2.1. Defining equations

A vorticity–stream function ( ${\it\omega}$ – ${\it\psi}$ ) method of solution is employed (see Batchelor Reference Batchelor1967, appendix 2), where the velocity and vorticity fields are defined by

(2.4a−c ) $$\begin{eqnarray}u_{r}=\frac{1}{r}\frac{\partial {\it\psi}}{\partial {\it\theta}},\quad u_{{\it\theta}}=-\frac{\partial {\it\psi}}{\partial r},\quad {\it\omega}=-\frac{1}{r}\frac{\partial }{\partial r}\left(r\frac{\partial {\it\psi}}{\partial r}\right)-\frac{1}{r^{2}}\frac{\partial ^{2}{\it\psi}}{\partial {\it\theta}^{2}}.\end{eqnarray}$$

The boundary conditions (2.2) impose significant constraints on the velocity near the boundary, mainly because $\partial u_{{\it\theta}}/\partial {\it\theta}=\partial u_{r}/\partial {\it\theta}=0$ . This means that

(2.5) $$\begin{eqnarray}\left.{\it\omega}=\frac{\partial u_{{\it\theta}}}{\partial r}\right|_{r=a}\end{eqnarray}$$

and

(2.6) $$\begin{eqnarray}\left.\frac{\partial u_{r}}{\partial r}\right|_{r=a}=-\frac{U_{b}}{a}\end{eqnarray}$$

(where mass conservation was used in (2.6)). From (2.1), the vorticity equation is

(2.7) $$\begin{eqnarray}U_{\infty }\frac{\partial {\it\omega}}{\partial x}+\frac{U_{b}a}{r}\frac{\partial {\it\omega}}{\partial r}={\it\nu}{\rm\nabla}^{2}{\it\omega}.\end{eqnarray}$$

Our starting point is quite similar to that of Lamb (Reference Lamb1932) and involves expressing the vorticity field $\tilde{{\it\omega}}$ ( $=2a{\it\omega}/U_{\infty }$ ) as

(2.8) $$\begin{eqnarray}\tilde{{\it\omega}}=\text{e}^{(Re/4)\tilde{r}\cos {\it\theta}}P,\end{eqnarray}$$

giving

(2.9) $$\begin{eqnarray}\frac{Re^{2}}{8}{\it\Lambda}\frac{1}{\tilde{r}}\cos {\it\theta}P+\frac{Re}{2}\frac{{\it\Lambda}}{\tilde{r}}\frac{\partial P}{\partial \tilde{r}}+\left(\frac{Re}{4}\right)^{2}P=\tilde{{\rm\nabla}}^{2}P,\end{eqnarray}$$

where $Re=2aU_{\infty }/{\it\nu}$ and $\tilde{r}=r/a$ . Following Dukowicz (Reference Dukowicz1982), we seek the leading-order symmetric solution, which is of the form

(2.10) $$\begin{eqnarray}P=P_{1}(\tilde{r})\sin {\it\theta},\end{eqnarray}$$

where $P_{1}$ satisfies

(2.11) $$\begin{eqnarray}P_{1}^{\prime \prime }+\frac{P_{1}^{\prime }}{\tilde{r}}(1-2{\it\beta})-\left(\left(\frac{Re}{4}\right)^{2}+\frac{1}{\tilde{r}^{2}}\right)P_{1}=0,\end{eqnarray}$$

where ${\it\beta}=Re\,{\it\Lambda}/4$ . This is valid provided that $Re^{2}|{\it\Lambda}|\ll 1$ and for ${\it\beta}\rightarrow -\infty$ because the flow is symmetric about ${\it\theta}={\rm\pi}/2$ , but not for ${\it\beta}\rightarrow \infty$ . The solution that satisfies $P_{1}\rightarrow 0$ as $\tilde{r}\rightarrow \infty$ is

(2.12) $$\begin{eqnarray}P_{1}=C_{1}\tilde{r}^{{\it\beta}}K_{(1+{\it\beta}^{2})^{1/2}}(Re\,\tilde{r}/4).\end{eqnarray}$$

The stream function, ${\it\psi}$ , can be constructed by writing it as the sum of the known blowing and free-stream components, together with a component to be determined. As such we write

(2.13) $$\begin{eqnarray}{\it\psi}=U_{\infty }a({\it\Lambda}{\it\theta}+\tilde{r}\sin {\it\theta}+f_{1}\sin {\it\theta}).\end{eqnarray}$$

Substitution of (2.13) into (2.4) gives

(2.14) $$\begin{eqnarray}f_{1}^{\prime \prime }+\frac{f_{1}^{\prime }}{r}-\frac{f_{1}}{r^{2}}=-\frac{1}{{\rm\pi}}\int _{0}^{{\rm\pi}}\sin {\it\theta}\tilde{{\it\omega}}\,\text{d}{\it\theta}=-\frac{1}{2{\rm\pi}}P_{1}(\tilde{r})\int _{0}^{2{\rm\pi}}\text{e}^{(Re\,\tilde{r}/4)\cos {\it\theta}}\sin ^{2}{\it\theta}\,\text{d}{\it\theta}.\end{eqnarray}$$

The boundary conditions for $f_{1}$ are

(2.15a−c ) $$\begin{eqnarray}f_{1}(1)=-1,\quad f_{1}^{\prime }(1)=-1,\quad \lim _{\tilde{r}\rightarrow \infty }f_{1}(\tilde{r})=0.\end{eqnarray}$$

The right-hand side of (2.14) is defined as $C_{1}p_{1}(\tilde{r})$ , where

(2.16) $$\begin{eqnarray}p_{1}(\tilde{r})=-\tilde{r}^{{\it\beta}}\frac{J_{1}(\text{i}Re\,\tilde{r}/4)}{\text{i}Re\,\tilde{r}/4}K_{(1+{\it\beta}^{2})^{1/2}}(Re\,\tilde{r}/4).\end{eqnarray}$$

We can solve (2.14) exactly by writing $f_{1}=\tilde{r}g_{1}$ , which transforms the ordinary differential equation to

(2.17) $$\begin{eqnarray}\frac{\text{d}(\tilde{r}^{3}g_{1}^{\prime })}{\text{d}\tilde{r}}=C_{1}p_{1}(\tilde{r})\tilde{r}^{2}.\end{eqnarray}$$

The boundary conditions on $g_{1}$ are $g_{1}(1)=-1$ , $g_{1}^{\prime }(1)=0$ and $g_{1}\rightarrow 0$ as $\tilde{r}\rightarrow \infty$ . Integrating twice, we find two results. The first is that

(2.18) $$\begin{eqnarray}f_{1}=C_{1}\tilde{r}\left(-\frac{1}{2\tilde{r}^{2}}G(1)-\int _{\infty }^{\tilde{r}}G(\tilde{r})\tilde{r}^{-3}\,\text{d}\tilde{r}\right),\end{eqnarray}$$

where

(2.19) $$\begin{eqnarray}G(\tilde{r})=\int _{\tilde{r}}^{\infty }p_{1}(\tilde{r})\tilde{r}^{2}\,\text{d}\tilde{r}.\end{eqnarray}$$

The second result (which ensures that the far-field boundary condition is satisfied) is

(2.20) $$\begin{eqnarray}C_{1}=\frac{2}{\displaystyle \int _{1}^{\infty }p_{1}(\tilde{r})\,\text{d}\tilde{r}}.\end{eqnarray}$$

The integrand scales as $\tilde{r}^{{\it\beta}-2}$ in the far field (using $K_{n}(z)\sim \text{exp}(-z)/z^{1/2}$ and $J_{1}(\text{i}z)\sim \text{exp}(z)/z^{1/2}$ ) and so the integral converges when $-\infty <{\it\beta}<1$ .

2.2. Diagnostics

The pressure and viscous drag coefficients for characterizing the force on a cylinder are

(2.21a,b ) $$\begin{eqnarray}C_{P}=\int _{0}^{2{\rm\pi}}\left(\frac{1}{Re}\frac{\partial \tilde{{\it\omega}}}{\partial \tilde{r}}-\frac{1}{2}{\it\Lambda}\tilde{{\it\omega}}\right)\sin {\it\theta}\,\text{d}{\it\theta},\quad C_{{\it\nu}}=-\frac{1}{Re}\int _{0}^{2{\rm\pi}}\tilde{{\it\omega}}\sin {\it\theta}\,\text{d}{\it\theta},\end{eqnarray}$$

which is an extension of the relationship given by Dennis & Shimshoni (Reference Dennis and Shimshoni1965) to include a through-surface flow. On substituting (2.8), (2.10), (2.12) into (2.21),

(2.22) $$\begin{eqnarray}\displaystyle \hspace{-8.0pt}\left.\begin{array}{@{}c@{}}\displaystyle C_{P}=-C_{1}\frac{{\rm\pi}}{Re}\left(({\it\beta}+(1+{\it\beta}^{2})^{1/2})K_{(1+{\it\beta}^{2})^{1/2}}(Re/4)+\frac{Re}{4}K_{(1+{\it\beta}^{2})^{1/2}-1}(Re/4)\right),\\ \displaystyle C_{{\it\nu}}=-C_{1}\frac{{\rm\pi}}{Re}K_{(1+{\it\beta}^{2})^{1/2}}(Re/4).\quad \end{array}\!\!\!\right\} & & \displaystyle\end{eqnarray}$$

The ratio of the viscous drag to the total drag coefficient is

(2.23) $$\begin{eqnarray}\frac{C_{{\it\nu}}}{C_{D}}=\frac{1}{{\it\beta}+(1+{\it\beta}^{2})^{1/2}+1+\displaystyle \frac{Re}{4}\displaystyle \frac{K_{(1+{\it\beta}^{2})^{1/2}-1}(Re/4)}{K_{(1+{\it\beta}^{2})^{1/2}}(Re/4)}},\end{eqnarray}$$

where $C_{D}=C_{P}+C_{{\it\nu}}$ . Since $Re$ is small, $C_{{\it\nu}}/C_{D}$ is effectively a function of $Re\,{\it\Lambda}$ . It should be noted that (2.23) does not depend on $C_{1}$ .

3.1. No-blow case ( ${\it\Lambda}=0$ )

The purpose here is to retrieve Lamb’s solution for the no-blow case. When ${\it\Lambda}=0$ , $p_{1}$ can be expressed exactly as

(3.1) $$\begin{eqnarray}p_{1}=-K_{1}(Re\,\tilde{r}/4)\frac{J_{1}(\text{i}Re\,\tilde{r}/4)}{\text{i}Re\,\tilde{r}/4}.\end{eqnarray}$$

Using the substitution $z=Re\,\tilde{r}/4$ , the integral in (2.20) can be written as

(3.2) $$\begin{eqnarray}\int _{1}^{\infty }p_{1}\,\text{d}\tilde{r}=\frac{4}{Re}\int _{Re/4}^{\infty }K_{1}(z)\frac{J_{1}(\text{i}z)}{\text{i}z}\,\text{d}z=\frac{4}{Re}\int _{Re/4}^{\infty }K_{1}(z)\left(\frac{1}{2}+\mathop{\sum }_{n=1}^{\infty }\frac{z^{2n}}{2^{2n+1}n!(n+1)!}\right)\,\text{d}z,\end{eqnarray}$$

such that

(3.3) $$\begin{eqnarray}\int _{1}^{\infty }p_{1}\,\text{d}\tilde{r}=\frac{2}{Re}\left(K_{0}(Re/4)+\mathop{\sum }_{n=1}^{\infty }\int _{Re/4}^{\infty }\frac{z^{2n}K_{1}(z)}{2^{2n}n!(n+1)!}\,\text{d}z\right).\end{eqnarray}$$

In the limit of $Re\ll 1$ , the lower limit is close to zero and it can be shown, using (A 4), that

(3.4) $$\begin{eqnarray}\mathop{\sum }_{n=1}^{\infty }\int _{0}^{\infty }\frac{z^{2n}K_{1}(z)}{2^{2n}n!(n+1)!}\,\text{d}z=\frac{1}{2},\end{eqnarray}$$

such that

(3.5) $$\begin{eqnarray}C_{1}=\frac{Re}{\frac{1}{2}+K_{0}(Re/4)}.\end{eqnarray}$$

The drag coefficient corresponding to (3.5) is

(3.6) $$\begin{eqnarray}C_{D}=-\frac{2{\rm\pi}}{\frac{1}{2}+K_{0}(Re/4)}\left(K_{1}(Re/4)+\frac{Re}{8}K_{0}(Re/4)\right)\cong -\frac{8{\rm\pi}}{Re(\frac{1}{2}+K_{0}(Re/4))},\end{eqnarray}$$

where use was made of (A 2). Now, Lamb (Reference Lamb1932, p. 616) derived the following expression for vorticity:

(3.7) $$\begin{eqnarray}{\it\omega}=C\text{e}^{(\tilde{r}Re/4)\cos {\it\theta}}\frac{\partial }{\partial y}K_{0}(Re\,\tilde{r}/4)=-\frac{C}{a}\frac{Re}{4}\text{e}^{(\tilde{r}Re/4)\cos {\it\theta}}K_{1}(Re\,\tilde{r}/4)\sin {\it\theta},\end{eqnarray}$$

which is the same expression as that derived here from the vorticity equation. (It should be noted that there is a typographical error in Lamb’s vorticity expression, where $\text{e}^{kz}$ should read $\text{e}^{kx}$ ; the correct analysis is given in Lamb (Reference Lamb1911) except for the typographical error of ‘sphere’ which should be ‘cylinder’ after equation (54).) A higher-order expansion of the stream function was determined,

(3.8) $$\begin{eqnarray}C=\frac{2U_{\infty }}{\frac{1}{2}+K_{0}(Re/4)},\end{eqnarray}$$

or equivalently

(3.9) $$\begin{eqnarray}C_{D}=-\frac{8{\rm\pi}}{Re\left(\frac{1}{2}+K_{0}(Re/4)\right)}.\end{eqnarray}$$

Therefore, the general expression for the drag coefficient agrees exactly with Lamb’s expression as $Re\rightarrow 0$ (the difference for $Re=1$ is less than 1 %).

3.2. Strongly sucking flow ( $-{\it\Lambda}\gg 1$ )

For strongly sucking flows where $|{\it\beta}|\gg 1$ and ${\it\beta}<0$ , we can write

(3.10) $$\begin{eqnarray}\frac{\displaystyle \int _{1}^{\infty }p_{1}\,\text{d}\tilde{r}}{K_{(1+{\it\beta}^{2})^{1/2}}(Re/4)}=-\frac{1}{2}\left(\frac{4}{Re}\right)^{{\it\beta}+1}\int _{Re/4}^{\infty }z^{{\it\beta}}\frac{K_{(1+{\it\beta}^{2})^{1/2}}(z)}{K_{(1+{\it\beta}^{2})^{1/2}}(Re/4)}\,\text{d}z\cong \frac{1}{2({\it\beta}-(1+{\it\beta}^{2})^{1/2}+1)},\end{eqnarray}$$

which can be substituted into (2.22), giving a drag coefficient of

(3.11) $$\begin{eqnarray}C_{D}\approx -\frac{4{\rm\pi}}{Re}({\it\beta}+(1+{\it\beta}^{2})^{1/2}+1)({\it\beta}-(1+{\it\beta}^{2})^{1/2}+1).\end{eqnarray}$$

This reduces to

(3.12) $$\begin{eqnarray}C_{D}\approx -2{\rm\pi}{\it\Lambda}.\end{eqnarray}$$

This approximation is appropriate when $|{\it\beta}|\geqslant 1$ or ${\it\Lambda}<-4/Re$ (so that the asymptotic approximation is valid). Equation (3.12) agrees with a global momentum analysis when the far-field downstream flow is irrotational, which was derived by Pankhurst & Thwaites (Reference Pankhurst and Thwaites1953, appendix I) for high- $Re$ flows. This is to be expected because in both cases, the boundary layer is thin compared with the size of the cylinder.

4.1. Solution technique

The Navier–Stokes equation was numerically solved using a finite-element method that employs a characteristic-based split (CBS) methodology (see Zienkiewicz, Taylor & Nithiarasu Reference Zienkiewicz, Taylor and Nithiarasu2005). The ACEsim code has been validated for two-dimensional flows (e.g. Nicolle & Eames Reference Nicolle and Eames2011; Klettner & Eames Reference Klettner and Eames2012). For low $Re$ , White (Reference White1945) suggests a domain width of 2000 $a$ for the no-blow case to be unaffected by boundedness. As the influence of boundedness is increased for strong blowing/sucking, the domain width was increased to 20 000 $a$ for the two cases of $|{\it\Lambda}|=5$ .