4.3.1 Entrained mass flow rate

The entrained mass flux is related to the local relative velocity of fluid elements with the TNTI. Thus, it is useful to start with examining the velocities associated with the entrainment. There are three different velocities related to the entrainment process (Turner Reference Turner1986). The first velocity is the rate at which irrotational fluid flows into the turbulent region across its boundaries, which is sometimes called induced velocity, $E_{v}$ , shown in figure 4. In the configuration studied here, this velocity is the transverse component of velocity vector, $v$ , at the TNTI. The second velocity considered is the rate at which the turbulent boundaries are spreading out into the non-turbulent flow. This velocity is called boundary entrainment velocity, $E_{b}$ , shown in figure 4. $E_{b}$ is the average transverse velocity component of the TNTI (da Silva et al. Reference da Silva, Hunt, Eames and Westerweel2014a ),

(4.6) $$\begin{eqnarray}\displaystyle E_{b}=\frac{\text{d}\langle Y_{I}\rangle }{\text{d}t}, & & \displaystyle\end{eqnarray}$$

where $Y_{I}$ is the transverse location of the TNTI, shown in figure 3. The third velocity is the velocity that determines the mass flow rate across the TNTI. This velocity, which is called local entrainment velocity (Holzner & Lüthi Reference Holzner and Lüthi2011), $v_{n}$ , is the relative velocity of the local fluid element and the TNTI.

The method used to calculate the local entrainment velocity, $v_{n}$ , is briefly presented. Detailed description can be found in Pope (Reference Pope1988) and Holzner & Lüthi (Reference Holzner and Lüthi2011). The first step is to define the TNTI, which is an enstrophy isosurface in this study. By definition, the total change of the enstrophy on the TNTI in a frame of reference moving with the velocity of the TNTI, $\boldsymbol{U}^{s}$ , is zero,

(4.7) $$\begin{eqnarray}\displaystyle \frac{\text{D}^{s}{\it\Omega}}{\text{D}^{s}t}=\frac{\partial {\it\Omega}}{\partial t}+u_{j}^{s}\frac{\partial {\it\Omega}}{\partial x_{j}}=0. & & \displaystyle\end{eqnarray}$$

For each point on the TNTI, the velocity of the interface is a summation of the local fluid element velocity, $\boldsymbol{U}$ , and the velocity of the area element on the TNTI relative to the fluid, $\boldsymbol{V}$ ; $u_{j}^{s}=u_{j}+V_{j}$ . Substituting this into (4.7),

(4.8) $$\begin{eqnarray}\displaystyle \frac{\text{D}{\it\Omega}}{\text{D}t}=\frac{\partial {\it\Omega}}{\partial t}+u_{j}\frac{\partial {\it\Omega}}{\partial x_{j}}=-V_{j}\frac{\partial {\it\Omega}}{\partial x_{j}}=-\boldsymbol{V}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}{\it\Omega}. & & \displaystyle\end{eqnarray}$$

The relative velocity vector, $\boldsymbol{V}$ , at each point is normal to the isosurface (Pope Reference Pope1988), $\boldsymbol{V}=v_{n}\hat{\boldsymbol{n}}$ , where $\hat{\boldsymbol{n}}$ is the unit vector normal to the isosurface shown in figure 4, and $v_{n}$ is the velocity component in the normal direction, and is called the local entrainment velocity. From (4.8),

(4.9) $$\begin{eqnarray}\displaystyle v_{n}=-\frac{\text{D}{\it\Omega}/\text{D}t}{|\boldsymbol{{\rm\nabla}}{\it\Omega}|}. & & \displaystyle\end{eqnarray}$$

The local entrained mass flux of the irrotational flow into the turbulent region at the TNTI can be written as,

(4.10) $$\begin{eqnarray}\displaystyle {\dot{m}}^{\prime \prime }=-{\it\rho}v_{n}. & & \displaystyle\end{eqnarray}$$

In this equation, positive ${\dot{m}}^{\prime \prime }$ corresponds to flux of the irrotational flow into the rotational region. In the rest of this work, ${\dot{m}}^{\prime \prime }>0$ is referred to as entrainment and ${\dot{m}}^{\prime \prime }<0$ is referred to as detrainment. Using (4.5) and (4.9), (4.10) can be rewritten as

(4.11) $$\begin{eqnarray}\displaystyle {\dot{m}}^{\prime \prime } & = & \displaystyle \underbrace{\frac{2{\it\rho}{\it\omega}_{i}{\it\omega}_{j}}{|\boldsymbol{{\rm\nabla}}{\it\Omega}|}\frac{\partial u_{i}}{\partial x_{j}}}_{{\dot{m}}_{{\mathcal{I}}{\mathcal{I}}}^{\prime \prime }}\underbrace{-\frac{2{\it\rho}{\it\Omega}}{|\boldsymbol{{\rm\nabla}}{\it\Omega}|}\frac{\partial u_{j}}{\partial x_{j}}}_{{\dot{m}}_{{\mathcal{I}}{\mathcal{I}}{\mathcal{I}}}^{\prime \prime }}\underbrace{+\frac{2{\it\omega}_{i}{\it\epsilon}_{ijk}}{{\it\rho}|\boldsymbol{{\rm\nabla}}{\it\Omega}|}\frac{\partial {\it\rho}}{\partial x_{j}}\frac{\partial p}{\partial x_{k}}}_{{\dot{m}}_{{\mathcal{I}}{\mathcal{V}}}^{\prime \prime }}\nonumber\\ \displaystyle & & \displaystyle \underbrace{+\,\frac{{\it\mu}}{Re|\boldsymbol{{\rm\nabla}}{\it\Omega}|}\frac{\partial ^{2}{\it\Omega}}{\partial x_{j}\partial x_{j}}}_{{\dot{m}}_{{\mathcal{V}}}^{\prime \prime }}\underbrace{-\frac{2{\it\mu}}{Re|\boldsymbol{{\rm\nabla}}{\it\Omega}|}\frac{\partial {\it\omega}_{i}}{\partial x_{j}}\frac{\partial {\it\omega}_{i}}{\partial x_{j}}}_{{\dot{m}}_{{\mathcal{V}}{\mathcal{I}}}^{\prime \prime }}\underbrace{-\frac{2{\it\omega}_{i}{\it\epsilon}_{ijk}}{{\it\rho}Re|\boldsymbol{{\rm\nabla}}{\it\Omega}|}\frac{\partial {\it\rho}}{\partial x_{j}}\frac{\partial {\it\tau}_{km}}{\partial x_{m}}}_{{\dot{m}}_{{\mathcal{V}}{\mathcal{I}}{\mathcal{I}}}^{\prime \prime }},\end{eqnarray}$$

where the effect of inviscid and viscous terms can be studied separately as

(4.12a,b ) $$\begin{eqnarray}\displaystyle {\dot{m}}_{inv}^{\prime \prime }={\dot{m}}_{{\mathcal{I}}{\mathcal{I}}}^{\prime \prime }+{\dot{m}}_{{\mathcal{I}}{\mathcal{I}}{\mathcal{I}}}^{\prime \prime }+{\dot{m}}_{{\mathcal{I}}{\mathcal{V}}}^{\prime \prime },\quad {\dot{m}}_{vis}^{\prime \prime }={\dot{m}}_{{\mathcal{V}}}^{\prime \prime }+{\dot{m}}_{{\mathcal{V}}{\mathcal{I}}}^{\prime \prime }+{\dot{m}}_{{\mathcal{V}}{\mathcal{I}}{\mathcal{I}}}^{\prime \prime }. & & \displaystyle\end{eqnarray}$$

Table 5.

Average values of induced velocity, $E_{v}$ , boundary entrainment velocity, $E_{b}$ , local entrainment velocity, $v_{n}$ , and the terms in (4.13) and (4.14), for different convective Mach numbers. ${\rm\Delta}E_{b}$ and $\langle {\rm\Delta}E_{v}\rangle$ are the relative velocities of the upper and lower interfaces, defined as $E_{b_{upper}}-E_{b_{lower}}$ and $\langle E_{v_{lower}}\rangle -\langle E_{v_{upper}}\rangle$ respectively. ${\rm\Delta}U$ is the velocity difference of upper and lower streams, $u_{{\it\eta}}$ is the Kolmogorov velocity, $Re_{{\mathcal{L}}}$ is the Reynolds number of the largest eddies in the flow, $\bar{{\it\rho}}$ is the average density of the flow on the midplane and ${\it\rho}_{0}$ is the density of the outside irrotational flow.

Table 6.

Average entrained mass flux, $\langle {\dot{m}}^{\prime \prime }\rangle$ , and contribution of different terms of (4.11) and (4.12) to this flux, ${\mathcal{E}}_{{\mathcal{N}}}=100\times \langle {\dot{m}}_{{\mathcal{N}}}^{\prime \prime }\rangle /\langle {\dot{m}}^{\prime \prime }\rangle$ , for different convective Mach numbers. ${\it\rho}_{0}$ is the reference value of density, and ${\rm\Delta}U$ is the velocity difference of upper and lower streams.

It is observed in § 4.3.3 and also is shown by Wolf et al. (Reference Wolf, Lüthi, Holzner, Krug, Kinzelbach and Tsinober2012, Reference Wolf, Holzner, Lüthi, Krug, Kinzelbach and Tsinober2013a ,Reference Wolf, Lüthi, Holzner, Krug, Kinzelbach and Tsinober b ) and Philip et al. (Reference Philip, Bermejo-Moreno, Chung and Marusic2015), that the local entrainment is a function of the geometrical shape of the TNTI. Therefore, the entrainment analysis in this study is conducted on the entire surface of the TNTI not just the interface envelope. The difference between these two surfaces is shown in figure 3.

Tables 5 and 6 summarize the results for the velocities associated with the entrainment and the entrained mass flux normalized with appropriate parameters for all the cases. It can be seen in table 5 that the values of all the velocities decrease as the convective Mach number increases. Turner (Reference Turner1986) estimated the boundary entrainment velocity to be twice the induced velocity for an incompressible jet in agreement with case $M02$ . Westerweel et al. (Reference Westerweel, Fukushima, Pedersen and Hunt2009) and Khashehchi et al. (Reference Khashehchi, Ooi, Soria and Marusic2013) respectively reported the values of 0.07 and 0.066 for $E_{b}/U_{m}$ from experimental results of an incompressible jet, which are close to the result for case $M02$ in table 5. The reported Reynolds numbers are $Re_{{\it\lambda}}\approx 95$ for Westerweel et al. (Reference Westerweel, Fukushima, Pedersen and Hunt2009) and $Re_{{\it\lambda}}\approx 122$ for Khashehchi et al. (Reference Khashehchi, Ooi, Soria and Marusic2013). Wolf et al. (Reference Wolf, Lüthi, Holzner, Krug, Kinzelbach and Tsinober2012, Reference Wolf, Holzner, Lüthi, Krug, Kinzelbach and Tsinober2013a ) reported the values $E_{b}/U_{m}\approx 0.08$ and $\langle -v_{n}\rangle /u_{{\it\eta}}\approx 0.40$ for an incompressible jet with $Re_{{\it\lambda}}\approx 100$ , which are of the same order as the results for case $M02$ in this study. It is discussed by Dimotakis (Reference Dimotakis2005) that for $Re_{{\it\lambda}}>100$ various flow phenomena are a weak function of the Reynolds number. The similarities between the results for case $M02$ with incompressible jets found in the literature with different Reynolds numbers (Westerweel et al. Reference Westerweel, Fukushima, Pedersen and Hunt2009; Wolf et al. Reference Wolf, Lüthi, Holzner, Krug, Kinzelbach and Tsinober2012, Reference Wolf, Holzner, Lüthi, Krug, Kinzelbach and Tsinober2013a ; Khashehchi et al. Reference Khashehchi, Ooi, Soria and Marusic2013) are in accordance with Dimotakis’ discussion. Similar to incompressible jets (Yule Reference Yule1978; Liepmann & Gharib Reference Liepmann and Gharib1992; Babu & Mahesh Reference Babu and Mahesh2004; Westerweel et al. Reference Westerweel, Fukushima, Pedersen and Hunt2009; Philip & Marusic Reference Philip and Marusic2012) and spatially developing shear layers (Brown & Roshko Reference Brown and Roshko1974; Dimotakis & Brown Reference Dimotakis and Brown1976; Bernal & Roshko Reference Bernal and Roshko1986; Attili & Bisetti Reference Attili and Bisetti2012), table 5 shows that an induced velocity is generated in compressible shear layers of current work. The conditional profile of transverse velocity component in irrotational region far from the TNTI, not shown here, resembles the potential flow being induced by larger vortices in the turbulent region. Therefore, similar to the previous works (Dimotakis & Brown Reference Dimotakis and Brown1976; Yule Reference Yule1978; Liepmann & Gharib Reference Liepmann and Gharib1992; Philip & Marusic Reference Philip and Marusic2012), $E_{v}$ is induced primarily by large-scale structures in the flow and helps the nibbling mechanism to become more effective by increasing the local entrainment velocity. In turbulent wakes and boundary layers, it is reported that there is either very small or no induced inward motion of the irrotational flow towards the TNTI (Hunt Reference Hunt1994; Bisset et al. Reference Bisset, Hunt and Rogers2002; Philip & Marusic Reference Philip and Marusic2012; Chauhan et al. Reference Chauhan, Philip and Marusic2014a ,Reference Chauhan, Philip, de Silva, Hutchins and Marusic b ). Table 5 shows that there is no significant change in $\langle -v_{n}\rangle /u_{{\it\eta}}$ between cases $M08$ , $M12$ and $M18$ . This result is analogous to the observation made by Krug et al. (Reference Krug, Holzner, Lüthi, Wolf, Kinzelbach and Tsinober2015) on the effect of stratification on local entrainment velocity. In their work, two flows with different initial stratification levels are reported to have very similar $\langle -v_{n}\rangle /u_{{\it\eta}}$ .

At large enough Reynolds numbers for an incompressible flow, it is suggested by Corrsin & Kistler (Reference Corrsin and Kistler1955) that local propagation velocity of the turbulence boundary, $v_{n}$ , is determined by kinematic viscosity, ${\it\nu}$ , and rate of dissipation of kinetic energy, ${\it\epsilon}$ . Therefore, $v_{n}$ is proportional to the Kolmogorov velocity scale, $u_{{\it\eta}}$ . Also a common assumption in the entrainment process is that $E_{b}$ is proportional to the typical large-scale velocity in the flow (Morton et al. Reference Morton, Taylor and Turner1956; Turner Reference Turner1986; van Reeuwijk & Holzner Reference van Reeuwijk and Holzner2014; da Silva et al. Reference da Silva, Hunt, Eames and Westerweel2014a ). The largest eddies in the flow are characterized by the length scale ${\mathcal{L}}$ , which is of the order of the flow scale, and their characteristic velocity, $u_{{\mathcal{L}}}$ , is of the order of the r.m.s. turbulence intensity, $\sqrt{2k/3}$ , where $k$ is the turbulent kinetic energy (Pope Reference Pope2000). Consequently, the ratio of the local entrainment velocity to the boundary entrainment velocity is given by (Pope Reference Pope2000; van Reeuwijk & Holzner Reference van Reeuwijk and Holzner2014)

(4.13) $$\begin{eqnarray}\displaystyle \frac{v_{n}}{E_{b}}\propto \frac{u_{{\it\eta}}}{u_{{\mathcal{L}}}}\sim Re_{{\mathcal{L}}}^{-1/4}, & & \displaystyle\end{eqnarray}$$

where $Re_{{\mathcal{L}}}\equiv (u_{{\mathcal{L}}}{\mathcal{L}})/{\it\nu}$ is the Reynolds number of the largest eddies in the flow. Therefore, the process of entrainment of the irrotational ambient fluid into the turbulent region is affected by the interaction of the structures of all sizes, from viscous eddies to the energy-containing eddies, however the entrained mass flow rate is determined by the large-scale structures of the flow (Townsend Reference Townsend1976; Tsinober Reference Tsinober2009; van Reeuwijk & Holzner Reference van Reeuwijk and Holzner2014). The proportionality shown in (4.13) holds well for case $M02$ , however as $M_{c}$ increases the ratio of the local entrainment velocity to the boundary entrainment velocity is no longer proportional to $Re_{{\mathcal{L}}}^{-1/4}$ . The notion of the universality of the small scales in compressible turbulence has been questioned in a number of previous works (Livescu & Madnia Reference Livescu and Madnia2004; Kritsuk et al. Reference Kritsuk, Norman, Padoan and Wagner2007; Schmidt, Federrath & Klessen Reference Schmidt, Federrath and Klessen2008; Aluie Reference Aluie2011, Reference Aluie2013; Wang et al. Reference Wang, Yang, Shi, Xiao, He and Chen2013). To extend Kolmogorov’s phenomenology for compressible flows, it is suggested that a density-weighted velocity, ${\it\rho}^{{\it\alpha}}\boldsymbol{u}$ with $0\leqslant {\it\alpha}\leqslant 1$ , provides scaling laws that take into account the compressibility (Kida & Orszag Reference Kida and Orszag1990a ; Kritsuk et al. Reference Kritsuk, Norman, Padoan and Wagner2007; Schmidt et al. Reference Schmidt, Federrath and Klessen2008; Wang et al. Reference Wang, Yang, Shi, Xiao, He and Chen2013; Aluie Reference Aluie2013). The reported values for ${\it\alpha}$ in different works seems to be a function of the level of compressibility and Reynolds number (Kritsuk et al. Reference Kritsuk, Norman, Padoan and Wagner2007; Schmidt et al. Reference Schmidt, Federrath and Klessen2008; Wang et al. Reference Wang, Yang, Shi, Xiao, He and Chen2013). ${\it\alpha}\sim 1$ seems to yield the best scaling for the range of compressibility of the flow in the current work. Using this scaling, we obtain

(4.14) $$\begin{eqnarray}\displaystyle \frac{v_{n}}{E_{b}}\propto \frac{\bar{{\it\rho}}}{{\it\rho}_{0}}Re_{{\mathcal{L}}}^{-1/4}, & & \displaystyle\end{eqnarray}$$

where $\bar{{\it\rho}}$ is the average density of the flow on the midplane and ${\it\rho}_{0}$ is the density of the outside irrotational flow. The proportionality in (4.14) holds for all the cases in the current work. Table 5 shows the values of the terms in (4.13) and (4.14). In this table, ${\mathcal{L}}$ is considered to be equal to $(l_{x}+l_{z}+{\it\delta}_{{\it\theta}})/3$ , where $l_{x}$ and $l_{z}$ are the integral length scales in $x$ and $z$ directions and ${\it\delta}_{{\it\theta}}$ is the momentum thickness. For this choice of ${\mathcal{L}}$ , $v_{n}/E_{b}$ and $(\bar{{\it\rho}}/{\it\rho}_{0})Re_{{\mathcal{L}}}^{-1/4}$ are approximately equal, however, they may not become equal for other choices of ${\mathcal{L}}$ . Thus, the proportionality symbol is used in (4.13) and (4.14).

The global entrained volume flow rate, $Q_{g}=E_{b}S$ , where $S$ is the surface area based on the average distance of the TNTI from midplane, is equal to local entrained volume flow rate, $Q_{l}=\langle -v_{n}\rangle A_{TNTI}$ , where $A_{TNTI}$ is the total surface area of the TNTI, for an incompressible turbulence (van Reeuwijk & Holzner Reference van Reeuwijk and Holzner2014). In the present study, the ratio of $Q_{l}/Q_{g}$ for cases $M02$ , $M08$ , $M12$ and $M18$ is equal to 0.98, 0.91, 0.84 and 0.72 respectively. The change of $Q_{l}/Q_{g}$ with $M_{c}$ indicates that for a compressible flow, volume is not conserved. In order to examine the deviation of this ratio from unity, the continuity equation, the Reynolds transport theorem, and the divergence theorem are used to find the change of mass inside the turbulent region, $m_{t}$ , as

(4.15) $$\begin{eqnarray}\displaystyle \frac{\text{d}m_{t}}{\text{d}t}=-\iiint _{V_{t}}\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }({\it\rho}\boldsymbol{U})\,\text{d}V-\iint _{TNTI}{\it\rho}(\boldsymbol{U}^{s}\boldsymbol{\cdot }\hat{\boldsymbol{n}})\,\text{d}A=-\iint _{TNTI}{\it\rho}v_{n}\,\text{d}A, & & \displaystyle\end{eqnarray}$$

where $V_{t}$ is the volume of the turbulent region, $\hat{\boldsymbol{n}}$ is the inward-pointing unit vector normal to the TNTI, shown in figure 4, and $\text{d}A$ is the surface area element on the TNTI. In (4.15), it is assumed that the contribution of engulfment to entrainment is small, discussed in § 4.4. In a temporal shear layer, on average the TNTI moves in the transverse direction, and assuming a constant density on the TNTI, (4.15) can be rewritten as

(4.16) $$\begin{eqnarray}\displaystyle \frac{\text{d}m_{t}}{\text{d}t}\approx -\iiint _{V_{t}}\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }({\it\rho}\boldsymbol{U})\,\text{d}V+{\it\rho}_{0}E_{b}S\approx {\it\rho}_{0}\langle -v_{n}\rangle A_{TNTI}. & & \displaystyle\end{eqnarray}$$

By replacing $E_{b}S$ and $\langle -v_{n}\rangle A_{TNTI}$ in (4.16) with $Q_{g}$ and $Q_{l}$ respectively, and dividing both sides of the equation by ${\it\rho}_{0}Q_{g}$ , we can obtain

(4.17) $$\begin{eqnarray}\displaystyle \frac{Q_{l}}{Q_{g}}\approx 1-\frac{1}{{\it\rho}_{0}Q_{g}}\iiint _{V_{t}}\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }({\it\rho}\boldsymbol{U})\,\text{d}V. & & \displaystyle\end{eqnarray}$$

As (4.17) shows, the ratio of local entrained volume flow rate to the global entrained volume flow rate is a function of temporal variation of density, $\partial {\it\rho}/\partial t=-\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }({\it\rho}\boldsymbol{U})$ , in the turbulent region. Using (4.17), $Q_{l}/Q_{g}$ is equal to 0.99, 0.94, 0.89 and 0.78 for cases $M02$ , $M08$ , $M12$ and $M18$ . These values are close to the values calculated by computing $Q_{l}=\langle -v_{n}\rangle A_{TNTI}$ and $Q_{g}=E_{b}S$ . The discrepancy between these two sets of values for $Q_{l}/Q_{g}$ is due to the assumptions made in deriving (4.17), and also the inaccuracies in finding the exact values of $A_{TNTI}$ .

Table 6 shows the values of the average entrained mass flux and contributions of different terms in (4.11) and (4.12) to this flux. The values reported by Wolf et al. (Reference Wolf, Holzner, Lüthi, Krug, Kinzelbach and Tsinober2013a ) for inviscid and viscous contributions to the entrained mass flux are 35 % and 65 % respectively, for an incompressible jet with $Re_{{\it\lambda}}\approx 100$ . This is in agreement with our results for case $M02$ . Chauhan et al. (Reference Chauhan, Philip, de Silva, Hutchins and Marusic2014b ) measured the entrained mass flux experimentally for a zero pressure gradient turbulent boundary layer with $Re_{{\it\lambda}}\approx 300$ and reported a value of 0.009 for the normalized entrained mass flux, $\langle {\dot{m}}^{\prime \prime }\rangle /({\it\rho}_{0}U)$ , which is comparable with the normalized entrained mass flux for case $M02$ in table 6. It can be seen in table 6 that the entrained mass flux decreases with increasing convective Mach number. The contribution of the inviscid terms to the entrainment increases for compressible cases as the viscous contribution decreases. Table 6 shows that the contribution of vortex stretching term, ${\mathcal{E}}_{{\mathcal{I}}{\mathcal{I}}}$ , increases from case $M02$ to case $M08$ . However, increasing the level of compressibility beyond $M_{c}=0.8$ , does not have a significant effect on the contribution of this term. As is discussed in § 4.3.3, this behaviour can be related to the effect of compressibility on the local curvatures of the TNTI, shown in figure 8. The main difference between the contribution of the inviscid terms to the entrainment for compressible cases is due to the change of the baroclinic, ${\dot{m}}_{{\mathcal{I}}{\mathcal{V}}}^{\prime \prime }$ , and the dilatation, ${\dot{m}}_{{\mathcal{I}}{\mathcal{I}}{\mathcal{I}}}^{\prime \prime }$ , terms in (4.11). These two terms are almost equal with opposite signs for cases $M08$ and $M12$ , however for case $M18$ , the baroclinic term becomes larger than the dilatation term. Table 6 also shows contribution of term ${\dot{m}}_{{\mathcal{V}}{\mathcal{I}}{\mathcal{I}}}^{\prime \prime }$ to the entrained mass flux increases with increasing $M_{c}$ .

The entrained mass flow rate due to nibbling can be calculated using the entrained mass flux and surface area of the TNTI, ${\dot{m}}={\dot{m}}^{\prime \prime }A_{TNTI}$ . $A_{TNTI}$ for each case is reported in § 4.1. In order to make a meaningful comparison between the cases, the total shear layer mass flow rate, $\dot{{\mathcal{Q}}}$ , is used to normalize the entrained mass flow rate. $\dot{{\mathcal{Q}}}$ , which is an estimate of the mass flow rate inside the turbulent region, is calculated for a compressible shear layer as

(4.18) $$\begin{eqnarray}\displaystyle \dot{{\mathcal{Q}}}=L_{z}\int _{-\infty }^{+\infty }{\mathcal{P}}(t,y)\bar{{\it\rho}}|\tilde{u} |\,\text{d}y, & & \displaystyle\end{eqnarray}$$

where ${\mathcal{P}}(t,y)$ is the probability of a given fluid element in location $y$ and time $t$ to be between the upper and lower interface envelopes, $L_{z}$ is the domain size in spanwise direction, $\bar{{\it\rho}}$ is the density average and $\tilde{u}$ is the density-weighted (Favre) average of the streamwise velocity. The entrained mass flow rate due to nibbling normalized by the total shear layer mass flow rate, ${\dot{m}}/\dot{{\mathcal{Q}}}$ , for cases $M02$ , $M08$ , $M12$ and $M18$ are 1.07, 0.65, 0.59 and 0.48 respectively.

The interface between turbulent and non-turbulent regions can also be detected with a conserved scalar convected by the turbulent flow, as usually done in experimental studies (Westerweel et al. Reference Westerweel, Fukushima, Pedersen and Hunt2009). Similar analysis is conducted using mixture fraction isosurface as the TNTI. The mixture fraction threshold is chosen using the method described by Prasad & Sreenivasan (Reference Prasad and Sreenivasan1989) and Westerweel et al. (Reference Westerweel, Hofmann, Fukushima and Hunt2002). Similar values of entrainment velocities and entrained mass flow rate are obtained using this analysis.

Table 7.

The correlation coefficient, $R$ , of the entrained mass flux, ${\dot{m}}^{\prime \prime }$ , with terms in (4.11) on the TNTI.

The parameter ${\mathcal{E}}_{{\mathcal{N}}}$ , shown in table 6, can only reveal the average contribution of different terms in (4.11) to the entrained mass flux across the TNTI. For more detailed study of the effect of each term in (4.11) on the local entrained mass flux, it is useful to examine the joint PDFs of these terms with ${\dot{m}}^{\prime \prime }$ (not shown here), and compute the correlation coefficient between them. Table 7 summarizes these correlation coefficients for all cases. This table reveals that for case $M02$ the entrained mass flux on the TNTI is highly correlated with ${\dot{m}}_{{\mathcal{V}}}^{\prime \prime }+{\dot{m}}_{{\mathcal{V}}{\mathcal{I}}}^{\prime \prime }$ . However, as the convective Mach number increases, ${\dot{m}}^{\prime \prime }$ becomes less correlated with the summation of viscous diffusion and viscous dissipation, and become more correlated with the mass fluxes due to the baroclinic, ${\dot{m}}_{{\mathcal{I}}{\mathcal{V}}}^{\prime \prime }$ , and the dilatation, ${\dot{m}}_{{\mathcal{I}}{\mathcal{I}}{\mathcal{I}}}^{\prime \prime }$ , terms. These results indicate that small values of ${\mathcal{E}}$ for the contribution of baroclinic and dilatation terms to the entrained mass flux reported in table 6, may be related to the cancellation of large positive and negative values. In order to further investigate the effect of compressibility on the entrainment, the entrained mass fluxes in equation (4.11) are conditioned on the dilatation in the next section.

4.3.2 Effect of local dilatation on the entrained mass flux

It is shown in previous works on compressible turbulent flows (Suman & Girimaji Reference Suman and Girimaji2010; Vaghefi & Madnia Reference Vaghefi and Madnia2015) that the dilatation, $\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\boldsymbol{U}$ , plays an important role in characterization of small-scale effects in turbulence. Therefore, in this section the contribution of the terms in (4.11) to the entrained mass flux is conditioned on the level of the dilatation. Figure 13 depicts the PDF of the divergence of the velocity on the TNTI for case $M18$ . As can be seen in this figure, the PDF is skewed towards the negative values, however the peak of the PDF is in the locally expanded region. As a result the percentage of the locally expanded regions, $\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\boldsymbol{U}>0$ , on the TNTI is more than locally compressed regions, $\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\boldsymbol{U}<0$ , for the compressible cases of this study, i.e. the expanded regions cover 51 %, 56 %, 57 % and 61 % of the surface area on the TNTI for cases $M02$ , $M08$ , $M12$ and $M18$ respectively.

Figure 13.

PDF of the dilatation, $\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\boldsymbol{U}$ , on the TNTI for case $M18$ .

Figure 14 shows the average of the entrained mass flux and the terms contributing to this flux on the TNTI conditioned on the divergence of the velocity for case $M18$ . As it can be seen in this figure, moving from the compressed regions towards expanded regions, the average entrained mass flux decreases on the TNTI. Also $\langle {\dot{m}}^{\prime \prime }\rangle$ becomes negative for $\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\boldsymbol{U}>0.012$ , which indicates detrainment on these regions of the TNTI. Thus, for case $M18$ most of the positive entrained mass flux are associated with the compressed regions. As the convective Mach number increases, the percentage of the area of the expanded regions on the TNTI increases, which results in a decrease of the average entrained mass across the TNTI. Figure 14 shows that the dilatation and the baroclinic terms are important in contributing to entrainment on compressed regions and to detrainment on expanded regions. Thus, the small values of ${\mathcal{E}}_{{\mathcal{I}}{\mathcal{I}}{\mathcal{I}}}$ and ${\mathcal{E}}_{{\mathcal{I}}{\mathcal{V}}}$ reported in table 6, are due to the cancellation of positive and negative values. Figure 14 also reveals that vortex stretching causes entrainment for $\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\boldsymbol{U}>-0.015$ . The average mass flux due to viscous diffusion plus viscous dissipation causes detrainment on highly compressed and expanded regions of the TNTI. For $-0.06<\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\boldsymbol{U}<0.06$ , $\langle {\dot{m}}_{{\mathcal{V}}}^{\prime \prime }+{\dot{m}}_{{\mathcal{V}}{\mathcal{I}}}^{\prime \prime }\rangle$ contributes to entrainment and is slightly higher on the expanded regions. The trends shown in figure 14 are also observed for other compressible cases in this study.

Figure 14.

Average entrained mass flux and different terms in (4.11) conditioned on the dilatation on the TNTI for case $M18$ .

4.3.3 Effect of local curvatures of the TNTI on the entrained mass flux

It is reported by Wolf et al. (Reference Wolf, Lüthi, Holzner, Krug, Kinzelbach and Tsinober2012, Reference Wolf, Holzner, Lüthi, Krug, Kinzelbach and Tsinober2013a ,Reference Wolf, Lüthi, Holzner, Krug, Kinzelbach and Tsinober b ) that in incompressible jets depending on the geometrical shape of the TNTI, looking from the turbulent region, different mechanisms are responsible for the local entrainment of the irrotational flow into the turbulent region, i.e. viscous effects for convex shaped surfaces and vortex stretching for concave shaped surfaces. Philip et al. (Reference Philip, Bermejo-Moreno, Chung and Marusic2015) also studied the dependence of local entrainment on the local geometry through the geometric parameters of curvedness and shape index on the TNTI of an incompressible turbulent wake. They found that the entrainment happens mostly through the concave shaped surfaces with lower curvedness (flatter surfaces). Figure 6 shows the different geometrical shapes of the TNTI, looking from the turbulent region, in mean, $K_{m}$ , and Gaussian, $K_{g}$ , curvatures plane. The effect of the local curvatures of the TNTI on the entrained mass flux is studied in this section.

Figure 15.

The average values of (a,b) entrained mass flux, (c,d) mass flux due to inviscid terms and (e,f) mass flux due to viscous terms, in (4.11) and (4.12) conditioned on mean curvature, $K_{m}$ , and Gaussian curvature, $K_{g}$ . (a,c,e) Correspond to case $M02$ , and (b,d,f) correspond to points on the TNTI with $|\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\boldsymbol{U}|<0.001$ for case $M18$ . ${\it\lambda}$ is the reference Taylor length scale, ${\it\rho}_{0}$ is the reference value of density, and ${\rm\Delta}U$ is the velocity difference of upper and lower streams. The contour levels are the same for all of the figures. (a) $M02,\langle {\dot{m}}^{\prime \prime }\rangle /({\it\rho}_{0}{\rm\Delta}U)$ , (b) $M18,\langle {\dot{m}}^{\prime \prime }\rangle /({\it\rho}_{0}{\rm\Delta}U)$ , (c) $M02,\langle {\dot{m}}_{inv}^{\prime \prime }\rangle /({\it\rho}_{0}{\rm\Delta}U)$ ,(d) $M18,\langle {\dot{m}}_{inv}^{\prime \prime }\rangle /({\it\rho}_{0}{\rm\Delta}U)$ , (e) $M02,\langle {\dot{m}}_{vis}^{\prime \prime }\rangle /({\it\rho}_{0}{\rm\Delta}U)$ , (f) $M18,\langle {\dot{m}}_{vis}^{\prime \prime }\rangle /({\it\rho}_{0}{\rm\Delta}U)$ .

In the present work, it is observed that the effect of the local curvatures on the general behaviour of mass fluxes contributing to entrainment are similar for case $M02$ and points on the TNTI of other cases with $\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\boldsymbol{U}\approx 0$ . Figure 15 shows the average entrained mass flux, and the mass fluxes due to inviscid and viscous terms in (4.11) and (4.12) conditioned on the mean and Gaussian curvatures for case $M02$ , and for points on the TNTI with $|\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\boldsymbol{U}|<0.001$ corresponding to case $M18$ . For $\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\boldsymbol{U}\approx 0$ , the contribution of the dilatation, the baroclinic and the shear stress/density gradient terms to the entrained mass flux are very small. Thus, ${\dot{m}}_{inv}^{\prime \prime }$ in figure 15 is approximately equal to the mass flux due to vortex stretching, and ${\dot{m}}_{vis}^{\prime \prime }$ is approximately equal to the mass flux due to viscous diffusion plus viscous dissipation.

The average values of the viscous mass flux on incompressible regions of the TNTI for case $M18$ , shown in figure 15(f), are smaller compared to $\langle {\dot{m}}_{vis}^{\prime \prime }\rangle$ for case $M02$ , shown in figure 15(e). However, as figure 15(c,d) reveals the mass flux due to inviscid terms is approximately the same for both cases. The decrease of the mass fluxes due to viscous effects as the inviscid mass flux remains approximately constant, results in a decrease in the entrained mass flux for case $M18$ compared to case $M02$ . This confirms the results of table 6. Figure 15(a,e) for case $M02$ , and figure 15(b,f) for case $M18$ show that on the average the entrained mass flux and the mass flux due to viscous terms on the concave shaped surfaces are larger than these fluxes on convex shaped surfaces. These results are not similar to the conditional averages of ${\dot{m}}^{\prime \prime }$ and ${\dot{m}}_{vis}^{\prime \prime }$ in incompressible jets (Wolf et al. Reference Wolf, Lüthi, Holzner, Krug, Kinzelbach and Tsinober2012, Reference Wolf, Holzner, Lüthi, Krug, Kinzelbach and Tsinober2013a ,Reference Wolf, Lüthi, Holzner, Krug, Kinzelbach and Tsinober b ), in which the reported entrained mass flux and mass flux due to viscous terms are larger on highly curved convex shaped surfaces. Similar to the present study, Philip et al. (Reference Philip, Bermejo-Moreno, Chung and Marusic2015) observed that on the TNTI of an incompressible wake, the entrained mass flux happens mostly on concave shaped surfaces. However, they found that the flatter surfaces (small $K_{m}$ and $K_{g}$ ) correspond to the larger ${\dot{m}}^{\prime \prime }$ for incompressible wakes. This is not similar to the results of this work, in which ${\dot{m}}^{\prime \prime }$ is larger on highly curved concave shaped surfaces. Figure 15(c,d) reveals that the mass flux due to inviscid terms, ${\dot{m}}_{inv}^{\prime \prime }$ , seems to have a weaker dependency on the local curvatures of the surfaces for both cases. As this figure shows, the mass flux due to inviscid terms is slightly larger on concave shaped surfaces. This behaviour is similar to the observations for the incompressible jets (Wolf et al. Reference Wolf, Lüthi, Holzner, Krug, Kinzelbach and Tsinober2012, Reference Wolf, Holzner, Lüthi, Krug, Kinzelbach and Tsinober2013a ,Reference Wolf, Lüthi, Holzner, Krug, Kinzelbach and Tsinober b ).

Figures 16 and 17 depict the effect of the local curvatures on the entrained mass fluxes on expanded and compressed regions of the TNTI. These figures show the average mass fluxes in (4.11) conditioned on mean, $K_{m}$ , and Gaussian, $K_{g}$ , curvatures for the points on the TNTI of case $M18$ with $\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\boldsymbol{U}>0.02$ and $\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\boldsymbol{U}<-0.02$ , respectively. These range of dilatation is chosen to exclude the effect of incompressible regions of the TNTI from the statistics. In accordance with figure 14, figures 16 and 17 reveal that the sign of the entrained mass flux is generally determined by the level of dilatation, i.e. $\langle {\dot{m}}^{\prime \prime }\rangle$ is predominately positive on compressed regions of the TNTI and mostly negative on the expanded regions.

Figure 16.

The average values of (a) entrained mass flux, along with the mass fluxes due to (b) vortex stretching, (c) dilatation, (d) baroclinic, (e) viscous diffusion plus viscous dissipation and (f) shear stress/density gradient, terms in (4.11) conditioned on mean, $K_{m}$ , and Gaussian curvatures, $K_{g}$ . These figures correspond to points on the TNTI with $\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\boldsymbol{U}>0.02$ for case $M18$ . ${\it\lambda}$ is the reference Taylor length scale, ${\it\rho}_{0}$ is the reference value of density, and ${\rm\Delta}U$ is the velocity difference of upper and lower streams. The contour levels are the same for all of the figures. (a) $\langle {\dot{m}}^{\prime \prime }\rangle /({\it\rho}_{0}{\rm\Delta}U)$ , (b) $\langle {\dot{m}}_{{\mathcal{I}}{\mathcal{I}}}^{\prime \prime }\rangle /({\it\rho}_{0}{\rm\Delta}U)$ , (c) $\langle {\dot{m}}_{{\mathcal{I}}{\mathcal{I}}{\mathcal{I}}}^{\prime \prime }\rangle /({\it\rho}_{0}{\rm\Delta}U)$ , (d) $\langle {\dot{m}}_{{\mathcal{I}}{\mathcal{V}}}^{\prime \prime }\rangle /({\it\rho}_{0}{\rm\Delta}U)$ , (e) $\langle {\dot{m}}_{{\mathcal{V}}}^{\prime \prime }+{\dot{m}}_{{\mathcal{V}}{\mathcal{I}}}^{\prime \prime }\rangle /({\it\rho}_{0}{\rm\Delta}U)$ ,(f) $\langle {\dot{m}}_{{\mathcal{V}}{\mathcal{I}}{\mathcal{I}}}^{\prime \prime }\rangle /({\it\rho}_{0}{\rm\Delta}U)$ .

Figure 16(a) depicts the effect of local curvatures on the entrained mass flux on expanded regions of the TNTI. The regions with positive average entrained mass flux in this figure is caused by mass fluxes due to vortex stretching and viscous terms. Figure 16(a) also reveals that large negative entrained mass flux on the TNTI occurs mostly on saddle shaped surfaces. Figure 16(b,e) shows that the effect of local curvatures on ${\dot{m}}_{{\mathcal{I}}{\mathcal{I}}}^{\prime \prime }$ and ${\dot{m}}_{{\mathcal{V}}}^{\prime \prime }+{\dot{m}}_{{\mathcal{V}}{\mathcal{I}}}^{\prime \prime }$ is similar to the incompressible regions of the TNTI. In these figures, both fluxes are on the average larger for concave shaped surfaces, and dependency of $\langle {\dot{m}}_{{\mathcal{I}}{\mathcal{I}}}^{\prime \prime }\rangle$ on the local curvatures is less than $\langle {\dot{m}}_{{\mathcal{V}}}^{\prime \prime }+{\dot{m}}_{{\mathcal{V}}{\mathcal{I}}}^{\prime \prime }\rangle$ . Figure 16(c) depicts that the largest negative $\langle {\dot{m}}_{{\mathcal{I}}{\mathcal{I}}{\mathcal{I}}}^{\prime \prime }\rangle$ occurs on saddle shaped surfaces and elliptic concave shaped surfaces. Figure 16(d) shows that the baroclinic torque predominantly causes detrainment on the expanded regions of the TNTI, and $\langle {\dot{m}}_{{\mathcal{I}}{\mathcal{V}}}^{\prime \prime }\rangle$ is a weak function of the local curvatures. Figure 16(f) reveals that the average contribution of the shear stress/density gradient torque to the entrainment is small everywhere on expanded regions of the TNTI.

Figure 17.

The average values of (a) entrained mass flux, along with the mass fluxes due to (b) vortex stretching, (c) dilatation, (d) baroclinic, (e) viscous diffusion plus viscous dissipation and (f) shear stress/density gradient, terms in (4.11) conditioned on mean, $K_{m}$ , and Gaussian curvatures, $K_{g}$ . These figures correspond to points on the TNTI with $\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\boldsymbol{U}<-0.02$ for case $M18$ . ${\it\lambda}$ is the reference Taylor length scale, ${\it\rho}_{0}$ is the reference value of density and ${\rm\Delta}U$ is the velocity difference of upper and lower streams. The contour levels are the same for all of the figures. (a) $\langle {\dot{m}}^{\prime \prime }\rangle /({\it\rho}_{0}{\rm\Delta}U)$ ,(b) $\langle {\dot{m}}_{{\mathcal{I}}{\mathcal{I}}}^{\prime \prime }\rangle /({\it\rho}_{0}{\rm\Delta}U)$ , (c) $\langle {\dot{m}}_{{\mathcal{I}}{\mathcal{I}}{\mathcal{I}}}^{\prime \prime }\rangle /({\it\rho}_{0}{\rm\Delta}U)$ , (d) $\langle {\dot{m}}_{{\mathcal{I}}{\mathcal{V}}}^{\prime \prime }\rangle /({\it\rho}_{0}{\rm\Delta}U)$ , (e) $\langle {\dot{m}}_{{\mathcal{V}}}^{\prime \prime }+{\dot{m}}_{{\mathcal{V}}{\mathcal{I}}}^{\prime \prime }\rangle /({\it\rho}_{0}{\rm\Delta}U)$ ,(f) $\langle {\dot{m}}_{{\mathcal{V}}{\mathcal{I}}{\mathcal{I}}}^{\prime \prime }\rangle /({\it\rho}_{0}{\rm\Delta}U)$ .

Figure 17(a) depicts the effect of the local curvatures on the average entrained mass flux on the compressed regions of the TNTI. This figure shows that elliptic concave shaped surfaces correspond to maximum entrainment. Figure 17(b) reveals that $\langle {\dot{m}}_{{\mathcal{I}}{\mathcal{I}}}^{\prime \prime }\rangle$ is mainly negative, and similar to expanded and incompressible regions of the TNTI, the dependency of this flux on the local curvatures is weak. Figure 17(c) shows that $\langle {\dot{m}}_{{\mathcal{I}}{\mathcal{I}}{\mathcal{I}}}^{\prime \prime }\rangle$ is the largest on elliptic concave shaped surfaces. As it can be seen in figure 17(d), the average entrained mass flux due to baroclinic torque is mostly positive, and is slightly larger on elliptic concave shaped surfaces. Figure 17(e) depicts that $\langle {\dot{m}}_{{\mathcal{V}}}^{\prime \prime }+{\dot{m}}_{{\mathcal{V}}{\mathcal{I}}}^{\prime \prime }\rangle$ is mostly negative on concave shaped surfaces and is positive on convex ones. Figure 17(f) reveals that the average contribution of the shear stress/density gradient torque to the entrainment is small everywhere.

The averages of the terms contributing to entrainment conditioned on local curvatures on the incompressible, figure 15, on the expanded, figure 16, and on the compressed, figure 17, regions of the TNTI, shows that in general mass fluxes due to vortex stretching and baroclinic terms are weak functions of the local curvatures. Thus, in order to determine the local contribution of these terms to the entrained mass flux, $\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\boldsymbol{U}$ is the main parameter which needs to be considered. However, mass fluxes due to dilatation and viscous terms have a stronger dependency on the local curvatures. Therefore, for finding the local contribution of these terms to the entrainment, $\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\boldsymbol{U}$ , $K_{m}$ and $K_{g}$ should be considered together. Figures 15–17 also reveal that regardless of the level of dilatation, the average entrained mass flux is larger on highly curved concave shaped surfaces. It is observed in figure 8(a) that as the convective Mach number increases, the probability of finding highly curved concave shaped surfaces decreases. Therefore, one of the reasons for the decrease of the average entrained mass flux in table 6 can be associated with the effect of the compressibility on the geometrical curving of the TNTI isosurface. The magnitudes of $\langle {\dot{m}}_{{\mathcal{I}}{\mathcal{I}}{\mathcal{I}}}^{\prime \prime }\rangle$ and $\langle {\dot{m}}_{{\mathcal{V}}}^{\prime \prime }+{\dot{m}}_{{\mathcal{V}}{\mathcal{I}}}^{\prime \prime }\rangle$ are also larger on concave shaped surfaces. As a result, the value of ${\mathcal{E}}_{vis}=\langle {\dot{m}}_{vis}^{\prime \prime }\rangle /\langle {\dot{m}}^{\prime \prime }\rangle$ is larger on the concave shaped surfaces. Thus, the decrease in the value of ${\mathcal{E}}_{vis}$ in table 6 can be attributed to the effect of compressibility on the geometrical shape of the TNTI.