3.1 Nearly inviscid flow

In the limit $\mathit{Re}\gg 1$ , the shear layers bounding the jets, of characteristic thickness $\mathit{Re}^{-1/2}H$ at distances of order $H$ , appear as infinitesimally thin surfaces. As indicated in figure 1, the two impinging streams are separated by the streamline $x=x_{s}(y)$ departing from the stagnation point $\boldsymbol{x}_{o}=(x_{o},0)$ . In addition, each jet is separated from the outer stagnant gas by the streamline $x=x_{1}(y)$ and $x=x_{2}(y)$ , respectively, each originating from their respective nozzle rim $(x,y)=(-L/H,1)$ and $(x,y)=(L/H,1)$ . These bounding surfaces are unknowns to be determined as part of the solution of a free-boundary problem, to be formulated below.

Consideration of the limit $\mathit{Re}\gg 1$ in (3.3) and (3.4) leads to $\boldsymbol{v}\boldsymbol{\cdot }\unicode[STIX]{x1D735}T\!=\!\boldsymbol{v}\boldsymbol{\cdot }\unicode[STIX]{x1D735}Y_{i}\!=\!0$ , indicating that the temperature and composition remain constant along any given streamline in the nearly inviscid jets outside the bounding mixing layers. Using these equations in (3.5) yields

a result that can be used in (3.1) to give

and in (3.2) to give $\boldsymbol{v}\boldsymbol{\cdot }\unicode[STIX]{x1D735}(p^{\prime }+\unicode[STIX]{x1D70C}\boldsymbol{v}^{2}/2)=0$ . The latter corresponds to the familiar condition of constant stagnation pressure $p^{\prime }+\unicode[STIX]{x1D70C}\boldsymbol{v}^{2}/2$ along streamlines, yielding in particular

along the centreline, where $p_{o}^{\prime }$ is the pressure at the stagnation point $\boldsymbol{x}_{o}$ , and

along the streamlines separating the jets from the stagnant air, where $p^{\prime }=0$ , with the constant taking in general different values on the surface of each jet.

According to (3.6), for inviscid incompressible flow each streamline carries the value of the density, so that the density field in the jet collision region is linked to the upstream boundary distributions of density in the feed streams. In configurations in which the density of the feed streams is non-uniform, the resulting density is different along different streamlines. In that case, vorticity is generated in the jet collision region as a result of the interaction of misaligned pressure and density gradients. This additional vorticity production supplements the vorticity already carried by the feed streams. The analysis below is restricted to configurations involving feed streams with uniform temperature and uniform composition, typically encountered in experiments, for which the density in each one of the jets is uniform, given by $\unicode[STIX]{x1D70C}=1$ in jet 1 and $\unicode[STIX]{x1D70C}=\unicode[STIX]{x1D70C}_{2}^{\ast }/\unicode[STIX]{x1D70C}_{1}^{\ast }$ in jet 2. Correspondingly, the baroclinic torque is negligible everywhere, except at the vortex sheet $x=x_{s}(y)$ , where it generates a velocity jump according to

with the subscripts $1$ and $2$ denoting the velocities on the left and right sides of the interface. The above equation follows from the condition of zero pressure jump across the interface separating the two jets, with the pressure evaluated with use made of $p^{\prime }+\unicode[STIX]{x1D70C}\boldsymbol{v}^{2}/2=$  const. on both sides of $x=x_{s}(y)$ . Along all other streamlines the vorticity magnitude $\unicode[STIX]{x1D714}=\unicode[STIX]{x2202}v_{y}/\unicode[STIX]{x2202}x-\unicode[STIX]{x2202}v_{x}/\unicode[STIX]{x2202}y$ remains constant, with a value equal to that found in the feed streams, to be determined from the boundary velocity distributions

where $u_{1}(y)$ and $u_{2}(y)$ represent non-dimensional shape functions satisfying $\int _{0}^{1}u_{1}\,\text{d}y=\int _{0}^{1}u_{2}\,\text{d}y=1$ . For example $u_{1}=u_{2}=1$ and $u_{1}=u_{2}=(3/2)(1-y^{2})$ for uniform and parabolic Poiseuille distributions, respectively.

In the vicinity of $\boldsymbol{x}_{o}=(x_{o},0)$ , the velocity is given by the well-known stagnation-point distribution

where the dimensionless strain rates $A_{1}=A_{1}^{\ast }/(Q_{1}/H^{2})$ and $A_{2}=A_{2}^{\ast }/(Q_{1}/H^{2})$ are related by

consistent with (3.10). As previously mentioned, this inviscid value of the strain rate, to be quantified below in terms of the parameters defining the opposed-nozzle flow (i.e. $L/H$ , $\unicode[STIX]{x1D70C}_{2}^{\ast }/\unicode[STIX]{x1D70C}_{1}^{\ast }$ , and $Q_{2}/Q_{1}$ ), determines the near-stagnation-point solution for the thin reactive mixing layer separating the two jet streams. The accuracy of the self-similar mixing-layer solution, which applies strictly at the centreline, can be expected to degrade with increasing distance from the stagnation point as a result of curvature effects and departures of the outer velocity from the linear distributions $v_{y}=A_{1}y$ and $v_{y}=A_{2}y$ . Curvature effects can be quantified by computing the shape of the separating streamline $x=x_{s}(y)$ away from the centreline, given by

with $r_{c}$ representing the local radius of curvature. On the other hand, the variations of the velocity on both sides of the separating vortex sheet are of the form

written with account taken of the result $\unicode[STIX]{x2202}^{2}v_{y}/\unicode[STIX]{x2202}y^{2}=0$ at $y=0$ , stemming from the symmetry condition $\unicode[STIX]{x2202}v_{x}/\unicode[STIX]{x2202}y=0$ at $y=0$ and the solenoidal character of the velocity field. Clearly, configurations with smaller values of $r_{c}^{-1}$ and $A_{1}^{\prime \prime }=(\unicode[STIX]{x1D70C}_{2}^{\ast }/\unicode[STIX]{x1D70C}_{1}^{\ast })^{1/2}A_{2}^{\prime \prime }$ can be expected to exhibit an extended domain where the mixing layer is planar and is subject to a constant strain rate. In that respect, quantifications of the near-stagnation-point region based on the inviscid solution can be useful in assessing the range of validity of the one-dimensional self-similar description for the reactive mixing layer.

3.2 Vorticity–streamfunction formulation

In view of (3.7) the problem can be formulated in terms of the standard streamfunction $\unicode[STIX]{x1D713}$ , related to the vorticity by

The vorticity is constant along the streamlines, so that $\unicode[STIX]{x1D714}=\unicode[STIX]{x1D6FA}(\unicode[STIX]{x1D713})$ , with the function $\unicode[STIX]{x1D6FA}(\unicode[STIX]{x1D713})$ determined implicitly through the expressions

derived with use of the boundary distributions (3.11). The problem reduces to the integration of

supplemented with the implicit definition of $\unicode[STIX]{x1D6FA}(\unicode[STIX]{x1D713})$ given in (3.17) and subject to the boundary conditions

The surfaces $x_{s}(y)$ , $x_{1}(y)$ and $x_{2}(y)$ are unknown free boundaries to be determined with use made of the additional boundary conditions (3.9) and (3.10), written in the form

and

respectively. In writing (3.22) from (3.9) we have used (3.8) to express the pressure for the parallel flow in the nozzles far upstream from the exit in terms of the stagnation-point pressure $p_{o}^{\prime }$ .

3.3 Reduction to the case of equal densities

The problem defined in (3.17)–(3.23) determines the streamfunction $\unicode[STIX]{x1D713}(x,y)$ along with the jet boundaries $x_{1}(y)$ , $x_{2}(y)$ and $x_{s}(y)$ , and the stagnation-point pressure $p_{o}^{\prime }$ . Besides the shapes of the velocity profiles in the feed streams, defined by the functions $u_{1}(y)$ and $u_{2}(y)$ , the solution depends on three parameters, namely, $L/H$ , $Q_{2}/Q_{1}$ and $\unicode[STIX]{x1D70C}_{2}^{\ast }/\unicode[STIX]{x1D70C}_{1}^{\ast }$ . As noted earlier in connection with axisymmetric jets (Carpio et al. Reference Carpio, Liñán, Sánchez and Williams2017), the solution can be simplified by incorporating a renormalization factor $(\unicode[STIX]{x1D70C}_{2}^{\ast }/\unicode[STIX]{x1D70C}_{1}^{\ast })^{1/2}$ in the definition of the kinematic variables for the fluid of density $\unicode[STIX]{x1D70C}_{2}^{\ast }$ . Specifically, we introduce new density-weighted functions $\hat{\unicode[STIX]{x1D713}}$ and $\hat{\unicode[STIX]{x1D6FA}}$ , defined by

to write (3.18) as

to be integrated with the boundary conditions

and

which follow from (3.19)–(3.22), respectively, whereas the dynamic condition (3.23) is automatically satisfied provided that $\unicode[STIX]{x1D735}\hat{\unicode[STIX]{x1D713}}$ is continuous, thereby removing the need to consider the separating surface $x_{s}(y)$ as a free boundary. In the reduced formulation the ratios $Q_{2}/Q_{1}$ and $\unicode[STIX]{x1D70C}_{2}^{\ast }/\unicode[STIX]{x1D70C}_{1}^{\ast }$ appear jointly in the new parameter

with $\unicode[STIX]{x1D6EC}^{2}$ representing a measure of the ratio of jet momentum fluxes. The function $\hat{\unicode[STIX]{x1D6FA}}(\hat{\unicode[STIX]{x1D713}})$ , identically zero for uniform velocity in the feed streams, can be determined in general from (3.17) written in the form

Inspection of (3.25)–(3.27) reveals that the transformation (3.24) effectively simplifies the problem to one involving constant density, removing the vortex-sheet character of the separating surface $x_{s}(y)$ , and concurrently reduces the number of controlling parameters from three to only two, namely, $L/H$ and $\unicode[STIX]{x1D6EC}$ . Since the streamline pattern found with $\unicode[STIX]{x1D6EC}$ is a mirror image about the plane $x=0$ of that found with $1/\unicode[STIX]{x1D6EC}$ , only flows with $\unicode[STIX]{x1D6EC}\geqslant 1$ need to be considered in the following.

The jet outer interfaces $x_{1}(y)$ and $x_{2}(y)$ remain as unknown free boundaries, to be determined along with the unknown value of the stagnation pressure $p_{o}^{\prime }$ as part of the computation of $\hat{\unicode[STIX]{x1D713}}(x,y)$ for given values of $u_{1}(y)$ , $u_{2}(y)$ , $\unicode[STIX]{x1D6EC}$ and $L/H$ . The solution provides the location of the stagnation point $\boldsymbol{x}_{o}=(x_{o},0)$ and also the associated local values of

with $A_{o}=A_{1}=(\unicode[STIX]{x1D70C}_{2}^{\ast }/\unicode[STIX]{x1D70C}_{1}^{\ast })^{1/2}A_{2}$ and $A_{o}^{\prime \prime }=A_{1}^{\prime \prime }=(\unicode[STIX]{x1D70C}_{2}^{\ast }/\unicode[STIX]{x1D70C}_{1}^{\ast })^{1/2}A_{2}^{\prime \prime }$ . The free-boundary problem formulated above has analytical solutions only for configurations with uniform velocity distributions in the feed streams, such that $\hat{\unicode[STIX]{x1D6FA}}$ is identically zero, that being the case considered in the following section, with rotational solutions addressed in § 5.

4.1 General considerations

When the velocity profiles in the feed streams are uniform, the vorticity function $\hat{\unicode[STIX]{x1D6FA}}$ in (3.25) is identically zero, so that the inviscid flow is irrotational. With $u_{1}=u_{2}=1$ , the stagnation pressure in both feed streams is uniform, equal to the stagnation-point pressure $p_{o}^{\prime }$ . The presence of the collision region generates an overpressure in the feed streams upstream from the nozzle exit, given by $p_{1}^{\prime }=p_{o}^{\prime }-1/2\geqslant 0$ and $p_{2}^{\prime }=p_{o}^{\prime }-\unicode[STIX]{x1D6EC}^{2}/2\geqslant 0$ , respectively. The boundary condition (3.27) on the jet surface reduces to

revealing in particular that the speed $|\unicode[STIX]{x1D735}\hat{\unicode[STIX]{x1D713}}|$ remains constant along these free surfaces, with a value given by $U=\sqrt{2p_{o}^{\prime }}$ , to be determined as part of the analysis.

The configuration depicted in figure 2 corresponds to $L\sim H$ , the case typically encountered in counterflow burners. Large and small values of $L/H$ , also considered below, are of interest in chemical engineering applications (Tamir Reference Tamir1994). For large separation distances, in the symmetric case $\unicode[STIX]{x1D6EC}=1$ the problem reduces to the classical problem of collision of two free jets (Birkhoff & Zarantonello Reference Birkhoff and Zarantonello1957; Gurevich Reference Gurevich1966; Milne-Thomson Reference Milne-Thomson1968), with the stagnation plane $x=0$ located far from both nozzle exits. This symmetric configuration is known to be prone to oscillatory instabilities (Li et al. Reference Li, Yao, Liu and Wang2011) that cause the stagnation plane to shift alternatively between both nozzle exits. Configurations with unbalanced momentum flux (i.e. $\unicode[STIX]{x1D6EC}\neq 1$ ) result in the collision region migrating to the vicinity of the nozzle carrying less momentum (the left nozzle in the cases $\unicode[STIX]{x1D6EC}>1$ considered here). In that case, the presence of a collision region has little effect on the flow at the outlet of nozzle 2, where the overpressure is $p_{2}^{\prime }=0$ and the velocity profile remains uniform with magnitude $\unicode[STIX]{x1D6EC}$ , so that for $L/H\gg 1$ and $\unicode[STIX]{x1D6EC}>1$ the constant free-streamline speed $U$ is simply equal to  $\unicode[STIX]{x1D6EC}$ .

The collision of the opposed jets results in two symmetric jets that emerge laterally, as shown in figure 2. At transverse distances large compared with $H$ the streamlines in these lateral jets become aligned as the pressure approaches the ambient pressure $p^{\prime }=0$ , with the velocity correspondingly approaching the unknown uniform value $U=\sqrt{2p_{o}^{\prime }}$ across the jet, as follows from conservation of stagnation pressure along the streamlines. Conservation of mass and of longitudinal momentum provides the two relationships

respectively, involving the thickness $h/H$ and deflection angle $\unicode[STIX]{x1D6FC}$ of the lateral jet far from the opening, the latter measured relative to the $y$ -axis. These two expressions can be combined with $U=\sqrt{2p_{o}^{\prime }}=\sqrt{\unicode[STIX]{x1D6EC}^{2}+2p_{2}^{\prime }}$ to give

involving the unknown overpressure $p_{2}^{\prime }\geqslant 0$ in the feed stream $2$ . As can be seen, the value of $\unicode[STIX]{x1D6FC}$ , identically zero in the symmetric case $\unicode[STIX]{x1D6EC}=1$ , increases for increasing values of $\unicode[STIX]{x1D6EC}$ . An interesting conclusion stemming from (4.3) is that for the opposed-jet arrangement investigated here (i.e. aligned jets issuing from nozzles of equal radius) the lateral-jet deflection is limited to a maximum value $\unicode[STIX]{x1D6FC}=\unicode[STIX]{x03C0}/6$ , achieved for $\unicode[STIX]{x1D6EC}\gg 1$ when the nozzles are placed far apart, so that the pressure in nozzle 2 equals the ambient value $p_{2}^{\prime }=0$ .

The thickness of the shear layers at the periphery of the lateral jets, small in the collision region, grow with distance, so that the potential description given here fails at distances of the order of $\mathit{Re}$ times the jet width $h$ . Consideration of viscous effects, which are significant all across the jet in this far-field region, would be needed to describe the transition of the jet velocity profile from the uniform value $U$ to the far-field Bickley profile, as done by Revuelta, Sánchez & Liñán (Reference Revuelta, Sánchez and Liñán2002).

Figure 2.

Schematic view of the collision region for $L/H\sim 1$ and uniform velocity profiles in the feed streams.

4.2 Selected potential-flow formulae

The solution simplifies for $L/H\ll 1$ , in that the flow in the nozzles away from the small opening, including the central region near the stagnation point, is independent of the lateral emerging jets, with the opening acting as an apparent point sink of strength $(1+\unicode[STIX]{x1D6EC})$ for the flow in the nozzles. The solution can be determined using superposition of a uniform flow and a sink in a channel (see Milne-Thomson Reference Milne-Thomson1968) to give

The above expression can be used to determine the stagnation-point axial location from the condition $\unicode[STIX]{x2202}\hat{\unicode[STIX]{x1D713}}/\unicode[STIX]{x2202}y=0$ at $y=0$ as well as the values of $A_{o}$ , $r_{c}^{-1}$ and $A_{o}^{\prime \prime }$ from (3.30), yielding

independent of $L/H$ . The high-speed flow at distances of order $L\ll H$ from the opening corresponds to the potential solution for the discharge of a pressurized container through an aperture, described for instance on pp. 310–311 of Milne-Thomson (Reference Milne-Thomson1968), including an emerging jet with

with the associated value of $h/(2L)=\unicode[STIX]{x03C0}/(2+\unicode[STIX]{x03C0})\simeq 0.61$ being the well-known coefficient of contraction of a planar jet. Because of its large velocity, the deflection of the emerging jet, required to accommodate the unbalanced momentum flux of the opposed streams, is very small, as can be seen by using (4.2) and (4.6) to write

The derivation of the corresponding analytic solution for the potential flow in the general case $L/H\sim 1$ requires use of conformal-mapping techniques. A full and systematic discussion of the needed analysis is presented in appendix A. The development provides, in particular, the equation

relating the three unknowns $\unicode[STIX]{x1D6FC}$ , $h/H$ and $U$ to the parameters $\unicode[STIX]{x1D6EC}$ and $L/H$ , as well as the expressions

for the stagnation-point properties in terms of $U$ and $\unicode[STIX]{x1D6EC}$ . Also, the conformal transformation provides integral expressions for the boundary surfaces $x_{1}(y)$ and $x_{2}(y)$ , given in (A 18) and (A 19), and for the separating streamline $x_{s}(y)$ departing from the stagnation point $\boldsymbol{x}_{o}=(x_{o},0)$ , obtained by integrating (A 16) along the contour $\unicode[STIX]{x1D6E4}$ defined by (A 24), with $x_{o}$ determined from (A 23).

Equations (4.9), valid in general for configurations with $L/H\sim 1$ , enable quantification of stagnation-point properties in the limiting cases $L/H\ll 1$ and $L/H\gg 1$ . For instance, the expressions given in (4.5) for $A_{o}$ , $r_{c}^{-1}$ and $A_{o}^{\prime \prime }$ when $L/H\ll 1$ may be recovered by letting $U\rightarrow \infty$ in (4.9), the appropriate limit as $L/H\rightarrow 0$ . In the opposite limit $L/H\gg 1$ the free-stream velocity reduces to $U=\unicode[STIX]{x1D6EC}$ , as discussed above, so that (4.9) gives

as the limiting values characterizing the stagnation point for distant nozzles with $L/H\gg 1$ .

4.3 General dependences on $L/H$ and $\unicode[STIX]{x1D6EC}$

Figure 3.

The variation with nozzle spacing $L/H$ of the deflection $\unicode[STIX]{x1D6FC}$ of the emerging jet ( $a$ ), surface speed $U$ ( $b$ ) and emerging-jet width $h/H$ ( $c$ ) for selected values of the momentum-flux ratio $\unicode[STIX]{x1D6EC}$ . The dashed lines in panels ( $a$ ) and ( $c$ ) correspond to $\unicode[STIX]{x1D6EC}\rightarrow \infty$ .

The formulas developed above can be employed to determine the dependence of different flow features on the inter-nozzle spacing $L/H$ and on the momentum-flux ratio $\unicode[STIX]{x1D6EC}$ . The free-surface velocity $U$ and the deflection angle $\unicode[STIX]{x1D6FC}$ and dimensionless width $h/H$ of the emerging jet are computed by solving the mass and momentum conservation equations given in (4.2) together with (4.8). For ease of calculation, it is convenient to select the values of $\unicode[STIX]{x1D6EC}$ and $U$ and then use (4.2) to determine $\unicode[STIX]{x1D6FC}$ and $h/H$ , with the corresponding value of $L/H$ finally computed from (4.8). Results are shown in figure 3 in the extended parametric ranges $0<L/H\leqslant 4$ and $1\leqslant \unicode[STIX]{x1D6EC}\leqslant 10$ .

The curves in figure 3 help to quantify the effect of the inter-nozzle spacing $L/H$ on the resulting emerging jet. For small values of $L/H$ the inter-nozzle opening appears as a small gap, so that a large stagnation pressure $p_{o}^{\prime }$ is needed to maintain the finite outflow rate $1+\unicode[STIX]{x1D6EC}$ . This results in emerging jets with larger speeds $U$ and smaller thicknesses $h/H$ , as described by the asymptotic behaviours given in (4.6) and (4.7), corresponding to a jet discharging from a pressurized container through an aperture of width $2L$ on a flat wall. The opposite limit $L/H\gg 1$ corresponds to two nozzles placed at large distances. In the symmetric case $\unicode[STIX]{x1D6EC}=1$ , the two jets collide at the midpoint between the two distant nozzles, giving rise to two identical transverse jets of width $h=2H$ and velocity $U=1$ . When $\unicode[STIX]{x1D6EC}>1$ , the collision region migrates to the vicinity of nozzle 1, where the outgoing stream encounters the opposed jet of stream 2, which effectively behaves as a free jet with speed $U=\unicode[STIX]{x1D6EC}$ . This is also the speed of the emerging jet for $L/H\gg 1$ , whereas its thickness approaches $h/H=(1+\unicode[STIX]{x1D6EC})/U=(1+\unicode[STIX]{x1D6EC})/\unicode[STIX]{x1D6EC}$ in this same limit, as follows from continuity. The jet deflection, identically zero in the symmetric case $\unicode[STIX]{x1D6EC}=1$ , is seen to increase with increasing $L/H$ for $\unicode[STIX]{x1D6EC}\neq 1$ to reach a maximum value $\unicode[STIX]{x1D6FC}=\arcsin [(\unicode[STIX]{x1D6EC}-1)/(2\unicode[STIX]{x1D6EC})]$ for $L/H\gg 1$ , as can be obtained from (4.3) with $U=\unicode[STIX]{x1D6EC}$ . As pointed out earlier, the deflection is limited to a maximum value $\unicode[STIX]{x1D6FC}=\unicode[STIX]{x03C0}/6$ , reached as $\unicode[STIX]{x1D6EC}\rightarrow \infty$ .

Figure 4.

Free-streamline patterns for different values of $\unicode[STIX]{x1D6EC}$ and $L/H$ , including the outer interfaces $\hat{\unicode[STIX]{x1D713}}=1$ and $\hat{\unicode[STIX]{x1D713}}=\unicode[STIX]{x1D6EC}$ (solid curves) and the separating interface $\hat{\unicode[STIX]{x1D713}}=0$ (dashed curves).

The interfaces $x=x_{1}(y)$ , $x=x_{2}(y)$ and $x=x_{s}(y)$ are shown in figure 4 for configurations with unbalanced momentum flux and different nozzle separation distances. As expected, a larger value of $\unicode[STIX]{x1D6EC}$ results in a displacement of the stagnation point towards the jet with smaller momentum and in a larger deflection of the emerging jet, the latter feature quantified in figure 3( $a$ ). The variation with $L/H$ of the distance $x_{o}+L/H$ between the stagnation point and the left nozzle is shown in figure 5( $a$ ), with negative values of $x_{o}+L/H$ corresponding to stagnation points lying inside the left nozzle. As can be seen, for a given momentum-flux ratio the stagnation point moves for increasing $L/H$ from the location $-\ln (\unicode[STIX]{x1D6EC})/\unicode[STIX]{x03C0}$ corresponding to $L/H=0$ to reach a finite distance from the nozzle as $L/H\rightarrow \infty$ , the only exception being the symmetric case $\unicode[STIX]{x1D6EC}=1$ , for which the stagnation point keeps getting farther according to $x_{o}+L/H=L/H$ . It is worth noting that for $\unicode[STIX]{x1D6EC}\gtrsim 8.4$ the stagnation point remains inside the nozzle regardless of the value of $L/H$ .

Figure 5.

The variation with $L/H$ of the distance of the stagnation point from the exit plane of the left nozzle $x_{o}+L/H$ ( $a$ ), the stagnation-point strain rate $A_{o}$ ( $b$ ), the local curvature of the separating interface at the stagnation point $1/r_{c}$ ( $c$ ) and the parameter $A_{o}^{\prime \prime }$ measuring the departures of the velocity from the stagnation-point solution ( $d$ ) for selected values of $\unicode[STIX]{x1D6EC}$ ; the dots along the vertical axes represent the limiting values given in (4.5) for $L/H=0$ whereas the dashed lines are the asymptotic values given in (4.10) for $L/H\gg 1$ .

As previously mentioned, the flame in combustion experiments is embedded in the mixing layer separating the two jets, centred about the inviscid interface $x=x_{s}(y)$ . Interest is focused on the near-stagnation-point region, where the solution for the inner structure of the reactive mixing layer is self-similar in the first approximation, determined by stagnation-point values of the strain rate $A_{1}$ and $A_{2}$ found in both colliding streams outside the mixing layer, related by $A_{o}=A_{1}=(\unicode[STIX]{x1D70C}_{2}^{\ast }/\unicode[STIX]{x1D70C}_{1}^{\ast })^{1/2}A_{2}$ . For slot nozzles with upstream uniform velocity profiles in the feed streams the analytic expression given in (4.9) can be employed to determine $A_{o}$ , with $\unicode[STIX]{x1D6EC}$ and $L/H$ entering as the only controlling parameters. The value of $A_{o}$ evaluated from (4.9) with the surface velocity $U$ obtained from solving (4.2) and (4.8) is shown in figure 5( $b$ ) as a function of $L/H$ for different $\unicode[STIX]{x1D6EC}$ . The results indicate that $A_{o}$ increases with increasing $\unicode[STIX]{x1D6EC}$ and with decreasing $L/H$ , so that the resulting values range between the minimum $A_{o}=\unicode[STIX]{x03C0}/8$ , corresponding to symmetric configurations with distant nozzles, to the maximum $A_{o}=\unicode[STIX]{x03C0}$ , found irrespective of the nozzle spacing when the momentum flux is severely unbalanced.

The self-similar solution for the counterflow mixing layer assumes a locally planar structure, with negligible curvature. In reality, however, the separating interface is curved, and there is interest in quantifying this curvature to assess departures from locally planar self-similar structures. This is done in figure 5( $c$ ) with use made of (4.9). The plots reveal that the curvature is always positive for $\unicode[STIX]{x1D6EC}>2$ , corresponding to convex interfaces. For values of the momentum-flux ratio in the range $1<\unicode[STIX]{x1D6EC}<2$ , however, the curvature becomes negative for sufficiently large values of $L/H$ , with a minimum value, reached as $L/H\rightarrow \infty$ , given in (4.10). This change of sign of the curvature from convex to concave as the inter-nozzle distance is increased is apparent in the separating interfaces shown in figure 4 for $\unicode[STIX]{x1D6EC}=1.5$ . The results indicate that for configurations with $L/H\sim 1$ and $\unicode[STIX]{x1D6EC}\lesssim 2$ , most often found in typical experimental arrangements, the separating streamline remains fairly flat, a finding that supports the neglect of curvature effects in analysing the structure of the reactive mixing layer.

Also central to the self-similar character of the counterflow mixing layer is the assumption that the velocity in the outer inviscid stream increases linearly with the streamwise distance, as occurs sufficiently close to the stagnation point. With increasing distances, however, the deviations from the linear acceleration become more noticeable, with the non-dimensional parameters $A_{1}^{\prime \prime }$ and $A_{2}^{\prime \prime }$ measuring the extent of the departures, as indicated in (3.15). The value of $A_{o}^{\prime \prime }=A_{1}^{\prime \prime }=(\unicode[STIX]{x1D70C}_{2}^{\ast }/\unicode[STIX]{x1D70C}_{1}^{\ast })^{1/2}A_{2}^{\prime \prime }$ evaluated from (4.9) is shown in figure 5( $d$ ). For $L/H\lesssim 1$ , the differences in values of $A_{o}^{\prime \prime }$ for different $\unicode[STIX]{x1D6EC}$ can be fairly large, with small values of $\unicode[STIX]{x1D6EC}$ resulting in accelerating flows with $A_{o}^{\prime \prime }>0$ whereas sufficiently large values yield decelerating velocities with negative $A_{o}^{\prime \prime }$ . The differences diminish as the nozzle spacing increases. As seen in figure 5(c,d), configurations with $1\leqslant \unicode[STIX]{x1D6EC}\leqslant 2$ tend to produce relatively small values of $A_{o}^{\prime \prime }$ and $1/r_{c}$ for $L/H\gtrsim 1.5$ , delineating an attractive parametric range for experimental designs aimed at minimizing departures of the mixing-layer structure from one-dimensional solutions.

5.1 Rotational flow with $L/H\ll 1$

For cases with $L\ll H$ , the velocity at distances of order $L$ from the opening becomes a factor $H/L$ larger than that found in the feed streams, as follows from a straightforward continuity balance. The associated point-sink singularities arising in the limit $L/H\rightarrow 0$ at the apparent opening locations $(x,y)=(0,\pm 1)$ can be effectively handled in the numerical description by expressing $\hat{\unicode[STIX]{x1D713}}$ as the sum of a potential streamfunction $\hat{\unicode[STIX]{x1D713}}_{p}$ carrying the volume flux of the two streams and a rotational streamfunction $\hat{\unicode[STIX]{x1D713}_{r}}$ with associated zero volume flux. The rotational streamfunction satisfies

where $\hat{\unicode[STIX]{x1D6FA}}(\hat{\unicode[STIX]{x1D713}})$ must be evaluated from (5.1) in terms of $\hat{\unicode[STIX]{x1D713}}=\hat{\unicode[STIX]{x1D713}_{p}}+\hat{\unicode[STIX]{x1D713}_{r}}$ with $\hat{\unicode[STIX]{x1D713}_{p}}$ given in (4.4). The boundary conditions for $\hat{\unicode[STIX]{x1D713}_{r}}$ reduce to

and

for $0\leqslant y\leqslant 1$ . The streamfunction approaches the Poiseuille distributions according to

where $\unicode[STIX]{x1D706}_{1}=2.59$ and $F_{1}$ are the smallest eigenvalue and corresponding eigenfunction of the homogeneous problem

obtained by linearizing (5.2) about the Poiseuille velocity distribution. The constant factors $C_{\pm \infty }$ in (5.5) are to be determined as part of the numerical integration of (5.2).

Figure 6.

Streamlines corresponding to $L/H=0$ for feed streams with Poiseuille velocity profiles (upper half of the plots) and with uniform velocity profiles (lower half of the plots). Besides the separating streamline $\hat{\unicode[STIX]{x1D713}}=0$ , denoted with a dashed curve, the plots show the streamlines $\hat{\unicode[STIX]{x1D713}}=(0.05,0.2,0.4,0.6,0.8)$ for stream 1 and the streamlines $\hat{\unicode[STIX]{x1D713}}=-\unicode[STIX]{x1D6EC}\times (0.05,0.2,0.4,0.6,0.8)$ for stream 2.

Equation (5.2) was solved iteratively on a fixed rectangular domain until convergence was achieved, determined by the condition $||\hat{\unicode[STIX]{x1D713}}_{r}^{k+1}-\hat{\unicode[STIX]{x1D713}}_{r}^{k}||_{max}<10^{-6}$ . A second-order central difference scheme was used to discretize (5.2) written at each iteration $k$ in the form $\unicode[STIX]{x1D6FB}^{2}\hat{\unicode[STIX]{x1D713}}_{r}^{k+1}=\hat{\unicode[STIX]{x1D6FA}}(\hat{\unicode[STIX]{x1D713}}^{k})$ , which includes a nonlinear solve for $\hat{\unicode[STIX]{x1D6FA}}(\hat{\unicode[STIX]{x1D713}}^{k})$ as defined by the implicit representation (5.1). To facilitate convergence of the integrations in the finite domain $x_{-\infty }\leqslant x\leqslant x_{+\infty }$ , the boundary conditions (5.4) as $x\rightarrow \pm \infty$ were replaced by

derived with account taken of the asymptotic behaviours (5.5). The solution was found to be independent of the longitudinal extent of the integration domain for sufficiently large values of $-x_{-\infty }$ and $x_{+\infty }$ , with the results shown in figures 6 and 7 corresponding to $-x_{-\infty }=x_{+\infty }=5$ .

Figure 7.

The variation with $\unicode[STIX]{x1D6EC}$ for $L/H=0$ of the distance of the stagnation point from the exit plane of the left nozzle $x_{o}$ ( $a$ ), the stagnation-point strain rate $A_{o}$ ( $b$ ), the local curvature of the separating interface at the stagnation point $1/r_{c}$ ( $c$ ) and the parameter $A_{o}^{\prime \prime }$ measuring the departures of the velocity from the stagnation-point solution ( $d$ ) as obtained numerically for selected values of $\unicode[STIX]{x1D6EC}$ with boundary Poiseuille velocity profiles (dashed curves) and evaluated from (4.5) for uniform velocity profiles (solid curves).

The streamlines $\hat{\unicode[STIX]{x1D713}}=\hat{\unicode[STIX]{x1D713}_{p}}+\hat{\unicode[STIX]{x1D713}_{r}}=$  const. are shown in figure 6 for different values of $\unicode[STIX]{x1D6EC}$ along with the corresponding potential-flow results, evaluated with use made of (4.4). As can be seen, the general morphology of the flow is not critically affected by the shape of the boundary velocity distribution. The resulting changes in connection with the stagnation point are further quantified in figure 7. In the range $1\leqslant \unicode[STIX]{x1D6EC}\leqslant 10$ , the relative changes in axial location $x_{o}$ and in the local curvature $1/r_{c}$ of the separating streamline $x=x_{o}+y^{2}/(2r_{c})$ are of the order of 5 %. By way of contrast, the local velocity field in the vicinity of $\boldsymbol{x}_{o}=(x_{o},0)$ is much more sensitive to the shape of the velocity in the feed streams, as seen in figure 7(b,d). The higher velocity of the Poiseuille distribution at the centreline produces a significantly higher value of $A_{o}$ . The parameter $A_{o}^{\prime \prime }$ , measuring the variation of the strain rate along the separating interface also exhibits pronounced differences. In particular, for $\unicode[STIX]{x1D6EC}\sim 1$ the rate of increase of the strain rate is considerably larger for potential flow.

5.2 Vortical flow with $L/H\simeq O(1)$

Figure 8.

Free and separating streamlines corresponding to uniform velocity in the feed streams and $L/H=2$ obtained from numerical integrations of the Navier–Stokes equations for $\mathit{Re}=500$ with $\unicode[STIX]{x1D70C}_{2}^{\ast }/\unicode[STIX]{x1D70C}_{1}^{\ast }=1$ and $Q_{2}/Q_{1}=3/2$ (solid curves), with $\unicode[STIX]{x1D70C}_{2}^{\ast }/\unicode[STIX]{x1D70C}_{1}^{\ast }=9/4$ and $Q_{2}/Q_{1}=1$ (dashed curves), and from evaluation of the potential-flow results of § 4 (dash-dotted). All three cases correspond to $\unicode[STIX]{x1D6EC}=3/2$ .

The computation of inviscid flows with free boundaries is in general a difficult task, especially with vorticity present. As shown by Bergthorson et al. (Reference Bergthorson, Sone, Mattner, Dimotakis, Goodwin and Meiron2005) when examining the related problem of an axisymmetric constant-density jet impinging on a perpendicular wall, the inviscid description can be approached by considering integrations of the Navier–Stokes equations for moderately large values of the Reynolds number, with associated departures from inviscid flow of order $\mathit{Re}^{-1/2}$ . This is the procedure followed below.

The previous results concerning the parametric dependence of the problem in the limit $\mathit{Re}\gg 1$ indicate that the specific selection of composition and temperature in the feed streams for the Navier–Stokes computations is not critical, in that different configurations with the same value of $\unicode[STIX]{x1D6EC}$ approach the same nearly inviscid flow structure in the limit $\mathit{Re}\gg 1$ , with differences arising only in the interior of the mixing layers surrounding the jets. For simplicity, the integrations below pertain to isothermal jets of two gases of different densities $\unicode[STIX]{x1D70C}_{2}^{\ast }$ and $\unicode[STIX]{x1D70C}_{1}^{\ast }$ but identical viscosities $\unicode[STIX]{x1D707}_{1}^{\ast }=\unicode[STIX]{x1D707}_{2}^{\ast }$ , so that $\unicode[STIX]{x1D707}=1$ in (3.2). The density of the ambient gas is taken to be equal to that of stream 1, so that the density at any point can be computed in terms of the mass fraction $Y_{2}$ by writing (3.5) in the form $\unicode[STIX]{x1D70C}=[1-Y_{2}+Y_{2}(\unicode[STIX]{x1D70C}_{1}^{\ast }/\unicode[STIX]{x1D70C}_{2}^{\ast })]^{-1}$ , which can be used in (3.4) to give

as a conservation equation for the gas density. Here $\mathit{Sc}=\unicode[STIX]{x1D707}_{1}^{\ast }/(\unicode[STIX]{x1D70C}_{1}^{\ast }D^{\ast })$ is the Schmidt number based on the binary diffusion coefficient $D^{\ast }$ of the two gases, with the value $\mathit{Sc}=0.7$ used in the numerical integrations reported below. The problem reduces to the integration of (3.1), (3.2), and (5.8) with boundary conditions in the feed streams $v_{x}-(3/2)(1-y^{2})=v_{y}=\unicode[STIX]{x1D70C}-1=0$ as $x\rightarrow -\infty$ and $v_{x}+(3/2)(Q_{2}/Q_{1})(1-y^{2})=v_{y}=\unicode[STIX]{x1D70C}-(\unicode[STIX]{x1D70C}_{2}^{\ast }/\unicode[STIX]{x1D70C}_{1}^{\ast })=0$ as $x\rightarrow +\infty$ . Non-permeable walls with a no-slip flow condition are employed in the description, along with the symmetry conditions $\unicode[STIX]{x2202}v_{x}/\unicode[STIX]{x2202}y=v_{y}=\unicode[STIX]{x2202}\unicode[STIX]{x1D70C}/\unicode[STIX]{x2202}y=0$ at $y=0$ and a condition of vanishing normal stress in the surrounding gas atmosphere, where $\unicode[STIX]{x1D70C}=1$ . The equations were integrated in the rectangular domain $x_{-\infty }\leqslant x\leqslant x_{+\infty }$ and $0\leqslant y\leqslant y_{+\infty }$ , with $-x_{-\infty }=x_{+\infty }=3L/H$ and $y_{+\infty }=7$ . The integrations employed a finite element method with $P1$ elements for the pressure field and $P2$ elements for the remaining variables, combined with a Newton–Raphson root-finding algorithm; details of the discretization method, used for instance by Moreno-Boza et al. (Reference Moreno-Boza, Coenen, Sevilla, Carpio and Sánchez2016), can be found in Hecht (Reference Hecht2012). An illustration of the resulting flow is given in figure 1, where the boundary streamlines $x_{1}(y)$ , $x_{2}(y)$ , and $x_{s}(y)$ are obtained by integrating the above problem for $L/H=2$ , $\unicode[STIX]{x1D70C}_{2}^{\ast }/\unicode[STIX]{x1D70C}_{1}^{\ast }=4$ , $Q_{2}/Q_{1}=1.5$ , $\mathit{Re}=500$ .

For the purpose of verification, a number of integrations were performed for uniform velocity in the feed streams, with a slip flow condition used at the wall to make the results of the integrations independent of the longitudinal extent of the integration domain. Figure 8 shows the separating streamlines (departing from the nozzle rims and from the stagnation point $\boldsymbol{x}_{o}$ ) obtained for $L/H=2$ from integrations of the Navier–Stokes equations with $\mathit{Re}=500$ for two different set of conditions, namely, $\unicode[STIX]{x1D70C}_{2}^{\ast }/\unicode[STIX]{x1D70C}_{1}^{\ast }-1=Q_{2}/Q_{1}-3/2=0$ and $\unicode[STIX]{x1D70C}_{2}^{\ast }/\unicode[STIX]{x1D70C}_{1}^{\ast }-9/4=Q_{2}/Q_{1}-1=0$ , resulting in the same momentum-flux parameter $\unicode[STIX]{x1D6EC}=3/2$ . The associated curves are practically indistinguishable, supporting the reduced parametric dependence predicted by the density-weighted formulation presented in § 3.3. The figure includes comparisons with the potential-flow solution derived in § 4, whose associated jet interfaces $x=x_{1}(y)$ , $x=x_{2}(y)$ and $x=x_{s}(y)$ are represented as dot-dashed curves. The close agreement of the potential flow with the Navier–Stokes results, expected in the limit $\mathit{Re}\gg 1$ (Bergthorson et al. Reference Bergthorson, Sone, Mattner, Dimotakis, Goodwin and Meiron2005), serves as verification for the numerical method.

Figure 9.

The distribution of longitudinal velocity along the centreline determined from the Navier–Stokes computations for $\mathit{Re}=500$ , $L/H=2$ and $Q_{2}/Q_{1}=1.5/(\unicode[STIX]{x1D70C}_{2}^{\ast }/\unicode[STIX]{x1D70C}_{1}^{\ast })^{1/2}$ with $\unicode[STIX]{x1D70C}_{2}^{\ast }/\unicode[STIX]{x1D70C}_{1}^{\ast }=(4/9,1,9/4)$ ; the symbols represent the rescaled velocity distributions $\unicode[STIX]{x1D70C}^{1/2}v_{x}$ for $\unicode[STIX]{x1D70C}_{2}^{\ast }/\unicode[STIX]{x1D70C}_{1}^{\ast }=4/9$ (squares) and for $\unicode[STIX]{x1D70C}_{2}^{\ast }/\unicode[STIX]{x1D70C}_{1}^{\ast }=9/4$ (circles).

To investigate the flow near the stagnation point the distribution of longitudinal velocity along the centreline obtained with Poiseuille velocity profiles in the feed streams is plotted in figure 9 for $\mathit{Re}=500$ , $L/H=2$ and three different density ratios, with corresponding volume-flux ratios $Q_{2}/Q_{1}$ selected to maintain in all three cases the same value $\unicode[STIX]{x1D6EC}=(\unicode[STIX]{x1D70C}_{2}^{\ast }/\unicode[STIX]{x1D70C}_{1}^{\ast })^{1/2}Q_{2}/Q_{1}=1.5$ . The velocity on each side of the stagnation point outside the mixing layer follows a linear distribution given by $v_{x}=-A_{1}(x-x_{o})$ in stream 1 and $v_{x}=-A_{2}(x-x_{o})$ in stream 2, in agreement with the local potential solution (3.12). The values of $A_{1}$ and $A_{2}$ , which can be obtained by extrapolating the numerical results near the stagnation point, are seen to satisfy the relationship $A_{o}=A_{1}=(\unicode[STIX]{x1D70C}_{2}^{\ast }/\unicode[STIX]{x1D70C}_{1}^{\ast })^{1/2}A_{2}$ stated in (3.13). The resulting values, $A_{o}=(1.58,1.59,1.60)$ for $\unicode[STIX]{x1D70C}_{2}^{\ast }/\unicode[STIX]{x1D70C}_{1}^{\ast }=(4/9,1,9/4)$ , are almost identical in all three cases, in agreement with the reduced inviscid formulation introduced earlier, which predicts that the strain rate $A_{1}=(\unicode[STIX]{x1D70C}_{2}^{\ast }/\unicode[STIX]{x1D70C}_{1}^{\ast })^{1/2}A_{2}$ depends on $\unicode[STIX]{x1D70C}_{2}^{\ast }/\unicode[STIX]{x1D70C}_{1}^{\ast }$ and $Q_{2}/Q_{1}$ through the single parameter $\unicode[STIX]{x1D6EC}=(\unicode[STIX]{x1D70C}_{2}^{\ast }/\unicode[STIX]{x1D70C}_{1}^{\ast })^{1/2}(Q_{2}/Q_{1})$ . The derivation of this reduced dependence is based on the density-weighted variables defined in (3.24), which are tested in figure 9 by representing with symbols the distribution of $\unicode[STIX]{x1D70C}^{1/2}v_{x}$ for the two cases with unequal jet densities. The resulting curves fall on top of the velocity distribution of the uniform-density case, further illustrating the applicability of the reduced inviscid formulation.

The dependence of the results on the Reynolds number was found to be relatively weak for moderately large values of $\mathit{Re}$ . Only small variations of a few per cent were found in values of the stagnation-point location $x_{o}$ and in the stagnation-point values of $A_{o}$ and $A_{o}^{\prime \prime }$ for $100\leqslant \mathit{Re}\leqslant 500$ , with somewhat larger variations affecting the curvature of the separating streamline $1/r_{c}$ . For instance, for $L/H=2$ , $\unicode[STIX]{x1D70C}_{2}^{\ast }/\unicode[STIX]{x1D70C}_{1}^{\ast }=1$ and $Q_{2}/Q_{1}=1.5$ the integrations provide $x_{o}=(-1.27,-1.29,-1.29)$ , $A_{o}=(1.55,1.58,1.59)$ , $A_{o}^{\prime \prime }=(1.07,1.07,1.07)$ and $1/r_{c}=(0.044,0.0561,0.0589)$ for $\mathit{Re}=(100,300,500)$ , respectively.

The computations for $\mathit{Re}=500$ and $\unicode[STIX]{x1D70C}_{2}^{\ast }/\unicode[STIX]{x1D70C}_{1}^{\ast }=1$ are used in figure 10 as a basis to investigate the dependence of $x_{o}$ , $A_{o}$ , $1/r_{c}$ and $A_{o}^{\prime \prime }$ on $L/H$ and $\unicode[STIX]{x1D6EC}$ . The numerical evaluation of $A_{o}$ involves a linear fit about $x=x_{o}$ of the longitudinal velocity distribution $v_{x}(x,0)$ . Similarly, $A_{o}^{\prime \prime }$ is obtained by computing the slope of a linear fit of the longitudinal distribution of $\unicode[STIX]{x2202}^{2}v_{x}/\unicode[STIX]{x2202}y^{2}(x,0)$ about $x=x_{o}$ , while the computation of $r_{c}$ requires a parabolic fit of the form (3.14) for the separating streamline departing from $\boldsymbol{x}_{o}$ . The resulting curves, which include the limiting results for $L/H=0$ computed earlier in § 5.1, exhibit trends that are qualitatively similar to those of the irrotational results in figure 5.

Figure 10.

The variation with $L/H$ of the distance of the stagnation point from the exit plane of the left nozzle $x_{o}+L/H$ ( $a$ ), the stagnation-point strain rate $A_{o}$ ( $b$ ), the local curvature of the separating interface at the stagnation point $1/r_{c}$ ( $c$ ) and the parameter $A_{o}^{\prime \prime }$ measuring the departures of the velocity from the stagnation-point solution ( $d$ ) obtained from the Navier–Stokes computations for $\mathit{Re}=500$ and $\unicode[STIX]{x1D70C}_{2}^{\ast }/\unicode[STIX]{x1D70C}_{1}^{\ast }=1$ ; the solid symbols along the vertical axes represent the limiting values given in figure 7.