Evolution of scalar and velocity dynamics in planar shock-turbulence interaction

Abstract

Due to the short residence time of air in supersonic combustors, achieving efficient mixing in compressible turbulent reactive flows is crucial for the design of supersonic ramjet (Scramjet) engines. In this respect, improving the understanding of shock-scalar mixing interactions is of fundamental importance for such supersonic combustion applications. In these compressible flows, the interaction between the turbulence and the shock wave is reciprocal, and the coupling between them is very strong. A basic understanding of the physics of such complex interactions has already been obtained through the analysis of relevant simplified flow configurations, including propagation of the shock wave in density-stratified media, shock-wave–mixing-layer interaction, and shock-wave–vortex interaction. Amplification of velocity fluctuations and substantial changes in turbulence characteristic length scales are the most well-known outcomes of shock–turbulence interaction, which may also deeply influence scalar mixing between fuel and oxidizer. The effects of the shock wave on the turbulence have been widely characterized through the use of so-called amplification factors, and similar quantities are introduced herein to characterize the influence of the shock wave on scalar mixing. One of the primary goals of the present study is indeed to extend previous analyses to the case of shock-scalar mixing interaction, which is directly relevant to supersonic combustion applications. It is expected that the shock wave will affect the scalar dissipation rate (SDR) dynamics. Special emphasis is placed on the modification of the so-called turbulence–scalar interaction as a leading-order contribution to the production of mean SDR, i.e., a quantity that defines the mixing rate and efficiency. To the best of the authors’ knowledge, this issue has never been addressed in detail in the literature, and the objective of the present study is to scrutinize this influence. The turbulent mixing of a passive (i.e., chemically inert) scalar in the presence of a shock wave is thus investigated using high-resolution numerical simulations. The starting point of the analysis relies on the transport equations of the variance of the mixture fraction, i.e., a fuel inlet tracer that quantifies the mixing between fuel and oxidizer. The influence of the shock wave is investigated for three distinct values of the shock Mach number M, and the obtained results are compared to reference solutions featuring no shock wave. The computed solutions show that the shock wave significantly modifies the scalar field topology. The larger the value of M, the stronger is the amplification of the alignment of the scalar gradient with the most compressive principal direction of the strain-rate tensor, which signifies the enhancement of scalar mixing with the shock Mach number.

Appendices

Appendix 1: Some results of order of magnitude analyses

In a first step of the order of magnitude analysis (OMA), the ratio $l_\mathrm {t} / \delta _\mathrm {S}$ can be recast as follows

$$\begin{aligned} \frac{l_\mathrm {t}}{\delta _\mathrm {S}}=\frac{l_\mathrm {t}}{\eta _\mathrm {K}} \cdot \frac{\eta _\mathrm {K}}{\delta _\mathrm {S}} \end{aligned}$$

(18)

where the order of magnitude (OM) of the first non-dimensional ratio present in the right-hand side (RHS), i.e., $l_\mathrm {t}/\eta _\mathrm {K}$, is known to be ${\mathcal {O}}\left( \text {Re}_\mathrm {t}^{3/4} \right) ={\mathcal {O}}\left( \text {Re}_{\uplambda }^{3/2} \right)$.

For moderate Mach number values, the shock wave thickness $\delta _\mathrm {S}$ scales as $\nu / (c \cdot \Delta {M})$, while the Kolmogorov length scale is given by ${\mathcal {O}}\left( \nu ^{3/4} / \varepsilon ^{1/4} \right)$, in such a manner that the ratio of the two length scales is

$$\begin{aligned} \frac{\eta _\mathrm {K}}{\delta _\mathrm {S}}= & {} {\mathcal {O}}\left( \frac{\nu ^{3/4}}{\varepsilon ^{1/4}} \cdot \frac{c}{\nu } \cdot \Delta {M} \right) \nonumber \\= & {} {\mathcal {O}}\left( \frac{u_\mathrm {rms}}{\nu ^{1/4} \cdot \varepsilon ^{1/4}} \cdot \frac{c}{u_\mathrm {rms}} \cdot \Delta {M} \right) \nonumber \\= & {} {\mathcal {O}}\left( \frac{u_\mathrm {rms}}{u_\mathrm {K}} \cdot \frac{c}{u_\mathrm {rms}} \cdot \Delta {M} \right) \end{aligned}$$

(19)

with $u_\mathrm {K}={\mathcal {O}}\left( \nu ^{1/4} \cdot \varepsilon ^{1/4} \right)$ the OM of the Kolmogorov velocity fluctuation, one may thus obtain

$$\begin{aligned} \frac{\eta _\mathrm {K}}{\delta _\mathrm {S}} = {\mathcal {O}}\left( \text {Re}_{\uplambda }^{1/2} \cdot \frac{\Delta {M}}{{M}_\mathrm {t}} \right) \end{aligned}$$

(20)

where the scaling of $u_\mathrm {rms}/u_\mathrm {K}={\mathcal {O}}\left( \text {Re}_\mathrm {t}^{1/4} \right) ={\mathcal {O}}\left( \text {Re}_{\uplambda }^{1/2} \right)$ has been used.

Finally, the above analysis leads to

$$\begin{aligned} \frac{l_\mathrm {t}}{\delta _\mathrm {S}} ={\mathcal {O}}\left( \frac{\Delta {M}}{{M}_\mathrm {t}} \cdot \text {Re}_{\uplambda }^2 \right) ={\mathcal {O}}\left( \frac{\Delta {M}}{{M}_\mathrm {t}} \cdot \text {Re}_\mathrm {t} \right) \end{aligned}$$

(21)

Appendix 2: Scalar variance and SDR transport

The transport equation for the scalar variance writes:

$$\begin{aligned} \frac{\partial }{\partial t} ( \overline{\rho \xi ^{\prime \prime 2}}) + \frac{\partial \overline{{F}_k^{\xi ^{\prime \prime 2}}}}{\partial x_k}= & {} -\, 2\overline{\rho D \frac{\partial \xi ^{\prime \prime }}{\partial x_k}\frac{\partial \xi ^{\prime \prime }}{\partial x_k}} \nonumber \\&-\, 2\overline{\rho u^{\prime \prime }_k \xi ^{\prime \prime }}\frac{\partial {\widetilde{\xi }}}{\partial x_k} \nonumber \\&+\, 2 \overline{\xi ^{\prime \prime } \frac{\partial }{\partial x_k}\left( \rho D \frac{\partial {\widetilde{\xi }}}{\partial x_k}\right) } \end{aligned}$$

(22)

where the second term in the left-hand side (LHS) does involve the quantity $\overline{{F}_k^{\xi ^{\prime \prime 2}}}$ which denotes the total variance flux $\overline{{F}_k^{\xi ^{\prime \prime 2}}}=(\overline{\rho u_k \xi ^{\prime \prime 2}} - \overline{\rho D \cdot {\partial \xi ^{\prime \prime 2}}/{\partial x_k}} )$. In this transport equation, the first term on the left-hand side is the accumulation term and the second is the (conservative) flux term (convection and diffusion). On the right-hand side of (22), the first term corresponds to mean SDR, while the second is the production associated with mean concentration gradients. The last term is a molecular contribution that is often neglected just for the sake of simplicity.

A standard (i.e., simplified) form of the transport equation of the scalar dissipation rate (SDR) may be written as follows:

$$\begin{aligned} \frac{\partial \overline{\rho \varepsilon _{\xi }}}{\partial t}+ {\frac{\partial \overline{{F}_k^{\varepsilon _{\xi }}}}{\partial x_{k}}}= & {} -\underset{\mathrm {(V)}}{\underbrace{2\overline{\rho D\frac{\partial \xi ^{\prime \prime }}{\partial x_{i}}\frac{\partial u_{k}^{\prime \prime }}{\partial x_{i}}}\frac{\partial {\widetilde{\xi }}}{\partial x_{k}}}}\nonumber \\&-\,\underset{\mathrm {(VI)}}{\underbrace{2\overline{\rho D\frac{\partial \xi ^{\prime \prime }}{\partial x_{i}}\frac{\partial \xi ^{\prime \prime }}{\partial x_{k}}}\frac{\partial {\widetilde{u}}_{k}}{\partial x_{i}}}} \nonumber \\&-\,\underset{\mathrm {(VII)}}{\underbrace{2\overline{\rho D\frac{\partial \xi ^{\prime \prime }}{\partial x_{i}}\frac{\partial \xi ^{\prime \prime }}{\partial x_{k}}\frac{\partial u_{k}^{\prime \prime }}{\partial x_{i}}}}} \nonumber \\&-\,\underset{\mathrm {(VIII)}}{\underbrace{2\overline{\rho D^{2}\frac{\partial ^{2}\xi ^{\prime \prime }}{\partial x_{i}\partial x_{k}}\frac{\partial ^{2}\xi ^{\prime \prime }}{\partial x_{i}\partial x_{k}}}}} \end{aligned}$$

(23)

where $\overline{{F}_k^{\varepsilon _{\xi }}}$ denotes the contribution of scalar dissipation fluxes. Term (V) represents the production due to the mean scalar gradient, term (VI) stands for the production due to the mean velocity gradient, term (VII) is the effect of stretching by turbulence, and term (VIII) is associated with the local curvature of the scalar field. It is noteworthy that the so-called turbulence scalar interaction (TSI) term corresponds to the sum of terms (V), (VI), and (VII). This transport equation and its modeling has been discussed in many references, see for instance references [73, 74] where the same notations are used.