# The near field of coaxial jets: a numerical study

Guillaume Balarac, Olivier Métais

► To cite this version:

Guillaume Balarac, Olivier Métais. The near field of coaxial jets: a numerical study. Physics of Fluids, 2005, 17, pp.65-102. 10.1063/1.1900786. hal-00265278

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# The near field of coaxial jets: A numerical study

Guillaume Balarac<sup>a)</sup> and Olivier Métais

Équipe MoST/LEGI, Boîte Postale 53, 38041 Grenoble Cedex 09, France

The near-field behavior of coaxial jets is studied through direct numerical simulation (DNS) with a particular focus on the influence of the inner shear layer steepness characterized by its momentum thickness $\theta_{01}$ thus mimicking the variation in the lip thickness of a real jet nozzle. We investigate the two distinct jet regimes $r_u > r_{uc}$ for which a recirculation bubble is present near the jet inlet and $r_u < r_{uc}$ without any recirculation bubble, $r_u$ being the velocity ratio between the outer jet and inner jet velocities. It is shown that small values of $\theta_{01}$ lead to a fast transition to turbulence. The various mechanisms leading to this transition are investigated in detail: the three-dimensionality growth, the appearance of secondary vortices superimposed on the main ring vortices, and the subsequent longitudinal stretching of streamwise vortices. This stretching mechanism is shown to play a dominant role in the transition processes towards a fully developed turbulent state. For high enough values of $r_u$, a pinching of the jet is observed near the inlet and it is shown that this pinching phenomenon lasts on a shorter downstream distance for small values of $\theta_{01}$ due to a more efficient turbulent mixing. In the $r_u > r_{uc}$ case, variations of $\theta_{01}$ strongly affect the shape and the downstream extent of the recirculation bubble. The DNS allow to show the strong dependency of the inner and outer potential core lengths and of the critical value $r_{uc}$ on the jet inlet velocity profile. We finally revisit the theoretical model originally proposed by Rehab, Villermaux, and Hopfinger ["Flow regimes of large-velocity-ratio coaxial jets," J. Fluid Mech. 345, 357 (1997)] first aimed at the prediction of the variations of various jet characteristics as a function of $r_u$. The model is extended to determine the dependency of the jet characteristics with $\theta_{01}$. A very good correspondence between the theoretical predictions and the numerical results is obtained.

## I. INTRODUCTION

Coaxial jets are present in various industrial devices: they effectively constitute an efficient way of mixing species for combustion applications. They are also used in aeroaeroacoustics: the noise produced by a single jet can indeed be reduced thanks to the addition of a surrounding coaxial stream (e.g., Williams, Ali, and Anderson¹).

Coaxial jets are composed of an inner jet issued from a nozzle of diameter $D_1$ and an outer annular jet issued from an outer annulus of diameter $D_2 (D_2 > D_1)$. $U_1$ and $U_2$ designate the respective velocities of the inner and outer jets. One of the important parameters characterizing the coaxial jet dynamics is the ratio between the outer to the inner jet momentum flux, $M = \rho_2 U_2^2 / \rho_1 U_1^2$ where $\rho_1$ and $\rho_2$ are, respectively, the inner and outer density (see Favre-Marinet and Camano Schettini²). For constant density jets ($\rho_1 = \rho_2$), the momentum flux ratio reduces to the velocity ratio, $r_u = U_2 / U_1$.

In the present study, we consider constant density coaxial jets for which the outer velocity is larger than the inner one ($r_u > 1$). These jets are situated in-between two limiting cases: a single round jet ($r_u = 1$) and a purely annular jet ($r_u \to \infty$). Purely annular jets are characterized by the presence of a big recirculation bubble near the jet axis (Ko and Chan³). This backflow region was experimentally observed by Rehab⁴ and numerically by da Silva, Balarac, and Métais⁵

for $r_u \gg 1$. Since this backflow is absent for small enough values of $r_u$, there exists a critical velocity ratio $r_{uc}$ which separates the two different main flow regimes, without recirculation bubble for $1 < r_u < r_{uc}$ and with recirculation bubble for $r_u > r_{uc}$.

Rehab, Villermaux, and Hopfinger⁶ experimentally showed that the coaxial jet dynamics and its vortex topology are strongly dependent on the shape of the inlet nozzle. Indeed, different shapes lead to significant variations of the two shear layers which are present in the coaxial jet: the inner shear layer at the interface between the inner and the outer jets and the outer shear layer on the external border of the outer jet. In the following, we will characterize the respective thicknesses of the inner and outer shear layers by their momentum thicknesses $\theta_{01}$ and $\theta_{02}$. The previous experimental works⁶ have mainly focused on the influence of $r_u$ considering very small values of $\theta_{01}$ and $\theta_{02}$. However, even for high Reynolds number jets, a variation of the inner lip thickness of the jet yields a significant variation of $\theta_{01}$. The present paper completes the previous experimental and numerical works on coaxial jets by investigating the influence of $\theta_{01}$ on the transitional processes in the jet near field. We also vary the ratio $r_u$ and the two distinct jet regimes are successively considered: the case without recirculation bubble ($r_u < r_{uc}$) and with recirculation bubble ($r_u > r_{uc}$). In particular, we will show that the value of $r_{uc}$ varies widely with $\theta_{01}$. This completes Rehab, Villermaux, and Hopfinger⁷ observations

<footnote>a) Electronic mail: guillaume.balarac@hmg.inpg.fr</footnote>
showing a strong dependence of $r_{uc}$ with the jet nozzle shape with values located between 7 and 9.

As far as numerical investigations of coaxial jets are concerned, da Silva, Balarac, and Métais⁵ (referred to as SBM) previously performed direct numerical simulations (DNS) of spatially growing coaxial jets for two values of $r_u: r_u=3.3$ and $r_u=23.5$. The ratio of the jet radius to the initial momentum thicknesses was fixed with $R_1/\theta_{01}=13$ and $\theta_{01}=\theta_{02}$. In the present study, we widely vary the values of $\theta_{01}/R_1$ and $r_u$. We furthermore develop simple theoretical models inspired by the ones originally proposed by Rehab, Villermaux, and Hopfinger⁶ allowing to predict the variation laws of various statistical quantities in function of $\theta_{01}/R_1$ and $r_u$.

In Sec. II, the numerical method, physical, and computational parameters of the DNS are detailed. Section III investigates how $\theta_{01}$ influences the near-field jet dynamics and the mechanisms of transition towards a fully developed turbulent state. The two flow regimes $r_u<r_{uc}$ and $r_u>r_{uc}$ are successively considered. Section IV extends the theoretical model originally proposed by Rehab, Villermaux, and Hopfinger⁶ to predict the variation laws of various jet statistics as a function of $\theta_{01}$ and $r_u$.

## II. NUMERICAL METHOD AND COMPUTATIONAL PARAMETERS

All the simulations presented here were performed with the same numerical code as the one used by SBM. The full three-dimensional incompressible Navier-Stokes equations, assuming constant density, are solved with a very precise numerical technique combining sixth-order “compact” (Lele⁸) scheme in the flow direction and pseudospectral (Canuto et al.⁹) schemes in the two-transverse directions. This code was originally developed by Gonze¹⁰ and has been intensively validated both for round jets and plane jets configurations (see da Silva and Métais,¹¹ da Silva and Métais¹²). The detailed discretization procedure is described by Gonze¹⁰ and by da Silva¹³.

The velocity vector is decomposed as $\vec{U}=(u_x, u_y, u_z)$, where $u_x$ designates the streamwise component, $u_y$ and $u_z$ are the two-transverse components. We also use a cylindrical coordinate system $(x, r, \theta)$ with $x$ the streamwise (longitudinal) distance along the jet axis, $r$ the distance from the jet axis, and $\theta$ the azimuthal angle. Any vector $\vec{a}$ will be decomposed, in this system, as $\vec{a}=(a_x, a_r, a_\theta)$. For all the simulated cases, the inlet velocity profile is given by

$$ \vec{U}(\vec{x}_0, t) = \vec{U}_{\text{med}}(r) + \vec{U}_{\text{noise}}(\vec{x}_0, t), \quad (1) $$

where $\vec{U}(\vec{x}_0, t)$ is the instantaneous inlet velocity vector, which is prescribed at each time step. The inlet velocity profile $\vec{U}_{\text{med}}(r)$ mimics a realistic experimental profile: it is constructed with two hyperbolic tangent velocity profiles as shown in Fig. 1. It consists in a one-dimensional flow oriented along the $x$ direction whose modulus is given by

<figure><img src="image_2.png" /><figcaption>FIG. 1. Sketch of the inlet velocity profile. $\theta_{01}$ and $\theta_{02}$ are the momentum thicknesses for the inner shear layer (velocity jump from $U_1$ to $U_2$) and for the outer shear layer (velocity jump from $U_2$ to $U_3$), respectively.</figcaption></figure>

$$ U_{\text{med}}(r) = \begin{cases} \frac{U_1+U_2}{2} + \frac{U_1-U_2}{2} \tanh\left(\frac{r-R_1}{2\theta_{01}}\right) & \text{for } r < R_m \\ \frac{U_2+U_3}{2} + \frac{U_2-U_3}{2} \tanh\left(\frac{r-R_2}{2\theta_{02}}\right) & \text{for } r > R_m, \end{cases} \quad (2) $$

where $U_1$ is the inner coaxial jet velocity, $U_2$ is the outer velocity, and $U_3$ is a very small coflow. $R_1$, $R_2$, and $R_m = (R_1+R_2)/2$ are the inner, outer, and averaged radii, and $\theta_{01}$ and $\theta_{02}$ are the inlet momentum thicknesses from the inner and outer shear layers, respectively. For each station $x$, the inner and outer momentum thicknesses are defined by (3) and (4),

$$ \theta_1(x) = \int_0^{R_m} \left[ \frac{U_x(x, r) - U_{\min}(x)}{U_{\max}(x) - U_{\min}(x)} \right] \times \left[ 1 - \frac{U_x(x, r) - U_{\min}(x)}{U_{\max}(x) - U_{\min}(x)} \right] dr, \quad (3) $$

$$ \theta_2(x) = \int_{R_m}^{\infty} \left[ \frac{U_x(x, r) - U_{\min}(x)}{U_{\max}(x) - U_{\min}(x)} \right] \times \left[ 1 - \frac{U_x(x, r) - U_{\min}(x)}{U_{\max}(x) - U_{\min}(x)} \right] dr, \quad (4) $$

where $U_{\max}(x)$ and $U_{\min}(x)$ are, respectively, the maximum and minimum mean streamwise velocities for the inner or outer streams and $U_x(x, r)$ is the local mean streamwise velocity. The mean normal and spanwise velocities are set to zero at the inlet

$$ V_{\text{med}}(\vec{x}_0) = W_{\text{med}}(\vec{x}_0) = 0. \quad (5) $$

Note that the mean values are obtained through a temporal averaging based on a time interval long enough to reach statistical convergence. In the following, $\langle f \rangle(x, r)$ designates
the mean value of any quantity $f(x, r, \theta, t): \langle f \rangle$ is independent of the azimuthal angle $\theta$ due to the axisymmetry of the flow configuration.

$\tilde{U}_{\text{noise}}(\vec{x}_0, t)$ is the noise superimposed on the inlet profile. It is given by

$$ \tilde{U}_{\text{noise}}(\vec{x}_0, t) = A_n U_{\text{base}}(\vec{x}_0) \tilde{f}' . \quad (6) $$

$A_n$ is the maximum amplitude of the incoming noise and $U_{\text{base}}(\vec{x}_0)$ is a function that mainly locates the noise in the shear layer gradients:

$$ U_{\text{base}}(\vec{x}_0) = \begin{cases} 0.5 & \text{if } 0.85 > r/R_1 \\ 1.0 & \text{if } 0.85 < r/R_1 < 1.15 \\ 1.0 & \text{if } 0.85 < r/R_2 < 1.15 \\ 0 & \text{otherwise,} \end{cases} $$

$\tilde{f}'$ is a random noise designed to satisfy a given energy spectrum (see SBM for details). The random noise is applied on the three velocity components.

All simulations are carried out on the same computational grid consisting in 231 × 384 × 384 points for a domain size of $10.8D_1 \times 10.65D_1 \times 10.65D_1$, along the streamwise ($x$) and the two transverse directions ($y, z$), the mesh size being uniform in all three directions. In all simulations, the Reynolds number and the ratio of the outer to the inner diameter are $Re_{D_1} = U_2D_1/v = 3000$ and $D_2/D_1 = 2$, respectively. The ratio of the jet outer radius to the outer initial shear layer momentum thickness is fixed with $R_2/\theta_{02} = 25$. The maximum noise amplitude is set to $A_n = 3.0\%$. A very small coflow is imposed such that $U_3/U_2 = 0.04$: it was shown not to influence the jet dynamics (see da Silva¹³).

As previously pointed out, the goal of the present study is mainly to look at the influence of the parameter $\theta_{01}$. $r_u$ is also varied to investigate the two flow regimes, with and without recirculation bubble. Table I synthesizes the various runs which were carried out: $\theta_{01}/R_1$ varies from 0.04 to 0.1 and $r_u$ widely ranges from 3 to 30. Note that each simulation series DNSi (with $i$ varying from 1 to 5) corresponds with a fixed value of the parameter $\theta_{01}/R_1$. For convenience, both parameters $\theta_{01}/R_1$ and its opposite $R_1/\theta_{01}$ will be successively used.

It is important to note that the potential sources of numerical and modeling errors have been minimized in the present study. Indeed, precise numerical methods, a well-resolved regular mesh and direct numerical simulations are used. The price to pay is that the Reynolds number for our simulations ($Re_{D_1} = 3000$) is lower than the values usually encountered in the experimental studies (see, i.e., Rehab et al.,⁶ Warda, Kassab, Elshorbagy, and Elsaadawy,¹⁴ Buresti, Petagna, and Talamelli¹⁵). Furthermore, to properly discretize the velocity gradient, relatively large values of $\theta_{01}$ and $\theta_{02}$ have to be considered. The values usually considered in previous experiments are usually much smaller: this renders any validation by comparison with experimental data difficult. Previous experimental studies (Ko and Kwan¹⁶ and Ko and Chan³) have however shown that in the far field of coaxial jets a self-similar state is reached similar to the one reached in a single jet. SBM recovered this result with the

<table><caption>TABLE I. Description of the various runs.</caption><thead><tr><th>Category</th><th>Name</th><th>r<sub>u</sub></th><th>R<sub>1</sub>/θ<sub>01</sub></th><th>θ<sub>01</sub>/R<sub>1</sub></th></tr></thead><tbody><tr><td rowspan="3">DNS1</td><td>DNS1_3</td><td>3</td><td>10</td><td>0.1</td></tr><tr><td>DNS1_10</td><td>10</td><td>10</td><td>0.1</td></tr><tr><td>DNS1_17</td><td>17</td><td>10</td><td>0.1</td></tr><tr><td rowspan="6">DNS2</td><td>DNS2_3</td><td>3</td><td>12.5</td><td>0.08</td></tr><tr><td>DNS2_5</td><td>5</td><td>12.5</td><td>0.08</td></tr><tr><td>DNS2_6</td><td>6</td><td>12.5</td><td>0.08</td></tr><tr><td>DNS2_7</td><td>7</td><td>12.5</td><td>0.08</td></tr><tr><td>DNS2_10</td><td>10</td><td>12.5</td><td>0.08</td></tr><tr><td>DNS2_13</td><td>13</td><td>12.5</td><td>0.08</td></tr><tr><td rowspan="3">DNS3</td><td>DNS2_17</td><td>17</td><td>12.5</td><td>0.08</td></tr><tr><td>DNS2_30</td><td>30</td><td>12.5</td><td>0.08</td></tr><tr><td>DNS3_11</td><td>11</td><td>15</td><td>0.0667</td></tr><tr><td rowspan="5">DNS4</td><td>DNS3_13</td><td>13</td><td>15</td><td>0.0667</td></tr><tr><td>DNS4_3</td><td>3</td><td>20</td><td>0.05</td></tr><tr><td>DNS4_10</td><td>10</td><td>20</td><td>0.05</td></tr><tr><td>DNS4_12</td><td>12</td><td>20</td><td>0.05</td></tr><tr><td>DNS4_17</td><td>17</td><td>20</td><td>0.05</td></tr><tr><td rowspan="3">DNS5</td><td>DNS5_10</td><td>10</td><td>25</td><td>0.04</td></tr><tr><td>DNS5_11</td><td>11</td><td>25</td><td>0.04</td></tr><tr><td>DNS5_17</td><td>17</td><td>25</td><td>0.04</td></tr></tbody></table>

present numerical code (see their Fig. 7). We here complete this previous validation by considering the near-field behavior in the vicinity of the jet inlet. Figure 2 displays the downstream evolution of the rms velocity component $\langle u'_x^2 \rangle^{1/2}(x, r)$ at the jet centerline for the two simulations DNS4_3 ($r_u = 3$; $R_1/\theta_{01} = 20$) and DNS5_17 ($r_u = 17$; $R_1/\theta_{01} = 25$). Note that the first simulation corresponds with the case with no recirculation bubble while a bubble is present in the second simulation. The numerical results are compared with two different experiments: the first by Buresti et al.¹⁵ corresponding to $r_u = 3$, $Re_{D_1} \approx 160 000$, $R_1/\theta_{01} \approx 70$, and $R_2/\theta_{02} \approx 130$, the second by Rehab et al.⁶ ($r_u = 9$, $Re_{D_1} \approx 200 000$, $R_1/\theta_{01} \approx 100$, and $R_2/\theta_{02} \approx 100$). In the first experiment, $r_u < r_{uc}$ while $r_u > r_{uc}$ in the second. Despite the difference in the Reynolds numbers and the steepness of the velocity gradients a good agreement is observed between the numerical and experimental data. Note that a virtual origin $x_0$, with $x_0 \approx 1D_1$ has been used for the two DNS to account for the Reynolds number difference with the experimental value: indeed, the turbulence develops at much shorter downstream distance in a high Reynolds number jet.

### III. INFLUENCE OF THE INNER SHEAR LAYER THICKNESS ON THE JET DYNAMICS

#### A. Coaxial jet without recirculation bubble $r_u<r_{uc}$

##### 1. Mechanisms leading to turbulent transition

We first consider the case where no recirculation bubble is present. The influence of the shear layer thickness $\theta_{01}$ is investigated by first focusing on the comparison of the three simulations DNS1_10 ($r_u = 10$; $\theta_{01}/R_1 = 0.1$), DNS2_10 ($r_u$
<figure><img src="image_3.png" /><figcaption>FIG. 2. Downstream evolution of the axial velocity component rms $\langle u'_{x}^2 \rangle^{1/2}(x, r=0)$ at the jet centerline: (a) DNS4_3 and (b) DNS5_17. The numerical results are compared with the measurements by Buresti et al. (Ref. 15) and Rehab et al. (Ref. 6).</figcaption></figure>

$=10; \theta_{01}/R_1=0.08)$, and DNS5_10 ($r_u=10; \theta_{01}/R_1=0.04$). To characterize, the transition from a quasilaminar regime near the inlet to a fully developed turbulent regime we first consider the downstream evolution of $\langle u'_{x}^2 \rangle^{1/2}(x, r)$ (Fig. 3). $r$ is here taken equal to $R_1$ corresponding to the radial location of the vorticity local maximum within the inner shear layer of the inlet profile. For the three considered cases, the

<figure><img src="image_4.png" /><figcaption>FIG. 3. Downstream evolution of the axial velocity component rms $\langle u'_{x}^2 \rangle^{1/2}(x, r=R_1)$ at the inner shear layer for the three simulations (DNS1_10, DNS2_10, and DNS5_10).</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>FIG. 4. Visualization of the coherent structures for the three simulations DNS1_10 (a), DNS2_10 (b), and DNS5_10 (c). Cut view of isosurfaces of positive $Q=0.5(U_2/D_1)^2$ colored by the azimuthal vorticity. Light gray corresponds with negative values and dark gray with positive values.</figcaption></figure>

turbulence first undergoes a slow downstream growth before exhibiting a sudden transition characterized by a much faster growth. The transition location is situated further and further downstream as $\theta_{01}$ increases: it is indeed located around $x/D_1 \approx 5$ for DNS1_10, $x/D_1 \approx 4$ for DNS2_10, and $x/D_1 \approx 2$ for DNS5_10. This faster transition can be explained by the fact that the instability of the inner shear layer leads to the generation of smaller scales for smaller values of $\theta_{01}$ and the transition towards turbulence is consequently faster.

This is clearly illustrated by Fig. 4 which shows isosurfaces of positive $Q$ for the three simulations DNS1_10, DNS2_10, and DNS5_10. We recall that $Q$ is the second invariant of the velocity gradient tensor defined as

$$Q = \frac{1}{2}(\Omega_{ij}\Omega_{ij} - S_{ij}S_{ij}), \qquad (7)$$

where $\Omega_{ij}$ and $S_{ij}$ are the antisymmetrical and symmetrical part of the velocity gradient tensor, respectively. The positive $Q$ criterion was first proposed by Hunt, Wray, and Moin¹⁷ and is now well recognized as a good way to identify the flow coherent vortices (see, e.g., Dubief and Delcayre¹⁸). In
<figure><img src="image_6.png" /><figcaption>FIG. 5. Downstream evolution of radial and azimuthal contributions for the turbulent kinetic energy $E_r$ and $E_\theta$ [see Eqs. (8)–(11) for definition] calculated in the inner and outer shear layer for the two simulations: (a) DNS2_10 and (b) DNS5_10.</figcaption></figure>

particular, it can been showed that the positive Q regions are in good correspondence with low-pressure regions and thus constitute good tracors of intense vortices (see Métais¹⁹). Figure 4 shows the coherent structures development near the jet inlet (0<x/D₁<7). As pointed out by SBM, we assist in the shedding of both outer and inner vortices, respectively, associated with the instability of the outer and inner shear layers. For $r_u<r_{uc}$, the outer vortices impose their shedding frequency to the inner vortices and the latter are trapped between two consecutive larger vortex rings.²⁰ As previously pointed out, the instability develops faster when $\theta_{01}$ is decreased and the inner layer vortices are shed at shorter and shorter locations when we go from DNS1_10 to DNS5_10 simulation. Although the value of $\theta_{02}$ is identical in the three simulations, it is interesting to note that the outer vortices are also shed at shorter downstream distances for the smaller values of $\theta_{01}$. This is attributable to the mutual interaction between the inner and outer shear layers: the faster growth of the disturbance within the inner shear layer induces an earlier destabilization of the outer layer. Since the DNS5_10 case is associated with an early development of the vortices, these are more rapidly subject to a three-dimensionalization process. It is clearly illustrated on Fig. 5 which shows the downstream evolution of $E_r$ and $E_\theta$ for the inner shear layer and

the outer shear layer. Only the two runs DNS2_10 and DNS5_10 are compared here. These quantities are the contributions of the radial ($E_r$) and azimuthal ($E_\theta$) Reynolds stresses to the turbulent kinetic energy at a given x location (see SBM). They are defined as

$$E_r(x) = \sqrt{\frac{2\pi}{L_y L_z}} \int_0^{R_m} \langle u_r'^2 \rangle (x, r) r \, dr, \quad (8)$$

$$E_\theta(x) = \sqrt{\frac{2\pi}{L_y L_z}} \int_0^{R_m} \langle u_\theta'^2 \rangle (x, r) r \, dr \quad (9)$$

for the inner shear layer and as

$$E_r(x) = \sqrt{\frac{2\pi}{L_y L_z}} \int_{R_m}^{\infty} \langle u_r'^2 \rangle (x, r) r \, dr, \quad (10)$$

$$E_\theta(x) = \sqrt{\frac{2\pi}{L_y L_z}} \int_{R_m}^{\infty} \langle u_\theta'^2 \rangle (x, r) r \, dr \quad (11)$$

for the outer shear layer. $u'_r$ and $u'_\theta$ are, respectively, the radial and azimuthal components of the fluctuating velocity field. Note that, for a purely axisymmetric vortex ring with no swirl, the component $E_\theta$ would be identically zero. $E_\theta$ then represents the departure from axisymmetry and therefore constitutes a measure of the three-dimensionality level. For both DNS2_10 and DNS5_10 cases, $E_r$, associated with the growth of the vortex rings, dominates $E_\theta$ during the first stage of the transition: this indicates a weak degree of three-dimensionalization in this early stage. Further downstream, $E_\theta$ catches up with $E_r$ showing a full three-dimensionalization of the jet (see da Silva and Métais¹¹). The comparison of Figs. 5(a) and 5(b) clearly indicates that the three-dimensionalization processes become dominant on a shorter downstream distance for the DNS5_10 simulation than for the DNS2_10 simulation. A fully developed turbulent state is reached at $x/D_1 \approx 8$ for DNS5_10 when the computational box is too short for such a state to be observed for DNS2_10.

It is clear from Fig. 4 that the three-dimensionalization process is strongly linked with the appearance of streamwise secondary vortices. These appear much sooner when $\theta_{01}$ is smaller and induce a strong three-dimensionalization of the inner and outer vortices between which they are stretched and consequently trigger an earlier transition towards a fully turbulent regime. Figures 6(a) and 6(b) display a zoom of the three-dimensional coherent structures in the transition region situated between $x/D_1=3$ and $x/D_1=7$ for DNS5_10 simulation. The secondary streamwise vortices appear and are stretched both between two consecutive inner vortices [Fig. 6(a)] and between two consecutive outer vortices [Fig. 6(b)]. Note that, for DNS5_10 simulation, the inner streamwise vortices form at a shorter downstream distance than the outer vortices which appear further downstream. This difference in behavior between the inner and outer shear layer is explained below.

The strong amplification of the streamwise vorticity as one moves downstream is illustrated in Fig. 7 which shows the profile of $\langle \omega_x' \rangle (x, r)$ as a function of r for several down-
<figure><img src="image_7.png" /></figure>

<figure><img src="image_8.png" /><figcaption>FIG. 6. Zoom showing the secondary streamwise vortices in simulation DNS5_10: (a) vortices stretched between two consecutive inner vortices (cut view); (b) vortices stretched between two consecutive outer vortices. Isosurfaces of positive $Q=0.5(U_2/D_1)^2$ colored by the axial vorticity. Light gray corresponds with negative values and dark gray with positive values.</figcaption></figure>

stream locations $x/D_1=1, 3, \text{ and } 5$. The two simulations DNS2_10 and DNS5_10 are here compared. DNS2_10 corresponds with a case where the momentum thicknesses of the inner and outer shear layers are identical ($\theta_{01}=\theta_{02}$). The growth of the three dimensionality takes place at a similar rate within both shear layers: this gives rise to two longitudinal vorticity peaks of comparable intensity, respectively, situated at $r/D_1 \approx 0.5$ and $r/D_1 \approx 1$. The longitudinal vorticity stretching between the main jet vortices, previously mentioned, yields a significant increase of the vorticity peak amplitude as we move downstream. Conversely, DNS5_10 corresponds to a case where the two momentum thicknesses differ: here $\theta_{01}=\theta_{02}/2$. In that case, the streamwise vortices are first stretched between two consecutive vortices of the inner layer yielding a strong amplification for $r/D_1 \le 0.5$ (see Fig. 7 at $x/D_1=3$). The stretching between two consecutive outer vortices takes place only further downstream leading to

<figure><img src="image_9.png" /></figure>

<figure><img src="image_10.png" /></figure>

<figure><img src="image_11.png" /><figcaption>FIG. 7. Radial profile of $\langle\omega_x'\rangle(x,r)$ at three downstream location ($x/D_1=1, 3, \text{ and } 5$) for the two simulations (DNS2_10 and DNS5_10).</figcaption></figure>

a later longitudinal vorticity enhancement in the outer shear layer region near $r/D_1 \approx 1$ (Fig. 7 at $x/D_1=5$).

Figure 8 compares the profiles of the three rms vorticity components $\langle\omega_x'\rangle(x,r)$, $\langle\omega_r'\rangle(x,r)$, and $\langle\omega_\theta'\rangle(x,r)$ for the two simulations DNS2_10 ($x/D_1=6$) and DNS5_10 ($x/D_1=3$). This figure confirms that longitudinal stretching is the dominant factor leading to transition to a fully developed turbulent regime: indeed, the longitudinal vorticity component clearly dominates the other two components. We have
<figure><img src="image_12.png" /></figure>

<figure><img src="image_13.png" /><figcaption>FIG. 8. Comparison between $\langle\omega_x'\rangle^2(x,r)$, $\langle\omega_r'\rangle^2(x,r)$, and $\langle\omega_\theta'\rangle^2(x,r)$ profiles: (a) simulation DNS2_10 at the location $x/D_1=6$; (b) simulation DNS5_10 at $x/D_1=3$.</figcaption></figure>

checked that the ratio between the peak values of $\omega_x'$ and $\omega_r'$ reaches a maximum of 4.5 for DNS2_10 and 5 for DNS5_10 as we move downstream: this maximum being respectively reached at $x/D_1=4.5$ (DNS2_10) and at $x/D_1=2.5$ (DNS5_10). Similarly, the corresponding maximal ratio between $\omega_x'$ and $\omega_\theta'$ are, respectively, 1.5 for DNS2_10 and 2 for DNS5_10 and take place at $x/D_1=6$ and $x/D_1=3.5$.

## 2. Jet spreading and potential cores

We now investigate the influence of $\theta_{01}$ on more global quantities. We first consider the jet spreading rate defined through a global shear layer thickness by analogy with the definition used for single jets. Let $r_{max}(x)$ be the radial distance for which the mean streamwise velocity reaches its maximum value, $\max[U_x(x,r)]=U_x(x,r_{max}(x))$. $\delta(x)$ is then chosen as the radial distance such that

$$U_x(x, \delta(x)) = \frac{1}{2}[U_x(x, r_{max}(x)) - U_3], \quad (12)$$

where $U_3$ is the streamwise velocity in the irrotational region. Since we are interested in the global spreading rate of the jet, we furthermore choose $\delta(x)$ such that $\delta(x) > r_{max}(x)$: $\delta(x)$ therefore characterizes the jet width. The downstream evolution of $\delta(x)$ for the three simulations DNS1_10, DNS2_10, and DNS5_10 is shown in Fig. 9. The simulation DNS2_3 corresponding to a smaller value of $r_u$ ($r_u=3$; $\theta_{01}/R_1=0.08$) is also shown for comparison. For

<figure><img src="image_14.png" /><figcaption>FIG. 9. Downstream evolution of jet spreading rate for the four simulations DNS1_10, DNS2_3, DNS2_10, and DNS5_10.</figcaption></figure>

the simulation with moderate $r_u$ (DNS2_3), $\delta(x)$ behaves similarly to a single jet: it is indeed almost constant for $x/D_1<7$ and undergoes a sudden transition with a rapid growth for $x/D_1>7$. In this second stage, transition to turbulence has taken place and the linear growth characteristic of the turbulent regime is observed. We have checked that the slope in the linear regime closely coincides with the slope for a single jet. For the cases with larger $r_u$ (DNS1_10, DNS2_10, and DNS5_10), a decrease of $\delta$ is observed before transition. This corresponds to a so-called pinching phenomenon of the coaxial jet which has previously been observed experimentally.⁴ For large enough values of $r_u$, the velocity difference between the outer jet and the inner jet induces a curvature of the outer mean streamlines towards the jet axis. This is directly related to the fluid entrainment by the outer coaxial jet: a limited amount of fluid can indeed be entrained from the slow part of the jet core and this entrainment eventually leads to a fluid depletion in the core region. This fluid depletion is accompanied by a pressure drop within the jet core leading to a curvature of the streamlines towards the jet axis. As previously mentioned, for $r_u$ larger than the critical value $r_{uc}$ it eventually yields a reverse backflow near the jet axis.

The comparison of DNS1_10, DNS2_10, and DNS5_10 allows to show that the pinching phenomenon is also strongly influenced by the value of the momentum thickness $\theta_{01}$. For a fixed value of $r_u$, the decrease of $\theta_{01}$ is indeed associated with a decrease of the inner jet bulk velocity. One can then anticipate that the fluid depletion leading to a low-pressure region within the jet core will be more pronounced for small values of $\theta_{01}$ since a smaller quantity of fluid is injected within the core by the inner jet. This is confirmed by Fig. 9 which shows that the initial decrease of $\delta$ is more and more pronounced as we go from DNS1_10 to DNS5_10. The transition towards a turbulent linear growth occurs however at a shorter downstream location when $\theta_{01}$ is small. This is due to a faster transition to turbulence in that case as shown in Fig. 3. It is important to remark that the linear growth rate in the turbulent regime is identical for all the considered cases and becomes independent of the initial transitional stage. Note that the comparison of Figs. 3 and 9 indicates
that the effect of the turbulence growth has a delayed effect on the spreading rate. Indeed, it was observed on Fig. 3 that the rapid growth of $\langle u_x^2 \rangle^{1/2}$ due to transition was taking place at $x/D_1 \approx 5$, $x/D_1 \approx 4$, and $x/D_1 \approx 2$ for DNS1_10, DNS2_10, and DNS5_10, respectively. As shown on Fig. 9, this turbulence amplification has a direct effect on the spreading rate only for $x/D_1 \ge 7$, $x/D_1 \ge 7$, and $x/D_1 \ge 5$ for DNS1_10, DNS2_10, and DNS5_10, respectively. This is attributable to the fact that the turbulence originates from the inner shear layer and requires some downstream distance to diffuse towards the jet exterior and efficiently affect the jet spreading rate.

We now investigate how the two potential cores of the jet are affected by a variation in the value of $\theta_{01}$. For a single jet, the potential core is the region where the flow remains free from the influence of viscosity. In this cone shaped region the flow is potential and conserves the streamwise velocity it had at the inlet. For the coaxial jet, we can define two potential cores, one for the inner round jet $0<r<R_1$ and another for the outer annular flow $R_1<r<R_2$. For these two inner and outer flow regions, the center of the respective potential cores can be identified as the radial locations at which the inlet streamwise velocity is conserved on the longest downstream distance. We call $r_{PC1}$ and $r_{PC2}$ these two radial locations for the inner and outer potential cores. Note that $r_{PC1}=0$. We define the length of the potential cores as the two longest distances we just defined: they are designated as $L_{PC1}$ with $i=1$ for the inner potential core and $i=2$ for the outer one. From a practical point of view, $L_{PC1}$ are determined as the two downstream distances from the inlet at which the mean streamwise velocity starts to differ by more than 5% from its value at the inlet.

The faster transition to turbulence associated with the decrease of $\theta_{01}$, is clearly illustrated by Fig. 10 which shows the variations with $R_1/\theta_{01}$ of $L_{PC1}$ and $L_{PC2}$ at fixed $r_u(r_u=10)$. The parameter $R_1/\theta_{01}$ is used here for a better readability of the figures: the simulations DNS1_10, DNS2_10, and DNS5_10 correspond to $R_1/\theta_{01}=10, 12.5$, and 25. The increasingly efficient mixing of momentum due to turbulence when $\theta_{01}$ decreases yields a diminution of both $L_{PC1}$ and $L_{PC2}$. Note that $L_{PC1}$ varies with $r_u$ (see below) while previous studies showed that $L_{PC2}$ is independent of $r_u$ (see Rehab, Villermaux, and Hopfinger$^7$).

## B. Recirculation bubble

We here consider the regime such that $r_u > r_{uc}$. To investigate the influence of $\theta_{01}$ two simulations are here compared DNS2_17 ($r_u=17$; $\theta_{01}/R_1=0.08$) and DNS5_17 ($r_u=17$; $\theta_{01}/R_1=0.04$). As $r_u$ increases the entrainment by the outer annular jet and the depletion of the inner fluid is more and more pronounced. When $r_u$ exceeds $r_{uc}$, the inner potential core breaks and a backflow region establishes associated with negative streamwise velocity near the jet axis. Figure 11 displays the downstream evolution of the mean axial velocity on the jet axis for the two simulations DNS2_17 and DNS5_17. In both cases, a region of negative velocity is apparent near the jet inlet. The longitudinal extent of the recirculation region varies with $\theta_{01}$ with a smaller value for

sharper velocity gradients: the end of the recirculation bubble is located around $x/D_1 \approx 2.5$ for DNS5_17 when it ends around $x/D_1 \approx 3.5$ for DNS2_17.

The variation in size of the recirculation bubble is confirmed by Fig. 12 which shows positive Q isosurfaces for DNS2_17 and DNS5_17. In both cases, a recirculation bubble is seen to appear near the jet inlet but its longitudinal extension is significantly reduced for DNS5_17. Figure 13 displays a zoom on the recirculation bubble for both simulations. In both cases, the backflow region is centred at $x/D_1 \approx 2$ but the shape of the bubble significantly varies in function of $\theta_{01}$: for DNS2_17 which corresponds to a relatively large value of $\theta_{01}$, the Kelvin-Helmholtz vortices of the inner shear layer have not yet developed at this downstream location. Conversely, DNS5_17 corresponds with a value of $\theta_{01}$ which is twice smaller yielding an earlier formation of the Kelvin-Helmholtz vortices. In the first case, the bubble is then surrounded by a quasilaminar flow and it exhibits a very smooth shape. This is no longer the case in the second case where the recirculation bubble is significantly perturbed by the surrounding vortices of the inner shear layer. This has a significant impact on the flow turbulent behavior. Figure 14 indeed shows the downstream evolution of the rms velocity component $\langle u_x^2 \rangle^{1/2}(x,r)$ at the jet centerline for both simulations DNS2_17 and DNS5_17, this rms quantity being a

<figure><img src="image_15.png" /><figcaption>FIG. 10. (a) Evolution of the inner potential core length $L_{PC1}$ with $R_1/\theta_{01}$ (at $r_u=10$). (b) Evolution of the outer potential core length $L_{PC2}$ with $R_1/\theta_{01}$ (at $r_u=10$).</figcaption></figure>
<figure><img src="image_16.png" /><figcaption>FIG. 11. Downstream evolution of the axial mean velocity at the centerline ($r=0$) for the two simulations DNS2_17 and DNS5_17.</figcaption></figure>

measure of the turbulent activity in the backflow region ($1 < x/D_1 < 4$). The recirculation bubble is seen to be laminar for DNS2_17 with $\langle u'_x^2 \rangle^{1/2} \approx 0$ within the bubble. Conversely, the turbulent activity in the bubble is important for DNS5_17. Note that, in DNS2_17, a peak is present at $x/D_1 \approx 0.75$ due to the turbulent production associated with the strong streamwise gradient of the mean streamwise velocity component corresponding to the flow deceleration in the stagnation region at the upstream part of the bubble (see SBM).

As already shown by SBM, Fig. 12 confirms that very elongated and intense longitudinal vortices form just down-

<figure><img src="image_17.png" /><figcaption>FIG. 12. Visualization of the coherent structures of DNS2_17 (a) and DNS5_17 (b) simulations near the jet inlet. Cut view of the isosurfaces of $Q=0.5(U_2/D_1)^2$ colored by the streamwise vorticity. Light gray corresponds with negative values and dark gray with positive values.</figcaption></figure>

<figure><img src="image_18.png" /><figcaption>FIG. 13. Isosurface of zero velocity (light gray) showing the recirculation bubble and contours of vorticity modulus in the jet central plane: (a) DNS2_17 and (b) DNS5_17.</figcaption></figure>

stream of the bubble. The downstream end of the recirculation bubble corresponds with an important longitudinal gradient of longitudinal velocity and consequently with a significant production of streamwise vorticity. This is confirmed by Figs. 15(a) and 15(b) which, respectively, represent the downstream evolution of the maximum value of the longitudinal vorticity rms $\langle u_x' \rangle$. The two simulations, DNS2_17 and DNS5_17, with a recirculation bubble are compared with the corresponding simulations without a recirculation bubble, DNS2_10 and DNS5_10. A strong increase of the longitudinal vorticity component in the flow region just downstream of the recirculation bubble is ob-

<figure><img src="image_19.png" /><figcaption>FIG. 14. Downstream evolution of the axial velocity component rms $\langle u_x' \rangle^{1/2}$ at $r=0$ at the jet centerline for the two simulations DNS2_17 and DNS5_17.</figcaption></figure>
<figure><img src="image_20.png" /></figure>

<figure><img src="image_21.png" /><figcaption>FIG. 15. Downstream evolution of the maximal value of $\langle\omega_x^2\rangle$: (a) DNS2_10 and DNS5_17; (b) DNS5_10 and DNS5_17.</figcaption></figure>

served for both simulations DNS2_17 and DNS5_17. The peak value reached by this maximum is much larger than for the corresponding cases without a recirculation bubble: this indicates that the vortex stretching mechanisms are much more efficient in the presence of the bubble.

The streamwise vorticity generation just downstream the recirculation bubble triggers a fast transition towards a fully developed turbulent regime. This is illustrated by Figs. 16(a) and 16(b) which show the downstream evolution of the jet spreading rate $\delta(x)$ [see Eq. (12) for definition]: DNS2_17 and DNS5_17 are, respectively, compared with DNS2_10 and DNS5_10. The previously mentioned pinching phenomenon is observed in the two cases with the recirculation bubble and is even more pronounced than for the cases where the bubble is not present. However, for the two values of $\theta_{01}$ here considered, the enhanced turbulent activity downstream of the recirculation bubble leads to an earlier linear turbulent growth. It is important to note that the pinching phenomenon reaches a limiting value for the $r_u > r_{uc}$: indeed the minimum value reached by $\delta$ is almost identical for DNS2_17 and for DNS5_17. This is confirmed by Fig. 17 which shows the variation with $r_u$ of the radial location of the outer potential core $r_{PC2}$ (see preceding section). The DNS2 series of simulations are here considered corresponding to $R_1/\theta_{01}=12.5$. Due to the pinching phenomenon previously mentioned, the radial extension of the outer potential

core is more and more reduced when $r_u$ is increased. This is confirmed by the experimental observations of Rehab.⁴ We remark that a plateau is reached when $r_u$ exceeds the critical value $r_{uc}$ for which a recirculation bubble appears (here $r_{uc} = 13$), showing a limitation of the pinching phenomenon in the presence of a reverse flow region.

<figure><img src="image_22.png" /></figure>

<figure><img src="image_23.png" /><figcaption>FIG. 16. Downstream evolution of the jet spreading rate $\delta$: (a) DNS2_10 and DNS2_17; (b) DNS5_10 and DNS5_17.</figcaption></figure>

<figure><img src="image_24.png" /><figcaption>FIG. 17. Evolution with $r_u$ of the radial localization $r_{PC2}$ of the outer potential (at $R_1/\theta_{01}=12.5$).</figcaption></figure>
<figure><img src="image_25.png" /><figcaption>FIG. 18. Schematic diagram of the inner potential core: $\bar{u}_1$ designates the inner jet bulk velocity and $u_e$ the entrainment velocity of the inner jet fluid into the mixing layer.</figcaption></figure>

## IV. THEORETICAL MODELS

Rehab et al.⁶ (referred to as RVH) proposed a theoretical model aimed at the prediction of the variation with $r_u$ of the inner potential core length $L_{PC1}$. Since for their experimental setup, the inner jet profile was very steep with $R_1/\theta_{01} \approx 100$ they restricted their model to the case $R_1/\theta_{01} \to \infty$. We here extend their model to predict the variations laws with $R_1/\theta_{01}$ of both $L_{PC1}$ and the critical ratio $r_{uc}$. We next validate the extended model by comparing our results with our numerical data.

### A. The inner potential core length $L_{PC1}$

The basic ideas of RVH's model to predict $L_{PC1}$ are the following: the potential core is assumed to be perfectly conical with a basis of radius $R_1$ and a length $L_{PC1}$. The entrainment velocity of the fluid within the inner core due to the shear layer on the jet border is assumed to be equal to $u_e$ (see Fig. 18). $u_e$ is assumed to have a constant intensity and to be normal to the side surface of the cone. The mass flux injected in the inner core is equalized with the mass flux crossing the side with the entrainment velocity $u_e$. For an incompressible fluid, the mass conservation equation can then be written as

$$ \frac{\pi D_1^2}{4} u_1 = \frac{\pi D_1}{2} \sqrt{\frac{D_1^2}{4} + (L_{PC1})^2} u_e, \quad (13) $$

which leads to

$$ \frac{L_{PC1}}{D_1} = \frac{1}{2} \sqrt{\left(\frac{u_1}{u_e}\right)^2 - 1}. \quad (14) $$

Here $\bar{u}_1$ stands for the inner jet bulk velocity. To evaluate the entrainment velocity, RVH used a so-called entrainment hypothesis assuming that the entrainment velocity is proportional

<figure><img src="image_26.png" /><figcaption>FIG. 19. Radial profiles of the mean axial velocity (continuous line) and of the rms of the longitudinal velocity fluctuation (dashed line) for DNS2_5 simulation. The profiles are computed at $x/D_1=1$.</figcaption></figure>

to the rms value of the streamwise velocity fluctuation $u'_{xrms}$ within the shear layer,

$$ u_e \approx C u'_{xrms}, \quad (15) $$

where C is a proportionality constant which depends on the shear flow under consideration. For the coaxial jet, we will determine C through fitting with our numerical results. RVH and Hussain and Zedan²¹ experiments showed that the $u'_{xrms}$ within a mixing layer is given by

$$ u'_{xrms}/U_1 \approx 0.17(r_u - 1). \quad (16) $$

A similar value is found in our numerical simulations (see Fig. 19). Equations (15) and (16) yield the following variation law for the entrainment velocity:

$$ u_e/U_1 \approx 0.17C(r_u - 1). \quad (17) $$

For the hyperbolic tangent profile given by Eq. (2), $\bar{u}_1$ is given by

$$ \begin{aligned} \frac{\pi D_1^2}{4U_1} u_1 &= \frac{2\pi}{U_1} \int_0^{R_1} U_{\text{med}}(r) r \, dr \\ &= 2\pi \int_0^{R_1} \left[ \frac{r_u + 1}{2} + \frac{r_u - 1}{2} \tanh\left(\frac{r - R_1}{2\theta_{01}}\right) \right] r \, dr. \end{aligned} \quad (18) $$

It gives

$$ \frac{u_1}{U_1} = \frac{r_u + 1}{2} + (r_u - 1)B\left(\frac{R_1}{\theta_{01}}\right), \quad (19) $$

with $B(R_1/\theta_{01})$ verifying

$$ B\left(\frac{R_1}{\theta_{01}}\right) = \frac{1}{R_1^2} \int_0^{R_1} r \tanh\left(\frac{r - R_1}{2\theta_{01}}\right) dr. \quad (20) $$

Equations (14), (17), and (19) provide a new formulation of RVH's model taking into account the variations of $L_{PC1}/D_1$ with $R_1/\theta_{01}$,
<figure><img src="image_27.png" /></figure>

<figure><img src="image_28.png" /><figcaption>FIG. 20. (a) Evolution with $r_u$ of the inner potential core length $L_{PC1}$ (for $R_1/\theta_{01}=12.5$). (b) Evolution with $R_1/\theta_{01}$ of the inner potential core length $L_{PC1}$ (for $r_u=10$). The points represent our simulations, the continuous lines the model given by Eq. (21) and the dashed lines RVH's model [see Eq. (22)].</figcaption></figure>

$$ \frac{L_{PC1}}{D_1} \approx \frac{1}{2} \sqrt{\frac{1}{(0.17C)^2} \left[ \frac{1}{2} \frac{r_u + 1}{r_u - 1} + B \left( \frac{R_1}{\theta_{01}} \right) \right]^2 - 1}. \quad (21) $$

Note that original RVH's model is based on a top-hat velocity profile at the inlet with $u_1 \approx U_1$ which leads to the following expression for $L_{PC1}$ which is independent of $R_1/\theta_{01}$:

$$ \frac{L_{PC1}}{D_1} = \frac{1}{2} \sqrt{\left(\frac{U_1}{u_e}\right)^2 - 2} \approx \frac{1}{2} \sqrt{\frac{1}{(0.17C)^2} \frac{1}{(r_u - 1)^2} - 1}. \quad (22) $$

Figure 20 compares the modified model and RVH's model predictions with the results of our numerical simulations. We found that $C=0.85$ corresponds to the best fit between the theoretical model and the numerical results. As far as the dependence of $L_{PC1}$ with $r_u$ is concerned, the modified model yields a better agreement with the numerical values than the original RVH's model. It reproduces the decrease of the inner potential core length with increasing $r_u$ due to the enhanced turbulence level and the amplified entrainment. The advantage of the modified model lies in its ability to reproduce the correct variations of $L_{PC1}$ with $\theta_{01}$ and in particular its decrease when the shear layer becomes steeper.

<figure><img src="image_29.png" /><figcaption>FIG. 21. Normalized mean pressure $(P-P_{ext})/(1/2)\rho u_{rms}'^2$ with $P_{ext}$ reference external pressure plotted in the jet symmetry plane.</figcaption></figure>

## B. The critical velocity ratio $r_u$

The theoretical determination of the critical ratio $r_{uc}$ proposed by RVH is based upon pressure arguments. Due to the curvature of the mean streamlines, the pressure within the inner jet core is lower than the outer pressure. Using phenomenological arguments, Villermaux, Rehab, and Hopfinger$^{52}$ estimate this pressure jump $\Delta P$ to be of the order of $1/2\rho u_{rms}'^2$ with $u_{rms}'$ the streamwise rms velocity fluctuation within the external shear layer of the inner core and $\rho$ the fluid density assumed to be constant. Figure 21 shows the isocontours of the mean pressure near the jet inlet for DNS1_13. The pressure is normalized by $1/2\rho u_{rms}'^2$ and confirms the above phenomenological arguments. The ambient pressure within the inner core is then of the order of $P_{ext} - 1/2\rho u_{rms}'^2$, where $P_{ext}$ is the reference external pressure. The apparition of a reversed flow is necessarily associated with a stagnation point near the end of the inner potential core. The pressure at this stagnation point will be of the order of $P_S=P_{ext}-1/2\rho u_{rms}'^2+1/2\rho u_1'^2$. If $P_S<P_{ext}$, a reverse flow takes place and the limiting case therefore corresponds to the balance:

$$ \frac{1}{2\rho u_{rms}'^2} = \frac{1}{2\rho u_1'^2}. \quad (23) $$

Using the estimation of $u_{rms}'$ given by Eq. (16), it yields

$$ 0.17(r_{uc} - 1)U_1 = u_1. \quad (24) $$

Replacing $\overline{u_1}$ by its expression given in Eq. (19), we obtain

$$ 0.17(r_{uc} - 1) = \frac{r_{uc} + 1}{2} + (r_{uc} - 1)B\left(\frac{R_1}{\theta_{01}}\right), \quad (25) $$

where $B(R_1/\theta_{01})$ is defined by Eq. (20). Therefore the model to evaluate $r_{uc}$ is
<figure><img src="image_30.png" /><figcaption>FIG. 22. Evolution with $R_1/\theta_{01}$ of the critical velocity ratio $r_{uc}$. The points represent the numerical simulations and the line the model given by Eq. (26).</figcaption></figure>

$$r_{uc} = \frac{B\left(\frac{R_1}{\theta_{01}}\right) - 0.67}{B\left(\frac{R_1}{\theta_{01}}\right) + 0.33} \qquad (26)$$

Figure 22 shows that the model is able to satisfactorily predict the variation of $r_{uc}$ with $R_1/\theta_{01}$ and confirms the importance of this parameter for the limit between the two flow regimes of coaxial jet. Note that the limiting case $R_1/\theta_{01} \to \infty$ leads to $r_{uc} \to 7$: our computations therefore indicate that no backflow region can exist for $r_u < 7$.

## V. CONCLUSION

The goal of the present study is to complete previous experimental and numerical works devoted to the study of coaxial jets. The influence of the velocity ratio between the outer jet and the inner jet $r_u = U_2/U_1$ has experimentally been investigated by Rehab et al.⁶ In these previous studies very small values of the inner and outer jet shear layers thicknesses were considered. However, Rehab et al. showed that the shape of the inlet nozzle strongly influences the near jet behavior and the dynamics of the coherent vortices, since different nozzle shapes induce a variation in the steepness of the inner and outer shear layers. Here, we characterize the respective thicknesses of the two shear layers by their momentum thicknesses called $\theta_{01}$ and $\theta_{02}$. Even for high Reynolds number jets, a variation of the inner lip thickness of the jet yields a significant variation of $\theta_{01}$. The present paper uses a series of direct numerical simulations to investigate the influence of $\theta_{01}$ on the transitional processes in the jet near-field. As shown by Rehab et al., when $r_u$ exceeds a critical value $r_{uc}$, the jet exhibits a reverse flow region in its inner part called a recirculation bubble. We therefore vary the ratio $r_u$ and the two distinct jet regimes are successively considered: the case without a recirculation bubble ($r_u < r_{uc}$) and with a recirculation bubble ($r_u > r_{uc}$).

We show that, both for $r_u < r_{uc}$ and for $r_u > r_{uc}$, small values of $\theta_{01}$ are associated with an earlier jet transition to turbulence due to a faster destabilization of the inner shear layer. We then study in detail the mechanisms responsible for

this transition towards a fully developed turbulent state. For $r_u < r_{uc}$, we show that the three dimensionality of the primary jet vortices is greatly enhanced for small $\theta_{01}$ and leads to the appearance of secondary streamwise vortices which are stretched between consecutive vortices of the inner and of the outer shear layer. We observe that the transition towards a fully developed turbulent state is strongly linked with the formation and stretching of these streamwise vortices. The critical value $r_{uc}$ separating the regime without and with recirculation bubble is found to be strongly dependent on $\theta_{01}$. For $r_u > r_{uc}$, the shape and the length of the recirculation bubble is seen to be strongly affected by the shape of the inlet profile: the bubble is indeed significantly shortened when $\theta_{01}$ is small. The downstream region of the bubble is the siege of intense longitudinal vorticity production yielding the generation of intense streamwise vortices. Again these vortices are associated with a fast flow three-dimensionalization and rapid transition to turbulence.

The thickness of the inner shear layer also affects the evolution of the global quantities such as the jet spreading rate and the length of its two potential cores. For large enough values of $r_u$, the entrainment by the outer jet leads to an important depletion of the fluid situated near the jet axis. This depletion is associated with a low-pressure region in the jet core and with a curvature of the main velocity streamlines towards the jet axis: this creates a diminution of the jet width corresponding to a pinching phenomenon. This pinching phenomenon is more and more pronounced when $\theta_{01}$ decreases. The initial pinching stage is followed by a stage of linear spreading characteristic of a turbulent regime. The linear growth is reached at shorter downstream distances when $\theta_{01}$ is small due to the faster turbulence growth within the jet core. The pinching phenomenon is seen to reach a saturation level for $r_u > r_{uc}$.

Rehab et al.⁶ proposed a theoretical model based on phenomenological arguments to predict the variations of the inner potential core length with $r_u$. The direct numerical simulations allow to verify that the main assumptions of Rehab's model are justified. This initial model was however limited to an inlet velocity profile with a top-hat shape corresponding with $\theta_{01} \to 0$. We here extend the model to nonzero values of $\theta_{01}$ to theoretically predict the variation laws with $\theta_{01}$ of the inner potential core length and of the critical ratio $r_{uc}$. The variation law of the inner potential length with $r_u$ was also determined for finite values of $\theta_{01}$.

The present numerical study allows one to reach a better understanding of the transitional mechanisms leading to turbulence in coaxial jets. This knowledge could be used for control purposes by making possible an efficient manipulation of the flow coherent vortices.

## ACKNOWLEDGMENTS

Patrick Begou's help with the numerical-code parallelization is greatly acknowledged. Moreover, numerous discussions with Carlos B. da Silva have contributed to the success of this work. Part of the computations were carried out at the Institut du Développement et des Ressources en Informatique Scientifique (IDRIS, France).
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