3.1 Flow field

Figure 2 displays isopycnals, streamlines and levels of the vertical velocity, $\boldsymbol{u}\boldsymbol{\cdot }\boldsymbol{e}_{z}-1$ , about the sphere for selected values of $Re,\,Fr$ and $Pr$ .

When viscous diffusion dominates momentum transport (figure 2 a–d), isopycnals are significantly distorted by the passing sphere only when $Pe\gg 1$ , i.e. for large enough $Pr$ (figure 2 b,d). In this case, the top-down symmetry of the velocity field is almost unaffected when $Fr\gg 1$ (figure 2 b), whereas no such symmetry subsists when $Fr\lesssim 1$ (figure 2 d). Instead, for low $Fr$ , a region with upward absolute velocities $\boldsymbol{u}\boldsymbol{\cdot }\boldsymbol{e}_{z}-1>0$ takes place at some distance downstream of the sphere. This specific structure is driven by the baroclinic torque which converts the positive radial density gradient encountered within the fluid column dragged by the sphere into a vortex ring with positive azimuthal vorticity, i.e. upward velocities near the symmetry axis. Decreasing $Fr$ and/or increasing $Pr$ increases the curvature of the streamlines (figures 2 b and 2 c), which yields closed flow regions (i.e. toroidal eddies) with size comparable to the body length when the vertical confinement imposed by the stratification is strong enough (figure 2 d).

In inertia-dominated situations (figure 2 e–l), the lower $Fr$ and the larger $Re$ and $Pr$ , the thinner the dragged fluid column and the larger the upward velocity on the wake axis. With $Fr=0.5$ and $Pr=700$ (figures 2 h and 2 l), the high- $Re$ wake structure is dominated by a thin upward jet (Torres et al. Reference Torres, Hanazaki, Ochoa, Castillo and Van Woert2000; Hanazaki et al. Reference Hanazaki, Kashimoto and Okamura2009a , Reference Hanazaki, Nakamura and Yoshikawa2015) with centreline velocities several times larger than the sphere speed. Internal waves (Mowbray & Rarity Reference Mowbray and Rarity1967) propagating upwards become salient when $Fr\lesssim 1$ (figure 2 g–h,k–l). For a given $Fr$ , the size of the closed regions is smaller than in the low- $Re$ regime; for low $Fr$ , they exhibit a V-shape due to the streamwise modulation of the vertical velocity by the internal waves (figure 2 g–h,k–l). The influence of $Pr$ is weaker than in the low- $Re$ regime, except within the jet region. In line with the findings of Torres et al. (Reference Torres, Hanazaki, Ochoa, Castillo and Van Woert2000), no standing eddy (which would correspond to vertical velocities less than $-1$ in figure 2) exists at the back of the sphere for $Re=100$ throughout the explored range of $Fr$ , in contrast with the homogeneous situation in which this structure is present for $Re\gtrsim 10$ (Batchelor Reference Batchelor1967).

3.2 Buoyancy-induced contributions to the drag

The various contributions to the drag acting on the sphere are shown in figure 3. The drag is only weakly affected by stratification effects for $Fr=\mathit{O}(10)$ , with less than $5\,\%$ (respectively $25\,\%$ ) increase compared to the ‘homogeneous’ drag at $Re=0.05$ (respectively $100$ ). In contrast, decreasing $Fr$ to lower values makes the drag increase dramatically whatever $Re$ , especially for large $Pr$ . With $Pr=700$ and $Fr=0.5$ , the drag almost doubles compared to its value in a homogeneous fluid when $Re=0.05$ , and is eight times larger when $Re=100$ . Decreasing $Pr$ to $0.7$ , while maintaining $Fr$ and $Re$ unchanged, reduces the drag enhancement by a factor of $5$ (respectively $2$ ) at low (respectively high) $Re$ .

Figure 3.

Contributions, as given in (2.16), to the vertical force on the sphere versus the Froude number for $Re=0.05$ (left) and $Re=100$ (right) with, from top to bottom, $Pr=0.7,\,7$ and $700$ . All contributions are normalized by the ‘homogeneous’ drag force, $F_{w}=\boldsymbol{F}_{w}\boldsymbol{\cdot }\boldsymbol{e}_{z}$ at the same $Re$ , so that $\boldsymbol{F}\boldsymbol{\cdot }\boldsymbol{e}_{z}/F_{w}=1+(\boldsymbol{F}_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}}+\boldsymbol{F}_{\unicode[STIX]{x1D70C}u}+\boldsymbol{F}_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D714}})\boldsymbol{\cdot }\boldsymbol{e}_{z}/F_{w}$ .

In all cases, the contribution $\boldsymbol{F}_{\unicode[STIX]{x1D70C}u}$ due to the inertial pressure correction is negative (i.e. it provides a downthrust) but is negligibly small in magnitude. The reason for this may be qualitatively understood by examining the behaviour of the right-hand side of (2.14), i.e. the momentum flux, in the vicinity of the body surface. As $\boldsymbol{u}_{w}$ and $\boldsymbol{u}_{\unicode[STIX]{x1D70C}}$ both obey a no-slip condition on $({\mathcal{S}})$ , continuity requires that their tangential (respectively normal) component varies linearly (respectively quadratically) with the distance $r-1$ to $({\mathcal{S}})$ in the immediate vicinity of the latter ( $r=||\boldsymbol{x}||$ ). Therefore $(\boldsymbol{u}_{\unicode[STIX]{x1D70C}}+\boldsymbol{u}_{w})\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}_{\unicode[STIX]{x1D70C}}+\boldsymbol{u}_{\unicode[STIX]{x1D70C}}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}_{w}$ varies as $(r-1)^{2}$ within this surface layer, forcing the momentum flux to vanish on $({\mathcal{S}})$ and be proportional to $r-1$ close to it. Consequently the source term in the Poisson equation (2.14) is nearly zero close to $({\mathcal{S}})$ , and variations of $p_{\unicode[STIX]{x1D70C}\text{u}}$ along the body surface essentially result from the distribution of the momentum flux at some distance from $({\mathcal{S}})$ . This non-locality limits the variations of $p_{\unicode[STIX]{x1D70C}\text{u}}$ along $({\mathcal{S}})$ , hence the magnitude of $\boldsymbol{F}_{\unicode[STIX]{x1D70C}\text{u}}$ . No such effect exists in the case of $\boldsymbol{F}_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}}$ , since the distribution of $p_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}}$ along $({\mathcal{S}})$ is mostly driven by the non-zero near-surface density gradients.

When $Re$ is low, the additional Archimedes force $\boldsymbol{F}_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}}$ is also negligible compared to the vortical contribution $\boldsymbol{F}_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D714}}$ , although its relative magnitude increases with $Pr$ . Both contributions are of the same order when $Re=100$ and $Pr$ is moderate. However, when $Pr$ is large, $\boldsymbol{F}_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D714}}$ provides again the major part of the extra drag at $Re=100$ , being roughly twice as large as $\boldsymbol{F}_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}}$ . These results reveal that modifications of the vorticity field resulting from the baroclinic torque play a pivotal role in the drag increase throughout the explored $Re$ -range. Conversely, density variations at the body surface play a negligible role at low $Re$ but their relative magnitude gradually increases with the Reynolds number. This state of affairs seriously questions the available attempts in which the stratification-induced drag has been modelled by evaluating the buoyancy provided by the volume of light fluid dragged by the body, especially in the low- $Re$ regime (Yick et al. Reference Yick, Torres, Peacock and Stocker2009), or for Reynolds numbers of some units (Srdic-Mitrovic et al. Reference Srdic-Mitrovic, Mohamed and Fernando1999).

4.1 Low-Reynolds-number range

We first consider the low- $Re$ range, in which $\ell _{\unicode[STIX]{x1D708}}=Re^{-1}$ . Depending on whether $Pe$ is small ( $Pr=\mathit{O}(1)$ ) or large ( $Pr\gg 1$ ), one then has $\ell _{\unicode[STIX]{x1D705}}=Pe^{-1}$ or $Pe^{-1/3}$ (Levich Reference Levich1962; Batchelor Reference Batchelor1980). If stratification is strong enough for the condition $\ell _{s}\ll \ell _{\unicode[STIX]{x1D708}}$ to be satisfied, inertial terms are negligible in (4.2). Then the balance between buoyancy and viscous effects reduces to

(4.3) $$\begin{eqnarray}Re^{-1}\underbrace{\unicode[STIX]{x1D6FB}^{2}\unicode[STIX]{x1D74E}_{\unicode[STIX]{x1D70C}}}_{\mathit{O}(u/\ell _{s}^{3})}\approx Fr^{-2}\underbrace{\unicode[STIX]{x1D735}\unicode[STIX]{x1D70C}\times \boldsymbol{e}_{z}}_{\mathit{O}(\unicode[STIX]{x1D70C}_{\ell }/\ell _{s})},\end{eqnarray}$$

so that the density disturbance, $\unicode[STIX]{x1D70C}_{\ell }$ , associated with the velocity disturbance, $u$ , obeys $\unicode[STIX]{x1D70C}_{\ell }\sim Fr^{2}Re^{-1}u/\ell _{s}^{2}$ . If moreover $\ell _{s}\ll \ell _{\unicode[STIX]{x1D705}}$ (which corresponds to configuration (a) in figure 4), advective effects are negligible in the linearized density balance (4.1) which then reduces to

(4.4) $$\begin{eqnarray}Pe^{-1}\underbrace{\unicode[STIX]{x1D6FB}^{2}\unicode[STIX]{x1D70C}}_{\mathit{O}(\unicode[STIX]{x1D70C}_{\ell }/\ell _{s}^{2})}\approx \underbrace{1-(\boldsymbol{u}_{\unicode[STIX]{x1D70C}}+\boldsymbol{u}_{w})\boldsymbol{\cdot }\boldsymbol{e}_{z}}_{\mathit{O}(u)}.\end{eqnarray}$$

Hence the driving term, $1-\boldsymbol{u}_{w}\boldsymbol{\cdot }\boldsymbol{e}_{z}$ , is balanced by diffusive effects, which requires $Pe^{-1}\unicode[STIX]{x1D70C}_{\ell }/\ell _{s}^{2}\sim u$ . Note that the buoyancy-induced advective term, $\boldsymbol{u}_{\unicode[STIX]{x1D70C}}\boldsymbol{\cdot }\boldsymbol{e}_{z}$ , subsists in (4.4), as it is assumed to have the same magnitude as the driving term, in contrast with the settling-induced advective contribution, $\boldsymbol{u}_{w}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D70C}$ . The disturbance $u$ being small but arbitrary, compatibility between (4.4) and (4.3) implies $\ell _{s}^{4}\approx Fr^{2}/(PeRe)$ . This defines a viscous–diffusive regime, which we refer to as R1, characterized by a stratification length scale

(4.5) $$\begin{eqnarray}\ell _{s}\approx \ell _{s_{1}}\equiv (Fr/Re)^{1/2}Pr^{-1/4}.\end{eqnarray}$$

The relevance of this viscous–diffusive length scale was first recognized by List (Reference List1971) and later rediscovered independently by Ardekani & Stocker (Reference Ardekani and Stocker2010).

To evaluate $\boldsymbol{F}_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D714}}$ and $\boldsymbol{F}_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}}$ under such conditions, the spatial distribution of the settling-induced velocity, $\boldsymbol{u}_{w}$ , about the body must be considered. At a distance $r$ from the sphere centre, the disturbance $\boldsymbol{u}_{w}-\boldsymbol{e}_{z}$ is dominated by the contribution of the Stokeslet, which makes it decay approximately as $r^{-1}$ for increasing $r$ . Departures from this dominant behaviour arise because of the presence of a dipole (required to satisfy the no-slip condition on $({\mathcal{S}})$ ), and of inertial corrections required for the solution to be valid throughout the fluid domain, including the outer region $r\gg Re^{-1}$ (Batchelor Reference Batchelor1967). These additional contributions make the settling-induced velocity field decay slightly faster than $\mathit{O}(r^{-1})$ , so that we may consider that, for $r\gg 1$ , $|\boldsymbol{u}_{w}\boldsymbol{\cdot }\boldsymbol{e}_{z}-1|$ approximately behaves as $r^{-(1+\unicode[STIX]{x1D6FC})}$ with $\unicode[STIX]{x1D6FC}\gtrsim 0$ . The vortical force, $\boldsymbol{F}_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D714}}$ , scales as the viscous traction at the body surface, $Re^{-1}\boldsymbol{n}\boldsymbol{\cdot }(\unicode[STIX]{x1D735}\boldsymbol{u}_{\unicode[STIX]{x1D70C}}+\unicode[STIX]{x1D735}\boldsymbol{u}_{\unicode[STIX]{x1D70C}}^{\text{T}})$ . On $({\mathcal{S}})$ , $||\boldsymbol{u}_{w}-\boldsymbol{e}_{z}||=1$ due to the no-slip condition, whereas at the radial position $r=1+\ell _{s1}$ , $||\boldsymbol{u}_{w}-\boldsymbol{e}_{z}||\ll 1$ provided $\ell _{s1}\gg 1$ , owing to the $r^{-(1+\unicode[STIX]{x1D6FC})}$ -decay. From a physical point of view, assuming $\ell _{s1}\gg 1$ implies that we are considering a characteristic stratification length scale much larger than the body size (or equivalently, that the sphere is seen as a point force). This condition is fulfilled throughout the range covered by the low- $Re$ computations. The variation $\unicode[STIX]{x1D6FF}u$ of $||\boldsymbol{u}_{w}-\boldsymbol{e}_{z}||$ from $r=1+\ell _{s1}$ to $r=1$ is then of $\mathit{O}(1)$ , and so is that of $||\boldsymbol{u}_{\unicode[STIX]{x1D70C}}||$ , due to our definition of the stratification length scale. Assuming that the above scaling for $\ell _{s1}$ , derived under the assumption $u\ll 1$ , holds up to $\unicode[STIX]{x1D6FF}u=\mathit{O}(1)$ leads to the conclusion that $\boldsymbol{n}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}_{\unicode[STIX]{x1D70C}}\sim 1/\ell _{s1}$ on $({\mathcal{S}})$ . Hence $\boldsymbol{F}_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D714}}\sim (Re\ell _{s})^{-1}$ , which in the viscous–diffusive R1 regime implies

(4.6) $$\begin{eqnarray}\boldsymbol{F}_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D714}}\equiv \boldsymbol{F}_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D714}1}\sim (FrRe)^{-1/2}Pr^{1/4}.\end{eqnarray}$$

The above result was obtained without considering the contribution of the pressure component $p_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D714}}$ . However the latter exists only to ensure that the velocity field $\boldsymbol{u}_{\unicode[STIX]{x1D70C}}$ is solenoidal (i.e. $p_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D714}}$ plays the role of a Lagrange multiplier), which implies that its variations round the body have the same order of magnitude as those of the viscous traction defined above. Hence the contributions of $p_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D714}}\boldsymbol{n}$ and $Re^{-1}\boldsymbol{n}\boldsymbol{\cdot }(\unicode[STIX]{x1D735}\boldsymbol{u}_{\unicode[STIX]{x1D70C}}+\unicode[STIX]{x1D735}\boldsymbol{u}_{\unicode[STIX]{x1D70C}}^{\text{T}})$ to $\boldsymbol{F}_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D714}}$ have the same magnitude whatever the Reynolds number (for the same reason, it is well known (Batchelor Reference Batchelor1967) that the surface shear stress (respectively pressure) provides $2/3$ (respectively $1/3$ ) of the drag force on a sphere in the low-Reynolds-number regime, and the same ratios hold in the limit of very large Reynolds numbers for a sphere obeying a shear-free condition, i.e. a bubble with a negligible inner viscosity (Kang & Leal Reference Kang and Leal1988)).

The force component $\boldsymbol{F}_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}}$ is directly proportional to the pressure component $p_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}}$ on $({\mathcal{S}})$ . From (2.12) and (2.13) one infers that $\unicode[STIX]{x1D735}p_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}}\sim -Fr^{-2}\unicode[STIX]{x1D70C}\boldsymbol{e}_{z}$ near the body, so that the surface value of $p_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}}$ scales as $Fr^{-2}\unicode[STIX]{x1D70C}_{\ell }z$ , with $z$ the vertical position with respect to the sphere centre. Hence one has $\boldsymbol{F}_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}}\sim Fr^{-2}\unicode[STIX]{x1D70C}_{\ell }$ , with $\unicode[STIX]{x1D70C}_{\ell }$ evaluated on $({\mathcal{S}})$ . To estimate $\unicode[STIX]{x1D70C}_{\ell }$ , one has to return to (4.4) and make use of the various estimates obtained above, which yields

(4.7) $$\begin{eqnarray}Pe^{-1}\underbrace{\unicode[STIX]{x1D6FB}^{2}\unicode[STIX]{x1D70C}}_{\mathit{O}(\unicode[STIX]{x1D70C}_{\ell }/\ell _{s1}^{2})}+\underbrace{\boldsymbol{u}_{\unicode[STIX]{x1D70C}}\boldsymbol{\cdot }\boldsymbol{e}_{z}}_{\mathit{O}(\unicode[STIX]{x1D70C}_{\ell }Re\,\ell _{s1}^{2}/Fr^{2})}\approx \underbrace{1-\boldsymbol{u}_{w}\boldsymbol{\cdot }\boldsymbol{e}_{z}}_{\mathit{O}(r^{-(1+\unicode[STIX]{x1D6FC})})}.\end{eqnarray}$$

Integrating the right-hand side twice and balancing with the diffusive term yields $\unicode[STIX]{x1D70C}_{\ell }\sim Pe\ell _{s1}$ (the non-zero, albeit small, $\unicode[STIX]{x1D6FC}$ avoids a logarithmic divergence during this integration). Similarly, provided $\ell _{s1}\gg 1$ , $\int _{r=1}^{r=1+\ell _{s1}}|\boldsymbol{u}_{w}\boldsymbol{\cdot }\boldsymbol{e}_{z}-1|dr\approx 1$ , so that on average $|\boldsymbol{u}_{w}\boldsymbol{\cdot }\boldsymbol{e}_{z}-1|$ is of $\mathit{O}(1/\ell _{s1})$ in between $r=1+\ell _{s1}$ and $r=1$ . Then, balancing $\boldsymbol{u}_{\unicode[STIX]{x1D70C}}\boldsymbol{\cdot }\boldsymbol{e}_{z}$ with this averaged driving term and assuming $\unicode[STIX]{x1D70C}\approx 0$ for $r\gtrsim 1+\ell _{s1}$ yields $\unicode[STIX]{x1D70C}_{\ell }\sim Fr^{2}/(Re\ell _{s1}^{3})$ at $r=1$ . Given (4.5), both estimates imply $\unicode[STIX]{x1D70C}_{\ell }\sim (ReFr)^{1/2}Pr^{3/4}$ near the sphere surface, from which one concludes that in the R1 regime

(4.8) $$\begin{eqnarray}\boldsymbol{F}_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}}\equiv \boldsymbol{F}_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}1}\sim Fr^{-3/2}Re^{1/2}Pr^{3/4}.\end{eqnarray}$$

Figure 5.

Variations of normalized force components $\boldsymbol{F}_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}}\boldsymbol{\cdot }\boldsymbol{e}_{z}/F_{w}$ (triangles, red online) and $\boldsymbol{F}_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D714}}\boldsymbol{\cdot }\boldsymbol{e}_{z}/F_{w}$ (circles, blue online) in the low- $Re$ regime ( $Re=0.05$ ); dash-dotted line: $Pr=0.7$ , solid line: $Pr=700$ .

Variations of the buoyancy-induced drag components with the Froude number predicted by (4.6) and (4.8) are compared with numerical data in figure 5 for $Re=0.05$ . Results (4.6) and (4.8) apply provided $\ell _{s_{1}}$ is much smaller than $\ell _{\unicode[STIX]{x1D708}}$ and $\ell _{\unicode[STIX]{x1D705}}$ . Hence if $Re$ and $Pe$ are both small, the R1 regime takes place provided that $Fr\ll \text{min}(Re^{-1}Pr^{1/2},Re^{-1}Pr^{-3/2})$ , which with $Re=0.05$ and $Pr=0.7$ implies approximately $Fr\ll 15$ . In contrast, if $Re$ is small but $Pe$ is large, the second of the above bounds changes into $Fr\ll Re^{1/3}Pr^{-1/6}$ , which with $Pr=700$ implies $Fr\ll 0.1$ at the same Reynolds number. Hence the R1 regime is only expected to exist for $Pr=0.7$ in figure 5. Panels (a) and (c) in figure 2 correspond to this regime. In figure 5, the $-3/2$ slope predicted by (4.8) covers the entire explored $Fr$ -range, whereas the $-1/2$ slope corresponding to (4.6) is only identified for $Fr\lesssim 1$ . Comparing predictions (4.6) and (4.8) indicates that, for a given $Fr$ , the ratio $\boldsymbol{F}_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}1}/\boldsymbol{F}_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D714}1}$ varies as $RePr^{1/2}$ . Hence $\boldsymbol{F}_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}}$ is expected to be negligibly small for $Re\ll 1$ , the buoyancy-induced drag being dominated by the vortical contribution in this limit. This is confirmed by figure 5 which shows that $\boldsymbol{F}_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}}$ is two orders of magnitude smaller than $\boldsymbol{F}_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D714}}$ throughout the R1 regime. At the moderate value $Pr=7$ , the low- $Fr$ low- $Re$ conditions considered in the DNS also belong to the R1 regime. These results may be used to check the $Pr^{1/4}$ -dependence predicted by (4.6), although the comparison is limited to one decade ( $0.7\leqslant Pr\leqslant 7$ ). As panels (a) and (b) in figure 3 indicate for $Fr=0.1$ , the ratio $\boldsymbol{F}_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D714}}(Pr=7)/\boldsymbol{F}_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D714}}(Pr=0.7)$ is close to 2.0, which compares reasonably well with the expected ratio $10^{1/4}\approx 1.8$ . The same check can be performed regarding the $Pr^{3/4}$ -dependence of $\boldsymbol{F}_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}}$ predicted by (4.8), although the numerical values are too small for the difference to be visible in figure 3. The corresponding ratio is found to be $5.6$ , which compares well with the predicted ratio $10^{3/4}=5.25$ . The scaling (4.6) is similar to that found by Candelier et al. (Reference Candelier, Mehaddi and Vauquelin2014) who computed the buoyancy-induced correction to the drag using matched asymptotic expansions. These authors made use of the viscous Richardson number, $Ri=Re/Fr^{2}$ , and found this correction, normalized by the Stokes drag, to be $0.66(PeRi)^{1/4}$ . Hence the corresponding force behaves as $Re^{-1}(PeRi)^{1/4}$ , in line with (4.6).

If stratification is such that $\ell _{\unicode[STIX]{x1D705}}\ll \ell _{s}\ll \ell _{\unicode[STIX]{x1D708}}$ (which corresponds to configuration (b) in figure 4), the transport of the density disturbance at distances of $\mathit{O}(\ell _{s})$ from the body is dominated by advective effects, be the Péclet number small or large. Thus, in the quasi-steady approximation, the mass balance (4.1) becomes at leading order

(4.9) $$\begin{eqnarray}\underbrace{\boldsymbol{u}_{w}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D70C}}_{\mathit{O}(\unicode[STIX]{x1D70C}_{\ell }/\ell _{s})}\approx \underbrace{(\boldsymbol{u}_{\unicode[STIX]{x1D70C}}+\boldsymbol{u}_{w})\boldsymbol{\cdot }\boldsymbol{e}_{z}-1}_{\mathit{O}(u)},\end{eqnarray}$$

so that $\unicode[STIX]{x1D70C}_{\ell }\sim \ell _{s}u$ , assuming that $||\boldsymbol{u}_{w}||\approx 1$ at a distance $r\approx 1+\ell _{s}$ from the sphere centre. Compatibility with the unchanged condition resulting from (4.3) then defines a viscous–advective regime, R2, characterized by (Chadwick & Zvirin Reference Chadwick and Zvirin1974; Zvirin & Chadwick Reference Zvirin and Chadwick1975)

(4.10) $$\begin{eqnarray}\ell _{s}\sim \ell _{s_{2}}\equiv (Fr^{2}/Re)^{1/3}.\end{eqnarray}$$

Still assuming $\ell _{s2}\gg 1$ , repeating the above reasoning implies that the scaling law for the buoyancy-induced vortical force is now

(4.11) $$\begin{eqnarray}\boldsymbol{F}_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D714}2}\sim (ReFr)^{-2/3}.\end{eqnarray}$$

Making again use of (4.3), the typical orders of magnitude in (4.9) are

(4.12) $$\begin{eqnarray}\underbrace{\boldsymbol{u}_{w}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D70C}}_{\mathit{O}(\unicode[STIX]{x1D70C}_{\ell }/\ell _{s2})}\,-\underbrace{\boldsymbol{u}_{\unicode[STIX]{x1D70C}}\boldsymbol{\cdot }\boldsymbol{e}_{z}}_{\mathit{O}(\unicode[STIX]{x1D70C}_{\ell }\,Re\,\ell _{s2}^{2}/Fr^{2})}\approx \,\underbrace{\boldsymbol{u}_{w}\boldsymbol{\cdot }\boldsymbol{e}_{z}-1}_{\mathit{O}(\ell _{s2}^{-(1+\unicode[STIX]{x1D6FC})})}.\end{eqnarray}$$

Given (4.10), both terms on the left-hand side are of $\mathit{O}(\unicode[STIX]{x1D70C}_{\ell }/\ell _{s2})$ . Hence with $\unicode[STIX]{x1D6FC}\rightarrow 0$ , the leading-order balance in (4.12) implies $\unicode[STIX]{x1D70C}_{\ell }\sim 1$ on $({\mathcal{S}})$ . The reasoning used in the R1 regime then immediately yields

(4.13) $$\begin{eqnarray}\boldsymbol{F}_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}2}\sim Fr^{-2}.\end{eqnarray}$$

If $Pe\ll 1$ but $Pr$ is large (a combination never met in present computations), the condition $\ell _{\unicode[STIX]{x1D705}}\ll \ell _{s}\ll \ell _{\unicode[STIX]{x1D708}}$ implies that the R2 regime takes place if the Froude number satisfies $Pr^{-3/2}Re^{-1}\ll Fr\ll Re^{-1}$ . In contrast, $\ell _{\unicode[STIX]{x1D705}}=Pe^{-1/3}$ if $Pe\gg 1$ (Levich Reference Levich1962; Batchelor Reference Batchelor1980), in which case the R2 regime takes place if the Froude number stands in the range $Pr^{-1/2}\ll Fr\ll Re^{-1}$ . With $Re=0.05$ and $Pr=700$ , hence $Pe=35$ , this condition corresponds to $0.04\ll Fr\ll 20$ . Variations of $\boldsymbol{F}_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D714}}$ with $Fr$ in figure 5 suggest that this regime actually takes place only up to $Fr\approx 1$ (panel d in figure 2 corresponds to this regime). Things are less clear at first glance with $\boldsymbol{F}_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}}$ for which the $-2$ slope is approximately found only for $Fr\gtrsim 5$ . The reason is that this contribution is affected by finite- $\ell _{s}$ corrections when $Fr$ is small enough because the estimate $\unicode[STIX]{x1D70C}_{\ell }\approx 1$ no longer holds. It may be shown that the most general prediction for the density disturbance obtained without assuming $\ell _{s2}\gg 1$ is actually $\unicode[STIX]{x1D70C}_{\ell }(r)\sim (\ell _{s2}/r)^{3}$ . Hence the general scaling for $\boldsymbol{F}_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}2}$ is $\boldsymbol{F}_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}2}\sim Re^{-1}(1+(Fr^{2}/Re)^{1/3})^{-3}$ , which reduces to (4.13) only in the limit $\ell _{s2}\gg 1$ . In contrast the general expression predicts $\boldsymbol{F}_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}2}\sim Re^{-1}$ when $\ell _{s2}$ is very small, i.e. when $Fr\ll 1$ . This is why in figure 5 the negative slope of the corresponding line is seen to decrease with $Fr$ (a qualitatively similar alteration of (4.8) is expected to take place when $\ell _{s1}\ll 1$ and may be computed with similar arguments; however this regime is not reached in present computations). Comparing (4.11) and (4.13) shows that $\boldsymbol{F}_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}2}/\boldsymbol{F}_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D714}2}$ varies as $Re^{2/3}$ at a given $Fr$ . Hence $\boldsymbol{F}_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}}$ is again expected to be negligibly small for $Re\ll 1$ in the R2 regime, but the ratio $\boldsymbol{F}_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}}/\boldsymbol{F}_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D714}}$ is larger than in the R1 regime, as confirmed by figure 5. Predictions corresponding to the R2 regime were obtained using matched asymptotic expansions by Zvirin & Chadwick (Reference Zvirin and Chadwick1975) who found the drag correction, normalized by the Stokes drag, to be $1.06Ri^{1/3}$ in the limit $Pe\rightarrow \infty$ . This implies a $Re^{-1}Ri^{1/3}$ -scaling of the buoyancy-induced drag, equivalent to (4.11).

Finally, an inertial–advective regime, R3, emerges if stratification is so weak that $\ell _{s}\gg \ell _{\unicode[STIX]{x1D708}}$ (hence $\ell _{s}\gg \ell _{\unicode[STIX]{x1D705}}$ , given the range of $Pr$ considered here). Under such circumstances, which correspond to configuration (c) in figure 4, advection dominates over viscous effects. Consequently, the relevant quasi-steady approximation of (4.2) to be considered is at leading order

(4.14) $$\begin{eqnarray}\underbrace{\boldsymbol{u}_{w}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D74E}_{\unicode[STIX]{x1D70C}}}_{\mathit{O}(u/\ell _{s}^{2})}+\underbrace{\boldsymbol{u}_{\unicode[STIX]{x1D70C}}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D74E}_{w}}_{\mathit{O}(u/\ell _{s}^{2})}-\underbrace{\unicode[STIX]{x1D74E}_{\unicode[STIX]{x1D70C}}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}_{w}}_{\mathit{O}(u/\ell _{s}^{2})}-\underbrace{\unicode[STIX]{x1D74E}_{w}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\boldsymbol{u}_{\unicode[STIX]{x1D70C}}}_{\mathit{O}(u/\ell _{s}^{2})}\approx Fr^{-2}\underbrace{\unicode[STIX]{x1D735}\unicode[STIX]{x1D70C}\times \boldsymbol{e}_{z}}_{\mathit{O}(\unicode[STIX]{x1D70C}_{\ell }/\ell _{s})},\end{eqnarray}$$

which implies $\unicode[STIX]{x1D70C}_{\ell }\sim Fr^{2}u/\ell _{s}$ . As the relevant density balance is still (4.9), which imposes $\unicode[STIX]{x1D70C}_{\ell }\sim \ell _{s}u$ , compatibility implies

(4.15) $$\begin{eqnarray}\ell _{s}\sim \ell _{s_{3}}\equiv Fr,\end{eqnarray}$$

and the reasoning that led to (4.6) immediately yields

(4.16) $$\begin{eqnarray}\boldsymbol{F}_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D714}3}\sim (FrRe)^{-1}.\end{eqnarray}$$

At $r=1+\ell _{s3}$ , the velocity disturbance due to the body motion is governed by the Oseen equation. Considering moderate Froude numbers such that $\ell _{s}\gtrsim Re^{-1}$ , we assume that $\boldsymbol{u}_{w}\boldsymbol{\cdot }\boldsymbol{e}_{z}-1$ still behaves approximately as $r^{-(1+\unicode[STIX]{x1D6FC})}$ with $\unicode[STIX]{x1D6FC}\gtrsim 0$ at such distances from the body, so that the typical orders of magnitude in (4.9) are

(4.17) $$\begin{eqnarray}\underbrace{\boldsymbol{u}_{w}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\unicode[STIX]{x1D70C}}_{\mathit{O}(\unicode[STIX]{x1D70C}_{\ell }/\ell _{s3})}\quad -\underbrace{\boldsymbol{u}_{\unicode[STIX]{x1D70C}}\boldsymbol{\cdot }\boldsymbol{e}_{z}}_{\mathit{O}(\unicode[STIX]{x1D70C}_{\ell }\,\ell _{s3}/Fr^{2})}\approx \,\underbrace{\boldsymbol{u}_{w}\boldsymbol{\cdot }\boldsymbol{e}_{z}-1}_{\mathit{O}(\ell _{s3}^{-(1+\unicode[STIX]{x1D6FC})})}.\end{eqnarray}$$

Again this yields $\unicode[STIX]{x1D70C}_{\ell }\sim 1$ , so that the scaling of $\boldsymbol{F}_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}3}$ is found to be similar to that of $\boldsymbol{F}_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}2}$ , viz.

(4.18) $$\begin{eqnarray}\boldsymbol{F}_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}3}\sim Fr^{-2}.\end{eqnarray}$$

Variations of $\boldsymbol{F}_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D714}}$ shown in figure 5 display the $-1$ slope predicted by (4.16) for both $Pr=0.7$ and $Pr=700$ when $Fr\gtrsim 5$ . Thus, the R1–R3 and R2–R3 transitions are found to take place for $Fr$ -values of some units in both cases (panel b in figure 2 corresponds to the lower limit of the R3 regime in the high- $Pr$ case). Mehaddi et al. (Reference Mehaddi, Candelier and Mehling2018) determined asymptotically the first-order drag corrections due to stratification or inertia effects in the limit $Re\ll 1$ and $\ell _{s_{1}}\gg 1$ but did not identify the R3 regime. This is because $Fr$ is much larger than $Re^{-1}$ there (provided $Pr\gtrsim 1$ ), which implies $\boldsymbol{F}_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D714}_{3}}\ll 1$ according to (4.16). Hence the corresponding buoyancy-induced drag correction is much smaller than the $\mathit{O}(1)$ Oseen inertial contribution, which was the leading-order correction computed in the limit $FrRe\gg 1$ by these authors.

To finish with the low- $Re$ range, it is interesting to have a look at the empirical low- $Re$ buoyancy-induced drag correction proposed by Yick et al. (Reference Yick, Torres, Peacock and Stocker2009). In this work, it was suggested that the relative increase of the drag coefficient behaves as $Ri^{1/2}$ provided $Re$ is small and $Pr$ is large (their equation (5.2)). This correlation was based on experimental and numerical results obtained for $Pr=700$ with two different Reynolds numbers, $Re=0.05$ and $0.5$ , and Froude numbers up to $Fr=2$ . With $Pr=700$ and $Re=0.05$ , one has $\ell _{\unicode[STIX]{x1D708}}\approx 20$ , $\ell _{\unicode[STIX]{x1D705}}\approx 0.3$ , $\ell _{s_{1}}\approx 0.9\,Fr^{1/2}$ and $\ell _{s_{2}}\approx 4.5Fr^{2/3}$ . Hence, according to the present analysis, the whole range $0.4\leqslant Fr\leqslant 2$ stands essentially in the R2 regime. Similarly, with $Pr=700$ and $Re=0.5$ , one has $\ell _{\unicode[STIX]{x1D708}}\approx 2$ , $\ell _{\unicode[STIX]{x1D705}}\approx 0.15$ , $\ell _{s_{1}}\approx 0.3\,Fr^{1/2}$ and $\ell _{s_{2}}\approx 2.1\,Fr^{2/3}$ . Hence data in these series approach the R1 regime for $Fr\approx 0.4$ (since $\ell _{s_{1}}\approx \ell _{\unicode[STIX]{x1D705}}$ there) and the R3 regime for $Fr\gtrsim 1$ (since $\ell _{s_{3}}\approx \ell _{\unicode[STIX]{x1D708}}$ there) but they also essentially stand in the intermediate R2 regime. As mentioned above, (4.11) implies that the relative drag increase behaves as $Ri^{1/3}$ in this regime, which suggests that, at a given $Fr$ , it should be approximately $2.2$ larger for $Re=0.5$ than for $Re=0.05$ when $Fr\gtrsim 1$ , in line with the ratio deduced from data reported by Yick et al. (Reference Yick, Torres, Peacock and Stocker2009). Conversely, in the R1 regime, equation (4.6) implies that the relative drag increase is proportional to $(Re/Fr)^{1/2}$ at a given $Pr$ , so that the ratio of the two relative drag increases should be close to $3.15$ for $Fr\approx 0.4$ , which is again in agreement with their data. Hence the correlation proposed in Yick et al. (Reference Yick, Torres, Peacock and Stocker2009) appears to be an ad hoc approximation which actually mixes two different asymptotic regimes (see § B.2 in appendix B for more details).

4.2 High-Reynolds-number range

When the Reynolds number is large, $\ell _{\unicode[STIX]{x1D708}}=Re^{-1/2}$ , owing to the presence of the viscous boundary layer (hereinafter abbreviated to VBL). In this $Re$ -range, the thickness of the diffusive boundary layer obeys $\ell _{\unicode[STIX]{x1D705}}=Re^{-1/2}Pr^{-1/3}$ (Acrivos Reference Acrivos1960; Levich Reference Levich1962), since we are only considering moderate-to-large Prandtl numbers. Although the scalings $\boldsymbol{F}_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D714}}\sim (Re\ell _{s})^{-1}\unicode[STIX]{x1D6FF}u$ and $\boldsymbol{F}_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}}\sim Fr^{-2}\unicode[STIX]{x1D70C}_{\ell }$ derived in § 4.1 still apply, the presence of the VBL, the thickness of which is much smaller than the body radius, prevents the existence of a simple expression for $\boldsymbol{u}_{w}$ about the sphere. Details of the $\boldsymbol{u}_{w}$ -distribution within the VBL are especially important in the strongly stratified R1 and R2 regimes, since the corresponding stratification length scales are by definition (much) smaller than $\ell _{\unicode[STIX]{x1D708}}$ . These features make the overall situation less tractable with qualitative scaling arguments than in the low- $Re$ case, restricting the validity range of the corresponding predictions, especially at low Froude number.

Let us first assume that the Froude number is large enough for the stratification length scale to be such that $\ell _{s}\gg \ell _{\unicode[STIX]{x1D708}}\,\text{Max}(1,Pr^{-1/3})$ . Under such conditions, neither molecular diffusion nor viscosity affects the buoyancy-induced flow, defining a high- $Re$ R3 regime. Panels $(i)$ and $(j)$ of figure 2 provide two examples of this regime. The corresponding situation is similar to the low- $Re$ R3 regime except that, close to the body, it involves much larger velocity gradients. This makes it necessary to determine the scaling of $\ell _{s}$ within the VBL. To this end, one has to recognize that, except in the top region of the sphere where the upward jet originates, variations in directions parallel to the body surface are much slower than in the radial direction. For this reason one has to distinguish between longitudinal ( $\unicode[STIX]{x1D735}_{\Vert }$ ) and radial ( $\unicode[STIX]{x1D735}_{\bot }$ ) gradient components, and longitudinal ( $L_{s}$ ) and radial ( $\ell _{s}$ ) buoyancy length scales. One also has to take into account the fact that $\boldsymbol{u}_{w}^{\Vert }$ and $\boldsymbol{u}_{w}^{\bot }$ are respectively of $\mathit{O}(1)$ and $\mathit{O}(\ell _{\unicode[STIX]{x1D708}})$ near the outer edge of the VBL, owing to continuity. The buoyancy-induced vorticity $\unicode[STIX]{x1D74E}_{\unicode[STIX]{x1D70C}}$ being of $\mathit{O}(u/\ell _{s})$ within the VBL, the leading-order approximations of (4.1) and (4.2) for $r=O(1+\ell _{\unicode[STIX]{x1D708}})$ respectively reduce to

(4.19) $$\begin{eqnarray}\displaystyle & \displaystyle \underbrace{\boldsymbol{u}_{w}^{\Vert }\boldsymbol{\cdot }\unicode[STIX]{x1D735}_{\Vert }\unicode[STIX]{x1D70C}}_{\mathit{O}(\unicode[STIX]{x1D70C}_{\ell }/L_{s})}\,+\,\underbrace{\boldsymbol{u}_{w}^{\bot }\boldsymbol{\cdot }\unicode[STIX]{x1D735}_{\bot }\unicode[STIX]{x1D70C}}_{\mathit{O}(\unicode[STIX]{x1D70C}_{\ell }\ell _{\unicode[STIX]{x1D708}}/\ell _{s})}\,\approx \,\underbrace{(\boldsymbol{u}_{\unicode[STIX]{x1D70C}}\,+\,\boldsymbol{u}_{w})\boldsymbol{\cdot }\boldsymbol{e}_{z}-1}_{\mathit{O}(u)}, & \displaystyle\end{eqnarray}$$

(4.20) $$\begin{eqnarray}\displaystyle & \displaystyle \underbrace{\boldsymbol{u}_{w}^{\Vert }\boldsymbol{\cdot }\unicode[STIX]{x1D735}_{\Vert }\unicode[STIX]{x1D74E}_{\unicode[STIX]{x1D70C}}}_{\mathit{O}(u/(\ell _{s}L_{s}))}\,+\,\underbrace{\boldsymbol{u}_{w}^{\bot }\boldsymbol{\cdot }\unicode[STIX]{x1D735}_{\bot }\unicode[STIX]{x1D74E}_{\unicode[STIX]{x1D70C}}}_{\mathit{O}(u\,\ell _{\unicode[STIX]{x1D708}}/\ell _{s}^{2})}\,+\cdots \approx Fr^{-2}\underbrace{\unicode[STIX]{x1D735}\unicode[STIX]{x1D70C}\times \boldsymbol{e}_{z}}_{\mathit{O}(\unicode[STIX]{x1D70C}_{\ell }/\ell _{s})}. & \displaystyle\end{eqnarray}$$

Terms on the left-hand side have the same magnitude provided that $L_{s}=\ell _{\unicode[STIX]{x1D708}}^{-1}\ell _{s}$ , and compatibility between (4.19) and (4.20) implies that, close to the body,

(4.21) $$\begin{eqnarray}\ell _{s}\sim \ell _{s3}\equiv \ell _{\unicode[STIX]{x1D708}}Fr=Re^{-1/2}Fr.\end{eqnarray}$$

It may then be concluded that the condition $\ell _{s}\gg \ell _{\unicode[STIX]{x1D708}}\,\text{Max}(1,Pr^{-1/3})$ defining the high- $Re$ R3 regime corresponds to values of the Froude number such that $Fr\gg \text{Max}(1,Pr^{-1/3})$ , i.e. $Fr\gg 1$ since we are not considering low Prandtl numbers. As $\ell _{s3}/\ell _{\unicode[STIX]{x1D708}}\gg 1$ , the driving term in (4.19) is small at $r\approx 1+\ell _{s3}$ , so that the approximation $\unicode[STIX]{x1D6FF}u\approx 1$ holds in between $({\mathcal{S}})$ and $r=1+\ell _{s3}$ . Hence the scaling $\boldsymbol{F}_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D714}}\sim (Re\ell _{s})^{-1}$ derived in § 4.1 is still valid, implying

(4.22) $$\begin{eqnarray}\boldsymbol{F}_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D714}3}\sim Fr^{-1}Re^{-1/2}.\end{eqnarray}$$

Within the VBL, spatial variations of the forcing term $1-\boldsymbol{u}_{w}\boldsymbol{\cdot }\boldsymbol{e}_{z}$ in (4.7) deeply differ from the low- $Re$ case. They now correspond to a boundary-layer-type solution, $F(\unicode[STIX]{x1D702})$ , with $\unicode[STIX]{x1D702}=(r-1)/\ell _{\unicode[STIX]{x1D708}}$ , supplemented by wake corrections. Unfortunately, the explicit form of $F(\unicode[STIX]{x1D702})$ is unknown except in the limit $\unicode[STIX]{x1D702}\ll 1$ , especially in the wake region where most of the distortion of the isopycnals takes place. For this reason, a direct qualitative integration of (4.19) from $r=1$ to $r=1+\ell _{s3}$ is not possible. This is why one has to resort to a different strategy to estimate $\boldsymbol{F}_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}3}$ . The alternative approach calls on a Lagrangian energy balance over an infinitesimal fluid element, from its initial arbitrary position to its final position close to the body surface. This balance is established in appendix C. Its final outcome is (C 6) which allows us to conclude that, close to the body but at distances from its surface larger than the VBL thickness, $\unicode[STIX]{x1D70C}_{\ell }\sim Fr(1+\mathit{O}(Re^{-1}))$ . Since $p_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}}$ scales as $Fr^{-2}\unicode[STIX]{x1D70C}_{\ell }\,z$ on $({\mathcal{S}})$ (see § 4.1), one has in this regime and provided $Re\gg 1$

(4.23) $$\begin{eqnarray}\boldsymbol{F}_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}3}\sim Fr^{-1}.\end{eqnarray}$$

Predictions (4.22) and (4.23) are confirmed in figure 6 in which both drag components are seen to exhibit a $-1$ slope beyond $Fr\approx 1$ for $Pr=0.7$ and, with some slight deviations, for $Pr=700$ . A crucial property revealed by (4.23) is that $\boldsymbol{F}_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}3}$ does not depend on $Re$ , while according to (4.22) $\boldsymbol{F}_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D714}3}$ still varies as $Re^{-1/2}$ . This is reflected in the relative magnitude of the two drag components which is now of $\mathit{O}(1)$ for $Re=100$ , although $\boldsymbol{F}_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D714}}$ is still typically twice as large as $\boldsymbol{F}_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}}$ as soon as $Fr$ is of $\mathit{O}(1)$ or larger (see also panels $d$ and $f$ in figure 3). This is in stark contrast with the low- $Re$ regime in which the vortical contribution is always responsible for almost the entire buoyancy-induced drag increase.

Figure 6.

Variations of normalized force components $\boldsymbol{F}_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}}\boldsymbol{\cdot }\boldsymbol{e}_{z}/F_{w}$ (triangles, red online) and $\boldsymbol{F}_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D714}}\boldsymbol{\cdot }\boldsymbol{e}_{z}/F_{w}$ (circles, blue online) in the high- $Re$ regime ( $Re=100$ ); dash-dotted line: $Pr=0.7$ , solid line: $Pr=700$ .

Let us now examine the two low- $Fr$ regimes. The viscous–diffusive R1 regime takes place if $\ell _{s_{1}}\ll ~\text{min}(\ell _{\unicode[STIX]{x1D708}},\,\ell _{\unicode[STIX]{x1D705}})$ , i.e. $Fr\ll ~\text{min}(1,Pr^{-1/6})$ , with $\ell _{s1}$ still given by (4.5). Since $Pr^{-1/6}\approx 0.3$ for $Pr=700$ , this regime may only be observed for $Pr=0.7$ and $Pr=7$ in present computations (figure 2 $(k)$ stands near the higher- $Fr$ bound of this regime). The reasoning that led to (4.6) remains unchanged. Despite the VBL structure, the estimate $\unicode[STIX]{x1D6FF}u=\mathit{O}(1)$ in between $r=1$ and $r=1+\ell _{s1}$ still holds, provided $\ell _{s1}$ , hence $Fr$ , is ‘not too small’. This may be seen by assuming that the flow near the sphere surface may locally be roughly approximated by Blasius’ boundary-layer solution over a flat plate. Then, considering for instance $Fr=0.1$ and $Pr=0.7$ , one has $\ell _{s1}/\ell _{\unicode[STIX]{x1D708}}=\unicode[STIX]{x1D702}\approx 0.35$ , a position at which the local tangential velocity predicted by Blasius’ solution is approximately $55\,\%$ of the free-stream velocity. Hence $u$ has changed from $u=1$ at the sphere surface to $u\approx 0.5$ at $r=1+\ell _{s1}$ for the shortest $\ell _{s1}$ , which makes the approximation $\unicode[STIX]{x1D6FF}u=\mathit{O}(1)$ still reasonable. Consequently the scaling law (4.6) for $\boldsymbol{F}_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D714}1}$ is unchanged at leading order. This is confirmed in figure 6 in the subrange $0.1\leqslant Fr\lesssim 0.3$ , in which the expected $-1/2$ slope is observed. This is also confirmed regarding the $Pr^{1/4}$ -dependence, by comparing results displayed in panels (d) and (e) of figure 3. Indeed, for $Fr=0.1$ , $\boldsymbol{F}_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D714}}(Pr=7)/\boldsymbol{F}_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D714}}(Pr=0.7)\approx 1.85$ , which is close to the expected ratio $10^{1/4}\approx 1.8$ . Similar to the R3 regime, one has to rely on the Lagrangian energy balance established in appendix C to estimate $\boldsymbol{F}_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}1}$ , since most of the distortion of the isopycnals takes place outside the sublayer $1<r\leqslant 1+\ell _{s1}$ . In this case, equation (C 6) implies $\unicode[STIX]{x1D70C}_{\ell }\sim Fr(1+\mathit{O}(Re^{-1/2}))$ near the body surface, which leaves the leading-order estimate for $\boldsymbol{F}_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}}$ unchanged, viz.

(4.24) $$\begin{eqnarray}\boldsymbol{F}_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}1}\sim Fr^{-1}.\end{eqnarray}$$

The $Fr^{-1}$ -behaviour is observed in figure 6 for $Pr=0.7$ and $Fr\lesssim 0.3$ . Again, the high- $Re$ predictions suggest that the vortical component to the buoyancy-induced drag decreases as $Re^{-1/2}$ , and the Archimedes-like component is independent of the Reynolds number. However, while the former is still larger than the latter for $Re=100$ in the case of the R3 regime, the situation is reversed in the R1 regime, the ratio $\boldsymbol{F}_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D714}}/\boldsymbol{F}_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}}$ being close to $0.5$ for $Fr=0.1$ according to figure 6. This situation is also to be compared with the low- $Re$ R1 regime in which this ratio is typically two orders of magnitude larger. Finally, it must be noticed that, although $\boldsymbol{F}_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D714}1}$ varies with the Prandtl number according to (4.6), (4.24) predicts that $\boldsymbol{F}_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}1}$ does not. Comparing panels (d) and (e) in figure 3 confirms that increasing $Pr$ from $0.7$ to $7$ leaves $\boldsymbol{F}_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}}$ unchanged in the low- $Fr$ range.

The R2 regime also exists when $Pr\gg 1$ , provided the Froude number is such that $\ell _{\unicode[STIX]{x1D705}}\ll \ell _{s}\ll \ell _{\unicode[STIX]{x1D708}}$ . If these inequalities hold, the relevant orders of magnitude in the vorticity equation arise from (4.3), implying $\unicode[STIX]{x1D70C}_{\ell }\sim Fr^{2}Re^{-1}u/\ell _{s}^{2}$ . In the density equation, they would be similar to those resulting from (4.19) if $\ell _{s2}$ were only ‘slightly smaller’ than $\ell _{\unicode[STIX]{x1D708}}$ , since the two components of $\boldsymbol{u}_{w}$ were evaluated near the outer edge of the VBL in (4.19). If this were the case, the density equation would imply $\unicode[STIX]{x1D70C}_{\ell }\sim (\ell _{s}/\ell _{\unicode[STIX]{x1D708}})u$ . However the decrease of both components of $\boldsymbol{u}_{w}$ with $\unicode[STIX]{x1D702}$ cannot be ignored if $\ell _{s2}$ is significantly smaller than $\ell _{\unicode[STIX]{x1D708}}$ , i.e. $\unicode[STIX]{x1D702}(\ell _{s2})\ll 1$ . In particular, Blasius’ solution suggests that $||\boldsymbol{u}_{w}^{\bot }||$ varies approximately linearly (respectively quadratically) with $\unicode[STIX]{x1D702}$ in the range $0.2\leqslant \unicode[STIX]{x1D702}\leqslant 0.7$ (respectively $0<\unicode[STIX]{x1D702}<0.2$ ). Hence in the former range, $||\boldsymbol{u}_{w}^{\bot }||\sim \ell _{s}$ , so that the corresponding advective term in (4.19) is of $\mathit{O}(\unicode[STIX]{x1D70C}_{\ell })$ , and the balance with the right-hand side implies $\unicode[STIX]{x1D70C}_{\ell }\sim u$ . Although there is no reason to expect $\boldsymbol{u}_{w}$ to vary as a power law of $\unicode[STIX]{x1D702}$ for $\unicode[STIX]{x1D702}\geqslant 0.2$ (Blasius’ solution rather suggests a hyperbolic tangent behaviour), the above two distinct scalings arising from the density equation may be qualitatively gathered in the form $\unicode[STIX]{x1D70C}_{\ell }\sim (\ell _{s}/\ell _{\unicode[STIX]{x1D708}})^{\unicode[STIX]{x1D6FD}}u$ , with $\unicode[STIX]{x1D6FD}$ decreasing from 1 to 0 as $\ell _{s}/\ell _{\unicode[STIX]{x1D708}}$ decreases from $\mathit{O}(1)$ - to $\mathit{O}(10^{-1})$ -values. Compatibility between the leading-order balances provided by (4.3) and (4.19) then yields

(4.25) $$\begin{eqnarray}\ell _{s}\sim \ell _{s2}\equiv Re^{-1/2}Fr^{2/(2+\unicode[STIX]{x1D6FD})},\end{eqnarray}$$

which predicts the R2 regime to take place in the $Fr$ -range $Pr^{-1/3}\ll Fr\ll 1$ , since the lower bound of $\ell _{s2}$ is reached with $\unicode[STIX]{x1D6FD}\approx 0$ . We previously showed that the estimate $\unicode[STIX]{x1D6FF}u=\mathit{O}(1)$ is still legitimate for stratification lengths significantly smaller than $\ell _{\unicode[STIX]{x1D708}}$ . So, in the above range of $\ell _{s}/\ell _{\unicode[STIX]{x1D708}}$ , the vortical component of the buoyancy-induced drag may still be considered to scale as $\boldsymbol{F}_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D714}2}\sim (Re\ell _{s2})^{-1}$ , i.e.

(4.26) $$\begin{eqnarray}\boldsymbol{F}_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D714}2}\sim Re^{-1/2}Fr^{-2/(2+\unicode[STIX]{x1D6FD})}.\end{eqnarray}$$

Therefore the above reasoning predicts that variations of $\boldsymbol{F}_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D714}}$ in the high- $Re$ R2 regime should be close to a power law with an exponent standing in between $-2/3$ and $-1$ , depending on the actual ratio $\ell _{s2}/\ell _{\unicode[STIX]{x1D708}}$ . For $Re=100$ and $Pr=700$ , $\ell _{\unicode[STIX]{x1D705}}\approx 0.1\ell _{\unicode[STIX]{x1D708}}$ and, with $\unicode[STIX]{x1D6FD}=0$ , $Pr^{-(2+\unicode[STIX]{x1D6FD})/6}=Pr^{-1/3}\approx 0.11$ . Consequently the R2 regime is expected to cover a limited extent in the range $0.1\ll Fr\ll 1$ . A subrange with a slope corresponding to an exponent in between $-2/3$ and $-1$ is observed in the corresponding plot of figure 6, confirming the above predictions.

The situation regarding the determination of the Archimedes-like component is similar to that discussed in the R1 regime, so that the scaling (4.24) applies to $\boldsymbol{F}_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}2}$ . In figure 6, the slope of the $\boldsymbol{F}_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}}$ -curve corresponding to $Pr=700$ is observed to change for $Fr\approx 0.3$ , similar to that of $\boldsymbol{F}_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D714}}$ . For lower $Fr$ , this slope is close to $-1$ as predicted by (4.24). Similar to the other two high- $Re$ regimes, the scaling (4.26) predicts that $\boldsymbol{F}_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D714}2}$ decreases as $Re^{-1/2}$ , while $\boldsymbol{F}_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}2}$ no longer varies with the Reynolds number. However, figure 6 shows that the former is still approximately twice as large as the latter for $Re=100$ , similar to the behaviour observed in the R3 regime.

Table 1.

Domains of existence of the various stratification regimes. Numbers in each row refer to the equation providing the corresponding scaling law for $\boldsymbol{F}_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D714}}$ and $\boldsymbol{F}_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}}$ , respectively. The bounds of some regimes have been simplified by considering only $Pr$ in the range $[1,\,\infty ]$ .