPsi }}_2&={-}\tilde{g}^0(\theta) \tilde{x} + \gamma \frac{1}{h_0}\frac{\partial}{\partial \theta}\left(\frac{\sin(\theta- \alpha)}{h_0}\right) \frac{\sin(\theta- \alpha)}{h_0} \sum_{k=1}^\infty \textrm{Re}\left\{\tilde{p}^0_k \,\textrm{e}^{\textrm{i}kt}\right\}, \end{align}

(3.36)\begin{align} \tilde{g}^0(\theta)&={-}\lambda \,\textrm{e}^{r_0}\lambda^{2} \cosh r_0 \sinh r_0\frac{\sin(\theta-\alpha)}{2 h_0^4}\nonumber\\ &\quad +\lambda \,\textrm{e}^{r_0} h_0^{{-}1} \left(\sum_{j=1}^\infty j\left(A_j\cos \alpha \sin (\kern1.5pt j\theta) - B_j\sin \alpha \cos (\kern1.5pt j\theta) \right) \right). \end{align}

3.6. Determination of $f_2$

To determine the third component of the force $f_2$ in the general case according to formula (3.14a–c) one needs to solve the sub-problems for $\bar {\varPsi }_{1}, \tilde {\varPsi }_{2}$. The sub-problem for $\bar {\varPsi }_{1}$ is strongly nonlinear, so its solution (for $\gamma >0$) can be built only numerically. At the same time, using the method proposed by Nuriev & Egorov (Reference Nuriev and Egorov2019), it is possible to determine the force $f_2$ in projection onto the axis of oscillation (in-line component of force) using only the value of ${\tilde {\varPsi }}_{{2}}$ at the cylinder boundary that was calculated in the previous section (see (3.35)). To obtain the corresponding formula, we multiply the equation by the function $S$, defined according to (3.15a,b), and integrate the result over the flow region. Finally, we get

(3.37)\begin{align} &\lambda \frac{\partial }{\partial t} \int _{0}^{2{\rm \pi} }\left.\left(\frac{\partial\tilde{\varPsi }_{2}}{\partial r} \,\textrm{e}^r -\tilde{\varPsi }_{2} \,\textrm{e}^r \right)\right| _{r \to \infty }\sin(\theta-\alpha) \,\textrm{d}\theta\nonumber\\ &\qquad -\gamma \tilde{u}(t) \frac{\textrm{e}^{2r_0}}{h^2}\int _{0}^{2{\rm \pi} }\left. \left(\Delta \bar{\varPsi }_{1} \sinh^2(r-r_0) \right) \right|_{r \to \infty} \sin(\theta-\alpha) \cos(\theta-\alpha) \,\textrm{d}\theta\nonumber\\ &\quad ={-}\lambda \,\textrm{e}^{r_0} \frac{\partial }{\partial t}\int _{0}^{2{\rm \pi} }\left. \tilde{\varPsi }_{2} \right|_{r \to r_0 } \sin(\theta-\alpha) \,\textrm{d}\theta. \end{align}

As can be seen, the left side of (3.37) is equal to the projection of the force $f_2$ vector on the oscillation axis

(3.38)\begin{align} f_{\alpha,2}&=\,f_{x,2}\cos \alpha+\,f_{y,2}\sin \alpha ={-}\lambda \,\textrm{e}^{r_0} \frac{\partial }{\partial t}\int _{0}^{2{\rm \pi} }\left. \tilde{\varPsi }_{2} \right|_{r \to r_0 } \sin(\theta-\alpha) \,\textrm{d}\theta \nonumber\\ &= \cos \alpha \lambda \,\textrm{e}^{r_0} \tilde{u} \int _{0}^{2{\rm \pi} } \tilde{g}^0(\theta) \sin \theta \,\textrm{d}\theta - \sin \alpha \lambda \,\textrm{e}^{r_0} \tilde{u} \int _{0}^{2{\rm \pi} } \tilde{g}^0(\theta) \cos \theta \,\textrm{d}\theta. \end{align}

Note that the $\sin \theta , \cos \theta$ harmonics of the function ${\tilde {\varPsi }}_2$ at the boundary $r_0$ are contained only in the term $g^0(\theta )$. The final formula for $f_{\alpha ,2}$ can be written as

(3.39)\begin{gather} f_{\alpha,2}= \tilde{u} {\rm \pi}(1+b)^2 \left(\cos^2 \alpha\left(- \frac{1}{2}+\varSigma_A\right)+ \sin^2 \alpha \left(-\frac{1}{2 b^2}+\varSigma_B \right)\right),\end{gather}

(3.40a,b)\begin{gather} \varSigma_A = \sum_{j=1}^\infty j (A_j)^2, \quad \varSigma_B = \sum_{j=1}^\infty j (B_j)^2.\end{gather}

The third component of the force is proportional to the velocity $\tilde {u}$ of motion and expresses the Stokes viscous drag. The coefficients $A_j, B_j$ are determined by formula (3.24a,b). The corresponding series ($\varSigma _A, \varSigma _B$) in formula (3.39) converge for $b> 0$. In the range $0.05 <b <1$, we can approximate their values using the following rational functions:

(3.41)\begin{equation} \left.\begin{gathered} \varSigma_A\approx \frac{0.3135b^2+1.11b+0.0974}{b^2+0.4998b+0.02154}, \\ \varSigma_B\approx \frac{3.042b+0.9046}{b^3+3b^2-0.05713b+0.001518}. \end{gathered}\right\} \end{equation}

The error of the presented formulas does not exceed 0.7 %.

3.7. The resulting formula for the in-line force

Collecting all the expansion terms, we can obtain the following formula for the determination of the in-line force acting on the elliptic cylinder:

(3.42)\begin{align} f_{\alpha}&=(1+b) {\rm \pi}(b \cos^2 \alpha+ \sin^2 \alpha) \frac{\textrm{d}\tilde{u}(t)}{\textrm{d}t}\nonumber\\ &\quad +\varepsilon (1+b)^2 \sqrt{\rm \pi} (A_1 \cos^2 \alpha + B_1 \sin^2 \alpha)\int_{-\infty}^{t}\frac{\textrm{d} \tilde{u}}{\textrm{d}\tau}\frac{\textrm{d}\tau}{\sqrt{t-\tau}}\nonumber\\ &\quad + \varepsilon^2 {\rm \pi}(1+b)^2 \left(\cos^2 \alpha\left(- \frac{1}{2}+\varSigma_A\right)+ \sin^2 \alpha \left(-\frac{1}{2 b^2}+\varSigma_B\right)\right) \tilde{u}+O(\varepsilon^3). \end{align}

The coefficients $A_1, B_1$ and $\varSigma _A, \varSigma _B$ can be calculated directly using formulas (3.24a,b) and (3.40a,b), or using the approximations (3.29a,b) and (3.41).

To estimate the applicability range of the resulting formula (3.42) in the next section we will analyse the behaviour of the hydrodynamic forces for several special cases.

4.1. Harmonic oscillations

Let us consider the case of harmonic oscillations when the cylinder speed is expressed in the form

(4.1)\begin{equation} \tilde{u} = \cos (t). \end{equation}

In this case, formula (3.42) for the force calculation is rewritten as

(4.2)\begin{align} f_{\alpha}&={-}(1+b) {\rm \pi}(b \cos^2 \alpha+ \sin^2 \alpha) \sin(t)\nonumber\\ &\quad +\varepsilon (1+b)^2 {\rm \pi}(A_1 \cos^2 \alpha +B_1 \sin^2 \alpha )\frac{\cos(t)-\sin(t)}{\sqrt{2}}\nonumber\\ &\quad + \varepsilon^2 {\rm \pi}(1+b)^2 \left(\cos^2 \alpha\left(- \frac{1}{2}+\varSigma_A\right)+ \sin^2 \alpha \left(-\frac{1}{2 b^2}+\varSigma_B\right)\right) \cos(t)+O(\varepsilon^3). \end{align}

Assuming $b = 1$, we obtain the well-known formula for the case of a circular cylinder (see, e.g. Wang Reference Wang1968)

(4.3)\begin{equation} f_{\alpha}(b=1)={-}2 {\rm \pi}\sin(t)+ \varepsilon 2\sqrt{2} {\rm \pi}(\cos(t)-\sin(t))+ \varepsilon^2 2 {\rm \pi}\cos(t)+O(\varepsilon^3). \end{equation}

To determine the limits of applicability of the asymptotic theory, we use the numerical and experimental data available in the literature and the new numerical results obtained in this study. We conducted the numerical simulation of the flow around elliptic cylinder using the numerical model from Nuriev et al. (Reference Nuriev, Kamalutdinov and Egorov2019). The details of the simulations are described in appendix A.

The results of experimental and numerical studies (see, e.g. Shih & Buchanan Reference Shih and Buchanan1971; Singh Reference Singh1979; Bearman et al. Reference Bearman, Downie, Graham and Obasaju1985a; Sarpkaya Reference Sarpkaya1986; Justesen Reference Justesen1991; An et al. Reference An, Cheng and Zhao2015) of the hydrodynamic forces acting on a harmonically oscillating cylinder are usually represented in the literature as the coefficients of the Morison approximation (Morison, Johnson & Schaaf Reference Morison, Johnson and Schaaf1950). The Morison approximation is based on the representation of the total force as a sum of two components, one of which is proportional to the instantaneous acceleration of the body, the second being proportional to the square of the instantaneous velocity. In non-dimensional variables it can be written as follows:

(4.4)\begin{equation} f^M_{\alpha} ={-}{\rm \pi} C_M \sin t + \varepsilon \gamma C_D |\cos t| \cos t. \end{equation}

As can be seen from the structure of the asymptotic formula (3.42), the force depends not only on the instantaneous characteristics of the flow but also on the history of movement. Formally, for the case of a steady flow, if we assume that $\cos t |\cos t| \approx (8/3{\rm \pi} ) \cos t$, then equation (3.42) can be rewritten in the representation (4.4). In this case, the history force is decomposed into two components, one of which is included in the coefficient $C_M$, the other being included in the coefficient $C_D$

(4.5)\begin{gather} f_{\alpha} \approx{-}{\rm \pi} C^{(3)}_M \sin t + (8/3{\rm \pi}) \varepsilon \gamma C^{(3)}_D \cos t, \end{gather}

(4.6)\begin{gather}C^{(3)}_M=(1+b) (b \cos^2 \alpha+ \sin^2 \alpha)+ \varepsilon \frac{(1+b)^2}{\sqrt{2}} (A_1 \cos^2 \alpha +B_1 \sin^2 \alpha ), \end{gather}

(4.7)\begin{align} C^{(3)}_D&=\frac{3{\rm \pi}}{8\varepsilon \gamma}\left(\vphantom{\left(-\frac{1}{2 b^2}+\varSigma_B\right)} \varepsilon (1+b)^2 {\rm \pi}(A_1 \cos^2 \alpha +B_1 \sin^2 \alpha )/\sqrt{2}\right.\nonumber\\ &\quad + \varepsilon^2 {\rm \pi}(1+b)^2 \left.\left(\cos^2 \alpha\left(- \frac{1}{2}+\varSigma_A\right)+ \sin^2 \alpha \left(-\frac{1}{2 b^2}+\varSigma_B\right)\right)\right). \end{align}

Here, the superscript $(i)$ on the coefficients $C^{(i)}_M, C^{(i)}_D$ indicates the order of accuracy of the asymptotic formula for the hydrodynamic force relative to the parameter $\varepsilon$ (in this case the coefficients are defined by formula (4.2), which has an error of the order of $O (\varepsilon ^3)$). The force components grouped in this way have an obvious physical interpretation: the term proportional to the velocity is the damping force (see, e.g. Landau & Lifshitz Reference Landau and Lifshitz1987) and the term proportional to the acceleration is the inertial force that affects the change of the natural frequency in the case of resonant oscillations (see, e.g. experiments of Egorov, Kamalutdinov & Nuriev Reference Egorov, Kamalutdinov and Nuriev2018).

Due to the special practical importance, we also denote the relation between the non-dimensional parameters introduced in the present work (2.6a–c) and the classical experimental parameters that are usually used for high-amplitude harmonic oscillations as

(4.8a,b)\begin{equation} \beta=\frac{(2R_x)^2 \omega}{2{\rm \pi} \nu}=2/(\varepsilon^2{\rm \pi}),\quad KC=2{\rm \pi} \frac{A}{2R_x}={\rm \pi}\varepsilon\gamma. \end{equation}

Here $KC$ is the Keulegan–Carpenter number, or the dimensionless oscillation amplitude, and $\beta$ is the dimensionless oscillation frequency. The values of these parameters will be also presented in the text below for convenience when working with numerical and experimental data.

In figures 3 and 4 the comparison of the values of coefficients calculated by the asymptotic formulas (4.6) and various numerical and experimental data is presented. First of all, let us pay attention to the change of the coefficients $C_D, C_M$ depending on the ratio of the semi-axes $b$ for fixed values of the parameters $\gamma = 0.56, 1.12, \varepsilon =0.056, \alpha = {\rm \pi}/2$ ($\beta =200, KC=0.1, 0.2$). As seen in figure 3, the first two terms of the expansion describe the change of the coefficient $C_M$ in the entire considered range $0.05\leqslant b\leqslant 1$ with high accuracy.

Figure 3.

Harmonic oscillations. Dependences of the coefficients $C_D$ and $C_M$ on the parameter $b$. Here, $\gamma = 0.56, 1.12, \varepsilon =0.056, \alpha = {\rm \pi}/2$ ($\beta =200, KC=0.1, 0.2$)

Figure 4.

Harmonic oscillations. Dependences of $C_D, C_M$ on the parameter $KC={\rm \pi} \gamma \varepsilon$ for $\alpha = {\rm \pi}/2$ and different values of $\varepsilon$, $b$

The obtained asymptotic dependence for $C_D$ does not show such a good agreement with the numerical data for all values of the parameter $b$. The sum of the second and third terms of the expansion ($C^{(3)}_D$) allows us to get a good approximation in the range $0.2\leqslant b\leqslant 1$ for both $\gamma = 0.56$ and $\gamma = 1.12$. At lower values of $b$, when $b \sim \varepsilon$, the series in powers of $\varepsilon$ begins to diverge. In fact, the case of the smallness of $b$ was not taken into account when we built the expansion in the boundary layer; such an assumption requires a revision of the internal problem (3.11). At the same time, for the terms of the order $O(\varepsilon )$, the expansion remains valid. As can be seen for $\gamma = 0.56$ (and lower values) the second term of the expansion ($C^{(2)}_D$) gives a good approximation for $C_D$ in the range $0.05<b<0.2$, for $\gamma = 1.12$ the difference between the numerical data and the asymptotic solution does not exceed 15 %. The deviation of the numerical data from the asymptotic solution with increasing values of $\gamma$ for small values of $b$ is associated with the reaching of the upper limit of allowable oscillation amplitudes (for a given approximation).

Let us consider the applicability limits of the obtained asymptotic theory with respect to the amplitude of oscillations. In figure 4 the graphs of changes of the hydrodynamic coefficients depending on the dimensionless amplitude ($KC$) are presented (in this case, the change of amplitude is proportional to the parameter $\gamma$). In addition to the asymptotic results, numerical and experimental data, the higher-order asymptotic solution for a circular cylinder (obtained by Nuriev & Egorov (Reference Nuriev and Egorov2019) in the framework of the similar approximation $\varepsilon \ll 1, \gamma =O(1)$) is shown

(4.9)\begin{align} f_{\alpha}(b=1)&={-}2 {\rm \pi}\sin(t)+ \varepsilon 2\sqrt{2} {\rm \pi}(\cos(t)-\sin(t))+\varepsilon^2 2 {\rm \pi}\cos(t)\nonumber\\ &\quad +\varepsilon^3\left(-\frac{5 {\rm \pi}}{2\sqrt{2}}(\cos(t)+\sin(t))+2{\rm \pi}\gamma^2\right.\nonumber\\ &\quad \times\left. \left(\frac{1}{3 \sqrt{2}}(1-2\sqrt{2})\sin(t)+\frac{5}{12}(1+\sqrt{2})\cos(t)\right)\right)\nonumber\\ &\quad + \varepsilon^3 \gamma^2 3 {\rm \pi}(\sin(3t)-\cos(3t))(6\sqrt{2}+\sqrt{3}-5-2\sqrt{6})/4+O(\varepsilon^4), \end{align}

and approximate formulas for rectangular plates obtained by Phan et al. (Reference Phan, Aureli and Porfiri2013) for low-amplitude oscillations are presented

(4.10)\begin{align} f_{plate}(b^*)&\approx{-}\big((1.02+0.995b-0.496b^2) +\varepsilon(3.07+1.83b-1.245b^{0.5})\big){\rm \pi}\sin(t)\nonumber\\ &\quad +\big(\varepsilon(3.12+1.495b-0.520b^{0.5})\big){\rm \pi}\cos(t). \end{align}

The obtained theoretical estimates of the coefficient $C_M$ coincides well with numerical data in almost the entire considered range of amplitude variation, which corresponds to symmetric flow regimes with a predominant two-dimensional structure. In addition, it can be noted that the values of $C_M$ for the elliptical cylinder (4.6) and for the rectangular plate (4.10) are almost identical for small values of $b$ ($b <0.2$).

The theoretical formula (4.6) for $C_D$ has more strict constraints. As can be seen, with increasing $KC$, the numerical data deviate from the asymptotic behaviour. For the case of $b = 1$ the deviation of the numerical data from linear theory (where the total force does not depend on $\gamma$) is well described by a nonlinear correction of the order $O(\varepsilon ^3)$ that we can get from formula (4.9). The comparison of the asymptotic solution with the numerical and experimental data for this case is shown in figure 4 for $\varepsilon = 0.056$, 0.036, 0.025 ($\beta = 200, 483, 1035$). If we assume that the permissible error of the linear theory should not exceed 14 %–15 %, then we get from (4.9) the following limitation: $\gamma <\sqrt {0.21}/ \varepsilon$. In this case, the critical amplitude is independent of $\varepsilon$ and can approximately be estimated as $KC_{cr} \approx 1.44$.

Formula (4.9) also shows the presence in the hydrodynamic force of a time harmonic that is different from the main one. In other words, in addition to the damping and inertial terms, the force also contains other components. This is fully confirmed by the results of the numerical simulation. However, as can be seen, the relative contribution of these terms to the hydrodynamic force in the range of low oscillation amplitudes is small.

Based on numerical data, for cylinders with different semi-axis ratios ($b>\varepsilon$), it can be established that the limit of applicability of the asymptotic $C^{(3)}_D$ changes almost proportionally with the parameter $b$ (for $\alpha = {\rm \pi}/2$). In other words, the error of the linear theory (in per cent) can be defined as

(4.11)\begin{equation} err\approx0.74\left(\frac{KC}{b{\rm \pi}}\right)^2\times100. \end{equation}

This can be confirmed by the comparison of the asymptotic formula for $C_D$ with the results of numerical data for different values of $b = 1$, 0.4, 0.25 shown in figure 4, and by the direct comparison of the theoretical values of $err$ and the values of $err_{num}$ determined by numerical data at constant values of $KC$ (see figure 5)

(4.12)\begin{equation} err_{num}=(C_D-C^{(3)}_D)/C^{(3)}_D\times100. \end{equation}

Figure 5.

Harmonic oscillations. Error of the approximation $C^{(3)}_D$ as a function of the parameter $b$ for $\varepsilon = 0.056$ ($\beta = 200$).

For $b \sim \varepsilon$ the lower-order asymptotics $C^{(2)}_D$ works well. For the case $\varepsilon = 0.056$, $b = 0.1$ shown in figure 4, a difference of approximately $15\,\%$ between the numerical data and the asymptotic $C^{(2)}_D$ is achieved at $KC/b \approx 2$.

At the end of this section, we can also note that, although the dependences $C_D(b)$ for a plate and an elliptical cylinder have significantly different behaviour, in the range $b<0.2$ (this can easily be seen by comparison of the corresponding asymptotic formulas in the leading terms), the values of $C_D$ in the region of $0.05<b<0.2$ differ by no more than 10 %.

4.2. Multi-harmonic oscillations

As a second example, we consider the case when the law of the motion contains two harmonics: the first and the second. The speed of the oscillations in this case can be written in the form

(4.13)\begin{equation} \tilde{u} = \tilde{u}_1 \cos(t)+\tilde{u}_2 \cos(2t). \end{equation}

The asymptotic formula for determination of the force acting on the cylinder in a steady periodic flow regime for the given law of oscillations according to (3.42) is written as

(4.14)\begin{align} f_{\alpha}&={-}(1+b) {\rm \pi}(b \cos^2 \alpha+ \sin^2 \alpha) (\tilde{u}_1 \sin(t)+2\tilde{u}_2\sin(2t))\nonumber\\ &\quad + \varepsilon (1+b)^2 {\rm \pi}(A_1 \cos^2 \alpha +B_1 \sin^2 \alpha )\left(\tilde{u}_1 \frac{\cos(t)-\sin(t)}{\sqrt{2}}+\tilde{u}_2(\cos(2t)-\sin(2t))\right)\nonumber\\ &\quad \times\varepsilon^2 {\rm \pi}(1+b)^2 \left(\cos^2 \alpha\left(- \frac{1}{2}+\varSigma_A\right)+ \sin^2 \alpha \left(-\frac{1}{2 b^2}+\varSigma_B\right)\right)\cdot (\tilde{u}_1 \cos(t)\nonumber\\ &\quad +\tilde{u}_2 \cos(2t))+O(\varepsilon^3). \end{align}

For the analysis of the hydrodynamic force, it is again convenient to distinguish the damping and inertial components

(4.15)\begin{gather} f_{\alpha} \approx (8/3{\rm \pi}) \varepsilon \gamma \sum_{k=1}^{n} C^{(3)}_{Dk} u_k \cos kt - {\rm \pi}\sum_{k=1}^{n} C^{(3)}_{Mk} u_k k \sin kt, \quad n=2, \end{gather}

(4.16a,b)\begin{gather} C^{(3)}_{Mk}(\varepsilon) = C^{(3)}_{M}(\varepsilon/\sqrt{k}), \quad C^{(3)}_{Dk}(\varepsilon,\gamma) = C^{(3)}_{D}(\varepsilon/\sqrt{k},\gamma/\sqrt{k}). \end{gather}

Here, $C^{(3)}_{Mk}$ are the coefficients of the inertial forces and $C^{(3)}_{Dk}$ are the coefficients of the damping forces. Generally speaking, such a representation can also be used for an arbitrary periodic law of motion (in the approximation of the order of $O(\varepsilon ^3)$), if we assume that the flow around the cylinder induced by such motion is steady. As can be seen, the coefficients $C^{(3)}_{Mk}$ and $C^{(3)}_{Dk}$ do not depend on the structure of the law of motion (for example, $C^{(3)}_{M1}$ and $C^{(3)}_{D1}$ are the same as in the case of harmonic oscillations), since interaction between different time harmonics in the found terms of expansion does not occur (it will be observed in terms of the order of $O(\varepsilon ^3)$ and higher). The coefficients of the inertial forces of different time harmonics change only in the terms of the order of $O(\varepsilon )$, while the damping force of the $k$th harmonic increases compared to the first by $\sqrt {k}$ times (in the leading term).

Let us compare the theoretical values of the coefficients with the data from numerical simulation. The coefficients of the inertial and damping forces in the framework of numerical calculations were determined using integration

(4.17a,b)\begin{equation} C_{Mk} ={-}\frac{1}{{\rm \pi}^2 u_k k} \int_{2{\rm \pi} j}^{2{\rm \pi} (\kern1.5pt j+1)} \,f_\alpha \sin kt \,\textrm{d}t, \quad C_{Dk} = \frac{3}{8\varepsilon \gamma u_k} \int_{2{\rm \pi} j}^{2{\rm \pi} (\kern1.5pt j+1)} \,f_\alpha \cos kt \,\textrm{d}t. \end{equation}

In figure 6, the values of the coefficients for the case of $u_1=0.8$, $u_2=0.8485$, $\alpha ={\rm \pi} /2$ are shown. As can be seen, the numerical values of the coefficients $C_{Mk}$ in the studied range of $KC<3.2$ are in good agreement with the asymptotic approximation. The numerical values of the coefficients $C_{Dk}$, as in the case of harmonic oscillations, begin to noticeably deviate from the theoretical values for $KC/b>1$.

Figure 6.

Multi-harmonic oscillations. Dependencies of the coefficients $C_{Dk}, C_{Mk}$ on the dimensionless amplitude $KC/b$ for elliptical cylinders with different semi-axis ratios at $\varepsilon =0.056$, $\alpha = {\rm \pi}/2$, $(\beta =200)$.

Comparing the data of multi-harmonic and harmonic oscillations, we can evaluate the influence of the new effects on the change of $C_{Dk}$, which arises in particular as a result of the interaction of the time harmonics. In figure 7, the deviation of the numerical data from the asymptotic ($err_{num}$), as well as the theoretical error estimate ($err$) obtained in the previous section for the case of harmonic oscillations (see formula (4.11)) are shown. As can be seen for the presented cases of multi-harmonic oscillations, the deviation of the damping force from the asymptotic for the first harmonic occurs almost 2 times faster than for the monoharmonic case, and for the second harmonic almost 4 times faster (taking into account the change of the dimensionless frequency and amplitude of the oscillations). At the same time, in the range of $KC / b <1$ the influence of the new effects is relatively small and does not exceed 7 %–12 %. It can also be noted here that, according to the numerical results for the considered case of biharmonic oscillations, with an increase of dimensionless amplitude, asymmetry of the flow regime appears (see appendix B). This leads to the appearance of a constant force component (that is an order of magnitude smaller than the damping) acting along the axis of oscillation.

Figure 7.

Multi-harmonic oscillations. The deviation of the damping force from the asymptotics depending on the dimensionless amplitude $KC/b$ at $\varepsilon =0.056, \alpha = {\rm \pi}/2, (\beta =200)$ for elliptical cylinders with different semi-axis ratios.