3.1 Experimental measurements

Since the lee waves are stationary in the frame of reference moving with the bluff body, it is possible to do a temporal average in this frame of reference. This amounts to do a change of variable from $x$ to a new streamwise coordinate $X=x+U_{B}t$ . To do so, the instantaneous divergence fields are translated by a distance $x=U_{B}t$ and the mean field is obtained at each location $X$ from the average of all available instantaneous divergence fields. This type of reconstruction is convenient since it removes the wake waves which are not stationary in this frame of reference. An example is given in figure 4 for three different Froude numbers. The lee waves propagate from the position of the sphere and have a complicated structure with dislocations in the isophases. These dislocations were not present in the experimental results of Bonneton et al. (Reference Bonneton, Chomaz and Hopfinger1993) probably due to the fact that the height of fluid is here smaller than in their experiments. Their results were in good agreement with the theory in an infinite fluid. Here, the presence of dislocations suggest that only a finite number of vertical Fourier modes are excited as obtained experimentally by Robey (Reference Robey1997) and explained theoretically by Vasholz (Reference Vasholz2011).

Figure 4.

Mean vertical velocity gradient $\unicode[STIX]{x2202}w/\unicode[STIX]{x2202}z$ in the frame moving with the sphere measured two diameters above the centre of the sphere at (a) $F=1$ , $Re=600$ , (b) $F=2$ , $Re=1200$ and (c) $F=4$ , $Re=2400$ . $z/H=0.629$ and $z_{0}/H=0.435$ .

By comparing the divergence fields at different Froude numbers, it is clear that the wavelength (dimensionalized by $D$ ) increases with the Froude number and that the amplitude (dimensionalized by $U_{B}/D$ ) decreases with the Froude number. This can be quantified by extracting $X$ -profiles of divergence at a given position $y$ . Figure 5 shows such a profile at $y=-3D$ for two different Froude numbers. The wavelength is well defined on this type of profiles and clearly increases with the Froude number. For each Froude number, the wavelength has been measured over five wavelengths; it is plotted in figure 6(a) as a function of the Froude number. The wavelength increases linearly with the Froude number. This can be intuitively understood by the fact that a fluid particle oscillates at the frequency $N/(2\unicode[STIX]{x03C0})$ and is advected at the velocity $U_{B}$ . The wavelength is thus $\unicode[STIX]{x1D706}=2\unicode[STIX]{x03C0}U_{B}/N$ which can be rewritten $\unicode[STIX]{x1D706}/D=\unicode[STIX]{x03C0}F$ , in excellent quantitative agreement with the experimental measurements.

Figure 5.

Mean vertical velocity gradient $\unicode[STIX]{x2202}w/\unicode[STIX]{x2202}z$ as a function of the downstream distance $x$ measured at $z=2D$ and $y=-3D$ for $F=1$ , $Re=600$ (solid line) and for $F=4$ , $Re=2400$ (dotted line). $z/H=0.629$ and $z_{0}/H=0.435$ .

Figure 6.

(a) Wavelength of the lee waves measured at $z=2D$ and $y=-3D$ over the first five wavelengths. (b) Amplitude of the mean divergence at the first minimum located at $X=6FD\pm 10\,\%$ , $y=-3D$ and $z=2D$ . The Reynolds number is equal to $Re=600F$ . $h=H/D=10.2$ , $z/H=0.629$ and $z_{0}/H=0.435$ .

Figure 6(b) plots the amplitude of the lee waves. The maximum divergence is not measured at the first maximum which is too close to the sphere and is thus very noisy. This amplitude is rather measured at the first minimum of the $X$ -profiles, and is usually very close to the largest amplitude. This minimum is obtained close to $X/D=6F$ for all Froude numbers, which indicates that not only the wavelength but also the phase is proportional to the Froude number.

Figure 6(b) shows that the amplitude varies linearly with the inverse of the Froude number for $F\in [1,16]$ . Such a scaling seems similar to the scaling obtained by Bonneton et al. (Reference Bonneton, Chomaz and Hopfinger1993) for a Froude number ranging from 1 to 2. Indeed, they obtained that the displacement $\unicode[STIX]{x1D701}$ of the isopycnals is proportional to $F^{-1}$ . However, this would mean that the vertical velocity $w=\text{d}\unicode[STIX]{x1D701}/\text{d}t\sim (U_{B}/\unicode[STIX]{x1D706})\unicode[STIX]{x1D701}$ scales as $F^{-2}$ (since $\unicode[STIX]{x1D706}$ is proportional to $F$ ) together with the vertical divergence. Our scaling of the vertical divergence as $F^{-1}$ is thus in disagreement with their measurements and their theoretical explanation. We will show in the following that their scaling is only valid at low Froude numbers and that an additional term must be taken into account at larger Froude numbers.

The various scalings of the amplitude and of the wavelength indicate that the lee waves seem to be self-similar such that $\unicode[STIX]{x2202}_{z}w$ must be dimensionalized by $U_{B}/(FD)$ and plotted as a function of $X/(FD)$ . This is what has been done in figure 7 for four different Froude numbers. The structure of the lee waves is now roughly independent of the Froude numbers. Most of the lee waves are emitted between the two black lines which correspond to angles equal to $15^{\circ }$ and $40^{\circ }$ for $F=1$ . However, at $F=1$ there is an additional structure of internal waves emitted at a smaller angle (located between the two white lines). We will show in the next theoretical section that this structure is a subcritical mode which can occur only at low Froude numbers. It also seems that the amplitude is slightly weaker for $F=8$ than for $F=1$ . It indicates that it decays slightly faster than $F^{-1}$ , especially for $F<2$ .

Figure 7.

Mean vertical velocity gradient $\unicode[STIX]{x2202}w/\unicode[STIX]{x2202}z$ renormalized by the Froude number measured two diameters above the centre of the sphere at (a) $F=1$ , $Re=600$ , (b) $F=2$ , $Re=1200$ , (c) $F=4$ , $Re=2400$ and (d) $F=8$ , $Re=4800$ . $z/H=0.629$ and $z_{0}/H=0.435$ .

3.2 Lighthill theory

The calculation is adapted from the classical Lighthill theory and only the key points are given here. The details of the calculation are given in appendix A. For low amplitude, the structure of the lee waves is governed by the linearized Navier–Stokes equations for the velocity $\boldsymbol{u}=(u,v,w)$ in the Boussinesq approximation, together with the density equation and the incompressibility:

Here, ${\mathcal{Q}}$ and ${\mathcal{F}}$ are two forcing terms stationary in the frame moving at the bluff body velocity. ${\mathcal{Q}}$ is the volumetric forcing term created by the moving boundaries of the bluff body. It is taken as a dipole in $x$ with a strength taken such that the induced velocity is equal to $-U_{B}$ at the surface of the sphere:

where $z_{0}$ is the vertical position of the sphere. ${\mathcal{F}}$ corresponds to the force exerted by the sphere on the fluid, which has been often neglected in previous studies. We have assumed that the force term has only an $x$ component ${\mathcal{F}}_{x}$ (no lift term). It is taken as a point force of strength given by the drag coefficient:

Manipulating these equations to eliminate variables leads to a forced internal wave equation for the vertical velocity component:

In the $z$ direction, the lee waves propagate rapidly such that they quickly fill the small height $H$ of fluid. The velocity is thus decomposed in a discrete sum of modes in the vertical direction. Since the buoyancy frequency is nearly constant in the experiment, the orthonormal modes are simply sine functions:

Introducing this decomposition into (3.4) and taking the Fourier transform in $x$ and $y$ leads to an expression (see detailed calculation in appendix A) for the vertical velocity

where the volumetric mode is given by

and the drag mode is given by

with $\hat{n}=n\unicode[STIX]{x03C0}D/(2H)$ and $s_{n}=0$ for supercritical modes (defined by $\hat{n}F>1$ ) and $s_{n}=(1-\hat{n}^{2}F^{2})^{1/2}$ for subcritical modes (defined by $\hat{n}F<1$ ). It can be noted that the volumetric mode $I_{n}^{vol}(X,y)$ has a structure similar to the drag mode $I_{n}^{vol}(X,y)$ but in phase quadrature (see figure 22 in appendix A). The vertical velocity gradient $\unicode[STIX]{x2202}w/\unicode[STIX]{x2202}z$ is obtained from this formula by replacing $\sin (n\unicode[STIX]{x03C0}z/H)$ by $n\unicode[STIX]{x03C0}\cos (n\unicode[STIX]{x03C0}z/H)/H$ . In the volume and drag modes, the integrals are difficult to calculate since they strongly oscillate at large $X$ and $y$ . They can be approximated using the stationary phase method as shown in appendix B. However, this approximation fails at the cusps such that the full solution is used in the core of the paper. Numerically, the integrals were evaluated using the standard integration function ‘quad’ of Matlab which uses a recursive adaptive Simpson quadrature method. The tolerance was set to $10^{-7}$ . The inviscid solution for $\unicode[STIX]{x2202}w/\unicode[STIX]{x2202}z$ is plotted in figure 8(a) for $F=4$ with a drag coefficient given by Blevins (Reference Blevins1984) as $C_{D}=0.4$ for the sphere in a homogeneous fluid for Reynolds numbers between $10^{3}$ and $10^{5}$ . The drag coefficient is only weakly affected by the stratification for $F>2$ . This solution presents dislocations (near $y/D=-3$ for $X/(FD)$ between 10 and 30) as in the experiments, which are due to interference between the discrete modes.

Figure 8.

Theoretical prediction of the vertical velocity derivative $\unicode[STIX]{x2202}w/\unicode[STIX]{x2202}z/(U_{B}/FD)$ at (a) $Re=+\infty$ and (b) $Re=2400$ for a sphere with $F=4$ , $h=H/D=10.2$ . $z/H=0.629$ and $z_{0}/H=0.435$ . Black lines correspond to 15 and 40 degree angles which surround most of the lee waves measured experimentally (plotted in figure 7). Thirty modes have been used in order to decreases the error down to 20 % in the viscous case. Robey (Reference Robey1997) only observed 2–3 modes since he used a larger sphere with a diameter of the order of the pycnocline thickness, leading to strong finite size effects (see § C.1).

However, it is clear that this solution is very different from the experimental result of figure 7(c). Indeed, the theoretical lee waves extend very far downstream close to the $X$ axis (up to $X/(FD)=60$ ) whereas they quickly decay close to the $X$ axis in the PIV measurements. This is due to the presence of high modes in the theory (i.e. with high vertical wavenumber $n\unicode[STIX]{x03C0}/H$ ). In the experiment, these modes are viscously damped and the theory must be modified in order to add viscous effects as shown by Scase & Dalziel (Reference Scase and Dalziel2006).

This is done by multiplying the amplitude of the wave by a decay factor $\exp (-\unicode[STIX]{x1D708}k^{2}t)$ where $k$ is the norm of the wavevector and $t=X/U_{B}$ . This term can be calculated as $\exp (-\unicode[STIX]{x1D708}Xn^{2}\unicode[STIX]{x03C0}^{2}/(U_{B}H^{2}(1-s)^{2}))$ and must be multiplied to the integrand of (3.7) and (3.8). This leads to a very different theoretical solution, as shown in figure 8(b) for the same Reynolds number as in the experiment $Re=2400$ . The internal waves are strongly damped close to the $X$ -axis for $X/FD$ larger than 20. The structure is now closer to the structure found experimentally. It should be noted that all experiments have been done with the bluff body towed close to mid-height ( $z_{0}/H\sim 1/2$ ). As seen in (3.6), the amplitude of odd modes is close to zero since $w$ is proportional to $\cos (n\unicode[STIX]{x03C0}z_{0}/H)$ . The amplitude of these modes is thus highly sensitive to the position of the bluff body, to the exact profile of stratification and to other additional effects. Several perturbative effects have been added to the theory (non-Boussinesq effects, surface waves at the free surface and finite size of the sphere) in order to explain this discrepancy (see appendix C). But their influence on the lee waves is negligible. It is possible that the experimental attenuation of the lee waves comes from a loss of coherence far from the wake axis due to the turbulent motions of the wake or to the nonlinearity of the density profile at the bottom and at the free surface.

For a quantitative comparison, the $x$ -wavelength of the lee waves has been measured theoretically on the first five lobes at $y/D=-3$ as in the experiment. The wavelength is proportional to the Froude number as in the experiment. It is equal to

within 0.3 %, which slightly overestimates the experimental values by 30 %. Furthermore, the amplitude of the lee waves has been measured as the minimum (over $X$ ) of $\unicode[STIX]{x2202}w/\unicode[STIX]{x2202}z$ at $y/D=-3$ . This minimum is obtained close to $X/(FD)=7$ , which is close to the experimental position of the minimum ( $X/(FD)=6$ ).

The amplitude is plotted in figure 9 as a function of the Froude number. The amplitude is first calculated for the volume source term ${\mathcal{Q}}$ only, then for the drag term ${\mathcal{F}}_{x}$ only and finally for both source terms. It is clear from this figure that the volume source term is negligible compared to the drag term at moderate and large Froude numbers (for $F>2$ ). This can be explained by the fact that in (3.6) the volume mode $I_{n}^{vol}$ is divided by $F^{2}$ whereas the drag mode $I_{n}^{drag}$ is divided by $F$ . As a consequence, the amplitude of the drag (respectively volume) term scales as $F^{-1}$ (respectively $F^{-2}$ ) in figure 9. The total amplitude thus scales as $F^{-1}$ for $F>2$ , in excellent quantitative agreement with the experimental result. For Froude numbers smaller than 1, the volume term may become dominant, leading to a different scaling. This can explain the scaling measured by Bonneton et al. (Reference Bonneton, Chomaz and Hopfinger1993) for $1<F<2$ , which can be translated into $\unicode[STIX]{x2202}w/\unicode[STIX]{x2202}z\sim F^{-2}$ . For larger Froude numbers, Robey (Reference Robey1997) had noted that the amplitude was smaller in the model than in the experiments. He fixed the problem by taking into account the recirculation bubble (of finite downstream size) behind the bluff body. This translated the maximum of the amplitude toward larger Froude numbers but did not modify the scaling. Here, we have considered the drag which creates a recirculating jet (of infinite downstream size) behind the bluff body, leading to the separation of the flow. This extension of Robey’s ideas gives a better agreement on a larger range of Froude numbers.

Figure 9.

Amplitude of the vertical velocity derivative $\unicode[STIX]{x2202}w/\unicode[STIX]{x2202}z$ at $y/D=-3$ and $z/D=2$ . The theory is obtained for the volume source term only (dotted line), for the drag source term only (dashed line) and for both source terms (solid line). $Re=600F$ , $h=H/D=10.2$ , $z/H=0.629$ and $z_{0}/H=0.435$ .

It is very interesting to see that the drag term is in fact the dominant term at moderate and large Froude numbers. Indeed, most of the theoretical studies have been done by considering the volume source term only. This simple and poorly known contribution should probably be taken into account in most geophysical stratified wakes.

3.3 Influence of the shape of the bluff body

Experiments using two different bluff bodies have been done in order to quantify the influence of the shape of the bluff body. First, a 6 : 1 cylinder has been translated horizontally along its axis. The rescaled divergence has been measured by PIV and averaged in the frame of reference of the bluff body as done with the sphere. This quantity (which reveals the lee waves) is plotted in figure 10 for 4 different Froude numbers. As for the sphere, the lee waves contain a subcritical mode for $F=1$ only, which propagates at an angle between $6^{\circ }$ and $12^{\circ }$ (plotted as white lines). The lee waves also contain supercritical modes for all Froude numbers, which propagate at an angle between $15^{\circ }$ and $40^{\circ }$ (plotted as black lines) exactly as for the sphere. The isophases exhibit dislocations which reveal the interference of several supercritical modes.

Figure 10.

Mean vertical velocity gradient $\unicode[STIX]{x2202}w/\unicode[STIX]{x2202}z$ renormalized by the Froude number measured two diameters above the centre of the 6 : 1 cylinder at (a) $F=1$ , $Re=740$ , (b) $F=2$ , $Re=1480$ , (c) $F=4$ , $Re=2960$ and (d) $F=8$ , $Re=5920$ . $z/H=0.606$ and $z_{0}/H=0.403$ .

Second, a 6 : 1 spheroid has been translated along its major axis. The lee waves of the spheroid are shown in figure 11 for four different Froude numbers. They are exactly similar to the lee waves of the cylinder and the sphere. They exhibit a subcritical mode for $F=1$ only and several supercritical modes for all Froude numbers.

Figure 11.

Mean vertical velocity gradient $\unicode[STIX]{x2202}w/\unicode[STIX]{x2202}z$ renormalized by the Froude number measured two diameters above the centre of the 6 : 1 spheroid at (a) $F=1$ , $Re=680$ , (b) $F=2$ , $Re=1360$ , (c) $F=4$ , $Re=2720$ and (d) $F=8$ , $Re=5440$ . $z/H=0.596$ and $z_{0}/H=0.401$ .

The wavelength of the lee waves have been measured for the cylinder and the spheroid. They are plotted in figure 12(a) and show a linear variation with the Froude number for $1<F<8$ . They are slightly larger than the wavelength of the sphere (by approximately 5 % for the cylinder and by approximately 15 % for the spheroid). The model obtained from the Lighthill theory (plotted as a solid line in figure 12 a) is in fair agreement with these experimental measurements although overpredicting the wavelength by 20 %.

Figure 12.

(a) Wavelength of the lee waves measured at $z=2D$ and $y=-3D$ over the first five wavelengths. (b) Amplitude of the mean divergence at the first minimum located at $x=6FD\pm 20\,\%$ , $y=-3D$ and $z=2D$ . Symbols correspond the experimental measurements for a sphere (○), a 6 : 1 cylinder (▫) and a 6 : 1 spheroid (▵). Solid lines correspond to the theory for a sphere with the volumetric and the drag terms. The Reynolds number is approximately equal to $Re=600F$ .

The amplitude $\unicode[STIX]{x2202}w/\unicode[STIX]{x2202}z$ has also been measured at the first minimum of the $x$ -profiles at $y=-3D$ . For the 6 : 1 cylinder, the amplitude seems to scale as $F^{-1}$ as for the sphere. This indicates that the lee waves are created by the drag term rather than the volumetric term. Moreover, the amplitude is approximately twice as large for the cylinder as for the sphere. This is consistent with the fact that the drag coefficient is approximately twice as large for a cylinder as for a sphere. It thus seems that the Lighthill theory developed in the last subsection successfully predicts the amplitude of the lee waves of a cylinder since the drag term in (3.6) is proportional to the drag coefficient. For the spheroid, the amplitude is smaller than the amplitude of the sphere for $F>5$ . This is consistent with the fact that the drag coefficient is approximately twice smaller for a spheroid than for a sphere. However, for $F<5$ , the amplitude is larger for the spheroid than for the sphere. This could be explained by the fact that the volumetric term (which becomes dominant at small Froude numbers) may be larger for the spheroid than for the sphere. As a consequence, it seems that the amplitude of the lee waves scale as $F^{-2}$ for the spheroid.

4.1 Phase velocity of the wake waves

To do so, the phase velocities $U_{\unicode[STIX]{x1D719}}$ of the wake waves need to be determined accurately. They have been measured on spatio-temporal diagrams similar to the one in figure 3. They are reported in figure 13 for $F=8$ , where each symbol corresponds to a line identified in the spatio-temporal diagram. The phase velocity $U_{\unicode[STIX]{x1D719}}$ corresponds to the inverse of the slope and the time is taken at the middle of the line. Figure 13 indicates that the phase velocity is nearly constant in time although the waves are measured on a large period (between $Nt=10$ and $Nt=100$ ). Moreover, the phase velocity is independent of the location at which it is measured (in $y$ and in $z$ ) and also independent of the Reynolds number.

Figure 13.

Phase velocity of the wake waves as a function of time measured at $z=1D$ (circles) and at $z=2D$ (triangles). Filled symbols are obtained at $y=1.5D$ and open symbols at $y=5D$ . $F=8$ , $Re=4850$ (except for ▿: $Re=1260$ ).

Figure 14.

Phase velocity of the wake waves (a) as a function of the Froude number and (b) as a function of time rescaled by $F^{2/3}$ . Dotted symbols correspond to the experimental results of Robey (Reference Robey1997). The solid line corresponds to the wake velocity defect given by Spedding (Reference Spedding1997), taken from (4.1) at $Nt=2$ in (a). Symbols in (b) correspond to the same Froude numbers as in (a). $z=2D$ .

However, the phase velocity $U_{\unicode[STIX]{x1D719}}$ strongly depends on the Froude number, as shown in figure 14(a). These experimental measurements are compared to the phase velocities measured by Robey (Reference Robey1997) (small black round symbols) which are larger by up to a factor 2. This is probably because the results of Robey (Reference Robey1997) are obtained for a thin thermocline (approximately ten times smaller than the sphere) whereas our results are obtained for a thick thermocline (about ten times larger than the sphere). The phase velocity scales as $F^{-2/3}$ and corresponds exactly to the velocity defect of the wake at $Nt=2$ (shown as a solid line in figure 14 a) and given by Spedding (Reference Spedding1997) as

As already explained by Robey (Reference Robey1997), this indicates that the waves are probably emitted by the coherent structures (pancake vortices) which move at the wake velocity. For $Nt\ll 2$ , the local Froude number of the vortices is larger than 1 such that the waves emitted have a very small amplitude. At $Nt=2$ , the local Froude number is equal to 1 such that the amplitude of the emitted waves is maximum. This creates a burst of internal waves which propagate slowly in the vertical direction for $Nt>2$ . These waves are then observed between $Nt=10$ and $Nt=100$ when they reach the measurement plane (see figure 14). As time increases, the velocity defect decreases such that the amplitude of the emitted waves decreases. This probably creates slower and less energetic waves that may be observed later. This effect could explain the weak decay (by approximately 30 %) observed in figure 14(b) between the early waves ( $Nt=10$ ) and the late waves ( $Nt=100$ ). This effect is clearly visible in the simulations of Zhou & Diamessis (Reference Zhou and Diamessis2016) since the measurement plane is farther ( $z/D=9$ ). It should be noted that there are large fluctuations (of the order of 50 %) of the phase velocity in a single experiment. These fluctuations are much larger than the noise in the measurement (of the order of 10 %) and reflect the strong fluctuations of the turbulent wake itself.

Figure 15.

Velocity divergence (a–c) and corresponding $x$ -spectra (d–f) of the wake waves. The fields are obtained at $Nt=16.3$ (a,d), 60 (b,e), 130 (c,f), averaged over six velocity fields, i.e.  $N\unicode[STIX]{x0394}t=16$ . $F=8$ , $Re=4850$ , $z=2D$ .

4.2 Divergence fields

This accurate characterization of the phase velocity is now used to average the velocity fields in the frame of reference moving with the wake waves at $U_{\unicode[STIX]{x1D719}}$ . An example of the divergence $\unicode[STIX]{x2202}w/\unicode[STIX]{x2202}z$ is plotted in figure 15(a–c) averaged over an optimized duration ( $\unicode[STIX]{x0394}t=8.5~\text{sec}$ ). At $Nt=16$ , the wake waves are clearly visible with isophases oriented at $50^{\circ }$ with respect to the $x$ -axis. At early times, the waves are located directly above the sphere wake (around $y=0$ ). At later times, the waves propagate away from the wake centreline, creating two bands filled with internal waves on each side of the centreline (for $2<|y|/D<8$ at $Nt=60$ ). At very late stages, some waves are visible very far from the centreline (for $6<|y|/D<18$ at $Nt=130$ ). However, a dislocation at $|y|/D=10$ tends to indicate that the waves at $|y|/D>10$ are due to a reflection of the waves at the free surface.

These internal waves are similar to the wake waves obtained numerically by Abdilghanie & Diamessis (Reference Abdilghanie and Diamessis2013) and observed experimentally by Robey (Reference Robey1997). They also bear some similarities with the ‘random waves’ observed by Bonneton et al. (Reference Bonneton, Chomaz and Hopfinger1993). However, the ‘semi-circular concentric waves’ observed at a smaller wavelength by Bonneton et al. (Reference Bonneton, Chomaz and Hopfinger1993) are not visible here.

The velocity divergence fields have been Fourier transformed in $x$ in order to measure the $x$ -wavelength. At each cross-stream position $y$ , the $x$ -Fourier transforms are calculated and then plotted as a stack in figure 15(d–f). The spectra exhibit a peak which propagates towards large $y$ while remaining at a fixed wavenumber ( $K_{x}/D\approx 1.2$ ). The peak is relatively broad ( $1<K_{x}D<2$ ) probably due to the irregularity of the von Kármán wake.

Figure 16.

(a) Temporal evolution of the maximum streamwise wavenumber. (b) Wavelength of the wake waves as a function of the Froude number. Lines correspond to the wavelength of the von Kármán vortex street (i.e. the distance between same sign vortices) given by (4.2) at $Nt=2$ (solid line) and at $Nt=30$ (dotted line). $z=2D$ .

4.3 Characteristics of the wake waves

The $x$ -wavenumber $K_{x}$ corresponding to this peak is plotted in figure 16(a) as a function of time. It is nearly constant up to $Nt=160$ and can be extracted for each run. This wavenumber is not modified when the measurement plane is located one diameter above the sphere. The corresponding $x$ -wavelength $\unicode[STIX]{x1D706}_{x}^{w}=2\unicode[STIX]{x03C0}/K_{x}$ is plotted in figure 16(b) as a function of the Froude number. It scales as $F^{1/3}$ which is similar to the wake horizontal width and to the distance between same sign vortices. This is coherent with a burst of internal waves emitted by the von Kármán vortex street whose wavelength sets the wavelength of the emitted waves. The distance between same sign vortices has been found experimentally by Spedding (Reference Spedding2002) to be equal to

This prediction is plotted as a solid line in figure 16(b) assuming that the waves are emitted at the beginning of the NEQ regime (i.e. for $Nt=2$ ). Unfortunately, this prediction is twice smaller than the experimental results. An excellent agreement is obtained by assuming that the internal waves are emitted by the von Kármán vortices at $Nt=30$ (plotted as a dotted line) rather than at $Nt=2$ . However, this is in disagreement with the phase velocity of the waves measured previously which indicated that the waves are emitted at $Nt=2$ . It should be noted that the wavelength depends very weakly on the Reynolds number: it decreases by only 10 % when the Reynolds number increases by a factor 4. This is slightly smaller than the Reynolds dependence obtained by Zhou & Diamessis (Reference Zhou and Diamessis2016) far above the wake. This suggests that it could be due to an attenuation of the small wavelengths during their propagation. However, it should be noted that this $x$ -wavenumber does not correspond to the horizontal wavenumber $k_{H}$ measured by Abdilghanie & Diamessis (Reference Abdilghanie and Diamessis2013) and Zhou & Diamessis (Reference Zhou and Diamessis2016) which is a two-dimensional (2-D) wavenumber $k_{H}=\sqrt{k_{x}^{2}+k_{y}^{2}}$ .

The $y$ -location of the maximum of the spectrum has also been measured for each spectrum. This position $y^{w}$ , which corresponds to the distance between the wake centreline and the band of internal waves, is plotted in figure 17(a) as a function of time. It increases linearly with time. This is consistent with a burst of internal waves emitted at early times, which propagates with a constant group velocity. The horizontal velocity of the wavepacket has been extracted for all Froude numbers. It seems to be well fitted by the dimensional law:

This velocity is smaller by a factor 2 to 6 to the streamwise velocity of the waves $U_{\unicode[STIX]{x1D719}}$ . Moreover, it does not have the same scaling than $U_{\unicode[STIX]{x1D719}}$ which scales as $F^{1/3}ND$ . It should be noted that the position $y^{w}$ is identical at $z=D$ and $z=2D$ .

Figure 17.

Cross-stream position $y_{w}$ (a) and cross-stream velocity $\text{d}y_{w}/\text{d}t$ (b) of the maximum of the wake waves. In (a), $F=8$ , $Re=4850$ . $z=2D$ .

The angle $\unicode[STIX]{x1D6FC}^{w}$ of the isophases with respect to the centreline (see schematic definition in figure 15 b) can be extracted by calculating the 2-D Fourier spectrum of each instantaneous divergence field (not shown here). The angle $\unicode[STIX]{x1D6FC}^{w}$ is simply given by $\tan (\unicode[STIX]{x1D6FC}^{w})=K_{x}/K_{y}$ where $(K_{x},K_{y})$ correspond to the position of the maximum of the 2-D spectrum. It is plotted as a function of time in figure 18(a) and seems to be roughly constant around $50^{\circ }$ . The small increase at late stages may not be relevant since it might be related to waves reflected at the free surface. The mean angle is then plotted as a function of the Froude number in figure 18(b). The angle $\unicode[STIX]{x1D6FC}^{w}$ decreases from $55^{\circ }$ to $45^{\circ }$ when the Froude number increases from $F=4$ to $F=96$ . It should be noted that this tilt angle is independent of the Reynolds number and remains unchanged between $z=D$ and $z=2D$ .

Figure 18.

Angle of the horizontal wavevector (a) as a function of time and (b) as a function of $F$ . In (a), $F=8$ , $Re=4850$ . $z=2D$ .

4.4 Amplitude of the wake waves

Finally, the amplitude of the wake waves is measured as the root mean square (r.m.s.) value of the divergence of the wake waves on the line $y/D=-3$ . This amplitude strongly depends on time and is maximum around $Nt=25$ for any Froude and Reynolds number. This reflects the fact that the waves propagate horizontally at a velocity $\text{d}y^{w}/\text{d}t$ independent of $F$ and $Re$ . The maximum amplitude is plotted in figure 19(a) as a function of the Froude number. The amplitude seems to decrease as $F^{\unicode[STIX]{x1D6FE}}$ with an exponent $\unicode[STIX]{x1D6FE}$ between $-1/2$ and $-1/3$ . Such a scaling is consistent with the results by Bonneton et al. (Reference Bonneton, Chomaz and Hopfinger1993) and Robey (Reference Robey1997) who found that the displacement $\unicode[STIX]{x1D701}$ of the isopycnal seems proportional to the Froude number. Indeed, the velocity divergence $\unicode[STIX]{x2202}w/\unicode[STIX]{x2202}z$ is equal to $\text{i}k_{z}w$ where the vertical velocity $w=\text{d}\unicode[STIX]{x1D701}/\text{d}t$ is the product of the frequency of the wave $\unicode[STIX]{x1D714}=ik_{x}U_{\unicode[STIX]{x1D719}}$ (similar to a Doppler shift) times the displacement $\unicode[STIX]{x1D701}$ . It leads to an expression for the velocity divergence:

Assuming that the vertical wavenumber $k_{z}$ scales as the horizontal wavenumber $k_{x}\sim F^{-1/3}$ and taking into account the scaling $U_{\unicode[STIX]{x1D719}}\sim F^{-2/3}$ leads to a scaling for the velocity divergence as

A scaling of the velocity divergence $\unicode[STIX]{x2202}w/\unicode[STIX]{x2202}z\sim F^{-1/3}$ is thus consistent with a scaling of the displacement $\unicode[STIX]{x1D701}\sim F$ observed by Robey (Reference Robey1997) and Bonneton et al. (Reference Bonneton, Chomaz and Hopfinger1993). Moreover, the quantitative value $\unicode[STIX]{x1D701}=0.005F$ given by Bonneton et al. (Reference Bonneton, Chomaz and Hopfinger1993) is in quantitative agreement with the divergence obtained in our experiments.

Figure 19.

Maximum amplitude of the r.m.s. divergence as a function of the Froude number at $y/D=-3$ and $z/D=2$ . The dashed line corresponds to a fit of the experimental data scaling as $F^{-0.8}$ .

However, all these experiments are obtained with a varying Reynolds number which may drastically modify the scaling in Froude number at a fixed Reynolds number. Indeed, figure 19(a) indicates that the amplitude is approximately twice larger for the large Reynolds number experiments ( $D=2.52~\text{cm}$ , large open symbols) than for the small Reynolds number experiments ( $D=1.28~\text{cm}$ , small filled symbols). It means that the amplitude must be rescaled by $Re^{0.4}$ in order to collapse all data. Such a dependence on the Reynolds number has already been observed numerically by Abdilghanie & Diamessis (Reference Abdilghanie and Diamessis2013) although they obtained a weaker exponent for the wave momentum flux: $\langle uw\rangle \sim Re^{0.25}$ . The rescaled amplitude of the divergence is plotted in figure 19(b) and shows a reasonable collapse for the two series of Reynolds numbers. As a consequence, the scaling with the Froude number is modified and seems to be close to $F^{-0.8}$ . This scaling is close to the scaling of the velocity defect $U_{0}$ (and the phase velocity $U_{\unicode[STIX]{x1D719}}$ ) as $F^{-2/3}$ . This Froude dependence is in excellent agreement with the results by Abdilghanie & Diamessis (Reference Abdilghanie and Diamessis2013) who found that the momentum flux $\langle uw\rangle$ decreases slightly faster with respect to $F$ than the squared velocity defect $U_{0}^{2}$ at a fixed Reynolds number.

The best fit of the amplitude gives:

So far, there is no explanation for the dependence of the amplitude with the Reynolds number. It is possible that the wake becomes more turbulent and irregular at larger Reynolds numbers leading to a more energetic emission of internal waves on a longer duration. Further experiments at larger Reynolds numbers might be necessary in order to confirm the validity of the scalings at large Reynolds numbers.

Finally, it should be mentioned that the amplitude of the wake waves is smaller than the lee waves by an order of magnitude when extrapolating these results to $F=1$ . However, the wake waves become dominant for Froude numbers larger than 10, in agreement with the results of Bonneton et al. (Reference Bonneton, Chomaz and Hopfinger1993).

4.5 A simple model

It is possible to calculate theoretically the internal waves emitted by a von Kármán vortex street. The vortices are supposed to be separated by a wavelength $\unicode[STIX]{x1D706}_{x}=2\unicode[STIX]{x03C0}/k_{x}$ between same sign vortices given by (4.2). The distance between the two rows of vortices is taken as $d=0.4\unicode[STIX]{x1D706}_{x}$ which is characteristic of stratified wakes (Meunier & Spedding Reference Meunier and Spedding2004). The vortices are supposed to move at the velocity $U_{\unicode[STIX]{x1D719}}$ given by (4.1). The low pressures at the centre of each vortex creates a perturbation which is periodic in $x$ for each row of vortices. The pressure perturbation is thus taken at $z=0$ as

The internal waves emitted by this pressure perturbation are simply given for large $y$ and $z$ by

where $\text{Re}$ is the real part and $X=x+U_{\unicode[STIX]{x1D719}}t$ . The calculation, which makes use of the stationary phase method, is detailed in appendix D. This distribution is plotted in figure 20 two diameters above the wake for $F=8$ . It is composed of alternate lobes between $y/D=-1$ and $y/D=-4$ tilted at approximately $45^{\circ }$ with respect to the $x$ -axis. This structure is very similar to the internal waves measured experimentally at $Nt=16$ shown in figure 15(a). Moreover, the angle of $45^{\circ }$ is in excellent agreement with the measurements shown in figure 18. However, the theoretical solution does not spread in $y$ as much as in the experiments. This is probably because the pressure perturbation is broader in $y$ than the Dirac function taken theoretically in $y$ .

Figure 20.

Analytical solution (4.8) of the internal waves emitted by a von Kármán vortex street two diameters above the wake. $F=8$ .

This pattern is closer to the experimental observations than the pattern obtained by Dupont & Voisin (Reference Dupont and Voisin1996) and Robey (Reference Robey1997). Indeed, the pattern is here invariant in the streamwise direction (as in the experiments) whereas the pattern obtained by Dupont & Voisin (Reference Dupont and Voisin1996) and Robey (Reference Robey1997) are localized behind the emitting point source.

4.6 Influence of the shape of the bluff body

Finally, the experiments done with a cylinder and a 6:1 spheroid have been analysed in the same way as for the sphere. The instantaneous spectra (in the frame moving at $U_{\unicode[STIX]{x1D719}}$ ) have been used to extract the wavelength of the wake waves. It is plotted in figure 21(a) as a function of the Froude number and compared to the wavelength of the turbulent wake given by (4.2) at $Nt=30$ (dashed line). The wavelength of the cylinder is very close to the wavelength obtained for the sphere but it seems to be slightly larger (by approximately 20 %) for $F<20$ . As shown by Meunier & Spedding (Reference Meunier and Spedding2004), this can be explained for a self-preserved wake by the fact that the effective diameter of the turbulent wake is larger for a cylinder than for a sphere: it is 40 % larger when taken proportional to $\sqrt{C_{D}}$ . The wavelength should thus be 40 % larger but since the effective Froude number is also 30 % smaller (and since $\unicode[STIX]{x1D706}_{x}\sim F^{1/3}$ ), the data for the cylinder can be expected to be 25 % above the sphere’s data in figure 21. This is in good agreement with the experimental results.

Figure 21.

(a) Wavelength and (b) rescaled amplitude as a function of the Froude number. Symbols correspond to a sphere ( $\circ$ ), to a cylinder (▫) and to a 6:1 spheroid (▵). Large open symbols correspond to large bluff bodies ( $Re\sim 600F$ ) whereas small filled symbols correspond to small bluff bodies ( $Re\sim 160F$ ). In (a), the wavelength of the von Kármán vortex street (4.2) is plotted at $Nt=2$ (solid line) and at $Nt=30$ (dashed line). In (b), the dashed line corresponds to the prediction (4.6) for a sphere.

The wavelength of the slender spheroid is very close to the wavelength of the sphere for $F>20$ but it seems to be larger by approximately 40 % for $F<20$ . This result is counter-intuitive since the effective diameter of the spheroid is smaller than for the sphere and would give a wavelength approximately 20 % smaller. It is possible that the large aspect ratio of the spheroid creates initially a wake with a larger wavelength than for the sphere (in disagreement with the theory of self-preserved wakes). Meunier & Spedding (Reference Meunier and Spedding2004) indeed found experimentally that the wavelength of the spheroid’s vortex street is slightly larger (by approximately 15 %) at the beginning of their measurements (i.e.  $Nt=100$ ) but ultimately reaches the wavelength of the self-preserved turbulent wake at $Nt=200$ . This could explain the fact that the wavelength is 40 % larger than for the sphere at $Nt<30$ when the internal waves are emitted.

The amplitude of the internal waves is also measured for the cylinder and for the slender spheroid. It is plotted in figure 21(b). For the cylinder, the amplitude is very close to the amplitude obtained for the sphere on the whole range of Froude numbers. For the slender spheroid, the amplitude is approximately twice smaller for $F<20$ . This could be related to the fact that the wavelength is larger for these Froude numbers. Indeed, equation (4.4) indicates that the amplitude is proportional to the wavenumber squared (at constant isopycnal displacement $\unicode[STIX]{x1D701}$ ) such that an increase of 40 % of the wavelength leads to an amplitude twice larger.