On the influence of circulation on the linear stability of a system of a moving cylinder and two identical parallel vortex filaments

Abstract

The interaction of a moving circular cylinder of radius R with the arbitrary circulation $\gamma$ and two parallel vortex filaments of identical intensity are considered. The stability of a system of two vortex filaments and a cylinder located in the middle between them is studied. The filaments rotate around the cylinder at a constant angular velocity. The problem depends on three parameters $(q,a,\gamma )$: $q=R^2/R_0^2,$ where $2R_0$ is the distance between filaments, $0<q<1,$ the added mass of the cylinder $a>0$ and the circulation around cylinder $\gamma \ne 0.$ The case $\gamma =2$ was studied earlier. The purpose of this paper is to study the influence of circulation $\gamma$ in the problem under consideration. The eigenvalues of the linearization matrix are studied for all parameter values. For fixed values of $\gamma ,$ the parameter plane (q, a) is divided into areas of two types: the area of instability, when the linearization matrix has at least one eigenvalue in the right half-plane, and the linear stability area—all eigenvalues lie on the imaginary axis. The results of the study are consistent with the limiting case of a fixed cylinder (for $a\rightarrow \infty$).

Appendix: Explicit formulas characteristic polynomial \(\mathfrak {L}(\lambda )\)

Characteristic polynomial (18), (19) of matrix linearization (17) has the form

$$\begin{aligned}&\mathfrak {L}(\lambda )={\lambda ^2} (\lambda ^4+k_2 \lambda ^2+ k_0), \\&k_0 =32 k_{01}k_{02},\quad k_{01} = {\gamma }{b_2}^{2}+{a_1} {c_2},\quad k_{02} =2\,{\gamma } b_1^{2}+{a_1}({c_1}-{d_1}), \\&k_2 =-16\,{\gamma }\,{b_1}\,{b_2}-8\,{d_1}\,{c_2}+8\,{c_1}\,{c_2}+4a_1^{2} . \end{aligned}$$

Using formulas (16) we write coefficients of characteristic polynomial in explicit form as functions of the parameters q,  a and $\gamma$

$$\begin{aligned} k_{01}(q,a,\gamma )= & {} \Psi _{1} (q,\gamma ) + \frac{1}{a}\Psi _{2} (q,\gamma ) ,\nonumber \\ \Psi _{1} (q,\gamma )= & {} \frac{\left( 1-q \right) ^{2} \gamma }{16 \left( q +1\right) ^{2}}-\frac{\left( q^{2}+3\right) \left( 1-q \right) }{32 \left( q +1\right) ^{3}},\nonumber \\ \Psi _{2} (q,\gamma )= & {} \frac{\left( 4 q^{4}-9 q^{2}+2 q +3\right) \gamma }{16 \left( q +1\right) ^{2}}-\frac{\left( q^{2}+3\right) \left( 1-q \right) }{8(1+ q)}, \nonumber \\ k_{02}(q,a,\gamma )= & {} \Psi _{3}(q,\gamma ) + \frac{1}{a}\Psi _{4}(q,\gamma ) ,\nonumber \\ \Psi _3(q,\gamma )= & {} \frac{\gamma ^{2}}{2}+\frac{\left( 3 q^{3}-7 q^{2}+9 q -13\right) \gamma }{8 \left( q-1 \right) ^{2} \left( q +1\right) }+ \frac{\left( q^{2}+3\right) \left( q^{3}-5 q^{2}+3 q -7\right) }{16 \left( q +1\right) ^{2} \left( q-1\right) ^{3}},\nonumber \\ \Psi _4(q,\gamma )= & {} -\frac{\gamma ^{2}}{2}+\frac{\left( 4 q^{5}+4 q^{4}-9 q^{3}-3 q^{2}+q +11\right) \gamma }{8 \left( q-1 \right) ^{2} \left( q +1\right) }+\frac{\left( q +1\right) \left( q^{2}+3\right) }{4(q-1)},\nonumber \\ \end{aligned}$$

(A1)

$$\begin{aligned} k_2(q,a,\gamma )= & {} \Phi _{0}(q,\gamma ) +\frac{1}{a}\Phi _{1}(q,\gamma ) +\frac{1}{a^2}\Phi _{2}(q,\gamma ) ,\nonumber \\ \Phi _0(q,\gamma )= & {} \gamma ^{2}+\frac{\left( 2 q^{3}-2 q^{2}+6 q +2\right) \gamma }{\left( q +1\right) ^{2} \left( q-1 \right) }+\frac{q^{5}-3 q^{4}+10 q^{3}-6 q^{2}+13 q +1}{2 \left( q +1\right) ^{3} \left( q-1 \right) ^{2}},\nonumber \\ \Phi _1(q,\gamma )= & {} {-2 \gamma ^{2}+\frac{\left( 4 q^{4}-8 q^{3}-q^{2}+8 q -7\right) \gamma }{q^{2}-1}+\frac{2 q^{3}-6 q^{2}+2 q -6}{q +1}},\nonumber \\ \Phi _2(q,\gamma )= & {} \left( \gamma +2 q^{2} -2\right) ^{2}. \end{aligned}$$

(A2)