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# Bénard-von Kármán vortex street in a Kelvin-Helmholtz type confined geometry: wavelength selection by shear flow instabilities

Luc Lebon, Chi-Tuong Pham, Paul Boniface, Laurent Limat

► To cite this version:

Luc Lebon, Chi-Tuong Pham, Paul Boniface, Laurent Limat. Bénard-von Kármán vortex street in a Kelvin-Helmholtz type confined geometry: wavelength selection by shear flow instabilities. EPL - Europhysics Letters, 2025, 152 (4), pp.43002. 10.1209/0295-5075/ae2204. hal-04957697v2

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Submitted on 10 Nov 2025

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# Bénard-von Kármán vortex street in a Kelvin-Helmholtz type confined geometry: wavelength selection by shear flow instabilities.

L. LEBON¹, P. BONIFACE¹, C.-T. PHAM¹,² and L. LIMAT¹

¹ Matière et Systèmes Complexes, CNRS and Université Paris Cité, UMR 7057, Paris, France

² Laboratoire Interdisciplinaire des Sciences du Numérique, CNRS and Université Paris-Saclay, UMR 9015, Orsay, France

PACS 47.20.Ft – Instability of shear flows

PACS 47.32.ck – Vortex streets

PACS 47.20.-k – Flow instabilities

**Abstract** – We have reconsidered the formation and stability of a vortex street, induced in a rectangular container by a tape moving at high speed on its free surface. In a certain range of tape velocity and of geometrical aspect ratios, the liquid recirculates along the lateral sides of the pool, which induces two shear flows between the tape and these lateral sides, that undergo two coupled Kelvin-Helmholtz instabilities, giving rise to the vortex street. Contrary to the classical situation of a wake behind an obstacle, the double row remains static which allows one to study its absolute stability in a stationary framework. In the present paper we present a stability analysis around the mean flow that clarifies the wavelength selection problem, inside the stability tongue predicted long ago by Rosenhead, and reduced by steric arguments that we found in a previous paper. In summary the mean-wavelength favored by shear flow instabilities is given by $\lambda_{max} \approx \pi c$, where c is half the channel width, while the maximal wavelength predicted by marginal stability is equal to $\lambda_{c1} \approx 5.71 c$. Our available experimental data are in very good agreement with these results and with the resulting phase diagram.

**Introduction : the confined Bénard – von Kármán vortex street.** – The Bénard – von Kármán vortex street is a structure that can be widely observed in the wake of an obstacle inside a stationary parallel flow of large enough velocity : it has been investigated by Bénard [1] and Camichel *et al.* [2], then modeled by von Kármán [3, 4] more than a century ago : in a two-dimensional viscous flow a vortex street of point vortices are expected to be stable only if the wavelength of the street $2b$ is linked to the width of the street $2a$ by

$$ \frac{a}{b} = \frac{1}{\pi} \operatorname{argcosh}(\sqrt{2}) \approx 0.281. \quad (1) $$

Although this condition is often well satisfied in experiments, the precise nature of the stability has been widely discussed [5–8]. For instance, in the periodicity-spacing ratio plane for point vortex configurations, this expected stability is known to be a saddle point [5], the finite size of the vortices needing to be included to recover a "true" stability. More recently [9], it has been realized that for wakes behind an obstacle, the advection of the array by the

flow was moderating the instabilities growth, the stability being to be discussed in a absolute/convective framework.

When the entire flow is confined between two lateral walls, separated by a distance $2c$, the stability condition is deeply modified. In the 30’s theoretical works addressed the question of confined point vortices and in particular, Rosenhead [10] established that the stability condition (1) had to be replaced by a continuous stability tongue in the plane $(a/c, b/c)$ of finite extent.

In a previous paper [11], we have presented experiments exhibiting a confined Bénard – von Kármán vortex street induced by a confined shear flow : an endless tape was pulled with a constant speed at the free surface of a water layer in a long pool, of length 2 m, of variable width $2c$ ranging from a few centimeters to 70 cm and of depth $h$ of a few centimeters, see Fig. 1(a). Because of the mass conservation, a back flow is created in the pool along the lateral walls. Under suitable experimental conditions, shear produces a stable vortex street of wavelength $2b$ and width $2a$ as exhibited in Figs. 1(c) and 1(d) : two rows of
<figure><img src="image_2.png" /><figcaption>Fig. 1: Experimental study of confined Bénard - von Kármán vortex street (from [11]): a confined shear flow (a) produces a stable vortex street of wavelength $2b$ and width $2a$ (b). (c) : typical stream lines of the flow, (d) : typical vorticity distribution, ranging from $-4 \ s^{-1}$ to $4 \ s^{-1}$.</figcaption></figure>

vortices rotating in opposite directions. Contrary to the classical situation of a wake behind an obstacle, the double row remains nearly motionless which allows one to study a situation in which the array can be absolutely stable in the laboratory frame of reference.

In contrast with the well-known selection rule given by Eq. (1), we observed that the system explores a continuous band of stability, consistent with Rosenhead calculations [10]; we also observed that the experimental points do not cover the whole domain and remain distributed in a rather narrow area in the plane $(a/c, b/c)$.

**Basic steric model.** — In the same paper, we showed that a steric model based on the finite extent of the vortex cores, not taken into account in the Rosenhead point vortex model, can very correctly explain the restriction of the domain explored by the width of the street $2a$. Up to now, there is no equivalent of Rosenhead calculations for finite size vortices, but we proposed the following qualitative argument, based on the structure suggested in Fig. 2. If we consider the vortex cores as rigid disks that must tile the available space without overlapping, but keeping contacts between each other and with the two boundaries, it is easy to show that $b = 2\sqrt{R^2 - a^2}$ and $R = c - a$ so that the ratios $b/c$ and $a/c$ must satisfy the following relationships :

$$ \frac{b}{c} = 2\sqrt{1 - 2\frac{a}{c}}, \qquad (2) $$

and as $R \ge a$

$$ \frac{a}{c} \le \frac{1}{2}. \qquad (3) $$

If the vortices do not touch, the wavelength must be greater than the value corresponding to the Eq. (2) so that :

$$ \frac{b}{c} \ge 2\sqrt{1 - 2\frac{a}{c}} \qquad (4) $$

These two conditions (3) and (4) seem to explain remarkably well the observed confinement of the experimental points inside a narrow band of $a/c$ values.

<figure><img src="image_3.png" /><figcaption>Fig. 2: Steric model: vortex cores considered as rigid disks.</figcaption></figure>

On the other hand, we had no argument to explain the limitation of the domain explored by the wavelengths $2b$ of the streets, particularly the largest ones. In addition, the accumulation of points around mean values remained unexplained too.

In what follows, we propose a model solving these questions. We first remind the classical treatment of this kind of instabilities [12–14] and then show how to apply this formalism to our particular geometry.

**Kelvin–Helmholtz instability.** — The destabilization of the flow at the origin of the vortex street is presumably the Kelvin–Helmholtz instability: it is the inherent destabilization of sheared layers of fluid [15, 16]. This instability leads to vortices patterns in the flow, with a characteristic length [12, 17, 18] and it appears that this one can often be predicted by linear analysis studying the development of a small perturbation at short time. Therefore we will consider here a linear analysis of inviscid parallel shear flows.

**Rayleigh Equation for the linearized problem.** —

We consider a two-dimensional flow since the Squire theorem [19] shows that two-dimensional perturbations are suffisant to find an instability condition for a three-dimensional problem.

In the case of parallel flow along $x$, the basic velocity field is $V_o(y) = V_o(y)e_x$ and the pressure field $P_o(y)$. Let us introduce the perturbed fields $v(x, y, t)$ for the velocity and $p(x, y, t)$ for the pressure.

Also, we define $\psi(x, y, t)$ the stream function for the perturbed velocity : $v_x = \partial\psi/\partial y$ and $v_y = -\partial\psi/\partial x$.

After linearization, incompressibility and Euler equations lead directly to the following equation :

$$ \frac{\partial\Delta\psi}{\partial t} + V_o \frac{\partial\Delta\psi}{\partial x} = V_o'' \frac{\partial\psi}{\partial x} \qquad (5) $$

Let us remark that this equation is the transport equation of vorticity without diffusion.

We are searching normal mode solutions of this equation

$$ \psi(x, y, t) = \phi(y) e^{-i\omega t + ikx} \qquad (6) $$

where $k$ is the wave number and $\omega$ is the pulsation. Eq. (5) leads to the Rayleigh equation :

$$ \left(V_o - \frac{w}{k}\right) \left(\phi'' - k^2\phi\right) = V_o''\phi \qquad (7) $$
<figure><img src="image_4.png" /><figcaption>Fig. 3: (a) Basic velocity profile deduced from our confined vortex streets experiments. (b) Piecewise linear model with a boundary layer of thickness $\delta$.</figcaption></figure>

When the flow displays different layers, the normal velocity of the fluid is continuous at the interface between layers. After linearization, this condition leads to the continuity of the ratio $\phi/(V_o - w/k)$ through the interface [20].

We can remark here that, if the basic velocity field $V_o$ is continuous at the interface, then the amplitude of the stream function $\phi$ is also continuous.

The flow resolution also needs a stress expression. In this case, the stress is just the pressure and it can be derived from Euler equation; $p(x, y, t) = \pi(y) e^{-i\omega t+ikx}$, and we obtain:

$$ \pi = \rho \left( \frac{w}{k} - V_o \right) \phi' + \rho V_o' \phi \quad (8) $$

$\rho$ being the fluid density.

**Mean flow.** — We consider the confined flow described in the first part exhibiting vortex streets (Fig. 1 (a)). A mean flow profile experimentally obtained is represented in Fig. 3 (a) : a main flow in the center and a back flow on each side with zero velocity on the walls. We performed a time average when the vortices were slightly drifting or a streamwise average when they were rigorously motionless, leading to the same profile.

There is here an underlying unsolved issue because this mean flow is obtained in situations where vortices are developed and not before they appear ; there is at this stage no clear theoretical justification for the validity of linear analyses around mean flows. Nevertheless several works recently deal with the stability analyses around mean flows, this mean flow being directly available from experiments or simulations [21–24] or explicitly calculated from non-linear calculations [25, 26]. Most study the case of a wake in the flow past a cylinder and therefore used as mean flow a time averaged flow. In this case, the stability analysis based on mean flows give very accurate prediction of the frequency compared to DNS [21–23, 25] or experiments [21, 25]. Benneddine et al. [24] discuss the validity of mean flow stability analysis for turbulent flows and Turton et al. [26] in thermosolutal convection. We will apply here a mean flow stability analysis to our confined shear flow geometry.

We modelize the basic velocity profile including boundary layers as represented in Fig. 3 (b). At the center, the fluid flows at a maximum velocity $U_o$. On each side, the maximum velocity of the back flow $U_1$ occurs at a distance $\delta$ from the wall ; $\delta$ is the thickness of the boundary layer and the velocity is zero on the walls ($y = \pm c$). The mass conservation in the flow leads to:

$$ U_1 = U_o \left( 1 - \frac{\delta}{c} \right) \quad (9) $$

The basic flow is therefore a piecewise linear flow over four domains:

<table><thead><tr><th>j</th><th>interval</th><th>V<sub>0</sub>(y)</th><th>V'<sub>0</sub>(y)</th></tr></thead><tbody><tr><td>1</td><td>y ∈ [c − δ ; c]</td><td>U<sub>1</sub>/δ(y − c)</td><td>U<sub>1</sub>/δ</td></tr><tr><td>2</td><td>y ∈ [0 ; c − δ]</td><td>−U<sub>0</sub>+U<sub>1</sub>/(c − δ)y + U<sub>0</sub></td><td>−U<sub>0</sub>+U<sub>1</sub>/(c − δ)</td></tr><tr><td>3</td><td>y ∈ [−c + δ ; 0]</td><td>U<sub>0</sub>+U<sub>1</sub>/(c − δ)y + U<sub>0</sub></td><td>U<sub>0</sub>+U<sub>1</sub>/(c − δ)</td></tr><tr><td>4</td><td>y ∈ [−c ; δ − c + δ]</td><td>−U<sub>1</sub>/δ(y + c)</td><td>−U<sub>1</sub>/δ</td></tr></tbody></table>

**Linear analysis of the confined shear flow.** — In each domain, the Rayleigh equation (7) is solved by $\phi_j(y) = A_j e^{ky} + B_j e^{-ky}$, ($j = 1...4$).

We have five kinematic boundary conditions : the normal velocity $v_y$ is zero on the wall ; as $v_y = -\partial\psi/\partial x = -ik\phi(y)e^{-i\omega t+ikx}$, we obtain $\phi_1(c) = \phi_4(-c) = 0$:

$$ A_1 + \alpha B_1 = 0 \quad (10) $$

where $\alpha = e^{-2kc}$, and

$$ \alpha A_4 + B_4 = 0 \quad (11) $$

As pointed out previously, the continuity of the basic flow $V_o(y)$ at the boundary of each domain leads to the continuity of $\phi(y)$ : therefore $\phi_1(c - \delta) = \phi_2(c - \delta)$, $\phi_2(0) = \phi_3(0)$ and $\phi_3(-c + \delta) = \phi_4(-c + \delta)$ leading to the following relations:

$$ A_1 + \beta B_1 - A_2 - \beta B_2 = 0 \quad (12) $$

where $\beta = e^{-2k(c-\delta)}$,

$$ A_2 + B_2 - A_3 - B_3 = 0 \quad (13) $$

and

$$ \beta A_3 + B_3 - \beta A_4 - B_4 = 0 \quad (14) $$

The stress (reduced to the pressure in our problem) is also continuous at the boundary of each domain, which leads to three dynamic boundary conditions : $\pi_1(c - \delta) = \pi_2(c - \delta)$, $\pi_2(0) = \pi_3(0)$ and $\pi_3(-c + \delta) = \pi_4(-c + \delta)$ leading to the following relations:

$$ \begin{aligned} & (\omega + k U_1 + U_1/\delta)A_1 \\ & + \beta(-\omega - k U_1 + U_1/\delta)B_1 \\ & -[\omega - k U_1 + (U_1 + U_o)/(c - \delta)]A_2 \\ & + \beta[\omega + k U_1 + (U_1 + U_o)/(c - \delta)]B_2 = 0 \end{aligned} \quad (15) $$
<figure><img src="image_5.png" /></figure>

a

<figure><img src="image_6.png" /></figure>

b

<caption>Fig. 4: Growth rate of the instability $\sigma = \text{Im}(\omega c/U_0)$ as a function of $kc$ and $\delta/c$; unstable modes correspond therefore to the situations where $\sigma > 0$. (b) is a 3D enlargement of (a) for small values of $\delta/c$ corresponding to the range observed in experiments. Note that the color map differs between (a) and (b).</caption>

$$ \begin{aligned} & [\omega - k U_o - (U_1 + U_o)/(c - \delta)]A_2 \\ & +[-\omega + k U_o - (U_1 + U_o)/(c - \delta)]B_2 \\ & -[\omega - k U_1 + (U_1 + U_o)/(c - \delta)]A_3 \\ & +[\omega - k U_o - (U_1 + U_o)/(c - \delta)]B_3 = 0 \end{aligned} \quad (16) $$

$$ \begin{aligned} & \beta[\omega + k U_1 + (U_1 + U_o)/(c - \delta)]A_3 \\ & +[-\omega - k U_1 + (U_1 + U_o)/(c - \delta)]B_3 \\ & +\beta[-\omega - k U_1 + U_1/\delta]A_4 \\ & +[\omega + k U_1 + U_1/\delta]B_4 = 0 \end{aligned} \quad (17) $$

Eqs (10) to (17) form a system of eight linear homogeneous equations with eight unknowns. The corresponding system matrix $\mathbf{A}$ with dimensionless coefficients can be found in Appendix A where we used as length scale $c$ and as time scale $c/U_0$.

The non trivial solution of this system is obtained when the determinant of the system

$$ D(\omega c/U_0, kc, \delta/c) = \det(\mathbf{A}) = 0. \quad (18) $$

This equation was numerically solved and we present in figure 4 the growth rate of the instability $\sigma = \text{Im}(\omega c/U_0)$ as a function of $kc$ and $\delta/c$.

<figure><img src="image_7.png" /></figure>

<figure><img src="image_8.png" /><figcaption>Fig. 5: Dispersion relation for the confined shear flow for different values of the dimensionless thickness of the boundary layer $\delta/c$ (flow profile of figure 3(b)).</figcaption></figure>

It can be noted that the marginal stability limits are strongly dependent on the boundary layer thickness $\delta$: we present in figure 5 the dispersion relation for different values of $\delta/c$ ranging from 0.1 to 0.8. The shift of the most unstable eigenmode to higher wavenumbers seems consistent with the instability becoming localized around the increasingly narrow central jet that forms in the mean flow as $\delta/c$ tends to one.

**Comparison with our experiments.** — In our experiments, we observe typical value of $\delta/c$ between 0.1 and 0.2 (see Fig. 3(a)). For such thicknesses, the dispersion relation presents a maximum growth rate at $k_{\text{max}}c \approx 2$. The corresponding wavelength $\lambda_{\text{max}} = 2b$ is therefore related to $c$ by

$$ \frac{2b}{c} \approx \pi \quad (19) $$

The wavelength is controlled by the size of confinement $2c$. This theoretical prediction can be compared to our experiments. In figure 6, we plot $\lambda_{\text{max}}^* = 2b/c$, the dimensionless selected mean wavelength, measured for different values of the degree of confinement $c$; they are in good agreement with the predicted value of $\pi$.

The two relevant parameters of this model are the boundary layer thickness $\delta$ and the width of the water tank $2c$. This is consistent with our experiments where
<figure><img src="image_9.png" /><figcaption>Fig. 6: Dimensionless selected wavelength $\lambda_{\text{max}}^* = 2b/c$ : the mean experimental wavelengths (●) are in good agreement with the wavelength predicted by the linear analysis $\lambda_{\text{max}}^* = \pi$ (- -).</figcaption></figure>

the water height or the fluid velocity $U_o$ have no influence on observed wavelengths.

We plotted the mean experimental wavelength in Fig. 6 because the measured values are very dispersed around the mean value for a given confinement as plotted in Fig. 4 of our previous paper [11]. The origin of such a dispersion is unclear : the flow could be partially turbulent and particularly it is very difficult experimentally to avoid vertical beating of the tape, inducing fluctuations on the flow.

Nevertheless the dispersion relations plotted in Fig. 5 allow us to predict the marginal stability of the flow, corresponding to the points where the instability growth rate becomes zero. If $\delta/c = 0.2$, a typical value of the boundary layer thickness, the growth rate is positive for wavenumber between $k_{c1}$ and $k_{c2}$ where $k_{c1}c \approx 1.10$ and $k_{c2}c \approx 2.72$ ; the corresponding wavelengths and positions in the $(\pi a/2c, \pi b/2c)$ plane are :

$$ \begin{array}{lll} k_{c1}c \approx 1.10 & \lambda_{c1} \approx 5.71 c & \pi b_{c1}/2c \approx 4.48 \\ k_{\text{max}}c \approx 2.00 & \lambda_{\text{max}} \approx 3.14 c & \pi b_{\text{max}}/2c \approx 2.46 \\ k_{c2}c \approx 2.72 & \lambda_{c2} \approx 2.31 c & \pi b_{c2}/2c \approx 1.81 \end{array} \quad (20) $$

These values and boundaries predicted by the linear analysis can be compared to our experiments : in the $(\pi a/2c, \pi b/2c)$ plane, we plotted all our experimental data in Fig. 7. As previously pointed out [11], data satisfy the stability area predicted by Rosenhead and steric limitations induced by the rigid disk model. We can observe now that the experimental data points are also included between the marginal stability values $k_{c1}$ and $k_{c2}$, where the growth rate of the instability is positive, and as seen in Fig. 6, the value corresponding to the maximum growth rate passes through the middle of the experimental measurement set.

<figure><img src="image_10.png" /><figcaption>Fig. 7: Stability diagram : the experimental data set satisfy the stability area (■) predicted by Rosenhead inside the steric and shear flow marginal stability limitations.</figcaption></figure>

One remarkable result from this study is the stabilisation of the flow at small wavenumbers, for $k < k_{c1}$, both observed experimentally and predicted by linear analysis. Such stabilisation was not predicted for confined wakes or jets [27–30]. As $k_{c1} \sim 1/c$, this is clearly a consequence of the confinement that prevent the development of structures much larger than the width of the channel in the flow direction ; we can notice from figure 4 that this cut-off appears strongly dependent on the boundary layer thickness $\delta$.

**Conclusion.** – In summary, we reconsidered the formation of a Bénard–von Kármán street of vortices, induced in a pool of finite width $2c$ by a tape running at constant speed on its free surface. As expected for this kind of flow, shear flow instability leads to a typical wavelength triggered by the sole scale $c$, but the possible values observed are dispersed in a continuous domain limited by three effects : (1) this shear flow instability mechanism, (2) the stability of vortex points calculated by Rosenhead, (3) a possible effect due to the finite size of the vortex cores. We have thus for the first time a complete description of these phenomena.

This work is also a contribution to the validity of stability analyses around mean flows as such an analysis allow us to predict the wavelengths range experimentally observed
in our particular flow geometry.

It would be now interesting to study more complex arrangements of vortices, such as these encountered behind swimming bodies [31, 32] with for instance four rows and possible inversions of the rotation in each row. Also the case of a single vortex row could be studied, its mechanisms of destabilization remain disputed [33].

**Appendix A.** — Eqs (10) to (17) form a system of eight homogeneous linear equations with eight unknowns; we present here all the non-zero coefficients of corresponding matrix **A** used to find numerically the dispersion relation. The coefficients were adimensionalized using as length scale $c$ and as time scale $c/U_0$; $U_1$ was directly derived from $U_0$ and $\delta/c$ from eq. (9). Let us remind that $\alpha = e^{-2kc}$ and $\beta = e^{-2k(c-\delta)}$.

$$ 
\begin{aligned}
a_{11} &= 1 & a_{27} &= \alpha \\
a_{12} &= \alpha & a_{28} &= 1
\end{aligned}
 $$

$$ 
\begin{aligned}
a_{33} &= 1 & a_{35} &= -1 \\
a_{34} &= 1 & a_{36} &= -1
\end{aligned}
 $$

$$ 
\begin{aligned}
a_{41} &= 1 & a_{43} &= -1 \\
a_{42} &= \beta & a_{44} &= -\beta
\end{aligned}
 $$

$$ 
\begin{aligned}
a_{55} &= \beta & a_{57} &= -\beta \\
a_{56} &= 1 & a_{58} &= -1
\end{aligned}
 $$

$$ a_{61} = \omega c/U_0 + kc(1 - \delta/c) + (1 - \delta/c)/(\delta/c) $$

$$ a_{62} = \beta [-\omega c/U_0 - kc(1 - \delta/c) + (1 - \delta/c)/(\delta/c)] $$

$$ a_{63} = -\omega c/U_0 - kc(1 - \delta/c) + (2 - \delta/c)/(1 - \delta/c) $$

$$ a_{64} = \beta [\omega c/U_0 + kc(1 - \delta/c) + (2 - \delta/c)/(1 - \delta/c)] $$

$$ a_{73} = \omega c/U_0 - kc - (2 - \delta/c)/(1 - \delta/c) $$

$$ a_{74} = -\omega c/U_0 + kc - (2 - \delta/c)/(1 - \delta/c) $$

$$ a_{75} = -\omega c/U_0 + kc - (2 - \delta/c)/(1 - \delta/c) $$

$$ a_{76} = \omega c/U_0 - kc - (2 - \delta/c)/(1 - \delta/c) $$

$$ a_{85} = \beta [\omega c/U_0 + kc(1 - \delta/c) + (2 - \delta/c)/(1 - \delta/c)] $$

$$ a_{86} = -\omega c/U_0 - kc(1 - \delta/c) + (2 - \delta/c)/(1 - \delta/c) $$

$$ a_{87} = \beta [-\omega c/U_0 - kc(1 - \delta/c) + (1 - \delta/c)/(\delta/c)] $$

$$ a_{88} = \omega c/U_0 + kc(1 - \delta/c) + (1 - \delta/c)/(\delta/c) $$

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***

We are indebted to stimulating discussions with C. Arratia, F. Charru and J. E. Wesfreid about the instability mechanisms at play here.

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