3.1 Second-order terms

From (2.5) and (2.6) the second-order terms $\unicode[STIX]{x1D713}_{1}$ and $\unicode[STIX]{x1D70C}_{1}$ satisfy

and

with $\unicode[STIX]{x2202}\unicode[STIX]{x1D70C}_{1}/\unicode[STIX]{x2202}r=0$ and $\unicode[STIX]{x1D713}_{1}=\unicode[STIX]{x2202}\unicode[STIX]{x1D713}_{1}/\unicode[STIX]{x2202}r=0$ at $r=1$ . Substituting (3.4) and (3.3) into the right-hand side of (3.6), assuming $\unicode[STIX]{x1D70C}_{1}(r,\unicode[STIX]{x1D703})=G_{11}(r)\cos \unicode[STIX]{x1D703}+G_{13}(r)\cos 3\unicode[STIX]{x1D703}$ , leads to

where $d_{1}$ is an undetermined constant and the $r\cos \unicode[STIX]{x1D703}$ contributions have been set to zero as $\unicode[STIX]{x1D70C}_{0}$ has satisfied that far-field condition. Clearly, (3.7) does not satisfy $\unicode[STIX]{x1D70C}_{1}\rightarrow 0$ at large $r$ for any choice of $d_{1}$ , with its dominant behaviour of

arising from the $O(r^{2}\ln r)$ forcing term in (3.4). This apparent unsuitability of the form of $\unicode[STIX]{x1D70C}_{1}(r,\unicode[STIX]{x1D703})$ in (3.7) is resolved by matching it with the ‘outer-flow solution’ in § 4.2, and that process also enables the value of $d_{1}$ to be identified.

The second-order streamfunction $\unicode[STIX]{x1D713}_{1}(r,\unicode[STIX]{x1D703})$ is determined from (3.5) in a similar manner to $\unicode[STIX]{x1D70C}_{1}(r,\unicode[STIX]{x1D703})$ above, with contributions from the forcing terms that involve derivatives of both $\unicode[STIX]{x1D713}_{0}$ from (3.4) and $\unicode[STIX]{x1D70C}_{1}$ from (3.7). These lead to terms in $\unicode[STIX]{x1D713}_{1}$ that include terms multiplied by $c_{0}$ or $d_{1}$ , with those arising from $\unicode[STIX]{x1D70C}_{1}$ multiplied by $\unicode[STIX]{x1D70E}$ as well. Based on the right-hand side of (3.5), this solution is assumed to have the form $\unicode[STIX]{x1D713}_{1}(r,\unicode[STIX]{x1D703})=F_{12}(r)\sin 2\unicode[STIX]{x1D703}+F_{14}(r)\sin 4\unicode[STIX]{x1D703}+F_{16}(r)\sin 6\unicode[STIX]{x1D703}$ where each coefficient $F_{1n}(r)$ involves particular integer powers up to $r^{6}$ with some logarithmic terms. These $F_{1n}(r)$ include up to two undetermined multiples, say $c_{1n}$ and ${\hat{c}}_{1n}$ , of suitable solutions of the biharmonic equation with largest powers $r^{n}$ or $r^{n+2}$ , respectively. As applies for $\unicode[STIX]{x1D713}_{0}$ in (3.4), where the $O(r^{4})$ term was omitted, only terms in $\unicode[STIX]{x1D713}_{1}$ with powers of $r$ that are lower than or equal to those forced directly are retained, and hence $r^{8}$ terms are omitted from $F_{16}(r)$ .

The full solution for $\unicode[STIX]{x1D713}_{1}(r,\unicode[STIX]{x1D703})$ is not reproduced here but is available in Alias (Reference Alias2015). The pertinent detail is its behaviour for $r\gg 1$ , where

It will be shown in § 4.2.1 that $c_{0}$ , $d_{1}$ , ${\hat{c}}_{14}$ and $c_{16}$ are determined by matching the inner and the outer solutions, while the constants $c_{12}$ , ${\hat{c}}_{12}$ and $c_{14}$ that arise above remain unspecified. However, these constants are anticipated to be fixed by matching higher-order inner and outer solutions using the same principle as described in § 4.2.1. It is pertinent to recognise that all retained terms in (3.9) are multiples of $\unicode[STIX]{x1D70E}$ , and those which do not include that factor, for example arising from the second term on the right-hand side of (3.5), are not involved in the dominant behaviour of $\unicode[STIX]{x1D713}_{1}$ at this order.

3.2 Remarks on inner solutions as $r\rightarrow \infty$

Neglecting for the moment the terms arising from unknown constants, for $r\gg 1$ the largest terms in the expansion for the density perturbation $\bar{\unicode[STIX]{x1D70C}}=\unicode[STIX]{x1D70C}-\unicode[STIX]{x1D70C}_{b}$ are

and the largest terms of the streamfunction are

The second terms of these can become much larger than the leading terms $\unicode[STIX]{x1D713}_{0}$ and $\unicode[STIX]{x1D70C}_{0}$ for finite but small values of $\unicode[STIX]{x1D716}$ . Further, comparing the power of $r$ in these terms at each order and examining the size of forcing terms in the equations for $\unicode[STIX]{x1D70C}_{2}$ and $\unicode[STIX]{x1D713}_{2}$ based on (2.5) and (2.6), it is expected that $\unicode[STIX]{x1D70C}_{2}=O(r^{7}\ln r)$ and $\unicode[STIX]{x1D713}_{2}=O(r^{10}\ln r)$ for $r\gg 1$ . More generally, the largest forced power of $r$ in each term increases by four with each successive term of (3.1) and, although the higher-order terms are multiplied by powers of $\unicode[STIX]{x1D716}\ll 1$ , their sizes are comparable to the preceding terms when $r$ is $O(\unicode[STIX]{x1D70E}\unicode[STIX]{x1D716})^{-1/4}$ . This property is elucidated by introducing a scaled variable $R=\unicode[STIX]{x1D6FF}r=O(1)$ for large $r$ , where $\unicode[STIX]{x1D6FF}\ll 1$ , and comparing the sizes of the first and second-order terms in

Neglecting logarithmic quantities temporarily, the first two terms in both expansions have similar size when $\unicode[STIX]{x1D6FF}^{4}\sim \unicode[STIX]{x1D70E}\unicode[STIX]{x1D716}$ , or $\unicode[STIX]{x1D6FF}=(\unicode[STIX]{x1D70E}\unicode[STIX]{x1D716})^{1/4}$ . This scaling is examined in § 4 and it will be seen below to correspond to a distinguished limit of the governing equations, as was also noted by List (Reference List1971), and also more recently by Ardekani & Stocker (Reference Ardekani and Stocker2010) and Candelier et al. (Reference Candelier, Mehaddi and Vauquelin2014).

Before analysing the ‘outer region’ in § 4, for $r\gg 1$ it is worth noting that the leading term $(-\!\cos \unicode[STIX]{x1D703}/r)$ in (3.10) forces the first term in (3.11), with both also expected to be dominant terms at large $r$ in a similar analysis for symmetric objects of other shapes. For example, $\unicode[STIX]{x1D70C}_{0}(r,\unicode[STIX]{x1D703})$ at large $r$ for a slightly elliptical-shaped cylinder includes an $(\cos 3\unicode[STIX]{x1D703}/r^{3})$ term that forces a flow perturbation with $\unicode[STIX]{x1D713}_{0}(r,\unicode[STIX]{x1D703})=O(\ln r)$ for $r\gg 1$ . Objects with more complicated shapes force additional $(\cos k\unicode[STIX]{x1D703}/r^{k})$ terms with $k>3$ in the leading-order inner solution $\unicode[STIX]{x1D70C}_{0}$ , which lead to $O(r^{3-k}\ln r)$ terms in $\unicode[STIX]{x1D713}_{0}$ . The first terms of both (3.12) and (3.13) therefore correspond to the typical leading-order forcing by objects of almost any other closed shape in the $(x,z)$ plane, and not only the circular object here. The leading-order solution in § 4 may therefore be much more general than it may appear.

4.1 Numerical evaluation of inverse transforms

The inverse Fourier transforms solutions (4.20) and (4.21) can be approximated numerically, for example using Simpson’s rule in $\unicode[STIX]{x1D6FC}$ over a domain $(X,Z)$ . For the plots here the integral expressions for $\tilde{\unicode[STIX]{x1D713}}_{0}$ and $(\tilde{\unicode[STIX]{x1D70C}}_{0}+Z/R^{2})$ were evaluated using a uniform grid of $\unicode[STIX]{x1D6FC}$ with $\unicode[STIX]{x0394}\unicode[STIX]{x1D6FC}=0.01$ over $0\leqslant \unicode[STIX]{x1D6FC}\leqslant \unicode[STIX]{x1D6FC}_{max}$ , where $\unicode[STIX]{x1D6FC}_{max}$ is sufficiently large that all additional contributions are smaller than say $10^{-8}$ . Arguably, contour integration around branch cuts in the complex $\unicode[STIX]{x1D6FC}$ plane might yield more rapid convergence, but the approach here is simpler and not inconveniently time consuming.

Figure 2.

Contour plots of the leading-order streamfunction $\tilde{\unicode[STIX]{x1D713}}_{0}$ from (4.20) in the outer region $(X,Z)$ over (a) $-10\leqslant \{X,Z\}\leqslant 10$ and (b) for $-50\leqslant \{X,Z\}\leqslant 50$ . The contour intervals are 0.01 and 0.0025, respectively.

Figure 2 illustrates streamlines of the outer flow, and shows circulated fluid motion in each quadrant that is fastest close to the origin and directed in the same manner as near the cylinder in figure 1(b), but with no evidence of the flow reversal and a zero streamline at $r_{0}$ in this case. In effect, the cylindrical object in the outer region is represented by a point density disturbance at the origin. For the results shown here, the overall volume flux generated is $\max \{|\tilde{\unicode[STIX]{x1D713}}_{0}|\}\approx 0.1588$ , located near $(X,Z)=(\pm 2.36,\pm 2.12)$ , with maximum velocity components of $|\tilde{u} _{0}|\approx 0.1438$ and $|\tilde{w}_{0}|\approx 0.1469$ located nearby on the $X$ and $Z$ axes, respectively. One advantage of the integral form of the solutions (4.20) and (4.21) is that they can be evaluated accurately at individual points relatively easily, in contrast to a more typical approach of solving the governing equations (4.6) numerically. That said, to provide additional confidence that the analytical solutions presented here are correct, Alias (Reference Alias2015) also compared them with second-order finite-difference numerical solutions of (4.6) subject to (4.8b ) and with appropriate symmetry conditions on $X=0$ and $Z=0$ ; those results confirmed all of the key features noted above, including the contour shapes as well as the size and locations of the extrema.

From figure 2(b) it is clear that the motion in the outer-flow region extends far from its forcing at the origin, especially in the $X$ direction. The far-field motion decays rapidly as $|Z|$ increases but relatively slowly in $X$ , although note the small recirculating regions near $Z=\pm 10$ . Indeed, the broad horizontal orientation of the streamlines in figure 2(b) is consistent with the stratification constraining vertical fluid motion (Tritton Reference Tritton1988, p. 197). Similar horizontal-oriented contours have also been observed in a similar contexts by Zagumennyi & Chashechkin (Reference Zagumennyi and Chashechkin2013c ) for an unsteady flow, as well as the so-called $R^{1/3}$ layers described by Page (Reference Page2011b ) for example, although in those cases the region is much thinner than the size of the object.

Figure 3.

Contour plots of the leading-order density perturbation $\tilde{\unicode[STIX]{x1D70C}}_{0}$ from (4.21) in the outer region $(X,Z)$ over (a) $-10\leqslant \{X,Z\}\leqslant 10$ and (b) for $-50\leqslant \{X,Z\}\leqslant 50$ . The contour intervals are 0.01 and 0.0025, respectively (up to absolute values of one).

Figure 3 shows the density perturbation $\tilde{\unicode[STIX]{x1D70C}}_{0}$ , which is largest near the origin as a result of the inner matching condition (4.8b ). The perturbation density is consistent with bending of isopycnals that is evident near the cylinder for the inner solution $\unicode[STIX]{x1D70C}_{0}$ in figure 1(a), although recall the linear ‘background’ density profile is not part of $\tilde{\unicode[STIX]{x1D70C}}_{0}$ . For the results shown, the departure $|\tilde{\unicode[STIX]{x1D70C}}_{0}+Z/R^{2}|$ from the inner matching condition has a maximum of 0.2654 on the $Z$ axis near $(X,Z)=(0,\pm 3.24)$ , and a maximum $Z$ gradient of 0.1666 at the origin. As radius increases, the density perturbation $\tilde{\unicode[STIX]{x1D70C}}_{0}$ decreases in magnitude, again with relatively slow decay in the $X$ direction due to buoyancy forces that aim to restore the ‘deflected’ fluid to its neutral buoyant position.

The gradual spread in the $Z$ extent of $\tilde{\unicode[STIX]{x1D713}}_{0}$ in figure 2(b) and $\tilde{\unicode[STIX]{x1D70C}}_{0}$ in figure 3(b) as $X$ increases will be seen in § 4.3 to correspond to their far-field asymptotic behaviour, which can be represented in terms of the similarity variable $Z/X^{1/3}$ .

4.2 Asymptotic behaviour for $R\ll 1$

To complete and verify the expected matching between the inner and outer regions, asymptotic solutions at small outer radius $R=\sqrt{X^{2}+Z^{2}}\ll 1$ will be derived, assuming here that $\unicode[STIX]{x1D6FC}>0$ for simplicity. This matching also determines some of the unidentified constants in the inner solutions described in § 3. Asymptotic solutions of streamfunction and density perturbation for $R\ll 1$ are derived by finding approximations of $S$ and $T$ for $\unicode[STIX]{x1D6FC}\gg 1$ from (4.13), from which expansions for $\unicode[STIX]{x1D6FD}_{n}$ follow from (4.12). For example, the asymptotic behaviour of $F_{n}(\unicode[STIX]{x1D6FC})$ from (4.19) can be expanded in inverse powers in a Laurent series

with the equivalent for $G_{n}$ following from (4.15), and

with

It is anticipated that these expressions converge for real values of $\unicode[STIX]{x1D6FC}$ larger than $(4/27)^{1/4}$ , based on the location in the complex plane of zeroes of the denominator of all $F_{n}(\unicode[STIX]{x1D6FC})$ , taking into account branch points of $\unicode[STIX]{x1D706}_{n}(\unicode[STIX]{x1D6FC})$ within that radius.

Using the above, series expansions of $A_{0}$ and $B_{0}$ valid for $\unicode[STIX]{x1D6FC}\gg 1$ and $0<X\ll 1$ are

where $\text{e}^{-\unicode[STIX]{x1D6FC}X}$ (which arises from $\text{e}^{-\unicode[STIX]{x1D706}X}$ since $\unicode[STIX]{x1D706}\sim \unicode[STIX]{x1D6FC}$ ) is retained in these expansions to ensure uniformity and so that they match (4.17a,b ). Sufficient terms are included above to ensure accurate inversion up to $O(R^{6})$ when $R$ is small. Note that all expressions above assume $\unicode[STIX]{x1D6FC}>0$ and $X>0$ but can be extended to other values by inclusion of relevant factors of $\text{sgn}(\unicode[STIX]{x1D6FC})$ and/or $\text{sgn}(X)$ , noting that $A_{0}$ is odd in both variables while $B_{0}$ is odd in $\unicode[STIX]{x1D6FC}$ and even in $X$ .

To facilitate the exact inversion of these approximations for $A_{0}$ and $B_{0}$ , using (4.20) and (4.21), the improper integral

is introduced here for any $(X,Z)$ and integer values of $k$ . Although these integrals do not exist in the usual sense for $k\geqslant 2$ , and arguably the inverse transforms should also include contributions from $A_{0}$ and $B_{0}$ at smaller values of $\unicode[STIX]{x1D6FC}$ than where (4.26) and (4.27) apply, they can be interpreted and evaluated in the ‘generalised function’ sense of Lighthill (Reference Lighthill1958), for example. Further explanation of this approach, including an outline of how to derive expressions for (4.28) up to $k=4$ , is provided in the appendix. Values of $I_{k}(X,Z)$ for larger $k$ are listed in Alias (Reference Alias2015).

A series expansion for $\tilde{\unicode[STIX]{x1D713}}_{0}(X,Z)$ for $R\ll 1$ can then be obtained as

The first term in this matches with (4.8a ), even though that matching condition was not used directly in its determination. All of the $\arctan (Z/X)$ terms arising in the $I_{k}(X,Z)$ cancel here, so $\tilde{\unicode[STIX]{x1D713}}_{0}(X,Z)$ has no corresponding discontinuities in the $(X,Z)$ plane.

Similarly, $\tilde{\unicode[STIX]{x1D70C}}_{0}(X,Z)$ for $R\ll 1$ is given by

which matches with (4.8b ) as expected, again without any $\arctan (Z/X)$ terms present.

Here $C_{1}$ , $C_{2}$ and $C_{3}$ are constants that in principle are determined by the full exact expressions for $A_{0}$ and $B_{0}$ when (4.20) and (4.21) are evaluated over all $\unicode[STIX]{x1D6FC}$ , rather than based on (4.26) and (4.27) for $\unicode[STIX]{x1D6FC}>(4/27)^{1/4}$ . Indeed, one interpretation is that they arise from contributions to the Fourier inversion integral over $0<\unicode[STIX]{x1D6FC}<(4/27)^{1/4}$ that are not taken into account in individual terms of (4.26) and (4.27).

That said, the two constants $C_{2}$ and $C_{3}$ here can be identified by ensuring consistency between the asymptotic solutions, as for example substituting (4.29) and (4.30) into (4.6) requires that $C_{2}=C_{1}-5/6$ and $C_{3}=C_{1}-77/60$ . The value of $C_{1}$ cannot be identified in this manner although from (4.29) it clearly relates to $\unicode[STIX]{x2202}^{2}\tilde{\unicode[STIX]{x1D713}}_{0}/\unicode[STIX]{x2202}X\unicode[STIX]{x2202}Z$ when $R\ll 1$ . Differentiating (4.20) and (4.29) with respect to $X$ and $Z$ , and evaluating at $Z=0$ , gives

for $0<X\ll 1$ , which implies that

The estimated value of $C_{1}$ in (4.32) was found numerically using the full expressions for $\unicode[STIX]{x1D706}_{n}$ as a functions of $\unicode[STIX]{x1D6FC}$ as well as $F_{n}(\unicode[STIX]{x1D6FC})$ from (4.19), using a similar method to that described in § 4.1. This integral (4.31) converges slowly for large values of $\unicode[STIX]{x1D6FC}$ , arising from the $O(1/\unicode[STIX]{x1D6FC})$ behaviour of $\unicode[STIX]{x1D6FC}$ times the first term in (4.26), but this can be ameliorated by subtracting a corresponding multiple of $H(\unicode[STIX]{x1D6FC}-1)\exp (-\unicode[STIX]{x1D6FC}X)/\unicode[STIX]{x1D6FC}$ from the integrand, where $H$ is the Heaviside step function. This subtracted contribution can be evaluated exactly in terms of the exponential integral $\text{Ei}(-X)$ , and both significantly improves the decay rate of the integrand to $O(1/\unicode[STIX]{x1D6FC}^{5})$ when $\unicode[STIX]{x1D6FC}$ is large and accurately offsets the $\ln X$ term in the limit.

4.3 Asymptotic behaviour for $R\gg 1$

Using the same process as in § 4.2 but for $0<\unicode[STIX]{x1D6FC}\ll 1$ , the corresponding approximate forms for $\unicode[STIX]{x1D706}_{n}$ and $F_{n}$ when $X>0$ can be obtained as

and after simplification that

Based on these, the leading terms in $A_{0}$ and $B_{0}$ for small $\unicode[STIX]{x1D6FC}$ can be determined as

where terms involving $\exp (-\unicode[STIX]{x1D706}_{2}X)$ and $\exp (-\unicode[STIX]{x1D706}_{3}X)$ are neglected as they approach zero more rapidly when $X$ is large. Asymptotic approximations for the streamfunction (4.20) and density perturbation (4.21) at large values of $(X,Z)$ are therefore given by

and

Exact expressions for these integrals are not available, but they can be approximated numerically using the same approach as described in § 4.1. Plots of those show very similar features to figures 2(b) and 3(b), especially for $R\gg 1$ where they should correspond. This includes the narrow regions where $\tilde{\unicode[STIX]{x1D713}}_{0}$ and $\tilde{\unicode[STIX]{x1D70C}}_{0}$ change signs as $Z$ increases at fixed $X$ , indicating relatively minor reversals in the direction of the flow at the outer edge of the region along as well as small secondary peaks in the density perturbation.

Although the integrals (4.40) and (4.41) cannot be evaluated analytically, further information on their asymptotic properties can be identified based on Moore & Saffman (Reference Moore and Saffman1969). In the context of a related problem in rotating-fluid flow, they determined the asymptotic behaviour of a class of complex-valued integrals of the form

as $\unicode[STIX]{x1D70F}\rightarrow \pm \infty$ where $M<0$ relates to the power of $X$ that appears in the assumed similarity solutions. Substituting $\unicode[STIX]{x1D6FC}=pX^{-1/3}$ and $Z=\unicode[STIX]{x1D70F}X^{1/3}$ , (4.40) becomes

where $\text{Im}\{H_{-2/3}(\unicode[STIX]{x1D70F})\}$ is the imaginary part of (4.42) with $M=-2/3$ . Correspondingly,

for $M=-1$ , and the velocity component $\tilde{u} _{0}$ has the same form but involves $\cos$ rather than $\sin$ . Using (4.42), at the outer $Z$ extent of this region it can be shown that

for large $\unicode[STIX]{x1D70F}$ , where the higher-order expression for $\unicode[STIX]{x1D713}_{0}$ can be obtained by comparing the $X$ derivative of (4.40) with the second $Z$ derivative of (4.41). The expressions (4.45) clearly indicate that the far-field solutions decay algebraically for $Z\gg X^{1/3}$ , rather than exponentially.

More generally, the form of (4.43) and (4.44) verifies that all dependent variables in the far field, at large $(X,Z)$ , can be written as similarity solutions to leading order, in terms of $\unicode[STIX]{x1D70F}=Z/X^{1/3}$ . This behaviour is consistent with the gradual spreading apparent in figures 2(b) and 3(b). The form of $\unicode[STIX]{x1D70F}$ also accords with the large- $X$ case of (4.7) in which $\tilde{\unicode[STIX]{x1D6FB}}^{6}$ is replaced by $\unicode[STIX]{x2202}^{6}/\unicode[STIX]{x2202}Z^{6}$ . Further, as the asymptotic forms of $\tilde{\unicode[STIX]{x1D713}}_{0}$ and $\tilde{\unicode[STIX]{x1D70C}}_{0}$ are derived here from the exact solution in the outer region, both the correct values of $M$ and the relevant multiples of $H_{M}(\unicode[STIX]{x1D70F})$ can be identified unambiguously.

Related similarity solutions in $\unicode[STIX]{x1D70F}$ were described by Koh (Reference Koh1966) for a point source in a viscous stratified fluid, corresponding to $M=-1/3$ for both the horizontal velocity and density, and in that case there was a small region of reversed flow direction at the outer edge of the layer, as also observed above. The same solution was used to describe the so-called ‘ $R^{1/3}$ layers’ in diffusion-driven motion for contained flows at high Rayleigh numbers (Page & Johnson Reference Page and Johnson2008, Reference Page and Johnson2009), for which $\unicode[STIX]{x1D6FF}={\mathcal{R}}^{1/2}$ . The $M=-2/3$ case arose in Page (Reference Page2011b ) when describing high-order effects that occur for ‘corner-induced flows’, and more recently has been shown to be relevant to external corners in a similar context. This latter type of flow also accords with the large $X$ case of the two-dimensional ‘momentum jet’ considered by List (Reference List1971), for which the axisymmetric equivalent was described further by Ardekani & Stocker (Reference Ardekani and Stocker2010) and referred to as a ‘Stratlet’. Finally, Alias (Reference Alias2015) shows that far-field horizontal velocity and density perturbations due to the slow vertical motion of a cylinder can be represented using $M=-4/3$ in (4.42).